Page 149 - Dynamics of Mechanical Systems

0593_C05_fm Page 130 Monday, May 6, 2002 2:15 PM

130 Dynamics of Mechanical Systems

v = v = v n and a = a = a n (5.3.7)
P
Q
P
Q
where n is a unit vector parallel to the line of motion.

5.3.2 Rotation

If a body is in pure rotation, with its particles moving in circles, the kinematics can be
developed using the procedures of Section 3.7 describing the motion of points on circles.
To illustrate this, consider the body B depicted in Figure 5.3.2. Let P be a typical particle
of B. Let P move on a circle with radius r and center O as shown. Then, O has zero
P
velocity (otherwise, P would not move in a circle). From Eqs. (3.7.6) and (3.7.7), the velocity
and acceleration of P may then be expressed as:

v = ω n and a = α n − ω 2 n (5.3.8)
P
P
r
r
r
p θ p θ p r
where ω and α represent the angular velocity and angular acceleration, respectively, of
B and where n and n are radial and tangential unit vectors, respectively, as shown in
θ
r
Figure 5.3.2.
Let Q be any other point of B. If Q is located a distance r from O, then the velocity and
Q
acceleration of Q can be expressed as:
v = ω n and a = α n − ω 2 n (5.3.9)
r
r
r
Q Q θ Q Q θ Q r
By comparing Eqs. (5.3.8) and (5.3.9), we see that the magnitudes of the velocities and
accelerations are directly proportional to the radii of the circles on which the particles
move. Also, we see that if we know the velocity and acceleration of one point, say P, we
can find the angular velocity and angular acceleration of the body and, hence, the velocity
and acceleration of any and all other points.
5.3.3 General Plane Motion

General plane motion may be considered as a superposition of translation and rotation.
The velocity and acceleration of a typical particle P of a body in general plane motion
may be obtained from Eqs. (4.9.4) and (4.9.6). Specifically, if Q is a particle of B in the
plane of motion of P, the velocity and acceleration of P may be expressed as:

v = v + ωω + r (5.3.10)
Q
P

n θ
Q
ω α
n
r P r
P
O

FIGURE 5.3.2 B
A body B in pure rotation.