摘要
Quasi-periodic solutions with multiple base frequencies exhibit the feature of 2π-periodicity with respect to each of the hyper-time variables.However,it remains a challenge work,due to the lack of effective solution methods,to solve and track the quasi-periodic solutions with multiple base frequencies until now.In this work,a multi-steps variable-coefficient formulation is proposed,which provides a unified framework to enable either harmonic balance method or collocation method or finite difference method to solve quasi-periodic solutions with multiple base frequencies.For this purpose,a method of alternating U and S domain is also developed to efficiently evaluate the nonlinear force terms.Furthermore,a new robust phase condition is presented for all of the three methods to make them track the quasi-periodic solutions with prior unknown multiple base frequencies,while the stability of the quasi-periodic solutions is assessed by mean of Lyapunov exponents.The feasibility of the constructed methods under the above framework is verified by application to three nonlinear systems.
含有多个基频的准周期解,在超时域的各分量上都具有2π的周期特性.然而,求解并追踪完整的多基频准周期解至今仍非常具有挑战性.为此,本文提出多步变系数求解的通用公式(m-VCF),可基于谐波平衡法、配点法、有限差分法三种常规周期解数值方法来构建准周期解的求解方案,并形成统一的求解框架.此外,为高效计算系统的非线性项,本文创新地提出交替U域和S域方法,并给出了一种简单高效且鲁棒性强的相位条件,实现了对具有多个先验未知基频拟周期解的有效追踪.最后,借助计算李雅普诺夫指数对准周期解的稳定性进行了分析.通过对多自由度非线性动力学系统的数值计算,验证了所提方法的有效性.
基金
supported by the National Natural Science Foundation of China(Grant Nos.12172267 and 12302014).
