Research On Volterra Series Theory And Its Application

As we all know, the actual system always contain a variety of nonlinear factors, such as the gap, dry friction, oil film in mechanical system; large deformation in structural system, the nonlinear constitutive relationship relating stresses and strains, nonlinear control strategy in control system, and so on. Generally speaking, linear analysis can be adopted in such a case where the system requirements of accuracy is low or the nonlinearity has little effect on the performance of system. So, in essence, linear analysis is simplified from the actual system. In general, the linear model can almost capture the actual behavior of a system, however, in recent years, with the development and progress of science and technology, the linear approximation is not always reliable for the increasing requirements. If the nonlinear factors are overlooked in the analysis and calculation, it can result in unacceptable error. Moreover, more and more nonlinear phenomena have attracted people's attention in the project, nonlinear problem has become a hot research topic . Therefore, we need to study nonlinear systems, reveal the nature of nonlinear system, which have a great significance for analysis and design of nonlinear system.In recent decades, a number of nonlinear analytical methods, such as average, KBM method, perturbation method, multiple scales, harmonic balance method , have been developed under the efforts of many researchers. However, on the using of Volterra series for analyzing nonlinear systems is relatively new, in addition, this method has many advantages over other nonlinear analytical methods. Based on the above issue, this article will deal with how to use the Volterra series analyzing the nonlinear system.Volterra series which is a description of the nonlinear relationship between system input and output, is a mathematical function. Volterra series is a powerful mathematical tool for the analysis of nonlinear system, which is essentially an extension of the standard convolution description of linear systems. Meanwhile, The Volterra series can be regarded accordingly as a Taylor series with memory, it can be used to describe a class of nonlinear systems. Volterra series is a infinite series based on kernel function, is the output which is the result of kernel function high-order convolution with the input of system. Although the Volterra series is a infinite series, studies have shown that there is a large class of nonlinear systems which can be represented by finite Volterra series in engineering practices, so only if the Volterra series of nonlinear systems is convergence, we can use a truncated Volterra series to approximate the nonlinear systems.The main contents of this article are as follows: Chapter one describes the researching significance of Volterra series, the purpose of the research and the present situation of Volterra series and damage identification around domestic and abroad. Chapter two describes the completed and truncated expression of Volterra series, the definition of generalized frequency response function, the generalized frequency response function's solution method based on the harmonic probing, the general recursive algorithm of generalized frequency response function, the definition of nonlinear output frequency response function and its numerical solution algorithm, the definition of the output frequency response function and its solving methods, theory of NARMAX model, including the expression of NARMAX model, selection of key items based on forward recursive orthogonal least squares algorithm, model validation and so on. Chapter three introduces random vibration frequency response of nonlinear system based on Volterra series , this chapter includes three parts: first, derives the general expression of the output power spectrum of nonlinear system excited by non-determined signal based on Volterra series and the generalized frequency response function; second, researches the effect of excitation intensity on system output power spectrum based on the general expression of nonlinear system output power spectrum; third, researches the effect of nonlinear parameters in nonlinear systems on the system output power spectrum based on the general expression of nonlinear system output power spectrum .Chapter four introduces the theory of damage detection which is based on NARMAX model and nonlinear output frequency response function. Numerical simulation and experimental results show that the damage detection method can effectively detect the existence of structural damage, which has an important significance for the health monitoring of engineering structures. Chapter five introduces how to locate the nonlinear component in the periodic structure based on nonlinear output frequency response function. In addition, in general , if structure occurs damage, it will produce nonlinear characteristics, so if we can locate the nonlinear situation in structure, then we can determine the damage situation, which has great significance for engineering application. Chapter six summarizes the major work in this article and the outlook for future work.