重心 質心與形心 Center of Gravity and Centroidcct.me.ntut.edu.tw/.../statics/lecture2010-2/centroid.pdfChen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University
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重心、質心與形心 Center of Gravity and Centroid Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected]CCT Group 助教: 吳穎彥 May 20, 2011 Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (http 重心、質心與形心Center of Gravity and Centroid May 20, 2011 1 / 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 1 45
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課程大綱
1 重心質心與形心
2 混合結構體
3 Pappus and Guldinus定理
4 三維分佈負載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 2 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 3 45
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重重重心心心質質質心心心與與與形形形心心心Center of Gravity and Centroid
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 4 45
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重心(Center of Gravity)
重心(Center of Gravity)簡單來說乃指代表一物體受重力吸引的受力點任何一物體均可視為許多微小物體的組合其受力吸引方向均指向地球這些微小物體之受力大小為dW則
~FR = minussum
Fz~k (1)
W =
ˆdW (2)
假設微小物體(dW)之位置座標為(x y z)則討論微小物體(dW)分別在xyz軸上的扭矩(MR)x (MR)y (MR)z如下
(MR)y =sum
My xW = x
ˆdW =
ˆxdW x =
acutexdWacutedW
(3)
(MR)x =sum
Mx yW = y
ˆdW =
ˆydW y =
acuteydWacutedW
(4)
(MR)y =sum
My zW = z
ˆdW =
ˆzdW z =
acutezdWacutedW
(5)
其中(x y z)為物體重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 5 45
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質心(Center of Mass)
當物體置於重力場中或受力作用時物體將產生加速度
~W = m~g (6)
~F = m~a (7)
dW = g middot dm (8)
質心(Center of Mass)簡單來說乃指代表一物體受力作用質量代表位置點(Cm)任何一物體均可視為許多微小物體的組合這些微小物體為dm則
x =
acutexdWacutedW
=
acutexg middot dmacuteg middot dm
=
acutexdmacutedm
(9)
y =
acuteydWacutedW
=
acuteyg middot dmacuteg middot dm
=
acuteydmacutedm
(10)
z =
acutezdWacutedW
=
acutezg middot dmacuteg middot dm
=
acutezdmacutedm
(11)
其中(x y z)為微小物體(dm)的位置座標(x y z)為物體質
心(Cm)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 6 45
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體心(Centroid of Volume)
一均質的物體之密度為ρ = const時代表物體之體積位置座標稱為體心或幾何中心(Centroid of Volume or Geometric Center C)任何一物體均可視為許多微小物體的組合這些微小物體的體積為dV質量為dm = ρ middot dV則
x =
acutexdmacutedm
=
acuteV xρ middot dVacuteV ρ middot dV
=
acuteV xdVacuteV dV
(12)
y =
acuteydmacutedm
=
acuteV yρ middot dVacuteV ρ middot dV
=
acuteV ydVacuteV dV
(13)
z =
acutezdmacutedm
=
acuteV zρ middot dVacuteV ρ middot dV
=
acuteV zdVacuteV dV
(14)
其中(x y z)為微小物體之體心的位置座標(x y z)為物體之體心(C)的位置座標左圖角堆取y軸上之薄圓片的微小體積
dV = π middot r2dy = π middot z2dy (15)
其中微小圓片半徑r=z微小圓片體心(x = 0 y = y z = 0)角
堆體心(x = 0 y z = 0)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 7 45
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面心(Centroid of Area)
代表面積之位置座標稱為面心(Centroid of Area C)任何一面積均可視為許多微小面積的組合這些微小面積為dA則
x =
acuteA xdAacuteA dA
(16)
y =
acuteA ydAacuteA dA
(17)
其中(x y)為微小面積之面心的位置座標(x y)為面心(C)的位置座標
左圖曲線圍成之面積取x軸上之微小曲段dx並對y軸方向畫出矩形微小
面積dA=ydx(x = x y = y2
)為微小面積之面心的位置座標再取y軸上之
微小曲段dy並對x軸方向畫出矩形微小面積dA=xdy(x = x2 y = y)為微
小面積之面心的位置座標(x y)為面心(C)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 8 45
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線心(Centroid of Line)代表曲線之位置座標稱為線心(Centroid of Line C)任何一曲線均可視為許多微小曲線的組合這些微小曲線為dL則
x =
acuteL xdLacuteL dL
(18)
y =
acuteL ydLacuteL dL
(19)
其中(x y)為微小曲段之線心的位置座標(x y)為線心(C)的位置
座標由畢氏定理
dL =radic
(dx)2 + (dy)2 =
radic(dx
dx)2dx2 + (
dy
dx)2dx2 (20)
=
(radic1 + (
dy
dx)2
)dx (21)
=radic
(dx)2 + (dy)2 =
radic(dx
dy)2dy2 + (
dy
dy)2dy2 (22)
=
(radic(dx
dy)2 + 1
)dy (23)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 9 45
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線心(Centroid of Line) - Continued
左圖曲線
y = 2x2 (24)
dy
dx= 4x (25)
dL =
(radic1 + (
dy
dx)2
)dx (26)
= (radic
1 + (4x)2)dx (27)
x =
acuteL xdLacuteL dL
=
acute 10 x(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx(28)
y =
acuteL ydLacuteL dL
=
acute 10 y(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx=
acute 10 2x2(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx
其中(x = x y = y)為微小曲段之線心的位置座標(x y)為線心(C)的
位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 10 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 11 45
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 13 45
Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 14 45
Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 17 45
Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
課程大綱
1 重心質心與形心
2 混合結構體
3 Pappus and Guldinus定理
4 三維分佈負載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 2 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 3 45
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重重重心心心質質質心心心與與與形形形心心心Center of Gravity and Centroid
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 4 45
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重心(Center of Gravity)
重心(Center of Gravity)簡單來說乃指代表一物體受重力吸引的受力點任何一物體均可視為許多微小物體的組合其受力吸引方向均指向地球這些微小物體之受力大小為dW則
~FR = minussum
Fz~k (1)
W =
ˆdW (2)
假設微小物體(dW)之位置座標為(x y z)則討論微小物體(dW)分別在xyz軸上的扭矩(MR)x (MR)y (MR)z如下
