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NASA NTRS1985

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 NTRS1985

On the spectrum of some singular equations in magnetohydrodynamics

A class of singular differential equations that often occurs in problems in magnetohydrodynamics, plasma physics, and astrophysics is studied; first by means of the specific and fairly straightforward example of motions in a stratified magnetofluid, and secondly in terms of the nature of the eigenvalue spectrum for the more general class of problems. It is shown that in the absence of discontinuities in the basic profiles, the eigenvalue spectrum is real and continuous, and no discrete spectrum exists. Bounds are obtained on the location of the continuous spectrum.

Adam, J. A.↗

NASA NTRS1985

A method for determining acoustic-liner admittance in ducts with sheared flow in two-cross-sectional directions

A method is developed for determining the acoustic admittance of a test liner installed in the wall of a grazing flow impedance tube. The mean flow is permitted flow gradients in both cross-sectional directions of the tube. The unknown admittance value is obtained by solving an eigenvalue problem. This eigenvalue problem results from the application of the finite-element method to the partial differential equation and boundary conditions governing the acoustic field. The credibility of the method is established by comparing results with exact solutions obtained for a constant mean-flow profile and with previous results for cases involving shear in only one cross-sectional direction. Excellent comparisons are obtained in both cases. The analysis is used in conjunction with a limited amount of experimental data and shows that the flow must be accurately modeled in order to determine the acoustic-liner properties.

Watson, W. R.↗

NASA NTRS1985

Instability analysis procedure for 3-level multi-bearing rotor-foundation systems

A procedure for the instability analysis of a three-level multispan rotor systems is described. This procedure is based on a distributed mass elastic representation of the rotor system in several eight-coefficient bearings. Each bearing is supported from an elastic foundation on damped, elastic pedestals. The foundation is represented as a general distributed mass elastic structure on discrete supports, which may have different stiffness and damping properties in the horizontal and vertical directions. This system model is suited to studies of instability threshold conditions for multirotor turbomachines on either massive or flexible foundations. The instability conditions is found by obtaining the eigenvalues of the system determinant, which is obtained by the transfer matrix method from the three-level system model. The stability determinant is solved for the lowest rotational speed at which the system damping becomes zero in the complex eigenvalue, and for the whirl frequency corresponding to the natural frequency of the unstable mode. An efficient algorithm for achieving this is described. Application of this procedure to a rigid rotor in two damped-elastic bearings and flexible supports is described. A second example discusses a flexible rotor with four damped-elastic bearings. The third case compares the stability of a six-bearing 300 Mw turbine generator unit, using two different bearing types. These applications validate the computer program and various aspects of the analysis.

Zhou, S.↗

NASA NTRS1986

Inclusion of elastically connected members in exact buckling and frequency calculations

A standard stiffness matrix procedure which permits any combination of rigid, elastic, pinned or sliding connections of the degrees of freedom at the ends of a member to the nodes of its parent structure is described, in order to show how easily it can be extended to allow an existing algorithm to be used to ensure that no eigenvalues of the parent structure can be missed even when 'exact' member theory is used. The eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems. The method preserves the exactness of the member theory and an efficient method for computer application is indicated. The theory also permits any combination of rigid, elastic, pinned or sliding connections between the freedoms of a substructure and those of its parent structure.

Williams, F. W.↗

NASA NTRS1985

The influence of sampling rate on the dynamics and control of aeronautical systems

It is shown in the design of aeronautical control systems that sampling rate variations give rise to migrations of the discrete eigenvalues for both the plant and the controllers. In general, these migrations tend to degrade simulation fidelity and the overall performance of Ultrareliable Fault Tolerant Control Systems (UFTCS). These systems consist of a redundant asynchronous network of microprocessors which use sensor information to generate the servo commands that drive actuators, thrust linkages, etc. A change in the sampling period of a single microprocessor will, in addition to causing an eigenvalue shift, alter the magnitude of the servo command issued by the controller. The influence of these changes and the effect of initial conditions and skew are considered.

