Stability and Convergence Analysis of the Harmonic Balance Method for a Duffing Oscillator with Free Play Nonlinearity

Abstract

In this work, we determine the quality of the harmonic balance method (HBM) using a single degree-of-freedom forced Duffing oscillator with free play. HBM results are compared to results obtained using direct time integration with an event location procedure to properly capture contact behavior and identify nonperiodic motions. The comparison facilitates an evaluation of the accuracy of nonlinear, periodic responses computed with HBM, specifically by comparing super- and subharmonic resonances, regions of periodic and nonperiodic (i.e., quasiperiodic or chaotic) responses, and discontinuity-induced bifurcations, such as grazing bifurcations. Convergence analysis of HBM determines the appropriate number of harmonics required to capture nonlinear contact behavior, while satisfying the governing equations. Hill’s method and Floquet theory are used to compute the stability of periodic solutions and identify the types of bifurcations in the system. Extensions to multi- degree-of-freedom oscillators will be discussed as well.