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NASA NTRS1990

Robust eigensystem assignment for second-order dynamic systems

A novel approach for the robust eigensystem assignment of flexible structures using full state or output feedback is developed. Using the second-order dynamic equations, the approach can assign the eigenvalues of the system via velocity and displacement feedbacks, or acceleration and velocity feedbacks. The eigenvalues and eigenvectors of the system are assigned, via the second-order eigenvalue problem for the structural system, in two steps. First, an orthonormal basis spanning the attainable closed-loop eigenvector space corresponding to each desired closed-loop eigenvalue is generated using the Singular Value or QR decompositions. Second, a sequential procedure is used to choose a set of closed-loop eigenvectors that are as close as possible to the column space of a well-conditioned target matrix. Among the possible choices of the target matrix, the closest unitary matrix to the open-loop eigenvector matrix appears to be a suitable choice. A numerical example is given to illustrate the proposed algorithm.

Juang, Jer-Nan↗

NASA NTRS1990

Improved solution for system identification equations by Epsilon-Decomposition

Matrix eigenvalue theory is used to examine the source of ill-conditioning in linear algebraic equations. This approach highlights the crucial role played by the zero and near-zero eigenvalues and corresponding eigenvectors of poorly conditioned systems. Insight gained from this approach is used to significantly improve a recently developed solution procedure called Epsilon-Decomposition (E-D). E-D is an efficient alternative to Singular Value Decomposition (SVD) for ill-conditioned systems arising in parameter estimation and system identification studies. The efficiency of the improved E-D over SVD resides in the need to only obtain the zero and near-zero eigenvalues of the coefficient matrix as opposed to all of its eigenvalues and vectors (as required by SVD). Thus, the efficiency of E-D is significant for large matrices with small rank deficiency.

Ojalvo, Irving U.↗

NASA NTRS1990

Spurious frequencies as a result of numerical boundary treatments

The stability theory for finite difference Initial Boundary-Value approximations to systems of hyperbolic partial differential equations states that the exclusion of eigenvalues and generalized eigenvalues is a sufficient condition for stability. The theory, however, does not discuss the nature of numerical approximations in the presence of such eigenvalues. In fact, as was shown previously, for the problem of vortex shedding by a 2-D cylinder in subsonic flow, stating boundary conditions in terms of the primitive (non-characteristic) variables may lead to such eigenvalues, causing perturbations that decay slowly in space and remain periodic time. Characteristic formulation of the boundary conditions avoided this problem. A more systematic study of the behavior of the (linearized) one-dimensional gas dynamic equations under various sets of oscillation-inducing legal boundary conditions is reported.

Abarbanel, Saul↗

NASA NTRS1972

Asymptotic theory of a slender rotating beam with end masses.

The method of matched asymptotic expansions is employed to solve the singular perturbation problem of the vibrations of a rotating beam of small flexural rigidity with concentrated end masses. The problem is complicated by the appearance of the eigenvalue in the boundary conditions. Eigenfunctions and eigenvalues are developed as power series in the perturbation parameter beta to the 1/2 power, and results are given for mode shapes and eigenvalues through terms of the order of beta.

Whitman, A. M.↗

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 NTRS1975

Numerical solution of quadratic matrix equations for free vibration analysis of structures

This paper is concerned with the efficient and accurate solution of the eigenvalue problem represented by quadratic matrix equations. Such matrix forms are obtained in connection with the free vibration analysis of structures, discretized by finite 'dynamic' elements, resulting in frequency-dependent stiffness and inertia matrices. The paper presents a new numerical solution procedure of the quadratic matrix equations, based on a combined Sturm sequence and inverse iteration technique enabling economical and accurate determination of a few required eigenvalues and associated vectors. An alternative procedure based on a simultaneous iteration procedure is also described when only the first few modes are the usual requirement. The employment of finite dynamic elements in conjunction with the presently developed eigenvalue routines results in a most significant economy in the dynamic analysis of structures.

Gupta, K. K.↗

NASA NTRS1977

Determination of pole sensitivities by Danilevskii's method

In control theory, a synonymous term for pole sensitivity is eigenvalue sensitivity. Existing methods of calculating eigenvalues are cumbersome, and cannot be trusted for systems roughly greater than tenth order. The method proposed in the present paper is applicable to high-order system. (It has been routinely used to generate eigenvalue sensitivities for systems up to 26th order, using a UNIVAC 1106.) Danilevskii's method is shown to be suitable for performing the necessary evaluations. The result is a rational function that can be used to evaluate the sensitivities for all distinct poles.

