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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 397 records · Page 22
All sourcesNASA NTRSDOE OSTI
NASA NTRS1977

An equation that describes material outgassing for contamination modeling

A generalization of the Clausius-Claperon equation for vapor pressure is made for an outgassing material. The expression is derived using Langmuir's equation for the outgassing rate of a material and using an empirical equation for the vapor pressure of a material as a function of its molecular weight and temperature. Also, outgassing rate equations are derived in terms of the vapor pressure of the outgassing material for three general geometries.

Heslin, T. M.↗
NASA NTRS1977

Some MACSYMA program for solving difference equations

A set of MACSYMA programs are described for finding closed form solutions to linear recurrence relations in equations having either constant or variable coefficients. In the homogenous case, a polymonial equation is obtained and the solution to the recurrence relation can be written as a linear combination of the roots of the polynomial. Exponential generating functions are used to solve variable coefficient relations. Taking successive derivatives and using the recurrence relation, an ordinary differential equation is obtained. Expanding the solution to the differential equation in a Taylor series, shows that the nth term of the series is the solution to the recurrence relation. For second order recurrences, a check is made for those that can be solved in terms of Bessel functions.

Ivie, J.↗
NASA NTRS1964

Analytical Solutions for Non-Linear Differential Equations with the Help of a Digital Computer

A technique was developed with the help of a digital computer for analytic (algebraic) solutions of autonomous and nonautonomous equations. Two operational transform techniques have been programmed for the solution of these equations. Only relatively simple nonlinear differential equations have been considered. In the cases considered it has been possible to assimilate the secular terms into the solutions. For cases where f(t) is not a bounded function, a direct series solution is developed which can be shown to be an analytic function. All solutions have been checked against results obtained by numerical integration for given initial conditions and constants. It is evident that certain nonlinear differential equations can be solved with the help of a digital computer.

Cromwell, P. C.↗
NASA NTRS1976

A method for the selection of a functional form for a thermodynamic equation of state using weighted linear least squares stepwise regression

A method was developed for establishing a rational choice of the terms to be included in an equation of state with a large number of adjustable coefficients. The methods presented were developed for use in the determination of an equation of state for oxygen and nitrogen. However, a general application of the methods is possible in studies involving the determination of an optimum polynomial equation for fitting a large number of data points. The data considered in the least squares problem are experimental thermodynamic pressure-density-temperature data. Attention is given to a description of stepwise multiple regression and the use of stepwise regression in the determination of an equation of state for oxygen and nitrogen.

Jacobsen, R. T.↗
NASA NTRS1977

Use of rate equations to describe laser excitation in flames

The region of validity of conventional rate equation formulations used to describe the excitation of flame gases for spectroscopic diagnostic purposes is examined by direct comparison with the density matrix formulation. The density matrix equations are formulated for an ensemble of atoms or molecules in the gas state. These equations are reduced to the steady-state limit for a two-level system under the effect of single-mode and multimode optical fields. The steady-state equations are shown to be valid only when the laser pulse rises slowly with respect to the collisional dephasing rate for single-mode excitation, and for certain special cases under multimode excitation.

Daily, J. W.↗
NASA NTRS1976

Influence of drag coefficient equations on particle motion calculations

An important phase of laser velocimetry investigations in gas flow fields is an analysis of the particle motion in the gas. The present paper examines three important aspects of particle motion calculations. A comparison of various drag coefficient equations with available experimental sphere drag data is made to determine the relative accuracy of the various empirical expressions available. Then, the most accurate drag coefficient equation is used to determine the limitations of Stokes drag equation for calculating relaxation lengths behind normal shocks, and percent velocity lags in one-dimensional constant velocity gradient regions. Finally, a two-dimensional constant velocity gradient gas flow field is examined to determine the importance of the coupling between the governing equations for the components of particle velocity.

