Page 388 - Dynamics of Mechanical Systems

0593_C11_fm Page 369 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 369

ˆ
The contribution of the spring force to the generalized force F on S is then:
F
r r
F = () n v − () ⋅ P 2
⋅
ˆ
P 1
r fx ˙ q r fx n v ˙ q r
n⋅∂
= fx () ( v − v P 2 )
P 1
˙ q r ˙ q r
= fx () n⋅∂ ( v P 1 ˙ q ∂ − ∂ v P 2 ˙ q ∂ ) (11.6.6)
r r
= fx () n⋅ ∂ ( v − ) ( r =…1 , ,n)
v
P 1
P 2
˙ q ∂
r
Observe that the direction of the unit vector n will be a function of the coordinates q but
r
˙ q
not the coordinate derivative . Hence, in Eq. (11.6.6) n may be taken inside the partial
r
ˆ
˙ q F
derivative with respect to the . may thus be written as:
r r
⋅( [
F = () ∂ n v − )]
ˆ
fx
v
P 1
P 2
r ˙ q ∂
r

= fx () ∂  n⋅− dx  
n
˙ q ∂ r    dt   
=− fx () ∂  dx
˙ q ∂  dt 
r
n
=− fx () ∂   x ∂ + ∑ x ∂ ˙ q  
˙ q ∂ r   t ∂ s=1 q ∂ s s  
or
F =− () x ∂
ˆ
r fx q ∂ (11.6.7)
r

11.7 Example: Spring-Supported Particles in a Rotating Tube
For an example illustrating these concepts, consider a cylindri-
O
cal tube T with mass M and length L. Let T be pinned at one
of its ends O, and let T rotate in a vertical plane with the angle
θ L
of rotation being as in Figure 11.7.1. Let T contain three identical
spring-supported particles (or small spheres or balls) each hav-
T
ing mass m as in Figure 11.7.2. Let the four connecting springs
also be identical. Let the springs be linear with each having
modulus k and natural length . Let the interior of T be smooth FIGURE 11.7.1
so that the particles can move freely along the axis of T. Let the A rotating cylindrical tube.
movement of the particles within T be defined by the coordi-
nates x , x , and x shown in Figure 11.7.3. These coordinates measure the displacements of
1 2 3
the particles away from their static equilibrium position where the tube is horizontal.