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NASA NTRS1976

Computation and sensitivity considerations of the Ricatti equation in spacecraft CMG steering laws

The steady-state matrix Ricatti equation is studied with respect to sensitivity and computational burden for a linear regulator controller in spacecraft. The class of systems studied use control moment gyros (CMGs) as the torque driver, assuming that the CMGs are free of gimbal stops. Different solution techniques are evaluated relative to computation time and accuracy to determine which solution method is most acceptable with respect to the prescribed requirements. It is shown that it is not feasible to eliminate off-diagonal Ricatti equation terms in the relevent updata equation due to their high sensitivity, that Potter's algebraic technique requires the least computation time of the methods studied, and that the simple Euler method yields the best results when accuracy and computation time are equally weighted. If an optimal linear regulator formulation is employed, a full Ricatti equation solution of n(n + 1)2 elements of the symmetric positive definite gain matrix which is the solution to the matrix Ricatti equation must be computed at each update cycle.

Colburn, B. K.↗

NASA NTRS1975

Attenuation of sound in ducts with acoustic treatment - A generalized approximate equation

A generalized approximate equation for duct lining sound attenuation is presented. The specification of two parameters, the maximum possible attenuation and the optimum wall acoustic impedance is shown to completely determine the sound attenuation for any acoustic mode at any selected wall impedance. The equation is based on the nearly circular shape of the constant attenuation contours in the wall acoustic impedance plane. For impedances far from the optimum, the equation reduces to Morse's approximate expression. The equation can be used for initial acoustic liner design. Not least important is the illustrative nature of the solutions which provide an understanding of the duct propagation problem usually obscured in the exact calculations. Sample calculations using the approximate attenuation equation show that the peak and the bandwidth of the sound attenuation spectrum can be represented by quite simple functions of the ratio of actual wall acoustic resistance to optimum resistance.

Rice, E. J.↗

NASA NTRS1976

Numerical solution of nonlinear partial differential equations of mixed type

A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.

Jameson, A.↗

NASA NTRS1976

Stellar convection theory. I - The anelastic modal equations

Methods are developed for dealing with the various dynamical problems that arise because of convective zones in stars. A system of equations for stellar convection is derived from the full equations of compressible fluid dynamics with the aid of two major approximations. The first of these is the anelastic approximation, which involves both the filtering out of acoustic waves and a suitable linearization of the fluctuating thermodynamic variables. The second one approximates the horizontal structure of convection by expanding the motion in a set of horizontal cellular platforms and severely truncating the expansion. The resulting system of partial differential equations, referred to as the anelastic modal equations, is outlined along with suggested boundary conditions and techniques for solving the equations. Ways of assessing the overall validity of the present treatment are discussed.

Latour, J.↗

NASA NTRS1977

Exact solution of some linear matrix equations using algebraic methods

A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.

Djaferis, T. E.↗

NASA NTRS1977

A study of the effect of radical load distributions on calibrated strain gage load equations

For several decades, calibrated strain gages have been used to measure loads on airplanes. The accuracy of the equations used to relate the strain gage measurements to the applied loads has been based primarily on the results of the load calibration. An approach is presented for studying the effect of widely varying load distributions on strain gage load equations. The computational procedure provides a link between the load calibration and the load to be measured in flight. A matrix approach to equation selection is presented, which is based on equation standard error, load distribution, and influence coefficient plots of the strain gage equations, and is applied to a complex, delta-wing structure.

Jenkins, J. M.↗

NASA NTRS1976

Elementary derivation of the perturbation equations of celestial mechanics

The equations of celestial mechanics that govern the temporal rates of change of orbital elements are completely derived using elementary dynamics and proceeding only from Newton's equation and its solution. Two orbital equations and the four most meaningful orbital elements - semimajor axis, eccentricity, inclination, and longitude of pericenter - are written in terms of the orbital energy (E) and angular momentum (H) per unit mass. The six resulting equations are differentiated with respect to time to see the effect on the orbital elements of small changes in E and H. The usual perturbation equations in terms of disturbing-force components are then derived by computing the manner in which perturbing forces change E and H. The results are applied in a qualitative discussion of the orbital evolution of particles in nonspherical gravitational fields, through atmospheres, and under the action of tides.

