Design of nonlinear springs for attaining a linear response in gap-closing electrostatic actuators

Gap-closing electrostatic actuators are inherently nonlinear and their dynamic range is often limited by the pull-in instability. To overcome this, we propose a nonlinear spring that counteracts the nonlinear effects of electrostatic attraction. The nonlinear spring is designed to extend the stable range of the actuator and to enforce a linear electromechanical response. We present a method for designing elastic springs with monotonically increasing stiffness. The mechanism we propose is effective shortening of a straight clamed-guided beam flexure, by wrapping it over a cam. We consider two specific cases. The first case assumes the wrapped section of the beam flexure fully conforms to the cam shape. The second case assumes that there is a single contact point between the beam flexure and the cam. To validate the concept we have designed and measured the response of a nonlinear spring with a prescribed force-displacement law. Experimental measurements of a macro-scale spring are in good agreement with the model predictions.