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

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.