Howard, J. C.↗

NASA NTRS1986

Parameter sensitivity in the dynamics of rotor-bearing systems

When designing a rotor system it is frequently desirable to have at hand a set of design sensitivity coefficients which quantitatively predict a change in specific system characteristics to changes in design parameters. This paper presents eigenvalue sensitivity coefficients for the damped natural frequencies of whirl of general linear rotor system modelled by finite element discretization. In addition, a simple and direct method for calculation of the damped critical speeds is presented, which utilizes the eigenvalue sensitivity with respect to the spin speed. It is shown that the combination of design parameter and spin speed whirl frequency sensitivity coefficients may be used to also evaluate the damped critical speed sensitivity coefficients.

Rajan, M.↗

NASA NTRS1986

Design of a multivariable flutter control/gust load alleviation system

This paper discusses the use of eigenspace techniques for the design of an active flutter control/gust load alleviation system for a hypothetical research drone. One leading edge and two trailing edge aerodynamic surfaces are available for control. Full state control laws are designed for two combinations of control surfaces by selecting feedback gains which place closed loop eigenvalues and shape closed loop eigenvectors so as to stabilize wing flutter and reduce gust loads at the wing root while yielding acceptable robustness and satisfying constraints on rms control surface activity. These controllers are realized by state estimators designed using an eigenvalue placement/eigenvector shaping technique which results in recovery of the loop transfer characteristics of the full state feedback systems. The resulting feedback compensators are shown to perform almost as well as the full state designs. They also exhibit acceptable performance in situations in which the failure of an actuator is simulated.

Liebst, B. S.↗

NASA NTRS1987

Survey of methods for calculating sensitivity of general eigenproblems

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 NTRS1987

Eigenspace techniques for active flutter suppression

The use of eigenspace techniques for the design of an active flutter suppression system for a hypothetical research drone is discussed. One leading edge and two trailing edge aerodynamic control surfaces and four sensors (accelerometers) are available for each wing. Full state control laws are designed by selecting feedback gains which place closed loop eigenvalues and shape closed loop eigenvectors so as to stabilize wing flutter and reduce gust loads at the wing root while yielding accepatable robustness and satisfying constrains on rms control surface activity. These controllers are realized by state estimators designed using an eigenvalue placement/eigenvector shaping technique which results in recovery of the full state loop transfer characteristics. The resulting feedback compensators are shown to perform almost as well as the full state designs. They also exhibit acceptable performance in situations in which the failure of an actuator is simulated.

Garrard, William L.↗

NASA NTRS1987

Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations

An artificial dissipation model, including boundary treatment, that is employed in many central difference schemes for solving the Euler and Navier-Stokes equations is discussed. Modifications of this model such as the eigenvalue scaling suggested by upwind differencing are examined. Multistage time stepping schemes with and without a multigrid method are used to investigate the effects of changes in the dissipation model on accuracy and convergence. Improved accuracy for inviscid and viscous airfoil flow is obtained with the modified eigenvalue scaling. Slower convergence rates are experienced with the multigrid method using such scaling. The rate of convergence is improved by applying a dissipation scaling function that depends on mesh cell aspect ratio.

Swanson, R. C.↗

NASA NTRS1987

Precise computer controlled positioning of robot end effectors using force sensors

A major problem in space applications of robotics and docking of spacecraft is the development of technology for automated precise positioning of mating components with smooth motion and soft contact. To achieve the above objective, a design method was developed for optimally placing the closed-loop poles of a discretized robotic control system at exact prescribed locations inside the unit circle of the complex z-plane. The design method combines the merits of the pole placement and the linear quadratic design approaches. The proposed design procedure is based on the assignment of one real eigenvalue or two complex conjugate (or real) eigenvalues at each design step. The method involves solutions of simple algebraic equations and this is considered to be efficient for on-line or off-line computations. Also, two methods for the linearization of the nonlinear model of a robotic manipulator were presented. Since automatic control of multi-degree freedom robotic manipulators involves high nonlinear equations of systems, a pilot project was proposed involving the control of a one-dimensional system. This simple system can be readily implemented for testing the concepts and algorithms.

