Polynomial smoothing formulas and derivative formulas for one or two independent variables
Polynomial smoothing formulas and derivative formulas for one or two independent variables
Semtner, A. J.↗
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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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The Ultimate Solar Azimuth Formula: A Note on the Formula that Renders Circumstantial Treatment Unnecessary and an Update Of the Ephemerides to that of The Astronomical Almanac for the Year 2019
A conceptually and mathematically concise formula for computing the solar azimuth angle has been used by a subgroup of scientists, but for lack of documentation and publication, it has not been well circulated. This note introduces this formula which is based on the idea of a unit vector, 𝑺, originating from the observer’s location and pointing toward the center of the Sun. The vector is completely determined by the coordinates of the subsolar point and of the observer. The x-and y-components of the vector determine the solar azimuth angle, and their use along with the function atan2, which is available in a number of programming/scripting languages, including Fortran and Python, renders any circumstantial treatment absolutely unnecessary. The z-component of the vector, at the same time, determines the solar zenith angle.
Taiping Zhang↗
Development of a valid mathematical formula or group of formulas to establish within an accuracy of 5 percent the inductance audio range resulting in beryllium coil assemblies Final report
Mathematical formulas to determine net inductance in audio range of both hammer coil and cryogenic sealing coil
CRYOGENICS↗
A load factor formula
The ultimate test of a load factor formula is experience. The chief advantages of a semi rational formula over arbitrary factors are that it fairs in between points of experience and it differentiates according to variables within a type. Structural failure of an airplane apparently safe according to the formula would call for a specific change in the formula. The best class of airplanes with which to check a load factor formula seems to be those which have experienced structural failure. Table I comprises a list of the airplanes which have experienced failure in flight traceable to the wing structure. The load factor by formula is observed to be greater than the designed strength in each case, without a single exception. Table II comprises the load factor by formula with the designed strength of a number of well-known service types. The formula indicates that by far the majority of these have ample structural strength. One case considered here in deriving a suitable formula is that of a heavy load carrier of large size and practically no reserve power.
Miller, Roy G↗
The Kirchhoff Formulas for Moving Surfaces in Aeroacoustics - The Subsonic and Supersonic Cases
One of the active areas of computational aeroacoustics is the application of the Kirchhoff formulas to the problems of the rotating machinery noise predictions. The original Kirchhoff formula was derived for a stationary surface. In 1988, Farassat and Myers derived a Kirchhoff Formula obtained originally by Morgans using modem mathematics. These authors gave a formula particularly useful for applications in aeroacoustics. This formula is for a surface moving at subsonic speed. Later in 1995 these authors derived the Kirchhoff formula for a super-sonically moving surface. This technical memorandum presents the viewgraphs of a day long workshop by the author on the derivation of the Kirchhoff formulas. All necessary background mathematics such as differential geometry and multidimensional generalized function theory are discussed in these viewgraphs. Abstraction is kept at minimum level here. These viewgraphs are also suitable for understanding the derivation and obtaining the solutions of the Ffowcs Williams-Hawkings equation. In the first part of this memorandum, some introductory remarks are made on generalized functions, the derivation of the Kirchhoff formulas and the development and validation of Kirchhoff codes. Separate lists of references by Lyrintzis, Long, Strawn and their co-workers are given in this memorandum. This publication is aimed at graduate students, physicists and engineers who are in need of the understanding and applications of the Kirchhoff formulas in acoustics and electromagnetics.
Farassat, F.↗
Evaluating Daily Load Stimulus Formulas in Relating Bone Response to Exercise
Six formulas representing what is commonly referred to as "daily load stimulus" are identified, compared and tested in their ability to relate skeletal mechanical loading to bone maintenance and osteogenic response. Particular emphasis is placed on exercise- induced skeletal loading and whether or not the formulas can adequately capture the known experimental observations of saturation of continuous cyclic loading, rest insertion between repetitions (cycles), recovery of osteogenic potential following saturation, and multiple shorter bouts versus a single long bout of exercise. To evaluate the ability of the formulas to capture these characteristics, a set of exercise scenarios with type of exercise bout, specific duration, number of repetitions, and rest insertion between repetitions is defined. The daily load values obtained from the formulas for the loading conditions of the set of scenarios is illustrated. Not all of the formulas form estimates of daily load in units of stress or in terms of strain at a skeletal site due to the loading force from a specific exercise prescription. The comparative results show that none of the formulas are able to capture all of the experimentally observed characteristics of cyclic loading. However, the enhanced formula presented by Genc et al. does capture several characteristics of cyclic loading that the others do not, namely recovery of osteogenic potential and saturation. This could be a basis for further development of mathematical formulas that more adequately approximates the amount of daily stress at a skeletal site that contributes to bone adaptation.
