功能梯度Levinson梁自由振动响应的均匀化和经典化表示

Homogenized and classical expression for the response of free vibration of simply supported FGM Levinson beams

在两端简支边界条件下,给出了Levinson高阶剪切变形理论下功能梯度材料(FGM)梁的固有频率与参考均匀Euler-Bernoulli(E-B)梁的固有频率之间的解析转换关系。假设FGM梁的材料性质沿着梁的高度任意连续变化,通过分析FGM Levinson梁和均匀E-B梁的自由振动控制方程以及边界条件在数学上的相似性,推导出了用参考均匀E-B梁的固有频率表示的FGM Levinson梁的固有频率的解析式;从而,将复杂的耦合微分方程边值问题的求解简化为一些与梁的材料非均匀特性及几何特性有关的系数的计算问题。从而实现了Levinson剪切变形理论下FGM梁的振动响应的经典化和均匀化表示,可为工程应用提供便利。

An analytical transition relation between the natural frequencies of functionally graded material（ FGM）beam based on the Levinson beam theory and that of the reference homogenous Euler-Bernoulli（ E-B） beam was presented under simply supported boundary conditions. Material properties of the FGM beam were assumed to be varied continuously in the beam depth. By examining the mathematical similarity between the governing equations with boundary conditions of the two types of the beams for free vibrations,natural frequency of the FGM Levinson beam was expressed analytically in terms of that of the reference homogenous E-B beam. Consequently,solving the complex coupling differential equations with boundary conditions,or searching for the natural frequency of the FGM Levinson beam was simplified as the calculation of a serious of coefficients which can be easily determined by a specified gradient of the material properties and the geometry of the FGM beam. As a result,homogenized and classical expression for the response of free vibration of simply supported FGM Levinson beams was concluded,which can provide convenience for engineering applications.