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Nonlinear optical processes have been used for many years to cause light to behave in unique ways. These can influence various aspects of the light beams, including the spatial intensity profile. More specifically, a naturally divergent beam can maintain its initial beam profile when propagating through a nonlinear medium due to the increased refractive index causing focusing. This process is called self-trapping and can be elicited for both coherent and incoherent light in a number of nonlinear media. Analogous to this, modulation instability (MI) is a nonlinear process that causes a broad beam of light of break up into a large population of self-trapped filaments. When these processes occur in photopolymers, light guiding structures are inscribed within the material and persist even after the light is removed. Our group has previously studied the behaviour and interactions of light undergoing self-trapping and MI in photopolymer systems. The studies presented in this thesis show novel interactions between multiple incoherent beams undergoing MI, utilization of these interactions, as well as the development of new polymeric systems capable of inducing MI. The filaments produced when orthogonal beams underwent simultaneous MI would align with themselves, forming highly ordered structures within the material. These interactions were used as the basis of an encoding and computing system based on the specific ordering of the resulting filaments. The mechanical properties of the resulting polymers were improved and tuned by developing a long-chain organosiloxane based system. The embedded waveguide structures are capable of guiding light when significantly deformed and restore to their initial parameters. A hydrogel system was also developed that was capable of producing self-trapping and MI with incoherent light. The samples were tested with biological systems and were also used to produce dynamic lens samples with enhanced angle of view. / Thesis / Doctor of Philosophy (PhD)

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Aeroelastic scaling methodologies are developed for geometrically nonlinear applications. The new methods are demonstrated by designing an aeroelastically scaled model of a suitably nonlinear full-scale joined-wing aircraft. The best of the methods produce scaled models that closely replicate the target aeroelastic behavior. Internal loads sensitivity studies show that internal loads can be insensitive to axial stiffness, even for globally indeterminate structures. A derived transverse to axial stiffness ratio can be used as an indicator of axial stiffness importance. Two findings of the work extend to geometrically linear applications: new sources of local optima are identified, and modal mass is identified as a scaling parameter. Optimization procedures for addressing the multiple optima and modal mass matching are developed and demonstrated. Where justified, limitations of commercial software are avoided through development of custom tools for structural analysis and sensitivities, aerodynamic analysis, and nonlinear aeroelastic trim. / Ph. D.

The synthesis and characterization of phosphorus containing poly(arylene ether)s were investigated to determine the effect of phosphorus upon the thermal stability, mechanical strength, and fire resistance of thermoplastics. Phosphorus containing activated aromatic dihalides and bisphenols were synthesized in high purity. These monomers were successfully polymerized via nucleophilic aromatic substitution to afford high molecular weight polymers. It was determined that by incorporating the phosphine oxide moiety into the polymer backbone certain properties of the resulting poly(arylene ether)s were substantially improved, such as an increase in T_g, thermal stability in air, modulus, and char yield, compared with control poly(arylene ether sulfone)s. The high char yields obtained for these polymers in air, along with observed intumescence indicates that these materials have improved fire resistance. Preliminary cone calorimetry measurements support this conclusion. In addition, the phosphine oxide group in the backbone was reduced to a phosphine and successfully converted to a phosphonium bromide ionomer. The resulting system was further chemically modified to ionically bond second-order nonlinear optical chromophores to the backbone of selected poly(arylene ether)s. Initial results on corona poling of cast film at low temperature produced stable second harmonic generation in these materials, indicating that they may have promise in nonlinear optical applications. / Ph. D.

Electrically small, low profile antennas have become the new frontier in wireless communication research. With the pressure to miniaturize wireless communication devices, engineers are turning to small low profile antennas as a way to reduce their antennas and hence their devices. Ideally, one would also like to at least maintain antenna bandwidth and efficiency while reducing size. However, in theory, antenna performance degrades when it is miniaturized—impedance bandwidth decreases with the reduction in antenna size. This thesis investigates the possibility of increasing the input impedance bandwidth without enlarging the volume of the antenna. This thesis attempts to break the fundamental tradeoff between antenna size and bandwidth by loading it with a nonlinear element. First, a brief summary of antenna background definitions is presented. Next, the analytical framework of the thesis is presented using a model of a narrowband antenna. A literature review of various narrowband electrically small antennas is studied, including the pros and cons of the Inverted-F antenna (IFA), Inverted-L antenna (ILA), and the Planar Inverted-F antenna (PIFA).Next, the analysis and the methodology leading to results are discussed and simulated results are presented. Simulated results show that the PIFA is able to achieve a higher bandwidth with a loaded nonlinear element. However, it is difficult to sustain the efficiency of the antenna due to harmonics generated by nonlinearity in the antenna. Results indicate that an increase in nonlinearity tends to generate harmonics which leads to losses in the antenna. / Master of Science

This thesis explores the development of time-varying virtual constraint controllers that allow stable and agile gaits for full-order hybrid dynamical models of quadrupedal locomotion. Unlike time-invariant nonlinear controllers, time-varying controllers do not rely on sensor data for gait phasing and can initiate locomotion from zero velocity. Motivated by these properties, we investigate the stability guarantees that can be provided by the time-varying approach. More specifically, we systematically establish necessary and sufficient conditions that guarantee exponential stability of periodic orbits for time-varying hybrid dynamical systems utilizing the Poincar� return map. Leveraging the results of the presented proof, we develop time-varying virtual constraint controllers to stabilize bounding, trotting, and walking gaits of a 14 degree of freedom quadrupedal robot, Minitaur. A framework for selecting the parameters of virtual constraint controllers to achieve exponential stability is shown, and the feasibility of the analytical results is numerically validated in full-order model simulations of Minitaur. / Master of Science / This thesis extends a class of controllers designed to address the full dynamics of stable locomotion in quadrupedal robots. As of yet, there is no widely-accepted standard methodology for controlling the complex maneuvers of quadrupedal locomotion, as most strategies rely on simplified models to ease computational constraints. "Virtual constraint'' controllers - also known as Hybrid Zero Dynamics controllers - are a class of controllers designed to address the full dynamics of legged locomotion by coordinating the links of a legged robot model to follow a periodic trajectory representing the desired gait pattern. However, the formalized "time-invariant'' model of virtual constraint controllers relies on sensor data to track progress on the desired gait trajectory. This dependence on sensor data makes the resulting controllers unable to start from a state of zero velocity and sensitive to disturbances generated by high velocity impacts. The proposed "time-varying'' virtual constraints controllers utilize the elapsed time to track gait progress and do not have the previously mentioned limitations. Motivated by these benefits, we develop a formalized methodology for designing time-varying virtual constraint controllers for quadrupedal robots. This includes extending time-invariant means of mathematically validating the stability of the gait controllers to time-varying systems. With strategies of designing and validating time-varying virtual constraint controllers formalized, the methodology is implemented on numerical simulations of bounding, trotting, and walking gaits for the quadrupedal robot Minitaur which validates the stability and feasibility of the developed controllers.