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Articles 68941 - 68970 of 77678

Full-Text Articles in Engineering

Development Of Computer Software To Aid Environmental Decision Makers In The Education And Training Of Air Force Remedial Project Managers, Roger R. Ouellette, Bruce K. Lyman Dec 1993

Development Of Computer Software To Aid Environmental Decision Makers In The Education And Training Of Air Force Remedial Project Managers, Roger R. Ouellette, Bruce K. Lyman

Theses and Dissertations

By the end of Fiscal Year 1991, the Air Force had identified 4,354 contaminated sites. Much like the 177 toxic sites at McClellan AFB, bases across the country and throughput the world are filled with chrome, lead, paint solvents and many other toxics that are rendering water supplies unusable and endangering the health of millions of people living in and around these bases. Air Force officials continue to identify contaminated sites with no end in sight. Since the cleanup of these toxic materials involves diverse, complex activities, the Air Force created the duty position of Remedial Project Manager to manage …

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Simultaneous Design Of Active Vibration Control And Passive Damping, Michele Lynn Devereaux Gaudreault Dec 1993

Simultaneous Design Of Active Vibration Control And Passive Damping, Michele Lynn Devereaux Gaudreault

Theses and Dissertations

A modified form of the standard linear quadratic regulator LQR cost functional is used to optimally blend active vibration control and passive structural damping, whether viscous or viscoelastic. Viscoelastic damping is first modelled by a standard linear model, and then by a fractional derivative model. For the viscous damping case and the classically modelled viscoelastic damping case, a sub-optimal closed form solution is derived that is independent of the initial conditions. An iterative technique that minimizes the average value of the cost functional and one that minimizes the maximum value are developed. Both techniques are applicable to viscous and viscoelastic …

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Development And Implementation Of A Scramjet Cycle Analysis Code With A Finite-Rate-Chemistry Combustion Model For Use On A Personal Computer, Clarence F. Chenault Dec 1993

Development And Implementation Of A Scramjet Cycle Analysis Code With A Finite-Rate-Chemistry Combustion Model For Use On A Personal Computer, Clarence F. Chenault

Theses and Dissertations

This study compared the performance of an equilibrium combustion model to a finite-rate-chemistry combustion model for a fixed geometry Scramjet flying at the flight conditions of Mach 12, 15, and 18 with a constant dynamic pressure of 50,000 Pa. An integrated PC-based code, developed specifically for this study, models the combustor as an equilibrium combustion process or as finite-rate-chemistry combustion process. This integrated program is based on two existing programs, Ramjet Performance Analysis (RJPA) and 3 STREAM. The effects of mixing schedule, combustor length, and combustor exit area were investigated. The results of this study indicate that inefficient mixing is …

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Investigation Of Compressive Behavior For Metal Matrix Composites With A Circular Hole, Janet L. Gooder Dec 1993

Investigation Of Compressive Behavior For Metal Matrix Composites With A Circular Hole, Janet L. Gooder

Theses and Dissertations

This study investigated the notched and unnotched macromechanical response due to micromechanical behavior of [0/±45/90]S SCS-6/β 21 S and [0/90] 2S Sigma/β 21S subjected to static compressive loads at room temperature. The stress-strain behavior and mechanical response in the presence of a circular hole was examined. A clear description of the progressive nature of damage, which ultimately resulted in failure, was provided. The effect of elevated temperature was shown for SCS-6/β 21S. The ultimate compressive strength and elastic modulus were reported for both MMCs. Sigma/β 21S and SCS-6/β 21S were mildly notch sensitive at room temperature. Notch sensitivity of …

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Fatigue Behavior Of A Cross-Ply Ceramic Matrix Composite Under Strain Controlled Tension-Tension And Tension-Compression Loading, James J. Gudaitis Dec 1993

Fatigue Behavior Of A Cross-Ply Ceramic Matrix Composite Under Strain Controlled Tension-Tension And Tension-Compression Loading, James J. Gudaitis

Theses and Dissertations

This stuidy investigates the fatigue response of a cross-ply [0/90]2s Nicalon/Calcium Aluminosillicate (Nicalcc/CAS) ceramic matrix composite at room temperature under strain controlled tension-tension and tension-compression fatigue loading. The primary objectives were to determine strain fatigue limits for both loading cases and to complete a fatigue life diagram (Δe-N) for all tests. Failure mechanisms were also studied. The average initial modulus value for this lay-up was 120 GPa. The ultimate tensile strain was 0.00988 mn/mm (285 MPa). Eight tests were conducted with maximum strains that ranged from 0.00300 mm/mm to 0.00722 mm/mm. The tension-tension fatigue limit was found to be …

