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At least 685 records · Page 38

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NASA NTRS1986

Some path-following techniques for solution of nonlinear equations and comparison with parametric differentiation

Some path-following techniques are described and compared with other methods. Use of multipurpose techniques that can be used at more than one stage of the path-following computation results in a system that is relatively simple to understand, program, and use. Comparison of path-following methods with the method of parametric differentiation reveals definite advantages for the path-following methods. The fact that parametric differentiation has found a broader range of applications indicates that path-following methods have been underutilized.

Barger, R. L.↗

NASA NTRS1988

Minimal parameter solution of the orthogonal matrix differential equation

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed employing the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

Bar-Itzhack, Itzhack Y.↗

NASA NTRS1987

Factorization and the synthesis of optimal feedback kernels for differential-delay systems

A combination of ideas from the theories of operator Riccati equations and Volterra factorizations leads to the derivation of a novel, relatively simple set of hyperbolic equations which characterize the optimal feedback kernel for the finite-time regulator problem for autonomous differential-delay systems. Analysis of these equations elucidates the underlying structure of the feedback kernel and leads to the development of fast and accurate numerical methods for its computation. Unlike traditional formulations based on the operator Riccati equation, the gain is characterized by means of classical solutions of the derived set of equations. This leads to the development of approximation schemes which are analogous to what has been accomplished for systems of ordinary differential equations with given initial conditions.

Milman, Mark M.↗

NASA NTRS1987

Mixing-length, shears, and differential rotation

Three aspects of convection in the solar envelope, the mixing length, the shears, and the differential rotation, are considered. Numerical experiments show that the mixing length is scaled by the pressure scale height rather than by the density scale height. Results indicate that the effective viscosity of convective turbulence on large scale shears may be approximated as 1/3 the product of the rms vertical velocity and the pressure scale height. It is suggested that the differential rotation of the solar convection zone may be viewed as an unstable axisymmetric mode which produces slower rotation at the surface.

Chan, K. L.↗

NASA NTRS1988

Real-time optical laboratory solution of parabolic differential equations

An optical laboratory matrix-vector processor is used to solve parabolic differential equations (the transient diffusion equation with two space variables and time) by an explicit algorithm. This includes optical matrix-vector nonbase-2 encoded laboratory data, the combination of nonbase-2 and frequency-multiplexed data on such processors, a high-accuracy optical laboratory solution of a partial differential equation, new data partitioning techniques, and a discussion of a multiprocessor optical matrix-vector architecture.

Casasent, David↗

NASA NTRS1988

Validating the airborne and ground based components of a differential GPS system

A differential GPS mechanization of an external data link concept under development at NASA Ames Research Center is discussed. The system is intended to evaluate the use of differential GPS to support helicopter approach operation. Analytical techniques for static testing and evaluation of the system hardware and software performance in real time are described. Several techniques that have been used to detect and rectify a specific problem encountered during the tests are dicussed. An example of flight test results using the validated system is provided.

Edwards, F. G.↗

NASA NTRS1988

Minimal parameter solution of the orthogonal matrix differential equation

Baritzhack, Itzhack Y.↗

NASA NTRS1988

Discriminating late volcanic differentiates commonly associated with precious metal deposits, using Landsat Thematic Mapper imagery

The magmatic differentiates emplaced late in volcanotectonic cycles (which when highly evolved commonly occur as domes, flow-dome complexes, and shallow intrusive porphyries) are typically more silicic and felsic than their earlier counterparts and exhibit unusually steep spectral curves from about 1.5 to 2.2 microns. This spectral characteristic emerges in the form of relatively high Landsat TM band 7 DN values and low 5/7 values, as well as dark-contrast or enhanced 5/7 images. These evolved late-phase differentiates are commonly associated both temporally and spatially with precious metal deposits, furnishing site-specific exploration targets as well as guides to caldera margins and other late-stage volcanotectonic structures.

Spatz, David M.↗

NASA NTRS1989

Classical geometries defined by exterior differential systems on higher frame bundles

Exterior differential ideals are discussed, and sets of invariant generators presented, for Reimannian, conformal and projective geometries, and for specializations such as Ricci-flat, self-dual and Einstein-Maxwell theories. The Cartan characteristic integers are explicitly calculated, and involutory basis forms found, for each of these (specialized to four dimensions), exposing their algebraic structure and showing how they generate well-posed sets of partial differential equations.

Estabrook, Frank B.↗

NASA NTRS1989

Photoacoustic measurement of differential broadening of the Lambda doublets in NO(X 2Pi 1/2,v = 2-0) by Ar

A differential broadening of the Lambda doublets in the v = 2-0 overtone band of the 2pi1/2 ground electronic state of NO in an Ar buffer gas has been observed by photoacoustic spectroscopy using a tunable color-center laser. The broadening coefficients for the f symmetry components are larger than for the e symmetry components by up to about 6 percent for J of about 16.5. This differential depends on J and vanishes at low J, implicating the anisotropy of the unpaired electron Pi orbital in the plane of rotation. The 2Pi3/2 transitions are slightly broader than the 2Pi1/2 as a result of spin-flipping collisional relaxation. The observed line shapes also exhibit collisional or Dicke narrowing due to velocity-changing collisions.

