An integral transform with a Whittaker function kernel Interim report, 20 Jun. - 19 Aug. 1968
Integral transformation with Whittaker kernel functions
Wimp, J.↗
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Longitudinal dynamic stability of a shuttle vehicle.
Analytical study of the longitudinal dynamic stability of a nonrolling, lifting vehicle gliding at hypersonic speeds. The analysis applies to shuttle vehicles designed for operating up to the rim of a planetary atmosphere. A general nondimensional time transformation is introduced to derive a unified second-order linear differential equation for the angle of attack, valid for all types of reentry of a general type of vehicle. The stability of motion is discussed for two fundamental regimes of flight that are based on widely different assumptions. For near ballistic entry along a straight line trajectory, the equation reduces to a confluent hypergeometric equation, the solution of which can be expressed in terms of Whittaker's function. Using a theorem in the theory of stability of differential equations, criteria for damped oscillations are derived. It is shown that the aerodynamic criteria for stability are the same as for the case of ballistic entry. In addition, for each vehicle configuration, and specified planetary atmosphere, there exists an altitude range where the angle of attack frequency is nearly equal to the orbital frequency causing instability in pitch. This resonance instability is due to the ellipticity of the orbit. Criteria for eccentricity instability are derived.
Vinh, N. X.↗
Acoustic resonances and sound scattering by a shear layer
The reflection and transmission characteristics of plane waves scattered by a finite-thickness shear layer having a linear velocity profile and bounded by two otherwise uniform parallel flows is examined using the pressure perturbation equation solutions in the shear layer shown previously to be in terms of Whittaker M functions. In addition to the angle of plane wave incidence and the relative Mach number of the flows bounding the shear layer, it is found that the scattering properties of the shear layer depend crucially upon a parameter tau in such a manner that the case tau approaching zero characterizes the long wavelength properties of the layer and the case tau approaching infinity characterizes the short wavelength properties of the layer. In contrast to the region of ordinary reflection in the cases where the corresponding vortex sheet does not have a Brewster angle, the values of the reflection coefficient up to tau of 2 follow those of the vortex sheet; for the case for which the corresponding vortex sheet has a Brewster angle, the magnitude of the reflection coefficient may be sensitive to even small changes in tau in certain cases.
Koutsoyannis, S. P.↗
Characterization of acoustic disturbances in linearly sheared flows
The equation describing the plane wave propagation, the stability, or the rectangular duct mode characteristics in a compressible inviscid linearly sheared parallel, but otherwise homogeneous flow, is shown to be reducible to Whittaker's equation. The resulting solutions, which are real, viewed as functions of two variables, depend on a parameter and an argument, the values of which have precise physical meanings depending on the problem. The exact solutions in terms of Whittaker functions are used to obtain a number of known results of plane wave propagation and stability in linearly sheared flows as limiting cases in which the speed of sound goes to infinity (incompressible limit) or the shear layer thickness, or wave number, goes to zero (vortex sheet limit). The usefulness of the exact solutions is then discussed in connection with the problems of plane wave propagation and the stability of a finite thickness layer with a linear velocity profile.
Koutsoyannis, S. P.↗
Features of sound propagation through and stability of a finite shear layer
The plane wave propagation, the stability and the rectangular duct mode problems of a compressible inviscid linearly sheared parallel, but otherwise homogeneous flow, are shown to be governed by Whittaker's equation. The exact solutions for the perturbation quantities are essentially Whittaker M-functions. A number of known results are obtained as limiting cases of exact solutions. For the compressible finite thickness shear layer it is shown that no resonances and no critical angles exist for all Mach numbers, frequencies and shear layer velocity profile slopes except in the singular case of the vortex sheet.
Koutsoyannis, S. P.↗
Features of sound propagation through and stability of a finite shear layer
The plane wave propagation, the stability, and the rectangular duct mode problems of a compressible, inviscid, linearly sheared, parallel, homogeneous flow are shown to be governed by Whittaker's equation. The exact solutions for the perturbation quantities are essentially the Whittaker M-functions where the nondimensional quantities have precise physical meanings. A number of known results are obtained as limiting cases of the exact solutions. For the compressible finite thickness shear layer it is shown that no resonances and no critical angles exist for all Mach numbers, frequencies, and shear layer velocity profile slopes except in the singular case of the vortex sheet.
Koutsoyannis, S. P.↗
Multigrid-sinc methods
A Galerkin method using Whittaker cardinal or 'sinc' functions as basis functions is described for the solution of boundary-value problems. When the solution is analytic in the interior of the domain, the error of approximation using 2N + 1 points is O(e exp /-gamma sq rt N/) even if derivatives of the solution are singular at the boundaries. A multigrid method with overall complexity O(N log N) is used to solve the discrete equations. This paper contains a description of the multigrid-sinc algorithm along with some preliminary numerical results for two-point boundary-value problems.
