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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
NASA NTRS1945

Numerical Evaluation by Harmonic Analysis of the (Epsilon)-Function of the Theodorsen Arbitrary-Airfoil Potential Theory

Finite trigonometric series is fitted by harmonic analysis as an approximation function to the psi function of the Theodorsen arbitrary-airfoil potential theory. By harmonic synthesis, the corresponding conjugate trigonometric series is used as an approximation to the epsilon function. A set of coefficients of particularly simple form is obtained algebraically for direct calculation of the epsilon values from the corresponding set of psi values. Complete derivation of this process is presented.

Naiman, Irven↗
NASA NTRS2005

Full-Circle Resolver-to-Linear-Analog Converter

A circuit generates sinusoidal excitation signals for a shaft-angle resolver and, like the arctangent circuit described in the preceding article, generates an analog voltage proportional to the shaft angle. The disadvantages of the circuit described in the preceding article arise from the fact that it must be made from precise analog subcircuits, including a functional block capable of implementing some trigonometric identities; this circuitry tends to be expensive, sensitive to noise, and susceptible to errors caused by temperature-induced drifts and imprecise matching of gains and phases. These disadvantages are overcome by the design of the present circuit. The present circuit (see figure) includes an excitation circuit, which generates signals Ksin(Omega(t)) and Kcos(Omega(t)) [where K is an amplitude, Omega denotes 2(pi)x a carrier frequency (the design value of which is 10 kHz), and t denotes time]. These signals are applied to the excitation terminals of a shaft-angle resolver, causing the resolver to put out signals C sin(Omega(t)-Theta) and C cos(Omega(t)-Theta). The cosine excitation signal and the cosine resolver output signal are processed through inverting comparator circuits, which are configured to function as inverting squarers, to obtain logic-level or square-wave signals .-LL[cos(Omega(t)] and -LL[cos(Omega(t)-Theta)], respectively. These signals are fed as inputs to a block containing digital logic circuits that effectively measure the phase difference (which equals Theta between the two logic-level signals). The output of this block is a pulse-width-modulated signal, PWM(Theta), the time-averaged value of which ranges from 0 to 5 VDC as Theta ranges from .180 to +180deg. PWM(Theta) is fed to a block of amplifying and level-shifting circuitry, which converts the input PWM waveform to an output waveform that switches between precise reference voltage levels of +10 and -10 V. This waveform is processed by a two-pole, low-pass filter, which removes the carrier-frequency component. The final output signal is a DC potential, proportional to Theta that ranges continuously from -10 V at Theta = -180deg to +10 V at Theta = +180deg..

Alhorn, Dean C.↗
NASA NTRS1984

Analysis of lateral and torsional vibration characteristics of beams and shafts with end located rotational masses

Partial differential equations are derived for free lateral and torsional vibration of a uniform free-free beam with a rotational mass attached to each extremity. For appropriate boundary conditions, nonlinear algebraic equations are obtained using a symbolic manipulation computer program, the solutions of which enable the computation of the neutral frequencies and mode-shapes. The mode-shapes are linear combinations of trigonometric and hyperbolic sine and cosine functions. A computer program is written for the numerical solution of the algebraic equations mentioned above, which can compute the natural frequencies, mode-shapes, and node points for any given set of parameters, for any given number of modes.

Robertson, D. K.↗
NASA NTRS1985

Three-dimensional vibration analysis of a uniform beam with offset inertial masses at the ends

Analysis of a flexible beam with displaced end-located inertial masses is presented. The resulting three-dimensional mode shape is shown to consist of two one-plane bending modes and one torsional mode. These three components of the mode shapes are shown to be linear combinations of trigonometric and hyperbolic sine and cosine functions. Boundary conditions are derived to obtain nonlinear algebraic equations through kinematic coupling of the general solutions of the three governing partial differential equations. A method of solution which takes these boundary conditions into account is also presented. A computer program has been written to obtain unique solutions to the resulting nonlinear algebraic equations. This program, which calculates natural frequencies and three-dimensional mode shapes for any number of modes, is presented and discussed.

