# 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 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.