In this investigations, a nonlinear model for electrically actuated micro-bridge gyroscope with proof- mass is developed to track the dynamic pull-in instability of the system. The model accounts for the geometric nonlinearities, base rotation excitations, harmonic base excitations and instantaneous application of DC voltage. To avoid resonance behavior due to harmonic base excitations, viscous damping is also taken into consideration. Hamilton’s principle is employed to obtain the partial differential equations of motion. Afterward, A Galerkin based reduce order model is utilized to convert the partial equations of motion to ordinary differential equations of motion. A single-mode approximation is made to predict the behavior of micro-bridge gyroscopes. To this end, half of the micro-beam is modeled due to the symmetry. A mode shape for half of the beam is developed afterwards with the corresponding frequency equation. Then the fourth-order Runge-Kutta method is used to achieve the solution of the nonlinear governing equations of motion. The results are then validated through comparison with previous studies. Time response graphs for drive and sense motion are then presented for cases of stable and unstable behavior. The dynamic pull-in voltage of the system is calculated from the critical value of voltage for which the unstable behavior occurs. The results show that application of harmonic excitation will excite the system on two different frequencies. These frequencies are linear combinations of base rotation frequency and harmonic excitation frequency. The plots also show that if these frequencies are far from the natural frequency, the system will face quasi-static loading. Otherwise, loading will be exerted to system as dynamic loadings. The alteration of base rotation frequency has direct impact on the natural frequency so, the effect of base excitation frequency is investigated next. The results clarifies that increasing the base excitation frequency will end in reduction of pull-in voltage. Then the phase plot of the drive motion is presented and the pull-in voltage is obtained from phase plot. The effect of damping on the dynamic pull-in behavior of the system is also investigated.

Keywords