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NASA NTRS1984

Transonic calculation of airfoil stability and response with active controls

Transonic aeroelastic stability and response analyses are performed for the MBB A-3 supercritical airfoil. Three degrees of freedom are considered: plunge, pitch, and aileron pitch. The control of airfoil stability and response in transonic flow are studied. Stability analyses are performed using a Pade aeroelastic model based on the use of LTRAN2-NLR transonic small disturbance finite difference computer code. Response analyses are performed by coupling the structural equations of motion to the unsteady aerodynamic forces of LTRAN2-NLR. The focus is on transonic time marching transient response solutions using modal identification to determine stability. Frequency and damping of these modes are directly compared in the complex s-plane with Pade model eigenvalues. Transonic stability and response characteristics of 2-D airfoils are discussed and comparisons are made. Application of the Pade aeroelastic model and time marching analyses to flutter suppression using active controls is demonstrated.

Batina, J. T.↗

NASA NTRS1984

Optical systolic solutions of linear algebraic equations

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Neuman, C. P.↗

NASA NTRS1984

Matrix partitioning and EOF/principal component analysis of Antarctic Sea ice brightness temperatures

A field of measured anomalies of some physical variable relative to their time averages, is partitioned in either the space domain or the time domain. Eigenvectors and corresponding principal components of the smaller dimensioned covariance matrices associated with the partitioned data sets are calculated independently, then joined to approximate the eigenstructure of the larger covariance matrix associated with the unpartitioned data set. The accuracy of the approximation (fraction of the total variance in the field) and the magnitudes of the largest eigenvalues from the partitioned covariance matrices together determine the number of local EOF's and principal components to be joined by any particular level. The space-time distribution of Nimbus-5 ESMR sea ice measurement is analyzed.

Murray, C. W., Jr.↗

NASA NTRS1984

The ERODYN and QRPIG computer programs

The role of the ERODYN computer program in providing error analyses involving orbital, geodetic, and geophysical parameters is discussed. It was designed to operate as a companion program to the GEODYN orbit determination and parameter estimating program. The Q R Partitioned Eigenvalue/Eigenvector analysis program (QRPIG) is designed to process symmetric matrices with an out-of-core partitioning algorithm for the eigenvectors and eigenvalues of the matrices.

Felsentreger, T. L.↗

NASA NTRS1984

Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Ito, K.↗

NASA NTRS1984

Design Spectrum Analysis in NASTRAN

The utility of Design Spectrum Analysis is to give a mode by mode characterization of the behavior of a design under a given loading. The theory of design spectrum is discussed after operations are explained. User instructions are taken up here in three parts: Transient Preface, Maximum Envelope Spectrum, and RMS Average Spectrum followed by a Summary Table. A single DMAP ALTER packet will provide for all parts of the design spectrum operations. The starting point for getting a modal break-down of the response to acceleration loading is the Modal Transient rigid format. After eigenvalue extraction, modal vectors need to be isolated in the full set of physical coordinates (P-sized as opposed to the D-sized vectors in RF 12). After integration for transient response the results are scanned over the solution time interval for the peak values and for the times that they occur. A module called SCAN was written to do this job, that organizes these maxima into a diagonal output matrix. The maximum amplifier in each mode is applied to the eigenvector of each mode which then reveals the maximum displacements, stresses, forces and boundary reactions that the structure will experience for a load history, mode by mode. The standard NASTRAN output processors have been modified for this task. It is required that modes be normalized to mass.

Butler, T. G.↗

NASA NTRS1982

Interactive digital signal processor

The Interactive Digital Signal Processor (IDSP) is examined. It consists of a set of time series analysis Operators each of which operates on an input file to produce an output file. The operators can be executed in any order that makes sense and recursively, if desired. The operators are the various algorithms used in digital time series analysis work. User written operators can be easily interfaced to the sysatem. The system can be operated both interactively and in batch mode. In IDSP a file can consist of up to n (currently n=8) simultaneous time series. IDSP currently includes over thirty standard operators that range from Fourier transform operations, design and application of digital filters, eigenvalue analysis, to operators that provide graphical output, allow batch operation, editing and display information.