Schaffer, Steve↗
Sampling Theorem in Terms of the Bandwidth and Sampling Interval
An approach has been developed for interpolating non-uniformly sampled data, with applications in signal and image reconstruction. This innovation generalizes the Whittaker-Shannon sampling theorem by emphasizing two assumptions explicitly (definition of a band-limited function and construction by periodic extension). The Whittaker- Shannon sampling theorem is thus expressed in terms of two fundamental length scales that are derived from these assumptions. The result is more general than what is usually reported, and contains the Whittaker- Shannon form as a special case corresponding to Nyquist-sampled data. The approach also shows that the preferred basis set for interpolation is found by varying the frequency component of the basis functions in an optimal way.
Dean, Bruce H.↗
Water-Searchers: A Reconfigurable and Self Sustaining Army of Subsurface Exploration Robots Searching for Water/Ice Using Multiple Sensors
Perhaps the most promising site for extant life on Mars today is where subsurface water has been maintained. Therefore, searching for underground water will provide a good chance to find evidence of life on Mars. The following are scientific/engineering questions that we want to answer using our approach: (1) Is there subsurface water/ice? How deep is it? How much is there? Is it frozen? (2) What kinds of underground layers exist in the Martian crust? (3) What is the density of Martian soil or regolith? Can we dig into it? Should we drill into it? (4) Can a sudden release of underground water occur if a big asteroid hits Mars? Our approach provides essential information to answer these questions. Moreover, dependence on the water content and depth in soil, not only resultant scientific conclusions but also proper digging/drilling methods, are suggested. 'How much water is in the Martian soil?' There can be several possibilities: (1) high water content that is enough to form permafrost; (2) low water content that is not enough to form permafrost; or (3) different layers with different moisture contents. 'How deep should a rover dig into soil to find water/ice?' The exact size-frequency distribution has not been measured for the soil particles. On-board sensors can provide not only the water content but also the density (or porosity) of Martian soil as a function of depth.
Youk, G. U.↗
Sampling and Reconstruction of the Pupil and Electric Field for Phase Retrieval
This technology is based on sampling considerations for a band-limited function, which has application to optical estimation generally, and to phase retrieval specifically. The analysis begins with the observation that the Fourier transform of an optical aperture function (pupil) can be implemented with minimal aliasing for Q values down to Q = 1. The sampling ratio, Q, is defined as the ratio of the sampling frequency to the band-limited cut-off frequency. The analytical results are given using a 1-d aperture function, and with the electric field defined by the band-limited sinc(x) function. Perfect reconstruction of the Fourier transform (electric field) is derived using the Whittaker-Shannon sampling theorem for 1<Q<2. The Fourier transform is constructed by periodic extension, i.e., by spacing copies of the transform in a definite way, recognizing that no aliasing occurs for values of the sampling ratio such that 1<Q<2, which can be used to advantage in the application of phase retrieval estimation. A method was developed for propagating the electro magnetic field with no aliasing, which has been extended to 2-d optical apertures.
Dean, Bruce↗
Design of a walking robot
Carnegie Mellon University's Autonomous Planetary Exploration Program (APEX) is currently building the Daedalus robot; a system capable of performing extended autonomous planetary exploration missions. Extended autonomy is an important capability because the continued exploration of the Moon, Mars and other solid bodies within the solar system will probably be carried out by autonomous robotic systems. There are a number of reasons for this - the most important of which are the high cost of placing a man in space, the high risk associated with human exploration and communication delays that make teleoperation infeasible. The Daedalus robot represents an evolutionary approach to robot mechanism design and software system architecture. Daedalus incorporates key features from a number of predecessor systems. Using previously proven technologies, the Apex project endeavors to encompass all of the capabilities necessary for robust planetary exploration. The Ambler, a six-legged walking machine was developed by CMU for demonstration of technologies required for planetary exploration. In its five years of life, the Ambler project brought major breakthroughs in various areas of robotic technology. Significant progress was made in: mechanism and control, by introducing a novel gait pattern (circulating gait) and use of orthogonal legs; perception, by developing sophisticated algorithms for map building; and planning, by developing and implementing the Task Control Architecture to coordinate tasks and control complex system functions. The APEX project is the successor of the Ambler project.
Whittaker, William↗
Basic statistical analyses of candidate nickel-hydrogen cells for the Space Station Freedom
Nickel-Hydrogen (Ni/H2) secondary batteries will be implemented as a power source for the Space Station Freedom as well as for other NASA missions. Consequently, characterization tests of Ni/H2 cells from Eagle-Picher, Whittaker-Yardney, and Hughes were completed at the NASA Lewis Research Center. Watt-hour efficiencies of each Ni/H2 cell were measured for regulated charge and discharge cycles as a function of temperature, charge rate, discharge rate, and state of charge. Temperatures ranged from -5 C to 30 C, charge rates ranged from C/10 to 1C, discharge rates ranged from C/10 to 2C, and states of charge ranged from 20 percent to 100 percent. Results from regression analyses and analyses of mean watt-hour efficiencies demonstrated that overall performance was best at temperatures between 10 C and 20 C while the discharge rate correlated most strongly with watt-hour efficiency. In general, the cell with back-to-back electrode arrangement, single stack, 26 percent KOH, and serrated zircar separator and the cell with a recirculating electrode arrangement, unit stack, 31 percent KOH, zircar separators performed best.
