This research focuses on the development and analysis of predictive modelsfor frequency converters and frequency generators that are based on micro-electromechanical-system(MEMS) technology. In contrast to applications in which nonlinearityis sought to be avoided, frequency conversion and frequency generationnecessarily involve nonlinear processes, and while many existing technologies areavailable for realizing these operations, MEMS technology offers a potentially advantageouscombination of size, power requirements, and noise characteristics.This dissertation describes a series of investigations related to MEMS frequencyconversion and generation, including: (i) an analytical investigation of a class ofpassive multi-stage frequency dividers, (ii) the design and realization of this behaviorin a MEMS device, (iii) the development of a model for nonlinear modalinteractions in closed loop MEMS and (iv) the development of a computationalmethod for optimizing their nonlinear resonant response through shape optimization.Items (ii) and (iii) were carried out in close collaboration with experimentalgroups at the University of California at Santa Barbara and Argonne NationalLabs, respectively. Item (iv) was carried out in collaboration with the topologyoptimization group at the Technical University of Denmark.The subharmonic frequency divider is based on a class of mechanical structureswith nonlinearly coupled high Q vibration modes with sequential 2:1 internal resonances,for which sequential parametric resonances are used to transfer energyfrom a high frequency mode down to lower frequency modes. We analyze thenormal form for this subharmonic resonance cascade and predict the system re-sponse based on system and driving signal parameters. We then show how todesign and experimentally implement this subharmonic cascade in MEMS, andwe demonstrate frequency division by a factor of eight.The frequency generator model is based on a closed loop oscillator in whichthe resonator element has vibration modes with 1:3 frequency ratio and nonlinearintermodal coupling. Experimental observations have shown that the oscillatorphase noise performance is significantly improved when operating in a coupledmode regime, in which a flexural mode is nonlinearly coupled to a torsionalmode. The device is characterized by comparing its measured open loop responseagainst a model based on 1:3 internal resonance, demonstrating good agreement.The closed loop version of the model is analyzed with a focus on how noise sourcesare filtered through the system into phase noise. This model predicts the signifi-cant drop in phase noise observed when operating with internal resonance. Thispredictive model provides a basis for future designs that take full advantage ofthis nonlinear behavior, which has potential for commercialization in the growingarea of MEMS oscillators.Lastly, we describe the development of a computational tool that allows oneto tailor the nonlinear resonant response of mechanical structures using a combinationof normal forms and structural optimization tools. This approach is usedto improve a device's nonlinear modal coupling by nearly an order of magnitude.Such tools will be important for the continuing development of MEMS that utilizenonlinear resonant behavior.In summary, it is shown that internal resonance, in addition to offering interestingdynamic behavior, can be used to improve the performance of signal processingdevices. This work also demonstrates that devices that use internal resonancecan be analyzed with generic dynamic models, thereby providing a basis for understandingfundamental device characteristics and future design development.
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