A highly accurate analytical method for determination of the vibrational frequency of N/MEMS with electrostatic and van der Waals interaction forces

In this study, a novel approach based on the elliptic balance method (EBM) is proposed for the first time to find the approximate frequency of nano/micro-electromechanical systems modeled as Euler-Bernoulli beams under the effects of electrostatic and van der Waals interaction forces. Firstly, the governing equation of the beam is reduced to the single-mode vibration equation using the Galerkin method. A nonlinear differential equation for the time-dependent beam deflection is obtained. We present the approximate solution as an elliptic cosine function, which considers the free term contributing to the solution. This free term is relevant for vibrations with a non-zero mean in time, in which the beam is affected by a relatively large applied voltage. Via some manipulations, the obtained result is an algebraic equation with only one unknown in three unknowns: the free and vibration coefficient terms, and the modulus quantity of the elliptic cosine function. This nonlinear equation is solved using the Newton-Raphson method. The numerical results from the EBM show that the accuracy of the solution responses in time and approximate frequency is relatively accurate, almost coinciding with the results obtained from the numerical solution method using the Runge-Kutta algorithm. Our results also agree well with previously published experimental and simulation results. The results are meaningful when determining the frequency of the vibrating beam with high accuracy for micro/nano systems.