A generalized multiple scales approach to a class of linear differential equations
Generalized multiple scales method for solving linear differential equations with variable coefficients applied to Liouville-Green approximation
Ramnath, R. V.↗
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On differential transformations between Cartesian and curvilinear (geodetic) coordinates
Differential transformations are developed between Cartesian and curvilinear orthogonal coordinates. Only matrix algebra is used for the presentation of the basic concepts. After defining the reference systems used the rotation (R), metric (H), and Jacobian (J) matrices of the transformations between cartesian and curvilinear coordinate systems are introduced. A value of R as a function of H and J is presented. Likewise an analytical expression for J(-1) as a function of H(-2) and R is obtained. Emphasis is placed on showing that differential equations are equivalent to conventional similarity transformations. Scaling methods are discussed along with ellipsoidal coordinates. Differential transformations between elipsoidal and geodetic coordinates are established.
Soler, T.↗
Aerodynamic heating to representative SRB and ET protuberances
Heating data and data scaling methods which can be used on representative solid rocket booster and external tank (ET) protuberances are described. Topics covered include (1) ET geometry and heating points; (2) interference heating test data (51A); (3) heat transfer data from tests FH-15 and FH-16; (4) individual protuberance data; and (5) interference heating of paint data from test IH-42. A set of drawings of the ET moldline and protuberances is included.
Engel, C. D.↗
Lunar rock surfaces as detectors of solar processes
Lunar rock surfaces exposed at or just below the lunar surface are considered as detectors of the solar wind, solar flares and solar-derived magnetic fields through their interactions with galactic cosmic rays. The degradation of the solar detector capabilities of lunar surface rocks by meteoroid impact erosion, accreta deposition, loose dust, and sputtering, amorphous layer formation and accelerated diffusion due to solar particles and illumination is discussed, and it is noted that the complex interactions of factors affecting the outer micron of exposed surface material has so far prevented the development of a satisfactory model for a particle detector on the submicron scale. Methods for the determination of surface exposure ages based on the accumulation of light solar wind noble gases, Fe and Mg, impact craters, solar flare tracks, and cosmogenic Kr isotopes are examined, and the systematic variations in the ages determined by the various clocks are discussed. It is concluded that a means of obtaining satisfactory quantitative rate or flux data has not yet been established.
Hartung, J. B.↗
Structural dynamics payload loads estimates
Present analytical techniques by which design loads are predicted are very costly and time consuming. Chapter I presents the standard techniques used to analyze payload/booster systems. They are full scale methods in the sense that they all require the solution of the coupled equations of motion of the payload/booster system. Chapter II identifies several short cut methodologies. These already existing techniques do not require the solution of the coupled system equations. The potentials and shortcomings of each of these methods are discussed. Chapter III covers the favored methods accompanied by recommendations for further development, refinement, and demonstrations. An outline of a new approach is also included.
Engels, R. C.↗
The evolution of clusters of galaxies. I - Very rich clusters
A multiple one-body Monte Carlo code is used to investigate the evolution of galaxies in a steady cluster potential under the influence of dynamical friction, two-body relaxation, tidal stripping, and galaxy mergers. The basic assumptions, estimated time scales, method and computer program, and effects of uncertainties in galaxy encounter physics are addressed. Numerical experiments in which the mass initially carried by galaxies and the mass function are varied are reported. It is shown that the formation of a very massive cluster galaxy depends critically on the number of galaxies, the initial division of cluster material between galaxies and a smooth intracluster medium, the mass spectrum of the galaxies, and chance. By the end of all simulations, less than half of the mass in the central regions of the cluster is bound to galaxies. It appears possible to produce any of the Bautz-Morgan classes from very similar initial conditions.
Richstone, D. O.↗
Localized blocking accounted for by slow variations of the zonal driving
The incorporation of observed zonal topography in a barotropic nonlinear channel model is discussed. The model is then extended to explain the regional features of blocking by employing a two scale method to allow small slow variations of the momentum (zonal) driving.
Mo, K. C.↗
Accuracy of the energy-corrected sudden (ECS) scaling procedure for rotational excitation of CO by collisions with Ar
The utility of the energy-corrected sudden (ECS) scaling method is evaluated on the basis of how accurately it predicts the entire matrix of state-to-state rate constants, when the fundamental rate constants are independently known. It is shown for the case of Ar-CO collisions at 500 K that when a critical impact parameter is about 1.75-2.0 A, the ECS method yields excellent excited state rates on the average and has an rms error of less than 20 percent.
Green, S.↗
The interference of the model support mast with measurements of the longitudinal and lateral aerodynamic coefficients
The effects the single-bottom support masts used in the ONERA S1 and S4 wind tunnels have on aerodynamic data collected with scale model aircraft were examined experimentally and analytically. Systematic studies were performed on the flow characteristics around different diameters for the mounts. Scaling methods used to make data from one wind tunnel correspond to data from the other are described. Airbus 320 models were introduced into the tests and mast-body flow interactions were observed. A summary is presented of restrictions on the mast diameters, relative to cylindrical model diameters, which will minimize the effects the masts have on longitudinal and lateral aerodynamic stability data.
Vandekreeke, C.↗
Accretion rates of protoplanets
Projections are made of the collision/accretion rates of planetesimals in a uniform surface-density disk with a massive protoplanet, whose collision cross-section is enhanced, relative to its geometric cross-section, by a gravitational focusing of the planetesimals' trajectories. A scaling method is presented which generalizes the results obtained to protoplanets of arbitrary mass, radius, and orbital semimajor axis. It is established that, in the case where one protoplanet dominates gravitational scattering in a given protoplanetary disk region, planetesimal inclinations are substantially less rapidly excited than eccentrocities; the skewed velocity dispersion thus generated allows for a more rapid rate of protoplanet growth.