Walsh, M. J.↗
NASA NTRS1976

Characteristics of the boundary-layer equations of the minimum time-to-climb problem

In many singular perturbation solutions of optimal control problems, the most difficult numerical task is to solve the boundary-layer equations. However, these equations have a special structure that may often be used to expedite their solution. This paper begins by noting the general nature of the boundary-layer equations for optimal control problems. These results are then applied to the aircraft minimum time-to-climb problem. A specific numerical example is considered to illustrate the characteristics of the solution of the boundary-layer equations for this problem.

Ardema, M. D.↗
NASA NTRS1977

Equations and closure methods

Basic differential equations governing compressible turbulent boundary layer flow are reviewed, including conservation of mass and energy, momentum equations derived from Navier-Stokes equations, and equations of state. Closure procedures were broken down into: (1) simple or zeroth-order methods, (2) first-order or mean field closure methods, and (3) second-order or mean turbulence field methods.

Source record↗
NASA NTRS1978

Aeromechanical stability of helicopters with a bearingless main rotor. Part 1: Equations of motion

Equations of motion for a coupled rotor-body system were derived for the purpose of studying air and ground resonance characteristics of helicopters that have bearingless main rotors. For the fuselage, only four rigid body degrees of freedom are considered; longitudinal and lateral translations, pitch, and roll. The rotor is assumed to consist of three or more rigid blades. Each blade is joined to the hub by means of a flexible beam segment (flexbeam or strap). Pitch change is accomplished by twisting the flexbeam with the pitch-control system, the characteristics of which are variable. Thus, the analysis is capable of implicitly treating aeroelastic couplings generated by the flexbeam elastic deflections, the pitch-control system, and the angular offsets of the blade and flexbeam. The linearized equations are written in the nonrotating system retaining only the cyclic rotor modes; thus, they comprise a system of homogeneous ordinary differential equations with constant coefficients. All contributions to the linearized perturbation equations from inertia, gravity, quasi-steady aerodynamics, and the flexbeam equilibrium deflections are retained exactly.

Hodges, D. H.↗
NASA NTRS1978

Fixed-base and two-body equations of motion for an Annular Momentum Control Device (AMCD)

Fixed base and two body equations of motion for an Annular Momentum Control Device (AMCD) are presented. An AMCD consists of a spinning annular rim which is suspended by noncontacting magnetic bearings and powered by a noncontacting linear electromagnetic motor. The fixed base equations are for a rigid AMCD rim suspended by magnetic bearings attached to a rigid fixed base. The two body equations are for a rigid AMCD rim suspended by magnetic bearings attached to a rigid body spacecraft. The fixed base equations are applicable to any potential ground based AMCD application such as energy storage.

Groom, N. J.↗
NASA NTRS1978

Transitional Boundary-Layer Solutions Using a Mixing-Length and a Two-Equation Turbulence Model

Boundary-layer solutions were obtained using the conventional two-layer mixing-length turbulence model and the Wilcox-Traci two-equation model of turbulence. Both flatplate and blunt-body geometries were considered. The most significant result of the study is development of approximations for the two-equation model which permit streamwise stepsize comparable to that used in mixing-length computations. Additionally, a set of model-equation boundary conditions derived which apply equally well to both flat-plate and blunt-body geometries. Solutions obtained with the two-equations turbulence model are compared with experimental data and/or corresponding solutions obtained using the mixing-length model. Agreement is satisfactory for flat-plate boundary layers but not for blunt body boundary layers.

Anderson, E. C.↗
NASA NTRS1978

Computerized power supply analysis: State equation generation and terminal models

To aid engineers that design power supply systems two analysis tools that can be used with the state equation analysis package were developed. These tools include integration routines that start with the description of a power supply in state equation form and yield analytical results. The first tool uses a computer program that works with the SUPER SCEPTRE circuit analysis program and prints the state equation for an electrical network. The state equations developed automatically by the computer program are used to develop an algorithm for reducing the number of state variables required to describe an electrical network. In this way a second tool is obtained in which the order of the network is reduced and a simpler terminal model is obtained.

