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 引用本文: 代洪华, 王其偲, 严子朴, 岳晓奎. 重构谐波平衡法及其求解复杂非线性问题应用. 力学学报, 2024, 56(1): 212-224.
Dai Honghua, Wang Qisi, Yan Zipu, Yue Xiaokui. Reconstruction harmonic balance method and its application in solving complex nonlinear dynamical systems. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(1): 212-224.
 Citation: Dai Honghua, Wang Qisi, Yan Zipu, Yue Xiaokui. Reconstruction harmonic balance method and its application in solving complex nonlinear dynamical systems. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(1): 212-224.

## RECONSTRUCTION HARMONIC BALANCE METHOD AND ITS APPLICATION IN SOLVING COMPLEX NONLINEAR DYNAMICAL SYSTEMS

• 摘要: 谐波平衡法是求解非线性动力学系统周期解的最常用方法, 但对非线性项进行高阶近似需要庞杂的公式推导, 限制了该方法的超高精度解算. 通过对频域非线性量的时域等价重构, 提出了重构谐波平衡法(RHB法), 解决了经典谐波平衡法超高阶次计算难题. 然而, 上述两种方法均要求动力学系统为多项式型非线性, 且无法直接用来求解非线性系统的拟周期解. 针对上述问题, 文章提出一种将RHB法和复杂非线性系统等价重铸法相结合的计算方法, 首先将一般非线性问题无损重铸为多项式型非线性系统, 然后用RHB法进行高精度求解; 针对拟周期响应求解问题, 提出基于“补频”思想的RHB方法, 通过基频的优化筛选, 实现拟周期响应的快速精准捕捉. 选取非线性单摆、相对论谐振子和非线性耦合非对称摆等典型系统进行仿真计算, 仿真结果表明, 所提出的RHB-重铸法在解非多项式型非线性系统的稳态响应时精度保持为10^ - 12量级, 达计算机精度, 远超现有方法水平. 补频RHB法则实现了对拟周期问题的高效解算, 拓宽了方法对真实物理响应的求解范围.

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 derivations, which limits its ultra-high accuracy computation. 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 successfully conquered the problem of ultra-high-order calculation of the classical harmonic balance method. However, both two 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 novel 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, which is hard to obtain relatively accurate results by considering only one base frequency, 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 in this paper. The typical systems such as the nonlinear pendulum, and the nonlinear coupling asymmetric pendulum are selected for simulation and algorithm verification. Simulation results show that the accuracy of the proposed RHB-recast method achieves an accuracy up to 10−12 when solving the non-polynomial type nonlinear systems, reaching the computer round-off-error accuracy, far exceeding the state-of-the-art methods. The supplemental frequency RHB method has been shown to produce efficient solutions for quasi-periodic problems, expanding the scope of the RHB-like methods for solving complex physical responses.

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