Path following and numerical continuation methods for non-linear MEMS and NEMS

TL;DR

This paper discusses how path following and numerical continuation methods can be used to analyze and simulate the equilibrium and stability of non-linear MEMS and NEMS devices, which is crucial for their design.

Contribution

It demonstrates the application of path following techniques to model and analyze non-linear MEMS/NEMS systems, including derivation of equations and practical simulation examples.

Findings

Path following methods effectively identify equilibrium solutions.

Simulations in Mathematica, Matcont, and Comsol validate the approach.

The methods assist in stability analysis of complex electromechanical systems.

Abstract

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.