# Browsing by Subject "Nonlinear"

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Item Open Access A Study of Non-Smooth Impacting Behaviors(2015) George, Christopher MichaelThe dynamics of impacting components is of particular interest to engineers due to concerns about noise and wear, but is particularly difficult to study due to impact's non-linear nature. To begin transferring concepts studied purely analytically to the world of physical mechanisms, four experiments are outlined, and important non-linear concepts highlighted with these systems. A linear oscillator with a kicked impact, an impacting forced pendulum, two impacting forced pendulums, and a cam follower pair are studied experimentally, with complementary numerical results.

Some important ideas highlighted are limit cycles, basins of attraction with many wells, grazing, various forms of coexistence, super-persistent chaotic transients, and liftoff. These concepts are explored using a variety of non-linear tools such as time lag embedding and stochastic interrogation, and discussions of their intricacies when used in non-smooth systems yield important observations for the experimentalist studying impacting systems.

The focus is on experimental results with numerical validation, and spends much time discussing identification of these concepts from an experiment-first mindset, rather than the more traditional analytical-first approach. As such a large volume of experimentally important information on topics such as transducers and forcing mechanism construction are included in the appendices.

Item Open Access An Aeroelastic Evaluation of the Flexible Thermal Protection System for an Inflatable Aerodynamic Decelerator(2015) Goldman, Benjamin DouglasThe purpose of this dissertation is to study the aeroelastic stability of a proposed flexible thermal protection system (FTPS) for the NASA Hypersonic Inflatable Aerodynamic Decelerator (HIAD). A flat, square FTPS coupon exhibits violent oscillations during experimental aerothermal testing in NASA's 8 Foot High Temperature Tunnel, leading to catastrophic failure. The behavior of the structural response suggested that aeroelastic flutter may be the primary instability mechanism, prompting further experimental investigation and theoretical model development. Using Von Karman's plate theory for the panel-like structure and piston theory aerodynamics, a set of aeroelastic models were developed and limit cycle oscillations (LCOs) were calculated at the tunnel flow conditions. Similarities in frequency content of the theoretical and experimental responses indicated that the observed FTPS oscillations were likely aeroelastic in nature, specifically LCO/flutter.

While the coupon models can be used for comparison with tunnel tests, they cannot predict accurately the aeroelastic behavior of the FTPS in atmospheric flight. This is because the geometry of the flight vehicle is no longer a flat plate, but rather (approximately) a conical shell. In the second phase of this work, linearized Donnell conical shell theory and piston theory aerodynamics are used to calculate natural modes of vibration and flutter dynamic pressures for various structural models composed of one or more conical shells resting on several circumferential elastic supports. When the flight vehicle is approximated as a single conical shell without elastic supports, asymmetric flutter in many circumferential waves is observed. When the elastic supports are included, the shell flutters symmetrically in zero circumferential waves. Structural damping is found to be important in this case, as "hump-mode" flutter is possible. Aeroelastic models that consider the individual FTPS layers as separate shells exhibit asymmetric flutter at high dynamic pressures relative to the single shell models. Parameter studies also examine the effects of tension, shear modulus reduction, and elastic support stiffness.

Limitations of a linear structural model and piston theory aerodynamics prompted a more elaborate evaluation of the flight configuration. Using nonlinear Donnell conical shell theory for the FTPS structure, the pressure buckling and aeroelastic limit cycle oscillations were studied for a single elastically-supported conical shell. While piston theory was used initially, a time-dependent correction factor was derived using transform methods and potential flow theory to calculate more accurately the low Mach number supersonic flow. Three conical shell geometries were considered: a 3-meter diameter 70 degree shell, a 3.7-meter 70 degree shell, and a 6-meter diameter 70 degree shell. The 6-meter configuration was loaded statically and the results were compared with an experimental load test of a 6-meter HIAD vehicle. Though agreement between theoretical and experimental strains was poor, circumferential wrinkling phenomena observed during the experiments was captured by the theory and axial deformations were qualitatively similar in shape. With piston theory aerodynamics, the nonlinear flutter dynamic pressures of the 3-meter configuration were in agreement with the values calculated using linear theory, and the limit cycle amplitudes were generally on the order of the shell thickness. Pre-buckling pressure loads and the aerodynamic pressure correction factor were studied for all geometries, and these effects resulted in significantly lower flutter boundaries compared with piston theory alone.