(MR)y =sum
My xW = x
ˆdW =
ˆxdW x =
acutexdWacutedW
(3)
(MR)x =sum
Mx yW = y
ˆdW =
ˆydW y =
acuteydWacutedW
(4)
(MR)y =sum
My zW = z
ˆdW =
ˆzdW z =
acutezdWacutedW
(5)
其中(x y z)為物體重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 5 45
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質心(Center of Mass)
當物體置於重力場中或受力作用時物體將產生加速度
~W = m~g (6)
~F = m~a (7)
dW = g middot dm (8)
質心(Center of Mass)簡單來說乃指代表一物體受力作用質量代表位置點(Cm)任何一物體均可視為許多微小物體的組合這些微小物體為dm則
x =
acutexdWacutedW
=
acutexg middot dmacuteg middot dm
=
acutexdmacutedm
(9)
y =
acuteydWacutedW
=
acuteyg middot dmacuteg middot dm
=
acuteydmacutedm
(10)
z =
acutezdWacutedW
=
acutezg middot dmacuteg middot dm
=
acutezdmacutedm
(11)
其中(x y z)為微小物體(dm)的位置座標(x y z)為物體質
心(Cm)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 6 45
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體心(Centroid of Volume)
一均質的物體之密度為ρ = const時代表物體之體積位置座標稱為體心或幾何中心(Centroid of Volume or Geometric Center C)任何一物體均可視為許多微小物體的組合這些微小物體的體積為dV質量為dm = ρ middot dV則
x =
acutexdmacutedm
=
acuteV xρ middot dVacuteV ρ middot dV
=
acuteV xdVacuteV dV
(12)
y =
acuteydmacutedm
=
acuteV yρ middot dVacuteV ρ middot dV
=
acuteV ydVacuteV dV
(13)
z =
acutezdmacutedm
=
acuteV zρ middot dVacuteV ρ middot dV
=
acuteV zdVacuteV dV
(14)
其中(x y z)為微小物體之體心的位置座標(x y z)為物體之體心(C)的位置座標左圖角堆取y軸上之薄圓片的微小體積
dV = π middot r2dy = π middot z2dy (15)
其中微小圓片半徑r=z微小圓片體心(x = 0 y = y z = 0)角
堆體心(x = 0 y z = 0)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 7 45
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面心(Centroid of Area)
代表面積之位置座標稱為面心(Centroid of Area C)任何一面積均可視為許多微小面積的組合這些微小面積為dA則
x =
acuteA xdAacuteA dA
(16)
y =
acuteA ydAacuteA dA
(17)
其中(x y)為微小面積之面心的位置座標(x y)為面心(C)的位置座標
左圖曲線圍成之面積取x軸上之微小曲段dx並對y軸方向畫出矩形微小
面積dA=ydx(x = x y = y2
)為微小面積之面心的位置座標再取y軸上之
微小曲段dy並對x軸方向畫出矩形微小面積dA=xdy(x = x2 y = y)為微
小面積之面心的位置座標(x y)為面心(C)的位置座標
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線心(Centroid of Line)代表曲線之位置座標稱為線心(Centroid of Line C)任何一曲線均可視為許多微小曲線的組合這些微小曲線為dL則
x =
acuteL xdLacuteL dL
(18)
y =
acuteL ydLacuteL dL
(19)
其中(x y)為微小曲段之線心的位置座標(x y)為線心(C)的位置
座標由畢氏定理
dL =radic
(dx)2 + (dy)2 =
radic(dx
dx)2dx2 + (
dy
dx)2dx2 (20)
=
(radic1 + (
dy
dx)2
)dx (21)
=radic
(dx)2 + (dy)2 =
radic(dx
dy)2dy2 + (
dy
dy)2dy2 (22)
=
(radic(dx
dy)2 + 1
)dy (23)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 9 45
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線心(Centroid of Line) - Continued
左圖曲線
y = 2x2 (24)
dy
dx= 4x (25)
dL =
(radic1 + (
dy
dx)2
)dx (26)
= (radic
1 + (4x)2)dx (27)
x =
acuteL xdLacuteL dL
=
acute 10 x(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx(28)
y =
acuteL ydLacuteL dL
=
acute 10 y(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx=
acute 10 2x2(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx
其中(x = x y = y)為微小曲段之線心的位置座標(x y)為線心(C)的
位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 10 45
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課課課堂堂堂練練練習習習
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 13 45
Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 14 45
Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 17 45
Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 3 45
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重重重心心心質質質心心心與與與形形形心心心Center of Gravity and Centroid
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重心(Center of Gravity)
重心(Center of Gravity)簡單來說乃指代表一物體受重力吸引的受力點任何一物體均可視為許多微小物體的組合其受力吸引方向均指向地球這些微小物體之受力大小為dW則
~FR = minussum
Fz~k (1)
W =
ˆdW (2)
假設微小物體(dW)之位置座標為(x y z)則討論微小物體(dW)分別在xyz軸上的扭矩(MR)x (MR)y (MR)z如下
(MR)y =sum
My xW = x
ˆdW =
ˆxdW x =
acutexdWacutedW
(3)
(MR)x =sum
Mx yW = y
ˆdW =
ˆydW y =
acuteydWacutedW
(4)
(MR)y =sum
My zW = z
ˆdW =
ˆzdW z =
acutezdWacutedW
(5)
其中(x y z)為物體重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 5 45
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質心(Center of Mass)
當物體置於重力場中或受力作用時物體將產生加速度
~W = m~g (6)
~F = m~a (7)
dW = g middot dm (8)
質心(Center of Mass)簡單來說乃指代表一物體受力作用質量代表位置點(Cm)任何一物體均可視為許多微小物體的組合這些微小物體為dm則
x =
acutexdWacutedW
=
acutexg middot dmacuteg middot dm
=
acutexdmacutedm
(9)
y =
acuteydWacutedW
=
acuteyg middot dmacuteg middot dm
=
acuteydmacutedm
(10)
z =
acutezdWacutedW
=
acutezg middot dmacuteg middot dm
=
acutezdmacutedm
(11)
其中(x y z)為微小物體(dm)的位置座標(x y z)為物體質
心(Cm)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 6 45
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體心(Centroid of Volume)
一均質的物體之密度為ρ = const時代表物體之體積位置座標稱為體心或幾何中心(Centroid of Volume or Geometric Center C)任何一物體均可視為許多微小物體的組合這些微小物體的體積為dV質量為dm = ρ middot dV則
x =
acutexdmacutedm
=
acuteV xρ middot dVacuteV ρ middot dV
=
acuteV xdVacuteV dV
(12)
y =
acuteydmacutedm
=
acuteV yρ middot dVacuteV ρ middot dV
=
acuteV ydVacuteV dV
(13)
z =
acutezdmacutedm
=
acuteV zρ middot dVacuteV ρ middot dV
=
acuteV zdVacuteV dV
(14)
其中(x y z)為微小物體之體心的位置座標(x y z)為物體之體心(C)的位置座標左圖角堆取y軸上之薄圓片的微小體積
dV = π middot r2dy = π middot z2dy (15)
其中微小圓片半徑r=z微小圓片體心(x = 0 y = y z = 0)角
堆體心(x = 0 y z = 0)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 7 45
