应变梯度高阶剪切变形微板的自由振动特性研究

Research on vibration characteristics of strain gradient higher-order shear deformable microplates

摘要

基于修正的应变梯度理论和精化高阶剪切变形理论，建立了包含 3 个材料尺度参数和 2 个位移场变量的矩形微板自由振动模型，推导了对应的控制微分方程。 利用 Navier 法获得了四边简支矩形微板自由振动的解析解。 融合 Gauss-Lobatto 求积准则和微分求积准则，构造了一种 4 节点 72 自由度的微分求积有限元，以求解微板在一般边界条件下的自由振动。 通过典型数值算例，验证了本研究模型的有效性，探讨了边界条件、材料尺度参数、长宽比、长厚比等对微板振动频率及模态振型的影响。 结果表明，矩形微板各阶振动频率和部分模态振型呈现出尺度效应，且其强弱受到边界条件和几何尺寸的影响。

Abstract

This paper develops a free vibration model of rectangular microplates including three material length scale parameters and two displacement field variables using the modified strain gradient theory and a refined higher-order shear deformation theory，and presented the related governing differential equations. The analytical vibration frequencies of a four-edge supported rectangular microplate were obtained via the Navier method.Combining the Gauss-Lobatto quadrature and differential quadrature rules，a four-node seventy-two-DOF differential quadrature finite element was constructed to solve the free vibration of rectangular microplates with general boundary conditions.Through typical numerical examples，the effectiveness of the present model was established，and the effects of boundary conditions，material length scale parameters，aspect ratio and length-thickness ratio on the vibration frequencies and mode shapes of rectangular microplates were revealed.The results indicate that the vibration frequencies and some mode shapes of rectangular microplates exhibit significant size effect，and its intensity is associated with the boundary conditions and geometric dimensions.

关键词

修正的应变梯度理论 / 精化高阶剪切变形理论 / 矩形微板 / 自由振动 / 微分求积有限元

Key words

modified strain gradient theory / refined higher-order shear deformation theory / rectangular microplate / free vibration / differential quadrature finite element

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基金资助

国家自然科学青年基金资助项目(11602204)

四川省自然科学基金资助项目(23NSFSC0849)