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Fatigue Behavior Of A Cross-Ply Ceramic Matrix Composite At Elevated Temperatures Under Tension-Tension Loading, Richard J. Tuznik Dec 1993

Fatigue Behavior Of A Cross-Ply Ceramic Matrix Composite At Elevated Temperatures Under Tension-Tension Loading, Richard J. Tuznik

Theses and Dissertations

A study was conducted which investigated the behavior of a Nicalon/ Calcium-Aluminosilicate (Nicalon/CAS) cross-ply [0/90]2S ceramic matrix composite at elevated temperatures under tension-tension fatigue loading. Tension-tension fatigue tests were performed with a load ratio of R = 0.10 and a frequency of 10 Hertz at 700°C and 850°C. These results were compared to results from a previous study conducted at room temperature. Material behavior and damage was recorded by fatigue life curves, elastic modulus' maximum and minimum strain, acetate replication techniques, and post-mortem fractography. Analysis of these results showed identical matrix dominated damage behavior at RT and 700°C. Ultimate …

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Radar Cross Section Visualization Using Sample Buffer Progressive Refinement Volume Rendering, Alain L. Jones Dec 1993

Radar Cross Section Visualization Using Sample Buffer Progressive Refinement Volume Rendering, Alain L. Jones

Theses and Dissertations

This study developed a prototype for an interactive radar cross section visualization software system. The system, hosted on a Silicon Graphics workstation, is intended to support aircrews, mission planners, aircraft designers, and others who require an understanding of aircraft radar cross section characteristics. The input to the system is a set of radar cross section samples taken at various aspect angles. A pre-processor developed as part of this study transforms the input radar cross section data into a three-dimensional cuberille data volume. This data volume is then visualized using an interactive volume renderer. The interactive volume renderer implements progressive image …

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Synthetic Battlebridge: Information Visualization And User Interface Design Applications In A Large Virtual Reality Environment, Kirk G. Wilson Dec 1993

Synthetic Battlebridge: Information Visualization And User Interface Design Applications In A Large Virtual Reality Environment, Kirk G. Wilson

Theses and Dissertations

With shrinking budgets and fewer personnel, future military training will rely heavily on simulated environments. The goal for this training is to reduce cost while maintaining readiness and unparalleled capability for all levels of military command. This thesis effort, the Synthetic Battle Bridge SBB , provides a real-time simulated environment for military commanders to observe on-going computer simulations of varying participation levels and helps them wring the most from a simulation. The SBB, designed for higher ranking personnel with little time to spend learning how to run the system, must exhibit three capabilities ease-of-use, long-term retention, and adaptability. Based on …

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Enhanced Tracking Of Airborne Targets Using Forward Looking Infrared And Laser Return Measurements, Patrick J. Grondin Dec 1993

Enhanced Tracking Of Airborne Targets Using Forward Looking Infrared And Laser Return Measurements, Patrick J. Grondin

Theses and Dissertations

The Air Force Institute of Technology has been involved in developing Kalman filter based trackers of ballistic missiles for 15 years. The goal of this thesis is to develop a Multiple Model Adaptive Estimator MMAE that tracks the missile plume using a forward looking infrared sensor and the missile hardbody center-of-mass additionally using low energy laser returns for the purpose of directing a high power laser to incapacitate the missile. The missile plume pogos about an offset equilibrium point relative to the hardbody center-of-mass with an amplitude and frequency of oscillation that are not precisely known a priori. The MMAE …

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Filtering, Coding, And Compression With Malvar Wavelets, Stephen R. Hall Dec 1993

Filtering, Coding, And Compression With Malvar Wavelets, Stephen R. Hall

Theses and Dissertations

This thesis develops and evaluates a number of new concepts and tools for the analysis of signals using Malvar wavelets lapped orthogonal transforms . Windowing, often employed as a spectral estimation technique, can result in irreparable distortions in the transformed signal. By utilizing the Malvar wavelet, any signal distortion resulting from the transformation can be eliminated or cancelled during reconstruction. This is accomplished by placing conditions on the window and the basis function and then incorporating the window into the orthonormal representation. Paradigms for both a complex-valued and a real-valued Malvar wavelet are summarized. I-lie algorithms for the wavelets were …