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面心(Centroid of Area)
代表面積之位置座標稱為面心(Centroid of Area C)任何一面積均可視為許多微小面積的組合這些微小面積為dA則
x =
acuteA xdAacuteA dA
(16)
y =
acuteA ydAacuteA dA
(17)
其中(x y)為微小面積之面心的位置座標(x y)為面心(C)的位置座標
左圖曲線圍成之面積取x軸上之微小曲段dx並對y軸方向畫出矩形微小
面積dA=ydx(x = x y = y2
)為微小面積之面心的位置座標再取y軸上之
微小曲段dy並對x軸方向畫出矩形微小面積dA=xdy(x = x2 y = y)為微
小面積之面心的位置座標(x y)為面心(C)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 8 45
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線心(Centroid of Line)代表曲線之位置座標稱為線心(Centroid of Line C)任何一曲線均可視為許多微小曲線的組合這些微小曲線為dL則
x =
acuteL xdLacuteL dL
(18)
y =
acuteL ydLacuteL dL
(19)
其中(x y)為微小曲段之線心的位置座標(x y)為線心(C)的位置
座標由畢氏定理
dL =radic
(dx)2 + (dy)2 =
radic(dx
dx)2dx2 + (
dy
dx)2dx2 (20)
=
(radic1 + (
dy
dx)2
)dx (21)
=radic
(dx)2 + (dy)2 =
radic(dx
dy)2dy2 + (
dy
dy)2dy2 (22)
=
(radic(dx
dy)2 + 1
)dy (23)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 9 45
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線心(Centroid of Line) - Continued
左圖曲線
y = 2x2 (24)
dy
dx= 4x (25)
dL =
(radic1 + (
dy
dx)2
)dx (26)
= (radic
1 + (4x)2)dx (27)
x =
acuteL xdLacuteL dL
=
acute 10 x(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx(28)
y =
acuteL ydLacuteL dL
=
acute 10 y(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx=
acute 10 2x2(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx
其中(x = x y = y)為微小曲段之線心的位置座標(x y)為線心(C)的
位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 10 45
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課課課堂堂堂練練練習習習
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 13 45
Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 14 45
Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 17 45
Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
重重重心心心質質質心心心與與與形形形心心心Center of Gravity and Centroid
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重心(Center of Gravity)
重心(Center of Gravity)簡單來說乃指代表一物體受重力吸引的受力點任何一物體均可視為許多微小物體的組合其受力吸引方向均指向地球這些微小物體之受力大小為dW則
~FR = minussum
Fz~k (1)
W =
ˆdW (2)
假設微小物體(dW)之位置座標為(x y z)則討論微小物體(dW)分別在xyz軸上的扭矩(MR)x (MR)y (MR)z如下
(MR)y =sum
My xW = x
ˆdW =
ˆxdW x =
acutexdWacutedW
(3)
(MR)x =sum
Mx yW = y
ˆdW =
ˆydW y =
acuteydWacutedW
(4)
(MR)y =sum
My zW = z
ˆdW =
ˆzdW z =
acutezdWacutedW
(5)
其中(x y z)為物體重心(G)的位置座標
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質心(Center of Mass)
當物體置於重力場中或受力作用時物體將產生加速度
~W = m~g (6)
~F = m~a (7)
dW = g middot dm (8)
質心(Center of Mass)簡單來說乃指代表一物體受力作用質量代表位置點(Cm)任何一物體均可視為許多微小物體的組合這些微小物體為dm則
x =
acutexdWacutedW
=
acutexg middot dmacuteg middot dm
=
acutexdmacutedm
(9)
y =
acuteydWacutedW
=
acuteyg middot dmacuteg middot dm
=
acuteydmacutedm
(10)
z =
acutezdWacutedW
=
acutezg middot dmacuteg middot dm
=
acutezdmacutedm
(11)
其中(x y z)為微小物體(dm)的位置座標(x y z)為物體質
心(Cm)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 6 45
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體心(Centroid of Volume)
一均質的物體之密度為ρ = const時代表物體之體積位置座標稱為體心或幾何中心(Centroid of Volume or Geometric Center C)任何一物體均可視為許多微小物體的組合這些微小物體的體積為dV質量為dm = ρ middot dV則
x =
acutexdmacutedm
=
acuteV xρ middot dVacuteV ρ middot dV
=
acuteV xdVacuteV dV
(12)
y =
acuteydmacutedm
=
acuteV yρ middot dVacuteV ρ middot dV
=
acuteV ydVacuteV dV
(13)
z =
acutezdmacutedm
=
acuteV zρ middot dVacuteV ρ middot dV
=
acuteV zdVacuteV dV
(14)
其中(x y z)為微小物體之體心的位置座標(x y z)為物體之體心(C)的位置座標左圖角堆取y軸上之薄圓片的微小體積
dV = π middot r2dy = π middot z2dy (15)
其中微小圓片半徑r=z微小圓片體心(x = 0 y = y z = 0)角
堆體心(x = 0 y z = 0)
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面心(Centroid of Area)
代表面積之位置座標稱為面心(Centroid of Area C)任何一面積均可視為許多微小面積的組合這些微小面積為dA則
x =
acuteA xdAacuteA dA
(16)
y =
acuteA ydAacuteA dA
(17)
其中(x y)為微小面積之面心的位置座標(x y)為面心(C)的位置座標
左圖曲線圍成之面積取x軸上之微小曲段dx並對y軸方向畫出矩形微小
面積dA=ydx(x = x y = y2
)為微小面積之面心的位置座標再取y軸上之
微小曲段dy並對x軸方向畫出矩形微小面積dA=xdy(x = x2 y = y)為微
小面積之面心的位置座標(x y)為面心(C)的位置座標
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線心(Centroid of Line)代表曲線之位置座標稱為線心(Centroid of Line C)任何一曲線均可視為許多微小曲線的組合這些微小曲線為dL則
x =
acuteL xdLacuteL dL
(18)
y =
acuteL ydLacuteL dL
(19)
其中(x y)為微小曲段之線心的位置座標(x y)為線心(C)的位置
座標由畢氏定理
dL =radic
(dx)2 + (dy)2 =
radic(dx
dx)2dx2 + (
dy
dx)2dx2 (20)
=
(radic1 + (
dy
dx)2
)dx (21)
=radic
(dx)2 + (dy)2 =
radic(dx
dy)2dy2 + (
dy
dy)2dy2 (22)
=
(radic(dx
dy)2 + 1
)dy (23)
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線心(Centroid of Line) - Continued
左圖曲線
y = 2x2 (24)
dy
dx= 4x (25)
dL =
(radic1 + (
dy
dx)2
)dx (26)
= (radic
1 + (4x)2)dx (27)
x =
acuteL xdLacuteL dL
=
acute 10 x(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx(28)
y =
acuteL ydLacuteL dL
=
acute 10 y(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx=
acute 10 2x2(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx
其中(x = x y = y)為微小曲段之線心的位置座標(x y)為線心(C)的
位置座標
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課課課堂堂堂練練練習習習
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 13 45
Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 14 45
Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 17 45
Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
重心(Center of Gravity)
重心(Center of Gravity)簡單來說乃指代表一物體受重力吸引的受力點任何一物體均可視為許多微小物體的組合其受力吸引方向均指向地球這些微小物體之受力大小為dW則
~FR = minussum
Fz~k (1)
W =
ˆdW (2)
假設微小物體(dW)之位置座標為(x y z)則討論微小物體(dW)分別在xyz軸上的扭矩(MR)x (MR)y (MR)z如下
(MR)y =sum
My xW = x
ˆdW =
ˆxdW x =
acutexdWacutedW
(3)
(MR)x =sum
Mx yW = y
ˆdW =
ˆydW y =
acuteydWacutedW
(4)
(MR)y =sum
My zW = z
ˆdW =
ˆzdW z =
acutezdWacutedW
(5)
其中(x y z)為物體重心(G)的位置座標
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質心(Center of Mass)
當物體置於重力場中或受力作用時物體將產生加速度
~W = m~g (6)
~F = m~a (7)
dW = g middot dm (8)
質心(Center of Mass)簡單來說乃指代表一物體受力作用質量代表位置點(Cm)任何一物體均可視為許多微小物體的組合這些微小物體為dm則
x =
acutexdWacutedW
=
acutexg middot dmacuteg middot dm
=
acutexdmacutedm
(9)
y =
acuteydWacutedW
=
acuteyg middot dmacuteg middot dm
=
acuteydmacutedm
(10)
z =
acutezdWacutedW
=
acutezg middot dmacuteg middot dm
=
acutezdmacutedm
(11)
其中(x y z)為微小物體(dm)的位置座標(x y z)為物體質
心(Cm)的位置座標
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體心(Centroid of Volume)
一均質的物體之密度為ρ = const時代表物體之體積位置座標稱為體心或幾何中心(Centroid of Volume or Geometric Center C)任何一物體均可視為許多微小物體的組合這些微小物體的體積為dV質量為dm = ρ middot dV則
x =
acutexdmacutedm
=
acuteV xρ middot dVacuteV ρ middot dV
=
acuteV xdVacuteV dV
(12)
y =
acuteydmacutedm
=
acuteV yρ middot dVacuteV ρ middot dV
=
acuteV ydVacuteV dV
(13)
z =
acutezdmacutedm
=
acuteV zρ middot dVacuteV ρ middot dV
=
acuteV zdVacuteV dV
(14)
其中(x y z)為微小物體之體心的位置座標(x y z)為物體之體心(C)的位置座標左圖角堆取y軸上之薄圓片的微小體積
dV = π middot r2dy = π middot z2dy (15)
其中微小圓片半徑r=z微小圓片體心(x = 0 y = y z = 0)角
堆體心(x = 0 y z = 0)
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面心(Centroid of Area)
代表面積之位置座標稱為面心(Centroid of Area C)任何一面積均可視為許多微小面積的組合這些微小面積為dA則
x =
acuteA xdAacuteA dA
(16)
y =
acuteA ydAacuteA dA
(17)
其中(x y)為微小面積之面心的位置座標(x y)為面心(C)的位置座標
左圖曲線圍成之面積取x軸上之微小曲段dx並對y軸方向畫出矩形微小
面積dA=ydx(x = x y = y2
)為微小面積之面心的位置座標再取y軸上之
微小曲段dy並對x軸方向畫出矩形微小面積dA=xdy(x = x2 y = y)為微
小面積之面心的位置座標(x y)為面心(C)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 8 45
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線心(Centroid of Line)代表曲線之位置座標稱為線心(Centroid of Line C)任何一曲線均可視為許多微小曲線的組合這些微小曲線為dL則
x =
acuteL xdLacuteL dL
(18)
y =
acuteL ydLacuteL dL
(19)
其中(x y)為微小曲段之線心的位置座標(x y)為線心(C)的位置
座標由畢氏定理
dL =radic
(dx)2 + (dy)2 =
radic(dx
dx)2dx2 + (
dy
dx)2dx2 (20)
=
(radic1 + (
dy
dx)2
)dx (21)
=radic
(dx)2 + (dy)2 =
radic(dx
dy)2dy2 + (
dy
dy)2dy2 (22)
=
(radic(dx
dy)2 + 1
)dy (23)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 9 45
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線心(Centroid of Line) - Continued
左圖曲線
y = 2x2 (24)
dy
dx= 4x (25)
dL =
(radic1 + (
dy
dx)2
)dx (26)
= (radic
1 + (4x)2)dx (27)
x =
acuteL xdLacuteL dL
=
acute 10 x(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx(28)
y =
acuteL ydLacuteL dL
=
acute 10 y(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx=
acute 10 2x2(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx
其中(x = x y = y)為微小曲段之線心的位置座標(x y)為線心(C)的
位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 10 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 11 45
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 13 45
Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 14 45
Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 17 45
Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
質心(Center of Mass)
當物體置於重力場中或受力作用時物體將產生加速度
~W = m~g (6)
~F = m~a (7)
dW = g middot dm (8)
質心(Center of Mass)簡單來說乃指代表一物體受力作用質量代表位置點(Cm)任何一物體均可視為許多微小物體的組合這些微小物體為dm則
x =
acutexdWacutedW
=
acutexg middot dmacuteg middot dm
=
acutexdmacutedm
(9)
y =
acuteydWacutedW
=
acuteyg middot dmacuteg middot dm
=
acuteydmacutedm
(10)
z =
acutezdWacutedW
=
acutezg middot dmacuteg middot dm
=
acutezdmacutedm
(11)
其中(x y z)為微小物體(dm)的位置座標(x y z)為物體質
心(Cm)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 6 45
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體心(Centroid of Volume)
一均質的物體之密度為ρ = const時代表物體之體積位置座標稱為體心或幾何中心(Centroid of Volume or Geometric Center C)任何一物體均可視為許多微小物體的組合這些微小物體的體積為dV質量為dm = ρ middot dV則
x =
acutexdmacutedm
=
acuteV xρ middot dVacuteV ρ middot dV
=
acuteV xdVacuteV dV
(12)
y =
acuteydmacutedm
=
acuteV yρ middot dVacuteV ρ middot dV
=
acuteV ydVacuteV dV
(13)
z =
acutezdmacutedm
=
acuteV zρ middot dVacuteV ρ middot dV
=
acuteV zdVacuteV dV
(14)
其中(x y z)為微小物體之體心的位置座標(x y z)為物體之體心(C)的位置座標左圖角堆取y軸上之薄圓片的微小體積
dV = π middot r2dy = π middot z2dy (15)
其中微小圓片半徑r=z微小圓片體心(x = 0 y = y z = 0)角
堆體心(x = 0 y z = 0)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 7 45
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面心(Centroid of Area)
代表面積之位置座標稱為面心(Centroid of Area C)任何一面積均可視為許多微小面積的組合這些微小面積為dA則
x =
acuteA xdAacuteA dA
(16)
y =
acuteA ydAacuteA dA
(17)
其中(x y)為微小面積之面心的位置座標(x y)為面心(C)的位置座標
左圖曲線圍成之面積取x軸上之微小曲段dx並對y軸方向畫出矩形微小
面積dA=ydx(x = x y = y2
)為微小面積之面心的位置座標再取y軸上之
微小曲段dy並對x軸方向畫出矩形微小面積dA=xdy(x = x2 y = y)為微
小面積之面心的位置座標(x y)為面心(C)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 8 45
