A comparative study on multi- and variable-coefficient harmonic balance methods for quasi-periodic solutions

Abstract

Quasi-periodic solutions widely exist in nonlinear dynamical systems, which are the one of the most important dynamical behaviors but much more difficult to obtain full- or semi-analytically than equilibria and periodic solutions. Recently, multi-harmonic balance method (MHBM) and variable-coefficient harmonic balance method (VCHBM) have been proposed to determine the quasi-periodic solutions semi-analytically. In this paper, after a brief review on the two methods, that is, MHBM uses a multi-dimensional Fourier series and VCHBM a variable-coefficient Fourier series, the relationship between them is unveiled particularly for the case of quasi-periodic solutions consisting of two frequency components. The transform matrices between the algebraic equations of the Fourier coefficients in the two methods are derived. Furthermore, a new formulation for alternating Frequency-Time method (AFT) and a phase condition for arc-length continuation method (ALC) in VCHBM are proposed to make the method more efficient and robust. Through the application of two examples, it is found that when the two methods choose equal harmonic order and thus have the same number of unknowns, MHBM demonstrates higher computational efficiency but VCHBM shows more robust and accurate, especially for the quasi-periodic solutions in the nonlinear dynamical systems with a single-frequency excitation. This paper provides insight views into the features of MHBM and VCHBM..