Model Analysis Of Steady State Vibration Viscoelastic Materials Under Fractional Derivative

In the traditional integer-order vibration equation,it is impossible to accurately describe the properties of viscoelastic materials.We use the fractional derivative form to describe viscoelastic materials.The application of fractional order to the vibration of viscoelastic materials makes the constitutive relationship simple.The first chapter expounds the relevant knowledge of fractional derivative theory and its development and application,and the development status at home and abroad.The second chapter introduces some of the mathematical knowledge used in the article,which is the basis of this study.It includes the definition of fractional calculus,the vibration of fractional order,and the definition of distribution order.Chapter 3 studies the steady-state response of a fractional order vibration system.By using the fractional derivative operator,we derive the contribution of the fractional derivative viscoelasticity.The effects of viscosity contribution coefficient,elastic contribution coefficient,inertial contribution coefficient,amplitude-frequency relationship,phase-frequency relationship and order are discussed in detail.The results show that the fractional derivative can be used to characterize the viscoelasticity and viscous-inertia of the material.The fourth chapter studies the derivative containing distribution of steady-state vibration,the introduction of distribution in order to do further expansion of fractional order derivative,discusses the distribution of derivative of the contribution in the vibration condition,and introduced a coefficient,more concise discussion of viscosity,elasticity,and inertia.Curves of amplitude-frequency relationship are plotted.The amplitude is related to the excitation frequency and weight distribution function.The final chapter summarizes the full text and looks forward to the vibration of viscoelastic materials with fractional and distributed derivatives.