Closed-form relation to predict static pull-in voltage of an electrostatically actuated clamped–clamped microbeam under the effect of Casimir force

Abstract

The current work presents an accurate closed-form model to Microelectro Mechanical System designers for computing static pull-in voltage of electrostatically actuated microbeams with clamped–clamped end condition. The model incorporates the effect of Casimir force including correction for finite conductivity. The Euler–Bernoulli beam equation is adapted considering the effects of mid-plane stretching, residual stress, fringing field and Casimir force to derive the governing differential equation for electrostatically actuated microbeams. The Galerkin method is used with a multimodes reduced-order model to solve the governing differential equation of microbeams. The results obtained using the reduced-order model are further compared with the solution of the boundary value problem and are validated with published numerical and experimental results. The results of the current work indicate that at least three modes in reduced-order model are essential for the prediction of pull-in voltage of microbeams which have a large value of mid-plane stretching parameter. In order to develop a closed-form relation, dimensionless parameters are used to plot the curves of pull-in voltage versus various parameters such as axial force due to residual stress, Casimir force, fringing field, Casimir force including finite conductivity correction, and mid-plane stretching. Based on the relationship observed in the plotted curves for the independent effect and interaction effects of these parameters on static pull-in voltage, a closed-form model is proposed for the computation of static pull-in voltage. Optimised coefficients of the proposed model are determined using nonlinear regression analysis. An adjusted R $$^2$$ value equal to 0.99909, a P value equal to zero, and $${\chi }^2$$ tolerance equal to $$1\times 10^{-9}$$ obtained by statistical analysis exhibit the precision of fitted data, significance of model, and convergence of the fit, respectively. The proposed model is validated by comparing the results of the model with results of boundary value problem solutions, results predicted from reduced-order model and other several reported numerical and experimental results. The proposed model is robust enough for calculating the static pull-in voltage under different conditions with maximum error of 3% when compared to reported experimental and numerical results.