A method for nonlinear modal analysis and synthesis: Application to harmonically forced and self-excited mechanical systems

Abstract

The recently developed generalized Fourier-Galerkin method is complemented by a numerical continuation with respect to the kinetic energy, which extends the framework to the investigation of modal interactions resulting in folds of the nonlinear modes. In order to enhance the practicability regarding the investigation of complex large-scale systems, it is proposed to provide analytical gradients and exploit sparsity of the nonlinear part of the governing algebraic equations. A novel reduced order model (ROM) is developed for those regimes where internal resonances are absent. The approach allows for an accurate approximation of the multi-harmonic content of the resonant mode and accounts for the contributions of the off-resonant modes in their linearized forms. The ROM facilitates the efficient analysis of self-excited limit cycle oscillations, frequency response functions and the direct tracing of forced resonances. The ROM is equipped with a large parameter space including parameters associated with linear damping and near-resonant harmonic forcing terms. An important objective of this paper is to demonstrate the broad applicability of the proposed overall methodology. This is achieved by selected numerical examples including finite element models of structures with strongly nonlinear, non-conservative contact constraints.