Noise contribution in nonlinear systems is very different from that in linear systems. The noise effects in nonlinear systems can be complicated and not obvious to predict. In this thesis, the focus was on the noise contribution in nonlinear systems when measuring with the nonlinear vector network analyzer (NVNA). An additional noise source together with a single sinewave signal was fed into the input of the amplifier and the performance was studied. The input power of the amplifier is considered to be the sum of the noise power and the signal power. The variation of the 1 dB compression point and the third order interception point as functions of the added noise power were studied. From the measured results in this thesis, the 1 dB compression point referred to the output power will decrease when increasing the added noise power at the input of the amplifier. The contribution of the added noise to the 1 dB compression point of an amplifier is considered dual: with the added noise the linear regression lines of the AM/AM curves are changed, and due to hard clipping the useful output power is reduced. As a result of those two effects, the added noise made the compression start at a lower power level. When the added noise reaches a certain level, the 1 dB compression point is hard to measure. Thus when performing nonlinear measurements, the noise effects should be taken into considerations and further studies are required to get better understanding of the system’s behavior in noisy environment.

<p>Within the past decade, advances in small-scale electronics have reduced power consumption requirements such that mechanisms for harnessing ambient kinetic energy for self-sustenance are a viable technology. Such devices, known as energy harvesters, may enable self-sustaining wireless sensor networks for applications ranging from Tsunami warning detection to environmental monitoring to cost-effective structural health diagnostics in bridges and buildings. In particular, flexible electroelastic materials such as lead-zirconate-titanate (PZT) are sought after in designing such devices due to their superior efficiency in transforming mechanical energy into the electrical domain in comparison to induction methods. To date, however, material and dynamic nonlinearities within the most popular type of energy harvester, an electroelastically laminated cantilever beam, has received minimal attention in the literature despite being readily observed in laboratory experiments. </p><p>In the first part of this dissertation, an experimentally validated first-principles based modeling framework for quantitatively characterizing the intrinsic nonlinearities and moderately large amplitude response of a cantilevered electroelastic generator is developed. Nonlinear parameter identification is facilitated by an analytic solution for the generator's dynamic response alongside experimental data. The model is shown to accurately describe amplitude dependent frequency responses in both the mechanical and electrical domains and implications concerning the conventional approach to resonant generator design are discussed. Higher order elasticity and nonlinear damping are found to be critical for correctly modeling the harvester response while inclusion of a proof mass is shown to invigorate nonlinearities a much lower driving amplitudes in comparison to electroelastic harvesters without a tuning mass.</p><p>The second part of the dissertation concerns dynamical systems design to purposefully engage nonlinear phenomena in the mechanical domain. In particular, two devices, one exploiting hysteretic nonlinearities and the second featuring homoclinic bifurcation are investigated. Both devices exploit nonlinear magnet interactions with piezoelectric cantilever beams and a first principles modeling approach is applied throughout. The first device is designed such that both softening and hardening nonlinear resonance curves produces a broader response in comparison to the linear equivalent oscillator. The second device makes use of a supercritical pitchfork bifurcation wrought by nonlinear magnetic repelling forces to achieve a bistable electroelastic dynamical system. This system is also analytically modeled, numerically simulated, and experimentally realized to demonstrate enhanced capabilities and new challenges. In addition, a bifurcation parameter within the design is examined as a either a fixed or adaptable tuning mechanism for enhanced sensitivity to ambient excitation. Analytical methodologies to include the method of Harmonic Balance and Melnikov Theory are shown to provide superior insight into the complex dynamics of the bistable system in response to deterministic and stochastic excitation.</p> / Dissertation

