Nonlinear electroelastic resonators
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[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] In broad terms, this dissertation concerns the physics of electroelastic continua and its application to piezoelectric plate resonators, with a specific emphasis on nonlinear phenomena. The gradual reduction of device size, coupled with higher driving amplitudes, have revealed the need for a general framework to predict nonlinear behavior in resonant piezoelectric devices. Not only must a suitable theoretical model exist, but it should be amenable to analytical and numerical methods that give physical insight into the underlying nonlinear interactions. This work addresses the need for such a framework by extending existing theory and applying analytical and numerical methods in new ways. In the first chapter linear piezoelectric device physics is reviewed and a method for improving the quality factor of mass sensors in a liquid environment is presented. However, the limitations of this preliminary work motivate the focus on nonlinear behavior in the following chapters. After a brief review of electroelastic continua in chapter 2, the following chapter details the use of nonlinear Duffing behavior in a quartz resonator to improve mass sensitivity. A 1-DOF model is used but coupling with other modes is observed at higher driving amplitudes, showing the limits of the model. Thus, in chapter 4 a quantitative model is developed to predict the effect of modal interactions in nonlinear piezoelectric crystals. Perturbation analysis and numerical continuation methods are applied to reveal previously unwitnessed multistable behavior, which is then experimentally verified. As a result, the global bifurcation structure of quartz plates is mapped out in terms of the driving frequency and amplitude. The final chapter derives approximate nonlinear plate equations for electroelastic crystals, using both power series and trigonometric expansions. A very general reduced-order model of the governing PDEs is derived and used to quantify the effect of coupling to lateral eigenmodes on the Duffing behavior of the frequency response.
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