In the final phase of this work, the existing linear and nonlinear FTPS shell models were coupled with NASA's FUN3D Reynolds Averaged Navier Stokes CFD code, allowing for the most physically realistic flight predictions. For the linear shell structural model, the elastically-supported shell natural modes were mapped to a CFD grid of a 6-meter HIAD vehicle, and a linear structural dynamics solver internal to the CFD code was used to compute the aeroelastic response. Aerodynamic parameters for a proposed HIAD re-entry trajectory were obtained, and aeroelastic solutions were calculated at three points in the trajectory: Mach 1, Mach 2, and Mach 11 (peak dynamic pressure). No flutter was found at any of these conditions using the linear method, though oscillations (of uncertain origin) on the order of the shell thickness may be possible in the transonic regime. For the nonlinear shell structural model, a set of assumed sinusoidal modes were mapped to the CFD grid, and the linear structural dynamics equations were replaced by a nonlinear ODE solver for the conical shell equations. Successful calculation and restart of the nonlinear dynamic aeroelastic solutions was demonstrated. Preliminary results indicated that dynamic instabilities may be possible at Mach 1 and 2, with a completely stable solution at Mach 11, though further study is needed. A major benefit of this implementation is that the coefficients and mode shapes for the nonlinear conical shell may be replaced with those of other types of structures, greatly expanding the aeroelastic capabilities of FUN3D.

Item Open Access An Investigation of Sensitivity to Initial Conditions in an Experimental Structural System(2013) Waite, Joshua JosephThis thesis characterizes the nonlinear behavior of an experimental system that exhibits snap-through buckling behavior. A single-degree-of-freedom snap-through link model is harmonically forced using a Scotch yoke mechanism. In order to establish the sensitivity to initial conditions, experimental basins of attraction are constructed using the stochastic interrogation method. After, frequency sweeps are performed on the system to identify regions of interesting behavior. Then, time series data is collected at specific frequencies of interest to highlight the broad phenomenological behavior of the structural system.

A useful tool when modeling structural systems is numerical analysis. An equation of motion is developed to numerically simulate all experimentally observed results. The numerical results include snap-through boundaries, bifurcation diagrams, full initial condition grid basins of attraction, time-lag embedded basins of attraction, frequency sweeps, and time series of regions of pathological behavior.

Item Open Access Dynamics of Electromagnetic Systems for Energy Harvesting and Filtering(2014) Owens, Benjamin Andrew MichaelThe focus of this dissertation is on the dynamics of electromagnetic systems for energy harvesting and filtering applications. The inclusion of magnets into systems generates nonlinearity due to the nature of electromagnetic interactions. In this work, magnetic nonlinearity manifests in tip interactions for cantilever beams, coupling effects for electromagnetic transduction, and bistable potential wells for a two beam system. These electromagnetic interactions are used to add non-contact coupling effects for the creation of bistable oscillators or arrays of coupled beams for energy filtering.

Nonlinearity at the tip of cantilever beams acts to change the dynamic and static behavior of the system. In this dissertation, these interactions are analyzed both with and without the nonlinear tip interactions. A linear analysis of the system without the tip interaction first provides insight into the shifting frequencies of the first four natural oscillation modes when considering a rigid body tip mass with rotational inertia and a center of mass that is offset from the tip of the beam. Then, the characterization of the nonlinearities in the beam stiffness and magnetic interaction provide insight into the static and dynamic behavior of the beam. The analytical and numerical investigations, using Rayleigh-Ritz methods and an assumed static deflection, are shown to be consistent with experimental tests. These methods provide a framework for theoretically establishing nonlinear static modes and small-amplitude linear modes that are consistent with physical behavior.

In electromagnetic coupling, the role of nonlinearity can have a detrimental or beneficial effect on energy harvesting. This work includes an investigation of the response of an energy harvester that uses electromagnetic induction to convert ambient vibration into electrical energy. The system's response behavior with linear coupling or a physically motivated form of nonlinear coupling is compared with single and multi-frequency base excitation. This analysis is performed with combined theoretical and numerical studies.

The ability of magnets to add nonlinearity to a system allows for the expansion of the phenomenological behavior of said system and potential advantages and disadvantages for energy harvesting. This work studies a two beam system made up of carbon fiber cantilever beams and attached magnetic tip masses with a focus on energy harvesting potential. Numerical and experimental investigations reveal an array of phenomena from static bifurcations, chaotic oscillations, and sub-harmonic orbits. These features are used to highlight the harvesting prospects for a similarly coupled system.

