Computer generation of arbitrary functions
Computer generation of data matrixes for arbitrary functions by interpolation series
Fogarty, L. E.↗
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Results for “Arbitrary functions”
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High precision moments for the Fourier coefficients of arbitrary functions
High precision moments for fourier coefficients of arbitrary functions
Zohar, S.↗
Three-dimensional addition theorem for arbitrary functions involving expansions in spherical harmonics
Three-dimensional addition theorem for arbitrary functions involving expansions in spherical harmonics
ADDITION THEOREM↗
Separation of variables in the special diagonal Hamilton-Jacobi equation: Application to the dynamical problem of a particle constrained on a moving surface
For a time-dependent, n-dimensional, special diagonal Hamilton-Jacobi equation a necessary and sufficient condition for the separation of variables to yield a complete integral of the form was established by specifying the admissible forms in terms of arbitrary functions. A complete integral was then expressed in terms of these arbitrary functions and also the n irreducible constants. As an application of the results obtained for the two-dimensional Hamilton-Jacobi equation, analysis was made for a comparatively wide class of dynamical problems involving a particle moving in Euclidean three-dimensional space under the action of external forces but constrained on a moving surface. All the possible cases in which this equation had a complete integral of the form were obtained and these are tubulated for reference.
Blanchard, D. L.↗
A contribution to the controllability of time-lag systems.
Consideration of a property of 'complete controllability' which is a particular case of complete controllability with respect to an arbitrary function. The latter concept has been studied by Popov (1970), who gave a necessary and sufficient condition of controllability in algebraic form. It is shown that the necessary condition of complete controllability with respect to an arbitrary function is also a necessary condition of the particular property of complete controllability under certain given conditions.
Choudhury, A. K.↗
Gravity gradient torque for an arbitrary potential function
To permit investigation of the relative effects of higher-order terms in the potential function and allow computation of the gravity gradient torques consistent with the gravitational acceleration for a given potential function, a simple formula is derived for computing the gravity gradient torque on a vehicle in terms of the eigenvalues and eigenvectors of the gravity gradient matrix and the inertia properties of the vehicle. The formulation is applicable to any potential function for which the gravity matrix can be computed for a given point in the potential field.
Glandorf, D. R.↗
An efficient algorithm for choosing scattering directions in Monte Carlo work with arbitrary phase functions
This paper describes an efficient Monte Carlo algorithm for choosing a new direction of a photon after a scattering interaction. The algorithm chooses a scattering angle by linear interpolation in a table of the inverse cumulative scattering probability. A Legendre expansion of the phase function makes it easy to apply Clenshaw's algorithm to build the interpolation table. The points in the table are close enough together that linear interpolation is accurate. With a table of 100,000 entries, we can keep the absolute and relative errors in matching the probability distribution below 10(exp -5).
Barkstrom, Bruce R.↗
Localized orbitals for arbitrary molecular wave functions.
Localized orbitals defined for arbitrary wave functions for atoms and molecules with even or odd number of electrons, noting positive trihydrogen ion
Certain, P. R.↗
The algebra of probability density functions
Algebra of probability density functions of variable known as arbitrary function of one or more random variables
Grunebaum, B. G.↗
Cratering and cosmogenic nuclides
A simple probabilistic model was constructed for the average value of a cosmogenic nuclide as a function of depth in a regolith. An arbitrary function was chosen for the size distribution of craters. The resulting integro-differential equation was found to reduce in limiting cases to the marching equation with a characteristic residence time and to the diffusion equation. The regolith diffusion constant is shown to be a simple integral of the cratering rate weighted by geometrical terms. This formal treatment provides a direct and general connection between cosmogenic nuclides and cratering rates and crater population in a simple analytical form. The validity of this model remains to be tested.
Blake, M. L.↗
Symmetric basis functions for networks with arbitrary geometrical symmetries
Basis functions for arbitrary time variable network symmetries for large scale integration, using group theory
Rubin, H.↗
An approximate buckling analysis for rectangular orthotropic plates with centrally located cutouts
An approximate analysis for predicting buckling of rectangular orthotropic composite plates with centrally located cutouts is presented. In this analysis, prebuckling and buckling problems are converted from a two-dimensional to a one-dimensional system of linear differential equations with variable coefficients. The conversion is accomplished by expressing the displacements as series with each element containing a trigonometric function of one coordinate and a coefficient that is an arbitrary function of the other coordinate. Ordinary differential equations are then obtained from a variational principle. Analytical results obtained from the approximate analysis are compared with finite element analyses for isotropic plates and for specially orthotropic plates with central circular cutouts of various sizes. Experimental results for the specially orthotropic plates are also presented. In nearly all cases, the approximate analysis predicts the buckling mode shapes correctly and predicts the buckling loads to within a few percent of the finite element and experimental results.
