Modal subspaces and normal mode vibrations

Abstract

Trajectories of non-linear conservative systems having n-degrees of freedom can be identified with the motions of a unit mass in an n-dimensional Euclidean space under a force derivable from a potential function. Normal modes of vibrations of such systems have been defined and discussed by R.M. Rosenberg and others. This paper deals with special classes of motions of the system for which the trajectories of the unit mass remain in subspaces of the Euclidean n-space; these are called modal subspaces. Using the symmetry and other properties of the potential function, methods are developed to find these modal subspaces and to systematically reduce the dimensions of the modal subspaces. In many cases, this procedure leads to finding one-dimensional modal subspaces which represent normal modes of vibration. The methods determine the modes geometrically without having to integrate the differential equations. Examples are given to demonstrate the applications.