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At least 541 records · Page 30
All sourcesNASA NTRSDOE OSTI
NASA NTRS1982

Integrating matrix formulations for vibrations of rotating beams including the effects of concentrated masses

By expressing partial differential equations of motion in matrix notation, utilizing the integrating matrix as a spatial operator, and applying the boundary conditions, the resulting ordinary differential equations can be cast into standard eigenvalue form upon assumption of the usual time dependence. As originally developed, the technique was limited to beams having continuous mass and stiffness properties along their lengths. Integrating matrix methods are extended to treat the differential equations governing the flap, lag, or axial vibrations of rotating beams having concentrated masses. Inclusion of concentrated masses is shown to lead to the same kind of standard eigenvalue problem as before, but with slightly modified matrices.

Lakin, W. D.↗
NASA NTRS1981

Numerical analysis of spectral properties of coupled oscillator Schroedinger operators. I - Single and double well anharmonic oscillators

Several methods for computing many eigenvalues and eigenfunctions of a single anharmonic oscillator Schroedinger operator whose potential may have one or two minima are described. One of the methods requires the solution of an ill-conditioned generalized eigenvalue problem. This method has the virtue of using a bounded amount of work to achieve a given accuracy in both the single and double well regions. Rigorous bounds are given, and it is proved that the approximations converge faster than any inverse power of the size of the matrices needed to compute them. The results of computations for the g:phi(4):1 theory are presented. These results indicate that the methods actually converge exponentially fast.

Isaacson, D.↗
NASA NTRS1981

Methods for the calculation of axial wave numbers in lined ducts with mean flow

A survey is made of the methods available for the calculation of axial wave numbers in lined ducts. Rectangular and circular ducts with both uniform and non-uniform flow are considered as are ducts with peripherally varying liners. A historical perspective is provided by a discussion of the classical methods for computing attenuation when no mean flow is present. When flow is present these techniques become either impractical or impossible. A number of direct eigenvalue determination schemes which have been used when flow is present are discussed. Methods described are extensions of the classical no-flow technique, perturbation methods based on the no-flow technique, direct integration methods for solution of the eigenvalue equation, an integration-iteration method based on the governing differential equation for acoustic transmission, Galerkin methods, finite difference methods, and finite element methods.

Eversman, W.↗
NASA NTRS1982

Application of a systematic finite-element model modification technique to dynamic analysis of structures

A systematic finite-element model modification technique has been applied to two small problems and a model of the main wing box of a research drone aircraft. The procedure determines the sensitivity of the eigenvalues and eigenvector components to specific structural changes, calculates the required changes and modifies the finite-element model. Good results were obtained where large stiffness modifications were required to satisfy large eigenvalue changes. Sensitivity matrix conditioning problems required the development of techniques to insure existence of a solution and accelerate its convergence. A method is proposed to assist the analyst in selecting stiffness parameters for modification.

Robinson, J. C.↗
NASA NTRS1981

Spectral decomposition of a matrix using the generalized sign matrix

An algorithm for spectral decomposition is presented which does not require knowledge of eigenvalues and eigenvectors. A set of eigenprojectors are defined which covers the entire spectrum of a matrix, and special attention is given to the projection on the zero eigenvalue. Some useful applications are discussed in the paper.

Denman, E. D.↗
NASA NTRS1981

The algebraic theory of latent projectors in lambda matrices

Multivariable systems such as a finite-element model of vibrating structures, control systems, and large-scale systems are often formulated in terms of differential equations which give rise to lambda matrices. The present investigation is concerned with the formulation of the algebraic theory of lambda matrices and the relationship of latent roots, latent vectors, and latent projectors to the eigenvalues, eigenvectors, and eigenprojectors of the companion form. The chain rule for latent projectors and eigenprojectors for the repeated latent root or eigenvalues is given.

Denman, E. D.↗
NASA NTRS1982

Evaluation and reduction of errors induced by the Guyan transformation

The Guyan reduction refers to a method used to reduce the number of degrees of freedom in a structural model for dynamic analysis. Experience shows that, if the method is properly employed, then this reduction method does in fact provide a "reasonably" accurate approximation of the dynamic characteristics of the unreduced model. A method that provides an accurate estimate of the error in each mode of the reduced eigenvalue problem and, in addition, gives first order correction terms that greatly improve the accuracy of the eigenvectors and eigenvalues is presented. The method is demonstrated by standard COSMIC NASTRAN DAMP alters to rigid format 3, real normal mode analysis. Comparative computer time of modal extraction versus error analysis is discussed for the VAX-11/780 version of COSMIC NASTRAN.

Fox, G. L.↗
NASA NTRS1983

Accelerating an iterative process by explicit annihilation

A slowly convergent stationary iterative process can be accelerated by explicitly annihilating (i.e., eliminating) the dominant eigenvector component of the error. The dominant eigenvalue or complex pair of eigenvalues can be estimated from the solution during the iteration. The corresponding eigenvector or complex pair of eigenvectors can then be annihilated by applying an explicit Richardson process over the basic iterative method. This can be done entirely in real arithmetic by analytically combining the complex conjugate annihilation steps. The technique is applied to an implicit algorithm for the calculation of two dimensional steady transonic flow over a circular cylinder using the equations of compressible inviscid gas dynamics. This demonstrates the use of explicit annihilation on a nonlinear problem.

