chap16 notes

8/12/2019 CHAP16 Notes

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Chapter 16 PLANE MOTION OF RIGID BODIES:

FORCES AND ACCELERATIONS

1. A Relation exists between

the forces acting on a rigid

body, the shape and massof the body.

2. Study of these and themotion produced is knownas the kinetics of rigid

bodies.

3. Analysis is restricted to theplane motion of rigid slabs

and rigid bodiessymmetrical with respect tothe reference plane.

G

F1

F2

F3

F4

HG

ma

G

.


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The two equations for themotion of a system of

particles apply to the most

general case of the motionof a rigid body.

The first equation defines

the motion of the masscenter G of the body.

G

F1

F2

F3

F4

HG

ma

G

.

F = ma

2. The second is related to the motion of the body relative to acentroidal frame of reference.

MG = HG

.

where m is the mass of

the body, and a theacceleration ofG.


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G

F1

F2

F3

F4

HG

ma

G

.

F = ma

MG = HG

.

where HG is the rate of

change of the angularmomentum HG of the

body about its mass

centerG.

These equations express that the system of the external forces

is equipollent to the system consisting of the vector ma attached

at G and the couple of moment HG..

.

.


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G

F1

F2

F3

F4

HG

ma

G

.

HG = I

For the plane motion of

rigid slabs and rigid

bodies symmetrical withrespect to the reference

plane, the angular

momentum of the body is

expressed as

whereIis the moment of inertia of the body about a centroidal

axis perpendicular to the reference plane and is the angular

velocity of the body. Differentiating both members of thisequation

HG = I = I. .


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G

F1

F2

F3

F4For the restricted case

considered here, the rate

of change of the angular

momentum of the rigid

body can be represented

by a vector of the same

direction as

(i.e.

The plane motion of a rigid body symmetrical with respect to

the reference plane is defined by the three scalar equations

maG

I

perpendicular to the plane of reference) and of magnitudeI.

Fx = max Fy = may MG =I

The external forces acting on a rigid body are actually equivalent

to the effective forces of the various particles forming the body.

This statement is known as dAlemberts principle.


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Copyright 1997 by The McGraw-Hill Companies, Inc. All rights reserved.

Fig. 16.7


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G

F1

F2

F3

F4 dAlemberts principle can

be expressed in the form

of a vector diagram, wherethe effective forces are

represented by a vectorma attached at G and a

coupleI. In the case of a

slab in translation, theeffective forces (partb of

the figure) reduce to a

(a) (b)

single vector ma ; while in the particular case of a slab in

centroidal rotation, they reduce to the single coupleI ; in any

other case of plane motion, both the vector ma andI shouldbe included.

maG

I


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Fig. 16.10



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Fig. 16.15



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Fig. 16.19

Copyright1997

byTheMcGraw-HillCompa

nies,Inc.Allrightsreserved

.


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G

F1

F2

F3

F4 Any problem involving the

plane motion of a rigid slab

may be solved by drawing afree-body-diagram equation

similar to that shown. Three

equations of of motion can

then be obtained by

equating thex components,y components, and moments about an arbitrary pointA, of the

forces and vectors involved.

This method can be used to solve problems involving the

plane motion of several connected rigid bodies.

Some problems, such as noncentroidal rotation of rods and

plates, the rolling motion of spheres and wheels, and the plane

motion of various types of linkages, which move under

constraints, must be supplemented by kinematic analysis.

ma

G

I


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Fig. 16.18

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