Burns, J. A.↗

NASA NTRS1978

Numerical solution of the Navier-Stokes equations for super-sonic flows with strong shocks

The numerical solution of the full Navier-Stokes Equations for viscous flows with high Mach numbers and a strong detached bow shock was obtained. Two dimensional flows around a circular cylinder, and a circular cylinder with an aft-body in the form of a fairing, were considered. The solution of the compressible N.S. equations was accomplished by the method of finite differences. An implicit scheme of solution, the S.O.R., was used with the optimum acceleration parameters determined by trial and error. The tensor notation was used in writing the N-S Equations transformed into general curvilinear coordinates. The equations for the generation of the coordinate system were solved, followed by the solution of the N.S. equations, at the end of a set of given number of time steps. "Wiggles", constituted the one major problem that needed to be overcome. These oscillations give rise to quantities such as negative temperatures, which ultimately caused the computational program to break down. Certain dissipative finite-difference schemes damped these oscillations.

Devarayalu, K.↗

NASA NTRS1979

Equation modifying program, L219 (EQMOD). Volume 1: Engineering and usage

The analysis and use of the Equation Modifying Program (EQMOD) L219, digital computer program which modifies matrices according to specific instructions was described. The program modifies the theoretical equation of motion and load equations generated by the DYLOFLEX programs Equation of Motion L217 (EOM), and Load Equations, L218 (LOADS), respectively.

Miller, R. D.↗

NASA NTRS1979

Aeroelastic equations of motion of a Darrieus vertical-axis wind-turbine blade

The second-degree nonlinear aeroelastic equations of motion for a slender, flexible, nonuniform, Darrieus vertical-axis wind turbine blade which is undergoing combined flatwise bending, edgewise bending, torsion, and extension are developed using Hamilton's principle. The blade aerodynamic loading is obtained from strip theory based on a quasi-steady approximation of two-dimensional incompressible unsteady airfoil theory. The derivation of the equations has its basis in the geometric nonlinear theory of elasticity and the resulting equations are consistent with the small deformation approximation in which the elongations and shears are negligible compared to unity. These equations are suitable for studying vibrations, static and dynamic aeroelastic instabilities, and dynamic response. Several possible methods of solution of the equations, which have periodic coefficients, are discussed.

Kaza, K. R. V.↗

NASA NTRS1979

Computational simulation of corotational vortex merger using 0, 1, and 2 equation turbulence models

A computational study has been conducted to gain a better understanding of the role of turbulence in the merger of corotating vortices and in the decay of a single vortex. The flow field is simulated by using an inviscid flow model and zero-, one-, and two-equation turbulent-flow models. An implicit finite-difference procedure is employed to integrate the unsteady, two-dimensional, incompressible flow equations in a cross-plane. In the zero- and one-equation models, a spatially varying mixing-length model that simulates the streamline curvature effect is used to define an eddy viscosity. The computational merging distance criterion is in good agreement with the experimental one. In the case of the two-equation model, the turbulent kinetic energy and a modified rate of dissipation equation incorporating a curvature correction term are solved. A comparative study of the computational results corresponding to the various flow models and the available experimental data is presented.

Raj, P.↗

NASA NTRS1981

Time and frequency domain analysis of sampled data controllers via mixed operation equations

Specification of the mathematical equations required to define the dynamic response of a linear continuous plant, subject to sampled data control, is complicated by the fact that the digital components of the control system cannot be modeled via linear ordinary differential equations. This complication can be overcome by introducing two new mathematical operations; namely, the operation of zero order hold and digial delay. It is shown that by direct utilization of these operations, a set of linear mixed operation equations can be written and used to define the dynamic response characteristics of the controlled system. It also is shown how these linear mixed operation equations lead, in an automatable manner, directly to a set of finite difference equations which are in a format compatible with follow on time and frequency domain analysis methods.