Robertson, D. K.↗
NASA NTRS1988

Flaw characterization in structural ceramics using scanning laser acoustic microscopy

The ability of scanning laser acoustic microscopy (SLAM) to characterize artifically seeded voids in sintered silicon nitride structural ceramic specimens was investigated. The voids ranged from 20 to 430 microns in diameter and were embedded up to 2 mm beneath the surface of the specimens. Probability of detection was determined as a function of void depth and size. Trigonometric relationships and Airy's diffraction theory were used to obtain predictions of void depth and size from acoustic diffraction patterns produced by the voids. Agreement was observed between actual and predicted void depths. However, predicted void diameters were generally much greater than actual diameters. Precise diameter predictions are difficult to obtain because of measurement uncertainty and the limitations of the 100 MHz SLAM applied to typical ceramic specimens.

Roth, Don J.↗
NASA NTRS1993

On the Gibbs phenomenon 3: Recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function

The investigation of overcoming Gibbs phenomenon was continued, i.e., obtaining exponential accuracy at all points including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. It was shown that if we are given the first N expansion coefficients of an L(sub 2) function f(x) in terms of either the trigonometrical polynomials or the Chebyshev or Legendre polynomials, an exponentially convergent approximation to the point values of f(x) in any sub-interval in which it is analytic can be constructed.

Gottlieb, David↗
NASA NTRS1993

Rational trigonometric approximations using Fourier series partial sums

A class of approximations (S(sub N,M)) to a periodic function f which uses the ideas of Pade, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each approximation S(sub N,M) is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of S(sub N,M) agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these 'Fourier-Pade' approximations converge point-wise to (f(x(exp +))+f(x(exp -)))/2 more rapidly (in some cases by a factor of 1/k(exp 2M)) than the Fourier series partial sums on which they are based. The approximations are illustrated by several examples and an application to the solution of an initial, boundary value problem for the simple heat equation is presented.

Geer, James F.↗
NASA NTRS1972

On the analytic lunar and solar perturbations of a near earth satellite

The disturbing function of the moon (sun) is expanded as a sum of products of two harmonic functions, one depending on the position of the satellite and the other on the position of the moon (sun). The harmonic functions depending on the position of the perturbing body are developed into trigonometric series with the ecliptic elements l, l', F, D, and Gamma of the lunar theory which are nearly linear with respect to time. Perturbation of elements are in the form of trigonometric series with the ecliptic lunar elements and the equatorial elements omega and Omega of the satellite so that analytic integration is simple and the results accurate over a long period of time.

Estes, R. H.↗
NASA NTRS1974

On the analytic lunar and solar perturbations of a near earth satellite

The disturbing function of the moon (sun) is expanded as a sum of products of two harmonic functions, one depending on the position of the satellite and the other on the position of the moon (sun). The harmonic functions depending on the position of the perturbing body are developed into trigonometric series with the ecliptic elements l, l prime, F, D and Gamma of the lunar theory which are nearly linear with respect to time. Perturbation of elements are in the form of trigonometric series with the ecliptic lunar elements and the equatorial elements omega and Omega of the satellite, so that analytic integration is simple and the results are accurate over a long period of time.-

Estes, R. H.↗
NASA NTRS1994

On the Gibbs phenomenon 5: Recovering exponential accuracy from collocation point values of a piecewise analytic function

The paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point values (or a highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct a uniform exponentially convergent approximation to the function f(x) in any sub-interval of analyticity. The proof covers the cases of Fourier, Chebyshev, Legendre, and more general Gegenbauer collocation methods.

Gottlieb, David↗
NASA NTRS2021

The Field Substellar Mass Function Based on the Full-sky 20 pc Census of 525 L, T, and Y Dwarfs

We present final Spitzer trigonometric parallaxes for 361 L, T, and Y dwarfs. We combine these with prior studies to build a list of 525 known L, T, and Y dwarfs within 20 pc of the Sun, 38 of which are presented here for the first time. Using published photometry and spectroscopy as well as our own follow-up, we present an array of color–magnitude and color–color diagrams to further characterize census members, and we provide polynomial fits to the bulk trends. Using these characterizations, we assign each object a T(eff) value and judge sample completeness over bins of T(eff) and spectral type. Except for types ≥T8 and T(eff) < 600 K, our census is statistically complete to the 20 pc limit. We compare our measured space densities to simulated density distributions and find that the best fit is a power law (dN/dM ∝ M^(-α) with α = 0.6 ± 0.1. We find that the evolutionary models of Saumon & Marley correctly predict the observed magnitude of the space density spike seen at 1200 K < T(eff) < 1350 K, believed to be caused by an increase in the cooling timescale across the L/T transition. Defining the low-mass terminus using this sample requires a more statistically robust and complete sample of dwarfs ≥Y0.5 and with T(eff) < 400 K. We conclude that such frigid objects must exist in substantial numbers, despite the fact that few have so far been identified, and we discuss possible reasons why they have largely eluded detection.

