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NASA NTRS1983

Dynamics of momentum biased spacecraft in a near-polar orbit

The equations of motion of a momentum biased spacecraft are derived in a general form. The spacecraft is assumed to be orbiting in a near-elliptical orbit. An aerodynamic torque model which accounts for the atmospheric superrotation is assumed. The equilibrum attitude angles are obtained in terms of modified Bessel functions. Analytic expressions for the long-term motion of the momentum biased axis are derived for special cases. The analysis is applicable to the Magsat mission.

Sellappan, R. G.↗

NASA NTRS1983

Analytical study of the twin-jet shielding

The development of the analytical model of twin-jet shielding is summarized. The models consist of a point noise source impinging on a cylinder of heated flow in which the temperature and flow velocity are uniform cross the cross section. In the formulation of the model, the wave equations are written for the regions outside the flow and within the flow cylinder. The solutions to the wave equations are matched at the jet boundary under the conditions of continuity of pressure and continuity of the vortex sheet. The solution reduces to an indefinite integral involving Bessel functions. The integral is solved approximately using a saddle point method.

Gerhold, C. H.↗

NASA NTRS1984

Neutron and gamma-ray signatures for particle acceleration in solar flares

The production of neutrons, 4 to 7 MeW nuclear gamma rays, positrons and pions resulting from the interaction of flare accelerated particles with the solar atmosphere was calculated. For the energy spectra of these particles the Bessel function predicted by stochastic acceleration and power laws which could result from acceleration at large-scale planar shocks was used, with the thick-target model. It is assumed that in the interaction region the accelerated particles are isotropic. The neutron-to-2.223 MeV photon conversion factors for various flare locations on the Sun are derived by averaging conversion factors over these spectra. Comparing calculations with data confirms that for most gamma-ray flares the bulk of the accelerated particles remain trapped at the Sun and that these particles have spectra similar to the spectra of flare particles observed in interplanetary space.

Ramaty, R.↗

NASA NTRS1985

Numerical solution of the vertical structure equation in the normal mode method

In the present model of multilayered stability stratification, aimed at obtaining the analytic eigensolutions of the vertical structure equation, each layer is characterized by its own static stability value. By requiring continuity of pressure and vertical velocity across each interface level, and by imposing suitable upper and lower boundary conditions, matching eigensolutions are obtained in terms of the Bessel functions. Attention is given to an explicit example of a double-layered stratified atmosphere which demonstrates the mathematical manipulations involved; the resultant vertical structure functions are used to check the accuracy of the numerical solutions by the finite difference and finite element methods.

Sasaki, Y. K.↗

NASA NTRS1985

The acoustic gravity wave induced by a point source in the middle atmosphere

Acoustic gravity wave (AGW) results computed for a stationary impulsive point source and a moving point source in the middle atmosphere are presented. For a stationary impulsive point Row's far field formula of the AGW was extended into the near field one, which comprises the Zeroth order Bessel function and its derivative terms. When (t-t sub o) is not large, the contribution of the derivative terms is important. The computed results agree with the experimental ones. For a moving point source with supersonic velocity, AGW is calculated using the moving point theory. Two solar eclipses that occurred in the lower latitude and over the ocean on Feb. 16, 1980, and June 11, 1983, were compared. The results show that the theoretical curve of AGW is fairly consistent with the observed ones.

Zhang, X. J.↗

NASA NTRS1987

Solar neutron emissivity during the large flare on 1982 June 3

For the solar neutron event on June 3, 1982, it is shown here that the combined SMM Gamma Ray Spectrometer and Jungfraujoch neutron monitor data require a time-extended emission of neutrons at the sun with energies of 100 MeV to about 2 GeV. The solar neutron emissivity spectrum is shown to have a strong downward curvature or truncation between 2 and 4 GeV. A Bessel function and truncated power law give acceptable fits to the observational data, but only the power law can explain the rapid rise of the neutron monitor count rate. The integrated emissivity of neutrons above E(n) of 100 MeV is strongly constrained at 8 x 10 to the 28th neutrons/sr and is essentially independent of neutron spectral shape. At neutron energies of about 100 MeV, good agreement is found for both spectral forms with observations of neutron decay protons.

Chupp, E. L.↗

NASA NTRS1988

Eigenfunctions of the magnetospheric radial-diffusion operator

The applicability of eigenfunction methods to the theory of magnetospheric radial diffusion is investigated analytically, with a focus on the operator (Lambda) for radiation-belt and ring-current particles. It is shown that the eigenfunctions of Lambda can be expressed in terms of a linear combination of ordinary Bessel functions of the first and second kinds, and that the solution for the drift-averaged phase-space density can be expressed as the sum of a quasi-static solution and a time-dependent superposition of the eigenfunctions of Lambda. Results for illustrative and practical sample problems are presented in extensive tables and graphs and discussed in detail.