Mish, W. H.↗

NASA NTRS1983

Optimal trajectories for aeroassisted orbital transfer

Consideration is given to classical and minimax problems involved in aeroassisted transfer from high earth orbit (HEO) to low earth orbit (LEO). The transfer is restricted to coplanar operation, with trajectory control effected by means of lift modulation. The performance of the maneuver is indexed to the energy expenditure or, alternatively, the time integral of the heating rate. Firist-order optimality conditions are defined for the classical approach, as are a sequential gradient-restoration algorithm and a combined gradient-restoration algorithm. Minimization techniques are presented for the aeroassisted transfer energy consumption and time-delay integral of the heating rate, as well as minimization of the pressure. It is shown that the eigenvalues of the Jacobian matrix of the differential system is both stiff and unstable, implying that the sequential gradient restoration algorithm in its present version is unsuitable. A new method, involving a multipoint approach to the two-poing boundary value problem, is recommended.

Miele, A.↗

NASA NTRS1983

Feature selection methodologies using simulated Thematic Mapper data

The present investigation is concerned with the determination of the intrinsic dimensionality of a simulated Thematic Mapper data set. In addition, the effectiveness and sensitivity of 'standard' statistics separability measures (i.e., transformed divergence) is evaluated in comparison to eigenvectors for identifying the optimum subset of the original Thematic Mapper Simulator (TMS) bands for classifying the various cover types. TMS data were collected on May 2, 1979 by NASA's NS001 aircraft multispectral scanner over a bottomland forested area in South Carolina near the city of Camden. It is found that the eigenvectors and eigenvalues of a covariance matrix from a multispectral scanner system (MSS) data set can be obtained without having to actually transform the data.

Dean, M. E.↗

NASA NTRS1983

Direct and implicit optical matrix-vector algorithms

New direct and implicit algorithms for optical matrix-vector and systolic array processors are considered. Direct rather than indirect algorithms to solve linear systems and implicit rather than explicit solutions to solve second-order partial differential equations are discussed. In many cases, such approaches more properly utilize the advantageous features of optical systolic array processors. The matrix-decomposition operation (rather than solution of the simplified matrix-vector equation that results) is recognized as the computationally burdensome aspect of such problems that should be computed on an optical system. The Householder QR matrix-decomposition algorithm is considered as a specific example of a direct solution. Extensions to eigenvalue computation and formation of matrices of special structure are also noted.

Casasent, D.↗

NASA NTRS1983

Numerical approximations to nonlinear conservation laws with locally varying time and space grids

Numerical approximations to the initial value problem for nonlinear systems of conservation laws are considered. The considered system is said to be hyperbolic when all eigenvalues of every real linear combination of the Jacobian matrices are real. Solutions may develop discontinuities in finite time, even when the initial data are smooth. In the investigation, explicit finite difference methods which use locally varying time grids are considered. The global CFL restriction is replaced by a local restriction. The numerical flux function is studied from a finite volume viewpoint, and a differencing technique is developed at interface points between regions of distinct time increments.

Osher, S.↗

NASA NTRS1983

Large amplitude MHD waves upstream of the Jovian bow shock

Observations of large amplitude MHD waves upstream of Jupiter's bow shock are analyzed. The waves are found to be right circularly polarized in the solar wind frame, which suggests that they are propagating in the fast magnetosonic mode. A complete spectral and minimum variance eigenvalue analysis of the data was performed. The power spectrum of the magnetic fluctuations contains several peaks. The fluctuations at 2.3 MHz have a direction of minimum variance anti-parallel to the direction of the average magnetic field. Several harmonics at 6, 9, and 12 MHz are also present. The direction of minimum variance of these fluctuations lies at approximately 40 deg to the magnetic field. It is argued that these fluctuations are waves excited by protons reflected off the Jovian bow shock. The inferred speed of the reflected protons is about two times the solar wind speed in the solar wind frame. A linear instability analysis is presented that suggests an explanation for many of the observed features of the observations. The fluctuations apparently contain a significant fraction of magnetic energy that is linearly polarized and in the Alfven mode.

