基于Volterra级数的非线性系统动力学分析的广义极点留数法
Generalized Pole-Residue Method for Dynamic Analysis of Nonlinear Systems based on Volterra Series
摘要：
以二阶非线性常微分方程为特征的动力系统在物理和工程的许多领域中都有出现。为了解决这类问题，耗时的逐步数值积分方法和基于时域Volterra级数的卷积方法已被广泛使用。与此相反，这项工作开发了一个有效的广义极点残基方法的基础上进行拉普拉斯域的Volterra级数。所提出的方法包括两个步骤：（1）Volterra核的Laguerre多项式解耦，（2）部分响应的Laguerre多项式解析获得的条款的极点留数方法。与传统的线性系极点留数法相比，本文的极点留数法的一个创新之处在于如何处理高阶极点及其相应的系数。由于所提出的方法推导出一个显式的，连续的时间响应函数，它是更有效的比传统的数值方法。与传统的Laplace域方法不同，该方法适用于任意不规则激励。由于在求解过程中自然地获得了固有响应、强迫响应和交叉响应，因此获得了有意义的数学和物理见解。在数值研究中，研究了具有已知运动方程和未知运动方程的系统。对于每个系统，研究了规则激励和不同参数的复杂非规则激励。通过与四阶龙格-库塔法的比较，验证了该方法的高精度和高效率。
英文摘要：
Dynamic systems characterized by second-order nonlinear ordinary differential equations appear in many fields of physics and engineering. To solve these kinds of problems, time-consuming step-by-step numerical integration methods and convolution methods based on Volterra series in the time domain have been widely used. In contrast, this work develops an efficient generalized pole-residue method based on the Volterra series performed in the Laplace domain. The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method. Compared to the traditional pole-residue method for a linear system, one of the novelties of the pole-residue method in this paper is how to deal with the higher-order poles and their corresponding coefficients. Because the proposed method derives an explicit, continuous response function of time, it is much more efficient than traditional numerical methods. Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations. Because the natural response, forced response and cross response are naturally obtained in the solution procedure, meaningful mathematical and physical insights are gained. In numerical studies, systems with a known equation of motion and an unknown equation of motion are investigated. For each system, regular excitations and complex irregular excitations with different parameters are studied. Numerical studies validate the good accuracy and high efficiency of the proposed method by comparing it with the fourth-order Runge--Kutta method.
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