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0593_C11_fm Page 383 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 383

v G 1 = (l 2) n =−(l 2) sinθ 1 n +(l 2) cosθ 1 n 2
θ 1
1
˙ θ 1
v G 2 = l n = −l sinθ n + l cosθ n
˙ θ 1 θ 1 1 1 1 2
(11.10.19)
v G 1 = 0
˙ θ 2
v G 2 = (l 2) n θ =−(l 2) sinθ n +(l 2) cosθ n
˙ θ 2 2 2 1 2 2

and

1 B
B
1 B
B
ωω = n , ωω = 0 , ωω = 0 , ωω = n (11.10.20)
2
2
˙ θ 1 3 ˙ θ 1 ˙ θ 2 ˙ θ 2 3
Finally, from Eq. (11.9.6) the generalized inertia forces are:
θ cos(
θ −(
F =−( 4 3)ml 2 ˙˙ 12)ml 2 ˙˙ θ − ) +( 12)ml 2 2 ˙ θ sin( θ − ) (11.10.21)
θ
θ
*
θ 1 1 2 2 1 2 2 1
and
F =−( 13)ml 2 ˙˙ θ −( 12)ml 2 ˙˙ θ − ) −( 12)ml 2 2 ˙ θ sin( θ − ) (11.10.22)
θ cos(
θ
θ
θ 2 * 2 1 2 1 1 2 1
Observe how routine the computation is; the principal difficulty is the detail. We will
discuss this later.
Example 11.10.4: Spring-Supported Particles in a Rotating Tube
Consider again the system of Section 11.7 consisting of a cylindrical tube T containing
three spring-supported particles P , P , and P (or small spheres) as in Figure 11.10.5. As
1
2
3
before, T has mass M and length L and it rotates in a vertical plane with the angle of
rotation being θ as shown. The particles each have mass m, and their positions within T
are defined by the coordinates x , x , and x as in Figure 11.10.6, where is the natural
2
3
1
length of each of the springs.
j
n
2 n 3
i
P 3 x n 2
1 3

P n
2 1 2 x
2
P n x n
3 3 1 1
T
R
P P P T
1 2 3
FIGURE 11.10.5 FIGURE 11.10.6
A rotating tube containing spring-supported Coordinates of particles within the tube.
particles.