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線心(Centroid of Line)代表曲線之位置座標稱為線心(Centroid of Line C)任何一曲線均可視為許多微小曲線的組合這些微小曲線為dL則
x =
acuteL xdLacuteL dL
(18)
y =
acuteL ydLacuteL dL
(19)
其中(x y)為微小曲段之線心的位置座標(x y)為線心(C)的位置
座標由畢氏定理
dL =radic
(dx)2 + (dy)2 =
radic(dx
dx)2dx2 + (
dy
dx)2dx2 (20)
=
(radic1 + (
dy
dx)2
)dx (21)
=radic
(dx)2 + (dy)2 =
radic(dx
dy)2dy2 + (
dy
dy)2dy2 (22)
=
(radic(dx
dy)2 + 1
)dy (23)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 9 45
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線心(Centroid of Line) - Continued
左圖曲線
y = 2x2 (24)
dy
dx= 4x (25)
dL =
(radic1 + (
dy
dx)2
)dx (26)
= (radic
1 + (4x)2)dx (27)
x =
acuteL xdLacuteL dL
=
acute 10 x(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx(28)
y =
acuteL ydLacuteL dL
=
acute 10 y(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx=
acute 10 2x2(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx
其中(x = x y = y)為微小曲段之線心的位置座標(x y)為線心(C)的
位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 10 45
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課課課堂堂堂練練練習習習
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 13 45
Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
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Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
體心(Centroid of Volume)
一均質的物體之密度為ρ = const時代表物體之體積位置座標稱為體心或幾何中心(Centroid of Volume or Geometric Center C)任何一物體均可視為許多微小物體的組合這些微小物體的體積為dV質量為dm = ρ middot dV則
x =
acutexdmacutedm
=
acuteV xρ middot dVacuteV ρ middot dV
=
acuteV xdVacuteV dV
(12)
y =
acuteydmacutedm
=
acuteV yρ middot dVacuteV ρ middot dV
=
acuteV ydVacuteV dV
(13)
z =
acutezdmacutedm
=
acuteV zρ middot dVacuteV ρ middot dV
=
acuteV zdVacuteV dV
(14)
其中(x y z)為微小物體之體心的位置座標(x y z)為物體之體心(C)的位置座標左圖角堆取y軸上之薄圓片的微小體積
dV = π middot r2dy = π middot z2dy (15)
其中微小圓片半徑r=z微小圓片體心(x = 0 y = y z = 0)角
堆體心(x = 0 y z = 0)
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面心(Centroid of Area)
代表面積之位置座標稱為面心(Centroid of Area C)任何一面積均可視為許多微小面積的組合這些微小面積為dA則
x =
acuteA xdAacuteA dA
(16)
y =
acuteA ydAacuteA dA
(17)
其中(x y)為微小面積之面心的位置座標(x y)為面心(C)的位置座標
左圖曲線圍成之面積取x軸上之微小曲段dx並對y軸方向畫出矩形微小
面積dA=ydx(x = x y = y2
)為微小面積之面心的位置座標再取y軸上之
微小曲段dy並對x軸方向畫出矩形微小面積dA=xdy(x = x2 y = y)為微
小面積之面心的位置座標(x y)為面心(C)的位置座標
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線心(Centroid of Line)代表曲線之位置座標稱為線心(Centroid of Line C)任何一曲線均可視為許多微小曲線的組合這些微小曲線為dL則
x =
acuteL xdLacuteL dL
(18)
y =
acuteL ydLacuteL dL
(19)
其中(x y)為微小曲段之線心的位置座標(x y)為線心(C)的位置
座標由畢氏定理
dL =radic
(dx)2 + (dy)2 =
radic(dx
dx)2dx2 + (
dy
dx)2dx2 (20)
=
(radic1 + (
dy
dx)2
)dx (21)
=radic
(dx)2 + (dy)2 =
radic(dx
dy)2dy2 + (
dy
dy)2dy2 (22)
=
(radic(dx
dy)2 + 1
)dy (23)
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線心(Centroid of Line) - Continued
左圖曲線
y = 2x2 (24)
dy
dx= 4x (25)
dL =
(radic1 + (
dy
dx)2
)dx (26)
= (radic
1 + (4x)2)dx (27)
x =
acuteL xdLacuteL dL
=
acute 10 x(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx(28)
y =
acuteL ydLacuteL dL
=
acute 10 y(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx=
acute 10 2x2(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx
其中(x = x y = y)為微小曲段之線心的位置座標(x y)為線心(C)的
位置座標
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課課課堂堂堂練練練習習習
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
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Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
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Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
面心(Centroid of Area)
代表面積之位置座標稱為面心(Centroid of Area C)任何一面積均可視為許多微小面積的組合這些微小面積為dA則
x =
acuteA xdAacuteA dA
(16)
y =
acuteA ydAacuteA dA
(17)
其中(x y)為微小面積之面心的位置座標(x y)為面心(C)的位置座標
左圖曲線圍成之面積取x軸上之微小曲段dx並對y軸方向畫出矩形微小
面積dA=ydx(x = x y = y2
)為微小面積之面心的位置座標再取y軸上之
微小曲段dy並對x軸方向畫出矩形微小面積dA=xdy(x = x2 y = y)為微
小面積之面心的位置座標(x y)為面心(C)的位置座標
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線心(Centroid of Line)代表曲線之位置座標稱為線心(Centroid of Line C)任何一曲線均可視為許多微小曲線的組合這些微小曲線為dL則
x =
acuteL xdLacuteL dL
(18)
y =
acuteL ydLacuteL dL
(19)
其中(x y)為微小曲段之線心的位置座標(x y)為線心(C)的位置
座標由畢氏定理
dL =radic
(dx)2 + (dy)2 =
radic(dx
dx)2dx2 + (
dy
dx)2dx2 (20)
=
(radic1 + (
dy
dx)2
)dx (21)
=radic
(dx)2 + (dy)2 =
radic(dx
dy)2dy2 + (
dy
dy)2dy2 (22)
=
(radic(dx
dy)2 + 1
)dy (23)
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線心(Centroid of Line) - Continued
左圖曲線
y = 2x2 (24)
dy
dx= 4x (25)
dL =
(radic1 + (
dy
dx)2
)dx (26)
= (radic
1 + (4x)2)dx (27)
x =
acuteL xdLacuteL dL
=
acute 10 x(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx(28)
y =
acuteL ydLacuteL dL
=
acute 10 y(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx=
acute 10 2x2(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx
其中(x = x y = y)為微小曲段之線心的位置座標(x y)為線心(C)的
位置座標
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課課課堂堂堂練練練習習習
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
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Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
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Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