Pennline, James A.↗
Classical seventh-, sixth-, and fifth-order Runge-Kutta-Nystrom formulas with stepsize control for general second-order differential equations
Runge-Kutta-Nystrom formulas of the seventh, sixth, and fifth order were derived for the general second order (vector) differential equation written as the second derivative of x = f(t, x, the first derivative of x). The formulas include a stepsize control procedure, based on a complete coverage of the leading term of the local truncation error in x, and they require no more evaluations per step than the earlier Runge-Kutta formulas for the first derivative of x = f(t, x). The developed formulas are expected to be time saving in comparison to the Runge-Kutta formulas for first-order differential equations, since it is not necessary to convert the second-order differential equations into twice as many first-order differential equations. The examples shown saved from 25 percent to 60 percent more computer time than the earlier formulas for first-order differential equations, and are comparable in accuracy.
Fehlberg, E.↗
The Adams formulas for numerical integration of differential equations from 1st to 20th order
The Adams Bashforth predictor coefficients and the Adams Moulton corrector coefficients for the integration of differential equations are presented for methods of 1st to 20th order. The order of the method as presented refers to the highest order difference formula used in Newton's backward difference interpolation formula, on which the Adams method is based. The Adams method is a polynomial approximation method derived from Newton's backward difference interpolation formula. The Newton formula is derived and expanded to 20th order. The Adams predictor and corrector formulas are derived and expressed in terms of differences of the derivatives, as well as in terms of the derivatives themselves. All coefficients are given to 18 significant digits. For the difference formula only, the ratio coefficients are given to 10th order.
Kirkpatrick, J. C.↗
Extension of Kirchhoff's formula to radiation from moving surfaces
Kirchhoff's formula for radiation from a closed surface has been used recently for prediction of the noise of high speed rotors and propellers. Because the closed surface on which the boundary data are prescribed in these cases is in motion, an extension of Kirchhoff's formula to this condition is required. In this paper such a formula, obtained originally by Morgans for the interior problem, is derived for regions exterior to surfaces moving at speeds below the wave propagation speed by making use of some results of generalized function theory. It is shown that the usual Kirchhoff formula is a special case of the main result of the paper. The general result applies to a deformable surface. However, the special form it assumes for a rigid surface in motion is also noted. In addition, Morgans' result is further extended by showing that edge line integrals appear in the formula when applied to a surface that is piecewise smooth. Some possible areas of application of the formula to problems of current interest in aeroacoustics are discussed.
Farassat, F.↗
Extension of Kirchhoff's formula to radiation from moving surfaces
Kirchhoff's formula for radiation from a closed surface has been used recently for prediction of the noise of high speed rotors and propellers. Because the closed surface on which the boundary data are prescribed in these cases is in motion, an extension of Kirchhoff's formula to this condition is required. In this paper such a formula, obtained originally by Morgans for the interior problem, is derived for regions exterior to surfaces moving at speeds below the wave propagation speed by making use of some results of generalized function theory. It is shown that the usual Kirchhoff formula is a special case of the main result of the paper. The general result applies to a deformable surface. However, the special form it assumes for a rigid surface in motion is also noted. In addition, Morgans' result is further extended by showing that edge line integrals appear in the formula when applied to a surface that is piecewise smooth. Some possible areas of application of the formula to problems of current interest in aeroacoustics are discussed.
Farassat, F.↗
The Kirchhoff Formula for a Supersonically Moving Surface
The Kirchhoff formula for radiation from stationary surfaces first appeared in 1882, and it has since found many applications in wave propagation theory. In 1930, Morgans extended the formula to apply to surfaces moving at speeds below the wave propagation speed; we refer to Morgans formula as the subsonic formulation. A modern derivation of Morgans result was published by Farassat and Myers in 1988, and it has now been used extensively in acoustics, particularly for high speed helicopter rotor noise prediction. Under some common conditions in this application, however, the appropriate Kirchhoff surface must be chosen such that portions of it travel at supersonic speed. The available Kirchhoff formula for moving surfaces is not suitable for this situation. In the current paper we derive the Kirchhoff formula applicable to a supersonically moving surface using some results from generalized function theory. The new formula requires knowledge of the same surface data as in the subsonic case. Complications that arise from apparent singularities in the new formulation are discussed briefly in the paper.
Farassat, F.↗
Classical eight- and lower-order Runge-Kutta-Nystroem formulas with stepsize control for special second-order differential equations
The formulas include a stepsize control procedure, based on a complete coverage of the leading term of the truncation error in x. The formulas require fewer evaluations per stop than other Runge-Kutta-Nystrom formulas if the latter are operated by using the standard procedure for stepsize control. An example is presented. With results being of the same accuracy, Runge-Kutta-Nystrom formulas discussed save 50 percent or more computer time compared with other Runge-Kutta-Nystrom formulas.
