Vibration of Continuous Systems, 2nd Edition

4Derivation of Equations: Variational Approach

4.1 INTRODUCTION

As stated earlier, vibration problems can be formulated using an equilibrium, a variational, or an integral equation approach. The variational approach is considered in this chapter. In the variational approach, the conditions of the extremization of a functional are used to derive the equations of motion. The variational methods offer the following advantages:

1. Forces that do no work, such as forces of constraint on masses, need not be considered.
2. Accelerations of masses need not be considered; only velocities are needed.
3. Mathematical operations are to be performed on scalars, not on vectors, in deriving the equations of motion.

Since the variational methods make use of the principles of calculus of variations, the basic concepts of calculus of variations are presented. However, a brief review of the calculus of a single variable is given first to indicate the similarity of the concepts.

4.2 CALCULUS OF A SINGLE VARIABLE

To understand the principles of calculus of variations, we start with the extremization of a function of a single variable from elementary calculus [2]. For this, consider a continuous and differentiable function of one variable, defined in the interval , with extreme points at a, b, and c as shown in Fig. 4.1. In Fig. 4.1, the point denotes a local minimum with for all x in the neighborhood ...