線心(Centroid of Line)代表曲線之位置座標稱為線心(Centroid of Line C)任何一曲線均可視為許多微小曲線的組合這些微小曲線為dL則
x =
acuteL xdLacuteL dL
(18)
y =
acuteL ydLacuteL dL
(19)
其中(x y)為微小曲段之線心的位置座標(x y)為線心(C)的位置
座標由畢氏定理
dL =radic
(dx)2 + (dy)2 =
radic(dx
dx)2dx2 + (
dy
dx)2dx2 (20)
=
(radic1 + (
dy
dx)2
)dx (21)
=radic
(dx)2 + (dy)2 =
radic(dx
dy)2dy2 + (
dy
dy)2dy2 (22)
=
(radic(dx
dy)2 + 1
)dy (23)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 9 45
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線心(Centroid of Line) - Continued
左圖曲線
y = 2x2 (24)
dy
dx= 4x (25)
dL =
(radic1 + (
dy
dx)2
)dx (26)
= (radic
1 + (4x)2)dx (27)
x =
acuteL xdLacuteL dL
=
acute 10 x(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx(28)
y =
acuteL ydLacuteL dL
=
acute 10 y(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx=
acute 10 2x2(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx
其中(x = x y = y)為微小曲段之線心的位置座標(x y)為線心(C)的
位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 10 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 11 45
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 13 45
Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 14 45
Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 17 45
Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
線心(Centroid of Line) - Continued
左圖曲線
y = 2x2 (24)
dy
dx= 4x (25)
dL =
(radic1 + (
dy
dx)2
)dx (26)
= (radic
1 + (4x)2)dx (27)
x =
acuteL xdLacuteL dL
=
acute 10 x(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx(28)
y =
acuteL ydLacuteL dL
=
acute 10 y(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx=
acute 10 2x2(
radic1 + (4x)2)dxacute 1
0 (radic
1 + (4x)2)dx
其中(x = x y = y)為微小曲段之線心的位置座標(x y)為線心(C)的
位置座標
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 11 45
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 12 45
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
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Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 14 45
Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 11 45
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解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
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Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
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Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
解題技巧
找尋線心面心體心等座標之方法首先確立一微小量(dLdAdV)接著確立微小量(dLdAdV)之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式確立微小量及其位置座標時記得分別取xyz軸的扭矩
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
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Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
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Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
畢氏定理
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
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Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
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Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
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Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
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Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-1 解答 I
題目欲求曲線之線心座標(x y)
x =
acutexdLacutedL
(29)
y =
acuteydLacutedL
(30)
取微小曲段dL其線心座標為C(x = x y = y)
dL =radic
(dx)2 + (dy)2 (31)
=
(radic(dxdy
)2+ 1
)dy (32)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 13 45
Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 14 45
Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 15 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 17 45
Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Pappus and Guldinus定定定理理理
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-1 解答 II
由於曲線方程
x = y2 (33)
dx
dy= 2y (34)
dL =(radic
(2y)2 + 1)dy (35)
帶入式(29)與式(30)得
x =
acuteL xdLacuteL dL
=
acute 10 x(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y2
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(36)
y =
acuteL ydLacuteL dL
=
acute 10 y(radic
(2y)2 + 1)dy
acute 10
(radic(2y)2 + 1
)dy
=
acute 10 2y
(radicy2 + ( 1
2)2)dy
acute 10 2(radic
y2 + ( 12
)2)dy
(37)
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Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-1 解答 III
其中積分式
ˆ(radic
x2 plusmn a2)dx =1
2[xradic
x2 plusmn a2 plusmn a2ln(x +radic
x2 plusmn a2)] + C (38)
ˆx(radic
x2 plusmn a2)dx =1
3
radic(x2 plusmn a2)3 + C (39)
ˆx2(radic
x2 plusmn a2)dx =x
4
radic(x2 plusmn a2)3 ∓
a2
8xradic
x2 plusmn a2 minusa4
8ln(x +
radicx2 plusmn a2) + C (40)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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求混合曲線之線心座標
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A曲線
B曲線
C曲線
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求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Pappus and Guldinus定定定理理理
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課課課堂堂堂練練練習習習
