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0593_C04*_fm Page 85 Monday, May 6, 2002 2:06 PM

Kinematics of a Rigid Body 85

Example 4.4.1: Revolutions Turned Through During Braking
Suppose a rotor is rotating at an angular speed of 100 rpm. Suppose further that the rotor
2
is braked to rest with a constant angular deceleration of 5 rad/sec . Find the number N
of revolutions turned through during braking.
Solution: From Eq. (4.4.8), when the rotor is braked to rest, its angular speed ω is zero.
The angle θ turned through during braking is, then,
) ] () − ( )]
θ =− ω 2 α =− ( [ 2 π)(100 60 2 [ 2 5
2
o
= 10 966. radians = 628 3. degrees
Hence, the number of revolutions turned through is:

.
N = 1 754

4.5 General Angular Velocity
Angular velocity may be defined intuitively as the time rate of change of orientation.
Generally, however, no single quantity defines the orientation for a rigid body. Hence,
unlike velocity, angular velocity cannot be considered as the derivative of a single quantity.
Nevertheless, it is possible to define the angular velocity in terms of derivatives of a set
of unit vectors fixed in the body. These unit vector derivatives thus provide a measure of
the rate of change of orientation of the body.
Specifically, let B be a body whose orientation is changing in a reference frame R, as
depicted in Figure 4.5.1. Let n , n , and n be mutually perpendicular unit vectors as shown.
1
3
2
R
B
Then, the angular velocity of B in R, written as ω , is defined as:
ωω = ⋅nn + ⋅nn + ⋅nn (4.5.1)
R B D  dn 2   dn 3   dn 1 
 dt 3  1  dt 1  2  dt 2  3
The angular velocity vector has several properties that are useful in dynamical analyses.
Perhaps the most important is the property of producing derivatives by vector multipli-
cation: specifically, let c be any vector fixed in B. Then, the derivative of c in R is given
by the single expression:

ddt= ωω B × c (4.5.2)
R
c

B
n
3
n
2

n
1
FIGURE 4.5.1 R
A rigid body changing orientation in
a reference frame.