Numerical Methods for Assessing Response Metrics

Abstract

Due to the high expense associated with evaluating high-fidelity models of nonlinear systems, reduced order models (ROMs) are typically used. One consequence of this, however, is that the quality of the ROM must be assessed. When evaluating the quality of a ROM, it is common to compare the time response of the model to that of the full order system, but the time response is a complicated function that depends on a predetermined set of initial conditions or external force. This is difficult to use as a metric to measure convergence of a ROM, particularly for systems with strong, non-smooth nonlinearities, for two reasons: first, the accuracy of the response depends directly on the amplitude of the load/initial conditions, and second, small differences between two signals can become large over time. Here, a validation metric is proposed that is based solely on the ROM’s equations of motion. The nonlinear normal modes (NNMs) of the ROMs are computed and tracked as modes are added to the basis set. The NNMs are expected to converge to the true NNMs of the full order system with a sufficient set of basis vectors. This comparison captures the effect of the nonlinearity through a range of amplitudes of the system and is akin to comparing natural frequencies and mode shapes for a linear structure. In this chapter, the convergence metric is evaluated on a simply supported beam with a contacting nonlinearity modeled as a unilateral piecewise-linear function. Various time responses are compared to show that the NNMs provide a good measure of the accuracy of the ROM. The results suggest the feasibility of using NNMs as a convergence metric for reduced order modeling of systems with various types of nonlinearities.