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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 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

Derivatives of eigenvalues and eigenvectors of a general complex matrix

A survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non-Hermitian matrices is presented. In addition, a modification of one method based on a better normalizing condition is proposed. Methods are classified as Direct or Adjoint and are evaluated for efficiency. Operation counts are presented in terms of matrix size, number of design variables and number of eigenvalues and eigenvectors of interest. The effect of the sparsity of the matrix and its derivatives is also considered, and typical solution times are given. General guidelines are established for the selection of the most efficient method.

Murthy, Durbha V.↗
NASA NTRS1978

The stability of an asymmetric rotor in damped supports

A general-purpose method of evaluating the stability of an asymmetric flexible rotor, mounted in symmetric damped bearings, is defined. This method evaluates the complex eigenvalues of the rotor system by solving the equations of motion in a rotating coordinate frame. The application of this method to a rotor mounted in tilting-pad bearings is demonstrated. The observed behavior of a number of different rotor configurations is compared with corresponding predictions of stability. For the configurations predicted to be unstable, a distinct and unnegotiable threshold of instability is encountered. The sharpness of this threshold is emphasized by careful balancing at speeds fractionally below the threshold. In a final configuration predicted to be marginally stable, lightly damped resonant behavior, negotiable by balancing, is encountered in the region of the first critical speed.

Smalley, A. J.↗
NASA NTRS1987

Improvement of modal scaling factors using mass additive technique

A general investigation into the improvement of modal scaling factors of an experimental modal model using additive technique is discussed. Data base required by the proposed method consists of an experimental modal model (a set of complex eigenvalues and eigenvectors) of the original structure and a corresponding set of complex eigenvalues of the mass-added structure. Three analytical methods,i.e., first order and second order perturbation methods, and local eigenvalue modification technique, are proposed to predict the improved modal scaling factors. Difficulties encountered in scaling closely spaced modes are discussed. Methods to compute the necessary rotational modal vectors at the mass additive points are also proposed to increase the accuracy of the analytical prediction.

Zhang, Qiang↗
NASA NTRS1993

Eigenvalue routines in NASTRAN: A comparison with the Block Lanczos method

The NASA STRuctural ANalysis (NASTRAN) program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this paper is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the Block Lanczos algorithm by Boeing Computer Services (BCS). This eigenvalue routine is now available in the BCS mathematics library as well as in several commercial versions of NASTRAN. In addition, CRAY maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The Block Lanczos algorithm was found to be extremely efficient, in particular, for very large size problems.

Tischler, V. A.↗
NASA NTRS1965

Lower Bounds to Energy Eigenvalues for the Stark Effect in a Rigid Rotator

Upper and lower bounds have been calculated for the energy levels of a rigid rotator in an electric field, in order to study the problems associated with the use of the partitioning method for bracketing an eigenvalue of the Schrödinger equation. Results of arbitrarily high accuracy are possible in this example.

Choi, Jong H.↗
NASA NTRS1973

Eigenvalue uncertainty in stressed structures.

A method is presented for calculating the statistics of the natural frequencies and mode shapes of vibration for a structure acted upon by an external static loading which results in the structure being stressed for eigenvalue analysis. The analytical tools presented apply to the probabilistic eigenvalue problem, and it is apparent that structural parameter uncertainty will significantly influence all aspects of the structure's response. The treatment of a sample problem serves the purpose of furthering understanding for the importance of considering structural parameters as random variables.

Hart, G. C.↗
NASA NTRS1989

Accurate calculation of control-augmented structural eigenvalue sensitivities using reduced-order models

A method is presented for generating mode shapes for model order reduction in a way that leads to accurate calculation of eigenvalue derivatives and eigenvalues for a class of control augmented structures. The method is based on treating degrees of freedom where control forces act or masses are changed in a manner analogous to that used for boundary degrees of freedom in component mode synthesis. It is especially suited for structures controlled by a small number of actuators and/or tuned by a small number of concentrated masses whose positions are predetermined. A control augmented multispan beam with closely spaced natural frequencies is used for numerical experimentation. A comparison with reduced-order eigenvalue sensitivity calculations based on the normal modes of the structure shows that the method presented produces significant improvements in accuracy.

