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<p>  The wave dynamics within a rotating detonation engine were investigated using a combustor where the fuel injector was varied continuously relative to the oxidizer throat. Both natural gas and a hydrocarbon fuel blend containing the major components of a "cracked" kerosene fuel were characterized using high speed imaging, pressure sensors, and photomultiplier tubes. Major detonation features were visualized with high-speed cameras through a 360 optical outerbody. The detonation region, oblique shock, contact surface where fresh reactants mixed with products of a previous wave, and burning above the fuel injectors in a stratified zone beneath the detonation wave were studied as fuel conditions and fuel injector position were changed. As the inner body of the engine was translated away from the oxidizer throat, or started at a position far from the oxidizer throat, the combustor was not able to support coherent detonation behavior. At these points, the region of highest heat release remained close to the fuel injectors, and there was very little heat release processed behind the front edge of the wave compared to the level of deflagrative combustion occurring inside the chamber. The surrogate hydrocarbon blend is more representative of a composition that high speed vehicles would use, so the operability limits of the fuel and the fuel with nitrogen dilution were characterized using a metal and an optical outerbody on the combustor. With a larger amount of ethylene in the fuel composition compared to the amount of methane, the chamber tended towards slower waves and higher wave modes, and the combustor was able to sustain a coherent detonative mode with up to 40% nitrogen. When all chosen fuel blend components were present in the fuel except ethane, the combustion kinetics of the fuel was slowed significantly, and there was a measured decrease in thrust. No fuel tested was able to support coherent detonative modes with 50% nitrogen in the oxidizer.  </p>

The problem of finding the stress distribution about circular, noncentral holes in a spinning disk is discussed. A theoretical approximation is first developed using the methods of elasticity and combining solutions to similar problems from Ling (reference 6) and Timoshenko and Goodier (reference 2). Results are then found in the form of a maximum stress concentration factor which occurs at the edges of the holes. The accuracy of the derived solution is checked by obtaining values photoelastically from a similar spinning disk. Stress patterns are studied in photographs obtained with a strobeflash and camera. The results are presented in a usable form for application to design. / Master of Science

A vehicle has several rotating components such as a traction electric motor, the driveline, and the wheels and tires. The rotating inertia of these components is important in vehicle performance analyses. However, in many studies, the rotating inertias are typically lumped into an equivalent inertial mass to simplify the analysis, making it difficult to investigate the effect of those components and losses for vehicle energy use. In this study, a backward-tracking model from the wheels and tires to the power source (battery or fuel cell) is developed to estimate the effect of rotating inertias for each component during propulsion and regenerative braking of a vehicle. This paper presents the effect of rotating inertias on the power and energy for propulsion and regenerative braking for two-wheel drive (either front or rear) and all-wheel drive (AWD) cases. On-road driving and dynamometer tests are different since only one axle (two wheels) is rotating in the latter case, instead of two axles (four wheels). The differences between an on-road test and a dynamometer test are estimated using the developed model. The results show that the rotating inertias can contribute a significant fraction (8 -13 %) of the energy recovered during deceleration due to the relatively lower losses of rotating components compared to vehicle inertia, where a large fraction is dissipated in friction braking. In a dynamometer test, the amount of energy captured from available energy in wheel/tire assemblies is slightly less than that of the AWD case in on-road test. The total regenerative brake energy capture is significantly higher (> 70 %) for a FWD vehicle on a dynamometer compared to an on-road case. The rest of inertial energy is lost by inefficiencies in components, regenerative brake fraction, and friction braking on the un-driven axle. / Master of Science

