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Linear theory is useful in determining small static and dynamic deflections. However, to characterize large static and dynamic deflections, it is no longer useful or accurate, and more sophisticated analysis methods are necessary. In the case of beam deflections, linear beam theory makes use of an approximate curvature expression. Here, the exact curvature expression is used to derive the governing partial differential equations that describe the in-plane equilibrium and dynamics of a long, thin, inextensible beam, where the self-weight of the beam is included in the analysis. These beam equations are expressed in terms of arclength, and the resulting equilibrium shape is called the elastica. The analysis gives solutions that are accurate for any deflection size, and the method can be used to characterize the behavior of many structural systems. Numerical and analytical methods are used to solve or to approximate solutions to the governing equations. Both a shooting method and a finite difference, time-stepping algorithm are developed and implemented to find numerical solutions and these solutions are compared with some analytical approximation method results. The elastica equations are first used to determine both linear and nonlinear equilibrium configurations for a number of boundary conditions and loading types. In the case of a beam with a significant self-weight, the system can exhibit nonlinear static behavior even in the absence of external loading, and the elastica equations are used to determine the weight corresponding to the onset of instability (or self-weight buckling). The equations are also used to characterize linear and nonlinear vibrations of some structural systems, and experimental tests are conducted to verify the numerical results. The linear vibration analysis is applied to a vibration isolator system, where a postbuckled clamped-clamped beam or otherwise highly-deformed structure is used (in place of a conventional spring) to reduce system motion. The method is also used to characterize nonlinear dynamic behavior, and the resulting frequency-response curves are compared with those in the literature. Finally, the method is used to investigate the dynamics of subsea risers, where the effects of gravity, buoyancy, and the current velocity are considered. / Dissertation

While nonlinear behavior in mechanical systems typically degrades the behavior and performance the devices, the presence of system nonlinearities can sometimes improve the quality of the system. A reason for avoiding nonlinearities within a device is the difficulty in controlling the device due to the effects of the nonlinearities on system behavior. However, careful analysis of nonlinear systems can allow for one to take advantage of the nonlinear behavior to improve system performance. The objective of this thesis is to exploit the use of nonlinearities to enhance system performance, specifically the sensitivity of a micromachined magnetic sensor. A device design will be presented that is similar to a prototype that has been fabricated by a student within the Electrical and Computer Engineering Department at Georgia Tech. The operating principle of the device is that changes in the orientation and the strength of an external magnetic field will result in changes in the dynamic behavior of the sensor. While previous device provided a proof of the design concept, it was unable to achieve a sensitivity that would allow for its use as a compass. Improvements in the sensitivity of the sensor are achieved through the modeling and optimization of the magnetic sensor. The optimization and redesign of the magnetic sensor will improve the quality of the device and provide another step towards sensor commercialization. A new design that incorporates the use of variable force comb drives will be proposed that will further improve the sensitivity of the device by modifying the dynamic behavior of the sensor. Another approach that is presented to exploit the nonlinear behavior of the magnetic sensor involves a frequency detection scheme that uses nonlinear vibrations to characterize sensor behavior. Some benefits of this detection technique are that it is insensitive to noise in the vibration of the sensor and is also independent of the damping present within the system. In addition, the implementation of this sensing technique can be readily applied to variety of sensors types without the redesign of a system or the addition of complex components such as vacuum packaging or signal processing electronics.

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Receptor-based biological detection techniques often suffer from the problem of non-specific interactions. This is largely due to the presence of weak electrostatic and Van der Waals forces between the receptor and the non-target substances in the analyte that are not easily dissociated in practice. Most existing detection techniques are unable to probe the interaction between the bound entity and the surface and differentiate between specific and non-specific interactions in terms of bond strength or activation energy. The resulting false positive responses lead to various issues, such as misdiagnosis and mistreatment in clinical diagnostics and false alarms in biosecurity. The problem is even more significant with direct direction techniques, such as the resonant frequency shift based detection using quartz crystal microbalance (QCM) or micro-cantilevers, which involve minimal sample processing and washing steps. The work presented in this thesis investigates, through modeling and experiments, the mechanical interactions of a resonator with microparticles attached via biomolecular linkers and analyses the resulting nonlinear acoustic modulation of the resonator from the transduced electrical signal. Physisorbed and specific interactions both in air and liquid medium are studied using thickness shear mode quartz crystal resonators and streptavidin-coated polystyrene microbeads (SCPM) of various sizes. It is found that the modification in the transduced electrical signal measured at the third harmonic (3f), or three times the driving frequency f, is significant in presence of the attached particles and approximately proportional to the number of particles. A detection limit of approximately 2 SCPM of 5.6 µm diameter in air and 6700 SCPM of 0.39 µm diameter in liquid is demonstrated, which corresponds to a mass detection limit of ~200 pg. Most interestingly, the deviation in the magnitude of the 3f signal as a function of the resonator oscillation amplitude is found to hold a distinct relationship with the type of particle-surface interaction. This provides a basis for selectivity in detection over and above the efficacy of the receptor. The function is also found to correlate well with the event of SCPM diffusion on the surface. This detection technique, based on the measurement of deviation in magnitude of the transduced electrical signal measured at a higher odd harmonic of the drive frequency due to the presence of surface-bound particles on a resonator, is termed as the anharmonic detection technique (ADT). A feasibility study with Bacillus subtilis spores in phosphate buffer saline (PBS) is carried out successfully where the modeling and experimental results with SCPM are successfully reproduced. A detection limit of 430 spores is demonstrated, which corresponds to a mass detection limit of ~650 pg. Capability for differentiation of the specifically-captured spores from unwashed physisorbed SCPM of similar dimensions is demonstrated using the shape of the ADT signal. These results indicate that the spore immobilization step may be directly followed by the detection step, which are 9 mins and 2 mins respectively in these experiments. ADT thus potentially enables a rapid, sensitive, reliable and direct detection without the need for any sample processing. Moreover, being an entirely electronic technique, ADT suitably lends itself to multiplexing, large scale fabrication and implementation on a miniaturized low-cost point-of-care detection platform that is of immense need in clinical diagnostics, food and environmental monitoring and biosecurity. Furthermore, fitting the experimental results with modeling estimates enables ADT to determine the force-extension characteristics of the binding biomolecular linker. The force-extension characteristics and the estimated unbinding force for a streptavidin-biotin complex estimated using ADT agrees well with those computed using molecular dynamics (MD) simulation at similar loading rates. Thus ADT contributes a unique force-spectroscopic method, which unlike conventional techniques such as the atomic force microscopy (AFM) provides statistically averaged data for multiple biomolecules in a relatively quicker and simpler experimental format. A method for determination of activation energy of the interaction is also proposed using ADT. This potentially enables a method for rapid and large scale biomolecular screening and studying of interaction networks, which have important applications in drug discovery and individualized therapy.

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