Reconstruction Harmonic Balance Method and its Application in Solving Complex Nonlinear Dynamical Systems

TL;DR

This paper introduces an enhanced reconstruction harmonic balance (RHB) method combined with recast techniques to accurately solve complex nonlinear dynamical systems, including quasi-periodic responses, surpassing existing methods in precision and efficiency.

Contribution

The authors develop a novel RHB-recast method that non-destructively transforms nonlinear systems into polynomial form and efficiently computes quasi-periodic solutions with ultra-high accuracy.

Findings

Achieves solution accuracy on the order of 10^(-12).

Successfully captures quasi-periodic responses using supplemental frequency.

Outperforms state-of-the-art methods in accuracy and efficiency.

Abstract

The harmonic balance method is the most commonly used method for solving periodic solutions of nonlinear dynamic systems, but the high-order approximation of nonlinear terms requires sophisticated formula derivation, which limits its ultra-high accuracy. The authors' team proposed the reconstruction harmonic balance (RHB) method through the equivalent reconstruction of the frequency domain nonlinear quantity in the time domain, which settled the problem of ultra-high-order calculation of the classical harmonic balance method. However, both methods require the dynamical system to be polynomial nonlinear, and cannot be directly used to solve the quasi-periodic solution of the nonlinear system. In view of the above problems, this paper proposes a computational method that combines the RHB method and the recast technique for complex nonlinear systems. First, the general nonlinear problem is non-destructively recast into a polynomial nonlinear system, and then the RHB method is used for high-precision solutions. Aiming at computing the quasi-periodic response, the RHB method based on the idea of "supplemental frequency" is derived. By optimizing and selecting base frequencies, the fast and accurate capture of quasi-periodic response is achieved. The typical systems such as nonlinear pendulum, relativistic harmonic oscillator, and nonlinear coupling asymmetric pendulum are selected for simulation. The simulation results show that the accuracy of the RHB-recast method for solving nonpolynomial nonlinear systems is on the order of 10^(-12), reaching the computer accuracy, far exceeding state-of-the-art methods. The supplemental frequency RHB method achieves the efficient solution of quasi-periodic problems.