Results for “Eigenvalue problems” — Page 4

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 NTRS1970

Nonuniqueness of the energy correction in application of the WKB approximation to radial problems

Radial energy eigenvalue problem conversion into one dimensional by infinite set of transformations, considering WKB quantization condition

Beckel, C. L.↗

NASA NTRS1967

Integrating Matrix Method for Determining the Natural Vibrations of a Rotating, Unsymmetrical Beam with Application to Twisted Propeller Blades

A numerical method is Presented for determining the natural vibration frequencies, and the corresponding mode shapes, of a rotating cantilever beam which has a nonuniform, unsymmetrical cross section. Two coupled fourth-order differential equations of motion with variable coefficients are derived which govern the motion of such a beam having deformations in two directions. Through the development and utilization of the integrating matrix, the solution of the differential equations is obtained in the form of an eigenvalue problem. The solutions to the eigenvalue problem are determined by an iteration method based upon a special orthogonality relationship which is derived. Numerical examples, including an application to a twisted propeller blade, are presented with the results of the integrating matrix solutions being compared to exact solutions and experimental data.

Hunter, William F.↗

NASA NTRS1986

Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: An eigenvalue analysis

A hyperbolic initial-boundary-value problem can be approximated by a system of ordinary differential equations (ODEs) by replacing the spatial derivatives by finite-difference approximations. The resulting system of ODEs is called a semidiscrete approximation. A complication is the fact that more boundary conditions are required for the spatially discrete approximation than are specified for the partial differential equation. Consequently, additional numerical boundary conditions are required and improper treatment of these additional conditions can lead to instability. For a linear initial-boundary-value problem (IBVP) with homogeneous analytical boundary conditions, the semidiscrete approximation results in a system of ODEs of the form du/dt = Au whose solution can be written as u(t) = exp(At)u(O). Lax-Richtmyer stability requires that the matrix norm of exp(At) be uniformly bounded for O less than or = t less than or = T independent of the spatial mesh size. Although the classical Lax-Richtmyer stability definition involves a conventional vector norm, there is no known algebraic test for the uniform boundedness of the matrix norm of exp(At) for hyperbolic IBVPs. An alternative but more complicated stability definition is used in the theory developed by Gustafsson, Kreiss, and Sundstrom (GKS). The two methods are compared.

Warming, Robert F.↗

NASA NTRS1992

Crossflow effects on the growth rate of inviscid Goertler vortices in a hypersonic boundary layer

The effects of crossflow on the growth rate of inviscid Goertler vortices in a hypersonic boundary layer with pressure gradient are studied. Attention is focused on the inviscid mode trapped in the temperature adjustment layer; this mode has greater growth rate than any other mode. The eigenvalue problem which governs the relationship between the growth rate, the crossflow amplitude, and the wavenumber is solved numerically, and the results are then used to clarify the effects of crossflow on the growth rate of inviscid Goertler vortices. It is shown that crossflow effects on Goertler vortices are fundamentally different for incompressible and hypersonic flows. The neutral mode eigenvalue problem is found to have an exact solution, and as a by-product, we have also found the exact solution to a neutral mode eigenvalue problem which was formulated, but unsolved before, by Bassom and Hall (1991).

Fu, Yibin↗

NASA NTRS1989

Solution and sensitivity analysis of a complex transcendental eigenproblem with pairs of real eigenvalues

This paper considers complex transcendental eigenvalue problems where one is interested in pairs of eigenvalues that are restricted to take real values only. Such eigenvalue problems arise in dynamic stability analysis of nonconservative physical systems, i.e., flutter analysis of aeroelastic systems. Some available solution methods are discussed and a new method is presented. Two computational approaches are described for analytical evaluation of the sensitivities of these eigenvalues when they are dependent on other parameters. The algorithms presented are illustrated through examples.

