In this work, two widely studied nonlinear dynamical systems that describe the governing dynamical response of most resonant MEMS are investigated, both numerically and experimentally. Both the linear and nonlinear stiffness modulations are of interest in this study. A majority of MEMS resonators in applications are operated in the linear regime, however, the usable linear dynamical range of a linear resonator is limited by noise on the lower end, and by nonlinearity on the upper end. Hence, tuning the nonlinearities plays an important role in improving the linear dynamical range. First, we studied the effect of shape optimization on tuning the geometric/mechanical nonlinearity resulting from flexure mode mid-plane stretching.

By using a finite element normal form model in combination with optimization algorithms, we showed that distributing materials at critical locations where the slope of the resonator mode shape is at extrema results in continuous increase or decrease in the Duffing nonlinearity. We experimentally verified the theory and demonstrated a three-fold magnification or reduction in the Duffing nonlinearity through shape optimization alone. Though nonlinearity maximization is a potentially powerful tool, the case of nonlinearity minimization is of special interest to the MEMS community. The ability to reduce the Duffing nonlinearity is especially useful for increasing the linear dynamical range of resonators, which plays a pivotal role in reducing phase noise in MEMS oscillators and increasing the signal-to-noise ratio in resonant sensors.

The experimental results shown in this work are in good agreement with numerical predictions (within 8% error for the important case of minimizing the nonlinearity), and demonstrate that one can successfully utilize shape optimization methods for adjusting the resonator nonlinearity in a well-controlled manner. These results provide strong confidence that shape optimization methods can be applied in the design of nonlinear MEMS resonators with more complicated geometries and multi-physics effects, as well as for different nonlinear parameters, including hard-to-control modal coupling coefficients.

Next, we show that even though most MEMS are designed to operate in the linear region, in other applications utilizing nonlinear dynamics in MEMS can actually enhance their performance. We demonstrate that by periodically modulating the linear stiffness of the system, one can achieve nonlinear parametric resonance, which can be used as a superior sensing method in atmospheric pressure compared to the conventional linear harmonic sensing. System parameters are experimentally mapped, and excellent agreement between numerical and experimental models is established. Finally a robust amplitude controller using a systematic McFarlane and Glover Hinfinity loop shaping procedure is implemented in LabView's FPGA platform. The experimental results successfully demonstrate a microbeam sensor capable of real-time tracking on the order of microseconds with femtogram mass sensitivity. When applied to the detection of nerve gas this translates to a 13 parts-per-trillion limit of detection at atmospheric pressure.