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At least 343 records · Page 19

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NASA NTRS1975

Coordinate transformation and truncation for rotating spacecraft with flexible appendages

The analytical structures of alternative coordinate transformations for the variables that characterize the deformations of finite element models of flexible appendages on rotating spacecraft are examined. Particular emphasis is given to the truncation operations that are essential for efficient numerical simulations of flexible spacecraft, due to the necessarily large number of coordinates employed in a finite element description. A theorem is developed which establishes necessary conditions for the commutativity of the required truncation and inversion operations. The recommended alternative for the rotating elastic appendage is a complex transformation with previously published orthogonality properties, which permit the derivation in this paper of a relatively simple explicit set of transformed and truncated equations of modal vibration in real variables.

Likins, P.↗

NASA NTRS1976

Numerical solutions for laminar and turbulent viscous flow over single and multi-element airfoils using body-fitted coordinate systems

The technique of body-fitted coordinate systems is applied in numerical solutions of the complete time-dependent compressible and incompressible Navier-Stokes equations for laminar flow and to the time-dependent mean turbulent equations closed by modified Kolmogorov hypotheses for turbulent flow. Coordinate lines are automatically concentrated near to the bodies at higher Reynolds number so that accurate resolution of the large gradients near the solid boundaries is achieved. Two-dimensional bodies of arbitrary shapes are treated, the body contour(s) being simply input to the program. The complication of the body shape is thus removed from the problem.

Thompson, J. F.↗

NASA NTRS1977

Computer transformation of partial differential equations into any coordinate system

The use of tensors to provide a compact way of writing partial differential equations in a form valid in all coordinate systems is discussed. In order to find solutions to the equations with their boundary conditions they must be expressed in terms of the coordinate system under consideration. The process of arriving at these expressions from the tensor formulation was automated by a software system, TENSR. An allied system that analyzes the resulting expressions term by term and drops those that are negligible is also described.

Sullivan, R. D.↗

NASA NTRS1977

Coordinated rocket and satellite measurements of an auroral event. I - Satellite observations and analysis

Results of a coordinated auroral experiment involving the Atmosphere Explorer C satellite and a sounding rocket are reported. Auroral primary electron fluxes and neutral gas densities measured by instruments on the satellite are used in a model calculation of the thermospheric manifestation of the aurora. There is encouraging agreement between the calculated and measured electron density, electron temperature, secondary electron flux, and O I emissions at 5577 and 6300 A. A discrepancy between the calculated and the rocket-measured 3914-A emission profile is discussed in terms of experiment geometry and auroral physics. The coordinated measurements are used to infer vertical fluxes of ionization and of electron thermal energy at high altitudes

Rees, M. H.↗

NASA NTRS1977

Numerical solutions for viscous and potential flow about arbitrary two-dimensional bodies using body-fitted coordinate systems

A procedure for numerical solution of the time-dependent, incompressible Navier-Stokes equations for the flow about arbitrarily shaped two-dimensional bodies is given. This solution is based on a technique of automatic numerical generation of a curvilinear coordinate system having a coordinate line coincident with the body contour regardless of its shape. The implicit solution utilizes the vorticity-stream function formulation with a false-position iterative adjustment of the surface vorticity in satisfaction of the no-slip boundary condition. Excellent agreement with the Blasius boundary layer solution is obtained for a semi-infinite flat plate. Results are presented for Reynolds numbers up to 2000 for several airfoils and a cambered rock.

Thames, F. C.↗

NASA NTRS1978

Body-fitted coordinates systems transformations

Two computer programs generate two-dimensional body-fitted coordinate systems and coordinate transformation. Programs are useful in fields requiring accurate numerical representation of boundary conditions and accurate numerical solutions of partial differential equations.

