功能梯度板与均匀板固有频率之间的相似转换

摘要

Based on the classical plate theory, the free vibration response of functionally graded material (FGM) plates was studied. By eliminating the in-plane displacements, a similarity of the governing equation of free vibration of FGM plates to that of homogenous plates was found, from which an analogous transformation between the natural frequencies of the FGM plate and homogenous one was obtained. By assuming that the material properties of FGM plate would vary arbitrarily and continuously in lateral direction, the dimensionless analytical expression of analogous transition factor was derived. Concentrically reflecting the influence of the non-uniform material properties of the FGM plates on the vibrational frequency. Therefore, the solutions of free vibration and the evaluation of analogous transformation factors of the FGM plate could be transformed into that of homogenous plate with identical geometric dimension and boundary condition. This method could provide a simple and convenient approach in analysis of non-uniform plate, which would be very easy to be used in engineering.%基于经典板理论,研究功能梯度材料板的自由振动响应.通过消去功能梯度材料板的自由振动控制微分方程中的面内位移,发现功能梯度板与均匀板的控制方程的相似性,由此得到功能梯度材料板与均匀板固有频率之间的相似转换关系.在给定功能梯度材料板的材料性质在横向任意连续变化的情况下,给出无量纲相似转换系数的解析表达武.该系数集中反映功能梯度板的材料非均匀性对振动频率的影响.因此,可将功能梯度材料板的自由振动问题的求解转换为同样几何尺寸和边界条件下均匀板的振动问题的求解以及相似转换系数的计算问题.这一方法可为非均匀板的分析和求解提供便捷途径,便于在工程中应用.