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Eight-page report describes systems of notation used most commonly to represent tensors of various ranks, with emphasis on tensors in Cartesian coordinate systems. Serves as introductory or refresher text for scientists, engineers, and others familiar with basic concepts of coordinate systems, vectors, and partial derivatives. Indicial tensor, vector, dyadic, and matrix notations, and relationships among them described.
The scattering of the sound of a jet engine by an airplane fuselage is modeled by solving the axially symmetric Helmholtz equation exterior to a long thin ellipsoid. The integral equation method based on the single layer potential formulation is used. A family of coordinate systems on the body is introduced and an algorithm is presented to determine the optimal coordinate system. Numerical results verify that the optimal choice enables the solution to be computed with a grid that is coarse relative to the wavelength.
A simple matrix expression is obtained for the strain components of a beam in which the magnitudes of neither beam displacements nor rotations are explicitly restricted. The only kinematical restrictions are that: (1) strains are small compared to unity and (2) components of local rotation, a newly identified kinematical quantity, are of the order of the strains to a fractional power equal to at least one half. Local rotations are defined as the change of orientation of material elements off the beam reference axis relative to those on the beam reference axis. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of both open- and closed-section beams in applications of the theory. The resulting strain components are expressed in a local Cartesian coordinate system and can be calculated directly in that way. Thus, one can use a curvilinear coordinate system that is natural to the beam problem without the complications that usually surround such an approach. Examples show the simplicity and the generality of the present approach as well as why previously published results differ among themselves concerning tension-torsion coupling.
Method for computing relationships between coordinate axes of two vector-measuring instruments not aligned with each other, one need only measure two nonparallel vectors u and v simultaneously in both coordinate systems. These measurements provide all data needed to compute transformation of any vector from one coordinate system to other. Particularly useful in computing alignments and transformations between accelerometers, gyroscopes, and instruments mounted on platforms subjected to vibrations, thermal strains, and other distortions causing alignments to vary unpredictably with time.
An accurate and efficient computational method is needed for three-dimensional incompressible viscous flows in engineering applications. On solving the turbulent shear flows directly or using the subgrid scale model, it is indispensable to resolve the small scale fluid motions as well as the large scale motions. From this point of view, the pseudo-spectral method is used so far as the computational method. However, the finite difference or the finite element methods are widely applied for computing the flow with practical importance since these methods are easily applied to the flows with complex geometric configurations. However, there exist several problems in applying the finite difference method to direct and large eddy simulations. Accuracy is one of most important problems. This point was already addressed by the present author on the direct simulations on the instability of the plane Poiseuille flow and also on the transition to turbulence. In order to obtain high efficiency, the multi-grid Poisson solver is combined with the higher-order, accurate finite difference method. The formulation method is also one of the most important problems in applying the finite difference method to the incompressible turbulent flows. The three-dimensional Navier-Stokes equations have been solved so far in the primitive variables formulation. One of the major difficulties of this method is the rigorous satisfaction of the equation of continuity. In general, the staggered grid is used for the satisfaction of the solenoidal condition for the velocity field at the wall boundary. However, the velocity field satisfies the equation of continuity automatically in the vorticity-vector potential formulation. From this point of view, the vorticity-vector potential method was extended to the generalized coordinate system. In the present article, we adopt the vorticity-vector potential formulation, the generalized coordinate system, and the 4th-order accurate difference method as the computational method. We present the computational method and apply the present method to computations of flows in a square cavity at large Reynolds number in order to investigate its effectiveness.
We have extended our previous efforts to derive through controlled photoclinometry, accurate, calibrated, high-resolution topographic models of the martian surface. The process involves combining MGS MOLA topographic profiles and MGS MOC Narrow Angle images. The earlier work utilized, along with a particular MOC NA image, the MOLA topographic profile that was acquired simultaneously, in order to derive photometric and scattering properties of the surface and atmosphere so as to force the low spatial frequencies of a one-dimensional MOC photoclinometric model to match the MOLA profile. Both that work and the new results reported here depend heavily on successful efforts to: 1) refine the radiometric calibration of MOC NA; 2) register the MOC to MOLA coordinate systems and refine the pointing; and 3) provide the ability to project into a common coordinate system, simultaneously acquired MOC and MOLA with a single set of SPICE kernels utilizing the USGS ISIS cartographic image processing tools. The approach described in this paper extends the MOC-MOLA integration and cross-calibration procedures from one-dimensional profiles to full two-dimensional photoclinometry and image simulations. Included are methods to account for low-frequency albedo variations within the scene.
