Geometrical Nonlinearity of Circular Plates and Membranes: an Alternative Method

TL;DR

This paper introduces an analytical method to evaluate geometrical nonlinearities in circular micro/nano-mechanical structures, extending existing beam theories to non-axisymmetric modes without complex numerical computations.

Contribution

It develops a simple, analytical approach for nonlinear modeling of circular plates and membranes, applicable to non-axisymmetric modes and independent of drive scheme assumptions.

Findings

Analytic expression for Duffing nonlinear coefficient derived

Method validated against experimental and theoretical results

Framework applicable to mode-coupling in nonlinear dynamics

Abstract

We apply the well-established theoretical method developed for geometrical nonlinearities of micro/nano-mechanical clamped beams to circular drums. The calculation is performed under the same hypotheses, the extra difficulty being to analytically describe the (coordinate-dependent) additional stress generated in the structure by the motion. Specifically, the model applies to non-axisymmetric mode shapes. An analytic expression is produced for the Duffing (hardening) nonlinear coefficient, which requires only the knowledge of the mode shape functions to be evaluated. This formulation is simple to handle, and does not rely on complex numerical methods. Moreover, no hypotheses are made on the drive scheme and the nature of the in-plane stress: it is not required to be of electrostatic origin. We confront our predictions with both typical experimental devices and relevant theoretical results from the literature. Generalization of the presented method to Duffing-type mode-coupling should be a straightforward extension of this work. We believe that the presented modeling will contribute to the development of nonlinear physics implemented in 2D micro/nano-mechanical structures.