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NASA NTRS1982

The origin of the Kirkwood gaps - A mapping for asteroidal motion near the 3/1 commensurability

A mapping of the phase space onto itself with the same low-order resonance structure as the 3/1 commensurability in the planar-elliptic restricted three-body problem is obtained. This mapping is about 1,000 times faster than the usual method of numerically integrating the averaged equations of motion. It exhibits some surprising behavior that might provide a key to the origin of the Kirkwood gaps. It is noted that a test asteroid placed in the gap may evolve for a million years with low eccentricity (less than 0.05) and then suddenly jump to large eccentricity (greater than 0.3), becoming a Mars crosser. The removal of the asteroid by a close encounter with Mars would then be possible. As a first test of this hypothesis, a distribution of 300 test asteroids in the area of the 3/1 commensurability was evolved for two million years. When the Mars crossers are removed, the distribution of initial conditions reveals a gap at the location of the 3/1 Kirkwood gap.

Wisdom, J.↗

NASA NTRS1982

The tailward magnetopause field beyond 10 RE

The geomagnetic field in the near-earth magnetotail is made up of three major contributions related to the dipole field, the magnetotail current system, and the magnetopause current system exclusive of the return currents in the magnetotail current system. The present investigation is concerned with the third contribution. The calculation of the magnetospheric field is discussed, taking into account the calculation of the field inside the magnetopause on the basis of the numerical integration of the Biot Savart integral. The representation of the field for distances greater than 10 earth radii on the dark side of the magnetosphere is considered. The field internal to the magnetopause is expressed as the negative gradient of a scalar potential which is expressed as a sum of solutions to Laplace's equation in cylindrical coordinates. The origin of coordinates is the earth.

Beard, D. B.↗

NASA NTRS1982

Nonlinear analysis of rotor-bearing systems using component mode synthesis

The method of component mode synthesis is developed to determine the forced response of nonlinear, multishaft, rotor-bearing systems. The formulation allows for simulation of system response due to blade loss, distributed unbalance, base shock, maneuver loads, and specified fixed frame forces. The motion of each rotating component of the system is described by superposing constraint modes associated with boundary coordinates and constrained precessional modes associated with internal coordinates. The precessional modes are truncated for each component and the reduced component equations are assembled with the nonlinear supports and interconnections to form a set of nonlinear system equations of reduced order. These equations are then numerically integrated to obtain the system response. A computer program, which is presently restricted to single shaft systems, has been written and results are presented for transient system response associated with blade loss dynamics with squeeze film dampers, and with interference rubs.

Nelson, H. D.↗

NASA NTRS1982

Recent advances in electromagnetic synthesis and analysis of dual-shaped reflector antennas

Dual-shaped reflectors have been used for many years. Thus, these reflectors have been used as high gain antennas on Voyagers 1 and 2. The objectives of the geometrical optics (GO) dual shaped synthesis are considered. Concerning the synthesis of dual shaped reflectors, it has been shown for circular symmetric reflectors that an exact GO solution can be found to the problem of transforming, by two reflections, any feed pattern into any aperture distribution. This problem involves solving two simultaneous nonlinear ordinary differential equations. The same approach for offset geometry leads to two simultaneous nonlinear partial differential equations. It is shown that these equations could also be integrated numerically, except that in general these equations are not total and therefore, in general, they do not have a 'smooth' solution. It is further shown that the offset partial differentials often very nearly form a total differential in many cases of practical importance.

Galindo-Israel, V.↗

NASA NTRS1982

Symmetric baroclinic instability of a Hadley cell

A symmetric baroclinic instability is examined in terms of a Boussinesq fluid contained between two horizontal plates to determine the effects of the Ekman and thermal layers. Governing equations are written for a rotating reference frame, taking into account the Rossby, Ekman, and Prandtl numbers. Equations are defined for the perturbation functions, treated as an eigenvalue problem, and a numerical integration of the full eighth order differential system is performed by a shooting technique. An instability is found to occur in the Hadley cell containing both Ekman and thermal boundary layers when the Richardson number is close to unity. If the Prandtl number is fixed the critical Richardson number decreases with an increasing Ekman number until the Ekman number reaches a certain value, at which time the fluid is stable.

Antar, B. N.↗

NASA NTRS1984

Improvements in Vibration-Analysis Technique

Two reports present details of vibration-analysis technique. Technique scheme for accurate and efficient numerical integration of coupled equations of motion of vehicle and its payload(s). Applicable to large stationary structures, land vehicles, airplanes, ships, or wherever vibration analysis done repeatedly after each of many design changes.