Maloney, Thomas M.↗
Optimal trajectories for hypersonic launch vehicles
In this paper, we derive a near-optimal guidance law for the ascent trajectory from earth surface to earth orbit of a hypersonic, dual-mode propulsion, lifting vehicle. Of interest are both the optical flight path and the optimal operation of the propulsion system. The guidance law is developed from the energy-state approximation of the equations of motion. Because liquid hydrogen fueled hypersonic aircraft are volume sensitive, as well as weight sensitive, the cost functional is a weighted sum of fuel mass and volume; the weighting factor is chosen to minimize gross take-off weight for a given payload mass and volume in orbit.
Ardema, Mark D.↗
A walking prescription for statically-stable walkers based on walker/terrain interaction
The walker/terrain interaction phenomena for the control of a statically stable walking machine are described. The algorithms, measures, and knowledge of walker/terrain interaction phenomena are then combined to form a prescription for how to walk on general terrain. This prescription consists of two parts: nominal control and reactive control. The function of nominal control is the evaluation and execution of planned motions, based on predicted foot force redistributions, to achieve reliable locomotion. The function of reactive control is the monitoring of walker/terrain interaction in real-time to detect anomalous conditions and then respond with the appropriate reflexive actions. Simulations and experiments have been used to test and verify various aspects of the walking prescription.
Nagy, Peter V.↗
Light-Induced Alterations in Striatal Neurochemical Profiles
Much of our present knowledge regarding circadian rhythms and biological activity during space flight has been derived from those missions orbiting the Earth. During space missions, astronauts can become exposed to bright/dark cycles that vary considerably from those that entrain the mammalian biological timing system to the 24-hour cycle found on Earth. As a spacecraft orbits the Earth, the duration of the light/dark period experienced becomes a function of the time it takes to circumnavigate the planet which in turn depends upon the altitude of the craft. Orbiting the Earth at an altitude of 200-800 km provides a light/dark cycle lasting between 80 and 140 minutes, whereas a voyage to the moon or even another planet would provide a light condition of constant light. Currently, little is known regarding the effects of altered light/dark cycles on neurochemical levels within the central nervous system (CNS). Many biochemical, physiological and behavioral phenomena are under circadian control, governed primarily by the hypothalamic suprachiasmatic nucleus. As such, these phenomena are subject to influence by the environmental light/dark cycle. Circadian variations in locomotor and behavioral activities have been correlated to both the environmental light/dark cycle and to dopamine (DA) levels within the CNS. It has been postulated by Martin-Iverson et al. that DA's role in the control of motor activity is subject to modulation by circadian rhythms (CR), environmental lighting and excitatory amino acids (EAAs). In addition, DA and EAA receptor regulated pathways are involved in both the photic entrainment of CR and the control of motor activity. The cellular mechanisms by which DA and EAA-receptor ligands execute these functions, is still unclear. In order to help elucidate these mechanisms, we set out to determine the effects of altered environmental light/dark cycles on CNS neurotransmitter levels. In this study, we focused on the striatum, a region of the brain that receives a number of dopaminergic and glutamatergic input and is known to be involved in the modulation of locomotor and behavioral responses.
Sroufe, Angela E.↗
Light-Induced Alterations in Basil Ganglia Kynurenic Acid Levels
The metabolic synthesis, release and breakdown of several known CNS neurotransmitters have been shown to follow a circadian pattern entrained to the environmental light/dark cycle. The levels of excitatory amino acid (EAA) transmitters such as glutamate, have been shown to vary with environmental lighting conditions. Kynurenic Acid (KA), an endogenous tryptophan metabolite and glutamate receptor antagonist, has been reported to have neuroprotective effects against EAA-induced excitotoxic cell damage. Changes in KA's activity within the mammalian basal ganglia has been proposed as being contributory to neurotoxicity in Huntington's Disease. It is not known whether CNS KA levels follow a circadian pattern or exhibit light-induced fluctuations. However, because the symptoms of certain degenerative motor disorders seem to fluctuate with daily 24 hour rhythm, we initiated studies to determine if basal ganglia KA were influenced by the daily light/dark cycle and could influence motor function. Therefore in this study, HPLC-EC was utilized to determine if basal ganglia KA levels in tissue extracts from adult male Long-Evans rats (200-250g) entrained to 24 and 48 hours constant light and dark conditions, respectively. Samples were taken one hour before the onset of the subjective day and one hour prior to the onset of the subjective night in order to detect possible phase differences in KA levels and to allow for accumulation of factors expressed in association with the light or dark phase. Data analysis revealed that KA levels in the basal ganglia vary with environmental lighting conditions; being elevated generally during the dark. Circadian phase differences in KA levels were also evident during the subjective night and subjective day, respectively. Results from these studies are discussed with respect to potential cyclic changes in neuronal susceptibility to excitotoxic damage during the daily 24 hour cycle and its possible relevance to future therapeutic approaches in treating neurodegenerative disorders.
Sroufe, Angela E.↗