A High-Order Harmonic Balance Method for Systems With Distinct States

TL;DR

This paper introduces a high-order harmonic balance method tailored for nonlinear systems with distinct states, utilizing an event-driven approach for accurate periodic solution computation in piecewise-defined ODEs.

Contribution

It develops a novel frequency domain method with analytical derivatives and an event-driven scheme for systems with piecewise polynomial nonlinearities, enhancing robustness and efficiency.

Findings

Effective for systems with piecewise nonlinearities

Analytical derivatives up to second order are obtainable

Demonstrated robustness on structural dynamical systems

Abstract

A pure frequency domain method for the computation of periodic solutions of nonlinear ordinary differential equations (ODEs) is proposed in this study. The method is particularly suitable for the analysis of systems that feature distinct states, i.e. where the ODEs involve piecewise defined functions. An event-driven scheme is used which is based on the direct calculation of the state transition time instants between these states. An analytical formulation of the governing nonlinear algebraic system of equations is developed for the case of piecewise polynomial systems. Moreover, it is shown that derivatives of the solution of up to second order can be calculated analytically, making the method especially attractive for design studies. The methodology is applied to several structural dynamical systems with conservative and dissipative nonlinearities in externally excited and autonomous configurations. Great performance and robustness of the proposed procedure was ascertained.