Eigenvector Derivatives Research Articles

Calculation of eigenvalue and eigenvector derivatives with the improved Kron's substructuring method

For large-scale structures, the calculation of the eigensolution and the eigensensitivity is usually very time-consuming. This paper develops the Kron\'s substructuring method to compute the firstorder derivatives of the eigenvalues and eigenvectors with respect to the structural parameters. The global structure is divided into several substructures. The eigensensitivity of the substructures are calculated via the conventional manner, and then assembled into the eigensensitivity of the global structure by performing some constraints on the derivative matrices of the substructures. With the proposed substructuring method, the eigenvalue and eigenvector derivatives with respect to an elemental parameter are computed within the substructure solely which contains the element, while the derivative matrices of all other substructures with respect to the parameter are zero. Consequently this can reduce the computation cost significantly. The proposed substructuring method is applied to the GARTEUR AG-11 frame and a highway bridge, which is proved to be computationally efficient and accurate for calculation of the eigensensitivity. The influence of the master modes and the division formations are also discussed.

IRAM-based method for eigenpairs and their derivatives of large matrix-valued functions

Based on the implicitly restarted Arnoldi method for eigenpairs of large matrix, a new method is presented for the computation of a few eigenpairs and their derivatives of large matrix-valued functions. Eigenpairs and their derivatives are calculated simultaneously. Equation systems that are solved for eigenvector derivatives are greatly reduced from the original matrix size. The left eigenvectors are not required. Hence, the computational cost is saved. The convergence theory of the proposed method is established. Finally, numerical experiments are given to illustrate the efficiency of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.

Sensitivity of General Compound Planetary Gear Natural Frequencies and Vibration Modes to Model Parameters

Abstract This paper studies the sensitivity of general compound planetary gear natural frequencies and vibration modes to inertia and stiffness parameters. The model admits planetary gears having any combination of stepped-planet, meshed-planet, and multiple stage arrangements. Eigensensitivities in terms of eigenvalue and eigenvector derivatives are analytically derived for both tuned (i.e., cyclically symmetric) and mistuned systems. The results are expressed in compact closed-form formulas. The well-defined modal properties of general compound planetary gears simplify the expressions of eigenvalue sensitivities to ones that are proportional to modal strain/kinetic energies. Inspection of the modal strain/kinetic energy distribution plots provides an effective way to quantitatively and qualitatively determine the parameters that have the largest impact on a certain mode. For parameter perturbations that preserve the system symmetry, the structured modal properties imply that the modes of the same type are independent of the same group of system parameters. Parameter mistuning, with a few exceptions, splits a degenerate natural frequency of the unperturbed system into two frequencies; one frequency keeps its original value and retains its well-defined modal properties, while the other frequency changes and its associated mode lose its structured modal properties.

A test of the hypothesis of partial common principal components

Abstract A test of the equality of the first h eigenvectors of covariance matrices of several populations is constructed without the assumption that the sampled distributions are Gaussian. It is proved that the test statistic is asymptotically chi-square distributed. In this general setting, an explicit formula for column space of the asymptotic covariance matrix of the sample eigenvectors is derived and the rank of this matrix is computed. An essential assumption in deriving the asymptotic distribution of the presented test statistic is the existence of the finite fourth moments and the simplicity of the h largest eigenvalues of population covariance matrices, which makes possible to use the formulas for derivatives of eigenvectors of symmetric matrices.

On Veering of Eigenvalue Loci

Eigenvalue veering is studied in the context of two simple oscillators coupled by a (presumed weak) spring, variants of which have been considered by several authors. The concept of a center of veering is introduced, leading to a coordinate translation; a subsequent coordinate rotation, dependent on the degree of asymmetry of the system, reduces the frequency equation to a standard north-south opening hyperbola. Thus veering occurs even when coupling is strong, and may be characterized by these coordinate transformations and geometric features of the hyperbola, rather than eigenvalue and eigenvector derivatives.

