应变梯度双层微梁的动力学建模与振动特性研究

Dynamic modeling and vibration characteristics of strain gradient double-layered microbeam

  • 摘要: 融合修正应变梯度理论与精化高阶剪切变形理论,提出了由Winkler-Pasternak弹性夹层连接的双层微梁动力学模型,并推导了其控制微分方程及边界条件。针对上下层均为两端简支的情形,利用Navier法求解了系统自由振动的解析解。为有效求解复杂边值问题,发展了一种结合Gauss-Lobatto求积准则与微分求积法的3节点弱形式求积单元,该单元具备C2连续性。在充分验证模型有效性的基础上,引入能量参与准则与模态置信准则,深入探究了材料尺度参数及弹性夹层刚度对系统振动频率与模态振型的影响规律。研究表明:剪切变形效应对高阶振型的影响显著,且随长细比增大而减弱;边界约束较弱的层通常率先主导系统振动,随后与约束较强的层呈现交替主导特性;弹性夹层在系统呈现异步振动模态时作用显著,其刚度变化可诱发模态跃迁现象;模态置信准则能够有效表征夹层刚度与材料尺度参数对系统模态局部化及跃迁行为的影响机制。本文结果有望为微机电系统中多层微梁器件的动力学行为预测提供有效的理论支撑和数据积累。

     

    Abstract: By integrating the modified strain gradient theory (MSGT) and the refined higher-order shear deformation theory (RHSDT), this study proposes a dynamic model for a double-layered microbeam connected by a Winkler-Pasternak elastic interlayer. The governing differential equations and boundary conditions were derived. For the case where both upper and lower beams are simply supported at both ends, the analytical solution for the system’s free vibration was obtained using Navier’s method. To efficiently solve complex boundary value problems, a 3-node weak-form quadrature element incorporating the Gauss-Lobatto quadrature rule and the differential quadrature rule was developed, featuring C2; continuity. Following rigorous validation of the model’s effectiveness, the energy participation criterion (EPC) and the modal assurance criterion (MAC) were introduced to investigate the influence of material scale parameters and elastic interlayer stiffness on the system’s vibration frequencies and mode shapes. It is revealed that: shear deformation effects significantly impact higher-order modes, with this influence diminishing as the slenderness ratio increases; using the EPC revealed that the layer with weaker boundary constraints typically dominates the vibration initially, followed by an alternating dominance pattern with the more constrained layer; the elastic interlayer (particularly the Pasternak layer) plays a critical role specifically during asynchronous vibration modes; changes in its stiffness can induce mode transition phenomena; the MAC quantitatively characterized the mechanisms by which interlayer stiffness and material scale parameters influence system mode localization and transition behavior. The developed model and methodology provide an effective theoretical framework and benchmark data for predicting and analyzing the dynamic behavior of multi-layer microbeam devices in micro-electro-mechanical systems (MEMS), such as sensors, actuators, and resonators, offering valuable insights for their optimized design.

     

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