Large-amplitude non-linear normal modes of piecewise linear systems

A numerical method for constructing non-linear normal modes (NNMs) for piecewise linear autonomous systems is presented. These NNMs are based on the concept of invariant manifolds, and are obtained using a Galerkin-based solution of the invariant manifoldʹs non-linear partial differential equations. The accuracy of the constructed non-linear modes is checked by the comparison of the motion on the invariant manifold to the exact solution, in both time and frequency domains. It is found that this construction approach can accurately capture the NNMs over a wide range of amplitudes, including those with strong non-linear effects. Several interesting dynamic characteristics of the non-linear modal motion are found and compared to those of linear modes. A two-degree-of-freedom example is used to illustrate the technique. The existence, stability and bifurcations of the NNMs for this example are investigated.