Eigenvalue and eigenvector derivatives of second-order systems using structure-preserving equivalences

A “structure-preserving equivalence” in the sense intended here is a mapping between the stiffness, damping and mass matrices describing some initial second-order system and the corresponding three matrices of another second-order system having identical spectrum. Most second-order systems can be “diagonalised” through a mapping of this sort. The mapping provides a new approach to the evaluation and the understanding of eigenvalue and eigenvector derivatives. In place of pairs of eigenvalues, we think of real scalar stiffness, damping and mass quantities representing decoupled single-degree-of-freedom systems. In place of pairs of eigenvectors, we think of individual columns of the matrices involved in the mapping. This approach resolves the completely artificial phenomenon that the eigenvalue and eigenvector derivatives become “undefined” at instants when modification of, say, a damping parameter causes a pair of complex eigenvalues to turn into a pair of real eigenvalues or vice-versa. It also has the advantage of being applicable to cases where any one or more of the system matrices are singular.

Behavior modes, pathways and overall trajectories: eigenvector and eigenvalue analysis of dynamic systems

AbstractOne of the most fundamental principles in system dynamics is the premise that the structure of the system will generate its behavior. Such a philosophical position has fostered the development of a number of formal methods aimed at understanding the causes of model behavior. Behavior, to most in the field of system dynamics, is commonly interpreted as modes of behavior (e.g., exponential growth, exponential decay, and oscillation) because of their direct association with the feedback loops (e.g., reinforcing, balancing, and balancing with delays, respectively) that generate them. Hence, traditional research on formal model analysis has emphasized which loops cause a particular “mode” of behavior, with eigenvalues representing the most important link between structure and behavior. The main contribution of this work arises from a choice to focus our analysis on the overall trajectory of a state variable, instead of only a specific behavior mode. Since the overall behavior trajectory of state variable xi(t) is determined by a linear combination of the product of eigenvector components (rji) and behavior modes ($e^{l_jt}$) generated by eigenvalues (λj), contributions from both eigenvalues and eigenvectors are important. By studying how the overall trajectory changes due to changes in link (or loop) gains, we observe that the derivatives of eigenvectors are more closely associated with the short‐term transient impact of those changes, whereas derivatives of eigenvalues are associated with the long‐term impact. Since we care deeply about both the short‐ and the long‐term impact of those changes, there is value in looking at the contributions from both eigenvalues and eigenvectors. Copyright © 2009 John Wiley & Sons, Ltd.

Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems

One of the most fundamental principles in system dynamics is the premise that the structure of the system will generate its behavior. Such philosophical position has fostered the development of a number of formal methods aimed at understanding the causes of model behavior. To most in the field of system dynamics, behavior is commonly understood as modes of behavior (e.g., exponential growth, exponential decay, and oscillation) because of their direct association with the feedback loops (e.g., reinforcing, balancing, and balancing with delays, respectively) that generate them. Hence, traditional research on formal model analysis has emphasized which loops cause a particular mode of behavior, with eigenvalues representing the most important link between structure and behavior. The main contribution of this work arises from a choice to focus our analysis in the overall trajectory of a state variable - a broader definition of behavior than that of a specific behavior mode. When we consider overall behavior trajectories, contributions from eigenvectors are just as central as those from eigenvalues. Our approach to understanding model behavior derives an equation describing overall behavior trajectories in terms of both eigenvalues and eigenvectors. We then use the derivatives of both eigenvalues and eigenvectors with respect to link (or loop) gains to measure how they affect overall behavior trajectories over time. The direct consequence of focusing on behavior trajectories is that system dynamics researchers' reliance on eigenvalue elasticities can be seen as too-narrow a focus on model behavior - a focus that has excluded the short term impact of a change in loop (or link) gain in its analysis.

A new method for finding the first‐ and second‐order eigenderivatives of asymmetric non‐conservative systems with application to an FGM plate actively controlled by piezoelectric sensor/actuators

AbstractIn this paper, a new method for computing eigenvalue and eigenvector derivatives of asymmetric non‐conservative systems with distinct eigenvalues is presented. Several approaches have been proposed for eigenderivative analysis of systems with asymmetric and non‐positive‐definite mass, damping and stiffness matrices. The proposed formulation that is developed by combining the modal and algebraic methods neither have the complications of modal methods in calculating the complex left and right eigenvector derivatives nor suffer from numerical instability problems usually associated with algebraic methods. The method is applied to a functionally graded material (FGM) plate actively controlled by piezoelectric sensor/actuators. In this system, the feedback signal applied to each actuator patch is implemented as a function of the electric potential in its corresponding sensor patch. The use of this closed‐loop controlling system leads to a non‐self‐adjoint system with complex eigenvalues and eigenvectors. A finite element model is developed for static and dynamic analysis of closed‐loop controlled FGM plate. The first‐ and second‐order approximations of Taylor expansion are used to estimate the corresponding changes in the plate modal properties due to change in design parameters (the displacement feedback gains and the piezoelectric layer thickness in each S/A pair). Copyright © 2008 John Wiley & Sons, Ltd.

