Efficient energy balancing across multiple harmonics of nonlinear normal modes

Predicting the forced responses of nonlinear systems is a topic that attracts extensive studies. The energy balancing method considers the net energy transfer in and out of the system over one period and establishes connections between forced responses and nonlinear normal modes (NNMs). In this paper, we consider the energy balancing across multiple harmonics of NNMs for predicting forced resonances. This technique is constructed by combining the energy balancing mechanism with restrictions (established via excitation scenarios) on external forcing and harmonic phase-shifts; a semi-analytical framework is derived to achieve both accurate/robust results and efficient computations. With known inputs from NNM solutions, the required forcing amplitudes to reach NNMs at resonances, along with their discrepancy, i.e. the harmonic phase-shifts, are computed via a one-step scheme. Several examples are presented for different excitation scenarios to demonstrate the applicability of this method and to show its capability in accurately predicting the existence of an isola where multiple harmonics play a significant part in the response.

Automated summary

This paper presents a method for predicting forced resonances in nonlinear systems by considering energy balancing and harmonic phase shifts. By combining the energy balancing mechanism with restrictions on external forcing and harmonic phase shifts, a semi-analytical framework is derived to achieve accurate, robust, and efficient computations. Several examples are provided to demonstrate the applicability of this method and the significance of accurately predicting the role of multiple harmonics in the response.