Winkler弹性地基板梁的自由振动分析

Free Vibration Analysis of a Mindlin Plate-beam on a Winkler Elastic Foundation

中短型轨道板的几何构型介于梁、板之间,属于宽梁结构。从Mindlin板理论出发,退化得到适用于宽梁的Mindlin板梁控制方程;引入Winkler地基刚度系数,推导得到位移和转角的模态函数表达式。考虑两端简支的边界条件,得到弹性地基板梁的自由振动特征方程。通过无量纲数值算例求解出弹性地基板梁的自振频率,并与Timoshenko梁理论和Mindlin板理论进行对比。研究高跨比、泊松比和弹性地基刚度等参数对结构自振特性的影响,总结出弹性地基板梁方程的特点及适用范围,即宽度效应显著且泊松比较大的宽梁结构。

In recent years,short-and medium-length floating-slab tracks have become commonly used in railway engineering.These tracks are of a model type between a Winkler foundation beam and a Winkler foundation plate.For this type of wide-beam structure,a more suitable theory is required that both ensures calculation accuracy,as does the plate theory but also provides a simple analytical process,such as that of beam equations.In this study,the governing equations of the Mindlin plate are degraded and the dynamic equations for wide-beam structures are obtained：this is called the Mindlin plate-beam theory.Although the equations of the Mindlin plate-beam theory appear similar to the equations of the Timoshenko beam theory,the coefficient of bending stiffness is different and retains the direct influence of the Poisson＇s ratio parameters.This means that lateral deformation can be considered in the Mindlin plate-beam theory.The stiffness of the elastic foundation is also considered,and the equations are extended accordingly.For general analysis,the variables and parameters in this study are normalized and the expressions for both the vertical displacement and flexural rotation angle of a wide beam are obtained.Based on the boundary conditions,the dynamic characteristic equation for a simply supported wide beam is finally derived,and the normalized frequencies for the wide beam can be calculated through certain root searching programs.In this study,different kinds of boundary conditions are considered using the same procedure.To illustrate the wide-beam theory described herein,several numerical examples are used and the natural frequencies of a Mindlin plate-beam on a Winkler elastic foundation are compared with the results of a Timoshenko elastic foundation beam and a Mindlin elastic foundation plate.The results demonstrate the accuracy of the present equations.The effects of the height-to-length Poisson＇s ratios and elastic foundation stiffness are considered and the following conclusions are obtained from the numerical results：（1）For a simply supported wide beam,the first three frequencies of the plate-beam model show good agreement with those based on the Mindlin plate model.The discrepancy of the results from the two beam models increases with the width of the beam as compared to those of the Mindlin plate.However,the fundamental frequency of the Mindlin plate-beam theory still remains in good agreement with the plate theory;（2）the natural frequencies obtained from the Mindlin plate and Mindlin plate-beam theories will increase with the Poisson＇s ratio for a wide beam on an elastic foundation and those obtained from the Timoshenko beam theory will decrease.This result means that the relative error of the beam-plate and Mindlin plate will further reduce for a larger Poisson＇s ratio;and（3）the equations derived here are suitable for wide-beam analysis,can incorporate the effect of beam width,and are especially suitable for a wide foundation beam with a relatively large Poisson＇s ratio.The numerical examples based on this approximation theory are in good agreement with the Mindlin plate theory,while the equations and calculation process are much simpler.