Analysis Of Elastic Wave Propagation Characteristics In Periodic Rod And Beam Structures With Variable Cross Sections/Non-uniform Materials

The periodic rod and beam structures can be used to control the propagation of longitudinal waves and bending waves because of their band gap characteristics.The band gap characteristics means that the elastic waves in the pass band can propagate in the structure without attenuation,while the elastic waves in the band gap will be forbidden in the structure.Currently in most studies,the unit cell of periodic rod and beam structures are composed of rod and beam members with uniform cross-sections and homogeneous material properties,which no longer can satisfy the development needs for vibration and noise reduction,high-precision mechanical design and manufacturing,and signal monitoring.Geometric parameters and material properties due to non-uniform change of rod,beam element of periodic structure can change the uniform the uniform beam class period of the structure of the original band structure,and the ability to increase the number of band gap and control the width of band gap,make the class more and more get of mechanical design and manufacturing periodic structure and damping noise reduction,etc.In this thesis,the periodic rod and beam structures with the unit cell composed of members with variable（non-uniform）cross-sections and/or variable（inhomogeneous）material parameters are taken as the research object,and the propagation characteristics of longitudinal waves and bending waves in these periodic structure are analyzed as the alterations of cross-sections/material parameters along the rod or beam axes are regarded as continuous functions.The specific research work includes:（1）The models of rod or beam variable section and/or non-uniform with continuous variation of geometric parameters and/or material properties along the axis of the rod or beam are established.Based on classical rod theory,Euler-Bernoulli beam theory and Rayleigh-Timoshenko beam theory respectively,analytical solutions of the corresponding bar or beam governing equations and their closed forms are obtained.Based on Floquet-Bloch theorem,periodic boundary conditions are introduced into the structures with variable cross sections and/or non-uniform beams,and a unified expression of the transfer matrix method（TMM）for analyzing the propagation characteristics of elastic waves in the structures with periodic rods or beams is derived based on the transfer matrix method.（2）Unity based on the derived the transfer matrix method of column type,the compiled representation periodic variable cross-section rod/beam structure of elastic wave propagation in the dispersion curve of FORTRAN program,combined with a numerical example to verify the validity of the theoretical deduction and programming,and the correctness of the finite element software operation.（3）The influence of core parameters on elastic wave frequency band structure of periodic variable cross section bar or beam structure is deeply analyzed and summarized according to the dispersion curves of periodic variable cross section bar or beam structure under different cell configurations calculated by the program.From these investigations,the following conclusions are drawn:（1）In the process of solving the governing equation of the periodic structure with variable cross section and/or non-uniform beam,if the governing equation of the continuous function of cross section or material change is linear ordinary differential equation with constant coefficient,the analytic solution of closed form can be obtained by solving the characteristic equation.If the governing equation is a linear ordinary differential equation with variable coefficients,the solution can be given in the form of Bessel function if the equation has a solution.（2）The longitudinal waves frequency band structure of the periodic rod structure with variable cross-section and continuous cross-section is essentially different from that of the periodic rod structure consisting of several constant cross-section rod elements after equivalence: If the periodic element of the latter is equivalent to a constant section rods,the longitudinal waves frequency band structure of the latter has a bandgap at phase 0 followed by a passband,while the longitudinal waves frequency band structure of the former has a bandgap at phase 0,and a bandgap is generated in the frequency band corresponding to a passband of the latter.That is,the former has at least one more band gap than the latter.In the periodic beam structure based on Timoshenko beam theory,due to the coupling of shear mode wave and pure bending mode wave in the bending wave,the frequency band structure is relatively complex and the law similar to that in the periodic beam structure cannot be obtained.However,the above two results are also fundamentally different.（3）Continuous cross section changes will open up all the pass bands with phase at 0 and phase at π the frequency band structures of periodic rods,Euler-Bernoulli beams and Timoshenko beams,generating several new bandgaps.The continuous variation of the cross section also regulates the bandwidth of the band gap caused by different material parameters.（4）Increasing the absolute value of the cross section variation coefficient will increase the band gap width and attenuation wave propagation effect.However,the larger the cross section variation coefficient is not the better,when it reaches a certain value,the bandgap width will not increase.These characteristics are found in periodic variable section rods,Euler-Bernoulli beams and Timoshenko beam structures.In particular,the coefficient of cross section variation affects the number of shear mode waves and the number of pure bending mode waves or cutoff frequencies in the characteristic bending waves of Timoshenko beams with variable cross-section.