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 引用本文: 张波, 张文博, 彭志科. 一类非线性系统的广义伴随线性方程分析研究. 力学学报, 2024, 56(3): 1-15.
Zhang Bo, Zhang Wenbo, Peng Zhike. An analytical study of generalized associated linear equations for a class of nonlinear systems. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(3): 1-15.
 Citation: Zhang Bo, Zhang Wenbo, Peng Zhike. An analytical study of generalized associated linear equations for a class of nonlinear systems. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(3): 1-15.

## AN ANALYTICAL STUDY OF GENERALIZED ASSOCIATED LINEAR EQUATIONS FOR A CLASS OF NONLINEAR SYSTEMS

• 摘要: 非线性输出频率响应函数是线性系统理论中频率响应函数在非线性系统中的一种推广, 越来越多的学者已将其应用在结构损伤检测及故障诊断中. 基于Volterra级数理论与由非线性微分方程描述的单输入单输出非线性系统的广义频率响应函数递归计算公式, 利用多重积分性质和多维傅里叶变换, 将广义频率响应函数映射到了一维频域中, 推导出了一类非线性系统的广义伴随线性方程计算公式. 研究表明, 这类系统的第n阶非线性输出响应是以系统输入激励和前n−1阶非线性输出响应的组合函数作为广义激励作用到系统各阶广义伴随线性方程中的输出响应, 最后通过求解一系列线性微分方程可得到这类非线性系统的任意阶非线性输出响应, 其结果弥补了伴随线性方程无法求解这类非线性系统的不足. 同时, 针对广义伴随线性方程的数值计算问题, 论文提出了一种耦合计算法, 提高了计算非线性输出响应的精度, 为非线性输出频率响应函数的计算提供了一种新思路. 最后利用广义伴随线性方程与线性算子理论研究了两种典型非线性系统中非线性现象产生的原因, 研究结果为非线性系统的分析与设计提供了一种有效途径.

Abstract: The nonlinear output frequency response function, which is suitable for modeling nonlinear system and has been applied for structural damage detection and fault diagnosis, is a generalization of the frequency response function in linear system theory. Based on the Volterra series theory and the recursive formulation of the generalized frequency response function of an SISO (single-input/single-output) nonlinear system described by nonlinear differential equations, this study maps the generalized frequency response function into a one-dimensional frequency domain by means of utilizing property of multiple integration and multi-dimensional Fourier transform. After that, a formula that can be used to solve generalized associated linear equations for a class of nonlinear systems is deduced. It is shown in this work that the nth-order nonlinear output response of this kind of system is the result of a generalized excitation that is a combination of the excitation and the first n−1 orders of nonlinear output response. Letting the above-mentioned generalized excitation be the input of each order of the generalized associated linear equations, the arbitrary order nonlinear output response of this kind of nonlinear system can be obtained by solving a series of linear differential equations; clearly, this presented approach overcomes the shortcoming that accompanying linear equation was not used for solving such nonlinear system. Meanwhile, a coupled computational method is proposed for the numerical calculation of the generalized associated linear equations. It is found out that this method can improve calculation accuracy of nonlinear output response and would be a new alternative for calculating nonlinear output frequency response function. Finally, as examples, emerging causes of nonlinear phenomena of two typical nonlinear systems are investigated by using the generalized associated linear equations and linear operator theory, and the corresponding results offer an effective reference for the analysis and design of some nonlinear systems.

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