Search NASA⌕ Search

SEARCH · Search NASA

Results for “ELLIPTIC INTEGRAL”

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 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1965

Tables of elliptic integrals

Elliptic integral tables for evaluation of integrals with algebraic or trigonometric integrands

ELLIPTIC INTEGRAL↗
NASA NTRS1976

Elliptic integral solutions to a class of space flight optimization problems

This paper is initially concerned with the minimum-time, exoatmospheric flight of a rocket with constant thrust acceleration magnitude, as in the cases of nuclear and solar electric propulsion. Gravitational acceleration is assumed to be a constant scalar multiple of the radius vector, plus a correction term which is a given function of time. The solution to the state equations is obtained in terms of elliptic integrals. A method is presented for the solution of the two-point boundary-condition problem associated with orbital transfer. At most, the latter method requires iteration upon final time, angle of injection, and two other parameters which are bounded. An example problem is provided which involves a rocket with very low thrust and a spiraling trajectory of many revolutions, but an altitude change of only several hundred miles above the earth. Finally, the original elliptic integral solution is extended to a larger class of low and intermediate thrust problems with constant thrust magnitude, mass decreasing with time, and an inverse square gravitational force.

Andrus, J. F.↗
NASA NTRS1986

Improved Numerical Evaluation Method for Elliptic Integrals

Two modifications to Bartky's method for numerical evaluation of complete elliptic integrals increase rate of convergence and eliminate loss of accuracy due to roundoff errors. Other integrals (nonelliptic) evaluated by manipulating integrand into form F(R)/R required for evaluation by this method. Improved method used to map upper half of complex plane conformally onto interior or exterior of round-nosed wedges of various angles.

Rossow, V. J.↗
NASA NTRS1992

Transverse Mercator Projection Via Elliptic Integrals

Improved method of construction of U.S. Army's universal transverse Mercator grid system based on Gauss-Kruger transverse Mercator projection and on use of elliptic integrals of second kind. Method can be used to map entire northern or southern hemisphere with respect to single principal meridian.

Wallis, David E.↗
NASA NTRS1986

Vortex conception of rotor and mutual effect of screw/propellers

A vortex theory of screw/propellers with variable circulation according to the blade and its azimuth is proposed, the problem is formulated and circulation is expanded in a Fourier series. Equations are given for inductive velocities in space for crews, including those with an infinitely large number of blades and expansion of the inductive velocity by blade azimuth of a second screw. Multiparameter improper integrals are given as a combination of elliptical integrals and elementary functions, and it is shown how to reduce elliptical integrals of the third kind with a complex parameter to integrals with a real parameter.

Lepilkin, A. M.↗
NASA NTRS1985

On Bartky's method for evaluation of integrals of elliptic type with application to round-nosed wedges

Bartky's method for the numerical evluation of integrals of the complete elliptic type is modified to improve its accuracy. The resulting modified method is then applied to integrals involving complex arithmetic which map the upper half of one plane onto either the interior or exterior of round-nosed wedges. These solutions have engineering applications to such problems as the design of magnets and the flow of fluids in and around corners.

Rossow, V. J.↗
NASA NTRS1958

Elliptic Functions and Integrals with Real Modulus in Fluid Mechanics

Advantage of the elliptic functions and of the more general functions of Schwarz for fluid mechanics. Flows outside and inside polygons. Application to the calculation of an elbow diffuser for a wind tunnel. Properties of the elliptic integrals of the first kind and of the elliptic functions. Properties of the theta functions and decomposition of the elliptic functions into products of theta functions. Properties of the zeta functions. Decomposition of the elliptic functions into sums of zeta functions and calculations of the elliptic integrals. Applications to the calculation of wing profiles, of compressor profiles, and to the study of the vibrations of airplane wings and of compressor vanes. The manuscript of the present paper was checked by Mr. Eichelbrenner who corrected several imperfections and suggested numerous improvements to make reading of the paper easier. However, the limited subject does not permit filling in more than an incomplete knowledge of the properties of analytic functions.

Legendre, Robert↗
NASA NTRS1926

Inertia Factors of Ellipsoids for Use in Airship Design

This report is based on a study made by the writer as a member of the Special Committee on Design of Army Semirigid Airship RS-1 appointed by the National Advisory Committee for Aeronautics. The increasing interest in airships has made the problem of the potential flow of a fluid about an ellipsoid of considerable practical importance. In 1833 George Green, in discussing the effect of the surrounding medium upon the period of a pendulum, derived three elliptic integrals, in terms of which practically all the characteristics of this type of motion can be expressed. The theory of this type of motion is very fully given by Horace Lamb in his "Hydrodynamics," and applications to the theory of airships by many other writers. Tables of the inertia coefficients derived from these integrals are available for the most important special cases. These tables are adequate for most purposes, but occasionally it is desirable to know the values of these integrals in other cases where tabulated values are not available. For this reason it seems worth while to assemble a collection of formulae which would enable them to be computed directly from standard tables of elliptic integrals, circular and hyperbolic functions and logarithms without the need of intermediate transformations. Some of the formulae for special cases (elliptic cylinder, prolate spheroid, oblate spheroid, etc.) have been published before, but the general forms and some special cases have not been found in previous publications. (author)

Tuckerman, L. B.↗
NASA NTRS1977

Simplified solution for elliptical-contact deformation between two elastic solids

A linear regression by the method of least squares is made on the geometric variables that occur in the equation for elliptical-contact deformation. The ellipticity and the complete elliptic integrals of the first and second kind are expressed as a function of the x,y-plane principal radii. The ellipticity was varied from 1 (circular contact) to 10 (a configuration approaching line contact). The procedure for solving these variables without the use of charts or a high-speed computer would be quite tedious. These simplified equations enable one to calculate easily the elliptical-contact deformation to within 3 percent accuracy without resorting to charts or numerical methods.

Brewe, D. E.↗
NASA NTRS1981

Simplified solution for stresses and deformations

A shortcut is presented to the classical Hertzian solution for local stress and deformation of two elastic bodies in contact, which employs simplified forms for the ellipticity and for the complete elliptic integrals of the first and second kinds as a function of the geometry. Unlike previously proposed simplified formulas whose range of applicability was limited to ellipticities equal to, or greater than one, the present method extends the range of validity to include ellipticities less than one; in which the semimajor axis in the elliptical contact lies in a direction parallel to the rolling, rather than being perpendicular, as in previous studies. An auxiliary shear stress parameter is also expressed in simplified form as a function of the geometry, permitting a shortcut calculation to be made for the location and magnitude of the maximum subsurface shear stress.

Hamrock, B. J.↗
NASA NTRS1973

Transmission of isotropic light across a dielectric surface in two and three dimensions.

Average transmittance of polarized diffuse light across a dielectric surface is calculated in both two and three dimensions. The incident light in both cases is confined to an angular range measured from the surface normal. Limiting values in three dimensions correspond to known results for two cases, (1) normal incidence, and (2) diffuse light incident from a 180 deg cone. The two-dimensional formulation is solvable in terms of elliptic functions and incomplete elliptic integrals of the first, second, and third kinds. Results are displayed graphically for values of transmittances in excess of 0.9 associated with relative indices of refraction in the range m = 1.0 to m = 2.6.

Allen, W. A.↗