基于Markov跳变的变质量微悬臂梁系统的随机动力学研究

Microcantilever beam is one of the most typical structures of micro-electro-mechanical devices. The random adsorption and desorption of the surrounding particles could lead to the variable mass of microcantilever beam. This will change the dynamic behavior of microcantilever beam and damage the system performance. Therefore, the accurate modeling of variable mass, and the study of the influences of variable mass on microcantilever dynamics have important theoretical meaning and urgent demands. At present, there are few studies on variable mass. The existing Gauss white noise model is imperfect due to the lack of the discreteness of variable mass. In this project, a variable mass stochastic process with Gauss white noise is constructed by using Markov jump to describe the discreteness. Firstly, a microcantilever system model with Markov-jump-based variable mass is proposed. Then, stochastic response and bifurcation of the microcantilever system with Markov-jump-based variable mass are studied, which can reveal the influences of parameters of variable mass on microcantilever beam. Finally, stochastic stability of the microcantilever system with Markov-jump-based variable mass is analyzed, which can reveal the failure mechanism of microcantilever beam and improve the reliability of micro-electro-mechanical devices. This project aims to develop the stochastic averaging method of Markov-jump-based variable mass systems under Gaussian white noise excitation. Besides, the project plans to study the stochastic response, bifurcation, stability of microcantilever systems with Markov-jump-based variable mass, such that it can bring out the mechanism of variable mass on microcantilever beam, and provide theoretical basis for the design and control of micro-electro-mechanical devices.

微悬臂梁是微机电器件的典型结构，其在随机吸附与解吸附作用下会产生随机变质量问题。这会改变微悬臂梁的动力学行为，破坏微器件的性能。因此，实现变质量的准确建模，并研究变质量对微悬臂梁动力学行为的影响机制具有重要的理论意义和迫切的现实需求。目前，变质量的相关研究较少，现有的高斯白噪声模型尚不完善，无法准确刻画变质量的离散性。本项目首先采用Markov跳变描述变质量离散性，结合高斯白噪声构建变质量随机过程，进一步建立基于Markov跳变的变质量微悬臂梁模型。其次，研究建模系统的随机响应和分岔，揭示变质量参数对微悬臂梁的影响机制。最后，研究建模系统的随机稳定性，探究变质量微悬臂梁的失效机理，给出微器件可靠性的提升方法。本项目拟发展高斯白噪声激励下Markov跳变质量系统的随机平均法，借助建模系统随机响应、分岔、稳定性等动力学研究，明确变质量对微悬臂梁的作用机制，为微器件的设计和控制提供理论依据。

微悬臂梁是微机电系统机电耦合的关键元件，其质量不断减小，尺寸仅在微米甚至纳米量级。周围微粒的随机吸附与解吸附作用会显著增加或者减少微悬臂梁的质量，产生变质量现象。这种现象会改变微悬臂梁的随机动力学特性，造成器件性能的破坏甚至失效。本项目通过Markov跳变过程描述了微悬臂梁系统中质量随机跳变的离散性，建立了更加精确的变质量微悬臂梁模型，研究了变质量微悬臂梁系统的随机响应、分岔、稳定性等随机动力学行为，探讨了变质量现象对微悬臂梁系统的影响机制，丰富了微悬臂梁系统设计和控制的理论依据。具体开展了如下研究工作：. 1、借助Markov跳变过程刻画了质量随机跳变的离散性，结合高斯白噪声，构造了变质量的数学模型，并将该模型引入到经典微悬臂梁的一维集总模型中，实现了基于Markov跳变的变质量微悬臂梁系统的建模。此外，采用SINDy-LM算法发现并描述了系统的非线性特性，进一步改进了完善微悬臂梁系统的动力学建模方法，实现了精确的动力学建模和预测。. 2、推导了所建模型系统稳态概率密度函数的近似解析解，分析了系统的稳态随机响应，研究了变质量对微悬臂梁吸合效应的影响，探究了变质量参数中Markov跳变转移率和高斯白噪声强度诱导的随机分岔现象，揭示了变质量现象对微悬臂梁系统响应和分岔的作用机制。. 3、推导了基于Markov跳变的变质量微悬臂梁系统概率为1渐近稳定的充要条件，结合数值模拟分析了变质量参数对微悬臂梁系统随机稳定性的影响，确定了系统的稳定性边界。此外，根据系统的随机稳定性结论分析了变质量微悬臂梁的失效机理，研究了微悬臂梁可靠性的提升方法，设计了可用于微悬臂梁系统控制的分数阶控制方法，并验证了该控制方法的鲁棒性。. 本项目促进了随机动力学和微机电系统的结合，提升了对微机电系统中变质量现象的认识，能够为微器件的设计和控制提供理论指导。