A Complete Parametric Study of Pull-In Voltage by Nonlinear Differential Equation

Abstract

Micro-cantilever beams are interested structures in MEMS because of their fabrication is very easy and its versatility. The importance of micro-cantilevers beam in MEMS has driven various investigations like static and dynamic performances under different loading such as potential fields. In this research the non-linear differential equation which models dynamics of a micro-cantilever beams vibration subjected to electrostatic field has been studied. The model which has one degree of freedom is used to calculate the pull-in voltage. This model adopted based on different method of calculating stiffness of micro-cantilever beam. The nonlinear ordinary differential equation which used to model the dynamics of the cantilever subjected to electric field close to snap on is highly stiff. Investigation on solving of nonlinear stiff ordinary equation showed that only Lsode algorithm yield to correct solution to the problem. Lsode is equipped with a robust adaptive time step selection mechanism that enables solutions to very stiff problems, as the one under discussion. The best match in the resonant frequency for equivalent stiffness based on four different models was considered. The stiffness model suitable for the best match in deflection is proved to be different from the model that yields. Pull-in voltage under electric field was studied. Pull-in voltage has been investigated from the analytical and numerical perspective. A complete parametric study of structural damping effect on large deflection of micro-cantilever beam was studied was done numerically in this work. Different kind of impulse voltages were considered and effect of them on pulling voltage numerically was studied. A cumbersome mathematical method, Lie symmetry, was used to drive a closed from of time response to step voltage for undamped system and pull in voltage of such system was calculated. Finally, a closed form driven from the nonlinear ODE for calculating pulling voltage was presented.

Nonlinear Analysis of Pull-in Voltage of Twin Micro-Cantilever Beams with Different Dimensions

“…Figure 11 illustrates the phase portrait of the response of the system. The solution as obtained is identical with the results presented in [2] and [12]. Time [µSec] 116-7 …”

“…This represents the classical problem [1][2][3]. This case is verified assuming that the bottom beam is extremely thick (2m).…”

“…The operation of micro-cantilever structures under the influence of different types of electric fields has been investigated. Modelling of the phenomenon as well as experimental works were reported [2,3,4]. The differential equation that describes the dynamic behaviour of the microsystem consisting of a micro-cantilever beam close to a rigid substrate charged at different potentials can lead to evaluate the pull-in voltage.…”