In order to find the attitude of a spacecraft with respect to a reference coordinate system, vector measurements are taken. The vectors are pairs of measurements of the same generalized vector, taken in the spacecraft body coordinates, as well as in the reference coordinate system. We are interested in finding the best estimate of the transformation between these coordinate system.s The algorithm called QUEST yields that estimate where attitude is expressed by a quarternion. Quest is an efficient algorithm which provides a least squares fit of the quaternion of rotation to the vector measurements. Quest however, is a single time point (single frame) batch algorithm, thus measurements that were taken at previous time points are discarded. The algorithm presented in this work provides a recursive routine which considers all past measurements. The algorithm is based on on the fact that the, so called, K matrix, one of whose eigenvectors is the sought quaternion, is linerly related to the measured pairs, and on the ability to propagate K. The extraction of the appropriate eigenvector is done according to the classical QUEST algorithm. This stage, however, can be eliminated, and the computation simplified, if a standard eigenvalue-eigenvector solver algorithm is used. The development of the recursive algorithm is presented and illustrated via a numerical example.
A 3-D Navier-Stokes code has been developed for analysis of turbomachinery blade rows and other internal flows. The Navier-Stokes equations are written in a Cartesian coordinate system rotating about the x-axis, and then mapped to a general body-fitted coordinate system. Streamwise viscous terms are neglected using the thin-layer assumption, and turbulence effects are modeled using the Baldwin-Lomax turbulence model. The equations are discretized using finite differences on stacked C-type grids and are solved using a multistage Runge-Kutta algorithm with a spatially-varying time step and implicit residual smoothing. Calculations have been made of a horseshoe vortex formed in front of a flat plate with a round leading edge standing in a turbulent endwall boundary layer. Comparisons are made with experimental data taken by Eckerle and Langston for a circular cylinder under similar conditions. Computer and measured results are compared in terms of endwall flow visualization pictures and total pressure loss contours and vector plots on the symmetry plane. Calculated details of the primary vortex show excellent agreement with the experimental data. The calculations also show a small secondary vortex not seen experimentally.
A 3-D Navier-Stokes code has been developed for analysis of turbomachinery blade rows and other internal flows. The Navier-Stokes equations are written in a Cartesian coordinate system rotating about the x-axis, and then mapped to a general body-fitted coordinate system. Streamwise viscous terms are neglected using the thin-layer assumption, and turbulence effects are modeled using the Baldwin-Lomax turbulence model. The equations are discretized using finite differences on stacked C-type grids and are solved using a multistage Runge-Kutta algorithm with a spatially-varying time step and implicit residual smoothing. Calculations have been made of a horseshoe vortex formed in front of a flat plate with a round leading edge standing in a turbulent endwall boundary layer. Comparisons are made with experimental data taken by Eckerle and Langston for a circular cylinder under similar conditions. Computer and measured results are compared in terms of endwall flow visualization pictures and total pressure loss contours and vector plots on the symmetry plane. Calculated details of the primary vortex show excellent agreement with the experimental data. The calculations also show a small secondary vortex not seen experimentally.
A 3-D Navier-Stokes code was developed for analysis of turbomachinery blade rows and other internal flows. The Navier-Stokes equations are written in a Cartesian coordinate system rotating about the x-axis, and then mapped to a general body-fitted coordinate system. Streamwise viscous terms are neglected using the thin layer assumption, and turbulence effects are modelled using the Baldwin-Lomax turbulence model. The equations are discretized using finite differences on stacked C-type grids and are solved using a multistage Runge-Kutta algorithm with a spatially varying time step and implicit residual smoothing. Calculations were made of the flow around a supersonic throughflow fan blade. The fan was designed as a key component in a supersonic cruise engine. The 3-D calculations were done on a 129x29x33 grid and took 50 minutes of cpu time. Comparisons with the quasi-3-D results show minor differences in loading due to 3-D effects. Particle traces show nearly 2-D flows near the pressure surface, but large secondary flows within the suction surface boundary layer. The horseshoe vortex ahead of the leading edge is clearly seen.
Visible Scalable Terrain (ViSTa) is a software format for production, interchange, and display of three-dimensional (3D) terrain data acquired by stereoscopic cameras of robotic vision systems. ViSTa is designed to support scalability of data, accuracy of displayed terrain images, and optimal utilization of computational resources. In a ViSTa file, an area of terrain is represented, at one or more levels of detail, by coordinates of isolated points and/or vertices of triangles derived from a texture map that, in turn, is derived from original terrain images. Unlike prior terrain-image software formats, ViSTa includes provisions to ensure accuracy of texture coordinates. Whereas many such formats are based on 2.5-dimensional terrain models and impose additional regularity constraints on data, ViSTa is based on a 3D model without regularity constraints. Whereas many prior formats require external data for specifying image-data coordinate systems, ViSTa provides for the inclusion of coordinate-system data within data files. ViSTa admits highspeed loading and display within a Java program. ViSTa is designed to minimize file sizes and maximize compressibility and to support straightforward reduction of resolution to reduce file size for Internet-based distribution.
