基于高阶梁理论的功能梯度材料自由振动分析

本文基于一种新型的高阶梁理论,研究了功能梯度材料梁的自由振动问题。首先对该新型高阶梁理论进行了介绍,然后对该理论进行了有限元实现,并利用Hamilton原理推导得到了离散的动力学平衡方程,构造了2节点8自由度的C1型高阶梁单元。参照文献作了均质悬臂梁的模态分析,验证了该梁单元的精度。然后利用该单元进行功能梯度梁的模态分析,并构造了一种材料相关性很弱的无量纲固有频率。由该无量纲固有频率引入了功能梯度梁与均质梁固有频率之间的转换关系,并通过算例分析了该转换关系的适用条件。

DYNAMIC RESPONSES OF BEAMS OF FUNCTIONALLY GRADED MATERIALS BASED ON HIGHER-ORDER BEAM THEORY

Based on a new type of the high-order beam theory, this paper studies the free vibration of beams of functionally graded materials (FGM). The finite element method is used for the new high-order beam theory. The discrete dynamic balance equation is derived by using the Hamilton principle, and the C1 type high-order beam element with 2 nodes and 8 degrees of freedom is constructed. The modal analysis of the homogeneous cantilever beam is carried out, and the accuracy of the beam element is verified. Then the modal analysis of the FGM beam is carried out by using the element, and a dimensionless natural frequency with a weak material correlation is formulated. Based on the dimensionless natural frequency, the transformation relationship between the natural frequency of the beam of the functionally graded materials and that of the homogeneous beam is established. The applicable conditions of the transformation relationship are analyzed by numerical simulations.