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由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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求面積與體積
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
取微小曲段 dL其線心座標
求曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 17 45
Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 16 45
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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求混合曲線之線心座標
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A曲線
B曲線
C曲線
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求混合區塊之面心座標
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求混合體之質心座標
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Pappus and Guldinus定定定理理理
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課課課堂堂堂練練練習習習
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由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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求面積與體積
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
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Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-3 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(45)
y =
acuteydAacutedA
(46)
取微小區塊dA其面心座標為C(x = x2 y = y)由於曲線方程
y =y
b(b minus x) (47)
x = b minus b
hy =
b
h(h minus y) (48)
dA = xdy =b
h(h minus y)dy (49)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 17 45
Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 18 45
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-3 解答 II
帶入式(45)與式(46)得
x =
acuteA xdAacuteA dA
=
acute h0
x2dAacute h
0 dA=
acute h0 [
bh
(hminusy)
2 ][bh (h minus y)]dyacute h0
bh (h minus y)dy
=b
3(50)
y =
acuteA ydAacuteA dA
=
acute h0 ydAacute h0 dA
=
acute h0 y [bh (h minus y)]dyacute h
0bh (h minus y)dy
=h
3(51)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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A曲線
B曲線
C曲線
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求混合區塊之面心座標
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求混合體之質心座標
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
近似為三角形
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
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Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-4 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(52)
y =
acuteydAacutedA
(53)
取微小區塊dA視為三角形面心座標為C(x = 2
3Rcosθ y = 23Rsinθ)d(sinθ) = cosθ d(cosθ) = minussinθ
dA =1
2(R)(Rdθ) =
R2
2dθ (54)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 19 45
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-4 解答 II
帶入式(52)與式(53)得
x =
acuteA xdAacuteA dA
=
acute π2
0 ( 23Rcosθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 cosθdθacute π2
0 dθ=
4R
3π(55)
y =
acuteA ydAacuteA dA
=
acute π2
0 ( 23Rsinθ)R
2
2 dθacute π2
0R2
2 dθ=
( 23R)acute π
20 sinθdθacute π2
0 dθ=
4R
3π(56)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 20 45
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 21 45
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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三三三維維維分分分佈佈佈負負負載載載
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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結結結論論論
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
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Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-5 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(57)
y =
acuteydAacutedA
(58)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
y = x2 (59)
dA = ydx = x2dx (60)
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Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-5 解答 II
帶入式(57)與式(58)得
x =
acuteA xdAacuteA dA
=
acute 10 xydxacute 10 ydx
=
acute 10 x3dxacute 10 x2dx
= 075 (61)
y =
acuteA ydAacuteA dA
=
acute 10 ( y2 )ydxacute 1
0 ydx=
acute 10
12x
4dxacute 10 x2dx
= 03 (62)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
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Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求區塊之面心座標
取微小區塊 dA其面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
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Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 22 45
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-6 解答 I
題目欲求區塊之面心座標(x y)
x =
acutexdAacutedA
(63)
y =
acuteydAacutedA
(64)
取微小區塊dA其面心座標為C(x = x y = y2 )由於曲線方程
x2
4+ y2 = 1 (65)
y =
radic1 minus x2
4(66)
dA = ydx =
(radic1 minus x2
4
)dx (67)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 23 45
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-6 解答 II
帶入式(63)與式(64)得
x =
acuteA xdAacuteA dA
=
acute 2minus2 xydxacute 2minus2 ydx
=
acute 2minus2 x
(radic1 minus x2
4
)dx
acute 2minus2
(radic1 minus x2
4
)dx
(68)
y =
acuteA ydAacuteA dA
=
acute 2minus2 ( y2 )ydxacute 2minus2 ydx
=12
acute 2minus2 (1 minus x2
4 )dxacute 2minus2
(radic1 minus x2
4
)dx
=4
3π(69)
其中積分式ˆ
(radica2 minus x2)dx =
1
2[xradic
a2 minus x2 + a2sinminus1 x
a] + C a gt 0 (70)
ˆx(radica2 minus x2)dx =
1
3
