chap16 notes
TRANSCRIPT
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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
Copyright 1997 by The McGraw-Hill Companies, Inc. All rights reserved.
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Fig. 16.15
Copyright 1997 by The McGraw-Hill Companies, Inc. All rights reserved.
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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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Copyright 1997 by The McGraw-Hill Companies, Inc. All rights reserved.
Fig. 16.18