1
ME 302 DYNAMICS OF ME 302 DYNAMICS OF MACHINERYMACHINERY
Dynamic Force Analysis V
Dr. Sadettin KAPUCU
© 2007 Sadettin Kapucu
2Gaziantep University
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
X
Z
Y
O
Inertial FrameX
Z
Y
O
X
Z
Y
O
X
Z
Y
O
X
Z
Y
O
X
Z
Y
O
axisYaboutrotationis oA 90axisXaboutrotationis o
B 90
axisYaboutrotationis oA 90 axisXaboutrotationis o
B 90BA
AB
3Gaziantep University
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
Inertial FrameX
Z
Y
O
X
Z
Y
O
BA AB
Finite Rotation
4Gaziantep University
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
X
Z
Y
O
ABBA
X
Z
Y
O
Xaboutthen
axisYabout
B
A
X
Z
Y
O
Yaboutthen
axisXabout
A
B
5Gaziantep University
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
tt
0lim
Infinitesimal rotation can be treated as vector
tt
0lim
6Gaziantep University
Rotation of a Rigid Body with Rotation of a Rigid Body with One Point FixedOne Point Fixed
http://hyperphysics.phy-astr.gsu.edu/HBASE/top.html
7Gaziantep University
PreliminaryPreliminary Kinematics of a Rigid Body Kinematics of a Rigid Body
2
1 21
8Gaziantep University
PreliminaryPreliminary Kinematics of a Rigid Body Kinematics of a Rigid BodyZ
Y
Xx
y
z Body coordinate
2
1
x2xyzXYZ dt
d
dt
d
21
21221 x
xyzxyz dt
d
dt
d
9Gaziantep University
Rotation of a Rigid Body with Rotation of a Rigid Body with One Point FixedOne Point Fixed
10Gaziantep University
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
X
Z
YP
x
z
y
O
O’
Inertial Frame
Body coordinate system it rotates at the same
angular velocity as the body
Arbitrary point in the body
Rigid body angular
velocity wrt inertial frame
11Gaziantep University
oR
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
x
Z
YP
x
z
y
O
O’
r
R
Position of P
kpjpipr zyx
�
The position of P wrt inertial coordinate frame
rRR o
The absolute velocity of P is
dt
rd
dt
Rd
dt
RdV o
12Gaziantep University
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
oR
X
Z
YP
x
z
y
O
O’
r
R
dt
kdp
dt
jdp
dt
idpk
dt
dpj
dt
dpi
dt
dp
dt
rdzyx
zyx
�
rRR o
The absolute velocity of P is
dt
rd
dt
Rd
dt
RdV o
dt
kdpk
dt
dp
dt
jdpj
dt
dp
dt
idpi
dt
dp
dt
rdz
zy
yx
x
Becomes zero because body is rigid
}}rx
rxdt
rd
13Gaziantep University
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
rRR o
The absolute velocity of P is
dt
rd
dt
Rd
dt
RdV o
oo V
dt
Rd
rxdt
rd
rxVV o
oR
X
Z
YP
x
z
y
O
O’
r
R
14Gaziantep University
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
rRR o
The absolute velocity of P is
rxVV o
Acceleration of P wrt inertial coordinate system is
dt
rxd
dt
Vd
dt
Vda o )(
dt
rdxrx
dt
daa o
rxxrxaa o
oR
X
Z
YP
x
z
y
O
O’
r
R
15Gaziantep University
Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample
The 0.8 m arm OA for a remote-control mechanism is pivoted about the horizontal x-axis of the clevis, and the entire assembly rotates about the z-axis with a constant speed N=60rev/min. Simultaneously the arm is being raised at the constant rate . For the position where =60o determine (a) angular velocity of OA, (b) the angular acceleration of OA, (c) the velocity of point A, and (d) the acceleration of point A.
srad /4
sradNz /283.660/)60(260/2
sradkizx /283.64
xx k
283.62/13.254283.6 sradjixk
z
x
kjr
4.0693.0
smkji
kji
rxV
/77.260.135.4
4.0693.00
283.604
sradx /4
16Gaziantep University
Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample
The 0.8 m arm OA for a remote-control mechanism is pivoted about the horizontal x-axis of the clevis, and the entire assembly rotates about the z-axis with a constant speed N=60rev/min. Simultaneously the arm is being raised at the constant rate . For the position where =60o determine (a) angular velocity of OA, (b) the angular acceleration of OA, (c) the velocity of point A, and (d) the acceleration of point A.
srad /4
sradkizx /283.64
2/13.254283.6 sradjixk
z
x
kjr
4.0693.0
Vxrx
rxxrxa
)(
2/40.644.3811.20
77.260.135.4
283.604
4.0693.00
013.250
smkji
kjikji
a
17Gaziantep University
Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample
The electric motor with an attached disk is running at a constant low speed of 120 rev/mm in the direction shown. Its housing and mounting base are initially at rest. The entire assembly is next set in rotation about the vertical Z-axis at the constant rate N=60 rev/min with a fixed angle of 300. Determine (a) the angular velocity and angular acceleration of the disk, (b) the space and body cones, and (c) the velocity and acceleration of point A at the top of the disk for the instant shown.
