磁场中载流微型梁的力-电-磁耦合非线性动力学

Current-carrying microbeams subjected to magnetic fields have become one of the most important components in MEMS. The corresponding dynamical problem is fast becoming an attractive issue involving mechanical-electronic-magnetic coupling. As for current-carrying microbeams in external magnetic fields, there is a lack of unified conclusion regarding the dynamical behavior and its size effect, and hence quantitative theoretical models need to be developed. This project will deal with the following work: 1) According to the bending test of microbeams, the size effect of the microbeam’s flexural deflections will be examined. 2) Based on the experimental results obtained, by applying and developing beam theories with the aid of higher-order continuum mechanics, the nonlinear dynamical model for current-carrying microbeams subjected to magnetic fields will be established. 3) By solving eigenvalue problems, the stability and linear vibration characteristics of the microbeam will be analyzed. 4) Utilizing modern nonlinear dynamic theories, the buckled configuration, free vibration, forced vibration, internal resonance and other complex dynamics of the microbeam subjected to constant Lorentz force will be studied; the parametric resonance of the microbeam with time-dependent Lorentz force will also be investigated in detail. Emphasis will be placed on the possible new properties of such a micro-system different from those of macro-scale beams. The research results of this project are useful for better understanding the mechanical-electronic-magnetic coupled nonlinear dynamics of microbeams as well as its size effect. This work can expand the application field of the theory of continuum mechanics. The knowledge gained from this work is also expected for providing theoretical basis and technical reserve for analyzing the multi-field coupling problems and design in MEMS-based structures.

磁场中载有电流的微型梁结构在微机电系统中具有广泛而重要的应用，其动力学问题是涉及力、电、磁等多学科交叉的前沿课题。关于磁场中载流微型梁的动力学机理及其微尺度效应，至今尚无统一的结论和定量的非线性动力学理论模型。本项目拟开展以下工作：1）利用弯曲实验，测试微型梁弯曲变形的尺度效应；2）在前述实验基础上，应用和发展基于高阶连续介质力学的梁理论，建立磁场中载流微型梁振动的非线性理论模型；3）求解线性系统的特征值问题，分析微型梁的稳定性和振动特性；4）应用现代非线性动力学理论，研究非线性系统在恒定电磁力下的屈曲位形、自由振动、受迫振动、内共振和复杂动力响应，探讨非恒定电磁力下的参数共振规律，重点揭示有异于宏观尺度梁的新特性、新机理。研究结果将深化人们对微型梁力-电-磁耦合非线性动力学及其微尺度效应的理解，拓宽连续介质力学理论的应用范围，为工程MEMS结构的多场耦合分析与设计提供理论基础和技术储备。

微纳尺度梁结构在微机电系统中有广泛的应用，其力-电-磁耦合动力学及其调控已成为相关领域一个重要的基础性科学问题，是一个多学科交叉的前沿课题，已引起学术界和工业界的密切关注。然而，关于微纳尺度梁的多场耦合动力学机理及其尺度效应，至今尚缺少有普适意义的非线性动力学理论模型。本项目在前人已有工作的基础上，尝试从实验测试、理论建模、数值计算和机理分析等多方位探究微纳尺度梁的动力学机理及其调节方法，主要研究内容包括三部分。（1）对微尺度梁的振动特性进行了实验测试。这部分的研究目的是通过实验测试确定两种典型材料的内秉尺度参数值，使之与修正偶应力理论有定量的匹配，从而为后续理论预测微尺度梁的静动力学行为提供实验支持。（2）在非经典连续介质力学框架下，基于表面弹性理论和非局部应变梯度理论发展了微纳尺度梁动力学分析的尺度相关理论模型，推导了微纳尺度梁的非线性控制方程（组），探讨了微纳尺度梁的自由振动、受迫振动和参数振动等基本动力学问题。（3）重点研究了微纳尺度梁（和管）在电场/磁场/内流场作用下的稳定性、动力学特性及其调节。借助前述的尺度相关的非经典连续介质理论模型，导出了电场、磁场或（和）内流场作用下微纳尺度梁的几类非线性动力学方程（组），预测了电场、磁场或（和）内流场对微纳尺度梁稳定性和失稳后静力学和动力学特性的影响机制。通过本项目的研究，构建了微纳尺度梁的力–电–磁耦合动力学数学模型，明确了微纳尺度梁的力-电-磁耦合动力学机理及其小尺度效应，揭示了屈曲、共振、分岔和混沌等动力学行为；研究成果增强了人们对微纳尺度梁力–电–磁耦合非线性动力学、稳定性机理和小尺度效应的理解与认识。