Nemeth, M. P.↗
On two-dimensional flows of compressible fluids
This report is devoted to the study of two-dimensional steady motion of a compressible fluid. It is shown that the complete flow pattern around a closed obstacle cannot be obtained by the method of Chaplygin. In order to overcome this difficulty, a formula for the stream-function of a two-dimensional subsonic flow is derived. The formula involves an arbitrary function of a complex variable and yields all possible subsonic flow patterns of certain types. Conditions are given so that the flow pattern in the physical plane will represent a flow around a closed curve. The formula obtained can be employed for the approximate determination of a subsonic flow around an obstacle. The method can be extended to partially supersonic flows.
HODOGRAPHY METHOD↗
Orthogonality of spherical harmonic coefficients
Orthogonality relations are obtained for the spherical harmonic coefficients of functions defined on the surface of a sphere. Following a brief discussion of the orthogonality of Fourier series coefficients, consideration is given to the values averaged over all orientations of the coordinate system of the spherical harmonic coefficients of a function defined on the surface of a sphere that can be expressed in terms of Legendre polynomials for the special case where the function is the sum of two delta functions located at two different points on the sphere, and for the case of an essentially arbitrary function. It is noted that the orthogonality relations derived have found applications in statistical studies of the geomagnetic field.
Mcleod, M. G.↗
Generalization of laplace's expansion to arbitrary powers and functions of the distance between two points
Generalization of laplace expansion to arbitrary powers and functions of distance between two points
LAPLACE EQUATION↗
Stroboscopic Interferometer for Measuring Mirror Vibrations
Stroboscopic interferometry is a technique for measuring the modes of vibration of mirrors that are lightweight and, therefore, unavoidably flexible. The technique was conceived especially for modal characterization of lightweight focusing mirror segments to be deployed in outer space; however, the technique can be applied to lightweight mirrors designed for use on Earth as well as the modal investigation of other optical and mechanical structures. To determine the modal structure of vibration of a mirror, it is necessary to excite the mirror by applying a force that varies periodically with time at a controllable frequency. The excitation can utilize sinusoidal, square, triangular, or even asynchronous waveforms. Because vibrational modes occur at specific resonant frequencies, it is necessary to perform synchronous measurements and sweep the frequency to locate the significant resonant modes. For a given mode it is possible to step the phase of data acquisition in order to capture the modal behavior over a single cycle of the resonant frequency. In order to measure interferometrically the vibrational response of the mirror at a given frequency, an interferometer must be suitably aligned with the mirror and adjustably phase-locked with the excitation signal. As in conventional stroboscopic photography, the basic idea in stroboscopic interferometry is to capture an image of the shape of a moving object (in this case, the vibrating mirror) at a specified instant of time in the vibration cycle. Adjusting the phase difference over a full cycle causes the interference fringes to vary over the full range of motion for the mode at the excitation frequency. The interference-fringe pattern is recorded as a function of the phase difference, and, from the resulting data, the surface shape of the mirror for the given mode is extracted. In addition to the interferometer and the mirror to be tested, the equipment needed for stroboscopic interferometry includes an arbitrary-function generator (that is, a signal generator), an oscilloscope, a trigger filter, and an advanced charge-coupled-device (CCD) camera. The optical components are positioned to form a pupil image of the mirror under test on the CCD chip, so that the interference pattern representative of the instantaneous mirror shape is imaged on the CCD chip.
Stahl, H. Philip↗
Optimizations of Human Restraint Systems for Short-Period Acceleration
A restraint system's main function is to restrain its occupant when his vehicle is subjected to acceleration. If the restraint system is rigid and well-fitting (to eliminate slack) then it will transmit the vehicle acceleration to its occupant without modifying it in any way. Few present-day restraint systems are stiff enough to give this one-to-one transmission characteristic, and depending upon their dynamic characteristics and the nature of the vehicle's acceleration-time history, they will either magnify or attenuate the acceleration. Obviously an optimum restraint system will give maximum attenuation of an input acceleration. In the general case of an arbitrary acceleration input, a computer must be used to determine the optimum dynamic characteristics for the restraint system. Analytical solutions can be obtained for certain simple cases, however, and these cases are considered in this paper, after the concept of dynamic models of the human body is introduced. The paper concludes with a description of an analog computer specially developed for the Air Force to handle completely general mechanical restraint optimization programs of this type, where the acceleration input may be any arbitrary function of time.
Payne, P. R.↗
Computer program for solving compressible nonsimilar-boundary-layer equations for laminar, transitional, or turbulent flows of a perfect gas
A computer program is described which solves the compressible laminar, transitional, or turbulent boundary-layer equations for planar or axisymmetric flows. Three-point implicit difference relations are used to reduce the momentum and energy equations to finite-difference form. These equations are solved simultaneously without iteration. Turbulent flow is treated by the inclusion of either a two-layer eddy-viscosity model or a mixing-length formulation. The eddy conductivity is related to the eddy viscosity through a static turbulent Prandtl number which may be an arbitrary function of the distance from the wall boundary. The transitional boundary layer is treated by the inclusion of an intermittency function which modifies the fully turbulent model. The laminar-boundary-layer equations are recovered when the intermittency is zero, and the fully turbulent equations are solved when the intermittency is unity.
Price, J. M.↗