Approximation of potential function in the problem of forced escape

Abstract

The paper revisits a famous problem of escape of a classical particle from a potential well under a harmonic forcing. In particular, the most prominent escape dynamics appears at 1:1 resonance and can be described by means of an isolated resonant (IR) approximation. In this case, the problem has to be reformulated in terms of action–angle variables (AA), which are available for only a few of the model potentials. It is proposed in the paper that realistic generic potentials can be approximated by low-order polynomial functions, admissible to the AA transformation, with possible truncation. In order to illustrate the idea, we first formulate the AA transformation and then solve the escape problem in the IR approximation for a generic quartic potential. Then, we apply this approach to the study of pull-in instability in microelectromechanical systems (MEMS). In particular, we consider a SDOF approximate model of a capacitive micromachined ultrasonic transducer (CMUT) device. The resonant external excitation corresponds to the acoustical wave incidental on the diaphragm of the CMUT. In general, pull-in effect is observed in MEMS due the electrostatic force growing faster than the restoring mechanical force. We propose to study the pull-in (escape) in the model electrostatic potential by approximating it with quartic polynomials (globally and locally). The quality of predicting the escape threshold is assessed numerically. Most accurate predictions are delivered by global L2-optimal heuristic approximation.