Murthy, Durbha V.↗

NASA NTRS1989

Solution and sensitivity analysis of a complex transcendental eigenproblem with pairs of real eigenvalues

This paper considers complex transcendental eigenvalue problems where one is interested in pairs of eigenvalues that are restricted to take real values only. Such eigenvalue problems arise in dynamic stability analysis of nonconservative physical systems, i.e., flutter analysis of aeroelastic systems. Some available solution methods are discussed and a new method is presented. Two computational approaches are described for analytical evaluation of the sensitivities of these eigenvalues when they are dependent on other parameters. The algorithms presented are illustrated through examples.

Murthy, D. V.↗

NASA NTRS1975

Numerical techniques for the linear, nonadiabatic stellar pulsation problem

The linear, nonadiabatic eigenvalue problem is formulated using Castor's method for calculating radial pulsations of stellar models. Both left and right eigenvectors are calculated. Initial eigenvalues for the linear, nonadiabatic solutions are obtained from the adiabatic eigenvalues and left and right eigenvectors. The orthogonality relation is obtained. Simple formulas for the Newton method are given. The iteration procedure is constrained to improve convergence. The Newton method is less satisfactory than the secant method for difficult cases. The linear, nonadiabatic solutions are shown to be sensitive to the number of zones with tau smaller than 2/3, and the value of (P/P sub r) surface or tau (surface). Optimum values can be determined for the number of zones and tau (surface). Application of the method to Population II Cepheids is briefly presented.

Bednarek, T. A.↗

NASA NTRS1987

Boundary layer stability calculations

In this paper numerical calculation of the spatial stability of disturbances in the parallel and nonparallel Blasius boundary layers is considered. Chebyshev polynomials are used for discretization. The problem with the boundary condition at infinity is overcome, and the resulting nonlinear matrix eigenvalue problem is attacked directly. The secondary eigenvalue problem for three-dimensional disturbances is shown to be uniformly stable, and particular solutions of this problem generated by the Orr-Sommerfeld equation are shown. A numerical solution of the nonparallel problem is considered using Chebyshev polynomials. The matrix equations are analyzed directly and the problem of uniqueness of the nonparallel correction is settled by careful application of the Fredholm alternative. Nonparallel corrections to the streamwise eigenfunction are shown.

Bridges, Thomas J.↗

NASA NTRS1972

Buckling and vibration analysis for stiffened orthotropic shells of revolution.

Development of a numerical method, using the multistep integration approach, in which the buckling and vibration analyses are formulated as a succession of linear eigenvalue problems. The method does not require an estimate of the eigenvector, and once an eigenvalue has been converged good estimates for other eigenvalues are automatically available. This is accomplished through the use of an in-core Householder scheme for solution of the eigenvalue problem. Furthermore, since the method uses an eigenvalue solution, the possibility of missing modes is eliminated.

Svalbonas, V.↗

NASA NTRS1984

A new method for determining acoustic-liner admittance in a rectangular duct with grazing flow from experimental data

A method is developed for determining acoustic liner admittance in a rectangular duct with grazing flow. The axial propagation constant, cross mode order, and mean flow profile is measured. These measured data are then input into an analytical program which determines the unknown admittance value. The analytical program is based upon a finite element discretization of the acoustic field and a reposing of the unknown admittance value as a linear eigenvalue problem on the admittance value. Gaussian elimination is employed to solve this eigenvalue problem. The method used is extendable to grazing flows with boundary layers in both transverse directions of an impedance tube (or duct). Predicted admittance values are compared both with exact values that can be obtained for uniform mean flow profiles and with those from a Runge Kutta integration technique for cases involving a one dimensional boundary layer.

Watson, W. R.↗

NASA NTRS1988

Comparison results for ignition in conjugate systems

The exothermic reaction of an arbitrarily shaped solid in a well-mixed continuous flow of constant density and velocity is investigated analytically, with a focus on a zeroth-order reaction in a pellet. Comparison theorems for the basic problem and the eigenvalue problem of ignition are derived, and the domain dependence of the bifurcation point is explored using the method of Joseph (1976). Some typical analytical results are presented in tables and compared with published values obtained using numerical methods.

