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NASA NTRS1975

An optimal control of a flexible spinning satellite

A modification of optimal spinning satellite control theory is proposed, based on introducing an additional vector into the equation for control torque components. This modification essentially means that the amplitudes of the time-dependent satellite perturbation functions can be made small when the amplitudes of the control torque components are allowed to be different. An algorithm for computer simulation of the satellite control theory is provided. A singular perturbation scheme is used which greatly reduces the number of degrees of freedom in the flexible satellite motion equations.

Das, A.↗

NASA NTRS1980

A magnetohydrodynamic model for corotating interplanetary structures

A model is developed which treats interplanetary structures as a small-scale incompressible Alfvenic fluctuation superposed on a large-scale background flow, and uses the complete magnetohydrodynamic equations and the full energy equation to describe the interplanetary structures. For steady-state corotating structures the model assumes the background flow to be field aligned, and the nonlinear system is treated as being composed of two subsets: one dealing with the expansion of the solar wind in stream tubes, the other with the interaction between neighboring stream tubes. The expansion subset has a singularity at Parker's critical radius, and the interaction subset has a singularity at the Alfvenic point. Concerning the region where the flow velocity is super-Alfvenic, the characteristic equation is derived, and a general method of numerical solution for the three-dimensional problems based on the characteristics theory is discussed. In addition, a viscous WKB solution of the microscale system is presented.

Whang, Y. C.↗

NASA NTRS1975

Low-energy electron scattering from CO

The scattering of electrons from carbon monoxide has been studied in the framework of a previously proposed pseudopotential method. It is found that the single-center expansion of the continuum electron orbital about the center of mass of the molecule is very well converged when the two noncoincident nuclear singularities and the nonlocal exchange interactions are properly taken into account in the scattering equations. An analytical proof has been given to show that unlike the total cross section, the momentum-transfer cross section in the fixed-nuclei approximation is finite even for polar molecules. The calculated results of the momentum-transfer cross section for e(-)-CO scattering in the energy range between 0.1 and 10.0 eV are in good accord with the experimental measurements.

Chandra, N.↗

NASA NTRS1979

Dipole and quadrupole synthesis of electric potential fields

A general technique for expanding an unknown potential field in terms of a linear summation of weighted dipole or quadrupole fields is described. Computational methods were developed for the iterative addition of dipole fields. Various solution potentials were compared inside the boundary with a more precise calculation of the potential to derive optimal schemes for locating the singularities of the dipole fields. Then, the problem of determining solutions to Laplace's equation on an unbounded domain as constrained by pertinent electron trajectory data was considered.

Tilley, D. G.↗

NASA NTRS1989

The prediction of pseudo-resonance positions in the Schwinger variational principle

The Schwinger Variational Principle is appled to s-wave electron-hydrogen atom scattering. It is shown computationally, that, consistent with a paper by B. Apagyi, P. Levay, and K. Ladanyi, there are pseudo-resonances at the static exchange level of approximation, but not at the static level. The T-matrix was employed, as well as, the K-matrix version of the Schwinger Principle, with a real Slater basis, and obtained the same results in both. The origin of the pseudo-resonances as resulting from singularities in the separable potential that is effectively employed in the Lippmann-Schwinger equation from which the Schwinger Variational Principle can be derived. The determination of the pseudo-resonance parameters from the separable potential is computationally inexpensive and may be used to predict the pseudo-resonance parameters for the scattering calculations so that they may be avoided.

Weatherford, Charles A.↗

NASA NTRS1989

A study of pseudoresonances in the application of the Schwinger variational principle to electron scattering from atomic hydrogen

The Schwinger variational principle is applied to s-wave electron-hydrogen atom scattering. Computationally, consistent with a recent paper by Apagyi et al. (1988), there are pseudoresonances at the static exchange level of approximation, but not at the static level. The T-matrix as well as the K-matrix version of the Schwinger principle is used with a real Slater basis, and the same results are obtained in both. The origin of the pseudoresonances from singularities in the separable potential that is effectively employed in the Lippman-Schwinger equation from which the Schwinger variational principle can be derived. The determination of the pseudoresonance parameters from the separable potential is computationally inexpensive and may be used to predict the pseudoresonance parameters for the scattering calculations so as to avoid them.

Weatherford, Charles A.↗

NASA NTRS1991

On the nature of control algorithms for free-floating space manipulators

It is suggested that nearly any control algorithm that can be used for fixed-based manipulators also can be employed in the control of free-floating space manipulator systems, with the additional conditions of estimating or measuring a spacecraft's orientation and of avoiding dynamic singularities. This result is based on the structural similarities between the kinematic and dynamic equations for the same manipulator but with a fixed base. Barycenters are used to formulate the kinematic and dynamic equations of free-floating space manipulators. A control algorithm for a space manipulator system is designed to demonstrate the value of the analysis.

