The effect of finite electrical conductivity of small-scale beam resonators on their vibrational response under electrostatic fields

Abstract

Electrostatic actuation is one of the most commonly used methods for excitation and measurement in micro and nanoscale resonators. In the dynamical behavior analyses of such systems, the resonating beam is often assumed to be a perfect conductor. In this paper, the effect of electrical resistivity on the vibrational response of these systems, including the natural frequency and damping, is investigated. The governing coupled nonlinear partial differential equations of motion are derived and a new finite element method formulation is presented by developing a new electromechanical element. The numerical natural frequencies are compared with experimental measurements and the achieved correlation is better than that in the prior studies. Results indicate that there is a jump in the frequency and damping of the system at a critical resistivity. As the system size decreases and the applied voltage approaches the pull-in voltage, the electrical resistivity completely dominates the response nature of the system. An experiment is also conducted, and good agreement with the theory is observed regarding the effect of electrical resistivity.