局部支承功能梯度板的自由振动分析

Free Vibration Analysis on Locally Supported Functionally Graded Plates

基于Mindlin板理论和有限元法研究了局部支承功能梯度板的自由振动问题。假设材料由陶瓷和金属组成,且材料参数沿板厚度方向按照幂指数形式连续变化,利用虚功原理推导功能梯度板横向自由振动的有限元方程,分别在100%、50%和25%等3种支承条件下研究系统自由振动特性。算例部分首先求解了纯金属板和纯陶瓷板的前三阶固有频率,同时给出了ANSYS软件计算得到的前三阶频率及其模态,两者对比证实了本文方法的准确性;随后重点探讨了支承范围和梯度指数与前三阶固有频率的关系。结果显示:支承范围对模态有重要影响,可通过调整支承来控制振动;支承范围与固有频率正相关,梯度指数与固有频率负相关;为实现同等精度,支撑范围越小需要的单元数越多。

Based on Mindlin plate theory and finite element method, the free vibration of functionally graded plates with local supports was studied. It is assumed that the material is composed of ceramic and metal, and the material parameters change continuously along the direction of plate thickness in a power exponential form. The finite element equation of transverse free vibration of functionally graded thin plate was derived based on the principle of virtual work, and the free vibration characteristics of the system were studied under 100%, 50% and 25% support conditions respectively. In the example part, the first three order natural frequencies of pure metal plate and pure ceramic plate were solved, and the first three order frequencies and their modes were calculated by ANSYS software. The comparison between the two verifies the accuracy of the method in this paper. Then the relationship between support range and the gradient index and the first three natural frequencies was discussed. The results showed that the supporting range has an important effect on the mode, and the mode can be controlled by adjusting the support. The range of support is positively correlated with the natural frequency, while the gradient index is negatively correlated with the natural frequency. In order to achieve the same accuracy, the smaller the support range, the more elements are needed.