Fehlberg, E.↗
Classical eighth- and lower-order Runge-Kutta-Nystroem formulas with a new stepsize control procedure for special second-order differential equations
New Runge-Kutta-Nystrom formulas of the eighth, seventh, sixth, and fifth order are derived for the special second-order (vector) differential equation x = f (t,x). In contrast to Runge-Kutta-Nystrom formulas of an earlier NASA report, these formulas provide a stepsize control procedure based on the leading term of the local truncation error in x. This new procedure is more accurate than the earlier Runge-Kutta-Nystrom procedure (with stepsize control based on the leading term of the local truncation error in x) when integrating close to singularities. Two central orbits are presented as examples. For these orbits, the accuracy and speed of the formulas of this report are compared with those of Runge-Kutta-Nystrom and Runge-Kutta formulas of earlier NASA reports.
Fehlberg, E.↗
Formula for the rms blur circle radius of Wolter telescope based on aberration theory
A formula for the rms blur circle for Wolter telescopes has been derived using the transverse ray aberration expressions of Saha (1985), Saha (1984), and Saha (1986). The resulting formula for the rms blur circle radius over an image plane and a formula for the surface of best focus based on third-, fifth-, and seventh-order aberration theory predict results in good agreement with exact ray tracing. It has also been shown that one of the two terms in the empirical formula of VanSpeybroeck and Chase (1972), for the rms blur circle radius of a Wolter I telescope can be justified by the aberration theory results. Numerical results are given comparing the rms blur radius and the surface of best focus vs the half-field angle computed by skew ray tracing and from analytical formulas for grazing incidence Wolter I-II telescopes and a normal incidence Cassegrain telescope.
Shealy, David L.↗
A revised gust-load formula and a re-evaluation of v-g data taken on civil transport airplanes from 1933 to 1950
A revised gust-load formula with a new gust factor is derived to replace the gust-load formula and alleviation factor widely used in gust studies. The revised formula utilizes the same principles and retains the same simple form of the original formula but provides a more appropriate and acceptable basis for gust-load calculations. The gust factor is calculated on the basis of a one-minus-cosine gust shape and is presented as a function of a mass-ratio parameter in contrast to the ramp gust shape and wing loading, respectively, used for the alleviation factor. A summary of gust-velocity data from v-g records taken on civil transport airplanes from 1933 to 1950, reevaluated by the revised formula, is also presented.
Pratt, Kermit G↗
Synthesizing Dynamic Programming Algorithms from Linear Temporal Logic Formulae
The problem of testing a linear temporal logic (LTL) formula on a finite execution trace of events, generated by an executing program, occurs naturally in runtime analysis of software. We present an algorithm which takes an LTL formula and generates an efficient dynamic programming algorithm. The generated algorithm tests whether the LTL formula is satisfied by a finite trace of events given as input. The generated algorithm runs in linear time, its constant depending on the size of the LTL formula. The memory needed is constant, also depending on the size of the formula.
Rosu, Grigore↗
A Fast Method of Deriving the Kirchhoff Formula for Moving Surfaces
The Kirchhoff formula for a moving surface is very useful in many wave propagation problems, particularly in the prediction of noise from rotating machinery. Several publications in the last two decades have presented derivations of the Kirchhoff formula for moving surfaces in both time and frequency domains. Here we present a method originally developed by Farassat and Myers in time domain that is both simple and direct. It is based on generalized function theory and the useful concept of imbedding the problem in the unbounded three-dimensional space. We derive an inhomogeneous wave equation with the source terms that involve Dirac delta functions with their supports on the moving data surface. This wave equation is then solved using the simple free space Green's function of the wave equation resulting in the Kirchhoff formula. The algebraic manipulations are minimal and simple. We do not need the Green's theorem in four dimensions and there is no ambiguity in the interpretation of any terms in the final formulas. Furthermore, this method also gives the simplest derivation of the classical Kirchhoff formula which has a fairly lengthy derivation in physics and applied mathematics books. The Farassat-Myers method can be used easily in frequency domain.
Farassat, F.↗
Field dependence of gaseous-ion mobility - Theoretical tests of approximate formulas.
The approximate formulas considered include relations based on the Wannier free-flight theory, the Kihara medium-field expansion, and the Frost-Patterson interpolation formulas. A few accurate theoretical results are available for testing the foregoing formulas. Cases concerning high fields, intermediate fields, and resonant charge transfer are examined. It is found that of the formulas tested, the one based on the Wannier free-flight theory is the most flexible, since it can be used for all fields and all ion-neutral force laws and mass ratios.
Hahn, H.↗