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The solution of partial differential equations which describe physical phenomena by the use of coordinate transformations is described. The constraints of the problems are stated in geometric terms and include boundary constraints, uniformity constraints, and internal constraints. Algebraic mesh generations are very satisfactory for these constraints.
Topics include: (1) polar coordinate transformations (M83); (2) multispectral ratios (M82); (3) maximum entropy restoration (M87); (4) automated computation of stellar magnitudes in nebulosity; (5) color and polarization; (6) aliasing.
An implicit finite difference scheme for an efficient computation of unsteady potential flow about airfoils is presented. The formulation uses density and velocity potential as dependent variables, and is cast in conservation form to assure the theoretically correct determination of shockwave location and speed. To enable boundary conditions to be imposed directly on the airfoil surface, a time varying sheared rectilinear coordinate transformation is employed. Calculated time history solutions on a pulsating airfoil are compared with the results of another unsteady transonic code. It is concluded that the method has excellent numerical stability and gives accurate solutions with sharply resolved shocks.
A method is developed and illustrated which finds minimum changes in analytical mass and stiffness matrices to make them consistent with a set of measured normal modes and natural frequencies. The corrected model is an improved base for studies of physical changes, changes in boundary conditions, and for prediction of forced responses. Features of the method are: efficient procedures not requiring solutions of the eigenproblem; the model may have more degrees of freedom than the test data; modal displacements at all the analytical degrees of freedom are obtained; the frequency dependence of the coordinate transformations are properly treated.
An implicit finite-difference scheme is presented for the efficient computation of unsteady potential flow about airfoils. The formulation uses density and the velocity potential as dependent variables and is cast in conservation form to assure the theoretically correct determination of shockwave location and speed. To enable boundary conditions to be imposed directly on the airfoil surface, a time-varying sheared-rectilinear coordinate transformation is employed. Calculated time-history solutions on a pulsating airfoil are compared with the results of other unsteady transonic codes, including a previous method of the authors. The present method is demonstrated to be unconditionally stable and to give accurate solutions with sharply resolved shocks.
Techniques are investigated for on-line estimation of rotor states in the nonrotating frame from multiple, simultaneous measurements in the rotating frame. The multiblade coordinate transformation is first applied to transform both flapping and flapping rate measurements into the nonrotating frame. The 'observer' approach is then used to generate algorithms for estimating tip-path plane rate and attitude from transformed flapping and flapping rate measurements. A numerical evaluation using simulated measurements is conducted to evaluate the performance of the algorithms and recommendations are made.
A class of interactive control systems is derived by generalizing interactive manipulator control systems. The general structural properties of such systems are discussed and an appropriate general software implementation is proposed. This is based on the fact that tasks of interactive control systems can be represented as a network of a finite set of actions which have specific operational characteristics and specific resource requirements, and which are of limited duration. This has enabled the decomposition of the overall control algorithm into a set of subalgorithms, called subcontrollers, which can operate simultaneously and asynchronously. Coordinate transformations of sensor feedback data and actuator set-points have enabled the further simplification of the subcontrollers and have reduced their conflicting resource requirements. The modules of the decomposed control system are implemented as parallel processes with disjoint memory space communicating only by I/O. The synchronization mechanisms for dynamic resource allocation among subcontrollers and other synchronization mechanisms are also discussed in this paper. Such a software organization is suitable for the general form of multiprocessing using computer networks with distributed storage.
The software specifications for the MAGSAT data processing system (MDPS) are presented. The MDPS is divided functionally into preprocessing of primary input data, data management, chronicle processing, and postprocessing. Data organization and validity, and checks of spacecraft and instrumentation are dicussed. Output products of the MDPS, including various plots and data tapes, are described. Formats for important tapes are presented. Dicussions and mathematical formulations for coordinate transformations and field model coefficients are included.
Subprograms for transforming coordinates and time, for determining the position of the Moon and Sun, and for calculating the atmosphere and disturbances, which are specified by anomalies of the Earth's gravitational field are described. The subprograms are written in FORTRAN IV and form a major part of the package of applied programs for calculating the navigational parameters of artificial Earth satellites.
For the U.S. Geological Survey maps at 1:1,000,000-scale and larger, the most common projections are conformal, such as the Transverse Mercator and Lambert Conformal Conic. Projections for these scales should treat the Earth as an ellipsoid. In addition, the USGS has conceived and designed some new projections, including the Space Oblique Mercator, the first map projection designed to permit low-distortion mapping of the Earth from satellite imagery, continuously following the groundtrack. The USGS has programmed nearly all pertinent projection equations for inverse and forward calculations. These are used to plot maps or to transform coordinates from one projection to another. The projections in current use are described.