Livne, Eli↗
NASA NTRSDate not supplied

Towards Formalization of Advanced Linear Algebra with Applications to Dynamical Systems using PVS

Linear Algebra is essential for numerous aerospace problems of interest. Formal reasoning about hybrid systems that contain variables modeled by differential equations rely on concepts from Linear Algebra such as eigenvalues, matrix decompositions, and matrix valued functions. For example, the long-term dynamics of a system of differential equations depend on the stability/instability of its equilibrium points, which often reduces to an eigenvalue problem. This talk will embark on a quest to formalize theorems and results about eigenvalues and eigenvectors using PVS. We shall start our journey with 2 x 2 complex matrices, where we will apply our PVS code to a simple example of a dynamical system. Since it can be difficult or impossible to give simple expressions of eigenvalues for larger matrices (i.e. 5 x 5 or higher), we then move towards specifying the power method for verified computation of eigenvalue approximations in PVS. This effort requires development of multivariate complex arithmetic. At the end of the day, having such additions to the PVS NASA libraries will help move towards the use of formal methods to verify concepts of control theory and system level verification.

Linear Algebra↗
NASA NTRS1988

Multigrid method for stability problems

The problem of calculating the stability of steady state solutions of differential equations is addressed. Leading eigenvalues of large matrices that arise from discretization are calculated, and an efficient multigrid method for solving these problems is presented. The resulting grid functions are used as initial approximations for appropriate eigenvalue problems. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a nonstandard way in which the right-hand side of the coarse grid equations involves unknown parameters to be solved on the coarse grid. This leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem are presented which demonstrate the effectiveness of the method.

Ta'asan, Shlomo↗
NASA NTRS1972

Adjoint variational methods in nonconservative stability problems.

A general nonself-adjoint eigenvalue problem is examined and it is shown that the commonly employed approximate methods, such as the Galerkin procedure, the method of weighted residuals and the least square technique lack variational descriptions. When used in their previously known forms they do not yield stationary eigenvalues and eigenfunctions. With the help of an adjoint system, however, several analogous variational descriptions may be developed and it is shown in the present study that by properly restating the method of least squares, stationary eigenvalues may be obtained. Several properties of the adjoint eigenvalue problem, known only for a restricted group, are shown to exist for the more general class selected for study.

Prasad, S. N.↗
NASA NTRS2011

Ares I-X In-Flight Modal Identification

Operational modal analysis is a procedure that allows the extraction of modal parameters of a structure in its operating environment. It is based on the idealized premise that input to the structure is white noise. In some cases, when free decay responses are corrupted by unmeasured random disturbances, the response data can be processed into cross-correlation functions that approximate free decay responses. Modal parameters can be computed from these functions by time domain identification methods such as the Eigenvalue Realization Algorithm (ERA). The extracted modal parameters have the same characteristics as impulse response functions of the original system. Operational modal analysis is performed on Ares I-X in-flight data. Since the dynamic system is not stationary due to propellant mass loss, modal identification is only possible by analyzing the system as a series of linearized models over short periods of time via a sliding time-window of short time intervals. A time-domain zooming technique was also employed to enhance the modal parameter extraction. Results of this study demonstrate that free-decay time domain modal identification methods can be successfully employed for in-flight launch vehicle modal extraction.

Bartkowicz, Theodore J.↗
NASA NTRS1987

Stability of streamwise vortices

A brief overview of some theoretical and computational studies of the stability of streamwise vortices is given. The local induction model and classical hydrodynamic vortex stability theories are discussed in some detail. The importance of the three-dimensionality of the mean velocity profile to the results of stability calculations is discussed briefly. The mean velocity profile is provided by employing the similarity solution of Donaldson and Sullivan. The global method of Bridges and Morris was chosen for the spatial stability calculations for the nonlinear eigenvalue problem. In order to test the numerical method, a second order accurate central difference scheme was used to obtain the coefficient matrices. It was shown that a second order finite difference method lacks the required accuracy for global eigenvalue calculations. Finally the problem was formulated using spectral methods and a truncated Chebyshev series.

