Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “eigenvalues”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
NASA NTRS1971

Eigenvalues of the Laplacian of a graph

Let G be a finite undirected graph with no loops or multiple edges. The Laplacian matrix of G, Delta(G), is defined by Delta sub ii = degree of vertex i and Delta sub ij = -1 if there is an edge between vertex i and vertex j. The structure of the graph G is related to the eigenvalues of Delta(G); in particular, it is proved that all the eigenvalues of Delta(G) are nonnegative, less than or equal to the number of vertices, and less than or equal to twice the maximum vertex degree. Precise conditions for equality are given.

Anderson, W. N., Jr.↗
NASA NTRS1972

Eigenvalues of singular differential operators by finite difference methods. I.

Approximation of the eigenvalues of certain self-adjoint operators defined by a formal differential operator in a Hilbert space. In general, two problems are studied. The first is the problem of defining a suitable Hilbert space operator that has eigenvalues. The second problem concerns the finite difference operators to be used.

Baxley, J. V.↗
NASA NTRS1972

Eigenvalues of singular differential operators by finite difference methods. II.

Note is made of an earlier paper which defined finite difference operators for the Hilbert space L2(m), and gave the eigenvalues for these operators. The present work examines eigenvalues for higher order singular differential operators by using finite difference methods. The two self-adjoint operators investigated are defined by a particular value in the same Hilbert space, L2(m), and are strictly positive with compact inverses. A class of finite difference operators is considered, with the idea of application to the theory of Toeplitz matrices. The approximating operators consist of a good approximation plus a perturbing operator.

Baxley, J. V.↗
NASA NTRS1972

Solution of eigenvalue problems by Sturm sequence method.

A generalized eigenvalue algorithm is presented herein along with the complete listing of the associated computer program, which may be conveniently utilized for the efficient solution of certain broad classes of eigenvalue problems. Extensive applications of the procedure are envisaged in the analysis of many important engineering problems, such as stability and natural frequency analysis of practical discrete structural systems, idealized by the finite element technique. The procedure based on the Sturm sequence method is accurate and fast, possessing several significant advantages over other known methods of such analysis. Numerical results are also presented for two representative structural engineering problems.

Gupta, K. K.↗
NASA NTRS1973

Eigenvalue routine by Sturm sequence method

Computer program has been generated for efficient solution of certain broad classes of eigenvalue problems. Procedure fully exploits banded nature of associated matrices and further enables user to compute either all roots or any specific ones desired. Special storage options enable storing only nonzero elements of associated main matrix of eigenvalue problem.

Gupta, K. K.↗
NASA NTRS1973

An automated procedure for computing flutter eigenvalues.

A new, fast and economical automated procedure for implementing the traditional V-g method of flutter solution is described. The procedure requires as input the generalized aerodynamic forces for a range of reduced frequencies obtained from an aerodynamic program. These aerodynamic forces are interpolated with respect to reduced frequency using a newly developed, partially tabulated cubic spline that is both fast in execution and economical in storage. The flutter solution is then obtained using an eigenvalue routine that has been developed to take advantage of the parametric nature of the V-g type of solution. Furthermore, the routine takes care of the fundamental and troublesome problem of properly sorting the output eigenvalues. By solving the root-sorting problem, the interpolation for flutter crossings and automatic plotting are accomplished efficiently. The computational techniques used in this new program are described and some sample results are given.

Desmarais, R. N.↗
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 NTRS1975

ALARM: A highly efficient eigenvalue extraction routine for very large matrixes

A highly efficient computer program, called ALARM, for the determination of eigenvalues and eigenvectors of large symmetric matrices (over 10,000 degrees of freedom), was described. As such, it is highly useful for analyzing complex finite element and finite difference idealizations encountered in structural dynamics and acoustic probelms. The program is based upon a scheme which reduces a large matrix to an equivalent tridiagonal one of much smaller size. The main strength of the algorithm lies in its ability to retain all the information required to obtain the eigenvalues at either end of the original matrix spectrum.

Ojalvo, I. U.↗
NASA NTRS1976

The eigenvalue spectrum of the Orr-Sommerfeld problem

A numerical investigation of the temporal eigenvalue spectrum of the ORR-Sommerfeld equation is presented. Two flow profiles are studied, the plane Poiseuille flow profile and the Blasius boundary layer (parallel): flow profile. In both cases a portion of the complex c-plane bounded by 0 less than or equal to CR sub r 1 and -1 less than or equal to ci sub i 0 is searched and the eigenvalues within it are identified. The spectra for the plane Poiseuille flow at alpha = 1.0 and R = 100, 1000, 6000, and 10000 are determined and compared with existing results where possible. The spectrum for the Blasius boundary layer flow at alpha = 0.308 and R = 998 was found to be infinite and discrete. Other spectra for the Blasius boundary layer at various Reynolds numbers seem to confirm this result. The eigenmodes belonging to these spectra were located and discussed.

