Some practical regards on the application of the harmonic balance method for hysteresis models

Abstract

Describing hysteretic systems with a closed-form solution is a challenging task due to some pitfalls regarding the non-smooth and memory effect mechanisms that do not permit, for example, to apply conventional frequency domain methods. Consequently, it is necessary to use some previous smoothing scheme to approximate the hysteresis loop. Thus, this work proposes a new way for approximating the hysteresis loops analytically using a truncated Taylor series as a simple and effective smoothing procedure to enable the use of the harmonic balance method. Two benchmark hysteretic systems, which were not addressed yet by closed-form solutions obtained by the harmonic balance method, are simulated to demonstrate the benefits of the proposed strategy. The first one is a Bouc-Wen oscillator and the second one is a LuGre model. The comparison with numerical integrations and other literature methods have shown that the obtained analytical solutions of the suggested smoothed hysteresis loops are adequate to describe the fundamental dynamics in both models using a feasible frequency domain approximation.