Advantageous Utilization of Nonlinear Phenomena in Micro-Structures and Macro-Structures: Applications to Micro-Resonators and Atomic Force Microscopy

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Within this work, the nonlinear oscillations of various beam-like structures are studied. Methods are developed to analyze systems of this type to better understand their behavior in order to utilize the nonlinear phenomena associated with them and to provide insights for device development. The specific applications explored within this study are piezoelectric micro-scale resonators, micro-resonator arrays, the cantilever probes of atomic force microscopes, and a macro-scale test apparatus for the AFM probe. In order to analytically, numerically, and experimentally study these systems, various methods are employed. Analytical models are developed, utilizing bending stiffness and axial stretching terms to explain nonlinear behavior. Reduced-order-modeling techniques are applied to develop single-mode and multi-mode approximations and study the dynamic behavior of these structures. Nonlinear analysis methods are used to study these systems and to determine approximate solutions. Discrete models are developed and utilized to conduct numerical simulations. Data collected through experimental observations are utilized to determine system parameters and verify simulation results.

Through this work, a multi-variable, parametric identification scheme is developed for characterizing nonlinear oscillators from frequency-response data with jumps in the amplitude values. Parameter values are identified for piezoelectric micro-scale resonators and good agreement is seen with the corresponding model predictions. By using multiple data sets, parameter trends are studied for changes in the input signal. In another nonlinear analysis, a relationship is identified between a nonlinear localization phenomena called Intrinsic Localized Modes (ILMs) and nonlinear vibration modes. A method is developed to derive equations for determining the spatial characteristics of the localizations and the profiles are used to conduct further studies. For a cantilever beam impactor system, a period doubling phenomenon is identified for off-resonance excitation conditions. Changes in this system's response are studied and a method is proposed to utilize this phenomenon to determine conditions for grazing. This work shows how important nonlinearities are in beam structures when oscillations exceed the linear range. More importantly, these studies show how an understanding of the nonlinearities can be used to the advantage of the system.