There are many limitations to the current methods used to measure vibration on rotating structures. These limitations include physical flow blockages, relating the measurement spot to the structure rotation, data processing issues, and having to physically alter the engine. This work further describes aspects of a self-tracking laser vibrometry system that can be used to measure the vibrations of rotating structures. This method, if setup correctly, has the capability to overcome many of the limitations listed above. A study of all misalignment effects is presented in this thesis. The study consists of a parametric sensitivity analysis of misalignment variables, a parametric Monte Carlo analysis of misalignment variables, and a full interaction Monte Carlo analysis of misalignment variables. In addition, the results of the misalignment variable analyses were used to develop a self-tracker test rig for obtaining fan vibration from a Pratt and Whitney JT15D turbofan engine. A prototype this test rig was designed, built, and tested on the turbofan. It was found that in order to achieve acceptable amounts of position and velocity error using the self-tracker LDV system, very strict alignment of the optical equipment is necessary. Additionally, the alignment criteria can likely be achieved with the use of digitally controlled high precision linear motion equipment. / Master of Science

Detonation cycles are identified as an efficient alternative to the Brayton cycles used in power and propulsion applications. Rotating Detonation Engine (RDE) operating on a detonation cycle works by compressing the working fluid across a detonation wave, thereby reducing the number of compressor stages required in the thermodynamic cycle. Numerical analyses of RDEs are flexible in understanding the flow field within the RDE, however, three-dimensional analyses are expensive due to the differences in time-scale required to resolve the combustion process and flow-field. The alternate two-dimensional analyses are generally modeled with perfectly premixed fuel injection and do not capture the effects of improper mixing arising due to discrete injection of fuel and oxidizer into the chamber. To model realistic injection in a 2-D analysis, the current work uses an approach in which, a Probability Density Function (PDF) of the fuel mass fraction at the chamber inlet is extracted from a 3-D, cold-flow simulation and is used as an inlet boundary condition for fuel mass fraction in the 2-D analysis. The 2-D simulation requires only 0.4% of the CPU hours for one revolution of the detonation compared to an equivalent 3-D simulation. Using this method, a perfectly premixed RDE is comparing with a non-premixed case. The performance is found to vary between the two cases. The mean detonation velocities, time-averaged static pressure profiles are found to be similar between the two cases, while the local detonation velocities and peak pressure values vary in the non-premixed case due to local pockets fuel rich/lean mixtures. The mean detonation cell sizes are similar, but the distribution in the non-premixed case is closer due to stronger shock structures. An analytical method is used to check the effects of fuel-product stratification and heat loss from the RDE and these effects adversely affect the local detonation velocity. Overall, this method of modeling captures the complex physics in an RDE with the advantage of reduced computational cost and therefore can be used for design and diagnostic purposes. / Master of Science / The conventional Brayton cycle used in power and propulsion applications is highly optimized, at cycle and component levels. In pursuit of higher thermodynamic efficiency, detonation cycles are identified as an efficient alternative and gained increased attention in the scientific community. In a Rotating Detonation Engine (RDE), which is based on the detonation cycle, the compression of gases occurs across a shock wave. This method of achieving high compression ratios reduces the number of compressor stages required for operation. In an RDE (where combustion occurs between two coaxial cylinders), the fuel and oxidizer are injected axially into the combustion chamber where the detonation is initiated. The resultant detonation wave spins continuously in the azimuthal direction, consuming fresh fuel mixture. The combustion products expand and exhaust axially providing thrust/mechanical energy when coupled with a turbine. Numerical analyses of RDEs are flexible over experimental analysis, in terms of understanding the flow physics and the physical/chemical processes occurring within the engine. However, three-dimensional numerical analyses are computationally expansive, and therefore demanding an equivalent, efficient two-dimensional analysis. In most RDEs, fuel and oxidizer are injected from separate plenums into the chamber. This type of injection leads to inhomogeneity of the fuel-air mixture within the RDE which adversely affects the performance of the engine. The current study uses a novel method to effectively capture these physics in a 2-D numerical analysis. Furthermore, the performance of the combustor is compared between perfectly premixed injection and discrete, non-premixed injection. The method used in this work can be used for any injector design and is a powerful/efficient way to numerically analyze a Rotating Detonation Engine.

