Page 389 - Dynamics of Mechanical Systems

0593_C11_fm Page 370 Monday, May 6, 2002 2:59 PM

370 Dynamics of Mechanical Systems

n
j 2
n i
O 2
3 x n
P 3 1
1
θ P 2 x
2 n 2
1 n n
P 3 x 1 3
3
T
P P P T
1 2 3
FIGURE 11.7.2 FIGURE 11.7.3
A rotating tube containing spring-connected Coordinates of the particles within the tube.
particles.
This system has four degrees of freedom described by the coordinates x , x , x , and θ.
2
3
1
Our objective is to determine the generalized forces applied to the system corresponding
to these coordinates. (This example is a modified and extended version of a similar
example appearing in References 11.1 and 11.2.)
To determine the generalized forces, it is helpful to first determine the partial velocities
of the particles, the partial velocities of the mass center G of T and the partial angular
velocities of T. By using the procedures of Chapter 4, we find the angular velocity of T,
its mass center velocity, and the particle velocities to be:

˙
˙
T
2
l
ωω = θn , v G = ( )θn (11.7.1)
3 2
and
v = ˙ x n +(l + x )θ ˙ n (11.7.2)
P 1
11 1 2
2 +
v = ˙ x n +( l x )θ ˙ n (11.7.3)
P 2
2 1 2 2
3 +
v = ˙ x n +( l x )θ ˙ n (11.7.4)
P 3
3 1 3 2
where n , n , and n are the unit vectors of Figures 11.7.2 and 11.7.3. From Eqs. (11.4.4)
1
3
2
and (11.4.15) we obtain the partial velocities and partial angular velocities:
ωω = ωω = ωω = 0 , ωω = n (11.7.5)
T
T
T
T
˙ x 1 ˙ x 2 ˙ x 3 ˙ θ 3
and
v = n , v P 1 = 0 , v = 0 , v = (l + x n )
P 1
P 1
P 1
˙ x 1 1 ˙ x 2 ˙ x 3 ˙ θ 1 2
v = 0 , v P 2 = n , v = 0 , v = ( l x n )
2 +
P 2
P 2
P 3
˙ x 1 ˙ x 2 1 ˙ x 3 ˙ θ 2 2
(11.7.6)
3 + )
v = 0 , v = 0 , v = n , v = ( l x n
P 3
P 3
3
P 3
P 3
˙ x 1 ˙ x 2 ˙ x 3 1 ˙ θ 3 2