# Browsing by Subject "Nonlinear dynamics"

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Item Bubble pulsation and translation near a soft tissue interface(2014-05) Tengelsen, Daniel R. (Daniel Ross), 1983-; Hamilton, Mark F.Show more A Lagrangian formalism presented by Hay, Ilinskii, Zabolotskaya, and Hamilton [J. Acoust. Soc. Am. 132, 124--137 (2012)] to calculate the pulsation of a spherical bubble, immersed in liquid and near one or two viscoelastic layers, is extended here to include bubble translation. The method presented here is simplified from that given by Hay et al. in that only a single interface between a liquid and a viscoelastic half-space is considered. In the present approach the force on the bubble due to the presence of the liquid-solid interface is calculated using a Green's function that takes into account elastic waves and viscosity in the layer, and the viscous boundary layer within the liquid near the interface. Previous models and experiments have shown that the direction of bubble translation near a viscoelastic layer is correlated with the direction of a liquid jet often produced by the bubble during collapse. In this dissertation an attempt is made to model the pulsation and translation of a spherical bubble near a liquid-solid interface to infer the direction of bubble translation in reference to material parameters of the liquid and viscoelastic medium, and the standoff distance of the bubble from the interface. The analysis is simplified by demonstrating that the direction of bubble translation can be inferred from the phase of the component of the Green's function associated with the reverberant pressure gradient. For linear bubble pulsation it is shown that the domain of material properties of the viscoelastic medium which generally corresponds to bubble translation away from the interface occurs when the effective stiffness of the viscoelastic medium is greater than the effective damping for both itself and the liquid. The analysis is performed assuming the viscoelastic medium is similar to soft tissue, and its dynamics are described by a Voigt, Kelvin, or Maxwell model. The simulations are compared with existing experimental data. Effects of high-amplitude bubble pulsation are explored in terms of how the simulations differ as the pulsation amplitude increases. At higher pulsation amplitudes, it is shown that bubble translation is still described qualitatively by analyzing the phase of the reverberant pressure gradient.Show more Item Nonlinear Dynamics of a Rotor Supported by Homopolar Magnetic Bearings with Saturation(2011-02-22) Kang, KyungdaeShow more An objective in the design of high performance machinery is to minimize weight so magnetic bearings are often designed to operate slightly lower than the magnetic material saturation. Further weight reduction in the bearings requires operation in the nonlinear portion of the B-H curve. This necessitates a more sophisticated analysis at the bearing and rotordynamic system levels during the design stage. This dissertation addresses this problem in a unique manner by developing a fully nonlinear homopolar magnetic bearing model. The nonlinear dynamics of permanent magnet-biased homopolar magnetic bearing (PMB HoMB) system with 2-dof rigid and 4-dof flexible rotor is analyzed. The dynamic behavior of the rotor-bearing system is examined in the feedback control loop that includes low pass filter effects. An analytical magnetization curve model is proposed to predict the nonlinear magnetic force under the influence of the magnetic flux saturation more accurately. The modified Langmuir method with the novel correction terms for the weak flux region is used to curve-fit the experimental magnetization data of Hiperco 50. A new curve fit model of the B-H curve is shown to have significantly better agreement with the measured counterpart than conventional piecewise linear and other models. PMB HoMB characteristics with flux saturation, such as forces depending on the rotor position and bearing stiffness, are compared with these other models. Frequency response curve, bifurcation diagram, Poincare plot, and orbit plot are utilized to demonstrate the effects of the nonlinearities included in the 2-dof rotorbearing system. Due to heavy static loads applied to the rotor, it operates within the magnetic flux saturation region at the bearing clearance. The voltage saturation in the power amplifier of the magnetic bearing introduces lag in the control loop and the response of the heavily loaded 4-dof rotor-bearing system shows that limit cycle stability can be achieved due to the magnetic flux saturation or current saturation in the amplifier; otherwise the system would experience a destructive instability. These simulation results provide the first explanation of this commonly observed limit cycle which is referred to as ?virtual catcher bearings?.Show more Item Nonlinear dynamics of hysteretic oscillators(2009-05-15) Shekhawat, AshivniShow more The dynamic response and bifurcations of a harmonic oscillator with a hysteretic restoring force and sinusoidal excitation are investigated. A multilinear model of hysteresis is presented. A hybrid system approach is used to formulate and study the problem. A novel method for obtaining exact transient and steady state response of the system is discussed. Simple periodic orbits of the system are analyzed using the KBM method and an analytic criterion for existence of bound and unbound resonance is derived. Results of KBM analysis are compared with those from numerical simulations. Stability and bifurcations of higher period orbits are studied using Poincar?e maps. The Poincar?e map for the system is constructed by composing the corresponding maps for the individual subsystems of the hybrid system. The novelty of this work lies in a.) the study of a multilinear model of hysteresis, and, b.) developing a methodology for obtaining the exact transient and steady state response of the system.Show more Item Scaling and instability of dynamic fracture(2014-05) Chen, Chih-Hung, active 21st century; Marder, Michael P., 1960-Show more This dissertation presents three inter-related studies. Chapter 2 presents a study of scaling of crack propagation in rubber sheets. Two different scaling laws for supersonic and subsonic cracks were discovered. Experiments and numerical simulations have been conducted to investigate subsonic and supersonic cracks. The experiments are performed at 85 °C to