Engels, R. C.↗

NASA NTRS1982

Tone-excited jet: Theory and experiments

A detailed study to understand the phenomenon of broadband jet-noise amplification produced by upstream discrete-tone sound excitation has been carried out. This has been achieved by simultaneous acquisition of the acoustic, mean velocity, turbulence intensities, and instability-wave pressure data. A 5.08 cm diameter jet has been tested for this purpose under static and also flight-simulation conditions. An open-jet wind tunnel has been used to simulate the flight effects. Limited data on heated jets have also been obtained. To improve the physical understanding of the flow modifications brought about by the upstream discrete-tone excitation, ensemble-averaged schlieren photographs of the jets have also been taken. Parallel to the experimental study, a mathematical model of the processes that lead to broadband-noise amplification by upstream tones has been developed. Excitation of large-scale turbulence by upstream tones is first calculated. A model to predict the changes in small-scale turbulence is then developed. By numerically integrating the resultant set of equations, the enhanced small-scale turbulence distribution in a jet under various excitation conditions is obtained. The resulting changes in small-scale turbulence have been attributed to broadband amplification of jet noise. Excellent agreement has been found between the theory and the experiments. It has also shown that the relative velocity effects are the same for the excited and the unexcited jets.

Ahuja, K. K.↗

NASA NTRS1982

Structural dynamics payload loads estimates

Methods for the prediction of loads on large space structures are discussed. Existing approaches to the problem of loads calculation are surveyed. A full scale version of an alternate numerical integration technique to solve the response part of a load cycle is presented, and a set of short cut versions of the algorithm developed. The implementation of these techniques using the software package developed is discussed.

Engels, R. C.↗

NASA NTRS1982

A wake traverse technique for use in a 2 dimensional transonic flexible walled test section

Reported two dimensional validation data from the Transonic Self-Streamlining Wind Tunnel (TSWT) concerns model lift. The models tested provided data on their pressure distributions. This information was numerically integrated over the model surface to determine lift, pressure drag and pitching moment. However, the pressure drag is only a small component of the total drag at nominal angles of attack and cannot be used to assess the quality of flow simulation. An intrusive technique for obtaining information on the total drag of a model in TSWT is described. The technique adopted is the wake traverse method. The associated tunnel hardware and control and data reduction software are outlined and some experimental results are presented for discussion.

Wolf, S. W. D.↗

NASA NTRS1982

A mechanism for plasma waves at the harmonics of the plasma frequency foreshock boundary

A bump-on-tail unstable reduced velocity distribution, constructed from data obtained at the upstream boundary of the electron foreshock by the GSFC electron spectrometer experiment on the ISEE-1 satellite, is used as the initial plasma state for a numerical integration of the 1D-Vlasov-Maxwell system of equations. The integration is carried through the growth of the instability, beyond its saturation, and well into the stabilized plasma regime. A power spectrum computed for the electric field of the stabilized plasma is dominated by a narrow peak at the Bohm-Gross frequency of the unstable field mode but also contains significant power at the harmonics of the Bohm-Gross frequency. The harmonic power is in sharp peaks which are split into closely spaced doublets. The fundamental peak at the Bohm-Gross frequency is split into a closely spaced triplet. The mechanism for excitation of the second harmonic is shown to be second order wave-wave coupling.

Klimas, A. J.↗

NASA NTRS1982

Study of tethered satellite active attitude control

Existing software was adapted for the study of tethered subsatellite rotational dynamics, an analytic solution for a stable configuration of a tethered subsatellite was developed, the analytic and numerical integrator (computer) solutions for this "test case' was compared in a two mass tether model program (DUMBEL), the existing multiple mass tether model (SKYHOOK) was modified to include subsatellite rotational dynamics, the analytic "test case,' was verified, and the use of the SKYHOOK rotational dynamics capability with a computer run showing the effect of a single off axis thruster on the behavior of the subsatellite was demonstrated. Subroutines for specific attitude control systems are developed and applied to the study of the behavior of the tethered subsatellite under realistic on orbit conditions. The effect of all tether "inputs,' including pendular oscillations, air drag, and electrodynamic interactions, on the dynamic behavior of the tether are included.

Colombo, G.↗

NASA NTRS1981

Unsteady boundary-layer injection

The boundary-layer equations for two-dimensional incompressible flow are integrated numerically for the flow over a flat plate and a Howarth body. Injection is introduced either impulsively or periodically along a narrow strip. Results indicate that injection perpendicular to the wall is transmitted instantly across the boundary layer and has little effect on the velocity profile parallel to the wall. The effect is a little more noticeable for flows with adverse pressure gradients. Injection parallel to the wall results in fuller velocity profiles. Parallel and oscillatory injection appears to influence the mean. The amplitude of oscillation decreases with distance from the injection strip but further downstream it increases again in a manner reminiscent of an unstable process.

