Page 117 - Dynamics of Mechanical Systems

0593_C04*_fm Page 98 Monday, May 6, 2002 2:06 PM

98 Dynamics of Mechanical Systems

Equations (4.9.4) and (4.9.6) are sometimes used to provide an interpretation of rigid
body motion as being a superposition of translation and rotation. Specifically, at any
instant, from Eq. (4.9.4) we may envision a body as translating with a velocity V and
Q
rotating about Q with an angular velocity ωω ωω .
B
R
Example 4.9.1: Relative Movement of a Sports Car Operator’s Hands During
a Simple Turn
To illustrate the application of Eqs. (4.9.4) and (4.9.5), suppose a sports car operator
traveling at a constant speed of 15 miles per hour, with hands on the steering wheel at 10
o’clock and 2 o’clock, is making a turn to the right with a turning radius of 25 feet. Suppose
the steering wheel has a diameter of 12 inches and is in the vertical plane. Suppose further
that the operator is turning the steering wheel at one revolution per second. Find the
velocity and acceleration of the motorist’s left hand relative to the right hand.
Solution: Let the movement of the car be represented as in Figure 4.9.2, and let the
steering wheel be represented as in Figure 4.9.3, where n , n , and n are mutually per-
y
z
x
pendicular unit vectors fixed in the car. Let n be forward; n be vertical, directed up; and
x
z
n be to the left. Let W represent the steering wheel, C represent the car, and S the road
y
surface (fixed frame). Let O represent the center of W, and let L and R represent the
motorist’s left and right hands. The desired relative velocity and acceleration ( V and
L/R
S
s L/R
a ) may be obtained directly from Eqs. (4.9.4) and (4.9.5) once the angular velocity and
the angular acceleration of W are known. From the addition theorem for angular velocity
(Eq. (4.7.6)), the angular velocity of the steering wheel W in S is:
ωω = ωω + ωω (4.9.7)
S W C W S C
From the data presented, the angular velocity of the car C in S is:
S ωω= −( V z ) ρ n =−(22 25 z )n rad sec (4.9.8)
C

where V is the speed of C (15 mph or 22 ft/sec) and ρ is the turn radius. Similarly, the
angular velocity of the steering wheel W in C is:

C W
ωω= 2πn rad sec (4.9.9)
x
W
S
Hence, ωω ωω is:
S W − z )
ωω= 2πn x (22 25 n rad sec (4.9.10)

n
x

n
y n
z
25 ft
25 ft
15 mph C
FIGURE 4.9.2
Sports car C entering a turn to the
right.