suppress strain-induced crystallites that complicate experiments at lower temperature. Calibration experiments were performed to obtain the parameters needed to compare with a theory including viscous dissipation. Both experiments and numerical simulations support supersonic cracks, and a transition from subsonic to supersonic is discovered in the plot of experimental crack speed curves versus extension ratio for different sized samples. Both experiments and simulations show two different scaling regimes: the speed of subsonic cracks scales with the elastic energy density while the speed of supersonic cracks scales with the extension ratio. Crack openings have qualitatively different shapes in the two scaling regimes. Chapter 3 describes a theory of oscillating cracks. Oscillating cracks are not seen very widely, but observed in rubber and gels. A theory has been proposed for the onset of oscillation in gels, but the oscillation of cracks in rubber has not been explained. This study provides a theory able to describe both rubber and gels and recover the experimental phase diagram for oscillating cracks in rubber. The main new idea is that the oscillations of cracks follow from basic features of fracture mechanics and are independent of details of the crack equation of motion. From the fact that oscillations exist, one can deduce some conditions on forms that equations of motion can take. A discrete model of hydraulic fracture is mentioned in Chapter 4. Hydraulic fracturing is a stimulation treatment wherein fluids are injected into reservoirs under high pressure to generate fractures in reservoirs. In this study, a lattice-based pseduo-3D model is developed to simulate hydraulic fracturing. This mode has been validated via a comparison with the KGD model. A series of pilot simulations was systematically tested for complex geometries under more realistic operation conditions, including flexible boundary conditions, randomness in elastic properties of shales and perforations. The simulation results confirm that perforation is likely to increase the complexity of fracture networks; the results also suggest that the interference between neighboring fractures is key to fracture network formation.Show more Item Stability and dynamics of systems of interacting bubbles with time-delay and self-action due to liquid compressibility(2012-08) Thomas, Derek Clyde; Hamilton, Mark F.; Fahrenthold, Eric P.; Ilinskii, Yurii A.; Kallivokas, Loukas F.; Wilson, Preston S.Show more A Hamiltonian model for the radial and translational dynamics of clusters of coupled bubbles in an incompressible liquid developed by Ilinskii, Hamilton, and Zabolotskaya [J. Acoust. Soc. Am. 121, 786-795 (2007)] is extended to included the effects of compressibility in the host liquid. The bubbles are assumed to remain spherical and translation is allowed. The two principal effects of liquid compressibility are time delay in bubble interaction due to the finite sound speed and radiation damping due to energy lost to acoustic radiation. The incorporation of time delays produces a system of delay differential equations of motion instead of the system of ordinary differential equations in models of bubble interaction in an incompressible medium. The form of the Hamiltonian equations of motion is significantly different from the commonly used models based on Rayleigh-Plesset equations for coupled bubble dynamics, and it provides certain advantages in numerical integration of the time-delayed equations of motion. Corrections for radiation damping in clusters of interacting bubbles are developed in the form of a time-delayed expression for bubble self-action following the method of Ilinskii and Zabolotskaya [J. Acoust. Soc. Am. 92, 2837-2841 (1992)]. A set of approximate series expansions of this delayed expression is calculated to first order in the ratio of bubble radius to the characteristic wavelength of acoustic radiation from the bubble, and to varying orders in the ratio of bubble radius to characteristic bubble separation distance. Stability of the delay differential equations of motion is analyzed with four successive levels of approximation for the effects of radiation damping and time delay. The stability is analyzed with and without the effects of viscous and thermal damping. The effect of time delay and radiation damping on the pressure radiated by small systems of bubbles is considered. An approximate method to account for the delays in bubble interaction in a weakly compressible liquid is presented. This method converts the system of delay differential equations into an approximate system of ordinary differential equations, which may simplify numerical integration. Several sets of model equations incorporating propagation time delay in bubble interactions are solved numerically with existing algorithms specialized for delay differential equations. Numerical simulations of the dynamics of single bubbles, pairs of bubbles, and clusters of bubbles are used to compare the different levels of approximation for compressibility effects for low- and high-amplitude radial motion in systems of bubbles under free response and pulsed excitation by an external pressure source.Show more Item Transport in higher dimensional phase spaces(2016-12) Curry, Christopher Timothy; Morrison, Philip J.; Horton, Jr., Claude W; Hazeltine, Richard; Matzner, Richard; Gamba, IreneShow more We use a four dimensional symplectic mapping, the coupled cubic-quadratic map, to provide evidence of Arnol’d Diffusion in phase space. We use the method of frequency analysis for dynamical systems to demonstrate the existence of regular orbits, and show that these orbits enclose weakly chaotic orbits which escape in finite time around the tori. A new collocation method for frequency analysis is employed by adapting it to allow for higher precision results. Arbitrary precision numerics are used to obtain highly accurate orbits for long timescales, and the adapted frequency method is used to obtain highly accurate frequencies of the mapping. We review the method of frequency analysis, demonstrate its effectiveness and accuracy in determining frequencies and finding tori in simple systems and low-dimensional mappings, and extend the results to higher dimensions. In the four dimensional mapping, we find several regular orbits with irrational frequency ratios, indicating the existence of tori in the phase space, as well as interior orbits that escape around these tori.Show more