Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “APPROXIMATIONS”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 199 records · Page 11
All sourcesNASA NTRSDOE OSTI
NASA NTRS1974

Some approximations to the flapping stability of helicopter rotors

The flapping equation for a helicopter in forward flight are reported which have coefficients that are periodic in time, and this effect complicates the calculation of stability. A constant coefficient approximation which will allow the use of all the well known methods for analyzing constant coefficient equations are presented. The flapping equation is first transformed into the nonrotating coordinate frame, where some of the periodic coefficients are transformed into constant terms. The constant coefficient approximation is then made by using time averaged coefficients in the nonrotating frame. Stability calculations based on the approximation are compared to results from a theory which correctly includes all of the periodicity. The comparison indicates that the approximation is reasonably accurate at advance ratios up to 0.5.

Biggers, J. C.↗
NASA NTRS1974

Some approximations to the flapping stability of helicopter rotors

The flapping equation for a helicopter in forward flight has coefficients which are periodic in time, and this effect complicates the calculation of stability. This paper presents a constant coefficient approximation which will allow the use of all the well known methods for analyzing constant coefficient equations. The flapping equation is first transformed into the nonrotating coordinate frame, where some of the periodic coefficients are transformed into constant terms. The constant coefficient approximation is then made by using time averaged coefficients in the nonrotating frame. Stability calculations based on the approximation are compared to results from a theory which correctly includes all of the periodicity. The comparison indicates that the approximation is reasonably accurate at advance ratios up to 0.5.

Biggers, J. C.↗
NASA NTRS1976

Approximation concepts for efficient structural synthesis

It is shown that efficient structural synthesis capabilities can be created by using approximation concepts to mesh finite element structural analysis methods with nonlinear mathematical programming techniques. The history of the application of mathematical programming techniques to structural design optimization problems is reviewed. Several rather general approximation concepts are described along with the technical foundations of the ACCESS 1 computer program, which implements several approximation concepts. A substantial collection of structural design problems involving truss and idealized wing structures is presented. It is concluded that since the basic ideas employed in creating the ACCESS 1 program are rather general, its successful development supports the contention that the introduction of approximation concepts will lead to the emergence of a new generation of practical and efficient, large scale, structural synthesis capabilities in which finite element analysis methods and mathematical programming algorithms will play a central role.

Schmit, L. A., Jr.↗
NASA NTRS1975

An approximation technique for predicting the transient response of a second order nonlinear equation

An approximation technique has been developed for determining the transient response of a nonlinear dynamic system. The nonlinearities in the system which has been considered appear in the system's dissipation function. This function was expressed as a second order polynomial in the system's velocity. The developed approximation is an extension of the classic Kryloff-Bogoliuboff technique. Two examples of the developed approximation are presented for comparative purposes with other approximation methods.

Laurenson, R. M.↗
NASA NTRS1976

Test particle propagation in magnetostatic turbulence. 1. Failure of the diffusion approximation

The equation which governs the quasi-linear approximation to the ensemble and gyro-phase averaged one-body probability distribution function is constructed from first principles. This derived equation is subjected to a thorough investigation in order to calculate the possible limitations of the quasi-linear approximation. It is shown that the reduction of this equation to a standard diffusion equation in the Markovian limit can be accomplished through the application of the adiabatic approximation. A numerical solution of the standard diffusion equation in the Markovian limit is obtained for the narrow parallel beam injection. Comparison of the diabatic and adiabatic results explicitly demonstrates the failure of the Markovian description of the probability distribution function. Through the use of a linear time-scale extension the failure of the adiabatic approximation, which leads to the Markovian limit, is shown to be due to mixing of the relaxation and interaction time scales in the presence of the strong mean field.

Klimas, A. J.↗
NASA NTRS1979

Linear Approximation SAR Azimuth Processing Study

A segmented linear approximation of the quadratic phase function that is used to focus the synthetic antenna of a SAR was studied. Ideal focusing, using a quadratic varying phase focusing function during the time radar target histories are gathered, requires a large number of complex multiplications. These can be largely eliminated by using linear approximation techniques. The result is a reduced processor size and chip count relative to ideally focussed processing and a correspondingly increased feasibility for spaceworthy implementation. A preliminary design and sizing for a spaceworthy linear approximation SAR azimuth processor meeting requirements similar to those of the SEASAT-A SAR was developed. The study resulted in a design with approximately 1500 IC's, 1.2 cubic feet of volume, and 350 watts of power for a single look, 4000 range cell azimuth processor with 25 meters resolution.

