Mixed Finite Volume Element Method For Vibration Equations Of Beam With Structural Damping

Beam is the most common component in mechanical equipment and building construction.Beam vibration equation is widely used in large engineering projects,bridge construction and aerospace.The beam vibration equation with damping and structural damping terms can describe the special properties of the object better.The viscoelastic constitutive relation described by fractional derivative has been paid more attention since the 1980 s,especially the time-fractional-order damping beam and time-fractional-order damping beam,which can more effectively describe the memory and time dependence of the vibration of the beam.The vibration equation of a beam is a kind of fourth order partial differential equation,and its main numerical solution is to construct a higher order trial function space,or to reduce the order of the equation,namely mixed finite element method.The mixed finite volume element method is an important method for solving partial differential equations.It can maintain the local conservation of physical quantities,has relatively low requirements for spatial smoothness,and can solve two physical quantities at the same time.Based on the above advantages,this method is applied to integer order and fractional order damping and structural damping beam vibration equations.The thesis is divided into six chapters.The first chapter is the introduction,which introduces the physical background and research status at home and abroad of damping beam vibration equation,and briefly describes the research content of this thesis.In Chapter 2,by introducing the velocity and bending moment of the beam transverse vibration as intermediate variables,the mixed finite volume meta-scheme of the vibration equation of the damped beam is constructed,and the convergence analysis is carried out by elliptic projection.In Chapter 3,the mixed finite volume element scheme of the vibration equation of structural damped beam is given,and the elliptical projection is also used for theoretical analysis.In Chapter4,a mixed finite volume element method is proposed for the vibration equation of fractionally damped beams.In Chapter 5,a mixed finite volume element method is proposed for vibration equations of damped beams with fractional order structures.Chapter 6 is the summary of the full text and the prospect of the future work.In this thesis,numerical examples of the four models are given to verify the feasibility and effectiveness of the proposed method,and the influence mechanism of damping coe cient on vibration magnitude is studied.