非线性MEMS和NEMS的路径跟踪和数值延拓方法
Path following and numerical continuation methods for non-linear MEMS and NEMS
摘要：
非线性在微纳机电系统（MEMS和NEMS）设计中起着重要的作用。在常见的静电和磁致动器中，力和电压可以以非线性方式取决于位置、电荷、电流和磁通量。机械弹簧结构可通过材料、几何和接触效应引起额外的非线性。对于非线性MEMS器件的设计和操作，能够对这种非线性进行建模和仿真是至关重要的。然而，当有许多自由度时，很难找到非线性方程的所有平衡解并确定其稳定性。在这些情况下，路径跟踪方法可以是一个强大的数学工具。在本文中，我们将展示如何路径跟踪方法可以用来确定机电设备的平衡和稳定性。从机电系统的能量、功和哈密顿量出发（第一节），导出了机电系统的运动方程（第二节）、平衡条件（第三节）和稳定性条件（第六节）。给出了Mathematica（第5节）、Mattech（第7节）中的路径跟踪仿真（第4节）和Comsol（第8节）中的FEM方法的示例来说明方法。
英文摘要：
Non-linearities play an important role in micro- and nano- electromechanical system (MEMS and NEMS) design. In common electrostatic and magnetic actuators, the forces and voltages can depend in a non-linear way on position, charge, current and magnetic flux. Mechanical spring structures can cause additional non-linearities via material, geometrical and contact effects. For the design and operation of non-linear MEMS devices it is essential to be able to model and simulate such non-linearities. However, when there are many degrees of freedom, it becomes difficult to find all equilibrium solutions of the non-linear equations and to determine their stability. In these cases path following methods can be a powerful mathematical tool. In this paper we will show how path following methods can be used to determine the equilibria and stability of electromechanical devices. Based on the energy, work and the Hamiltonian of electromechanical systems (section 1), the equations of motion (section 2), the equilibrium (section 3) and stability conditions (section 6) are derived. Examples of path following simulations (section 4) in Mathematica (section 5), Matcont (section 7) and using FEM methods in Comsol (section 8) are given to illustrate the methods.
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