| Fractional calculus provides an appropriate mathematical instrument for describing memory-dependent phenomena and intermediate processes in real world.In viscoelastic damped vibration system,the modeling of fractional differential equation shows the characteristics of clear physical significance,accuracy and simplicity.Distributed-order system identification is an important tool to determine the model and parameters of distributed-order system.Seeking the solution and analyzing the properties of the solution is helpful to master the law of motion change of the system.And the equations are especially difficult to solve due to the weak singularity and nonlocality of the fractional order operator.It is necessary to use the approximate method to find the approximate analytical solution.The first chapter surveys the research background,significance and current situation of this topic.In the second chapter,the definition and properties of fractional calculus and three basic fractional viscoelastic constitutive models are given.In the third chapter,the improved identification of the distributed-order system is studied.The linear interpolation is used to approximate the order-weight distribution function and the least square method is applied to identify the system.The Bode amplitude frequency response curve and satisfactory identification results are obtained.In the fourth chapter,the steady-state solutions of the fractional vibration equation with (?)and (?) are analyzed.By using Fourier transform,it is proved that the system under complex harmonic excitation is equivalent to a second-order integer order system.The viscoelasticity and viscous inertia changes caused by different fractional derivative terms are explained.With the help of the equivalence theorem,the contribution of fractional derivative terms to mass,stiffness and damping and the steady-state response of the system under harmonic and general periodic excitation are given conveniently.Fourier series and superposition principle are used in the process.Three contribution functions of ω,α and β,amplitude frequency relation and phase frequency relation of the system are discussed in detail.The results show that fractional derivative is very suitable for featuring viscoelasticity and viscous inertia of materials.In the fifth chapter,some solutions of fractional differential equations are studied analytically and approximately.Laplace transform method is used to solve the fractional vibration equation,the distributed-order equation and the viscoelastic constitutive equation.Adomian decomposition method is applied to solve the nonlinear fractional partial differential equation;Exp-function method is employed to find the solitary wave solution of the nonlinear fractional Fisher Equation.The final chapter summarizes the research results of this paper,and looks forward to the research of fractional differential equations. |