Papadopoulos, Evangelos↗

NASA NTRS1995

Optically thick winds: How is the mass flux determined?

Expanding envelopes of compact objects that possess a buring region at the base of the envelope are commonly observed and modeled. In many cases, such as the late stages of classical nova, the expanding envelope develops into a stationary, optically thick wind of matter escaping from the star. In the usual theoretical formalism there is one more unknown than equations, with closure being obtained by the requirement that the solution pass through the singularity at the sonic point. It is shown analytically that the mass flux, which is one of the unknowns, is almost completely determined by the physical conditions near the base of the envelope just above the burning zone. The sonic point closure relation determines whether the expanding solution can develop into a wind solution. For a given core mass the range of possible wind solutions is an outcome of the great sensitivity of the mass flux to the inner luminosity. Solutions determined through numerical integrations are shown to lie entirely within the narrow analytic boundaries.

Glasner, S. Ami↗

NASA NTRS1972

Some efficient methods for obtaining infinite series solutions of n-th order linear ordinary differential equations

The use of the theta-operator method and generalized hypergeometric functions in obtaining solutions to nth-order linear ordinary differential equations is explained. For completeness, the analysis of the differential equation to determine whether the point of expansion is an ordinary point or a regular singular point is included. The superiority of the two methods shown over the standard method is demonstrated by using all three of the methods to work out several examples. Also included is a compendium of formulae and properties of the theta operator and generalized hypergeometric functions which is complete enough to make the report self-contained.

Allen, G.↗

NASA NTRS1989

Nonlinear evolution of oblique waves on compressible shear layers

The effects of critical-layer nonlinearity on spatially growing oblique instability waves on compressible shear layers between two parallel streams are considered. The analysis shows that mean temperature nonuniformities cause nonlinearity to occur at much smaller amplitudes than it does when the flow is isothermal. The nonlinear instability wave growth rate effects are described by an integrodifferential equation which bears some resemblance to the Landau equation, in that it involves a cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. Inviscid solutions always end in a singularity at a finite downstream distance, but viscosity can eliminate this singularity for certain parameter ranges.

Goldstein, M. E.↗

NASA NTRS1989

Resonance in cylindrical-rectangular and wraparound microstrip structures

A rigorous analysis of the resonance frequency problem of both the cylindrical-rectangular and the wraparound microstrip structure is presented. The problem is formulated in terms of a set of vector integral equations. Using Galerkin's method to solve the integral equations, the complex resonance frequencies are studied with sinusoidal basis functions which incorporate the edge singularity. The complex resonance frequencies are computed using a perturbation approach. Modes suitable for resonator or antenna applications are investigated. The edge singularity of the patch current is shown to have no significant effect on the accuracy of the results. It is shown that the HE10 modes of the cylindrical-rectangular and wraparound patches are more appropriate for resonator applications. The HE01 and TE01 modes of the cylindrical-rectangular and wraparound patches, respectively, are efficient radiating modes.

Ali, Sami M.↗

NASA NTRS1991

Aerodynamic parameter studies and sensitivity analysis for rotor blades in axial flight

The analytical capability is offered for aerodynamic parametric studies and sensitivity analyses of rotary wings in axial flight by using a 3-D undistorted wake model in curved lifting line theory. The governing equations are solved by both the Multhopp Interpolation technique and the Vortex Lattice method. The singularity from the bound vortices is eliminated through the Hadamard's finite part concept. Good numerical agreement between both analytical methods and finite differences methods are found. Parametric studies were made to assess the effects of several shape variables on aerodynamic loads. It is found, e.g., that a rotor blade with out-of-plane and inplane curvature can theoretically increase lift in the inboard and outboard regions respectively without introducing an additional induced drag.

Chiu, Y. Danny↗

NASA NTRS1991

RWF rotor-wake-fuselage code software reference guide

The RWF (Rotor-Wake-Fuselage) code was developed from first principles to compute the aerodynamics associated with the complex flow field of helicopter configurations. The code is sized for a single, multi-bladed main rotor and any configuration of non-lifting fuselage. The mathematical model for the RWF code is based on the integration of the momentum equations and Green's theorem. The unknowns in the problem are the strengths of prescribed singularity distributions on the boundaries of the flow. For the body (fuselage) a surface of constant strength source panels is used. For the rotor blades and rotor wake a surface of constant strength doublet panels is used. The mean camber line of the rotor airfoil is partitioned into surface panels. The no-flow boundary condition at the panel centroids is modified at each azimuthal step to account for rotor blade cyclic pitch variation. The geometry of the rotor wake is computers at each time step of the solution. The code produces rotor and fuselage surface pressures, as well as the complex geometry of the evolving rotor wake.

