Vibration Theory III: Normal Modes

Abstract

In this chapter we will be concerned with the modal analysis of the response of lightly damped linear systems To begin with we wish to consider systems of such complexity that they cannot be regarded as a collection of concentrated masses, beams, shells, etc. whose partial differential equations can be formulated on the basis of geometric dimensions and material properties. Frequently it is found that the configuration of a machine — even as complicated as an aircraft — can be described adequately in terms of a certain set of n displacements or rotations q _j (j =1,2,...,n) at judiciously chosen points P _j . It is to be noted that in assuming that the q _j describe the configuration independent of time we are to a certain extent excluding systems with time varying parameters. For the sake of simplicity it will be assumed that no forced displacements are imposed on the system. Thus, in the parlance of higher dynamics, we will be considering scleronomic as opposed to rheonomic systems. After deciding on the n significant generalized coordinates q _j , the next step would be to conduct vibration tests on the given machine by applying sinusoidal forces at the points P _j . After considerable effort one will be in a position to set up a system of second-order differential equations governing the system.