The detonation-based engine has the key advantage of increased thermodynamic efficiency over the traditional constant pressure combustor. These detonation-based engines are also known as Pressure Gain Combustion systems (PGC) and Rotating Detonation Combustor (RDC) is a form of PGC, in which the detonation wave propagates azimuthally around an annular combustor. Prior researchers have performed a high fidelity 3-D numerical simulation of a rotating detonation combustor (RDC) to understand the flow physics such as detonation wave velocity, pressure profile, wave structure; however, performing these 3-D simulations is computationally expensive. 2-D simulations are a potential alternative to reduce computational cost. In most RDCs, fuel and oxidizer are injected discretely from separate plenums, and this discrete fuel/air injection results in inhomogeneous mixing within the domain. Due to the discrete fuel injection locations, fuel/oxidizer will stratify to form localized pockets of rich and lean mixtures. The motivation of the present study is to investigate the impact of unmixedness and stratification length scales on the performance of an RDC using a 2-D numerical approach. Unmixedness, which is defined as the standard deviation of equivalence ratio normalized by the mean global equivalence ratio, is a measure of the degree of fuel-oxidizer inhomogeneity. To model the effect of unmixedness in a 2-D domain, a lognormal distribution of the fuel mass fraction is generated with a mean equivalence ratio of 1 and varying standard deviations at the inlet boundary as a numerical source term. Moreover, to model the effects of stratification length scales, fuel mass fraction at the inlet boundary cells is bundled for a given length scale, and the mass fractions for these bundles are updated based on the lognormal distribution after every three-time steps. Using this methodology, 2-D numerical analyses are carried out to investigate the performance of an RDC for an H2-air mixture with varying unmixedness and stratification length scales. Results show that mean detonation velocity decreases and wave speed variation increases with an increase in unmixedness. However, with an increase in stratification length scale mean velocity remain relatively unchanged but variation in local velocity increases. The detonation wave front corrugation also increases with an increase in mixture inhomogeneity. The mean detonation cell size increases with an increase in unmixedness. The cell shape becomes more distorted and irregular with an increase in stratification length scale and unmixedness. The combined effect of unmixedness and stratification length scale leads to a decrease in pressure gain. Overall, this concept is able to elucidate the effects of varying unmixedness and stratification length scales on the performance of an RDC. / Master of Science / Pressure Gain Combustion (PGC) system has gained significant focus in recent years due to its increased thermodynamic efficiency over a constant pressure Brayton Cycle. Rotating Detonation Combustor (RDC) is a type of PGC system, which is thermodynamically more efficient than the conventional gas turbine combustor. One of the main aspects of the detonation process is the rapid burning of the fuel-oxidizer mixture, which occurs so fast that there is not enough time for pressure to equilibrate. Therefore, the process is thermodynamically closer to a constant volume process rather than a constant pressure process. A constant volume cycle is thermodynamically more efficient than a constant pressure Brayton cycle. In an RDC, a mixture of fuel and air is injected axially, and a detonation wave propagates continuously through the circumferential section. Numerical simulation of an RDC provides additional flexibility over experiments in understanding the flow physics, detonation wave structure, and analyzing the physical and chemical processes involved in the detonation cycle. Prior researchers have utilized a full-scale 3-D numerical simulation for understanding the performance of an RDC. However, the major challenge with 3-D analyses is the computational expense. Thus, to overcome this, an inexpensive 2-D simulation is used to model the flow physics of an RDC. In most RDCs, the fuel and oxidizer are injected discretely from separate plenums. Due to the discrete fuel injection, the fuel/air mixture is never perfectly premixed and results in a stratified flow field. The objective of the current work is to develop a novel approach to independently investigate the effects of varying unmixedness and stratification length scales on RDC performance using a 2-D simulation.

