  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.