Berry, John D.↗

NASA NTRS1991

Aerodynamic parametric studies and sensitivity analysis for rotor blades in axial flight

The analytical capability is offered for aerodynamic parametric studies and sensitivity analyses of rotary wings in axial flight by using a 3D undistorted wake model in curved lifting line theory. The governing equations are solved by both the Multhopp Interpolation technique and the Vortex Lattice method. The singularity from the bound vortices is eliminated through the Hadamard's finite part concept. Good numerical agreement between both analytical methods and finite differences methods are found. Parametric studies were made to assess the effects of several shape variables on aerodynamic loads. It is found, e.g., that a rotor blade with out-of-plane and inplane curvature can theoretically increase lift in the inboard and outboard regions respectively without introducing an additional induced drag.

Chiu, Y. D.↗

NASA NTRS1992

Validity of the field line resonance expansion

The field line resonance expansion is studied with a magnetohydrodynamic (MHD) model applicable to earth's magnetotail. Using a Frobenius series solution, this expansion is found to be inconsistent. The inconsistency is seen by evaluating the terms in the model equations that the expansion neglects as small. Their effect is to provide a solution more singular than that determined by the expansion procedure. A tentative alternative expansion procedure, retaining these neglected terms approximately, yields a nonsingular solution, suggesting that resonant mode conversion rather than field line resonance is the appropriate phenomenon.

Hansen, P. J.↗

NASA NTRS1993

Vortex/surface interaction

This paper considers the interaction of a vortex generated upstream in a flow field with a downstream aerodynamic surface that possesses a large chord. The flow is assumed to be steady, incompressible, inviscid and irrotational, and the surface to be semiinfinite. The vortex is considered to be a straight vortex filament. To lowest order the problem is modeled using potential theory, where the 3D Laplace's equation for the velocity potential on the surface is solved exactly. The closed-form equation for pressure distribution obtained from this theory is found to have a square root singularity at the leading-edge. It also converges, as x goes to infinity, to the solution of the 2D point-vortex/infinite plane problem. The pressure coefficient presents an anti-symmetric behavior, near the leading-edge and a symmetric behavior as x goes to infinity.

Bodstein, G. C. R.↗

NASA NTRS1992

Kinematic equations for control of the redundant eight-degree-of-freedom advanced research manipulator 2

The forward position and velocity kinematics for the redundant eight-degree-of-freedom Advanced Research Manipulator 2 (ARM2) are presented. Inverse position and velocity kinematic solutions are also presented. The approach in this paper is to specify two of the unknowns and solve for the remaining six unknowns. Two unknowns can be specified with two restrictions. First, the elbow joint angle and rate cannot be specified because they are known from the end-effector position and velocity. Second, one unknown must be specified from the four-jointed wrist, and the second from joints that translate the wrist, elbow joint excluded. There are eight solutions to the inverse position problem. The inverse velocity solution is unique, assuming the Jacobian matrix is not singular. A discussion of singularities is based on specifying two joint rates and analyzing the reduced Jacobian matrix. When this matrix is singular, the generalized inverse may be used as an alternate solution. Computer simulations were developed to verify the equations. Examples demonstrate agreement between forward and inverse solutions.

Williams, Robert L., II↗

NASA NTRS1991

A Fourier-Chebyshev pseudospectral method for solving steady 3-D Navier-Stokes and heat equations in cylindrical cavities

A Fourier-Chebyshev pseudospectral method for solving steady 3D Navier-Stokes equations in cylindrical cavities is presented and discussed. The general method is pseudo-unsteady and uses a semi-implicit finite difference scheme for the time discretization. The generalized ADI (Alternating Direction Implicit) procedure is then applied to reduce the problem to a successive solution of one-dimensional problems. The spatial approximation uses a Fourier-Galerkin approximation in the periodic direction and a Chebyshev-collocation approximation in the other directions. Difficulties related to the pressure are surmounted by using the artifical compressibility method. A suitable variable change was chosen to avoid the problem of singularity at the axis generated by cylindrical coordinates. The method is first tested on an advection-diffusion equation and then on the Navier-Stokes equations. Finally, the method is illustrated by a convection problem of a differentially heated fluid.

Pulicani, J. P.↗