Telionis, D. P.↗

NASA NTRS1982

Transmission of sound across a vortex layer enclosing a cylindrical column of jet

An approximate solution to the problem of transmission of sound across a cylindrical vortex was obtained. Results are considerably different from the plane vortex sheet case because of the added role played by the curvature of the jet. In comparison with the plane case, the specularly transmitted waves are more complex and require some numerical integration. Resonance waves are identically predicted for M 2, but there is also a wave field whose modified effect appears to extend the region of resonance just as the instability waves cover a region in space and time. The instability waves are predicted to exist for all Mach numbers but vanish for wavelengths that are large compared to the jet radius. The region of propagation is similarly wavelength dependent.

Luh, R.↗

NASA NTRS1983

Experimental results of a deflected thrust V/STOL nozzle research program

Four deflected thrust nozzle concepts, designed to operate at the low pressure ratio typical of high bypass-ratio turbofan engines for medium speed (subsonic) V/STOL aircraft, were studied. Maps of overall performance characteristics and exit velocity distributions are used to highlight similarities and differences between the four concepts. Analytically determined secondary flows at the exit of a 90 deg circular pipe bend are compared with the experimental results from the more complex three dimensional geometries. The relative impact of total-pressure losses and secondary flows on nozzle thrust coefficient is addressed by numerical integration of exit velocity measurements.

Burstadt, P. L.↗

NASA NTRS1981

A computer program for the simulation of heat and moisture flow in soils

A computer program that simulates the flow of heat and moisture in soils is described. The space-time dependence of temperature and moisture content is described by a set of diffusion-type partial differential equations. The simulator uses a predictor/corrector to numerically integrate them, giving wetness and temperature profiles as a function of time. The simulator was used to generate solutions to diffusion-type partial differential equations for which analytical solutions are known. These equations include both constant and variable diffusivities, and both flux and constant concentration boundary conditions. In all cases, the simulated and analytic solutions agreed to within the error bounds which were imposed on the integrator. Simulations of heat and moisture flow under actual field conditions were also performed. Ground truth data were used for the boundary conditions and soil transport properties. The qualitative agreement between simulated and measured profiles is an indication that the model equations are reasonably accurate representations of the physical processes involved.

Camillo, P.↗

NASA NTRS1982

Vibration in Planetary Gear Systems with Unequal Planet Stiffnesses

An algorithm suitable for a minicomputer was developed for finding the natural frequencies and mode shapes of a planetary gear system which has unequal stiffnesses between the Sun/planet and planet/ring gear meshes. Mode shapes are represented in the form of graphical computer output that illustrates the lateral and rotational motion of the three coaxial gears and the planet gears. This procedure permits the analysis of gear trains utilizing nonuniform mesh conditions and user specified masses, stiffnesses, and boundary conditions. Numerical integration of the equations of motion for planetary gear systems indicates that this algorithm offers an efficient means of predicting operating speeds which may result in high dynamic tooth loads.

Frater, J. L.↗

NASA NTRS1983

Study of multiple optical transitions in 87Rb using laser diodes

Recent work with laser diodes has indicated their potential usefulness in optically pumping atomic frequency standards. Various optical pumping schemes for (87)Rb incorporating such light sources at two D1 frequencies for experimental situations of either evacuated wall-coated cells or atomic beams are considered. Numerical integration of rate equations governing the level populations with arbitrary pumping light choices of intensity, D1 hyperfine transition(s), and polarization and subsequent calculation of scattered light provide a simplified 0-0 hyperfine signal analysis. By frequency modulating one laser diode, two optical transitions were excited in an evacuated wall-coated cell and a large delta m(F) = 1 hf signal was observed.

Newton, C. L. J.↗

NASA NTRS1982

Nonlinear dynamo oscillations

The stability of steady equilibrium amplitudes of magnetic fields generated by convection flows is investigated. Two cases are considered in detail, for which steady solutions are available from the previous work of Busse (1973) and Busse (1977). In the first case instability occurs primarily because of magnetic flux expulsion at high magnetic Reynolds numbers. In the second case the change in the velocity field caused by the Lorentz force enhances dynamo action. This subcritical finite amplitude dynamo is potentially unstable. In typical cases the nonlinear dynamo oscillations that replace the steady equilibrium solutions are investigated by numerical integration.

Busse, F. H.↗