J. Davy Kirkpatrick↗
NASA NTRS1973

Derivation of a tangent function using an integrated circuit four-quadrant multiplier

A tangent function generator is provided which utilizes input signals proportionate to the sine and cosine of a given angle. The equipment accomplishes a trigonometric manipulation which forms an output signal substantially independent of the input reference voltage but proportional to the tangent of the angle. The device uses a multiplier and operational amplifier.

Bregman, B. J.↗
NASA NTRS2004

Method of Calibration for a Large Cathetometer System

A method of calibration has been devised for a pair of mutually orthogonal two-axis cathetometers that, when used together, yield measurements of three-dimensional positions of objects mounted on an optical bench. Each cathetometer has a horizontal travel of 1.8 m and a vertical travel of 1.2 m. The cathetometers are required to measure X, Y, and Z coordinates (see figure) to within plus or minus 0.005 in. (plus or minus 0.127 mm). Each cathetometer consists of an alignment telescope on a platform mounted on a two-dimensional translation stage. The knowledge required for calibration of each cathetometer is (1) the two-dimensional position of the cathetometer platform as a function of the electronic readouts of position encoders on the translation stage and (2) the amount of any angular misalignment (roll, pitch, and/or yaw) of the cathetometer platform as a function of the two-dimensional coordinates or the position-encoder readouts. By use of three equations derived from the applicable trigonometric relationships, the calibrated X, Y, and Z coordinates can be computed from the raw encoder readouts.

Toland, Ronald↗
NASA NTRS2001

A Spectral Algorithm for Solving the Relativistic Vlasov-Maxwell Equations

A spectral method algorithm is developed for the numerical solution of the full six-dimensional Vlasov-Maxwell system of equations. Here, the focus is on the electron distribution function, with positive ions providing a constant background. The algorithm consists of a Jacobi polynomial-spherical harmonic formulation in velocity space and a trigonometric formulation in position space. A transform procedure is used to evaluate nonlinear terms. The algorithm is suitable for performing moderate resolution simulations on currently available supercomputers for both scientific and engineering applications.

Shebalin, John V.↗
NASA NTRS1990

Trigonometric Polynomials For Estimation Of Spectra

Orthogonal sets of trigonometric polynomials used as suboptimal substitutes for discrete prolate-spheroidal "windows" of Thomson method of estimation of spectra. As used here, "windows" denotes weighting functions used in sampling time series to obtain their power spectra within specified frequency bands. Simplified windows designed to require less computation than do discrete prolate-spheroidal windows, albeit at price of some loss of accuracy.

Greenhall, Charles A.↗
NASA NTRS1972

Cubic spline functions for curve fitting

FORTRAN cubic spline routine mathematically fits curve through given ordered set of points so that fitted curve nearly approximates curve generated by passing infinite thin spline through set of points. Generalized formulation includes trigonometric, hyperbolic, and damped cubic spline fits of third order.

Young, J. D.↗
NASA NTRS1991

Dynamical eigenfunction decomposition of turbulent channel flow

The results of an analysis of low-Reynolds-number turbulent channel flow based on the Karhunen-Loeve (K-L) expansion are presented. The turbulent flow field is generated by a direct numerical simulation of the Navier-Stokes equations at a Reynolds number Re(tau) = 80 (based on the wall shear velocity and channel half-width). The K-L procedure is then applied to determine the eigenvalues and eigenfunctions for this flow. The random coefficients of the K-L expansion are subsequently found by projecting the numerical flow field onto these eigenfunctions. The resulting expansion captures 90 percent of the turbulent energy with significantly fewer modes than the original trigonometric expansion. The eigenfunctions, which appear either as rolls or shearing motions, possess viscous boundary layers at the walls and are much richer in harmonics than the original basis functions.

Ball, K. S.↗