Electrostatic deflections of circular elastic membranes

Electrostatically deflected elastic systems are studied in areas as diverse as optical switching and cloud electrification. All such systems exhibit an instability commonly known as the “pull-in” or “snap-down” instability. In this instability, when applied voltages are increased beyond a certain critical voltage, there is no steady-state configuration of the system where mechanical members remain separate. The widespread use of electrostatic actuation in the design of micro- and nano-electromechanical systems (MEMS and NEMS) has rekindled an interest in this instability. Here, a mathematical model of a disk-shaped electrostatically actuated system is studied. This model represents a configuration typical of many MEMS/NEMS systems. Solutions to the model are examined for the case of a prescribed voltage drop across the system and for the case where the system is embedded in a capacitive control circuit. The curve of solutions, or bifurcation diagram, is shown to bend back upon itself repeatedly implying the existence of multiple steady-state configurations of the system. The effect of the control scheme is studied by examining the effect on the bifurcation diagram.