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Development Of Stereoscopic Particle Tracking Velocimetry For Turbulent Flow Field Diagnostics, Veera P. Rajendran Dec 1993

Development Of Stereoscopic Particle Tracking Velocimetry For Turbulent Flow Field Diagnostics, Veera P. Rajendran

Electronic Theses and Dissertations

Stereoscopic particle tracking velocimetry (SPTV) was used to analyze a turbulent submerged jet of Reynolds number (based on the diameter of the nozzle) 8000. SPTV involves tracking the motion of tracer particles over time in a flow in three dimensions. The first part of this work consisted of developing a stereoscopic particle tracking technique using two high-speed video cameras. This included calibration of the cameras with "known" points, and developing an algorithm to automatically match and track particles. The second part was to apply the same to a submerged water jet in order to measure velocities. The velocities were subsequently …

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Nitric Oxide Production: A Mechanism For Inhibition Of Chlamydia Trachomatis Replication, Bojun Chen Dec 1993

Nitric Oxide Production: A Mechanism For Inhibition Of Chlamydia Trachomatis Replication, Bojun Chen

Electronic Theses and Dissertations

Chlamydia trachomatis (CT) replicates in macrophages, but is inhibited by IFN-$\gamma$ or LPS. IFN-$\gamma$ and/or LPS induced nitrite production in mouse peritoneal macrophages, macrophage cell lines (RAW264.7 and J774A.1) and McCoy cells. Kinetic studies indicated that peak production occurred 48 hours post-treatment. CT infection itself was insufficient to induce nitrite production, but resulted in enhancement of nitrite production in IFN-$\gamma$-treated cells. Treatment with IFN-$\gamma$ or LPS resulted in significant inhibition of CT replication in these cells. Strong correlation between nitrite production and inhibition of CT replication was observed in RAW264.7 and J774A.1 cells (correlation coefficients: $-$0.93 and $-$0.94, p $<$ 0.001). N$\sp{\rm g}$- monomethyl-L-arginine (L-NMMA) specifically inhibited nitrite production and partially reversed inhibition of CT replication in macrophage cell lines. NOS mRNA was measured in RAW264.7 cells by Northern blot and Dot blot hybridization. Strong correlation between NOS mRNA expression and inhibition of CT replication (correlation coefficient: $-$0.97, p $<$ 0.05) was observed. Anti-TNF-$\alpha$ antibody completely neutralized the biological activity of TNF-$\alpha$ secreted by LPS-treated RAW264.7 cells, yet the antibody neither reduced nitrite production nor restored CT replication. Combination of the antibody and L-NMMA significantly enhanced restoration of CT replication. In peritoneal macrophages, inhibition of CT replication induced by IFN-$\gamma$ was partially restored by L-NMMA or anti-TNF-$\alpha$ antibody. In McCoy cells, inhibition of CT replication induced by IFN-$\gamma$ and LPS was not significantly restored by L-NMMA. Great restoration of CT replication by 1 mM L-NMMA was observed in LPS-treated J774A.1 cells (31%), but not in IFN-$\gamma$-treated cells (5%). Our data indicate that (1) NO production is one of the mechanisms for inhibition of CT replication in IFN-$\gamma$-activated peritoneal macrophages and RAW264.7 cells; (2) NO plays a significant role in CT inhibition in LPS-treated macrophage cell lines, but not peritoneal macrophages; (3) TNF-$\alpha$ may be associated with inhibition, but the mechanism(s) may not involve NO production; (4) NO production may not be the mechanism for CT inhibition in McCoy cells treated with IFN-$\gamma$ and LPS.