Khorrami, M. K.↗
NASA NTRS1981

Acoustic transmission in lined flow ducts - A finite element eigenvalue problem

The problem of acoustical transmission in lined ducts with subsonic mean flow is of considerable practical interest in the context of fan noise attenuation in the ducted inlet regions of turbofan aircraft engines. If nonaxisymmetric liners are present, a loss of axial symmetry results, and the study of acoustic transmission involves the solution of a full two-dimensional eigenvalue problem. The reported investigation is concerned with such an eigenvalue problem. The employed method of solution is effectively a two-dimensional analog of an approach considered by Astley and Eversman (1979). The approach makes use of a Galerkin Finite Element Method whereby the weighting and basis functions are generated automatically by the discretization.

Astley, R. J.↗
NASA NTRS1983

Direct structural parameter identification by modal test results

A direct identification procedure is proposed to obtain the mass and stiffness matrices based on the test measured eigenvalues and eigenvectors. The method is based on the theory of matrix perturbation in which the correct mass and stiffness matrices are expanded in terms of analytical values plus a modification matrix. The simplicity of the procedure enables real time operation during the structural testing.

Chen, J.-C.↗
NASA NTRS1984

Reduction of satellite magnetic anomaly data

Analysis of global magnetic anomaly maps derived from satellite data is facilitated by inversion to the equivalent magnetization in a constant thickness magnetic crust or, equivalently, by reduction to the pole. Previous inversions have proven unstable near the geomagnetic equator. The instability results from magnetic moment distributions which are admissible in the inversion solution but which make only small contribution to the computed values of anomaly field. Their admissibility in the solution could result from noisy or incomplete data or from small poorly resolved anomalies. The resulting magnetic moments are unrealistically large and oscillatory. Application of the method of principal components (e.g. eigenvalue decomposition and selective elimination of less significant eigenvectors) is proposed as a way of overcoming the instability and the method is demonstrated by applying it to the region around the Bangui anomaly in Central Africa.

Slud, E. V.↗
NASA NTRS1988

An improved error assessment for the GEM-T1 gravitational model

Several tests were designed to determine the correct error variances for the Goddard Earth Model (GEM)-T1 gravitational solution which was derived exclusively from satellite tracking data. The basic method employs both wholly independent and dependent subset data solutions and produces a full field coefficient estimate of the model uncertainties. The GEM-T1 errors were further analyzed using a method based upon eigenvalue-eigenvector analysis which calibrates the entire covariance matrix. Dependent satellite and independent altimetric and surface gravity data sets, as well as independent satellite deep resonance information, confirm essentially the same error assessment. These calibrations (utilizing each of the major data subsets within the solution) yield very stable calibration factors which vary by approximately 10 percent over the range of tests employed. Measurements of gravity anomalies obtained from altimetry were also used directly as observations to show that GEM-T1 is calibrated. The mathematical representation of the covariance error in the presence of unmodeled systematic error effects in the data is analyzed and an optimum weighting technique is developed for these conditions. This technique yields an internal self-calibration of the error model, a process which GEM-T1 is shown to approximate.

Lerch, F. J.↗
NASA NTRS1989

An improved error assessment for the GEM-T1 gravitational model

Several tests were designed to determine the correct error variances for the GEM-T1 gravitational solution which was derived exclusively from satellite tracking data. The basic method employs both wholly independent and dependent subset data solutions and produces a full field coefficient by coefficient estimate of the model uncertainties. The GEM-T1 errors were further analyzed using a method based upon eigenvalue-eigenvector analysis which calibrates the entire covariance matrix. Dependent satellite and independent altimetric and surface gravity data sets, as well as independent satellite deep resonance information, confirm essentially the same error assessment.

Lerch, F. J.↗
NASA NTRS1992

Solving Large Sparse Symmetric Eigenproblems

LANZ program implements sophisticated algorithm based on simple Lanczos method for solving generalized eigenvalue problem. Uses dynamic shifting to improve efficiency and reliability of basic Lanczos algorithm. Written in FORTRAN 77 and C language.

Jones, Mark T.↗