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基于 Green 函数法的微纳 Timoshenko曲梁强迫振动分析

FORCED VIBRATION ANALYSIS OF MICRO-NANO TIMOSHENKO CURVED BEAMS BASED ON THE GREEN'S FUNCTION METHOD

  • 摘要: 针对微纳尺度下经典连续介质力学难以精准刻画曲梁动力学行为、现有模型无法充分耦合曲率效应、小尺度效应与剪切变形的问题, 为揭示微纳 Timoshenko 曲梁 (MTCB) 强迫振动特性、为 Micro-Electro-Mechanical (MEMS) 谐振器件设计与性能评估提供理论依据, 本文开展系统解析研究与实验验证. 基于非局部弹性理论与 Timoshenko 曲梁 (TCB) 截面力学平衡关系, 引入特征尺度参数修正材料本构, 计入剪切变形与截面转动惯量, 推导建立 MTCB 强迫振动控制方程; 采用分离变量法与 Laplace 变换, 构建适用于简支、固支等多类边界约束的 Green 函数解析解; 通过退化算例验证理论一致性, 匹配单晶硅 MEMS 谐振器几何与材料参数, 经跨阻放大读出实验完成理论解与实测结果的对比验证. 结果表明: 本文解析解与已有文献结果误差合理, 固有频率理论值与实测值相对误差稳定在 0.46% 左右, 幅频响应曲线吻合良好; 小尺度参数增大削弱结构等效刚度, 其对固有频率的影响随长细比增大而增强; 边界约束、激励频率与小尺度效应存在显著耦合规律. 研究可为微纳曲梁振动分析与 MEMS 核心器件开发提供理论支撑.

     

    Abstract: To address the limitations that classical continuum mechanics fails to accurately characterize the dynamic behavior of curved beams at the micro-nano scale and existing models cannot fully couple the curvature effect, small-scale effect and shear deformation, this paper presents systematic analytical research and experimental verification, aiming to reveal the forced vibration characteristics of micro-nano Timoshenko curved beams and provide a theoretical basis for the design and performance evaluation of Micro-Electro-Mechanical (MEMS) resonant devices.Based on the nonlocal elasticity theory and the cross-sectional mechanical equilibrium of Timoshenko curved beams, a characteristic scale parameter is introduced to revise the material constitutive relation. With the consideration of shear deformation and rotary inertia of cross-sections, the forced vibration governing equation for micro-nano Timoshenko curved beams is derived and established. By adopting the separation of variables method and Laplace transform, the analytical solution of Green's function applicable to multiple boundary constraints (e.g., simply supported and fixed ends) is constructed. The theoretical consistency is verified through degradation cases. By matching the geometric and material parameters of a monocrystalline silicon MEMS resonator, comparative validation between theoretical solutions and experimental measurements is conducted via transimpedance amplification readout experiments.The results indicate that the error between the proposed analytical solution and existing literature results is within a reasonable range. The relative error between the theoretical and measured natural frequencies is stable at around 0.46%, and the amplitude-frequency response curves are in good agreement. An increase in the small-scale parameter weakens the equivalent structural stiffness, and its influence on natural frequency becomes more pronounced as the slenderness ratio rises. There exists a significant coupling law among boundary constraints, excitation frequency and the small-scale effect. This study can provide theoretical support for the vibration analysis of micro-nano curved beams and the development of core MEMS devices.

     

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