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.

翻译：在许多物理学和工程学领域中都出现了具有二阶非线性常微分方程特征的动态系统。为解决此类问题，传统上广泛采用两类方法：一是耗时的逐步数值积分方法，二是基于Volterra级数在时域内进行的卷积方法。相比之下，本研究基于Laplace域中的Volterra级数，提出了一种高效的广义极点-留数法。该方法包含两个步骤：（1）利用拉盖尔多项式对Volterra核进行解耦；（2）采用极点-留数法解析获得与单个拉盖尔多项式相关的部分响应。与线性系统传统的极点-留数法相比，本文所提出的极点-留数法的创新点之一在于如何处理高阶极点及其对应系数。由于该方法推导出了显式的连续时间响应函数，因此其计算效率远高于传统数值方法。与传统的Laplace域方法不同，本文提出的方法适用于任意不规则激励。由于在求解过程中自然获得了固有响应、受迫响应和交叉响应，因此能够获得有意义的数学和物理学见解。在数值研究中，分别对具有已知运动方程和未知运动方程的系统进行了研究。针对每个系统，研究了正则激励以及具有不同参数的复杂不规则激励。通过与四阶龙格-库塔法的比较，数值研究验证了所提方法具有良好的准确性和高效率。