Garrett, S. J.↗
NASA NTRS1978

Nonlinear flap-lag-axial equations of a rotating beam with arbitrary precone angle

In an attempt both to unify and extend the analytical basis of several aspects of the dynamic behavior of flexible rotating beams, the second-degree nonlinear equations of motion for the coupled flapwise bending, lagwise bending, and axial extension of an untwisted, torsionally rigid, nonuniform, rotating beam having an arbitrary angle of precone with the plane perpendicular to the axis of rotation are derived using Hamilton's principle. The derivation of the equations is based on the geometric nonlinear theory of elasticity and the resulting equations are consistent with the assumption that the strains are negligible compared to unity. No restrictions are imposed on the relative displacements or angular rotations of the cross sections of the beam other than those implied by the assumption of small strains. Illustrative numerical results, obtained by using an integrating matrix as the basis for the method of solution, are presented both for the purpose of validating the present method of solution and indicating the range of applicability of the equations of motion and the method of solution.

Kvaternik, R. G.↗
NASA NTRS1978

Coefficient matrices for implicit finite difference solution of the inviscid fluid conservation law equations

Although the Navier-Stokes equations describe most flows of interest in aerodynamics, the inviscid conservation law equations may be used for small regions with viscous forces. Thus, Euler equations and several time-accurate finite difference procedures, explicit and implicit, are discussed. Although implicit techniques require more computational work, they permit larger time steps to be taken without instability. It is noted that the Jacobian matrices for Euler equations in conservation-law form have certain eigenvalue-eigenvector properties which may be used to construct conservative-form coefficient matrices. This reduces the computation time of several implicit and semiimplicit schemes. Extensions of the basic approach to other areas are suggested.

Steger, J. L.↗
NASA NTRS1979

A review of at rest droplet growth equations for condensing nitrogen in transonic cryogenic wind tunnels

Droplet growth equations are reviewed in the free-molecular, transition, and continuum flow regimes with the assumption that the droplets are at rest with respect to the vapor. As comparison calculations showed, it was important to use a growth equation designed for the flow regime of interest. Otherwise, a serious over-prediction of droplet growth may result. The growth equation by Gyarmathy appeared to be applicable throughout the flow regimes and involved no iteration. His expression also avoided the uncertainty associated with selecting a mass accommodation coefficient and consequently involved less uncertainty in specifying adjustable parameters than many of the other growth equations.

Hall, R. M.↗
NASA NTRS1978

Nonlinear Equations of Equilibrium for Elastic Helicopter or Wind Turbine Blades Undergoing Moderate Deformation

A set of nonlinear equations of equilibrium for an elastic wind turbine or helicopter blades are presented. These equations are derived for the case of small strains and moderate rotations (slopes). The derivation includes several assumptions which are carefully stated. For the convenience of potential users the equations are developed with respect to two different systems of coordinates, the undeformed and the deformed coordinates of the blade. Furthermore, the loads acting on the blade are given in a general form so as to make them suitable for a variety of applications. The equations obtained in the study are compared with those obtained in previous studies.

Rosen, A.↗
NASA NTRS1979

Equation modifying program, L219 (EQMOD). Volume 2: Supplemental system design and maintenance document

A digital computer program, L219 (EQMOD), available for execution on the CDC 6600 is described. The program modifies matrices according to card input instructions and prepares magnetic files of matrices suitable for use in the linear systems analysis program (QR) and the random harmonic analysis program L221 (TEV156). The particular field of application of the program is the modification of the theoretical equations of motion and load equations generated in DYLOFLEX by the equation of motion program (L217) and the load equation program (L218), respectively.

Hirayama, M. Y.↗
NASA NTRS1978

An extension of MacCormack's method for flows with higher-order equations and in different configurations

The numerical scheme for the computation of a shock discontinuity developed by MacCormack has been extended to solve a number of differential equations, including cases explicitly containing higher-order derivatives: (1) Korteweg-de Vries equation with a term of third-order derivative, (2) a system of nonlinear equations governing nonsteady one-dimensional plasma flow in cylindrical coordinate, (3) equations of solar wind. Comparisons with previous results are made, if available, to illustrate the advantages of the present method. The question of convergence of the numerical calculation is discussed.

Ying, S. J.↗