radic(a2 minus x2)3 + C (71)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
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混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
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Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求錐體之體心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 24 45
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-7 解答 I
題目欲求錐體之體心座標(x = 0 y z = 0)
y =
acuteydVacutedV
(72)
取微小環狀區塊dV其體心座標為C(x = 0 y = y z = 0)由於曲線方程
z2 = 100y (73)
dV = πr2dy = πz2dy = 100πydy (74)
帶入式(72)得
y =
acuteV ydVacuteV dV
=
acute 1000 yπ100ydyacute 1000 π100ydy
=100π
acute 1000 y2dy
100πacute 100
0 ydy= 667 (75)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求圓柱體之質心座標
取微小環狀區塊 dV其體心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 25 45
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
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流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
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解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Example 9-8 解答 I
題目欲求圓柱體之質心座標(x = 0 y = 0 z)
z =
acutezρdVacuteρdV
(76)
取微小環狀區塊dV其體心座標為C(x = 0 y = 0 z = z)其密度為
ρ = 200z (77)
dV = π(05)2dz (78)
帶入式(76)得
z =
acuteV zρdVacuteV ρdV
=
acute 10 z(200z)π(05)2dzacute 10 (200z)π(05)2dz
=
acute 10 z2dzacute 10 zdz
= 0667 (79)
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 26 45
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
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混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
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解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
混混混合合合結結結構構構體體體Composite Body
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 27 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
混合結構體
由不同形狀的物體組成的結構體稱為混合結構體
x =
sumxLsumL
=
sumxAsumA
=
sumxWsumW
(80)
y =
sumyLsumL
=
sumyAsumA
=
sumyWsumW
(81)
z =
sumzLsumL
=
sumzAsumA
=
sumzWsumW
(82)
其中(x y z)為混合結構體之各別重心的位置座標
sumL為混合結構體之各別結構體的長度
和sum
A為混合結構體之各別結構體的面積和
sumW為混合結構體之各別結構體的重量
和(x y z)為混合結構體之線心面心重心(G)的位置座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 28 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 29 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
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Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
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重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
解題技巧
將混合結構體進行分析並把混合結構體進行分解成幾個簡單且熟悉的結構體分別找出分解後的結構體之線心面心體心等位置座標(x y)或(x y z)再帶入線心面心體心之換算公式
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合曲線之線心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
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Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
A曲線
B曲線
C曲線
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合區塊之面心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求混合體之質心座標
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 30 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Pappus and Guldinus定定定理理理
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 31 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 32 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 33 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 34 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
由 example 9-2得球之半圓之線心座標
由 example 9-4得球之半剖面之面心座標
求球之面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求面積與體積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 35 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
三三三維維維分分分佈佈佈負負負載載載
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 36 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 37 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
流體壓力
壓力的定義為單位面積承受的力(p = F
A )依據帕斯卡定律(Pascalrsquos Law)在液體中的任一點之壓力的作用方向為所有方向其值為
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 38 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 39 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
Wf = ρ middot g middot b middot ABDA = γ middot b middot ABDA (103)
~FR =sum
~F = ~FAD + ~FAB + ~Wf (104)
其中FAD等於梯形面積FAB為相同大小Wf為BDA面
積的力其力通過BDA面積的面心位置座
標ABDA為BDA面積
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 40 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 41 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 42 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
解題技巧
找出承受液體壓力之面積(A)與其面心座標(z)再帶入下式求得合力(FR)
FR = γzA (110)
或者利用分力之向量和求得合力(FR)接著以合扭矩(MRO)為零求出合力之作用點
合扭矩等於分扭矩之向量和
2L
3times FA =
2l
3times FB + (Lminus x) times FC (111)
其中FB為三角形OMB之合力FA為三角形ONA之合力FC為梯形BMNA之合力
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
求合力及壓力作用點
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 43 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
重心質心與形心
混合結構體
Pappus and Guldinus定理
三維分佈負載
結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 44 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()重心質心與形心Center of Gravity and Centroid May 20 2011 45 45
