Chapter Four - FREE VIBRATION

Abstract

When an elastic mechanical system is displaced from its equilibrium position and then released, it will oscillate about that position before returning to a state of rest. A car bounced on its springs or the vibrating string of a musical instrument are two examples of this oscillation, which is termed mechanical vibration. The phenomenon is present in all mechanical systems, more complex systems giving correspondingly complex vibration characteristics. This chapter presents a method of dealing with any linear vibrating system, from the elementary to the complex, and also describes the way in which a general approach capable of analyzing the dynamic behavior of any linear mechanical system may be adopted. The chapter presents the fundamentals of vibration analysis to lead into the more complex systems to which the matrix solutions can be applied. A system is said to be exhibiting free vibration when it is given an initial displacement from its equilibrium position and thereafter allowed to oscillate with no further imposed force. A practical condition for the analysis to be valid is that the initial displacement of the system be small compared with the system physical dimensions as the spring rate K is constant only for such displacements. These systems in which displacements are small and the stiffnesses constant are termed linear systems.