During the design and development stages of mechanical structures, after each modification made in order to satisfy design criteria, dynamic characteristics of the structure change and should be determined through reanalyzing the structure dynamically. Due to the significance of computational time and cost in design processes, it is inevitable for structural modification methods, especially for large systems, to become involved in predicting the dynamic behavior of modified structures from those of the original and modifying structures. Since most engineering structures are inherently nonlinear, linear approach may not be valid no more. Therefore, conventional structural modification methods can not be directly used, instead a nonlinear structural modification method needs to be employed. In this thesis, it is aimed to adapt an effective linear structural modification method to structures with nonlinear modification or coupling. The amplitude dependencies of nonlinearities are modeled by using describing function method. Mathematical formulations are embedded in a computer program developed in MATLAB&reg / with a graphical user interface. The software uses modal analysis results of ANSYS&reg / for the original structure and dynamic stiffness matrix and nonlinearity information that belong to the modifying structure in order to calculate dynamic response of the modified structure. The approach is verified by applying it to both discrete and real test structures previously studied in literature and generated discrete structures, then comparing the results with prior ones and ones obtained via time domain integration, respectively. Several other case studies are also included in order to demonstrate the applicability and to investigate the performance of the method. It is concluded in this study that the structural modification method proposed can be successfully and efficiently used for structures with nonlinear modification or coupling.

This dissertation investigates two long standing issues in nonlinear optics: complete characterization of the ultrafast dynamics of simple molecules, and the potential of a two-photon laser using a bulk semiconductor gain medium. Within the Born-Oppenheimer approximation, nonlinear refraction in molecular liquids and gases can arise from both bound-electronic and nuclear origins. Knowledge of the magnitudes, temporal dynamics, polarization and spectral dependences of each of these mechanisms is important for many applications including filamentation, white-light continuum generation, all-optical switching, and nonlinear spectroscopy. In this work the nonlinear dynamics of molecules are investigated in both liquid and gas phase with the recently developed beam deflection technique which measures nonlinear refraction directly in the time domain. Thanks to the utility of the beam deflection technique we are able to completely determine the third-order response function of one of the most important molecular liquids in nonlinear optics, carbon disulfide. This allows the prediction of essentially any nonlinear refraction or two-photon absorption experiment on CS2. Measurements conducted on air (N2 and O2) and gaseous CS2 reveal coherent rotational revivals in the degree of alignment of the ensemble at a period that depends on its moment of inertia. This allows measurement of the rotational and centrifugal distortion constants of the isolated molecules. Additionally, the rotational contribution to the beam deflection measurement can be eliminated thanks to the particular polarization dependence of the mechanism. At a specific polarization, the dominant remaining contribution is due to the bound-electrons. Thus both the bound-electronic nonlinear refractive index of air, and second hyperpolarizability of isolated CS2 molecules, are measured directly. The later agrees well with liquid CS2 measurements, where local field effects are significant. The second major portion of this dissertation addresses the possibility of using bulk semiconductors as a two-photon gain medium. A two-photon laser has been a goal of nonlinear optics since shortly after the original laser*s development. In this case, two-photons are emitted from a single electronic transition rather than only one. This processes is known as two-photon gain (2PG). Semiconductors have large two-photon absorption coefficients, which are enhanced by ~2 orders of magnitude when using photons of very different energies, e.g., ћωa≈10ћωb. This enhancement should translate into large 2PG coefficients as well, given the inverse relationship between absorption and gain. Here, we experimentally demonstrate both degenerate and nondegenerate 2PG in optically excited bulk GaAs via pump-probe experiments. This constitutes, to my knowledge, the first report of nondegenerate two-photon gain. Competition between 2PG and competing processes, namely intervalence band and nondegenerate three-photon absorption (ND-3PA), in both cases are theoretically analyzed. Experimental measurements of ND-3PA agree with this analysis and show that it is enhanced much more than ND-2PG. It is found for both degenerate and nondegenerate photon pairs that the losses dominate the two-photon gain, preventing the possibility of a two-photon semiconductor laser.