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Primary Coolant Flow Characteristics As It Exits The Palisades Reactor, Gregory Alan Baustian Dec 1993

Primary Coolant Flow Characteristics As It Exits The Palisades Reactor, Gregory Alan Baustian

Masters Theses

During fuel cycles 9 and 10 at the Palisades Nuclear Power Plant, new phenomena have been observed using the resistance temperature devices installed in the reactor coolant piping. Oscillations in the temperature readings have been observed leading to the assumptions that there is temperature stratification of the coolant in the reactor hot-legs and that the coolant rotates as it leaves the reactor. In addition, oscillations in the bulk coolant flow within the two hot legs have been postulated. This report outlines work performed using computational fluid dynamics modelling which predict the observed and assumed flow characteristics of the Palisades reactor …

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Evaluation And Implementation Of Asynchronous Adders, Poornima Y. Shankarreddy Dec 1993

Evaluation And Implementation Of Asynchronous Adders, Poornima Y. Shankarreddy

Masters Theses

The need for high speed addition is forcing digital system designers to trade off space. If speed is used as a criterion to judge adders, then the Ripple Carry Adder (RCA) and the Carry Look Ahead Adders (CLAAD) will rank last and first respectively. On the other hand, if space is used, then the order of these two is reversed. The rest of the adders rank between these two. This paper evaluates RCA, CLAAD, and some of the other well documented adders. Both space and speed are used to judge the merits of each of these, and whether or not …

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Modal Control Of A Satellite In An Unstable Periodic Orbit About The Earth-Sun Interior Lagrange Point, Douglas J. Hopper Dec 1993

Modal Control Of A Satellite In An Unstable Periodic Orbit About The Earth-Sun Interior Lagrange Point, Douglas J. Hopper

Theses and Dissertations

A periodic halo orbit which exists about the interior Lagrange point for the Earth-sun system was decomposed using Floquet theory into modal variables, which are dynamically decoupled subspaces for the six degree of freedom system. Modal control consisted of evaluating the diverging mode and maneuvering to counteract its divergence. In the unperturbed system, this was successful. Control costs were low, and the significance is that the controller did nothing to suppress modes that were oscillatory or converging. The effect of the moons motion allowed the scheme to operate with reasonable control costs, but the effect of eccentricity caused divergence in …

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A Study Of Graphite Peek Composite With Central Elliptical Holes Under Tension, Bill J. Lau Dec 1993

A Study Of Graphite Peek Composite With Central Elliptical Holes Under Tension, Bill J. Lau

Theses and Dissertations

This study investigated the stress-strain response and failure modes of Graphite/Polyetheretherketone (Gr/PEEK) tension specimens containing a central elliptical hole. Both experiments and finite element numerical analyses were conducted. Test specimens included seven layups and three hole geometries; three of each combination were tested. The holes consisted of an ellipse with major axis aligned with the load axis, an ellipse with major axis transverse to the load axis, and a circle. Strain gages were located adjacent to the hole and midway between the hole and tab on all specimens. Strain data was used to plot stress-strain response. The analysis utilized a …

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Using Discovery-Based Learning To Prove The Behavior Of An Autonomous Agent, David N. Mezera Dec 1993

Using Discovery-Based Learning To Prove The Behavior Of An Autonomous Agent, David N. Mezera

Theses and Dissertations

Computer-generated autonomous agents in simulation often behave predictably and unrealistically. These characteristics make them easy to spot and exploit by human participants in the simulation, when we would prefer the behavior of the agent to be indistinguishable from human behavior. An improvement in behavior might be possible by enlarging the library of responses, giving the agent a richer assortment of tactics to employ during a combat scenario. Machine learning offers an exciting alternative to constructing additional responses by hand by instead allowing the system to improve its own performance with experience. This thesis presents NOSTRUM, a discovery-based learning DBL system …

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Developing A Graphical User Interface To Support A Real-Time Digital Signal Processing System, Jeffrey C. Miller Dec 1993

Developing A Graphical User Interface To Support A Real-Time Digital Signal Processing System, Jeffrey C. Miller

Theses and Dissertations

A graphical software user interface for a VMEbus-based real-time digital signal processing system was designed. Use, requirements were defined and the Rumbaugh object-oriented analysis and design technique was applied to analyze the requirements and produce an object-oriented design. The software design includes a graphical, mouse- and keyboard driven user interface, specialized hardware driver modules, and operating system interfaces. An implementation plan was also developed to map the design into the C programming language using existing system code, automatically generated code, and newly written code. Based on the implementation plan, a limited software system prototype w successfully developed and demonstrated. The …

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Design And Construction Of The Aerobot Robotic Manipulator (Arm), William L. Cochran Dec 1993

Design And Construction Of The Aerobot Robotic Manipulator (Arm), William L. Cochran

