The strain gradient elasticity theory is adopted to capture the small-scale effect of micro/nanostructures to derive the governing equation by using Hamilton's principle. A mesh-free method on the basis of moving Kriging interpolation, of which shape function possesses C² continuum, is developed to investigate the vibration of the strain gradient plate. Owing to the inherent Kronecker delta function property of the moving Kriging interpolation, essential boundary conditions can be directly implemented. One superiority of the moving Kriging interpolation over the traditional moving least square approximation is that it can ensure the continuity and stability at the third order derivatives, which is confirmed by comparison with analytic solutions. Parametric studies in terms of small-scale parameters, boundary conditions and side length on the frequencies and mode shapes are carried out. Natural frequencies evaluated by strain gradient elasticity theory are found to be lower than those of classical plate model. Some lower natural frequencies corresponding to ultrahigh-order mode shapes appear. This demonstrates that the mesh-free moving Kriging interpolation method has high precision and good stability in the construction of the C² continuum shape functions.

采用应变梯度弹性理论捕捉微/纳米结构的小尺度效应，利用哈密顿原理推导出控制方程。一种基于移动克里金插值的无网格方法，其形状函数具有C ²连续体，用于研究应变梯度板的振动。由于移动克里金插值的固有 Kronecker delta 函数特性，可以直接实现基本边界条件。移动克里金插值相对于传统移动最小二乘近似的优势之一是它可以保证三阶导数的连续性和稳定性，这一点通过与解析解的比较得到了证实。对频率和模态振型的小尺度参数、边界条件和边长进行参数研究。发现由应变梯度弹性理论评估的固有频率低于经典板模型的固有频率。出现了一些对应于超高阶振型的较低固有频率。C ²连续体形状函数。