Goldstein, M. L.↗

NASA NTRS1984

The collisionless macroscopic Kelvin-Helmholtz instabiity. I Transverse electrostatic mode

The electrostatic limit of the Kelvin-Helmholtz instability in a collisionless plasma is investigated for a magnetic field perpendicular to the velocity flow. A two-fluid eigenvalue analysis of the transverse electrostatic Kelvin-Helmholtz instability is performed using the Chew-Goldberger-Low hydromagnetic equations with first-order finite Larmer radius corrections, establishing that the transverse electrostatic mode reduces to the incompressible transverse MHD mode in the limit of zero ion gyroradius. The linear growth rates are determined and a strong stabilization of the instability for (k-perpendicular) (rho sub i) of 0.2 or greater is observed. The nonlinear stage of the instability displays large vortices whose size is many times the initial velocity shear length. At saturation the simulation is dominated by the longest-wavelength mode permitted in the system. The simulation results can be understood in terms of a simple model which assumes that saturation occurs when the angular frequency of the vortex motion reaches the linear growth rate of the instability.

Prichett, P. L.↗

NASA NTRS1983

Stability analysis of a buck regulator employing input filter compensation

The interaction between the input filter and the regulator often causes serious degradation of performance. The reduction in loop gain due to input filter interaction can result in system instability. An exact stability analysis of the buck regulator system is presented. The input filter parameter values are varied and system instability is predicted for the case without feedforward. The eigenvalues of the system can be brought back into the unit circle and the system thus stabilized with the addition of the feedforward loop. Measurements made for the cases with and without feedforward confirm the analytical prediction.

Kelkar, S. S.↗

NASA NTRS1983

Light extinction by aerosols during summer air pollution

In order to utilize satellite measurements of optical thickness over land for estimating aerosol properties during air pollution episodes, the optical thickness was measured from the surface and investigated. Aerosol optical thicknesses have been derived from solar transmission measurements in eight spectral bands within the band lambda 440-870 nm during the summers of 1980 and 1981 near Washington, DC. The optical thicknesses for the eight bands are strongly correlated. It was found that first eigenvalue of the covariance matrix of all observations accounts for 99 percent of the trace of the matrix. Since the measured aerosol optical thickness was closely proportional to the wavelength raised to a power, the aerosol size distribution derived from it is proportional to the diameter (d) raised to a power for the range of diameters between 0.1 to 1.0 micron. This power is insensitive to the total optical thickness. Changes in the aerosol optical thickness depend on several aerosol parameters, but it is difficult to identify the dominant one. The effects of relative humidity and accumulation mode concentration on the optical thickness are analyzed theoretically, and compared with the measurements.

Kaufman, Y. J.↗

NASA NTRS1984

Stability and structure of stretched vortices

The linear stability of the symmetric Burgers vortex to a class of two-dimensional perturbations is demonstrated. The linear stability of the axially symmetric Burgers vortex to two-dimensional low-Reynolds number disturbances is first discussed. The analysis describes the connection of the normal modes with a classical set of special functions. An expansion in Reynolds number gives the variation of the temporal eigenvalues of the normal modes. Then, Burgers-type solutions are sought where the radially inward-straining field is not axially symmetric. A rational perturbation procedure is used to obtain a new class of nonsymmetric Burgers vortices valid when the Reynolds number and the asymmetry in the external strain are small. Finally, the perturbation solutions are numerically extended to larger values of the parameters.

Robinson, A. C.↗

NASA NTRS1984

Vibration and buckling of general periodic lattice structures

A method is presented for vibration and buckling analysis of arbitrary lattice structures having repetitive geometry in any combination of coordinate directions. The approach is based on exact member theory for representing the stiffness of an individual member subject to axial load, and in the case of vibration, undergoing harmonic oscillation. The method is an extension of previous work that was limited to specific geometries. The resulting eigenvalue problem is of the size associated with the repeating element of the structure. A computer program has been developed incorporating the theory and results are given for vibration of rectangular platforms and a large antenna structure having rotational symmetry. Buckling and vibration results for cable-stiffened rings are also given.

Anderson, M. S.↗

NASA NTRS1984

A generalized multiple-input, multiple-output modal parameter estimation algorithm

A new method for experimental determination of the modal parameters of a structure is presented. The method allows for multiple input forces to be applied simultaneously, and for an arbitrary number of acceleration response measurements to be employed. These data are used to form the equations of motion for a damped linear elastic structure. The modal parameters are then obtained through an eigenvalue technique. In conjunction with the development of the equations, an extensive computer simulation study was performed. The results of the study show a marked improvement in the mode shape identification for closely-spaced modes as the number of applied forces is increased. Also demonstrated is the influence of noise on the method's ability to identify accurate modal parameters. Here again, an increase in the number of exciters leads to a significant improvement in the identified parameters.

Craig, R. R., Jr.↗