Beyond nonlinearity, the non-contacting coupling effects of magnets allow for the hypothetical creation of energy filtering systems. In this work, the band structure of a two dimensional lattice of oscillating beams with magnetic tip masses is explored. The focus of the wave propagation analysis is primarily on regions in the band structure where propagation does not occur for the infinite construction of the system. These band gaps are created in this system of 2 x 2 repeating unit cells by periodically varying the mass properties and, for certain configurations, the frequency band gaps manifest in different size and band location. Uncertainty in these regions is analyzed using potential variations associated with specific physical parameters in order to elucidate their influence on the band gap regions. Boundary effects and damping are also investigated for a finite-dimensional array, revealing an erosion of band gaps that could limit the expected functionality.

Item Open Access Dynamics of Electronic Transport in Spatially-extended Systems with Negative Differential Conductivity(2010) Xu, HuidongNegative differential conductivity (NDC) is a nonlinear property of electronic transport for high electric field strength found in materials and devices such as semiconductor superlattices, bulk GaAs and Gunn diodes. In spatially extended systems, NDC can cause rich dynamics such as static and mobile field domains and moving charge fronts. In this thesis, these phenomena are studied theoretically and numerically for semiconductor superlattices. Two classes of models are considered: a discrete model based on sequential resonant tunneling between neighboring quantum wells is used to described charge transport in weakly-coupled superlattices, and a continuum model based on the miniband transport is used to describe charge transport strongly-coupled superlattices.

The superlattice is a spatially extended nonlinear system consisting a periodic arrangement of quantum wells (e.g., GaAs) and barriers (e.g., AlAs). Using a discrete model and only considering one spatial dimension, we find that the boundary condition at the injecting contact has a great influence on the dynamical behavior for both fixed voltage and transient response. Static or moving field domains are usually inevitable in this system. In order to suppress field domains, we add a side shunting layer parallel to the growth direction of the superlattice. In this case, the model includes both vertical and lateral spatial degrees of freedom. We first study a shunted weakly-coupled superlattice for a wide range of material parameters. The field domains are found to be suppressed for superlattices with small lateral size and good connection between the shunt and the quantum wells of the superlattice. As the lateral size of the superlattice increases, the uniform field configuration loses its stability to either static or dynamic field domains, regardless of shunt properties. A lower quality shunt generally leads to regular and chaotic current oscillations and complex spatio-temporal dynamics in the field profile. Bifurcations separating static and dynamic behaviors are characterized and found to be dependent on the shunt properties. Then we adopt the model to study the shunted strongly-coupled superlattice with the continuum model. Key structural parameters associated with both the shunt layer and SL are identified for which the shunt layer stabilizes a uniform electric field profile. These results support the possibility to realize a SL-based THz oscillator with a carefully designed structure.

Another important behavior of the static field domains in the weakly-coupled superlattice is bistability, i.e., two possible states (i.e., electric field configurations) for a single voltage. Noise can drive the system from one of these states (the metastable state) to the other one (the globally stable state). The process of escape from the metastable state can be viewed as a stochastic first-passage process in a high-dimensional system that possesses complex stability eigenvalues and for which a global potential energy function does not exist. This process is simulated using a stochastic differential equation system which incorporates shot noise. The mean switching time τ is fitted to an exponential expression e(Vth-V)α/D, where V

_{th}denotes the voltage at the end of the current branch. The exponent α in the fitting curve deviates from 1.5 which is predicted for a generic one dimensional system. We develop an algorithm to determine an effective locally valid potential. Principal component analysis is applied to find the most probable path for switching from the metastable current state.Item Open Access Numerical and Experimental Investigation of Multistable Systems(2013) Tweten, Dennis JeremyThe focus of this dissertation is on phenomena exhibited by multistable systems. Two phenomena of particular importance are chaos control and stochastic resonance. In this work, both models that can predict ordered responses and experiments in which ordered responses occur are explored. In addition, parameter identification methods are presented and improved.

Chaos control, when implemented with delays, can be an effective way to stabilize unstable periodic orbits within a multistable system experiencing a chaotic response. Delayed control is easy to implement physically but greatly increases the complexity of analyzing such systems. In this work, the spectral element method was adapted to evaluate unstable periodic orbits stabilized by feedback control implemented with delays. Examples are presented for Duffing systems in which the delay is equal to the forcing period. The spectral approach is also extended to analyze the control of chaos with arbitrary delays. Control with arbitrary delays can also be used to stabilize equilibria within the chaotic response. These methods for arbitrary delays are explored in self-excited, chaotic systems.