Derivatives of complex eigenvectors with distinct and repeated eigenvalues

AbstractIn this paper, we investigate the computation of the first‐order derivatives of complex eigenvectors for general non‐defective eigensystems. A new normalization condition is proposed, with which we can compute unique first‐order derivatives of arbitrary differentiable eigenvectors of systems with distinct and repeated eigenvalues. We also present an efficient algorithm to compute the particular solutions to the governing equations of the first‐order derivatives of eigenvectors. Finally, numerical examples are included to demonstrate the validity of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.

Adjoint Method for Design Sensitivity Analysis of Multiple Eigenvalues and Associated Eigenvectors

IGENPROBLEMS are commonly considered in structural stability, noise, and vibration analyses. Design sensitivity analysiscomputestherateofchangeofresponse-dependentfunction with respect to the change of design variables. Design sensitivity analysis is an essential step to improve systematically the existing design and to optimize a system with the aid of gradient-based optimization. Therefore an efficient and accurate method for design sensitivity analysis is necessary for diverse design optimizations. Fox and Kapoor have developed a general technique to compute design sensitivity of eigenvalues for symmetric matrices [1]. Plaut andHusseyin [2],Rudisill[3],andGodoyetal.[4]havedevelopeda formula for second-order design sensitivity analysis of eigenvalues. Forthedesignsensitivityanalysisofeigenvectors,themodalmethod [1,5], the modified modal method [6], and Nelson’s method [7] are oftenusedtocalculatethederivativesofeigenvectors.Acomparison of the above methods for calculating eigenvector derivatives has beencarriedout[8].Themodalmethodapproximatesthederivatives of eigenvectors as a linear combination of all eigenvectors. The modified modal method is developed to reduce the number of eigenvectors needed to represent the design derivative by including an additional term in the linear combination of eigenvectors. The modal and modified modal methods can be computationally expensive if large numbers of eigenvectors are needed to represent accurately the derivatives of eigenvectors. Nelson’s method is a direct differentiation method for calculating eigenvector derivatives, where it requires only the eigenvalue and eigenvectors for the mode being differentiated. Recently an adjoint method for calculating derivativesofdistincteigenvalueandtheirassociatedeigenvectorsis suggested, where an adjoint equation of simultaneous linear system equations requires only the eigenvalues and eigenvectors for the mode being differentiated [9]. Formanytypicalstructures,thereexistmultipleornearlyidentical eigenvalues due to structural symmetry. In the case of multiple eigenvalues, their associated eigenvectors are generally not differentiable. Therefore the methods for the design sensitivity analysis of distinct eigenvalues will no longer be valid. The eigenvectorscorrespondingtomultipleeigenvalueshaveagreatdeal of uncertainty compared with those associated with distinct eigenvalues because anylinear combination of eigenvectorsis also a valid eigenvector. For design sensitivity analysis of multiple eigenvalues, Ojalvo [10]hasprovided anequation tocalculate thederivatives of multiple eigenvaluesandtheirassociatedeigenvectorsbyextendingNelson’s method when the derivatives of the eigenvalues are distinct. MillsCurran [11] and Dailey [12] have independently modified and correctedOjalvo’smethodbydifferentiatingtheeigenvalueequation twice, that is, the second order design sensitivity analysis of eigenvalue. Friswell [13] has also extended the Nelson’s method to handle the multiple eigenvalue and their associated eigenvectors. Juang et al. [14], Bernard and Bronowicki [15], and Lee and Jung [16] have improved the modal method to evaluate the multiple eigenvalue and eigenvector sensitivity. Calculation of sensitivities for multiple eigenvalues has been obtained without using associated eigenvectors for the structural problem [17,18]. The proposed new method for design sensitivity analysis of multiple eigenvalues and their associated eigenvectors is an adjoint method that requires evaluation of the adjoint variables from the simultaneous system equation, the so-called adjoint equation with added side constraints. First we define the augmented function that consists ofthe response function andeigenvalue equations. Then the design derivatives of the response function are represented as an explicit design variation of the augmented function and an adjoint equation is obtained by requiring the implicit design variation of the augmented function to vanish. It is important to note that the adjoint equation is composed of eigenvalues and their associated eigenvectors of the mode being differentiated. Once we evaluate the adjoint variables, design sensitivity analysis of multiple eigenvalues and their associated eigenvectors can be computed directly. To verify the proposed method, an analytical example of a twodegree-of-freedom spring-mass system, planar grillage structure, anda finiteelementexampleofcantileverbeamareincluded.Further the developed method can be easily implemented into a commercial finite element program to carry out the design sensitivity analysis of eigenproblems needed for practical applications.