Pine, A. S.↗

NASA NTRS1989

Principles and procedures for determining absolute differential electron-molecule (atom) scattering cross sections

Procedures and calibration techniques for measuring the absolute elastic and inelastic differential cross sections (DCS) for electron impact on molecular (atomic) species are described and illustrated by examples. The elastic DCS for the molecule under study is first determined by calibration against helium using the relative flow technique. The second step involves the production of energy-loss spectra for the instrument response function, the unfolding of overlapping inelastic structures and the normalization of inelastic intensities to the elastic cross sections. It is concluded that this method of determining absolute differential electron-molecule (atom) scattering cross sections is generally applicable and provides reliable results.

Nickel, J. C.↗

NASA NTRS1989

A relationship between peak temperature drop and velocity differential in a microburst

Results from numerical microburst simulations using the Terminal Area Simulation System (Proctor, 1987) are used to develop a relationship between wind velocity differential and peak temperature drop. The numerical model and the relationships derived from the model are described. The relationship between peak temperature drop and differential wind velocity is shown to be valid during microburst development, for all precipitation shaft intensities and diameters. It is found that the relationship is not valid for low-reflectivity microburst events or in the presence of ground-based stable layers. The use of the relationship in IR wind shear detection systems is considered.

Proctor, Fred H.↗

NASA NTRS1987

Guidance simulation and test support for differential GPS flight experiment

Three separate tasks which supported the test preparation, test operations, and post test analysis of the NASA Ames flight test evaluation of the differential Global Positioning System (GPS) are presented. Task 1 consisted of a navigation filter design, coding, and testing to optimally make use of GPS in a differential mode. The filter can be configured to accept inputs from external censors such as an accelerometer and a barometric or radar altimeter. The filter runs in real time onboard a NASA helicopter. It processes raw pseudo and delta range measurements from a single channel sequential GPS receiver. The Kalman filter software interfaces are described in detail, followed by a description of the filter algorithm, including the basic propagation and measurement update equations. The performance during flight tests is reviewed and discussed. Task 2 describes a refinement performed on the lateral and vertical steering algorithms developed on a previous contract. The refinements include modification of the internal logic to allow more diverse inflight initialization procedures, further data smoothing and compensation for system induced time delays. Task 3 describes the TAU Corp participation in the analysis of the real time Kalman navigation filter. The performance was compared to that of the Z-set filter in flight and to the laser tracker position data during post test analysis. This analysis allowed a more optimum selection of the parameters of the filter.

Geier, G. J.↗

NASA NTRS1990

Differential equation based method for accurate approximations in optimization

A method to efficiently and accurately approximate the effect of design changes on structural response is described. The key to this method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses.

Pritchard, Jocelyn I.↗

NASA NTRS1990

Differential equation based method for accurate approximations in optimization

This paper describes a method to efficiently and accurately approximate the effect of design changes on structural response. The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in msot cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacement are used to approximate bending stresses.

Pritchard, Jocelyn I.↗

NASA NTRS1990

Minimal parameter solution of the orthogonal matrix differential equation

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed emplying the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

Bar-Itzhack, Itzhack Y.↗

NASA NTRS1990

Vertex implications of stability for a class of delay-differential interval system

Although vertex implication in parameter space does not hold for stability of general delay-differential interval systems, it is shown that in an important special case when the characteristic equation of the system is describable by certain quasi-polynomials the vertex implication for stability does hold. It is also pointed out that vertex implication for stability does not hold in general for delay-differential systems with interval delay.

Kim, K. D.↗

NASA NTRS1991

Asymptotic integration algorithms for nonhomogeneous, nonlinear, first order, ordinary differential equations

New methods for integrating systems of stiff, nonlinear, first order, ordinary differential equations are developed by casting the differential equations into integral form. Nonlinear recursive relations are obtained that allow the solution to a system of equations at time t plus delta t to be obtained in terms of the solution at time t in explicit and implicit forms. Examples of accuracy obtained with the new technique are given by considering systems of nonlinear, first order equations which arise in the study of unified models of viscoplastic behaviors, the spread of the AIDS virus, and predator-prey populations. In general, the new implicit algorithm is unconditionally stable, and has a Jacobian of smaller dimension than that which is acquired by current implicit methods, such as the Euler backward difference algorithm; yet, it gives superior accuracy. The asymptotic explicit and implicit algorithms are suitable for solutions that are of the growing and decaying exponential kinds, respectively, whilst the implicit Euler-Maclaurin algorithm is superior when the solution oscillates, i.e., when there are regions in which both growing and decaying exponential solutions exist.

Walker, K. P.↗