Stochastic resonance occurs in multistable systems when an increase in noise results in an ordered response. It is well known that noise excitation of multistable systems results in the system escaping from potential wells or switching between wells. In stochastic resonance, a small external signal is amplified due to these switching events. Methods for modeling stochastic resonance in both underdamped and overdamped systems are presented. In addition, stochastic resonance in a bistable, composite beam excited by colored noise is investigated experimentally. The experimental results are compared with analytical models, and the effect of modal masses on the analytical expressions is explored. Finally, an alternative approach for calculating the effect of colored noise excitation is proposed.

In order to implement analysis methods related to delay differential equations or stochastic resonance, the parameters of the system must be known in advance or determined experimentally. Parameter identification methods provide a natural connection between experiment and theory. In this work, the harmonic balance parameter identification method was applied to beam energy harvesters and is improved using weighting matrices. The method has been applied to a nonlinear, bistable, piezoelectric beam with a tip mass. Then, an experimental method of determining the number of restoring force coefficients necessary to accurately model the systems was demonstrated. The harmonic balance method was also applied to a bistable, beam system undergoing stochastic resonance. Finally, a new weighting strategy is presented based on the signal to noise ratio of each harmonic.

Item Open Access Using Reinforcement Learning and Bayesian Optimization on Problems in Vehicle Dynamics and Random Vibration Environmental Testing(2022) Manring, Levi HodgeTo accomplish the increasingly complex tasks that humans seek to achieve through technology, the advancement of the understanding and application of control systems is paramount for success. For relatively simple dynamic systems, model-based analytical control policies can be created without too much trouble (such as Proportional-Integral-Derivative (PID) or Linear-Quadratic-Regulator (LQR) controllers). However, for systems where the dynamics are very complex or even unknown, more advanced control techniques are necessary, especially when there is an interest in optimizing the control policy. This dissertation presents the application of nonlinear control methods to some challenging problems in vehicle automation and environmental testing.The first part of this dissertation presents the application of Reinforcement Learning (RL) to control a vehicle to get unstuck from a ditch. A simulation model of a vehicle moving on an arbitrary ditch surface was developed, with consideration of four different wheel-slip conditions. The transition between four state-spaces was developed as well as an integration routine to accurately integrate and switch between each of the four wheel-slip conditions. Two RL algorithms were applied to control the vehicle to escape the ditch – Probabilistic Inference for Learning COntrol (PILCO) and Deep Deterministic Policy Gradient (DDPG). PILCO was used to demonstrate the need of incorporating wheel-slip and the need for a neural network approach to capture all regions of the vehicle dynamic behavior. Reward functions were designed to incentivize the RL algorithms to achieve the desired goal. Both Rear-Wheel-Drive (RWD) and All-Wheel-Drive (AWD) simulation models were tested, and successful control policies achieved the goal of controlling the vehicle to get unstuck from the ditch while minimizing wheel-slip. Additionally, the control policies were tested over a wide range of ditch profile shapes, demonstrating a region of robustness. The second part of this dissertation presents a control solution in the area of environmental testing. In the area of environmental testing, there is an increasing demand for more challenging and aggressive environmental testing procedures. This dissertation presents a study on the convergence of the Matrix Power Control Algorithm (MPCA) for Random Vibration Control (RVC) testing, which is a particular type of environmental testing. A moving-average method was presented to reduce the control loop times and reduce the amplification of measurement noise. Additionally, Bayesian optimization was employed to optimize control parameters and the window size for the moving-average. An Euler-Bernoulli beam and the Box Assembly with Removable Component (BARC) structure were used in simulation and experiment, respectively, to demonstrate improvement in the convergence of MPCA over the baseline performance. In the experimental implementation, a LabVIEW controller was developed to implement the convergence improvements. This dissertation also presents a method for comparing Frequency Response Functions (FRFs), which is a data analysis problem in environmental testing. A Log-Frequency Shift (LFS) method was developed to shift a comparison FRF so that the dominant features (modes) of two FRFs were aligned. This then allowed the application of existing FRF comparison metrics with greater correlation with expert intuition. The Phase Similarity Metric (PSM) method was also introduced as an effective method for comparing the phases of two FRFs. These methods were demonstrated to be effective in simulation of an Euler-Bernoulli beam and validated using an experiment with random vibration applied to a thin beam.