Collocation‐based harmonic balance framework for highly accurate periodic solution of nonlinear dynamical system

Abstract

Periodic dynamical systems ubiquitously exist in science and engineering. The harmonic balance (HB) method and its variants have been the most widely‐used approaches for such systems, but are either confined to low‐order approximations or impaired by aliasing and improper‐sampling problems. Here we propose a collocation‐based harmonic balance framework to successfully unify and reconstruct the HB‐like methods. Under this framework a new conditional identity, which exactly bridges the gap between frequency‐domain and time‐domain harmonic analyses, is discovered by introducing a novel aliasing matrix. Upon enforcing the aliasing matrix to vanish, we propose a powerful reconstruction harmonic balance (RHB) method that obtains extremely high‐order (>100) nonaliasing solutions, previously deemed out‐of‐reach, for a range of complex nonlinear systems including the cavitation bubble dynamics, the three‐body problem and the two dimensional airfoil dynamics. We show that the present method is 2–3 orders of magnitude more accurate and simultaneously much faster than the state‐of‐the‐art method. Hence, it has immediate applications in multidisciplinary problems where highly accurate periodic solutions are sought.

Harmonic Balance Methods: A Review and Recent Developments

“…Based on the rules of aliasing, some recent studies have shown that the number of collocation points does not need to strictly follow the requirements of the sampling theorem. The HB method can also be reconstructed when fewer collocation points [37,73]. The sampling theorem makes the Fourier series formed by r not aliased, that is, harmonics in the range [−φN, φN] will not be aliased by highorder harmonics.…”

“…There is an important equivalent identity rN ≡ E * rM holds when M satisfies Eq. ( 35) [37], as described below: Theorem 3.4 Conditional equivalence. If number of collocation points M, truncated order of harmonics N and degree of nonlinearity φ satisfy M > (φ + 1)N, then the RHB method and the HB method are equivalent.…”

“…If number of collocation points M, truncated order of harmonics N and degree of nonlinearity φ satisfy M > (φ + 1)N, then the RHB method and the HB method are equivalent. Dai et al [37] found that the degree of aliasing is determined by an "aliasing matrix". As the number of collocation points increases, the non-zero elements in the aliasing matrix gradually decrease.…”

“…Dai et al [36] successfully uncovered the rules of aliasing and explained how the high-order harmonics mixed into the lower ones. Based on the above studies, a powerful reconstruction harmonic balance (RHB) method [37] was proposed by reconstructing the classical HB method via pure time domain collocations. The RHB method, which successfully avoids the aliasing phenomenon and oversampling problem, has been shown to be highly efficient for solving periodic solutions of nonlinear dynamic systems.…”

Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators

“…Analytical methods have been reviewed by Cvetićanin [2] for the unforced and undamped Duffing equation, presented in the form of some elliptic functions. In general, for nonlinear Duffing equations, there exists no analytical solution, and some semi-analytical methods such as as the power series and harmonic balance methods, have to be invoked [24][25][26]. In continuous works, Liu et al [27] developed the scaled power series techniques for solving the Duffing equation.…”