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0593_C11_fm Page 384 Monday, May 6, 2002 2:59 PM

384 Dynamics of Mechanical Systems

The velocities and accelerations of P , P , and P in the fixed or inertial frame R are (see
3
2
1
Eqs. (11.7.2), (11.7.3), and (11.7.4)):
2 ˙
θ
˙˙
θ
v = ˙ x n +(l + x )θ ˙ n , a = ˙˙ 1 [ + x ) ] n + ( [ l + x ) + 2θ ˙ x ˙ 1] n (11.10.23)
x −(l
P 1
P 1
11 1 2 1 1 1 2
θ
2 +
θ
˙˙
2 ˙
l
x −( l
v = ˙ x n +( l x )θ ˙ n , a = ˙˙ 2 [ 2 + x ) ] n + ( [ 2 + x ) + 2θ ˙ x ˙ 2] n (11.10.24)
P 2
P 2
2 1 2 2 2 1 2 2
2 ˙
θ
θ
3 +
˙˙
l
x −( l
v = ˙ x n +( l x )θ ˙ n , a = ˙˙ 3 [ 3 + x ) ] n + ( [ 3 + x ) + 2θ ˙ x ˙ 3] n (11.10.25)
P 3
P 3
3 1 3 2 3 1 3 2
where n , n , and n are the unit vectors shown in Figures 11.10.5 and 11.10.6. Let G be
2
1
3
the mass center of T. Then, from Eq. (11.7.1), the velocity and acceleration of G in R are:
v = ( ) 2 θ ˙ n , a =−( ) 2 θ 2 ˙ n +( ) 2 θ n (11.10.26)
˙˙
G
G
L
L
L
2 1 2
Finally, the angular velocity and the angular acceleration of T in R are:
˙˙
˙
ωω = θn and αα =θn (11.10.27)
3 3
As noted in Section 11.7, the system has four degrees of freedom represented by the
coordinates x , x , x , and θ. The corresponding partial velocities and partial angular
2
1
3
velocities are recorded in Eqs. (11.7.5), (11.7.6), and (11.7.7) as:
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
2 +
v = 0 , v P 2 = n , v = 0 , v = ( l x n )
P 2
P 2
P 2
˙ x 1 ˙ x 2 1 ˙ x 3 ˙ θ 2 2
3 + )
v = 0 , v P 3 = 0 , v = n , v = ( l x n (11.10.28)
P 3
P 3
P 3
3
˙ x 1 ˙ x 2 ˙ x 3 1 ˙ θ 3 2
v G = 0 , v G = 0 , v G = 0 , v G = (L 2)n
1 x ˙ 2 x ˙ 3 x ˙ ˙ θ 2
ωω = 0 , ωω = 0 , ωω = 0 , ωω = n
1 x ˙ 2 x ˙ 3 x ˙ ˙ θ 3
From Eqs. (11.10.23) to (11.10.27), the inertia forces on the particles and the tube may be
represented by:
1 [
˙˙
2 ˙
θ
F =− m a =− m x ˙˙ −(l + x ) ] n − ( [ + x ) + 2θ ˙ x ˙ 1] n
θ
*
m l
P 1
P 1 1 1 1 2
2 [
θ
m 2 + ) +
2 ˙
F =− m a =− m x ˙˙ −( 2 + x ) ] n − ( [ l x θ 2θ ˙ x ˙ 2] n
˙˙
*
l
P 2
P 2 2 1 2 2
3]
3) ]
[
− (
F =− m a =−mx ˙˙ 3 ( 3 + x θ n 1 [ 3 + x 3) +θ 2 x ˙ n (11.10.29)
θ
˙
˙˙
2 ˙
−
*
l
l
P 3
m
3
P 3 2
˙˙
2 ˙
F * =−Ma G = (L 2)θ n − (L 2)θn
M
M
T 1 2
˙˙
ααωω
ωω
T * =−I ⋅ − I ⋅ ) =−( ML 2 12) θn
T G ×( G 3