Jespersen, D. C.↗
NASA NTRS1983

Large space structure damping design

Several FORTRAN subroutines and programs were developed which compute complex eigenvalues of a damped system using different approaches, and which rescale mode shapes to unit generalized mass and make rigid bodies orthogonal to each other. An analytical proof of a Minimum Constrained Frequency Criterion (MCFC) for a single damper is presented. A method to minimize the effect of control spill-over for large space structures is proposed. The characteristic equation of an undamped system with a generalized control law is derived using reanalysis theory. This equation can be implemented in computer programs for efficient eigenvalue analysis or control quasi synthesis. Methods to control vibrations in large space structure are reviewed and analyzed. The resulting prototype, using electromagnetic actuator, is described.

Pilkey, W. D.↗
NASA NTRS1982

The baroclinic instability of highly structured one-dimensional basic states

A frictionless, adiabatic, quasi-geostrophic model on a beta-plane was employed to examine the baroclinic instability of highly structured one-dimensional basic states. Definitions of the model eigenvalue equation, the boundary conditions, and the fluxes and energetics are provided, and highly structured average wind speed and temperature profiles were selected to represent a wide range of latitudes. Attention was given to the instability source, wavelength, growth rate, phase velocity, steering levels, and the vertical structure of their amplitude, phase, meridional entropy transport, and conversion of potential-to-kinetic energy. Green modes, Charney-Eady modes, and the modes defined in the paper are examined in connection with the highly structured profiles. The eigenvalue approach is shown to be superior to the numerical time integration approach and necessary conditions for instabilities are defined.

Fullmer, J. W. A.↗
NASA NTRS1983

Flux vector splitting and approximate Newton methods

In the present investigation, the basic approach is employed to view an iterative scheme as Newton's method or as a modified Newton's method. Attention is given to various modified Newton methods which can arise from differencing schemes for the Euler equations. Flux vector splitting is considered as the basic spatial differencing technique. This technique is based on the partition of a flux vector into groups which have certain properties. The Euler equations fluxes can be split into two groups, the first group having a flux Jacobian with all positive eigenvalues, and the second group having a flux Jacobian with all negative eigenvalues. Flux vector splitting based on a velocity-sound speed split is considered along with the use of numerical techniques to analyze nonlinear systems, and the steady Euler equations for quasi-one-dimensional flow in a nozzle. Results are given for steady flows with shocks.

Jespersen, D. C.↗
NASA NTRS1981

Optimal damper location in the vibration control of large space structures

The problem of finding the optimal location of active dampers for the vibration control of space structures is investigated. The optimal location is where maximum damping can be introduced to a particular vibration mode. To this end, classical root locus techniques are used to study the effect on eigenvalues of placing a damper on an undamped structure. An efficient formulation is derived which avoids the solution of the damped eigenvalue problem. Based on the observation of the results for a free-free beam model, a Minimum Constrained Frequency Criterion (MCFC) for locating the optimal damper location is proposed.

Wang, B. P.↗
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 NTRS1982

COSAL: A black-box compressible stability analysis code for transition prediction in three-dimensional boundary layers

A fast computer code COSAL for transition prediction in three dimensional boundary layers using compressible stability analysis is described. The compressible stability eigenvalue problem is solved using a finite difference method, and the code is a black box in the sense that no guess of the eigenvalue is required from the user. Several optimization procedures were incorporated into COSAL to calculate integrated growth rates (N factor) for transition correlation for swept and tapered laminar flow control wings using the well known e to the Nth power method. A user's guide to the program is provided.

Malik, M. R.↗
NASA NTRS1984

Enthalpy damping for the steady Euler equations

For inviscid steady flow problems where the enthalpy is constant at steady state, it was previously proposed to use the difference between the local enthalpy and the steady state enthalpy as a driving term to accelerate convergence of iterative schemes. This idea is analyzed, both on the level of the partial differential equation and on the level of a particular finite difference scheme. It is shown that for the two-dimensional unsteady Euler equations, a hyperbolic system with eigenvalues on the imaginary axis, there is no enthalpy damping strategy which moves all the eigenvalues into the open left half plane. For the numerical scheme, however, the analysis shows and examples verify that enthalpy damping is potentially effective in accelerating convergence to steady state.

Jespersen, D. C.↗
NASA NTRS1983

On the identification of continuous vibrating systems modelled by hyperbolic partial differential equations

This paper deals with the identification of spatially varying parameters in systems of finite spatial extent which can be described by second order hyperbolic differential equations. Two questions have been addressed. The first deals with 'partial identification' and inquires into the possibility of retrieving all the eigenvalues of the system from response data obtained at one location x-asterisk epsilon (0, 1). The second deals with the identification of the distributed coefficients rho(x), a(x) and b(x). Sufficient conditions for unique identification of all the eigenvalues of the system are obtained, and conditions under which the coefficients can be uniquely identified using suitable response data obtained at one point in the spatial domain are determined. Application of the results and their usefulness is demonstrated in the identification of the properties of tall building structural systems subjected to dynamic load environments.

Udwadia, F. E.↗
NASA NTRS1982

A general algorithm for solving the algebraic Riccati equation

The generalized eigenvalue problem provides a suitable framework for reliable solutions of many system theoretic, control, and estimation problems. A general algorithm for solving the matrix algebraic Riccati equation (ARE) which utilizes a pencil structure is described here. This algorithm avoids unnecessary inversion of cost or transition matrices, making it a numerically sound way to solve for the gains and/or ARE with singular quadratic costs, for cases satisfying detectability and stabilizability conditions. Examples are solution with discrete dead-beat control, noiseless measurements in Kalman filters and time-delays in discrete-time systems, which cause difficulties in the Hamiltonian standard eigenvalue problem formulation. The ARE algorithm implementatiton and numerical examples are shown.

Walker, R. A.↗