In this paper, we apply the energy variational principle to arrive at the differential equation of motion and all appropriate boundary conditions for strain gradient Kirchhoff micro-plates. The resulting sixth-order boundary value problem of free vibration is solved by a thirty-six-DOF four-node differential quadrature plate finite element. The C²-continuity condition of the deflection is guaranteed by devising a sixth-order differential quadrature- based geometric mapping scheme that can transform the displacement parameters at Gauss-Lobatto quadrature points into those at element nodes. The total potential energy of a generic micro-plate element is firstly discretized in terms of nodal parameters. It is then minimized to obtain the formulation of element stiffness and mass matrices. For comparison reasons, a Hermite interpolation-based strain gradient finite element is provided. With the help of the symbolic computation system Maple, the explicit algebraic relationship between the stiffness (or mass) matrices of two types of elements is derived. Convergence and comparison studies are conducted to show the efficacy of our element in the free vibration analysis of macro/micro- plates. Finally, we apply the developed method to study the size-dependent vibration behavior of micro-plates with uniform or stepped thickness. Numerical examples reveal that strain gradient effects can change the vibration mode shapes, not the vibration frequencies alone.

中文翻译：

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自由振动应变梯度基尔霍夫板的变分公式和微分正交有限元

在本文中，我们应用能量变分原理来得出应变梯度基尔霍夫微板的运动微分方程和所有适当的边界条件。由此产生的自由振动的六阶边值问题由一个 36 自由度的四节点微分正交板有限元解决。的Ç ²-通过设计基于六阶微分正交的几何映射方案来保证偏转的连续性条件，该方案可以将高斯-洛巴托正交点处的位移参数转换为单元节点处的位移参数。通用微板单元的总势能首先根据节点参数进行离散化。然后将其最小化以获得单元刚度和质量矩阵的公式。出于比较原因，提供了基于 Hermite 插值的应变梯度有限元。借助符号计算系统 Maple，推导出两类单元的刚度（或质量）矩阵之间的显式代数关系。进行收敛和比较研究以显示我们的元件在宏观/微板自由振动分析中的功效。最后，我们应用开发的方法来研究具有均匀或阶梯厚度的微板的尺寸相关振动行为。数值例子表明应变梯度效应可以改变振动模式形状，而不仅仅是振动频率。