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Subtitle:
Article type: Research Article
Authors: Changizi, M. Amin | Stiharu, Ion*
Affiliations: Department of Mechanical and Industrial Engineering, CONCAVE Research Center, Concordia University, Montreal, PQ, Canada
Correspondence: [*] Corresponding author: Ion Stiharu, Department of Mechanical and Industrial Engineering, CONCAVE Research Center, Concordia University, Montreal, PQ, Canada. E-mail:istih@alcor.concordia.ca
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
DOI: 10.3233/JCM-150546
Journal: Journal of Computational Methods in Sciences and Engineering, vol. 15, no. 3, pp. 327-338, 2015
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