Analysis For Natural Vibration Characteristics Of Beams On Viscoelastic Foundation

Finite beams resting on viscoelastic foundations are widely used in enginnering structures,such as railways,highways,airports,ships,foundations of high-rise buildings,the vibration analysis of such a system has interested many researchers in science and engineering.The natural frequencies,attenuation coefficents and modes of viscoelastic structers in the process of free vibration are important and basic parameters to reflect the characteristics of the structures,through its efficient and accurate research,it not only provides a theoretical basis for the design of the structure and calculation of the construction,but also helps to avoid the resonance and fatigue problems and reduce the vibration of the structure.Since its first publication in 1998,the method of reverberation-ray matrix has been successfully applied to analyzing transient responses of embedded frame structure and isotropic layered media,the method of Newmann series expansion is used to eliminate singularity in the Fourier inverse transformation of frequency response,it has the advantages of clear physical meaning,uniform column style,easy programming,high accuracy,low consumption and good readability in structural dynamics analysis.Up to now,there are few relevant research on the natural vibration characteristics of the finite viscoelastic beam and the viscoelastic foundation beam,the existing research methods are mainly limited to the dynamic stiffness method,the transfer matrix method,the wave propagation method and the finite element method,there are many problems such as convergence,stability,accuracy and generality,lack of analytical or approximate analytic work.Because of this,the study is mainly focused on solving spectrum relation and natural vibration characteristics of the viscoelastic beam and the viscoelastic foundation beam by the reverberation-ray matrix method in this paper.Firstly,taking a viscoelastic cantilever beam without foundation as an example,according to the two typical damping characteristic models of materials,viscous damping and hysteretic damping,the analytical and numerical solutions of the natural frequency and attenuation coefficient of the viscoelastic beam under longitudinal free vibration and the numerical solution of the natural frequency of the transverse free vibration are obtained,the effect of material damping on the transient response,spectral relationship and free vibration characteristics of viscoelastic foundation beam is studied.The result shows that with the increase of material damping coefficient,the delay effect and energy dissipation is stronger in the process of wave propagation,when the axial viscous damping and hysteretic damping of materials increases,the attenuation coefficient of the viscoelastic structures increases,when the axial and transverse viscous damping of materials increases,the natural frequency of the viscoelastic structures decreases,when the axial and transverse hysteretic damping of materials increases,the natural frequency of the viscoelastic structures increases.Secondly,in order to study the influence of soil parameters on the spectral relationship of Euler beams on viscoelastic Pasternak foundation,the natural frequencies and vibration modes of any order are calculated by using MATLAB language,the results are compared with that of the finite element software SAP2000,which discretizes the beam of elastic foundation into 100 elements,the influence of soil parameters and boundary conditions on the natural vibration characteristics of Euler beams on viscoelastic foundation is analyzed.Result shows,soil spring coefficient and shear coefficient increases natural frequency,damping coefficient of soil decreases natural frequency and increases the attenuation coefficient of structure,however,the attenuation coefficient of the viscoelastic Pasternak foundation beam is constant when the soil damping coefficient is constant.Then,the influence of soil parameters on the spectral relationship of Timoshenko beams on viscoelastic Pasternak foundation is studied.Through specific examples,the effects of foundation type,bearing type,cross-section type and length of beam on the natural vibration characteristics of Euler beam and Timoshenko beam on viscoelastic Pasternak foundation are analyzed.The results show that the natural frequencies of each order under the simple support boundary on both sides are the smallest,and the natural frequencies of each under the fixed boundary on both sides are the largest,the natural frequencies of long beam relative to short beam are all decreased,the attenuation coefficient of long beam is larger than that of short beam,the simple support has no effect on the modes,the fixed support has a great influence on the modes.Finally,according to the equation of motion of longitudinal wave propagation in viscoelastic beam,and the equation of motion of shear wave propagation between Euler beam and Timoshenko beam on viscoelastic pasternak foundation,the Fourier transformation is carried out,combined with the coupling condition of nodes and the dual transformation of displacement and force in the local coordinate system of element,from the point of view of continuous model,the orthogonality of longitudinal and transverse vibration modes corresponding to different frequencies in dual local coordinate system is strictly proved,and an algorithm case is proposed to verify the correctness of the orthogonality of modes.