Pushpavanam, S.↗

NASA NTRS1989

Eigenvector derivatives of repeated eigenvalues using singular value decomposition

An explicit formula is obtained for the first-order eigenvector derivative that corresponds to the eigenvector of a repeated eigenvalue, in the case of the nonself-adjoint eigenvalue problem. This method applies to the class of nondefective problems whose first eigenvalue derivatives of the repeated eigenvalues are distinct. A singular-value decomposition approach is used to compute four requisite bases for eigenspaces, as well as to keep track of the dimensions of state variables and the conditioning of the state equations.

Lim, Kyong B.↗

NASA NTRS1993

On the cross-stream spectral method for the Orr-Sommerfeld equation

Cross-stream models are defined as solutions to the Orr-Sommerfeld equation which are propagating normal to the flow direction. These models are utilized as a basis for a Hilbert space to approximate the spectrum of the Orr-Sommerfeld equation with plane Poiseuille flow. The cross-stream basis leads to a standard eigenvalue problem for the frequencies of Poiseuille flow instability waves. The coefficient matrix in the eigenvalue problem is shown to be the sum of a real matrix and a negative-imaginary diagonal matrix which represents the frequencies of the cross-stream modes. The real coefficient matrix is shown to approach a Toeplitz matrix when the row and column indices are large. The Toeplitz matrix is diagonally dominant, and the diagonal elements vary inversely in magnitude with diagonal position. The Poiseuille flow eigenvalues are shown to lie within Gersgorin disks with radii bounded by the product of the average flow speed and the axial wavenumber. It is shown that the eigenvalues approach the Gersgorin disk centers when the mode index is large, so that the method may be used to compute spectra with an essentially unlimited number of elements. When the mode index is large, the real part of the eigenvalue is the product of the axial wavenumber and the average flow speed, and the imaginary part of the eigen value is identical to the corresponding cross-stream mode frequency. The cross-stream method is numerically well-conditioned in comparison to Chebyshev based methods, providing equivalent accuracy for small mode indices and superior accuracy for large indices.

Zorumski, William E.↗

NASA NTRS1990

Shape optimization for maximum stability and dynamic stiffness

Any optimization of structures for maximum stability or for maximum dynamic stiffness deals with an eigenvalue problem. The goal of this optimization is to raise the lowest eigenvalue (or eigenvalues) of the problem to its highest (optimal) level at a constant volume of the structure. Likely the lowest eigenvalue may be either inherently multi-modal or it can become multi-modal as a result of the optimization process. The multimodeness introduces some ambiguity to the eigenvalue problem and make the optimization difficult to handle. Thus far, only the simplest cases of multi-modal structures have been effectively optimized using rather elaborate analytical methods. Numerous publications report design of a minimum volume structure with different eigenvalues constraints, in which, however, the modality of the problem is assumed a priori. The method presented here utilizes a multi-modal optimality criteria and allows for inclusion of an arbitrary number of buckling or vibrations modes which might influence the optimization process. The real multi-modality of the problem, that is the number of modes participating in the final optimal design is determined iteratively. Because of a natural use of the FEM technique the method is easy to program and might be helpful in design of large flexible space structures.

Szyszkowski, W.↗

NASA NTRS2005

Linear Stability Analysis of Gravitational Effects on a Low-Density Gas Jet Injected into a High-Density Medium

The objective of this study was to determine the effects of buoyancy on the absolute instability of low-density gas jets injected into high-density gas mediums. Most of the existing analyses of low-density gas jets injected into a high-density ambient have been carried out neglecting effects of gravity. In order to investigate the influence of gravity on the near-injector development of the flow, a spatio-temporal stability analysis of a low-density round jet injected into a high-density ambient gas was performed. The flow was assumed to be isothermal and locally parallel; viscous and diffusive effects were ignored. The variables were represented as the sum of the mean value and a normal-mode small disturbance. An ordinary differential equation governing the amplitude of the pressure disturbance was derived. The velocity and density profiles in the shear layer, and the Froude number (signifying the effects of gravity) were the three important parameters in this equation. Together with the boundary conditions, an eigenvalue problem was formulated. Assuming that the velocity and density profiles in the shear layer to be represented by hyperbolic tangent functions, the eigenvalue problem was solved for various values of Froude number. The Briggs-Bers criterion was combined with the spatio-temporal stability analysis to determine the nature of the absolute instability of the jet whether absolutely or convectively unstable. The roles of the density ratio, Froude number, Schmidt number, and the lateral shift between the density and velocity profiles on the absolute instability of the jet were determined. Comparisons of the results with previous experimental studies show good agreement when the effects of these variables are combined together. Thus, the combination of these variables determines how absolutely unstable the jet will be.

