A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions

TL;DR

This paper introduces a high-order frequency-based harmonic balance method combined with continuation techniques, capable of efficiently analyzing a broad class of dynamical systems with smooth equations.

Contribution

It presents a novel approach that recasts dynamical systems into quadratic form to simplify the harmonic balance and continuation process, overcoming limitations of classical methods.

Findings

Applicable to a wide range of systems with smooth equations

Efficient derivation of algebraic systems for harmonic balance

Successful demonstration on classical examples

Abstract

Combinig the harmonic balance method (HBM) and a continuation method is a well-known technique to follow the periodic solutions of dynamical systems when a control parameter is varied. However, since deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, the classical HBM is often limited to polynomial (quadratic and cubic) nonlinearities and/or a few harmonics. Several variations on the classical HBM, such as the incremental HBM or the alternating frequency/time domain HBM, have been presented in the literature to overcome this shortcoming. Here, we present an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations. The main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the HBM. Once the equations have been rendered quadratic, it becomes obvious to derive the algebraic system and solve it by the so-called ANM continuation technique. Several classical examples are presented to illustrate the use of this numerical approach.