Nonlinear Frequency Tuning of Pre-curved Micro-scale Beams
Robatin, Mackenzie Rae
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Micro-electromechanical systems (MEMS) have many common advantages in applications due to their small size and ability to integrate with electronics. For MEMS with vibrating elements, other advantages include their tunable resonant frequencies and the frequency range in which the systems operate. Also, the damping in these devices is often so light that it is nearly negligible, resulting in sharp resonance peaks with good frequency selectivity. Most such devices are designed to operate in their linear range. However, large amplitudes are often desired in order to reduce the relative effects of noise. In such cases, due to mechanical and electrostatic nonlinear effects, higher vibration amplitudes can result in frequency shifts near resonance, generally hindering operation. In this thesis, the modeling of electrostatically actuated, initially curved, doubly-clamped micro beams is re-viewed and new theoretical and numerical results about their frequency response are obtained that are valuable for design. A special operating point is considered in which the frequency is locally independent of amplitude, a so-called zero dispersion (ZD) point, which has some advantages for low noise operation. The model investigated is a single degree of freedom, unforced, undamped system that specifically accounts for the first natural mode shape of the pre-curved beam with electrodes on both sides of the beam and which have different levels of DC bias voltages. Using an equivalent potential from the equation of motion, the energy-dependent period of vibration and the ZD point can be computed. Analytical and graphical results in various forms (including the phase plane) are presented and explored interms of parameters of interest, including the non-dimensional DC bias voltages and the symmetry breaking initial curvature parameter. The results obtained allow one to select parameters for beams that provide ZD operation at a desired vibration amplitude.