This thesis deals with the study of two problems in the area of nonlinear dynamics using the method of multiple scales. Accordingly, it consists of two parts. In the first part of the thesis, we explore the asymptotic stability of a planar two-degree- of-freedom robot with two rotary (R) joints following a desired trajectory under feedback control. Although such robots have been extensively studied and there exists stability and other results for position control, there are no analytical results for asymptotic stability when the end of the robot or its joints are made to follow a time dependent trajectory. The nonlinear dynamics of a 2R planar robot, under a proportional plus derivative (PD) and a model based computed torque control, is studied. The method of multiple scales is applied to the two nonlinear second-order ordinary deferential equations which describes the dynamics of the feedback controlled 2R robot. Amplitude modulation equations, as a set of four first order equations, are derived. At a fixed point, the Routh-Hurwitz criterion is used to obtain positive values of proportional and derivative gains at which the controller is asymptotically stable or indeterminate. For the model based control, a parameter representing model mismatch is incorporated and again controller gains are obtained, for a chosen mismatch parameter value, where the controller results in asymptotic stability or is indeterminate. From numerical simulations with gain values in the indeterminate region, it is shown that for some values and ranges of the gains, the non- linear dynamical equations are chaotic and hence the 2R robot cannot follow the desired trajectory and be asymptotically stable. The second part of the thesis deals with the study of the nonlinear dynamics of a rotating flexible link, modeled as a one dimensional beam, undergoing large deformation and with geometric nonlinearities. The partial deferential equation of motion is discretized using a finite element approach to yield four nonlinear, non-autonomous and coupled ordinary deferential equations. The equations are non-dimensional zed using two characteristic velocities – the speed of sound in the material and a speed associated with the trans- verse bending vibration of the beam. The method of multiple scales is used to perform a detailed study of the system. A set of four autonomous equations of the first-order are derived considering primary resonance of the external excitation with one of the natural frequencies of the model and one-to-one internal resonance between two different natural frequencies of the model. Numerical simulations show that for certain ranges of values of these characteristic velocities, the slow flow equations can exhibit chaotic motions. The numerical simulations and the results are related to a rotating wind turbine blade and the approach can be used for the study of the nonlinear dynamics of a single link flexible manipulator. The second part of the thesis also deals with the synchronization of chaos in the equations of motion of the flexible beam. A nonlinear control scheme via active nonlinear control and Lyapunov stability theory is proposed to synchronize the chaotic system. The proposed controller ensures that the error between the controlled and the original system asymptotically go to zero. A numerical example using parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.

<p dir="ltr">Rotating detonation combustors are a developing technology with the potential to successfully integrate pressure gain combustion in to modern propulsion devices. Utilization of propagating detonation waves could increase combustion cycle efficiency and reduce combustor size, resulting in an overall increase in system range or payload-carrying capabilities. However, the sensitivity of rotating detonation combustor operation and performance to geometric features, such as injector configuration or chamber length, still needs to be characterized over a wide range of operating conditions. In addition, the hardware configuration that promotes easy ignition into a coherent detonation operating mode does not always maximize combustor performance, especially at low-loss conditions where feedback between chamber and manifold dynamics can exist. Therefore, a rotating detonation combustor with continuously variable geometry capabilities was designed in order to continuously vary any number of hardware design parameters during combustor testing. Not only does the variable geometry combustor enable rapid characterization of operability sensitivity with minimal hardware swaps, it also enables exploration of hysteresis in performance as the combustor is ignited in one configuration and transitioned to a different geometry while maintaining detonative operation.