Frisch, H. P.↗

NASA NTRS1981

The nonlinear modified equation approach to analyzing finite difference schemes

The nonlinear modified equation approach is taken in this paper to analyze the generalized Lax-Wendroff explicit scheme approximation to the unsteady one- and two-dimensional equations of gas dynamics. Three important applications of the method are demonstrated. The nonlinear modified equation analysis is used to (1) generate higher order accurate schemes, (2) obtain more accurate estimates of the discretization error for nonlinear systems of partial differential equations, and (3) generate an adaptive mesh procedure for the unsteady gas dynamic equations. Results are obtained for all three areas. For the adaptive mesh procedure, mesh point requirements for equal resolution of discontinuities were reduced by a factor of five for a 1-D shock tube problem solved by the explicit MacCormack scheme.

Klopfer, G. H.↗

NASA NTRS1981

An empirical stress-intensity factor equation for the surface crack

This paper presents an empirical stress-intensity factor equation for a surface crack as a function of parametric angle, crack depth, crack length, plate thickness and plate width for tension and bending loads. The stress-intensity factors used to develop the equation were obtained from a previous three-dimensional, finite-element analysis of semielliptical surface cracks in finite elastic plates subjected to tension or bending loads. A wide range of configuration parameters was included in the equation. The ratios of crack length to plate thickness and the ratios of crack depth to crack length ranged from 0 to 1.0. The effects of plate width on stress-intensity variations along the crack front were also included. The equation was used to predict patterns of surface-crack growth under tension or bending fatigue loads. The equation was also used to correlate surface-crack fracture data for a brittle epoxy material within + or - 10 percent for a wide range of crack shapes and crack sizes.

Newman, J. C., Jr.↗

NASA NTRS1982

Transport equations for multicomponent anisotropic space plasmas - A review

An attempt is made to present a unified approach to the study of transport phenomena in multicomponent anisotropic space plasmas. In particular, a system of generalized transport equations is presented that can be applied to widely different plasma flow conditions. The generalized transport equations can describe subsonic and supersonic flows, collision-dominated and collisionless flows, plasma flows in rapidly changing magnetic field configurations, multicomponent plasma flows with large temperature differences between the interacting species, and plasma flows that contain anisotropic temperature distributions. In addition, if Maxwell's equations of electricity and magnetism are added to the system of transport equations, they can be used to model electrostatic shocks, double layers, and magnetic merging processes. These transport equations also contain terms which act to regulate both the heat flow and temperature anisotropy, processes which appear to be operating in the solar wind.

Barakat, A. R.↗

NASA NTRS1983

A study of dynamic energy equations for Stirling cycle analysis

An analytical and computer study of the dynamic energy equations that describe the physical phenomena that occurs in a Stirling cycle engine. The basic problem is set up in terms of a set o hyperbolic partial differential equations. The characteristic lines are determined. The equations are then transformed to ordinary differential equations that are valid along characteristic lines. Computer programs to solve the differential equations and to plot pertinent factors are described.

Larson, V. H.↗

NASA NTRS1983

Equations for the angles of arrival and departure for multivariable root loci using frequency-domain methods

Frequency domain methods are developed to obtain explicit equations for the angles of arrival and departure for multivariable root loci. The techniques involve an evaluation of polynomials formulated within the transfer function matrix. The equations defined require simpler computations than the state-space results of Shaked (1976). A class of higher order poles and zeros is formulated in terms of simpler equations than Shaked's, and the equations are shown to be generalizations of the single-input-single-output root locus equations.

Yagle, A. E.↗

NASA NTRS1983

A general perturbation approach for the equations of fluid dynamics

An efficient numerical technique to produce accurate solutions to the equations of fluid dynamics is presented where the governing equations are perturbed about an approximate solution and solved by finite-difference methods on a coarsened grid. The result is a scheme which substantially reduces the number of grid points necessary to accurately resolve the flow. Applications are presented for the two-dimensional Euler equations perturbed about a solution of the transonic full potential equation. However, the concept is applicable to arbitrary equation sets, higher dimensions and for a wide variety of applications.

Chow, L. J.↗