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0593_C11_fm Page 401 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 401

θ
θ
˙ ˙
*
32
F =−mr 2 [ ( ) ψ ˙˙ sin +( ) ˙˙ 2 θ +( ) φθsin cos θ
32
φsin
52
φ
(11.12.33)
+( ) ˙˙ θ 1 ψθcos θ]
˙ ˙
φcos −( ) 2
14
and
˙ ˙
˙˙
θ
*
F =−mr 2 [ ( ) ψ ˙˙ +( ) φsin +( ) θφcos θ ] (11.12.34)
32
52
32
ψ
While Eqs. (11.12.29), (11.12.30), and (11.12.31) are similar to Eqs. (11.12.32), (11.12.33),
and (11.12.34), they are not identical; therefore, Eq. (11.12.5) is not valid for nonholonomic
systems. It happens, however, that Eq. (11.12.5) may be modified and expanded to also
accommodate nonholonomic systems. Although the details of this expansion are beyond
the scope of this text, the interested reader is referred to References 11.2 and 11.4.
11.13 Closure
Our objective in this chapter has been to introduce the principles and procedures of
generalized dynamics. Our intention was to obtain a working knowledge of the elementary
procedures. More advanced procedures, such as those applicable with large multibody
systems and, to some extent, those concerned with nonholonomically constrained systems,
have not been discussed because they are beyond our scope at this time. The interested
reader may want to refer to References 11.2 and 11.4 and later chapters for a discussion
of these topics.
The examples are intended to demonstrate the principal advantages of the generalized
procedures over the procedures used in elementary mechanics, including: (1) non-working
constraint forces do not contribute to the generalized forces so these forces may be simply
ignored in the analysis; and (2) for holonomic systems (which include the vast majority
of systems of interest in machine dynamics), generalized inertia forces may be computed
from kinetic energy functions. This in turn means that vector acceleration need not be
computed, thus saving considerable analysis effort.
In addition to these advantages, if a system possesses a potential energy function, the
generalized active (or applied) forces may be obtained by a single derivative of the
potential energy function. Moreover, for gravity and spring forces (which are prevalent
in machine dynamics), the potential energy functions may often be directly obtained from
Eqs. (11.11.5) and (11.11.10). In the next chapter, we will consider the application of these
procedures in obtaining equations of motion.

References
11.1. Kane, T. R., Dynamics, Holt, Rinehart & Winston, New York, 1968, p. 78.
11.2. Kane, T. R., and Levinson, D. A., Dynamics: Theory and Applications, McGraw-Hill, New York,
1985, p. 100.