Greenzweig, Yuval↗
On WKB expansions for Alfven waves in the solar wind
The WKB expansion for 'toroidal' Alfven waves in solar wind, which is described by equations of Heinemann and Olbert (1980), is examined. In this case, the multiple scales method (Nayfeh, 1981) is used to obtain a uniform expansion. It is shown that the WKB expansion used by Belcher (1971) and Hollweg (1973) for Alfven waves in the solar wind is nonuniformly convergent.
Hollweg, Joseph V.↗
Iterative repair for scheduling and rescheduling
An iterative repair search method is described called constraint based simulated annealing. Simulated annealing is a hill climbing search technique capable of escaping local minima. The utility of the constraint based framework is shown by comparing search performance with and without the constraint framework on a suite of randomly generated problems. Results are also shown of applying the technique to the NASA Space Shuttle ground processing problem. These experiments show that the search methods scales to complex, real world problems and reflects interesting anytime behavior.
Zweben, Monte↗
Nonlinear response of infinitely long circular cylindrical shells to subharmonic radial loads
The nonlinear response of infinitely long circular cylindrical shells (thin circular rings) in the presence of a two-to-one internal (autoparametric) resonance to a subharmonic excitation of order one-half of the higher mode is analyzed with the multiple-scale method. Four autonomous first-order ordinary differential equations are derived for the modulation of the amplitudes and phases of the interacting models. These modulation equations are used to determine the fixed points and their stability. The fixed points correspond to periodic oscillations of the shell, whereas the limit-cycle solutions of the modulation equations correspond to amplitude and phase-modulated oscillations of the shell. The force response curves exhibit saturation, jumps, and Hopf bifurcation. As excitation frequency changes, all limit cycles deform and lose stability through either pitchfork or cyclic-fold (saddle-node) bifurcations. Some of these saddle-node bifurcations cause a transition to chaos. The pitchfork bifurcations break the symmetry of the limit cycles.
Nayfeh, Ali H.↗
A three-parameter approximation for radiative transfer in nonhomogeneous atmospheres - Application to the O3 9.6-micron band
A three-parameter scaling approximation is developed in conjunction with random models for the computation of transmittance along nonhomogeneous atmospheres. In addition to the scaled path length and pressure, a third parameter is introduced to account for intermediate absorption. With reference to the results derived from line-by-line calculations, the accuracy of this scaling approximation is assessed in different atmospheric conditions. For the O3 9.6-micron band where the Curtis-Godson and other conventional scaling methods usually produce large errors, the present model can achieve relative accuracies within about 1 percent in transmittance and the downward surface flux. Accuracy of about 0.15 K/d can be obtained in cooling rate calculations. The present three-parameter approximation is well suited for radiative transfer parameterizations in which the demand for high accuracy is required in the transmittance and flux calculations.
Fu, Qiang↗
On robustness analysis with non-diagonally structured uncertainty
The authors study the structured singular value for a class of robustness analysis problems in which the uncertainty is allowed to be nondiagonally structured, and the norm of each block is individually considered. The results extend the previous work conducted for problems where each uncertainty block is assumed to be a scalar. Computations of this generalized notion via both the similarity and nonsimilarity scaling methods are investigated. It is shown in particular that under certain circumstances the structured singular value defined here coincides with a vector-induced norm.
Chen, Jie↗
Measured acoustic characteristics of ducted supersonic jets at different model scales
A large-scale (about a 25x enlargement) model of the Georgia Tech Research Institute (GTRI) hardware was installed and tested in the Propulsion Systems Laboratory of the NASA Lewis Research Center. Acoustic measurements made in these two facilities are compared and the similarity in acoustic behavior over the scale range under consideration is highlighted. The study provide the acoustic data over a relatively large-scale range which may be used to demonstrate the validity of scaling methods employed in the investigation of this phenomena.
Jones, R. R., III↗
Digital implementation of vibration suppression controllers for LSS
A synthesis procedure for positivity controllers is described, and both simulation and experimental results for vibration suppression control experiments at the NASA Marshall ACES and the NASA Langley MiniMAST ground test facilities are presented. Positivity-designed controllers provide general stability robustness to modeling errors, coupled with controller phase compensation which provides stability margins required for actual implementation. A multivariable scaling method of positivity-designed controllers is used for the on-line tuning of nominal digital controllers. A phase compensation method uses model information to predict system measurements and controller states for compensating the nominal controllers for unmodeled time delays in the system.
Bosse, A.↗
Acoustical and Flowfield Characterization of a Scaled Tabletop Rocket
An analysis of the acoustical and flowfield environment for the scaled 1-pound-force (lbf) thrust tabletop motor was performed. This tabletop motor from NASA Stennis Space Center Is composed of Plexiglas burning In gaseous oxygen with a graphite insert for the nozzle portion. The nozzle has a throat diameter of 0.2 inch and an exit diameter of 0.38 Inch. With a chamber pressure at 55 pounds per square Inch absolute (psia), a normal shock is formed immediately downstream of the nozzle exit plane as the combustion products exhaust into the ambient at atmospheric pressure. The jet characterization Is based on computational fluid dynamics (CFD) in conjunction with Kirchhoff surface integral formulation and compared with correlations developed for measured rocket noise and a pressure fluctuation scaling (PFS) method. Predictions and comparisons are made for the overall sound pressure levels (OASPL's) and spectral dependence of sound pressure level (SPL). The overall objective of this effort is to develop methods for scaling the acoustic and flowfield environment of rockets with a wide range of thrust (1 lbf to 1 million lbf).
Kandula, Max↗