分数阶粘弹性地基上薄板波动和振动特性分析

本文研究了粘弹性地基上薄板的波动和振动问题．主要讨论了基于分数导数理论的粘弹性地基模型上 薄板弯曲波的传播特性以及固有频率对地基的依赖特性．推导了三种经典粘弹性地基模型的复模量．并利用分 数导数的性质得到分数阶粘弹性地基上 Kirchhoff板中弯曲波的传播速度、衰减系数以及自由振动的复固有频 率．数值算例表明粘弹性地基对弯曲波传播特性存在显著影响，不同粘弹性模型所对应的色散和衰减特性也存 在较大差别．分数阶导数可以实现相邻整数阶导数之间的光滑过渡．利用分数导数的本构关系可以更加真实地 描述粘弹性地基的历史依赖行为，更准确地表现出粘弹性地基板中弯曲波的色散和衰减特性．

Wave Propagation and Vibration in a Thin Plate Mounted on the Viscoelastic Foundation Modeled by Fraction Order Derivatives

The wave propagation and free vibration problems of a thin plate on the viscoelastic foundation are studied in this paper. The viscoelastic foundation is modelled by the fractional order derivative. Three kinds of viscoelastic models with fraction order derivatives are considered and the corresponding complex modulus are derived. The dispersion and attenuation rate of the flexural curves and the complex-valued natural frequencies are obtained for a Kirchhoff plate on the viscoelastic foundation and some numerical examples are provided. Based on the numerical examples, it is found that the dispersion and the attenuation of flexural waves are affected evidently by the viscoelastic feature of the foundation. Different kinds of viscoelastic models lead to different dispersion and attenuation characteristics. The fraction order derivative can bridge the gap of adjacent integer order derivatives and can model the history-dependent behavior of the viscoelastic foundation more realistically and therefore describe the dispersion and attenuation features of the flexural waves more accurately.