Schulz, Michael↗

NASA NTRS1991

Unified aeroacoustics analysis for high speed turboprop aerodynamics and noise. Volume 4: Computer user's manual for UAAP turboprop aeroacoustic code

The Unified AeroAcoustic Program (UAAP) code calculates the airloads on a single rotation prop-fan, or propeller, and couples these airloads with an acoustic radiation theory, to provide estimates of near-field or far-field noise levels. The steady airloads can also be used to calculate the nonuniform velocity components in the propeller wake. The airloads are calculated using a three dimensional compressible panel method which considers the effects of thin, cambered, multiple blades which may be highly swept. These airloads may be either steady or unsteady. The acoustic model uses the blade thickness distribution and the steady or unsteady aerodynamic loads to calculate the acoustic radiation. The users manual for the UAAP code is divided into five sections: general code description; input description; output description; system description; and error codes. The user must have access to IMSL10 libraries (MATH and SFUN) for numerous calls made for Bessel functions and matrix inversion. For plotted output users must modify the dummy calls to plotting routines included in the code to system-specific calls appropriate to the user's installation.

Menthe, R. W.↗

NASA NTRS1991

Feedback instability of the ionospheric resonant cavity

A model is developed that provides a theoretical basis for previous numerical results showing a feedback instability with frequencies characteristic of Alfven travel times within the region of the large increase of Alfven speed above the ionosphere. These results have been extended to arbitrary ionospheric conductivity by developing a numerical solution of the cavity dispersion relation that involves Bessel functions of complex order and argument. It is concluded that the large contrast between the magnetospheric and ionospheric Alfven speed leads to the formation of resonant cavity modes with frequencies ranging from 0.1 to 1 Hz. The presence of the cavity leads to a modification of the reflection characteristics of Alfven waves with frequencies that compare to the cavity's normal modes.

Lysak, Robert L.↗

NASA NTRS1991

Analytical solution for boundary heat fluxes from a radiating rectangular medium

Reference is made to the work of Shah (1979) which demonstrated the possibility of partially integrating the radiative equations analytically to obtain an 'exact' solution. Shah's solution was given as a double integration of the modified Bessel function of order zero. Here, it is shown that the 'exact' solution for a rectangular region radiating to cold black walls can be conveniently derived, and expressed in simple form, by using an integral function, Sn, analogous to the exponential integral function appearing in plane-layer solutions.

Siegel, R.↗

NASA NTRS1992

Hierarchical tapered bar elements undergoing axial deformation

A method is described to model the dynamics of tapered axial bars of various cross sections based on the well-known Craig/Bampton component mode synthesis technique. This element is formed in terms of the static constraint modes and interface restrained normal modes. This is in contrast with the finite elements as implemented in NASTRAN where the interface restrained normal modes are neglected. These normal modes are in terms of Bessel functions. Restoration of a few of these modes leads to higher accuracy with fewer generalized coordinates. The proposed models are hierarchical so that all lower order element matrices are embedded in higher order element matrices. The advantages of this formulation compared to standard NASTRAN truss element formulation are demonstrated through simple numerical examples.

Ganesan, N.↗

NASA NTRS1992

Gamma-rays and neutrons as a probe of the proton spectrum during the solar flare of 1988 December 16

We have previously reported on high-energy gamma-rays and neutrons from the flare of 1988 December 16 detected by the Gamma-Ray Spectrometer on the SMM satellite. In this paper, we present results on gamma-ray lines seen by the same detector during this flare. Together, these measurements constitute a powerful probe of the proton spectrum that produces the flare neutrals. Analysis of the data suggests a Bessel-function proton spectrum with a shape parameter (alpha T) of 0.054 +/- 0.004 and the number of protons above 30 MeV equal to (9.0 +/- 0.9) x 10 exp 32. The number of neutrons detected from this flare is much smaller than what is predicted from an isotropic distribution of the protons, indicating that the distribution may be nonisotropic.

Dunphy, P. P.↗

NASA NTRS1939

The Frequency of Torsional Vibration of a Tapered Beam

A solution for the equation of torsional vibration of tapered beams has been found in terms of Bessel functions for beams satisfying the following conditions: (a) the cross sections along the span are similar in shape; and (b) the torsional stiffness of a section can be expressed as a power of a linear function of distance along the span. The method of applying the analysis to actual cases has been described. Charts are given from which numerical values can be immediately obtained for most cases of practical importance. The theoretical values of the frequency ratio have been experimentally checked on five beams having different amounts of taper.

