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0593_C11_fm Page 398 Monday, May 6, 2002 2:59 PM

398 Dynamics of Mechanical Systems

j
O n
P i 3
1
n n
θ P 2 3 x 3 2
2
2 x
P n 3 2
3 n n
T 1 x 1 1
R
P P P T
1 2 3
FIGURE 11.12.5
Rotating tube containing spring-supported FIGURE 11.12.6
particles. Coordinates of particles in the tube.

Example 11.12.4: Spring-Supported Particles in a Rotating Tube
For another example illustrating the use of kinetic energy to obtain generalized inertia
forces, consider again the system of spring-supported particles in a rotating tube as in
Figure 11.12.5. (We considered this system in Sections 11.7 and 11.11.) Recall that this
system has four degrees of freedom represented by the coordinates x , x , x , and θ, as
2
1
3
shown in Figures 11.12.5 and 11.12.6.
The velocities of the particles, the tube mass center velocity, and the angular velocity of
the tube in the inertia frame R are (see Eqs. (11.10.23) through (11.10.27)):
v = ˙ x n +(l + x )θ ˙ n
P 1
11 1 2
2 +
v = ˙ x n +( l x )θ ˙ n
P 2
2 1 2 2
3 +
v = ˙ x n +( l x )θ ˙ n (11.12.16)
P 3
3 1 3 3
v = ( L 2)θ ˙ n
G
2
ωω = θ ˙ n
3
where the notation is the same as we used in Section 11.7 and 11.11.
The kinetic energy of the system is then:

2
2
K = 1 m v ( ) + 1 m v ( ) + 1 m v ( ) 2
1 P
2 P
3 P
2 2 2
1
()
2
+ 1 M v ( ) + I ωω 2
G
2 2
(11.12.17)
2
2
= ( )mx ˙ 2 +(l + ) θ 2 ˙ + x ˙ 2 +(2l + ) θ 2 ˙ + x ˙ 2
x
12
x
[ 1 1 2 2 3
2
2
θ
+(3l + ) θ 2 ˙ ] +( ) ( ) 2 θ +( )( 2 2 ˙ θ
˙ 2
1 2 M L
12 1 12) ML
x
3