A Statistical Linearization Method For Fractional Order Nonlinear Systems Under Combined Random And Harmonic Excitation

A statistical linearization method for calculating the response of nonlinear systems with fractional derivative damping under random and harmonic excitations is proposed.At present,the fractional derivative model is widely used to simulate the mechanical behavior of viscoelastic dampers.Although some progress has been made in solving the response of fractional order stochastic linear dynamical systems,there are few studies on the stochastic dynamic response of nonlinear systems under combined random and deterministic excitation.In this paper,a numerical method for the response of fractional order nonlinear systems under joint excitation is developed.At present,the numerical algorithm of the dynamic response of fractional linear system has been relatively mature.Several numerical integration methods are proposed to solve the dynamic response of linear single-degree-of-freedom systems and linear multi-degree-of-freedom systems.On this basis,the numerical integration methods for the response of fractional order linear dynamical systems are extended to fractional order nonlinear systems.According to the definition of fractional derivative(Grunwald and Riemann Liouville)and the discrete form of integral derivative(finite difference method or Newmark-? method)in the equation of motion,six different numerical integration algorithms are developed.Then,the displacement responses of single-degree-of-freedom linear or nonlinear systems and corresponding multi-degree-of-freedom systems under different excitations(including simple harmonic excitation and white noise excitation)are calculated by these methods.It is shown that although all schemes provide reasonably accurate results,but the efficiency of the finite difference method defined by Grunwald is the highest.In addition,the numerical analysis of fractional derivative requires the use of the response values of all previous times,which leads to a very time-consuming calculation;therefore,the finite difference method defined by Grunwald is preferred when the accuracy is satisfied.Next,this paper combines the statistical linearization method and harmonic balance method to solve the random dynamic response of fractional nonlinear system under joint excitations.First,the response has been expressed as the sum of a periodic(deterministic)component and of a zero-mean stochastic component.Next,ensemble averaging of the equation of motion has yielded a nonlinear(vector)ordinary differential equation(ODE)for the periodic component,and a nonlinear(vector)stochastic ODE for the zero mean stochastic component with the subtraction of the periodic component.Then,the nonlinear(vector)stochastic ODE are solved by using the statistical linearization method,and a set of equations,involving the amplitudes of the periodic component of the response and the statistical moments of the stochastic component of the response,has been derived.On the other hand,for the nonlinear(vector)ODE,a set of nonlinear algebraic equations can be derived by using the harmonic balance method.Similarly,this set of equations contains the amplitudes of the periodic component of the response and the statistical moments of the stochastic component of the response.Then,a set of algebraic equations areused to obtain the amplitudes of the period component and the statistical moments of the zero-mean stochastic component of the approximate system response.Finally,an assessment of the proposed formulation has been conducted through Monte Carlo simulations.