Bifurcation Behavior of a Capacitive Micro-Beam Suspended between Two Conductive Plates

Abstract

In this paper, bifurcation and pull-in phenomena of a capacitive micro switch suspended between two stationary plates have been studied. The governing dynamic equation of the switch has been attained using Euler Bernoulli beam theorem. Due to the nonlinearity of the electrostatic force, the analytical solution for the derived equation is not available. So the governing differential equation has been solved using combined Galerkin weighted residual and Step-By-Step Linearization Methods (SSLM). To obtain the fixed points and study the local and global bifurcational behavior of the switch, a mass-spring model has been utilized and adjusted so that to have similar static/dynamic characteristics with those of Euler-Bernoulli beam model (in the first mode). Using 1-DOF model, mathematical and physical equilibrium points of the switch have been obtained for three different cases. It is shown that the pull-in phenomenon in the present micro-switch can be occurred due to a pitchfork or transcritical bifurcations as well as saddle node bifurcation which are transpired in the classical micro-switches. And for some cases primary and secondary pull-in phenomena are observed where the first one is due to a transcritical bifurcation and the second one is due to a saddle node bifurcation. In addition the dynamic response of the switch to a step DC voltage has also been studied and the results show that in contrast to the classical microswitches, the ratio of the dynamic pull-in to the static one depends on the gaps and voltages ratio where for the classical one is approximately a constant value.