Lindquist, R. B.↗
NASA NTRS1979

Combined strength and aeroelastic wing synthesis via constraint approximation

A numerical optimization program using constraint approximations for preliminary sizing of wing structural design parameters to satisfy simultaneous strength and aeroelastic requirements is described. The effects of wing flexibility on loads and flutter are included. The iterative procedure approximates wing internal load distributions and flutter response in generating design constraints. Though a linear Taylor series approximation to the variation of flutter speed is made, constraints in general are formulated as nonlinear and solved in a minimum weight problem using a feasible directions search. The technique is demonstrated for the determination of the optimal ply orientations and for simultaneous sizing for strength and flutter with simple constraint approximations.

Mullen, J., Jr.↗
NASA NTRS1980

Evaluation of approximate methods for the prediction of noise shielding by airframe components

An evaluation of some approximate methods for the prediction of shielding of monochromatic sound and broadband noise by aircraft components is reported. Anechoic-chamber measurements of the shielding of a point source by various simple geometric shapes were made and the measured values compared with those calculated by the superposition of asymptotic closed-form solutions for the shielding by a semi-infinite plane barrier. The shields used in the measurements consisted of rectangular plates, a circular cylinder, and a rectangular plate attached to the cylinder to simulate a wing-body combination. The normalized frequency, defined as a product of the acoustic wave number and either the plate width or cylinder diameter, ranged from 4.6 to 114. Microphone traverses in front of the rectangular plates and cylinders generally showed a series of diffraction bands that matched those predicted by the approximate methods, except for differences in the magnitudes of the attenuation minima which can be attributed to experimental inaccuracies. The shielding of wing-body combinations was predicted by modifications of the approximations used for rectangular and cylindrical shielding. Although the approximations failed to predict diffraction patterns in certain regions, they did predict the average level of wing-body shielding with an average deviation of less than 3 dB.

Ahtye, W. F.↗
NASA NTRS1980

Two-stream approximations to radiative transfer in planetary atmospheres - A unified description of existing methods and a new improvement

Existing two-stream approximations to radiative transfer theory for particulate media are shown to be represented by identical forms of coupled differential equations if the intensity is replaced by integrals of the intensity over hemispheres. One set of solutions thus suffices for all methods and provides convenient analytical comparisons. The equations also suggest modifications of the standard techniques so as to duplicate exact solutions for thin atmospheres and thus permit accurate determinations of the effects of typical aerosol layers. Numerical results for the plane albedos of plane-parallel atmospheres are given for conventional and modified Eddington approximations, conventional and modified two-point quadrature schemes, the hemispheric-constant method and the delta-function method, all for comparison with accurate discrete-ordinate solutions. A new two-stream approximation is introduced that reduces to the modified Eddington approximation in the limit of isotropic phase functions and to the exact solution in the limit of extreme anisotropic scattering. Comparisons of plane albedos and transmittances show the new method to be generally superior over a wide range of atmospheric conditions (including cloud and aerosol layers), especially in the case of nonconservative scattering.

Meador, W. E.↗
NASA NTRS1980

Energy conservation - A test for scattering approximations

The roles of the extinction theorem and energy conservation in obtaining the scattering and absorption cross sections for several light scattering approximations are explored. It is shown that the Rayleigh, Rayleigh-Gans, anomalous diffraction, geometrical optics, and Shifrin approximations all lead to reasonable values of the cross sections, while the modified Mie approximation does not. Further examination of the modified Mie approximation for the ensembles of nonspherical particles reveals additional problems with that method.

Acquista, C.↗
NASA NTRS1980

An analytical technique for approximating unsteady aerodynamics in the time domain

An analytical technique is presented for approximating unsteady aerodynamic forces in the time domain. The order of elements of a matrix Pade approximation was postulated, and the resulting polynomial coefficients were determined through a combination of least squares estimates for the numerator coefficients and a constrained gradient search for the denominator coefficients which insures stable approximating functions. The number of differential equations required to represent the aerodynamic forces to a given accuracy tends to be smaller than that employed in certain existing techniques where the denominator coefficients are chosen a priori. Results are shown for an aeroelastic, cantilevered, semispan wing which indicate a good fit to the aerodynamic forces for oscillatory motion can be achieved with a matrix Pade approximation having fourth order numerator and second order denominator polynomials.