Nail, J. B.↗

NASA NTRS1978

The stability of numerical methods for second order ordinary differential equations

An important characterization of a numerical method for first order ODE's is the region of absolute stability. If all eigenvalues of the linear problem dy/dt = Ay are inside this region, the numerical method is stable. If the second order system d/dt(dy/dt) = 2Ady/dt - By is solved as a first order system, the same result applies to the eigenvalues of the generalized eigenvalue problem (lambda-squared)I 2(lambda)A + B. No such region exists for general methods for second order equations, but in some cases a region of absolute stability can be defined for methods for the single second order equation d/dt(dy/dt) = 2ady/dt - by. The absence of a region of absolute stability can occur when different members of a system of first order equations are solved by different methods.

Gear, C. W.↗

NASA NTRS1980

Preliminary demonstration of a robust controller design method

Alternative computational procedures for obtaining a feedback control law which yields a control signal based on measurable quantitites are evaluated. The three methods evaluated are: (1) the standard linear quadratic regulator design model; (2) minimization of the norm of the feedback matrix, k via nonlinear programming subject to the constraint that the closed loop eigenvalues be in a specified domain in the complex plane; and (3) maximize the angles between the closed loop eigenvectors in combination with minimizing the norm of K also via the constrained nonlinear programming. The third or robust design method was chosen to yield a closed loop system whose eigenvalues are insensitive to small changes in the A and B matrices. The relationship between orthogonality of closed loop eigenvectors and the sensitivity of closed loop eigenvalues is described. Computer programs are described.

Anderson, L. R.↗

NASA NTRS1982

Nonlinear Marangoni convection in bounded layers. I - Circular cylindrical containers. II - Rectangular cylindrical containers

Liquid undergoing nonlinear Marangoni instability in a circular cylinder is examined, with attention given to roll-cell development and interaction. Surface deflections are neglected and the side walls are considered as adiabatic and impenetrable, allowing the liquid to freely slip. The nonlinear convective states are calculated and their stability is defined. The behavior and amplitude of cells forming in the liquid, heated from below, are modeled in order to derive all the transport properties. A new small parameter is formulated which is related to the critical Marangoni number of the infinite matrix expressing the eigenvalue expansion of the problem. The observed roll cell amplitudes and transport properties are shown to be available from simple eigenvalues, with double eigenvalues, indicating the existence of two roll-states as predicted by linear theory, nonlinear theory indicates transitions from one steady convective state to another.

Rosenblat, S.↗

NASA NTRS1984

Design of multivariable feedback control systems via spectral assignment using reduced-order models and reduced-order observers

The feasibility of using reduced order models and reduced order observers with eigenvalue/eigenvector assignment procedures is investigated. A review of spectral assignment synthesis procedures is presented. Then, a reduced order model which retains essential system characteristics is formulated. A constant state feedback matrix which assigns desired closed loop eigenvalues and approximates specified closed loop eigenvectors is calculated for the reduced order model. It is shown that the eigenvalue and eigenvector assignments made in the reduced order system are retained when the feedback matrix is implemented about the full order system. In addition, those modes and associated eigenvectors which are not included in the reduced order model remain unchanged in the closed loop full order system. The full state feedback design is then implemented by using a reduced order observer. It is shown that the eigenvalue and eigenvector assignments of the closed loop full order system rmain unchanged when a reduced order observer is used. The design procedure is illustrated by an actual design problem.

Mielke, R. R.↗

NASA NTRS1982

Free vibration analysis of coupled fluid-structure systems

An efficient numerical technique for the eigenvalue solution in the free vibration analysis of compressible fluid-structure coupled systems is presented. The fluid is assumed to be compressible in nature and the incompressible problem is only a special case of the present generalized algorithm. A natural frequency analysis of the structure in the absence of any fluid is achieved by a combined Sturm sequence and inverse iteration technique that computes only the required eigenvalues and vectors. A special inverse iteration scheme is then developed for the coupled system that uses the computed eigenvalues as starting iteration values for convergence. Numerical results obtained by solving a number of standard test cases indicate the pattern of root convergence corresponding to various simplifying assumptions.

Gupta, K. K.↗

NASA NTRS1984

Stability of a laminar premixed supersonic free shear layer with chemical reactions

The stability of a two-dimensional compressible supersonic flow in the wake of a flat plate is discussed. The fluid is a multi-species mixture which is undergoing finite rate chemical reactions. The spatial stability of an infinitesimal disturbance in the fluid is considered. Numerical solutions of the eigenvalue stability equations for both reactive and nonreactive supersonic flows are presented and discussed. The chemical reactions have significant influence on the stability behavior. For instance, a neutral eigenvalue is observed near the freestream Mach number of 2.375 for the nonreactive case, but disappears when the reaction is turned on. For reactive flows, the eigenvalues are not very dependent on the free stream Mach number.

