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11

Generalized Dynamics: Kinematics and Kinetics

11.1 Introduction
Recall in the analysis of elementary statics problems we discover, after gaining experience,
that by making insightful choices about force directions and moment points, we can greatly
simplify the analysis. Indeed, with sufficient insight, we discover that we can often obtain
precisely the same number of equations as there are unknowns in the problem statement.
Moreover, these equations are often uncoupled, thus producing answers with little further
analysis.
In Chapter 10, we found that the work–energy principle, like clever statics solution
procedures, can often produce simple and direct solutions to dynamics problems. We also
found, however, that while the work–energy principle is simple and direct, it is also quite
restricted in its range of application. The work–energy principle leads to a single scalar
equation, thus enabling the determination of a single unknown. Hence, if two or more
unknowns are to be found, the work–energy principle is inadequate and is restricted to
relatively simple problems.
The objective of generalized dynamics is to extend the relatively simple analysis of the
work–energy principle to complex dynamics problems having a number of unknowns.
The intention is to equip the analyst with the means of determining unknowns with a
minimal effort — as with insightful solutions of statics problems.
In this chapter, we will introduce and discuss the elementary procedures of generalized
dynamics. These include the concepts of generalized coordinates, partial velocities and
partial angular velocities, generalized forces, and potential energy. In Chapter 12, we will
use these concepts to obtain equations of motion using Kane’s equations and Lagrange’s
equations.

11.2 Coordinates, Constraints, and Degrees of Freedom

In the context of generalized dynamics, a coordinate (or generalized coordinate) is a parameter
used to define the configuration of a mechanical system. Consider, for example, a particle
P moving on a straight line L as in Figure 11.2.1. Let x locate P relative to a fixed point O
on L. Specifically, let x be the distance between O and P. Then, x is said to be a coordinate of P.
Next, consider the simple pendulum of Figure 11.2.2. In this case, the configuration of
the system and, as a consequence, the location of the bob P are determined by the angle θ.
Thus, θ is a coordinate of the system.

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