</p><p dir="ltr">The operability of the variable geometry rotating detonation combustor was first characterized with variable fuel injector location. Higher wave speeds were observed at injector locations closest to the oxidizer throat, with decreased wave speed and eventual transition to deflagrative operation occurring at locations farther downstream due to increasing momentum flux ratio. Variation in fuel injection location induced bifurcations in the number of waves, resulting in corresponding changes in wave speed and gross thrust. Hysteresis was observed in these quantities as the direction of injector translation was reversed. Active translation promoted detonative operation of the experiment at conditions and configurations that hitherto operated only in a deflagrative mode with fixed combustor geometry. </p><p dir="ltr">Sensitivity of rotating detonation combustor operation and performance to oxidizer injector pressure drop was characterized using continuous variation of the injector area during combustor operation. Propulsive performance of the combustor was evaluated using thrust and equivalent available pressure, relating them back to reactant supply pressures for assessment of combustor pressure gain. An effective reactant supply pressure was developed in order to combine contributions of both fuel and oxidizer manifold pressures to the total pressure of the system so that pressure gain could be accurately calculated. Pressure gain increased during a test as oxidizer injector area was increased and the corresponding manifold pressure was decreased. At larger injector areas, pressure gain decreased as the operating mode of the combustor transitioned from detonation to deflagration, concomitant with reduction of gross thrust. Modeling of injector recovery time revealed that the injector operated in both choked and unchoked regimes, which was used to explain detonation wave number transitions in the experiment. A broadened range of detonative operability enabled by active variation of combustor geometry resulted in higher performance with lower injector pressure drop.</p><p dir="ltr">Sensitivity of rotating detonation combustor operation and performance to combustor chamber length was characterized using continuous variation of the chamber length during combustor operation. Specific impulse of the combustor remained relatively constant as chamber length was decreased from its maximum values, proving the practicality of efficient packaging for rotating detonation combustors. A limiting chamber length at which combustion could not longer be supported within the chamber was found to exist for every operating condition, resulting in flame blow-out and performance degradation. Modeling of detonation fill height revealed that relatively low specific impulse measurements could be attributed to unburned reactants exiting the chamber, and a more efficient use of reactants was potentially the cause for improved performance at higher mass flow rates as detonation wave number increased and reactant residence time decreased.</p><p dir="ltr">This experiment and the associated analysis has helped further characterize rotating detonation combustor sensitivity to hardware design parameters. The continuously variable geometry capabilities enabled precise identification of geometric parameters that resulted in operating mode transitions. Analysis and modeling of the flow processes within the injector and chamber were used to help explain why these mode transitions occurred, and can be used for future rotating detonation combustor development.</p>

Rotating Euler-Bernoulli beams and non-homogeneous Timoshenko beams are widely used to model important engineering structures. Hence the vibration analyses of these beams are an important problem from a structural dynamics point of view. The governing differential equations of both these type of beams do not yield any simple closed form solutions, hence we look for the inverse problem approach in determining the beam property variations given certain solutions. Firstly, we look for a rotating beam, with pinned-free boundary conditions, whose eigenpair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flexural stiffness function (FSF) for which the ith mode eigenpair of a rotating beam with uniform mass distribution, is identical to that of a corresponding non-rotating beam with same length and mass distribution. Inserting these derived FSF's in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realistic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams. Secondly, we study the free vibration of rotating Euler-Bernoulli beams, under cantilever boundary condition. For certain polynomial variations of the mass per unit length and the flexural stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation. It is found that there are an infinite number of rotating beams, with various mass per unit length variations and flexural stiffness distributions, which share the same fundamental frequency and mode shape. The derived flexural stiffness polynomial functions are used as test functions for rotating beam numerical codes. They are also used to design rotating cantilever beams which may be required to vibrate with a particular frequency. Thirdly, we study the free vibration of non-homogeneous Timoshenko beams, under fixed-fixed and fixed-hinged boundary conditions. For certain polynomial variations of the material mass density, elastic modulus and shear modulus, there exists a fundamental closed form solution to the coupled second order governing differential equations. It is found that there are an infinite number of non-homogeneous Timoshenko beams, with various material mass density, elastic modulus and shear modulus distributions, which share the same fundamental frequency and mode shape. They can be used to design non-homogeneous Timoshenko beams which may be required for certain engineering applications.