Wavelet-Galerkin Method For Stochastic Dynamic Response Of Fractional Nonlinear Systems

The dynamic analysis method of engineering structure is an important part of structural design,and the dynamic analysis of civil engineering structure has its particularity.In particular,not only the external excitation of the structure is nonstationary and random,but the structure itself has nonlinearity and complexity.For example,civil engineering researchers generally describe the seismic action and wind load by a stochastic process,in which the former has a strong time-frequency joint(time domain and frequency domain)non-stationary;The concrete structure is complex and its material properties cannot be precisely controlled,which leads to the complex nonlinear force state under the action of large earthquakes.Therefore,it is difficult to study the stochastic dynamic response of nonlinear structural systems under completely non-stationary random excitation.At present,the mature theory of random vibration of linear structural dynamic systems under stationary random excitation has been established.At the same time,a variety of analytical or numerical approximation methods have been developed to analyze stochastic dynamic systems with specific nonlinearities,including statistical linearization,stochastic averaging,and numerical simulation.The stochastic dynamic response of a general nonlinear dynamic system under completely non-stationary random excitation remains to be studied.Representative processing methods include statistical linearity based on wavelet analysis.On the other hand,considering that the fractional derivative model can easily model the frequency-dependent dynamic characteristics(such as the constitutive relation of viscoelastic materials),the focus of this paper is on the development of Dynamic response method of fractional-order nonlinear systems(such as the widely used viscoelastic passives Control structure)under completely non-stationary random excitation.Since the wavelet has joint resolution in the time-frequency domain,this paper intends to select the generalized harmonic wavelet which is widely used in engineering signal analysis to solve non-stationary random dynamic problems.At this time,the non-stationarity mainly comes from the non-stationarity of the excitation and the Time-varying of system.In terms of system time-varying,the actual damping and stiffness of the engineering structure will change slowly over time.Such a dynamic system is a linear time-varying system.The time-varying is assumed to be a slow-changing function,and the dynamic equation of the system can be transformed from a differential equation form to a linear(nonlinear)algebraic equation form through wavelet-Galerkin method.The system's response wavelet coefficients can be obtained by solving algebraic equations,and the system response displacement can be obtained by recombining wavelet coefficients.Combined with the relationship between wavelet coefficients and evolution power spectral density,the system response evolution power spectral density can be obtained.Numerical simulations show the applicability of the proposed method.In recent years,fractional derivative models have been widely used to simulate the constitutive relationship of viscoelastic materials.Therefore,the nonlinear structural system under viscoelastic damping control is taken as the research object.By introducing fractional-order nonlinear differential equations of motion and using wavelet-Galerkin method,the system' Non-stationary random dynamic response is obtained in the framework of frequency Monte Carlo simulation.Numerical simulations show the accuracy of the method.At the end of this paper,some prospects for the nonlinear stochastic dynamics of wavelet-Galerkin method are proposed.