Mastin, C. W.↗

NASA NTRS1978

Applications of the hybrid coordinate method to the TOPS autopilot

Preliminary results are presented from the application of the hybrid coordinate method to modeling TOPS (thermoelectric outer planet spacecraft) structural dynamics. Computer simulated responses of the vehicle are included which illustrate the interaction of relatively flexible appendages with an autopilot control system. Comparisons were made between simplified single-axis models of the control loop, with spacecraft flexibility represented by hinged rigid bodies, and a very detailed three-axis spacecraft model whose flexible portions are described by modal coordinates. While single-axis system, root loci provided reasonable qualitative indications of stability margins in this case, they were quantitatively optimistic when matched against responses of the detailed model.

Fleischer, G. E.↗

NASA NTRS1978

/U,B,K/ coordinates - A natural system for studying magnetospheric convection

The formulation of the magnetospheric convection problem involves a coordinate system consisting of the electric potential, the magnetic field intensity, and the modified longitudinal invariant. The mapping is extended to include all particles and not just those mirroring in the magnetic equator. Equations of motion for the particle mirror points in this representation are presented. The analysis results in a natural division of the magnetosphere into accelerator and dynamo regions and use of the coordinate system permits trajectory-dependent effects to be distinguished from source- or sink-dependent effects. The representation is illustrated by application to the analysis of the nose structure protons observed by Smith and Hoffman (1974).

Whipple, E. C., Jr.↗

NASA NTRS1979

The Look-point Aircraft Coordinate Estimator (LACE) and potential applications

A look-point aircraft coordinate estimator (LACE) consisting of a windshield runway symbol projector, pilot input controls, microprocessor, and eye-alignment device is described. The estimator is used by a pilot to determine his aircraft's position relative to a runway or other visible terrain or target. The pilot initially superimposes and then corrects the superposition of the runway symbol over the runway during approach during periods when the runway is visible. Using the pilot's inputs the microprocessor calculates the position of the aircraft in terms of runway coordinates, then generates an approach trajectory and issues instructions to an autopilot. The microprocessor contains a model of the aircraft's dynamics and calculates a theoretical aircraft trajectory. The theoretical position of the aircraft is then used to drive the runway symbol, with the pilot's input being additive. The system thus acts as an aid in making low visibility approaches and landings when only an occasional glimpse of the runway is possible and no ground referenced landing systems are available. The system can also be used as an independent landing monitor for ground referenced landing systems.

Anderson, W. W.↗

NASA NTRS1979

Tropospheric limitations on the accuracy of phase measurement of coordinates in astronomy

The effect of tropospheric fluctuation on the accuracy of phase measurements of coordinates is discussed. The nature of the averaging of the tropospheric effects, if N coordinate measurements of duration T with period mu are made, is investigated. Various averaging modes depending on the relation of the various time parameters are investigated. Equations taking into account the correlations between individual observations are presented. It is shown that the correlation interval between the individual observations is always greater than the fluctuation period of tropospheric inhomogeneities typical for a given baseline.

Dravskikh, A. F.↗

NASA NTRS1979

A computational fluid dynamic technique valid at the centerline for non-axisymmetric problems in cylindrical coordinates

A technique is described for the numerical solution of non-axisymmetric flow problems posed in cylindrical coordinates when the z-axis is included in the flowfield. The highlight of the technique is the manner in which the singularities at the centerline are handled. Specifically, the governing flowfield equations at r = 0 are put in a special form by applying L'Hospital's Rule. The required radial derivatives are evaluated using a one-sided, second-order accurate, first-difference. This leads to a smooth, convergent calculation of the flowfield at the centerline. This appears to be the first generally applicable numerical method for avoiding coordinate system singularities in the context of a finite-difference scheme, and could have application to many nonaxisymmetric flows. The technique is illustrated by specific results for the time-dependent flowfield inside an internal combustion engine.

Griffin, M. D.↗

NASA NTRS1980

A general time element using Cartesian coordinates: Eccentric orbit integration

A general time element, valid with any arbitrary independent variables, and used with Cartesian coordinates for the integration of the elliptic motion in orbits, is examined. The derivation of the time element from a set of canonical elements of the Delaunay type, developed in the extended phase space, is presented. The application of the method using an example of a transfer orbit for a geosynchronous mission is presented. The eccentric and elliptic anomaly are utilized as the independent variable. The reduction of the in track error resulting from using Cartesian coordinates with the time element is reported.

