Strong nonlinear dynamics of MEMS and NEMS structures based on semi-analytical approaches

The aim of this work is to derive a reduced order model (ROM) for an electrostatically actuated micro and nano structures which can be valid for relatively large displacements and high voltages. The study also presents a comparison between available semi-analytical ROMs based on Galerkin decomposition and Differential Quadrature Method (DQM) applied on micro and nano electrically actuated beams. The proposed ROM is an innovative approach to properly and accurately consider the electrostatic forces and account for the geometric nonlinearities in the derived equations of motion. The proposed ROM is derived by employing a mathematical model using the nonlinear Euler-Bernoulli theory, the DQM is employed to extract the mode shapes of the micro/nano beams. Discretized Galerkin approach in association with DQM is then utilized as a ROM and the Finite Difference Method (FDM) is employed to calculate limit-cycle solutions. The proposed ROM is used to investigate the effect of the strain's third order nonlinearities on the dynamic responses of micro/nano beams. A comparative study in terms of convergence and computational time is carried out on several ROMs based on modal decomposition for different mode shapes, and by performing or not the well known multiplication by the force denominator term in the equation of motion. As a benchmark, Local adaptive DQM (LaDQM) in combination with Long Time Integration are employed and their convergence is studied. Important outcomes of this study reveal the appropriateness of using only one mode in the discretized Galerkin approach for doubly clamped and cantilever at micro and nano scales, even when high voltages are applied. Besides, the results demonstrate that using high AC voltages for the actuation signal can lead to a non-negligible shifting effect regarding the equilibrium positions and the associated natural frequencies of a dynamic solution. An updated solution is proposed to predict this mismatch. (C) 2018 Elsevier B.V. All rights reserved.