A periodic structure consists of spatially repeating unit cells. From man-made multi-span bridges to naturally occurring atomic lattices, periodic structures are ubiquitous. The periodicity can be exploited to generate frequency bands within which elastic wave propagation is impeded. A limitation to the linear periodic structure is that the filtering properties depend only on the structural design and periodicity which implies that the dispersion characteristics are fixed unless the overall structure or the periodicity is altered. The current research focuses on wave propagation in nonlinear periodic structures to explore tunability in filtering properties such as bandgaps, cut-off frequencies and response directionality. The first part of the research documents amplitude-dependent dispersion properties of weakly nonlinear periodic media through a general perturbation approach. The perturbation approach allows closed-form estimation of the effects of weak nonlinearities on wave propagation. Variation in bandstructure and bandgaps lead to tunable filtering and directional behavior. The latter is due to anisotropy in nonlinear interaction that generates low response regions, or "dead zones," within the structure.The general perturbation approach developed has also been applied to evaluate dispersion in a complex nonlinear periodic structure which is discretized using Finite Elements. The second part of the research focuses on wave dispersion in strongly nonlinear periodic structures which includes pre-compressed granular media as an example. Plane wave dispersion is studied through the harmonic balance method and it is shown that the cut-off frequencies and bandgaps vary significantly with wave amplitude. Acoustic wave beaming phenomenon is also observed in pre-compressed two-dimensional hexagonally packed granular media. Numerical simulations of wave propagation in finite lattices also demonstrated amplitude-dependent bandstructures and directional behavior so far observed.

No description available.

Trial and error and the use of highly time-consuming methods are often necessary for modeling, simulating and characterizing nonlinear dynamical systems. However, for the rather common special case when a nonlinear system has linear relations between many of its degrees of freedom there are particularly interesting opportunities for more efficient approaches. The aim of this thesis is to develop and validate new efficient methods for the theoretical and experimental study of mechanical systems that include significant zero-memory or hysteretic nonlinearities related to only small parts of the whole system. The basic idea is to take advantage of the fact that most of the system is linear and to use much of the linear theories behind forced response simulations. This is made possible by modeling the nonlinearities as external forces acting on the underlying linear system. The result is very fast simulation routines where the model is based on the residues and poles of the underlying linear system. These residues and poles can be obtained analytically, from finite element models or from experimental measurements, making these forced response routines very versatile. Using this approach, a complete nonlinear model contains both linear and nonlinear parts. Thus, it is also important to have robust and accurate methods for estimating both the linear and nonlinear system parameters from experimental data. The results of this work include robust and user-friendly routines based on sinusoidal and random noise excitation signals for characterization and description of nonlinearities from experimental measurements. These routines are used to create models of the studied systems. When combined with efficient simulation routines, complete tools are created which are both versatile and computationally inexpensive. The developed methods have been tested both by simulations and with experimental test rigs with promising results. This indicates that they are useful in practice and can provide a basis for future research and development of methods capable of handling more complex nonlinear systems.

In the field of second harmonic (SH) generation most studies have been concerned with maximizing conversion efficiencies, generally achievable at the phase matching (PM) condition. Outside of the PM the conversion efficiency drastically decreases. This has caused that the possible working conditions out of PM to remain largely unexplored. In this thesis work we initiated a systematic study of the SH behavior in under conditions of large phase mismatch. When a pump pulse crosses an interface between a linear and a nonlinear medium there are always two generated SH components. These components may be understood on the basis of the mathematical solution of the inhomogeneous wave equations at the SH frequency. The homogeneous (HOM) solution is a component with wave-vector k(2¿) as expected from the dispersion relation and exchanges energy with the pump until the inevitable walk-off. The inhomogeneous (INH) solution is a component with a wave-vector 2k(¿), twice the pump wave-vector, and travels locked to the pump pulse. We divide our work in two parts, one for each generated component. Inhomogeneous component. We start a systematic study of the behavior of the generated INH component, phase locked to the pump. The consequences of phase locking (PL) can guide us towards new scenarios by allowing working conditions hitherto assumed inaccessible for absorbing materials. We show that while the HOM component travels with the group velocity given by material dispersion, the IHN component is captured by the pump pulse and experiences the same effective dispersion of the pump. It does not follow the PM condition. It naturally follows that the suppression of absorption at the SH wavelengths will occur if the pump is tuned to a region of transparency. We extended the same theory for the generated third harmonic (TH). We then studied the surprising behavior of SH and TH INH components with frequencies above the absorption edge when the material is placed inside a cavity resonant only at the fundamental frequency. We have shown that the PL mechanism not only inhibits absorption but also fosters the enhancement of harmonic generation by several orders of magnitude compared to the no-cavity case. Finally, we tested the INH SH and TH behaviors in metallic frequency regime of material. Homogeneous component. The techniques used to PM the nonlinear interaction enable efficient nonlinear interactions but drastically limit the spectral bandwidth of the nonlinear optical process, making the designed frequency converter only suitable for a fixed input wavelength and single interaction only. It has been shown that the use of disordered nonlinear media relaxes the PM condition thus allowing one to achieve relatively efficient broad bandwidth regime of the frequency conversion. An example of a quadratic nonlinear medium with a disordered domain structure is an un-poled Strontium Barium Niobate (SBN) crystal. It is composed of a system of random size anti parallel ferroelectric domains that allow to phase-match any second-order parametric process over a broad range of wavelengths without any poling. We have initiated an experimental and theoretical investigation of the properties of the SH waves generated in SBN crystals, with an extension to the generated TH. This study covers the coherence and polarization properties of the generated signal, as well as its spatial distribution. In addition, we have made an experimental study of the noncollinear interaction of short optical pulses in a SBN crystal by using two fundamental waves intersecting inside the crystal. We have shown that this effect may be employed as a simple tool for monitoring both the pulse duration and initial chirp. This method offers a simple and economic alternative to the existing methods for pulse characterization.

