Study On Wave Motion And Vibration Of Timoshenko Beam And Mindlin Plate On Fractional Viscoelastic Foundation

For a long time,beams and plates on viscoelastic foundations have been used as models for many engineering components,and their wave motion and vibration characteristics have been the focus of mechanical research and engineering applications.The study of vibration characteristics can better solve the problems of structural resonance and fatigue and other damages to engineering structures,and the study of wave propagation law and attenuation characteristics is of great significance to vibration and noise reduction in engineering.In order to better reveal the wave motion and vibration characteristics of beams and plates,this paper establishes viscoelastic foundation model and fractional viscoelastic foundation model to analyze the influence of relevant parameters on the transmission and vibration of flexural waves in Timoshenko beam and Mindlin plate.Based on the Timoshenko beam theory and Mindlin plate theory,the wave motion and vibration solutions of beams and plates are deduced in this paper,which lays a theoretical foundation for the dynamic calculation of beams and plates on various foundations.This paper introduces the knowledge of Winkler foundation model and viscoelastic foundation model.In the numerical example discussion,the wave velocity and natural frequency change of beam and plate on Winkler foundation are analyzed by taking different foundation reaction coefficients,and the wave velocity change and natural frequency change of beam and plate on viscoelastic foundation are analyzed by taking different viscosity coefficient.Secondly,on the basis of the viscoelastic foundation model,the constitutive equation of the fractional viscoelastic foundation model is established.By Laplace transform,the fractional complex modulus is obtained,and then substituted into the motion control equations of the beam and plate,the wave and vibration solutions are deduced.In a numerical example,the effects of the magnitude of the storage modulus,the dissipation modulus and the fractional derivative on the fluctuation and vibration are analyzed.Compared with the classical viscoelastic foundation model,the fractional viscoelastic foundation model can better describe the mechanical properties of the structure.In practical projects,due to the complex mechanical properties of viscoelastic foundations and the diversity of models of viscoelastic foundations,it is essential to perfect the dynamic analysis of Timoshenko beams and Mindlin plates on various foundations.By studying the vibration and fluctuation of the beams and plates on the viscoelastic foundation and the fractional viscoelastic foundation,the theoretical research on the beams and slabs on the viscoelastic foundation is greatly enriched and perfected in this paper,which has important guiding significance for engineering practice and beam and plate design.