Theses and Dissertations

This thesis designed, constructed, and tested a robotic arm for the Aerobot Aerial Robot. The main purpose of the ARM is to enable the Aerobot to retrieve objects for use in an annual robotics competition. Design of the ARM involved synthesizing the characteristics of simplicity, weight, strength, and size. The result was a three-degree-of-freedom manipulator that uses electric motors, cable linkages, and telescoping tubes to access a work space below the Aerobot. Forward and inverse kinematics were investigated to enable automation of the ARM. Data was collected from infrared sensors to validate the model. Manipulation of the ARM is presently …

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Analysis Of The Effects Of Wake Passage On Heat Transfer In A Linear Turbine Cascade, Kevin S. Allen Dec 1993

Analysis Of The Effects Of Wake Passage On Heat Transfer In A Linear Turbine Cascade, Kevin S. Allen

Theses and Dissertations

This study investigated the effect of wake passage on the heat transfer in a linear turbine cascade. The apparatus used was the Air-Force Institute of Technology linear Turbine Cascade Test Facility (TCTF). The TCTF was fitted with a be Pulley system to facilitate a series of translating bars. The bars, when passed d upstream of the cascade, created a series of wakes. The parameters varied were freestream model Reynolds number and bar conditions, i.e. , bars in, bars out, and bar passing frequency. One blade in the TCTF was instrumented to allow pressure and temperature measurements at discrete points on …

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Shock Wave Interaction With L-Shaped Structures, Richard C. Miller Dec 1993

Shock Wave Interaction With L-Shaped Structures, Richard C. Miller

Theses and Dissertations

This study investigated the interaction of shock waves with L-shaped structures using the CTH hydrodynamics code developed by Sandia National Laboratories. Computer models of shock waves traveling through air were developed using techniques similar to shock tube experiments. Models of L-shaped buildings were used to determine overpressures achieved by the reflecting shock versus angle of incidence of the shock front. An L-shaped building model rotated 45 degrees to the planar shock front produced the highest reflected overpressure of 9.73 atmospheres in the corner joining the two wings, a value 9.5 times the incident overpressure of 1.02 atmospheres. The same L-shaped …

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Interactive Control Of A Parallel Simulation From A Remote Graphics Workstation, Douglas C. Looney Dec 1993

Interactive Control Of A Parallel Simulation From A Remote Graphics Workstation, Douglas C. Looney

Theses and Dissertations

Modern military commanders are faced with an overwhelming amount of intelligence data concerning the disposition of engaging forces, The sheer volume of data produced for a single planning scenario is an obstacle to the user as well as the computer. Today’s commander requires a real-time, three- dimensional representation of the battlefield in order to assimilate the data for the management of a conflict. Parallel computation is required to complete the processing of this information in a timely manner. A network protocol is required to link the interface with the parallel simulation. The of this study is to improve user interaction …

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Design Of A Parallel Discrete Event Simulation Coprocessor, Jacob L. Berlin Dec 1993

Design Of A Parallel Discrete Event Simulation Coprocessor, Jacob L. Berlin

Theses and Dissertations

A Parallel Discrete Event Simulation Coprocessor was designed to off- load the synchronization overhead from the processors executing the application. In a multiprocessor architecture, one coprocessor executes the synchronization routines for each host processor. Speedup can be achieved when the host processor executes the application and the coprocessor concurrently executes synchronization routines. The coprocessor uses a programmable microcode control store to guarantee flexibility in the synchronization routines. The coprocessor uses an Extreme Search Associative Memory to support fast Next Event Queue NEQ management. This associative memory uses bit-serial word-parallel search logic to provide 01 insert and retrieval time of events …

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Automating Satellite Range Scheduling, S. Michael Schalck Dec 1993

Automating Satellite Range Scheduling, S. Michael Schalck

Theses and Dissertations

The objective of this study was to develop a computer based satellite range scheduling SRS algorithm to create a 24 hour satellite support schedule. The algorithms goal was to schedule as many satellite support requests as possible. An iterative heuristic approach was used to schedule satellite support requests in three successive sub problems. The first sub problem involves scheduling low altitude satellite support requests using a mixed integer programming approach. The next two sub problems each involve scheduling 12 hour blocks of medium and high altitude satellite support requests, again using a mixed integer programming approach. Fourteen 24 hour schedules …

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Spectroscopic And Vibrational Energy Transfer Studies In Molecular Bromine, Courtney D. Holmberg Dec 1993

Spectroscopic And Vibrational Energy Transfer Studies In Molecular Bromine, Courtney D. Holmberg

