Critical pull-in curves of MEMS actuators in presence of Casimir force

Brenda Mclellan, Luciano Medina, Chenmei Xu, Yisong Yang

Research output: Contribution to journalArticlepeer-review


We present analytic and computational studies of the dynamical behavior of an undamped electrostatic MEMS actuator with one-degree of freedom subject to a Casimir force. In such a situation, the well-known mathematical difficulty associated with an inverse quadratic term due to a Coulomb force is supplemented with an inverse quartic term due to the joint application of a Casimir force. We show that the small Coulomb and Casimir force situations, described by sufficiently low values of two positive parameters, λ and μ, respectively, are characterized by one-stagnation-point periodic motions and there exists a unique critical pull-in curve in the (λ, μ) coordinate quadrant beyond which a finite-time touch down or collapse of the actuator takes place. We demonstrate how to locate and approximate the pull-in curve. When mechanical nonlinearity such as that due to the presence of a cubic elastic force term is considered in the equation of motion, we show that a similar three-phase oscillation-pull-in-finite-time-touchdown phenomenon occurs and that pull-in curves are actually enhanced or elevated by nonlinear elasticity. Furthermore, we compute solutions of the MEMS wave equations and show that the same characteristic phenomena of subcritical periodic motions and loss of periodicity of motion and onset of a critical pull-in curve occur as one increases the levels of the Coulomb and Casimir forces as in the one-degree-of-freedom case.

Original languageEnglish (US)
Pages (from-to)1406-1422
Number of pages17
JournalZAMM Zeitschrift fur Angewandte Mathematik und Mechanik
Issue number12
StatePublished - Dec 1 2016


  • 02.60.Cb
  • 02.60.Jh
  • 34C15
  • 34C60
  • 37N15
  • 62.25.-g
  • 62.25.Jk
  • Casimir force
  • Coulomb force
  • MEMS
  • Pull-in curves
  • electrostatic actuation
  • periodic motion
  • stagnation point
  • touch down

ASJC Scopus subject areas

  • Computational Mechanics
  • Applied Mathematics


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