Numerical calculation of nonlinear normal modes in structural systems

Abstract

This paper presents two methods for numerical calculation of nonlinear normal modes (NNMs) in multi-degree-of-freedom, conservative, nonlinear structural dynamics models. The approaches used are briefly described as follows. Method 1: Starting with small amplitude initial conditions determined by a selected mode of the associated linear system, a small amount of negative damping is added in order to “artificially destabilize” the system; numerical integration of the system equations of motion then produces a simulated response in which orbits spiral outward essentially in the nonlinear modal manifold of interest, approximately generating this manifold for moderate to strong nonlinearity. Method 2: Starting with moderate to large amplitude initial conditions proportional to a selected linear mode shape, perform numerical integration with the coefficient e of the nonlinearity contrived to vary slowly from an initial value of zero; this simulation methodology gradually transforms the initially flat eigenspace for e = 0 into the manifold existing quasi-statically for instantaneous values of e. The two methods are efficient and reasonably accurate and are intended for use in finding NNMs, as well as interesting behavior associated with them, for moderately and strongly nonlinear systems with relatively many degrees of freedom (DOFs).