Shieh, Leang S.↗

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 NTRS1986

Preconditioning matrices for Chebyshev derivative operators

The problem of preconditioning the matrices arising from pseudo-spectral Chebyshev approximations of first order operators is considered in both one and two dimensions. In one dimension a preconditioner represented by a full matrix which leads to preconditioned eigenvalues that are real, positive, and lie between 1 and pi/2, is already available. Since there are cases in which it is not computationally convenient to work with such a preconditioner, a large number of preconditioners were studied which were more sparse (in particular three and four diagonal matrices). The eigenvalues of such preconditioned matrices are compared. The results were applied to the problem of finding the steady state solution to an equation of the type u sub t = u sub x + f, where the Chebyshev collocation is used for the spatial variable and time discretization is performed by the Richardson method. In two dimensions different preconditioners are proposed for the matrix which arises from the pseudo-spectral discretization of the steady state problem. Results are given for the CPU time and the number of iterations using a Richardson iteration method for the unpreconditioned and preconditioned cases.

Rothman, Ernest E.↗

NASA NTRS1986

A class of fast staircase algorithms for generalized state-space systems

Several methods are presented for defining generalized state space models (GSSM), with emphasis on 'fast' techniques for transforming GSSMs to condensed state space models (SSM). The fast forms are configured to yield invariant transfer functions. Details of the decomposition process are summarized in terms of the determination of eigenvalues which are separated within staircase matrices. Applications of the techniques are illustrated through implementation of a unimodular transformation which does not affect the finite eigenvalues, by demonstrating deadbeat control of a GSSM, which the definition of a reduced observer of a GSSM, and by embedding a polynomial matrix into a unimodular matrix.

Beelen, T.↗

NASA NTRS1986

Methods for the solution of radiative transfer equation

To obtain an exact solution of the radiative-transfer equation in media where both absorption and scattering are significant, the usual approach is to use a numerical method. Three methods are known in the literature: invariant imbedding, eigenvalue-eigenfunction, and matrix doubling. This paper examines the practical application of these methods to the problem of emission from an inhomogeneous (Rayleigh) layer, the effects of layer parameters on the stability. It is found that invariant imbedding is most suitable for computing emission from an inhomogeneous layer with a temperature profile but tends to be unstable as the optical thickness of the layer increases beyond 0.5. On the other hand, the matrix-doubling method is stable for arbitrary optical thickness but is not suitable for handling multilayers. The eigenvalue-eigenfunction method is more stable than the invariant imbedding as optical thickness increases up to 2.0. It also permits temperature profile in the layer, but the computation is much more complicated. It is less stable than the matrix-doubling method when optical thickness is larger than 2.0. In general, the choice of a method is dependent on the nature of the problem.

Chen, M. F.↗

NASA NTRS1987

Calculation of stresses and forces between the skin and stiffener in composite panels

A methodology for computing the stresses at the interface between the skin and stiffener in stiffened composite panels is described. The methodology is based on finite-element analyses and on elasticity solution. The finite-element analyses are standard, while the elasticity solution is based on eigenvalue expansions of the stress function. The eigenvalue expansions are assumed to be valid in the local region where the stiffener flange terminates. The local elasticity solution is coupled to the global finite-element analysis using collocation on the boundary of the local region. Accuracy and convergence of the method are discussed and several examples of its utility are presented.

Hyer, M. W.↗

NASA NTRS1987

Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations

An artificial dissipation model, including boundary treatment, that is employed in many central difference schemes for solving the Euler and Navier-Stokes equations is discussed. Modifications of this model such as the eigenvalue scaling suggested by upwind differencing are examined.Multistage time stepping schemes with and without a multigrid method are used to investigate the effects of changes in the dissipation model on accuracy and convergence. Improved accuracy for inviscid and viscous airfoil flows is obtained with the modified eigenvalue scaling. Slower convergence rates are experienced with the multigrid method using such scaling. The rate of convergence is improved by applying a dissipation scaling function that depends on mesh cell aspect ratio.

Swanson, R. C.↗