srad /4
)sin(cos kj
srado /460/)2(120
srad /260/)60(2
Kkoo
sradkj
kj
kj
kjk
oo
o
o
/)0.53(
)30sin24()30cos2(
)sin()cos(
)sin(cos
18Gaziantep University
Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample
The electric motor with an attached disk is running at a constant low speed of 120 rey/mm in the direction shown. Its housing and mounting base are initially at rest. The entire assembly is next set in rotation about the vertical Z-axis at the constant rate N=60 rev/min with a fixed angle of 300. Determine (a) the angular velocity and angular acceleration of the disk, (b) the space and body cones, and (c) the velocity and acceleration of point A at the top of the disk for the instant shown.
srado /460/)2(120 srad /260/)60(2
sradkj /)0.53(
19Gaziantep University
Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample
The electric motor with an attached disk is running at a constant low speed of 120 rey/mm in the direction shown. Its housing and mounting base are initially at rest. The entire assembly is next set in rotation about the vertical Z-axis at the constant rate N=60 rev/min with a fixed angle of 300. Determine (a) the angular velocity and angular acceleration of the disk, (b) the space and body cones, and (c) the velocity and acceleration of point A at the top of the disk for the instant shown.
srado /460/)2(120
srad /260/)60(2
x
sradiii
iio
o
o
/4.6830cos)4)(2()cos(
)cossin()cossincos( 2
kjxkj o
)sin()cos()sin(cos
20Gaziantep University
Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample
The electric motor with an attached disk is running at a constant low speed of 120 rey/mm in the direction shown. Its housing and mounting base are initially at rest. The entire assembly is next set in rotation about the vertical Z-axis at the constant rate N=60 rev/min with a fixed angle of 300. Determine (a) the angular velocity and angular acceleration of the disk, (b) the space and body cones, and (c) the velocity and acceleration of point A at the top of the disk for the instant shown.
srado /460/)2(120 srad /260/)60(2
sradkj /)0.53(
21Gaziantep University
Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample
The electric motor with an attached disk is running at a constant low speed of 120 rey/mm in the direction shown. Its housing and mounting base are initially at rest. The entire assembly is next set in rotation about the vertical Z-axis at the constant rate N=60 rev/min with a fixed angle of 300. Determine (a) the angular velocity and angular acceleration of the disk, (b) the space and body cones, and (c) the velocity and acceleration of point A at the top of the disk for the instant shown.
srad /4
kjr
250.0125.0
smi
kji
rxV /1920.0
250.0125.00
530
Vxrxrxxrxa )(
2/83.116.26
)192.0()53()250.0125.0(4.68
smkj
ixkjkjxia
22Gaziantep University
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
Br
X
Z
YA
x
z
y
O
B
BAr
Ar
coordinate system rotates with
this angular velocirty
Body coordinate frame rotates with this angular velocirty
BFLetting
rxrVV BA
Denote the angular velocity of the reference wrt the body frame, the angular
velocity of the body is related to that of the coordinate system
BF
The velocity of a point of the body may be represented by
rxVVrxrxVV relBB
FBA
23Gaziantep University
PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body
Br
X
Z
YA
x
z
y
O
B
BAr
Ar
coordinate system rotates with
this angular velocirty
Body coordinate frame rotates with this angular velocirty
BA
BA
BA
BABA rxxrxrxraa
2
Acceleration of a point of the body is obtained as:
The velocity of a point of the body may be represented by
BA
BF
BF
BA
BFrel rxxrxa
rel
BA
BF Vxrxx
22
BA
BA
BA
FB
BA
FB
FB
BA
FBBA
BA
BArelrelBA
rxxrxrxxrxxrxaa
rxxrxVxaaa
2
2
rxrVV BA
24Gaziantep University
Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample
The motor housing and its bracket rotate about the Z axis at the constant rate The motor shaft and disk have a constant angular velocity of spin with respect to the motor housing in the direction shown. If constant at 30o, determine the velocity and acceleration of point A at the top of the disk and angular acceleration of the disk.
srad /3
K
3
JrB
350.0
kjrB
A
120.0300.0
smiI
JxKrxV BB
/05.105.1
350.03
relBA VrxVV
smiiikjxKrxB
A /599.0)30sin36.0()30cos9.0()120.0300.0(3
smikjxjrxpVB
Arel /960.0)120.0300.0(8
smiiiiVA /689.0960.0599.005.1
sradp /8
25Gaziantep University
Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample
The motor housing and its bracket rotate about the Z axis at the constant rate The motor shaft and disk have a constant angular velocity of spin with respect to the motor housing in the direction shown. If constant at 30o, determine the velocity and acceleration of point A at the top of the disk and angular acceleration of the disk.
srad /3
2/899.073.2)30sin30cos(15.3
15.3)350.03(3)(
smkjkj
JJxKxKrxxa BB
2/899.0557.1)599.0(3
)120.0300.0(33)(
smkjixK
kjxKxKrxxB
A
2/88.299.4)30sin30cos(76.576.5960.0)3(22 smkjkjJixKVx rel
sradp /8
BA
BArelrelBA rxxrxVxaaa
2
0
2/68.7))120.0300.0(8(8)( smkkjxjxjrxpxpaB
Arel
2/086.8703.0 smkjaA
222 /12.8086.8703.0 smaA 2/8.20)30cos24(0 sradii
)83(3 jKxKx