This paper presents a new semi-analytical approach, namely the enhanced time-domain minimum residual method to solve the semi-analytical quasi-periodic solution for nonlinear system. This approach does not require multiple numerical integration and can be applied to strongly nonlinear systems. The approach is mainly three-fold. Firstly, the semi-analytical solution of the nonlinear quasi-periodic system is expanded into a set of trigonometric series with unknown coefficients, i.e., $\mathit{x}\left(\mathit{t}\right)\approx \sum _{k=1}^N[{\mathit{b}}_{\mathit{k}}\cos \left({\mathbf{\omega }}_{\mathit{k}}\mathit{t}\right)+{\mathit{c}}_{\mathit{k}}\sin \left({\mathbf{\omega }}_{\mathit{k}}\mathit{t}\right)]$. Then, the problem of solving quasi-periodic solution can be expressed as: determining the coefficients of the trigonometric series so that the residual objective function $\mathit{R}=\mathit{M}\ddot{\mathit{x}}+\mathit{C}\dot{\mathit{x}}+\mathit{Kx}+\mathit{N}(\ddot{\mathit{x}},\dot{\mathit{x}},\mathit{x},\mathit{t})-\mathit{F}\left(\mathit{t}\right)$ is minimum over a period, i.e., $\underset{\mathit{a}\in \mathcal{A}}{\min }{\int }_0^T\mathit{R}{(\mathit{a},t)}^T\mathit{R}(\mathit{a},t)\mathrm{d}t$. Finally, the nonlinear minimum optimization problem is solved iteratively through the enhanced response sensitivity approach. Moreover, the “adaptive zero-setting curve” is introduced to accelerate the convergence. Two numerical examples, a van der Pol-Duffing system and a nonlinear energy sink system are adopted to verify the feasibility of the proposed approach.

本文提出了一种新的半解析方法，即增强时域最小残差法来求解非线性系统的半解析准周期解。这种方法不需要多次数值积分，可以应用于强非线性系统。该方法主要是三方面的。首先将非线性拟周期系统的半解析解展开为一组未知系数的三角级数，即：$\mathit{X}\left(\mathit{吨}\right)\approx \sum _{克=1}^N[{\mathit{乙}}_{\mathit{克}}\cos \left({\mathbf{\omega }}_{\mathit{克}}\mathit{吨}\right)+{\mathit{C}}_{\mathit{克}}罪\left({\mathbf{\omega }}_{\mathit{克}}\mathit{吨}\right)]$. 那么，求解拟周期解的问题可以表示为：确定三角级数的系数，使得残差目标函数$\mathit{电阻}=\mathit{米}\ddot{\mathit{X}}+\mathit{C}\dot{\mathit{X}}+\mathit{克}+\mathit{N}(\ddot{\mathit{X}},\dot{\mathit{X}},\mathit{X},\mathit{吨})-\mathit{F}\left(\mathit{吨}\right)$ 在一段时间内是最小值，即 $\underset{\mathit{一种}\in \mathcal{一种}}{\mathrm{分钟}}{\int }_0^{吨}\mathit{电阻}{(\mathit{一种},吨)}^{吨}\mathit{电阻}(\mathit{一种},吨)\mathrm{d}吨$. 最后，通过增强响应灵敏度方法迭代求解非线性最小优化问题。此外，引入了“自适应置零曲线”来加速收敛。两个数值例子，一个van der Pol-Duffing系统和一个非线性能量汇系统被采用来验证所提出方法的可行性。