A range network is defined to consist of ground stations and targets where only distances between the two sets of points are observed. Such a network is said to be fundamental when only those six constraints are used that are needed to define the coordinate system for an adjustment. When ground stations or targets have certain configurations, a unique adjustment in terms of coordinates may be impossible, even when the number of observations is sufficient and the coordinate system is uniquely defined. Such configurations, resulting in the singular solutions, are said to be critical. Attention is therefore focused on singular solutions rather than on numerical problems in some ill-conditioned systems of fundamental range networks. The critical configurations are presented in two separate groups. The first deals with ground stations lying all in a plane and the second deals with ground stations generally distributed. The two kinds of problems require different mathematical treatments and lead to quite different conclusions.
Three dimensional surface measurements of large objects arc required in a variety of industrial processes. The nature of these measurements is changing as optical instruments arc beginning to replace conventional contact probes scanned over the objects. A common characteristic of the optical surface profilers is the trade off between measurement accuracy and field of view. In order to measure a large object with high accuracy, multiple views arc required. An accurate transformation between the different views is needed to bring about their registration. In this paper, we demonstrate how the transformation parameters can be obtained precisely by choosing control points which lie in the overlapping regions of the images. A good starting point for the transformation parameters is obtained by having a knowledge of the scanner position. The selection of the control points arc independent of the object geometry. By successively recording multiple views and obtaining transformation with respect to a single coordinate system, a complete physical model of an object can be obtained. Since all data arc in the same coordinate system, it can thus be used for building automatic models for free form surfaces.
The log-polar coordinate system, inherent in both polar exponential sensor arrays and log-polar remapped video imagery, is identical to the coordinate system of its corresponding Hough transform parameter space. The resulting unification of iconic and Hough domains simplifies computation for line recognition and eliminates the slope quantization problems inherent in the classical Cartesian Hough transform. The geometric organization of the algorithm is more amenable to massively parallel architectures than that of the Cartesian version. The neural architecture of the human visual cortex meets the geometric requirements to execute 'in-place' log-Hough algorithms of the kind described here.
An approximate method for calculating heating rates on three-dimensional vehicles at angle of attack is presented. The method is based on the axisymmetric analog for three-dimensional boundary layers and uses a generalized body fitted coordinate system. Edge conditions for the boundary layer solution are obtained from an inviscid flowfield solution, and because of the coordinate system used the method is applicable to any blunt body geometry for which a inviscid flowfield solution can be obtained. It is validated by comparing with experimental heating data and with Navier-Stokes calculations on the Shuttle orbiter at both wind tunnel and flight conditions and with Navier-Stokes calculations on the HL-20 at wind tunnel conditions.
Tensor analysis is one of the more abstruse, even if one of the more useful, higher math subjects enjoined by students of physics and engineering. It is abstruse because of the intellectual gap that exists between where most physics and engineering mathematics leave off and where tensor analysis traditionally begins. It is useful because of its great generality, computational power, and compact, easy to use, notation. This paper bridges the intellectual gap. It is divided into three parts: algebra, calculus, and relativity. Algebra: In tensor analysis, coordinate independent quantities are sought for applications in physics and engineering. Coordinate independence means that the quantities have such coordinate transformations as to leave them invariant relative to a particular observer s coordinate system. Calculus: Non-zero base vector derivatives contribute terms to dynamical equations that correspond to pseudoaccelerations in accelerated coordinate systems and to curvature or gravity in relativity. These derivatives have a specific general form in tensor analysis. Relativity: Spacetime has an intrinsic geometry. Light is the tool for investigating that geometry. Since the observed geometry of spacetime cannot be made to match the classical geometry of Euclid, Einstein applied another more general geometry differential geometry. The merger of differential geometry and cosmology was accomplished in the theory of relativity. In relativity, gravity is equivalent to curvature.
The author has identified the following significant results. Software development for a computer-aided crop and soil survey system is nearing completion. Computer-aided variety classification accuracies using LANDSAT-1 MSS data for a 600 hectare citrus farm were 83% for Redblush grapefruit and 91% for oranges. These accuracies indicate that there is good potential for computer-aided inventories of grapefruit and orange citrus orchards with LANDSAT-type MSS data. Mean digital values of clouds differed statistically from those for crop, soil, and water entities, and those for cloud shadows were enough lower than sunlit crop and soil to be distinguishable. The standard errors of estimate for the calibration of computer compatible tape coordinate system (pixel and record) to earth coordinate system (longitude and latitude) for 6 LANDSAT scenes ranged from 0.72 to 1.50 pixels and from 0.58 to 1.75 records.
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).