Lawson, Anthony L.↗

NASA NTRS2001

Instability Analysis of a Low-Density Gas Jet Injected into a High-Density Gas

The objective of this study was to determine the effects of buoyancy on the absolute instability of low-density gas jets injected into high-density gas mediums. Most of the existing analyses of low-density gas jets injected into a high-density ambient have been carried out neglecting effects of gravity. In order to investigate the influence of gravity on the near-injector development of the flow, a linear temporal stability analysis and a spatio-temporal stability analysis of a low-density round jet injected into a high-density ambient gas were performed. The flow was assumed to be isothermal and locally parallel; viscous and diffusive effects were ignored. The variables were represented as the sum of the mean value and a normal-mode small disturbance. An ordinary differential equation governing the amplitude of the pressure disturbance was derived. The velocity and density profiles in the shear layer, and the Froude number (signifying the effects of gravity) were the three important parameters in this equation. Together with the boundary conditions, an eigenvalue problem was formulated. Assuming that the velocity and density profiles in the shear layer to be represented by hyperbolic tangent functions, the eigenvalue problem was solved for various values of Froude number. The temporal growth rates and the phase velocity of the disturbances were obtained. It was found that the presence of variable density within the shear layer resulted in an increase in the temporal amplification rate of the disturbances and an increase in the range of unstable frequencies, accompanied by a reduction in the phase velocities of the disturbances. Also, the temporal growth rates of the disturbances were increased as the Froude number was reduced (i.e. gravitational effects increased), indicating the destabilizing role played by gravity. The spatio-temporal stability analysis was performed to determine the nature of the absolute instability of the jet. The roles of the density ratio, Froude number, Schmidt number, and the lateral shift between the density and velocity profiles on the jet s absolute instability were determined. Comparisons of the results with previous experimental studies show good agreement when the effects of these variables are combined together. Thus, the combination of these variables determines how absolutely unstable the jet will be. Experiments were carried out to observe the qualitative differences between a round low-density gas jet injected into a high-density gas (helium jet injected into air) and a round constant density jet (air jet injected into air). Flow visualizations and velocity measurements in the near-injector region of the helium jet show more mixing and spreading of the helium jet than the air jet. The vortex structures develop and contribute to the jet spreading causing the helium jet to oscillate.

Lawson, Anthony Layiwola↗

NASA NTRS2002

Linear Temporal Stability Analysis of a Low-Density Round Gas Jet Injected into a High-Density Gas

It has been observed in previous experimental studies that round helium jets injected into air display a repetitive structure for a long distance, somewhat similar to the buoyancy-induced flickering observed in diffusion flames. In order to investigate the influence of gravity on the near-injector development of the flow, a linear temporal stability analysis of a round helium jet injected into air was performed. The flow was assumed to be isothermal and locally parallel; viscous and diffusive effects were ignored. The variables were represented as the sum of the mean value and a normal-mode small disturbance. An ordinary differential equation governing the amplitude of the pressure disturbance was derived. The velocity and density profiles in the shear layer, and the Froude number (signifying the effects of gravity) were the three important parameters in this equation. Together with the boundary conditions, an eigenvalue problem was formulated. Assuming that the velocity and density profiles in the shear layer to be represented by hyperbolic tangent functions, the eigenvalue problem was solved for various values of Froude number. The temporal growth rates and the phase velocity of the disturbances were obtained. The temporal growth rates of the disturbances increased as the Froude number was reduced (i.e. gravitational effects increased), indicating the destabilizing role played by gravity.

Lawson, Anthony L.↗