# 3Derivation of Equations: Equilibrium Approach

## 3.1 INTRODUCTION

The equations of motion of a vibrating system can be derived by using the dynamic equilibrium approach, the variational method, or the integral equation formulation. The dynamic equilibrium approach is considered in this chapter. The variational and integral equation approaches are presented in Chapters 4 and 5, respectively. The dynamic equilibrium approach can be implemented by using either Newton's second law of motion or D'Alembert's principle.

## 3.2 NEWTON'S SECOND LAW OF MOTION

Newton's second law of motion can be used conveniently to derive the equations of motion of a system under the following conditions:

- The system undergoes either pure translation or pure rotation.
- The motion takes place in a single plane.
- The forces acting on the system either have a constant orientation or are oriented parallel to the direction along which the point of application moves.

If these conditions are not satisfied, application of Newton's second law of motion becomes complex, and other methods, such as the variational and integral equation approaches, can be used more conveniently. *Newton's second law of motion* can be stated as follows: The rate of change of the linear momentum of a system is equal to the net, if several forces act on the system, the resulting force acting on the system is given by and Newton's second ...

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