Page 147 - Dynamics of Mechanical Systems

0593_C05_fm Page 128 Monday, May 6, 2002 2:15 PM

128 Dynamics of Mechanical Systems

Equations (5.2.6) and (5.2.7) form a system of (3 + 3(N – 3)) or (3N – 6) constraint
equations. Because N unrestrained particles (in three dimensions) require 6N coordinates
to define their locations, the (3N – 6) constraint equations reduce the number of degrees
of freedom for the rigid body to 6N – (3N – 6), or six.
In like manner it is seen that, if a rigid body is restricted to planar movement, it has three
degrees of freedom.

5.3 Planar Motion of a Rigid Body
When a body has planar motion, each particle of the body moves in a plane; however, all
of the particles do not move in the same plane. Instead, they move in planes that are
parallel to each other, as depicted in Figure 5.3.1. In the figure, P and P are typical particles
1 2
of a body B. They move in parallel planes π and π . If the particles happen to be on the
1 2
same normal line N of the planes, they have identical motions. This means that if we
consider the movements of the particles of B in one of the planes (parallel to the planes
of motion), we are in effect considering the motion of all the particles of B. That is, any
particle of B not in our considered plane of motion can be identified with a particle in
that plane. Hence, the motion of B can be studied entirely in a plane.
Because a body with planar motion has at most three degrees of freedom, the kinematic
analysis is greatly simplified from that of general three-dimensional motion. Many of the
kinematical quantities are then more conveniently described with scalars than with vec-
tors. For example, with planar motion the angular velocity of a body is always directed
normal to the plane of motion. Hence, a vector is not needed to define its direction.
To demonstrate this, consider the definition of angular velocity (Eq. (4.5.1)):

ωω= ( [ ] + ( [ ] + ( [ ]
dn
2 dt )⋅n n 1 dn 3 dt ) ⋅n n 2 dn 1 dt )⋅n n 3 (5.3.1)
1
3
2
where n , n , and n are mutually perpendicular unit vectors fixed in the body. Let n be
1 2 3 3
parallel to line N and thus perpendicular to the plane of motion of the body B (see Figure
5.3.1). Let particles P and P of B lie on N, a distance d apart. Because the orientation of
1 2
N is fixed, n has fixed orientation and is thus constant. Hence, dn /dt is zero. Also, because
3 3
n and n are perpendicular to n they remain parallel to the planes of motion of B. Their
1 2 3
N
n 3
B π
1
P
1
π
d 2
P
2

FIGURE 5.3.1
A body with planar motion.