Theses and Dissertations

Vibrational transfer and electronic quenching in the lower vibrational levels (v' < 3) of the B3π(0u+) state of 79Br2 were investigated using spectrally resolved, temporally resolved laser induced fluorescence techniques. Spectrally resolved emissions from collisionally populated Br2(B) vibrational levels were observed for Br2 and rare gas collision partners. Vibrational transfer was efficient in the non-predissociative vibrational levels and was adequately described by the Montroll-Shuler model for harmonic oscillators. A single fundamental rate coefficient for vibrational transfer from v' = 1 to v' = 0, kv(1,0), characterizes vibrational relaxation. For Br2, the …

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Binaural Room Simulation, Brian A. Smith Dec 1993

Binaural Room Simulation, Brian A. Smith

Theses and Dissertations

Research in binaural and spatial hearing is of particular interest to the Air Force. Applications in cockpit communication, target recognition, and aircraft navigation are being explored. This thesis examines human auditory localization cues and develops a mathematical model for the transfer function of a sound signal traveling from an isotropic point source through a rectangular room to both ears of a listener. Using this model as a guide, non-head coupled binaural sound signals are generated in a binaural room simulation. Reflection and attenuation cues included in the computer generated signals are varied in order to determine which cues enhance the …

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Multiple Model Adaptive Control Of A Large Flexible Space Structure With Purposeful Dither For Enhanced Identifiability, James A. Fitch Dec 1993

Multiple Model Adaptive Control Of A Large Flexible Space Structure With Purposeful Dither For Enhanced Identifiability, James A. Fitch

Theses and Dissertations

Dithering techniques for enhancing uncertain parameter identification with moving-bank Multiple Model Adaptive Estimation MMAE and Control MMAC algorithms are analyzed in this thesis. The dithering techniques and multiple- model adaptive algorithms are applied to the SPICE 2 flexible space structure. The dithering techniques studied include purposefully constructed square wave, sine wave fixed and swept frequency and wide-band noise wave forms. Purposeful rigid-body slew commands are also used in order to excite the structures flexible bending modes. Dither inputs into the structure are performed in an effort to enhance the open-loop identifiability of the uncertain parameter, namely a scalar multiplier on …

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Orbital Applications Of Electrodynamic Propulsion, Troy Irwin Dec 1993

Orbital Applications Of Electrodynamic Propulsion, Troy Irwin

Theses and Dissertations

Electrodynamic propulsion (EDP) uses forces resulting from electric currents in conductors as a spacecraft travels through a magnetic field. A vehicle-independent expression for the specific power required for any maneuver is derived and used to assess EDP feasibility. Analytical expressions for the accelerations and combined current-conductor vector required to change the orbital plane or the argument of perigee are developed based on Lagranges planetary equations. Solutions to the forced Clohessy-Wiltshire equations are developed to study iii-plane rendezvous. Results show EDP can change inclination or right ascension of the ascending mode at approximately 0.4 degrees/day with current spacecraft specific power technology. …

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Modeling Of A Large Undamped Space Structure Using Time Domain Techniques, Anthony R. Nash Dec 1993

Modeling Of A Large Undamped Space Structure Using Time Domain Techniques, Anthony R. Nash

Theses and Dissertations

The first phase of this project was the selection of accurate time domain state space modeling techniques by comparing the models generated by various methods, against a known system. The techniques selected were the Eigensystem Realization Algorithm and the Backward Least Squares method. These modeling techniques were applied to the vibration data from the PACOSS dynamic test article. The resulting model transfer function was then compared to the frequency response function of the actual system, and was found to be an accurate match for the system poles, but not for the system zeros. Large Space structures, Dynamic Control, System Mode …

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Steady Motions Of Rigid Body Satellites In A Central Gravitational Field, Cynthia A. Provost Dec 1993

Steady Motions Of Rigid Body Satellites In A Central Gravitational Field, Cynthia A. Provost

Theses and Dissertations

Numerous studies have been conducted on equilibrium orientations of objects moving under the influence of a central gravitational field. The results of many of these studies conclude that equilibrium conditions exist only when one of the principal axes coincides with the radius vector. Furthermore, these results assume that the center of force is located within the orbit plane, thereby tracing a great circle orbit. While these previous works have approximated the gravitational potential, this study examines relative equilibrium obtained by retaining an exact expression for the potential of a spherical primary body, as shown in a recent paper by Wang, …

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