Unified nonlinear quasistatic and dynamic analysis of RF-MEMS switches

In this paper, the modified couple stress-based strain gradient theory is used to provide a unified nonlinear model of the quasistatic and dynamic behavior of an electrostatic microelectromechanical systems microbeam capacitive switch of the Euler–Bernoulli type. Our model not only accounts for the contact between the microbeam and the dielectric substrate using nonlinear springs and dampers, but also accounts for the system size by introducing an internal material length scale parameter. In view of the size of the microbeam and electrostatic gaps involved, Casimir and Van der Waals forces, damping force due to the squeeze membrane effect and electrostatic force with first-order fringing field effects were accounted for in our model. The resulting nonlinear system of PDEs was expanded into a coupled system using series expansion and integrated into ODEs using weighted residuals of the Galerkin type. To overcome the difficulties associated with the determination of the contact length, the Heaviside function for deflection was replaced with a Heaviside function for the contact length, and an iterative procedure was adopted to determine the contact length. To obtain the time variation of the microbeam, the dynamic system of equations was solved using Newmark’s integration scheme. The outcome of our work shows the dependence of the pull-in voltage upon the inertia force, slenderness ratios of the microbeam, the electrostatic gap and the initial boundary conditions of the switch. In addition, we were also able to provide the full history of the microbeam past the pull-in threshold.