Affiliations: Department of Mechanical and Industrial Engineering, CONCAVE Research Center, Concordia University, Montreal, PQ, Canada
Correspondence:
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Corresponding author: Ion Stiharu, Department of Mechanical and Industrial Engineering, CONCAVE Research Center, Concordia University, Montreal, PQ, Canada. E-mail:[email protected]
Abstract: This paper presents an analytical solution of nonlinear differential
equation of micro-structures subjected to electrostatic fields. The
constitutive equation of such a model is a second order differential
equation (ODE). The problem is solved when the assumption of linear
deflection is considered. However, deflection of micro cantilevers in
practical applications is non-linear. Moreover, the constitutive ODE is
stiff and various numerical algorithms used to solve it yield non-consistent
numerical solutions. A deduction order method - Lie group symmetry is
employed to reduce the order of the ODE. Although the resulting first order
ODE has no symmetry that would guarantee an explicit close form solution, it
enables an analytical formulation for the no-damping assumption only. The
restoring force term in the first order ODE reveals the pull-in voltage as
expressed in classical MEMS textbooks. It is shown that the numerical
solution for the second order ODE and the reduced first order ODE are same.
Finding any symmetry other than translation, scaling or rotation will enable
the reduction of the first order ODE and thus, the formulation of an
analytical solution to this highly non-linear problem.
Keywords: MEMS, electrostatic field in microstructures, snap-on critical voltage, nonlinear ODE, Lie group symmetry, reduction of the order of the ODE
