Uncertainty quantification in a resonant nonlinear MEMS structure

Abstract

This work is focused on Uncertainty Quantification (UQ) analysis on a nonlinear MEMS T-beam structure exhibiting a 1:2 autoparametric internal resonance. The study is presented by elaborating on the sources of uncertainty in fabrication and operation of the device and their quantification techniques. Nonlinear response of the system is formulated by using a two-mode model constructed with the lowest two linear modes of the structure in conjunction with nonlinear Lagrangian representing the dynamics of the beam structure as well as the excitation mechanism. The focus of system’s nonlinear dynamics is on the case when the two lowest modes are in 1:2 internal resonance. Thus, UQ is carried out in two steps. First, propagation and quantification of uncertainty in linear analysis outputs namely mode shapes, natural frequencies and tuning ratio is performed, and then a UQ analysis on the nonlinear response obtained from averaged equations determined by the averaged Lagrangian is carried out. Sensitivity analysis is performed of the nonlinear system to reduce the number of parameters by filtering out the non-critical parameters. Response surface analysis is performed using the generalized polynomial chaos (gPC) to generate an equivalent mathematical model. A comparison of response surface method with direct sampling is done to illustrate the efficiency and accuracy of gPC collocation technique for up to 5 uncertain parameters. The criticality of tuning, between the first two linear modes of vibration, for device operation is shown. Application of UQ techniques on the nonlinear 1:2 resonant system is presented. The performance of the device is characterized by steady-state amplitude of the lower mode. The influence of aleatoric uncertainties in system parameters is studied and quantification is carried out for a given range of parametric uncertainties.