Dunn, H. J.↗
NASA NTRS1981

Small-angle approximation to the transfer of narrow laser beams in anisotropic scattering media

The broadening and the signal power detected of a laser beam traversing an anisotropic scattering medium were examined using the small-angle approximation to the radiative transfer equation in which photons suffering large-angle deflections are neglected. To obtain tractable answers, simple Gaussian and non-Gaussian functions for the scattering phase functions are assumed. Two other approximate approaches employed in the field to further simplify the small-angle approximation solutions are described, and the results obtained by one of them are compared with those obtained using small-angle approximation. An exact method for obtaining the contribution of each higher order scattering to the radiance field is examined but no results are presented.

Box, M. A.↗
NASA NTRS1981

Approximate analysis of postbuckled through-width delaminations

An approximate analysis was developed to analyze the postbuckling behavior of through-width delaminations in a laminated coupon. The analysis contains two parameters which are determined using a finite element analysis. After calculating the parameters for a few configurations, the approximate analysis was used to analyze many other configurations. Lateral deflections and mode I strain-energy release rates obtained with the approximate analysis were compared with results from the finite element analysis. For the configurations analyzed, the approximate analysis agreed very well with the finite element results.

Whitcomb, J. D.↗
NASA NTRS1981

Approximate and exact numerical computation of supersonic flow over an airfoil

The computation of inviscid supersonic flow over a two-dimensional airfoil is considered. There are two main nonlinear approaches which lead to approximate solutions. Small-amplitude theory gives valid solutions provided the airfoil thickness is not too great and the Mach number is not too high. The second type of approximation, shock expansion theory, employs the fact that even for flows with strong shocks the effect of the down-running characteristics remains small. This leads to an analytic solution at the airfoil. It is pointed out that the approximate solution is accurate enough for many cases of interest. The numerical method furnishes a rapid correction to the solution in those cases where it is not. The characteristic-streamline coordinate system is useful both for the computation of the approximate solution and the corrections, and is also convenient for displaying and interpreting the results.

Lewis, T. S.↗
NASA NTRS1983

Approximation to the Probability Density at the Output of a Photmultiplier Tube

The probability density of the integrated output of a photomultiplier tube (PMT) is approximated by the Gaussian, Rayleigh, and Gamma probability densities. The accuracy of the approximations depends on the signal energy alpha: the Gamma distribution is accurate for all alpha, the Raleigh distribution is accurate for small alpha (approximate or less than 1 photon) and the Gaussian distribution is accurate for large alpha (approximate or greater than 10 photons).

Stokey, R. J.↗
NASA NTRS1983

Legendre-Tau approximations for functional differential equations

The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.

Ito, K.↗
NASA NTRS1985

Alternative approximation concepts for space frame synthesis

A structural synthesis methodology for the minimum mass design of 3-dimensionall frame-truss structures under multiple static loading conditions and subject to limits on displacements, rotations, stresses, local buckling, and element cross-sectional dimensions is presented. A variety of approximation concept options are employed to yield near optimum designs after no more than 10 structural analyses. Available options include: (A) formulation of the nonlinear mathematcal programming problem in either reciprocal section property (RSP) or cross-sectional dimension (CSD) space; (B) two alternative approximate problem structures in each design space; and (C) three distinct assumptions about element end-force variations. Fixed element, design element linking, and temporary constraint deletion features are also included. The solution of each approximate problem, in either its primal or dual form, is obtained using CONMIN, a feasible directions program. The frame-truss synthesis methodology is implemented in the COMPASS computer program and is used to solve a variety of problems. These problems were chosen so that, in addition to exercising the various approximation concepts options, the results could be compared with previously published work.

Lust, R. V.↗
NASA NTRS1985

Curved Finite Elements and Curve Approximation

The approximation of parameterized curves by segments of parabolas that pass through the endpoints of each curve segment arises naturally in all quadratic isoparametric transformations. While not as popular as cubics in curve design problems, the use of parabolas allows the introduction of a geometric measure of the discrepancy between given and approximating curves. The free parameters of the parabola may be used to optimize the fit, and constraints that prevent overspill and curve degeneracy are introduced. This leads to a constrained optimization problem in two varibles that can be solved quickly and reliably by a simple method that takes advantage of the special structure of the problem. For applications in the field of computer-aided design, the given curves are often cubic polynomials, and the coefficient may be calculated in closed form in terms of polynomial coefficients by using a symbolic machine language so that families of curves can be approximated with no further integration. For general curves, numerical quadrature may be used, as in the implementation where the Romberg quadrature is applied. The coefficient functions C sub 1 (gamma) and C sub 2 (gamma) are expanded as polynomials in gamma, so that for given A(s) and B(s) the integrations need only be done once. The method was used to find optimal constrained parabolic approximation to a wide variety of given curves.

Baart, M. L.↗