This thesis investigates the generation of second harmonic bulk waves in the presence of a free boundary. Second harmonic waves have proven to be useful in the field of nondestructive evaluation to detect fatigue in a material at an early stage. Since most experimental setups include a free surface, the influence of such a boundary is of significant practical interest. As a result, the objective of this research is to develop a quantitative understanding of the complete process of second harmonic generation at a free boundary. This research shows that the interaction of primary waves (with each other) in the nonlinear framework leads to the generation of second harmonic bulk waves. We distinguish between self-interaction of a single primary wave and the cross-interaction of two different primary waves. The proposed approach uses the perturbation method to solve the nonlinear equations of motion, and shows two fundamentally different solutions. In the case of resonance, the secondary waves grow with propagation distance. This is the most important practical case, since the growing amplitudes of these waves should be easier to experimentally measure. In the second, non-resonant case, the amplitudes of the secondary waves are constant. The complete process of second harmonic generation is analyzed for an incident Pand an incident SV-wave, with the primary and secondary fields given. Finally, the degenerate case of normal incidence is presented. Normal and oblique incidence are compared with regard to their feasibility in experimental setups. The specific behavior of second harmonic waves propagating in aluminum is numerically determined. These results enable a variety of physical insights and conclusions to be drawn from the analytical and numerical investigations.

This dissertation presents analog circuit analysis and design from a nonlinear dynamics perspective. An introduction to fundamental concepts of nonlinear dynamical systems theory is given. The procedure of nondimensionalization is used in order to derive the state space representation of circuits. Geometric tools are used to analyze nonlinear phenomena in circuits, and also to develop intuition about how to evoke certain desired behavior in the circuits. To predict and quantify non-ideal behavior, bifurcation analysis, stability analysis and perturbation methods are applied to the circuits. Experimental results from a reconfigurable analog integrated circuit chip are presented to illustrate the nonlinear dynamical systems theory concepts. Tools from nonlinear dynamical systems theory are used to develop a systematic method for designing a particular class of integrated circuit sinusoidal oscillators. This class of sinusoidal oscillators is power- and area-efficient, as it uses the inherent nonlinearity of circuit components to limit the oscillators' output signal amplitude. The novel design method that is presented is based on nonlinear systems analysis, which results in high-spectral purity oscillators. This design methodology is useful for applications that require integrated sinusoidal oscillators that have oscillation frequencies in the mid- to high- MHz range. A second circuit design example is presented, namely a bandpass filter for front end auditory processing. The bandpass filter mimics the nonlinear gain compression that the healthy cochlea performs on input sounds. The cochlea's gain compression is analyzed from a nonlinear dynamics perspective and the theoretical characteristics of the dynamical system that would yield such behavior are identified. The appropriate circuit for achieving the desired nonlinear characteristics are designed, and it is incorporated into a bandpass filter. The resulting nonlinear bandpass filter performs the gain compression as desired, while minimizing the amount of harmonic distortion. It is a practical component of an advanced auditory processor.