An instrumentation system to provide a noncontact method for measurement of target positions was evaluated. The system employs two electro-optic camera units which give stereo information for use in determining three dimensional target locations. Specially developed, infrared sensitive photodetectors are used in the cameras to sense radiation from light emitting diode targets. Up to 30 of these targets can be monitored with a sampling rate of 312 Hz per target. An important part of the system is a minicomputer which is used to collect the camera data, sort it, make corrections for distortions in the electro-optic system, and perform the necesssary coordinate transformations. If target motions are restricted to locations in a plane which is perpendicular to a camera's optical axis, the system can be used with just one camera. Calibrations performed in this mode characterize accuracies in single camera operation. This information is also useful in determination of single camera contributions to total system errors. For this reason the system was tested in both the single camera and two camera (stereo) modes of operation.
A system of computer programs was developed to model general three dimensional surfaces. Surfaces are modeled as sets of parametric bicubic patches. There are also capabilities to transform coordinates, to compute mesh/surface intersection normals, and to format input data for a transonic potential flow analysis. A graphical display of surface models and intersection normals is available. There are additional capabilities to regulate point spacing on input curves and to compute surface/surface intersection curves. Input and output data formats are described; detailed suggestions are given for user input. Instructions for execution are given, and examples are shown.
Techniques for the correction of errors in Doppler radar scans due to advection effects are presented. A moving frame of reference is shown to be useful in least-squares estimates with stationary observations expressed in scalars or Cartesian coordinates. For non-Cartesian coordinates, such as the deduction of radial velocities from triple Doppler radar data, an integral is defined for accounting for advection effects and consequent coordinate transformations. The multiple radars are necessary for unambiguous characterization of the horizontal wind velocity. A scale analysis is employed to estimate errors with and without the error correction procedure. Improvements in correlations between scans are demonstrated when the error correction method is used.
The significant changes that have been made in the Optical Alignment Facility at the Goddard Space Flight Center since it was last described (Appler and Howell, 1968) are discussed. The changes are reflected in both the hardware and the software involved in making alignment measurements. The data analysis procedures can now be done on a desktop computer with turn-around times of approximately 4 h. Arbitrary coordinate transformations and corrections for facility errors are also feasible while retaining absolute accuracies of less than 6 sec of arc.
A hybrid numerical algorithm, developed to solve the full three-dimensional Navier-Stokes equations, is applied to the computation of the flowfield in a simulated three-dimensional high speed aircraft inlet at a Mach number of 2.5 and Reynolds number of 1.4 x 10 to the 7th based on inlet length. The numerical algorithm incorporates a coordinate transformation in order to handle general flow geometries, and utilizes the algebraic turbulent eddy viscosity model of Baldwin and Lomax. The hybrid algorithm has been vectorized on the CDC CYBER 203 computer using the SL/1 vector programming language developed at NASA Langley. The computed results are compared with experimental measurements of the ramp and cowl static pressures, and boundary layer pitot profiles. The results are also compared with a previous two-dimensional Navier-Stokes computation of the same configuration. The agreement with the experimental data is generally good; however, additional improvements in turbulence modeling are needed.
The GEMPAK system, a general meteorological software package being developed at NASA/Goddard Space Flight Center to support mesoscale meteorological research programs, is described. The primary purpose of the system is to provide analysis support and data integration techniques for conventional and satellite derived data sets. Current capabilities of the system range from data listing and editing to interactive objective analysis procedures and coordinate transformations. Output graphics use a graphics subroutine package designed to support meteorological plotting functions. A flexible diagnostics package is currently under development.
The problem of minimum-time interception of a target flying in three dimensional space is analyzed with the interceptor aircraft modeled through energy-state approximation. A coordinate transformation that uncouples the interceptor's extremals from the target motion in an open-loop sense is introduced, and the necessary conditions for optimality and the optimal controls are derived. Example extremals are shown.
A procedure for analyzing the vibrations of rotating turbomachinery blades has been previously developed. This procedure is based upon shallow shell theory, and utilizes the Ritz method to determine frequencies and mode shapes. However, it has been limited heretofore to blades of uniform thickness, uniform curvature, and/or twist and rectangular planform. The present work shows how the procedure may be generalized to eliminate the aforementioned restrictions. Nonrectangular planforms are dealt with by a suitable coordinate transformation. This, as well as variable thickness, curvature and twist, require using numerical integration. The procedure is demonstrated on four examples of cantilevered blades for which theoretical and experimental data have been previously published: (1) flat plate with spanwise taper, (2) flat plate with chordwise taper, (3) twisted plate with chordwise taper, and (4) cylindrical shell with chordwise taper.