Research On Wave And Vibration Of Euler Beam And Kirchhoff Plate On Fractional Order Viscoelastic Foundation

The beam structure and plate structure on elastic foundation have been widely used in civil engineering.such as the foundation of building structures,airport runways,highway pavements,etc.The vibration problem has always been the focus of scientific theory and engineering applications.Considering the interaction of the soil-structure interaction,it is usually regarded as an elastic or viscoelastic model for mechanical calculation and analysis.The propagation characteristics of flexural waves and the free vibration are important references for reflecting the characteristics of structure.The analysis of vibration response is used as a reference for structural design.It has important theoretical significance and quite broad application prospects.Based on the classical beam and plate theory,this paper derives the control differential equations of Euler beam and Kirchhoff plate,solves the wave and vibration solutions of the differential equations and analyzes the wave propagation characteristics and the inherent characteristics of finite-length beams and plates under different boundary conditions.Finally,based on the theory of fractional derivative viscoelasticity,Discussing the influence of the order of the fractional differential operator on the dynamic characteristics of the beam and plate on the foundation.The main contents and conclusions are as follows:(1)Discussing the flexural waves of beams and plates supported on viscoelastic foundations.The propagation of flexural waves exhibits complex dispersion and attenuation properties due to the influence of viscous foundations.Two sets of very different attenuated traveling waves appear in viscoelastic foundation model.In the Kelvin model and the standard solid model,the propagation velocity curve has a peak,and the attenuation curve has a trough.And the peak frequency of the wave velocity is consistent with the valley frequency of the attenuation.It shows that the viscoelastic dissipation suppresses the flexural wave propagation velocity.(2)Analyzing the vibration characteristics of the beam and thin plate structure under different foundations,it is found that the natural frequencies of the beam and the plate on the Winkler foundation are both greater than the natural frequency without foundation.The specific difference is related to the elastic coefficient of the foundation,indicating that the elastic foundation improves the overall structure Thestiffness.In addition,the beams and slabs on the three viscoelastic foundations have complex frequencies.The real part is the true frequency,and the imaginary part is the time-dependent attenuation factor.(3)The propagation curve of bending waves in the beam and plate on the fractional derivative viscoelastic foundation model is basically the same as that of the beam and thin plate on the classical viscoelastic foundation model,but the order of the differential operator has a significant influence on the wave propagation speed and attenuation coefficient.Similarly,the order of the fractional derivative also has an effect on the attenuation of bending waves.The order of the fractional differential operator also has a significant effect on free vibration.The specific performance is that the larger the order,the smaller the natural frequency,and the more stable the time decay factor.It shows that the fractional derivative can describe the dynamic characteristics of the structure more flexibly and finely.