Menon, S.↗

NASA NTRS1984

Rotordynamic Analysis of the SSME Turbopumps Using Reduced Models

Alternative methods for the rotor-dynamic and sensitivity analysis of large rotor systems are examined. The methods are assessed for their ability to utilize accurate models of reduced size along with effective procedures for describing the dynamic behavior of the systems. Frequency response-based techniques are developed for determining the steady state response to imbalance of the space shuttle main engine (SSME) turbopumps and the related eigenvalue problem. The rotor and housing are represented by reduced receptances associated with their coupling points. The housing may be described by all of its normal modes within a frequency range of interest. The effects of truncated higher and lower modes are accounted for in an approximate manner. A procedure is described for determining the sensitivity of the coupling forces to changes in the coupling elements and rotor speed of the turbopump systems. In addition, an eigenvalue sensitivity analysis technique is adopted for application to the systems. Computer programs were developed for the numerical implementation of the impedance and eigenvalue sensitivity formulated in this study.

Noah, S. T.↗

NASA NTRS1985

Design of multivariable feedback control systems via spectral assignment using reduced-order models and reduced-order observers

The feasibility of using reduced order models and reduced order observers with eigenvalue/eigenvector assignment procedures is investigated. A review of spectral assignment synthesis procedures is presented. Then, a reduced order model which retains essential system characteristics is formulated. A constant state feedback matrix which assigns desired closed loop eigenvalues and approximates specified closed loop eigenvectors is calculated for the reduced order model. It is shown that the eigenvalue and eigenvector assignments made in the reduced order system are retained when the feedback matrix is implemented about the full order system. In addition, those modes and associated eigenvectors which are not included in the reduced order model remain unchanged in the closed loop full order system. The fulll state feedback design is then implemented by using a reduced order observer. It is shown that the eigenvalue and eigenvector assignments of the closed loop full order system remain unchanged when a reduced order observer is used. The design procedure is illustrated by an actual design problem.

Mielke, R. R.↗

NASA NTRS1986

Solution of the symmetric eigenproblem AX=lambda BX by delayed division

Delayed division is an iterative method for solving the linear eigenvalue problem AX = lambda BX for a limited number of small eigenvalues and their corresponding eigenvectors. The distinctive feature of the method is the reduction of the problem to an approximate triangular form by systematically dropping quadratic terms in the eigenvalue lambda. The report describes the pivoting strategy in the reduction and the method for preserving symmetry in submatrices at each reduction step. Along with the approximate triangular reduction, the report extends some techniques used in the method of inverse subspace iteration. Examples are included for problems of varying complexity.

Thurston, G. A.↗

NASA NTRS1985

Sum rules and other properties involving resonance projection operators

A sum rule is derived for the auxiliary eigenvalues of an equation whose eigenspectrum pertains to projection operators which describe electron scattering from multielectron atoms and ions. The sum rule's right-hand side depends on an integral involving the target system eigenfunctions. The sum rule is checked for several approximations of the two-electron target. It is shown that target functions which have a unit eigenvalue in their auxiliary eigenspectrum do not give rise to well-defined projection operators except through a limiting process. For Hylleraas target approximations, the auxiliary equations are shown to contain an infinite spectrum. However, using a Rayleigh-Ritz variational principle, it is shown that a comparatively simple aproximation can exhaust the sum rule to better than five significant figures. The auxiliary Hylleraas equation is greatly simplified by conversion to a square root equation containing the same eigenfunction spectrum and from which the required eigenvalues are trivially recovered by squaring.

Berk, A.↗

NASA NTRS1986

Implementation of the Lanczos method for structural vibration analysis on a parallel computer

The use of the Lanczos method in a parallel environment is investigated by implementing the algorithm for structural vibration problems on a parallel computer. It is shown that introducing shifts, assigning each processor a different region in the eigenvalue spectrum, and implementing the Lanczos method in parallel is an effective strategy for speeding up calculations. Test problem results include examples of the 'Lanczos phenomenon' where roundoff error in the vector orthogonalization can result in spurious eigenvalues which must be identified and discarded. The calculation strategy described here permits straightforward determination of these spurious eigenvalues. It is demonstrated that significant speedups in calculation time can be realized over traditional sequential methods.

Bostic, S. W.↗