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0593_C11_fm Page 393 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 393

FIGURE 11.11.5
A rotating tube containing spring-
connected particles.

˙ q
each of the q . Also, if the F involve derivatives of the (such as with “dissipation forces”),
r
r
r
the integrals will not exist, because P is to depend only on the q .
r
Further, observe that if knowledge of the generalized forces is needed to obtain a
potential energy function which in turn is to be used to obtain the generalized forces, little
progress has been made. Therefore, in the solution of practical problems as in machine
dynamics, potential energy is useful primarily for gravity and spring forces. Finally, it
should be noted that potential energy is not a unique function. Indeed, from Eq. (11.11.1),
we see that the addition of a constant to any valid potential energy function P also produces
a valid potential energy function.
It may be helpful to consider another illustration. Consider again the system of the
rotating tube containing three spring-supported particles as in Figures 11.11.5 and 11.11.6.
We discussed this system in Section 11.7, and we will use the same notation here without
repeating the description (see Section 11.7 for the details).
Recall from Eqs. (11.7.14) to (11.7.17) that the generalized forces for the coordinates x ,
1
x , x , and θ are:
2
3
F = mgcosθ + kx − 2 kx (11.11.20)
x 1 2 1
F = mgcosθ + kx + kx − 2 kx (11.11.21)
x 2 3 1 2
F = mgcosθ − 2 kx + kx (11.11.22)
x 3 3 2
(
( )
F =−Mg L 2 sin θ − mg 6l + x 1 + x 2 + )sin θ (11.11.23)
x
θ
3

FIGURE 11.11.6
Coordinates of the particles within
the tube.