Tutorial 13 Planar dynamics of

rigid bodyZhengjian, XU

DEC 3rd, 2008

Rotation about a fixed axis for a mass point

dt

dmm

vrarFr

vrv

rvr

mdt

d

dt

dm

dt

md

)(

MH

Fr dt

dAngular momentum:

Where H is the angular momentum with respect to the center of mass

Calculation of moment of inertia dmrrmI

m

ii 22

Circle:

Rectangle:

A bar:2

12

1mlIO

A BO

L

2

3

1mlII BA

x

y

O

2

12

1mhI x 2

12

1mbI y

)(12

1 22 bhmIO

O x

y

b

h

R2

4

1mRIO

2

8

1mRII yx

Another method: parallel-axis method

Example 1 A horizontal force F = 30 lb is applied to the 230-lb refrigerator as

shown. Friction is negligible. (a) What is the magnitude of the refrigerator’s acceleration? (b) What normal forces are exerted on the refrigerator by the floor at

A and B?

Solution: Assume the box doesnot tip over, then the box

has only horizontal velocity and acceleration.

NA NB

F

G28in

60in

28in

Vx, axxx

maF F

0 BAyNNGF

0141432 BAONNFM

Force equilibrium:

What’s the condition of F for tipping?

O

2ft/s2.4xa

lb7.80AN lb3.149BN

Example 2 Bar AB rotates with a constant angular velocity of 10 rad/s in the co

unterclockwise direction. The masses of the slender bars BC and CDE are 2 kg and 3.6 kg, respectively. The y axis points upward.

Determine the components of the forces exerted on bar BC by the pins at B and C at the instant shown.

Bx

By

Cy

Cx

B

C

C

D

ECx

Cy

1. Dynamics analysis:

xBCxx amCB G

yBCyy amCB

Moment equlibrium equations

BCBCyxyx ICCBB )35.02.0()35.02.0(

CDDCEy IC 40.0

Bx

By

Cy

Cx

2 Kinematics analysis

B

C

C

D

E

VBVC

At this instant, point A is the instantaneous center of BC.

ABABAB

BCABBCB

R

Rv

400

400

CDCDCE

BCACBCC

R

Rv

400

700

Cx

Cy

rad/s 10 ABBC

rad/s 5.174

7 BCCD

)(m/s jjja 2B 404.0100)( 2 ABAB R

BCBCBC2

BC RαR)(aa BCFrom AB-BC:

From CD: jijiaC CDCDCDCDCD α.RαR 405.122)( 2

From the kinematics analysis:

rad/s 10BC

rad/s 5.17CD