褚福运， 王莉华， 仲政^*， 何建彰

（  同济大学航空航天与力学学院；  ）

摘要： 基于基尔霍夫假设的经典薄板理论其控制方程是一个关于挠度四阶的偏微分方程，边界条件一般同时包含挠度及其导数项，所以对于基尔霍夫板其控制方程的数量和边界条件的数量是不一致的。如果直接采用径向基函数配点来求解此类问题，其系数矩阵不是方阵，会给特征值问题的求解带来很多不便，甚至无法求解。为求解此类特征值问题，将边界结点的法向导数作为一个基本未知量引入近似函数中，构造了一种艾米特径向基函数插值。它不仅可以直接应用于求解薄板特征值问题，而且有助于提高计算精度。变刚度薄板的控制方程是一个四阶变系数的偏微分方程，解析求解的难度非常大。利用径向基函数无穷可微的特点可以非常好的扑捉到变刚度板刚度梯度变化的性质。所以利用艾米特径向基函数配点法求解了刚度薄板的自由振动问题。数值算例表明该算法具有很高的近似精度高，同时变刚度的性质也会对板的自振频率和阵型等力学行为有显著影响。。
关键词： 艾米特径向基函数；功能梯度材料；薄板；面内变刚度；自由振动

CHU Fuyun， WANG Lihua， ZHONG Zheng^*， HE Jianzhang

（  School of Aerospace Engineering and Applied Mechanics, Tongji University；  ）

Abstract： The classical thin plate theory based on the Kirchhoff hypothesis leads to inconsistency in the number of boundary conditions and governing equations. If the radial basis function collocation method is directly used to solve this problem, the coefficient matrix is not square, which will bring about a lot of challenging problems for free vibration analysis of thin plates. Therefore, a Hermite radial basis function (HRBF) meshless method is formulated. The degree of freedom for the normal derivative is included in the approximation to make the proposed method effective in eliminating the hurdle in free vibration problems of thin plates. It is also helpful to improve the computational accuracy. As we known, the governing equation of FMGs plate with in-plane material inhomogeneity is variable coefficient, and it is very difficult to obtain the analytical solution for FGM plates. RBF can capture material inhomogeneity effectively, so the proposed method was used to solve the free vibration problem of thin plates with in-plane material inhomogeneity, in which the flexural rigidity is a function of space coordinates. A number of numerical examples have been considered and the results reveal that the method is accurate and material inhomogeneity has significant effects on the natural frequency and vibration mode shapes.(10 Points, Times New Roman)

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