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Free vibration analysis of porous functionally graded materials Timoshenko beam

DOI：10.7511/jslx20200804002

 作者 单位 E-mail 王伟斌 兰州理工大学 理学院, 兰州 730050 杨文秀 兰州理工大学 理学院, 兰州 730050 滕兆春 兰州理工大学 理学院, 兰州 730050 tengzc@lut.edu.cn

基于Timoshenko梁理论研究多孔功能梯度材料梁（FGMs）的自由振动问题。首先，考虑多孔功能梯度材料梁的孔隙率模型，建立了两种类型的孔隙分布。其次，基于Timoshenko梁变形理论，给出位移场方程、几何方程和本构方程，利用Hamilton原理推导多孔功能梯度材料梁的自由振动控制微分方程，并进行无量纲化，然后应用微分变换法（DTM）对无量纲控制微分方程及其边界条件进行变换，得到含有固有频率的等价代数特征方程。最后，计算了固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）三种不同边界下多孔功能梯度材料梁自由振动的无量纲固有频率。将其退化为均匀材料与已有文献数据结果对照，验证了正确性。讨论了孔隙率、细长比和梯度指数对多孔功能梯度材料梁无量纲固有频率的影响。

The free vibration of beam made from porous functionally graded materials (FGMs) is studied based on the theory of Timoshenko beam.Firstly, two types of pore distributions are established by considering the porosity model of the porous functionally graded material.Secondly, based on the theory of Timoshenko beam, displacement equation, geometric equation and constitutive equations, using the Hamilton principle, the governing differential equation of free vibration for porous functional gradient material beams is derived.Its dimensionless form is also obtained.Then through the application of differential transformation method (DTM) to the dimensionless governing differential equations and boundary conditions of transformation, the equivalent algebraic characteristics equations about the natural frequencies are obtained.Finally, the dimensionless natural frequencies of the free vibration of porous functionally graded material beams with three different boundaries, namely, clamped-clamped (C-C), clamped-simply supported (C-S) and simple supported-simple supported (S-S), are calculated.The results are compared with the data in the existing literature, and the results are verified.The effects of the porosity, slenderness ratio and gradient index on the dimensionless natural frequencies of the porous functionally graded beams are discussed.