kinematics - plane motion jacob y. kazakia © 20051 types of motion translation ( a straight line...

Kinematics - Plane motion

Jacob Y. Kazakia © 2005 1

Types of Motion

• Translation ( a straight line keeps its direction)

1. rectilinear translation2. curvilinear translation

• Rotation about a fixed axis ( particles remain in the same plane)

• General Plane Motion ( particles remain in the same plane)

• Motion about a fixed point

• General Motion



Translation

rA

rB

AB

x

z

y r B = r A + r B/A

r B/A has constant direction during translation &r B/A has constant length since we have a rigid body

consequently: v A = v B & a A = a B

rectilinear translation: curvilinear translation:



Rotation about a fixed axis

t

r

y

z

x O

= r sin 1 rev = 2 rad = 360o

the velocity vector v must be tangent to the circle

kwithrv

rdt

ds

rsdt

ds

t

sv

t

sin

sinlim0

kwhere

vrdt

rdr

dt

d

dt

vd

normal cmp.tangential cmp.



Example 1

300 mm

200 mm

120 mm

A B

C

D x

y

z

The bent rod ABCD rotates about AD with speed = 95 rad/s ( velocity of C downwards).Determine the velocity and acceleration of point B.

D(0.3, 0, 0) and A(0, 0.2, 0.12) hence

12.0,2.0,3.038.0

1Consequently:

0,0,3.0

0

30,50,7512.0,2.0,3.025095

/ ABr

And we can now calculate the velocity and acceleration vectors for point B.



Example 1 cont.

300 mm

200 mm

120 mm

A B

C

D x

y

z

0,0,3.0

0

30,50,75

/ ABr

kji

kji

vra

kjkji

kji

rv

BABB

ABB

67511251020

1590

305075

1591590

003.0

305075

/

/



General Plane Motion

A

B

A

B B

A

vA vA

vA

vB

vB/A

y’

x’

rB/A

Translation with A Rotation about A

rv

rkrv

vvv

AB

ABABAB

ABAB

/

///

/

The angular velocity of a rigid body in plane motion is independent of the reference point



Example 1

30 in

End A of the rod moves with speed vA= 25 in/s to the right. Angle = 250.Determine:1. The angular velocity of the rod2. The velocity of end B

Method 1:

A

B

A

B

A

B

A

B

vA

vB

vA

vA

vB/A

k

vA

vB vB/A

sradv

sinv

sinv

AB

AB

B

/55.030

/64.16

/5.23

/

/

40sin75sin65sin/ ABAB vvv



Example 1 cont.

30 in

End A of the rod moves with speed vA= 25 in/s to the right. Angle = 250.Determine:1. The angular velocity of the rod2. The velocity of end B

A

B

Method 2:x

y

0,25cos30,25sin30

0,25sin30,25cos30,,0,0

0,40sin,40cos40sin,40cos

unknowns are,0,0,250,25

/

/

AB

AB

BBBByBBxB

BAyAxA

r

r

vvvvvvv

vvvv

ABAABAB rvvvv//

From the relation:

We obtain:25cos30040sin

25sin302540cos

B

B

v

v

srad

sinvB/55.0

/5.23



Example 2B

C

C

vD

vB

vE

vC

r A = 120 mmr B = 60 mmr C = 45 mm

CB

AC

A

rpm

,

determine

48

0

/

BBBBDBD

D

CBACBAB

rvvvv

v

rrrvandrrv

/

Agear from0

2

rpmr

rr

B

BAB 1443

clockwise

BABBABBBE rrrrrrvv 43

Bgear from

ckwisecounterclorpm

r

rrrrr

r

vv

rvv

C

CBABA

C

CEC

CCCE

803

524

Cgear from



Example 380 mm

160 mm

180 mm 60 mm

a

B/A = 3 rad/s counterclockwise

Determine: C/B and D/C

A

B

C D

0

22

4.18240

80tan

25324080

mma

srad

sradvvv

vv

vv

vsmmv

CDCD

BCBCCDBCB

CDCDCDCD

BCBCBCBC

BAB

/401804.18cos759

/5.101604.18sin7590

:havemust we

1,0180180

0,1160160

4.18cos759,4.18sin759/7592533

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kinematics - plane motion jacob y. kazakia © 20051 types of motion translation ( a straight line...

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