Normal modes for piecewise linear vibratory systems

A method to construct the normal modes for a class of piecewise linear vibratory systems is developed in this study. The approach utilizes the concepts of Poincaré maps and invariant manifolds from the theory of dynamical systems. In contrast to conventional methods for smooth systems, which expand normal modes in a series form around an equilibrium point of interest, the present method expands the normal modes in a series form of polar coordinates in a neighborhood of an invariant disk of the system. It is found that the normal modes, modal dynamics and frequency-amplitude dependence relationship are all of piecewise type. A two degree of freedom example is used to demonstrate the method.