基于非线性理论的静电驱动微梁系统静动态设计与参数优化

The decreasing of characteristic scale in Micro-electro-mechanical systems (MEMS) can result in many complex static and dynamic problems. In this research, a coupled system, composed of an electrostatically actuated microbeam (the key sensing/resonant component in MEMS) and a detection/feedback circuit, is investigated in detail. By the application of microscale theory and surface elasticity theory, a coupled nonlinear dynamic model is established which concerns material property, structural property, microscale excitation and electromechanical coupling. A low dimension model, which can systematically describe the static/dynamic pull-in behaviors and properly reflect physical meanings of the system, is then extracted via the Finite Difference Method. Qualitative analysis is carried out to deduce the existence conditions for some typical nonlinear static/dynamic behaviors and grasp their mechanisms and change rules which exist in microbeam, detecting electrical signal and coupled system. Guided by qualitative results, quantitative analysis is carried out to investigate static/dynamic properties in original dynamic model and guide the modification of low dimension model. Inner relationships between static/dynamic behaviors and physical parameters are investigated in depth. Static/dynamic behaviors of the system are designed and then optimized by multi-objective optimization. COMSOL Multiphysics and LAMMPS are used to verify the effectiveness of optimal results. Finally, a scientific analysis and design method about “Qualitative analysis→ Quantitative analysis→ Design→ Optimization” of electrostatically actuated microbeam-based systems is proposed, which can provide important theoretical and technical supports in their static and dynamic design.

针对微机电系统（MEMS）随特征尺度减小而产生的复杂静动力学问题，以MEMS关键敏感/谐振元件静电驱动微梁及其检测/反馈电路所组成的耦合系统作为研究对象，结合尺度理论和表面弹性理论，建立可反应系统材料、结构、微载荷及机电耦合特性的非线性动力学模型；结合微分求积法获得既能描述吸合特性又能体现物理意义的低维模型，推导微梁结构、检测电信号及机电耦合系统中典型非线性行为的存在判定条件，定性揭示其产生机理及变化规律；以定性结果作为指导，定量挖掘原始模型的静动态特性并指导低维模型修正；深入探讨静动态特性与物理参数间的内在联系，实现系统的静动态设计，并运用优化算法进行多目标优化设计；结合数值模拟软件COMSOL Multiphysics和LAMMPS完成最优验证。最终，提出一套静电驱动微梁系统“定性→定量→设计→优化”分析设计方法，为其静动态设计及优化提供重要的理论及技术支撑。

本项目围绕静电驱动谐振结构所存在的复杂非线性力学行为展开研究，按照“定性分析、定量挖掘、静动态设计与参数优化”的研究思路，应用分岔理论、非线性振动近似解析方法和非线性数值分析方法，探索并挖掘结构典型静、动力学的全局演化规律及其产生机理，以期为微机械谐振器的优化设计与应用拓宽提供有益的理论指导。首先，应用高阶Galerkin离散模型、高阶微分求积离散模型、COMSOL有限元模型并结合已有文献的实验数据，证明了广义微分求积单自由度模型在定性研究物理参数对典型现象影响规律时的有效性。其次，研究了单极板微梁谐振器的静态吸合及主共振频率演化规律，推导得到吸合位置的理论上下极限及共振频率随偏置电压呈现单调或非单调行为的临界阈值，统一了当前相关研究所得的差异性结论。再次，应用同伦多尺度近似解析法探讨了双极板前屈曲微梁谐振器的主共振频响规律，综合多因素条件对影响主共振动频响的全参数域进行精细划分，实现了主共振全局动力学表征，揭示了高低能量频响曲线共存现象及交汇规律。此外，研究了非对称缺陷引发的静态分岔及主共振分岔规律，推导获得静态独立解分支的存在条件，并发现随偏置电压变化时主共振近似线性振动行为的高灵敏度特性（比传统无缺陷情况高出近40倍），为高灵敏度传感器的研发提供了借鉴思路。最后，通过非对称偏置电压加载操作实现了双极板微梁谐振器的多重工作模式切换，应用奇异性理论推导了静态分岔的转迁集，揭示了非对称偏置电压引发静态突跳及吸合行为的作用机理，识别了模型内部隐藏的“单向”与“双向”静动态突跳行为，并探讨了其在微机电逻辑运算领域中的应用前景。项目侧重理论推导与分析，强调厘清静电式谐振结构的静态与主共振动态行为的大范围或全局演化规律，旨在实现结构优化设计和反馈调控。上述研究成果为微机械谐振器的设计与应用提供了理论参考，也在一定程度上为新型非线性微机械谐振器的研发设计提供了思路。