A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities

TL;DR

This paper extends a frequency-based harmonic balance method to handle non-polynomial nonlinearities, enabling high-accuracy computation of periodic solutions in complex nonlinear dynamical systems.

Contribution

It introduces a novel transformation for non-polynomial terms, allowing the harmonic balance method to be applied to a broader class of nonlinear systems with high-order accuracy.

Findings

Successfully applied to systems with exponential nonlinearities

Achieved high-order solutions up to 1000 in complexity

Demonstrated accuracy with nonlinear free pendulum simulations

Abstract

In this paper, we extend the method proposed by Cochelin and Vergez [A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324 (2009) 243-262] to the case of non-polynomial nonlinearities. This extension allows for the computation of branches of periodic solutions of a broader class of nonlinear dynamical systems. The principle remains to transform the original ODE system into an extended polynomial quadratic system for an easy application of the harmonic balance method (HBM). The transformation of non-polynomial terms is based on the differentiation of state variables with respect to the time variable, shifting the nonlinear non-polynomial nonlinearity to a time-independent initial condition equation, not concerned with the HBM. The continuation of the resulting algebraic system is here performed by the asymptotic numerical method (high order Taylor series representation of the solution branch) using a further differentiation of the non-polynomial algebraic equation with respect to the path parameter. A one dof vibro-impact system is used to illustrate how an exponential nonlinearity is handled, showing that the method works at very high order, 1000 in that case. Various kinds of nonlinear functions are also treated, and finally the nonlinear free pendulum is addressed, showing that very accurate periodic solutions can be computed with the proposed method.