Model Reduction for Nonlinear Elastic Structures With Inertial Quadratic Nonlinearities

Abstract

In order to achieve accurate and high fidelity nonlinear response predictions, discrete models usually obtained through Galerkin approximation utilizing linear normal modes of the structure, need to retain a large number of degrees of freedom. This is specially the case if the structural response has the possibility of modal interactions. Then, a possible approach suggested in the literature to decrease the required degrees of freedom while retaining same accuracy is to use nonlinear normal modes of the structure to perform further model reduction. In this work, we discuss model reduction for nonlinear structural systems under harmonic excitations. The analysis needs to carefully consider the possibility of external resonances, parametric resonances, combination parametric resonances (the parametric excitation frequency being near the sum or difference of frequencies of two modes), and internal resonances. A master-slave separation of degrees of freedom is used, and a nonlinear relation between the slave coordinates and the master coordinates is constructed based on the multiple time scales approximation. More specifically, three cases are considered: external resonance of a mode without any internal resonance, and subharmonic as well as superharmonic excitation for systems with 1:2 internal resonance. The steady state periodic responses determined by the method of multiple time scales are compared to exact solutions of the discrete model computed by the bifurcation analysis and parameter continuation software AUTO. It is seen that for systems with essential inertial quadratic nonlinearities, the technique based on nonlinear model reduction through multiple time scales approximation over-predict the softening nonlinear response.