功能梯度斜板的自由振动分析

Free vibration analysis on functionally graded skew plates

斜板的材料密度和弹性模量假设按照幂指数形式沿板厚度方向连续变化,为克服常规单元不能适应复杂几何形状连续体的缺陷,采用8节点四边形等参单元(Q8)离散求解域,基于Reissner-Mindlin一阶剪切变形假设(first-order shear deformation theory, FSDT),利用虚功原理导出功能梯度斜板自由振动的有限元方程,通过四边简支板的振动响应特征验证模型的准确性。在梯度指数k=0和k=+∞两种情况下,对比了研究结果与ANSYS软件的计算结果,发现第4阶固有频率误差最大,为1.39%~2.53%,证明算法有效,随后考察了梯度指数和斜角对固有频率的影响。结果表明:功能梯度板振动频率介于两种均质材料板之间,且各阶固有频率随着梯度指数增大而降低;增大斜角使得板的相对厚度变大,导致各阶固有频率升高。

The material density and Young′s modulus of skew plate are assumed to change continuously along the direction of plate thickness in a power exponential form.In order to overcome the defect that conventional elements cannot adapt to the continuum of complex geometric shape,the 8-node quadrilateral isoparametric element(Q8)is used to discrete skew plate domain.Based on Reissner-Mindlin first-order shear deformation theory(FSDT),utilizing the principle of virtual work,the finite element equation of the free vibration of the functionally graded skew plate is derived,and the accuracy of the model is verified by the vibration response characteristics of the four-sided simply supported plate.In the two cases of gradient index k=0 and k=+∞,the results of this paper are compared with the results of ANSYS software,and the fourth order natural frequency error is the largest,which is 1.39%~2.53%,which proves that the algorithm is effective.Then the effects of gradient index and skew angle on natural frequency are investigated.The results show that the vibration frequencies of FGM plates are between the two kinds of pure materials,and the natural frequencies decrease with the increase of the gradient index.With the increase of the skew angle,the relative thickness of the plate increases which lead to the increase of natural frequency of each order.