Nonlinear and stochastic dynamics of MEMS-based angular rate sensing and switching systems

Instabilities in single-axis mass-spring type vibratory MEMS gyroscopes and impact dynamic behaviour (i.e. contact bounce) of cantilever type MEMS switching systems are investigated.;The aim of the investigation on the instability behaviour of this class of gyroscopes is to understand the dynamic effects due to periodic as well as stochastic fluctuations in the input angular speed. In practice, these fluctuations stem from the system itself or environment vibrations. In the MEMS gyroscope instability investigations, both nonlinear as well as stochastic formulations have been proposed. In the nonlinear study, a single-axis MEMS gyroscope that is subjected to periodic fluctuations in input angular rates is investigated considering the cubic nonlinearity in stiffness. For the purpose of characterizing the bifurcation behavior associated with the steady-state response, when the angular rate input is subject to small intensity periodic fluctuations, dynamic behavior of periodically perturbed nonlinear gyroscopic systems is studied in detail. An asymptotic approach based on the method of averaging has been employed for this purpose, and closed-form conditions for the frequency response due to parametric resonances have been obtained. This behavior has been illustrated via amplitude-frequency plots. In the stochastic study, instabilities in a vibratory MEMS gyroscope that is subject to stochastic fluctuations in input angular rates are investigated. An asymptotic approach based on the method of stochastic averaging has been employed for this purpose, and closed-form conditions for mean square stability of dynamic response are obtained for the case of exponentially correlated noise. Results are shown to depend only on those values of the excitation spectral density near twice or sum combination of the system natural frequencies. The results obtained remain valid if the stochastic parametric excitation has a small correlation time compared with the system relaxation time. Stability predictions have been illustrated via stability diagrams in the power-spectral-density---damping-ratio space.;As part of this thesis, the dynamic behaviour of a cantilever type MEMS switch has also been studied in detail. Micro-beam is known to be one of the widely used MEMS switch structures. Among these, the cantilever type can be considered as the most popular form used by the MEMS community, while its dynamic behavior is not adequately understood, especially the transient behavior. Typically, upon closing, the micro-beam bounces several times before making permanent contact with the adjoining structure. Bouncing due to switching can dramatically degrade the performance due to an increase in the transition time and the electrical breakdown. A transient contact bouncing analysis incorporating the asperity-based contact model is proposed in the present study together with the pull-in instability analysis. In addition, the dynamic effects of squeeze-film damping forces and electrostatic forces are considered simultaneously. An approximate analytical approach based on Galerkin method is suggested, since closed-form solutions are not achievable in this case. Prototype design and fabrication through UW-MEMS process and an experimental investigation via a micro-scanning laser Doppler vibrometer system are also performed to verify the analytical predictions.;The methodologies presented in this thesis can also be used for the analysis of other MEMS gyroscopic systems and switching systems, and the knowledge gained from the present study is envisaged to benefit the design and analysis of these classes of MEMS devices.;Keywords. Vibratory gyroscope, MEMS, Parametric resonance, Method of averaging, Method of stochastic averaging, Bifurcation, Stochastic fluctuation, Stability, Microscanning Laser Doppler Vibrometer, Contact bouncing, Asperity contact model, RF switch.