Coleman, Robert↗

NASA NTRS1949

General Theory of Aerodynamic Instability and the Mechanism of Flutter

The aerodynamic forces on an oscillating airfoil or airfoil-aileron combination of three independent degrees of freedom have been determined. The problem resolves itself into the solution of certain definite integrals, which have been identified as Bessel functions of the first and second kind and of zero and first order. The theory, being based on potential flow and the Kutta condition, is fundamentally equivalent to the conventional wing-section theory relating to the steady case. The air forces being known, the mechanism of aerodynamic instability has been analyzed in detail. An exact solution, involving potential flow and the adoption of the Kutta condition, has been analyzed in detail. An exact solution, involving potential flow and the adoption of the Kutta condition, has been arrived at. The solution is of a simple form and is expressed by means of an auxiliary parameter K.

Theodorsen, Theodore↗

NASA NTRS1994

Numerical formulation of composition segregation at curved solid-liquid interface during steady state solidification process

The lateral solute segregation that results from a curved solid-liquid interface shape during steady state unidirectional solidification of a binary alloy system has been studied both analytically and numerically by Coriell, Bosivert, Rehm, and Sekerka. The system under their study is a two dimensional rectangular system. However, most real growth systems are cylindrical systems. Thus, in a previous study, we have followed Coriell etc. formalism and obtained analytical results for lateral solute segregation for an azimuthal symmetric cylindrical binary melt system during steady state solidification process. The solid-liquid interface shape is expressed as a series combination of Bessel functions. In this study a computer program has been developed to simulate the lateral solute segregation.

Wang, Jai-Ching↗

NASA NTRS1994

Electromagnetic and corpuscular emission from the solar flare of 1991 June 15: Continuous acceleration of relativistic particles

Data on X-, gamma-ray, optical and radio emission from the 1991 June 15 solar flare are considered. We have calculated the spectrum of protons that produces gamma-rays during the gradual phase of the flare. The primary proton spectrum can be described as a Bessel-function-type up to 0.8 GeV and a power law with the spectral index approximately equal to 3 from 0.8 up to 10 GeV or above. We have also analyzed data on energetic particles near the Earth. Their spectrum differed from that of primary protons producing gamma-ray line emission. In the gradual phase of the flare additional pulses of energy release occurred and the time profiles of cm-radio emission and gamma-rays in the 0.8-10 MeV energy band and above 50 MeV coincided. A continuous and simultaneous stochastic acceleration of the protons and relativistic electrons at the gradual phase of the flare is considered as a natural explanation of the data.

Kocharov, L. G.↗

NASA NTRS1997

Analysis of ripple formation in single crystal spot welds

Stationary spot welds have been made at the (001) surface of Fe-l5%Ni-15%Cr single crystals using a Gas Tungsten Arc (GTA). On the top surface of the spot welds, very regular and concentric ripples were observed after solidification by differential interference color microscopy. Their height (typically 1--5 micrometers and spacing, typically approximately 60 micrometers) decreased with the radius of the pool. These ripples were successfully accounted for in terms of capillary-wave theory using the fundamental mode frequency f(sub 0) given by the first zero of the zero-order Bessel function. The spacing d between the ripples was then equated to v(sub s)/f(sub 0), where v(sub s) is the solidification rate. From the measured ripple spacing, the velocity of the pool was deduced as a function of the radius, and this velocity was in good agreement with the results of a heat-flow simulation.

Rappaz, M.↗

NASA NTRS2001

Controlling Fringe Sensitivity of Electro-Optic Holography Systems Using Laser Diode Current Modulation

Electro-Optic Holography (EOH) is a non-intrusive, laser-based, displacement measurement technique capable of static and dynamic displacement measurements. EOH is an optical interference technique in which fringe patterns that represent displacement contour maps are generated. At excessively large displacements the fringe density may be so great that individual fringes are not resolvable using typical EOH techniques. This thesis focuses on the development and implementation of a method for controlling the sensitivity of the EOH system. This method is known as Frequency Translated Electro-Optic Holography (FTEOH). It was determined that by modulating the current source of the laser diode at integer multiples of the object vibration, the fringe pattern is governed by higher order Bessel function of the first kind and the number of fringes that represent a given displacement can be controlled. The reduction of fringes is theoretically unlimited but physically limited by the frequency bandwidth of the signal generator, providing modulation to the laser diode. Although this research technique has been verified theoretically and experimentally in this thesis, due to the current laser diode capabilities it is a tedious and time consuming process to acquire data using the FTEOH technique.

Bybee, Shannon J.↗