弹性地基上转动FGM梁自由振动的DTM分析

DTM analysis for free vibration of rotating FGM beams resting on elastic foundations

基于Euler-Bernoulli梁理论,利用广义Hamilton原理推导得到弹性地基上转动功能梯度材料(FGM)梁横向自由振动的运动控制微分方程并进行无量纲化,采用微分变换法(DTM)对无量纲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FGM梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种边界条件下横向自由振动的无量纲固有频率,再将控制微分方程退化到无转动和地基时的FGM梁,计算其不同梯度指数时第一阶无量纲固有频率值,并和已有文献的FEM和Lagrange乘子法计算结果进行比较,数值完全吻合。计算结果表明,三种边界条件下FGM梁的无量纲固有频率随无量纲转速和无量纲弹性地基模量的增大而增大;在一定无量纲转速和无量纲弹性地基模量下,FGM梁的无量纲固有频率随着FGM梯度指数的增大而减小;但在夹紧-简支和夹紧-自由边界条件下,一阶无量纲固有频率几乎不变。

Based on Euler-Bernoulli beam theory,the governing differential equation of motion of the lateral free vibration a rotating functionally graded material( FGM) beam on elastic foundation is derived by using generalized Hamilton principle,and differential transform method( DTM) is used to transform the dimensionless governing differential equation and the boundary conditions. At the same time,the dimensionless natural frequencies of transverse free vibration of rotating FGM beam on elastic foundation at the clamped-clamped,clamped-simply supported and clamped-free three boundary conditions are determined,then the governing differential equation is degenerated to the FGM without rotation and elastic foundation. The values of first nondimensional natural frequency with different FGM gradient index are calculated and they are completely consistent with the results by either the FEM or the Lagrange multipliers method in the literature. The results show:at the above three kinds of the boundary conditions,the dimensionless natural frequencies increase with the growth of the dimensionless rotating speed and the dimensionless elastic foundation modulus. Under a certain dimensionless rotating speed and dimensionless elastic foundation modulus,the dimensionless natural frequencies decrease along with the growth of the FGM gradient index. However,when at clamped-simply supported and clamped-free boundary conditions,the first dimensionless natural frequency is almost constant.