Exact Solutions Of Fractional Order Osillation Equation With Two Fractional Derivative Terms

The phenomenon of vibration can be seen everywhere.The research and application of vibration bring great convenience to people’s life.At the same time,it may cause some harm.Therefore,it is very important to study the impact of vibration on industry and production equipment,reduce losses to the greatest extent,and provide safe production.When viscoelastic material or viscoelastic material is used as the damper of vibration system,it is not get good results to describe the mechanical characteristics of viscoelastic or viscoelastic material by integral calculus.Therefore,the introduction of fractional calculus can more accurately describe the mechanical characteristics of viscoelastic or viscoelastic material.The first chapter introduces the research background of fractional-order calculus and fractional-order vibration.The second chapter lists the basic knowledge of fractional-order calculus related to this research,including several special functions,integral transform,several fractional definitions and properties of fractional-order calculus and so on.The third chapter introduces the initial value problem of the fractional oscillation equation with two fractional derivative terms in the sense of Caputo,where the orders α and β satisfy 1<α ≤2 and 0<β≤1,is investigated,respectively.The system response is expressed by Laplace transform and inverse Laplace transform.The integer order cases,including overdamping,criticaldamping and underdamping oscillation systems are discussed first.In the chapter 4,using the method of inverse Laplace transform,the fractional order vibration equation of the unit step response is considered.The first method yields series solutions with the nonnegative powers of t,which converge fast for small t,via the series expansion to negative powers of s and the inverse Laplace transform term by term.In the chapter 5,where the complex path integral formula of the inverse Laplace transform is used,is our emphasis since more techniques are required and more information for the solution structures is educed.In order to determine singularities of integrand we first seek for the roots of the characteristic equation,which is a transcendental equation with four parameters,two coefficients and two noninteger power exponents.We give the existence conditions of the roots on the principal Riemann surface and prove that when the root exist,they are a pair of conjugated simple complex roots with a negative real part.We derive these responses as a sum of a classical exponentially damping oscillation,which vanishes in an indicated case,and an infinite integral of the Laplace type,which converges fast for large t,with a steady component for the unit step response.