A Compact Difference Scheme For The Vibration Of Damped Beam Equations

Beam vibration equation is widely used in bridge construction,civil engineering,noise control and aerospace.Beam vibration equation with damping term can better describe the special properties of the object,especially the fractional time damping term,which can more effectively depict the memory and time dependence of the object.Therefore,it has important theoretical significance and research value to study the damping beam vibration equation.The compact finite difference method uses fewer grid nodes to obtain higher accuracy,which greatly reduces the calculation amount.In this paper,a three-layer compact finite difference scheme is proposed for the vibration equation of integerorder damped beam,and a spatial compact Crank-Nicolson finite difference scheme is proposed for the vibration equation of fractional-order damped beam.This paper is divided into four chapters,the first chapter is the introduction,the background and research status of beam vibration equation and damping coefficient are introduced.The second chapter studies the initial boundary value problem of integral order damped beam vibration.By introducing a new variable,the fourthorder damped beam vibration equation is rewritten as two second-order equations.Then a compact finite difference operator and a central difference method are applied to construct a three-layer compact difference scheme.The presented difference scheme is unconditionally stable and convergent according to the energy method.Finally,the error orders of the numerical scheme of damping coefficient with different values are verified by numerical examples.The third chapter studies the initial boundary value problem of the vibration equation of fractional damped beam.By introducing two new variables,the vibration equation of fourth-order damped beam is rewritten as a second-order system of equations.A compact Crank-Nicolson scheme of fourth-order in space is constructed,where the fractional derivative is approximated by L1.By using the energy method,it is proved that the difference scheme is unconditionally stable and convergent.Finally,a numerical example is given to verify the effectiveness of this method.The fourth chapter is the summary of the full paper and the prospect of the future work.