弹性地基上转动功能梯度材料Timoshenko梁自由振动的微分变换法求解

基于Timoshenko梁理论研究弹性地基上转动功能梯度材料（FGM）梁的自由振动。首先确定功能梯度材料Timoshenko梁的物理中面，利用广义Hamilton原理推导出该梁在弹性地基上转动时横向自由振动的两个控制微分方程。其次采用微分变换法（DTM）对控制微分方程及其边界条件进行变换，计算了弹性地基上转动功能梯度材料Timoshenko梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种不同边界条件下横向自由振动的量纲一固有频率，与已有文献的计算结果进行比较，退化后结果一致。最后讨论了不同边界条件、转速、弹性地基模量和梯度指数对功能梯度材料Timoshenko梁自振频率的影响。结果表明：功能梯度材料Timoshenko梁的量纲一固有频率随量纲一转速和量纲一弹性地基模量的增大而增大；在量纲一转速和量纲一弹性地基模量一定的情况下，梁的量纲一固有频率随着功能梯度材料梯度指数的增大而减小。

DTM Analysis for Free Vibrations of Rotating Functionally Graded Material Timoshenko Beams on Elastic Foundations

Free vibrations of a rotating FGM beam on elastic foundation was investigated based on Timoshenko beam theory.Firstly， the physical neutral surface position for the FGM Timoshenko beam was determined， and two motion governing differential equations of the transverse free vibrations of the rotating beam on elastic foundation were derived by using generalized Hamilton principle.Secondly，DTM was used to transform the differential equations and the boundary conditions.At the same time， the dimensionless natural frequencies of transverse free vibrations of rotating FGM Timoshenko beam on elastic foundation with clamped-clamped， clamped-simply supported and clamped-free three boundary conditions were solved， then the governing differential equation was degenerated and the good agreement among the results of this paper and those available in the literatures validated the presented approach.Finally， the effects of different boundary conditions， different rotating speeds， different elastic modulus and different gradient indexs on the free vibration frequencies of FGM Timoshenko beam were discussed.The results show that the dimensionless natural frequencies of FGM Timoshenko beam increase with the growth of the dimensionless rotating speeds and the dimensionless elastic foundation modulus.Under a certain dimensionless rotating speeds and dimensionless elastic foundation modulus，the dimensionless natural frequencies decrease along with the growth of the FGM gradient indexes.