Stochastic analysis of pull-in instability of geometrically nonlinear size-dependent FGM micro beams with random material properties

Analyzing effects of uncertainty in material properties and geometric parameters in the manufacturing of MEMS is crucial for designing accurate tools with suitable performance. Pull-in voltage as a nonlinear characteristic of MEMS is affected significantly by these uncertainties. This paper investigates the randomness of the pull-in voltage of functionally graded material (FGM) micro-beams due to randomness in constituent material properties by presenting a reliable model for capturing uncertainties and effective procedures for solving stochastic nonlinear equations. The clamped-clamped FGM micro-beams, whose material properties vary smoothly based on the power law distribution in thickness, are under the influence of electrostatic and intermolecular forces. The governing equations are derived based on Euler-Bernoulli beam theory and modified couple stress theory considering nonlinear von Karman strain. The differential quadrature method (DQM) is used to solve the set of nonlinear ordinary differential equations. Monte Carlo simulation (MCS) is used to obtain the second order statistics (mean value and standard deviation) of the pull-in voltage. First order perturbation technique (FOPT) is also presented to investigate the randomness of the pull-in voltage, and the results are compared with those of independent MCS. It is shown that FOPT is fast and can predict mean value as accurate as MCS. The effects of volume fraction, the material length scale parameter, geometric parameters and the Casimir on the mean value and relative standard deviation (RSD) of the pull-in voltage are studied. Also, the effect of individual constituent materials and volume fraction index are investigated while the relative standard deviation changes.