Janin, G.↗

NASA NTRS1980

Application of a numerically generated orthogonal coordinate system to the solution of inviscid axisymmetric supersonic flow over blunt bodies

A numerically generated orthogonal coordinate system (with the body surface and shock wave as opposite boundaries) was applied with a time asymptotic method to obtain steady flow solutions for axisymmetric inviscid flow over several blunt bodies including spheres, paraboloids, ellipsoids, hyperboloids, hemisphere cylinders, spherically blunted cones, and a body with a concavity in the stagnation region. Comparisons with experimental data and with the results of other computational methods are discussed. The numerically generated orthogonal coordinate system is described and applications of the method to complex body shapes, particularly those with concave regions, are discussed.

Hamilton, H. H., II↗

NASA NTRS1979

Complete supersonic flowfields over blunt bodies in a generalized orthogonal coordinate system

A general orthogonal coordinate system is used to describe various axisymmetric and two-dimensional shapes. Close approximations to planetary probe configurations are possible. The full Navier-Stokes equations are discretized in this coordinate system in a manner based on Allen and Cheng's numerical procedure. The blow shock is treated as a discontinuity which floats between grid points. Completely coupled flows over the forebody, base, and near wake were calculated over a cylinder, sphere, and an approximation to the Viking Aeroshell. Some problem areas in determining the base flow for increasing Reynolds number are discussed. In particular, it is found that the mean free path of the fluid near the wall immediately below the corner of the Viking Aeroshell, which experiences a severe expansion, can become greater than the local mesh size required to resolve the boundary layer in the forebody.

Gnoffo, P. A.↗

NASA NTRS1980

Improved coordinates of features in the vicinity of the Viking lander site on Mars

The measurement of longitude of the Viking 1 landing site and the accuracy of the coordinates of features in the area around the landing site are discussed. The longitude must be measured photogrammatically from the small crater, Airy 0, which defines the 0 deg meridian on Mars. The computer program, GIANT, which was used to perform the analytical triangulations, and the photogrammetric computation of the longitude of the Viking 1 lander site are described. Improved coordinates of features in the vicinity of the Viking 1 lander site are presented.

Davies, M. E.↗

NASA NTRS1980

A comparison of finite difference methods for solving Laplace's equation on curvilinear coordinate systems

Various finite difference techniques used to solve Laplace's equation are compared. Curvilinear coordinate systems are used on two dimensional regions with irregular boundaries, specifically, regions around circles and airfoils. Truncation errors are analyzed for three different finite difference methods. The false boundary method and two point and three point extrapolation schemes, used when having the Neumann boundary condition are considered and the effects of spacing and nonorthogonality in the coordinate systems are studied.

Mccoy, M. J.↗

NASA NTRS1980

Computer coordination of limb motion for a three-legged walking robot

Coordination of the limb motion of a vehicle which could perform assembly and maintenance operations on large structures in space is described. Manipulator kinematics and walking robots are described. The basic control scheme of the robot is described. The control of the individual arms are described. Arm velocities are generally described in Cartesian coordinates. Cartesian velocities are converted to joint velocities using the Jacobian matrix. The calculation of a trajectory for an arm given a sequence of points through which it is to pass is described. The free gait algorithm which controls the lifting and placing of legs for the robot is described. The generation of commanded velocities for the robot, and the implementation of those velocities by the algorithm are discussed. Suggestions for further work in the area of robot legged locomotion are presented.

Klein, C. A.↗

NASA NTRS1980

Coordinates of features on the Galilean satellites

The coordinate systems of each of the Galilean satellites are defined and coordinates of features seen in the Voyager pictures of these satellites are presented. The control nets of the satellites were computed by means of single block analytical triangulations. The normal equations were solved by the conjugate iterative method which is convenient and which converges rapidly as the initial estimates of the parameters are very good.

Davies, M. E.↗