Antar, B. N.↗
NASA NTRS1976

Real eigenvalue analysis in NASTRAN by the tridiagonal reduction (FEER) method

Implementation of the tridiagonal reduction method for real eigenvalue extraction in structural vibration and buckling problems is described. The basic concepts underlying the method are summarized and special features, such as the computation of error bounds and default modes of operation are discussed. In addition, the new user information and error messages and optional diagnostic output relating to the tridiagonal reduction method are presented. Some numerical results and initial experiences relating to usage in the NASTRAN environment are provided, including comparisons with other existing NASTRAN eigenvalue methods.

Newman, M.↗
NASA NTRS1977

Initial values for the integration scheme to compute the eigenvalues for propagation in ducts

A scheme for the calculation of eigenvalues in the problem of acoustic propagation in a two-dimensional duct is described. The computation method involves changing the coupled transcendental nonlinear algebraic equations into an initial value problem involving a nonlinear ordinary differential equation. The simplest approach is to use as initial values the hardwall eigenvalues and to integrate away from these values as the admittance varies from zero to its actual value with a linear variation. The approach leads to a powerful root finding routine capable of computing the transverse and axial wave numbers for two-dimensional ducts for any frequency, lining, admittance and Mach number without requiring initial guesses or starting points.

Eversman, W.↗
NASA NTRS1978

Extension of the tridiagonal reduction (FEER) method for complex eigenvalue problems in NASTRAN

As in the case of real eigenvalue analysis, the eigensolutions closest to a selected point in the eigenspectrum were extracted from a reduced, symmetric, tridiagonal eigenmatrix whose order was much lower than that of the full size problem. The reduction process was effected automatically, and thus avoided the arbitrary lumping of masses and other physical quantities at selected grid points. The statement of the algebraic eigenvalue problem admitted mass, damping, and stiffness matrices which were unrestricted in character, i.e., they might be real, symmetric or nonsymmetric, singular or nonsingular.

Newman, M.↗
NASA NTRS1977

The role of eigenvalues in linear feature selection theory

The analysis concerns the role of eigenvalues in determining a particular measure of pattern class distinction called the divergence, which is the pairwise average of the expected interclass divergence derived from Hajek's two-class divergence. Decel and Quirein (1973) showed that there always exists a k x n real matrix B such that the transformation determined by B maximizes divergence in k-dimensional space, and that B can be written as a product involving an orthogonal n x n matrix U. In the present paper it is shown that divergence measure of pattern class distinction does not depend on the eigenvalues of U.

Brown, D. R.↗
NASA NTRS1979

Eigenvalues and eigenvectors for hybrid coordinate equations of motion for flexible spacecraft

The eigenvalues and eigenvectors of a system of linear time-invariant equations describing the attitude motion of flexible spacecraft in terms of hybrid coordinates are characterized in terms of literal expressions by using peculiar properties of the system parameter matrices. For the undamped case the eigenvalues are localized in terms of inertial matrices and modal parameters. A procedure for calculating the eigenvectors is proposed whereby the eigenproblem associated with the original system of dimension (2N + 6) is reduced to that of a symmetric and positive definite matrix of dimension N with the zero-damping assumption. The eigenvectors for systems of large dimension are obtained explicitly in terms of a 3x1 matrix whose elements are available from a system of three algebraic equations, which is provided.

Ohkami, Y.↗
NASA NTRS1982

Determination of eigenvalues of dynamical systems by symbolic computation

A symbolic computation technique for determining the eigenvalues of dynamical systems is described wherein algebraic operations, symbolic differentiation, matrix formulation and inversion, etc., can be performed on a digital computer equipped with a formula-manipulation compiler. An example is included that demonstrates the facility with which the system dynamics matrix and the control distribution matrix from the state space formulation of the equations of motion can be processed to obtain eigenvalue loci as a function of a system parameter. The example chosen to demonstrate the technique is a fourth-order system representing the longitudinal response of a DC 8 aircraft to elevator inputs. This simplified system has two dominant modes, one of which is lightly damped and the other well damped. The loci may be used to determine the value of the controlling parameter that satisfied design requirements. The results were obtained using the MACSYMA symbolic manipulation system.

Howard, J. C.↗
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 NTRS1987

Arbitrary eigenvalue assignments for linear time-varying multivariable control systems

The problem of eigenvalue assignments for a class of linear time-varying multivariable systems is considered. Using matrix operators and canonical transformations, it is shown that a time-varying system that is 'lexicography-fixedly controllable' can be made via state feedback to be equivalent to a time-invariant system whose eigenvalues are arbitrarily assignable. A simple algorithm for the design of the state feedback is provided.

Nguyen, Charles C.↗