Novel Modal Method for Efficient Calculation of Complex Eigenvector Derivatives

A novel modal superposition method is presented for calculating eigenvector derivatives in self-adjoint or non-self-adjoint systems. The method is especially appropriate for determination of derivatives of many eigenvectors and requires only one eigenvalue and its corresponding right and left eigenvectors. Two parameters for the eigenvalue under consideration are introduced to obtain a generic derivative formula, which contains most of the available modal superposition methods as a branch. Matrices of the modified system are decomposed with available lower modes to avoid repeatedly solving equations for different eigenvalues. The contribution of truncated higher modes is expanded in a continued product of a few matrices, polynomials, and one generalized power series, whereby rapid convergence can be achieved easily, even in the vicinity of the convergent boundary, without additional computation cost. All other modal superposition coefficients can be neglected intrinsically with the determination of introduced parameters, but a simple condition for coefficients associated with lower-order eigenvalues should be satisfied in a special case. Numerical examples show that the method is efficient and can give results comparable to the exact solutions. The method is applicable to various damped systems and closed-mode cases.

Improved Nelson's Method for Computing Eigenvector Derivatives with Distinct and Repeated Eigenvalues

T HE computation of derivatives of eigenvalues and eigenvectors with respect to model parameters is crucial for many applications [1]. Nelson [2] proposed an approach for general real eigensystemswith distinct eigenvalueswhere only the eigenvector of interest was required. Ojalvo [3], Mills-Curran [4], Dailey [5], and Shaw and Jayasuriya [6] developed Nelson’s method for real symmetric eigensystems with repeated eigenvalues. For computation of a particular solution, themethods described in [3,5]may fail in some circumstances. Other methods have been suggested by MillsCurran [4] and Song et al. [7], but they are difficult to implement. When a system of linear equations is solved, one hopes that the calculated solution is a close representation of the true solution. The computed solution for a system with a large condition number may be inexact due to extreme sensitivity of the solution to small changes in its coefficient matrix and right column vectors [8]. Because some diagonal elements in the coefficient matrix for particular solutions are set to units, its coefficients are not all of the same order of magnitude. So the coefficient matrix has a large condition number. In this Note, Nelson’s method is first extended to the case of repeated eigenvalues for the symmetric real eigensystems and its improvements are then presented. The condition number of the coefficient matrix of the improved method is considerably reduced.

A development of modal expansion method for eigensensitivity analysis of a defective matrix

In Zhang and Zhang (2006) [Z.Y. Zhang, H.S. Zhang, Calculation of eigenvalue and eigenvector derivatives of a defective matrix, Applied Mathematics and Computation 176 (2006) 7–26] a modal expansion method for eigensensitivity analysis of a defective matrix was developed, where all of the eigenvalues of the derived eigenvalue problem for the first-order eigenvalues derivatives are simple. In this paper the work is extended to the case where the derived eigenvalue problem has repeated eigenvalues. The formulas for calculating the differentiable eigenvectors, the first- to third-order eigenvalue derivatives and the first-order eigenvector derivatives are derived. A numerical example shows the validity of the method.

Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem

In many engineering applications, the physical quantities that have to be computedare obtained by solving a related eigenvalue problem. The matrix under consideration and thusits eigenvalues usually depend on some parameters. A natural question then is how sensitive thephysical quantity is with respect to (some of) these parameters, i.e., how it behaves for small changesin the parameters. To find this sensitivity, eigenvalue and/or eigenvector derivatives with respect tothose parameters need to be found. A method is provided to compute first order derivatives of theeigenvalues and eigenvectors for a general complex-valued, non-defective matrix.

Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA

The rigorous coupled-wave analysis (RCWA) is a method to compute diffraction of a field by a given grating structure. Within various applications, such as metrology, it is important to know how the field reacts to small perturbations in the grating. This behavior can be expressed by the field derivatives with respect to a certain parameter. Approximations of these derivatives can be found by using finite differences where the field is computed for a neighboring value of the parameter, and the difference gives the derivative. Unfortunately, RCWA involves solving eigenvalue systems that are computationally expensive. Therefore, a faster alternative is given that computes the derivatives by straightforward differentiation of the relations within RCWA. Solving additional eigensystems is replaced by finding derivatives of eigenvalues and eigenvectors, which is less computationally expensive.

A note on computing eigenvector derivatives with distinct and repeated eigenvalues

AbstractIn this paper, a new method for computing eigenvector derivatives with distinct and repeated eigenvalues for the real symmetric eigensystems is presented. Its main idea is to extend the governing equations of particular solutions for eigenvector derivatives. The extension is completed by requiring the solution to be mass orthogonal with respect to the distinct or repeated modes and adjusting the corresponding coefficients so that the coefficient matrix of the extended system is non‐singular and has smaller condition number. Numerical examples are included to demonstrate the validity of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.

Calculation of Eigensolution Derivatives for Nonviscously Damped Systems

A method to calculate the derivatives of the eigenvalues and eigenvectors of multiple-degree-of-freedom damped linear dynamic systems with respect to arbitrary design parameters is presented. In contrast to the traditional viscous damping model, a more general nonviscous damping model is considered. The nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over given kernel functions. Because of the general nature of the damping, eigensolutions are generally complex valued, and eigenvectors do not satisfy the classical orthogonality relationship. The proposed method to calculate the eigenvector derivative depends only on the eigenvector concerned. Numerical examples are provided to illustrate the derived results. Nomenclature c j = constant associated with the derivative of u j D(s) = dynamic stiffness matrix d j = constant associated with the derivative of v j G(s) = Laplace transform of G(t) G(t) = nonviscous damping matrix K = stiffness matrix L[ ] = Laplace transform of [ ] M = mass matrix m = total number of eigenvectors N = number of degrees of freedom of the system n = number of relaxation parameters p = design parameter rank () = rank of a matrix s = Laplace domain parameter u j = jth eigenvector (t) = displacement vector v j = jth adjoint (left) eigenvector x j = vector associated with the derivative of u j yj = vector associated with the derivative of v j θ j = normalization constant for the jth eigenvector λ j = jth eigenvalue μk = relaxation parameters Subscripts () T = matrix transpose () � = derivative with respect to s

Interative Computation of Modal Sensitivities

The computation of frequency and mode shape sensitivities with respect to design parameters is essential to many, structural optimization and finite element updating algorithms. Approximating the gradients may lead to poor estimates and loss of convergence, but on the other hand the cost of exact methods, such as Nelson's (Nelson, R.,, Simplified Calculations of Eigenvector Derivatives, A1AA Journal, Vol. 14, No. 9,1976, pp. 1201-1205), is often not acceptable for industrial-size models. The present study thus analyses existing approximation methods, suggests a first-order improvement of Fox and Kapoor's modal basis (Fox, R., and Kapoor, M., Rate of Change of Eigenvalues and Eigenvectors' A1AA Journal, Vol. 6, No. 12, 1968, pp. 2426-242, 9) and proposes a residual iteration technique allowing accurate computations of modeshape sensitivities. An engine cover model and a model of the Ariane 5 main cryogenic stage are used to analyze, for models of realistic sizes, the accuracy and computational costs of the proposed methods.

An approximate numerical method for the complex eigenproblem in systems characterised by a structural damping matrix

This paper presents an efficient numerical method that approximates the complex eigenvalues and eigenvectors in structural systems with viscoelastic damping materials, characterised by the complex structural damping matrix. This new method begins from the solution of the undamped system and approximates the complex eigenpair by finite increments using the eigenvector derivatives and the Rayleigh quotient. It is implemented in three different approaches: single-step, incremental and iterative schemes. The single-step technique is presented for systems with low and medium damping. From numerical examples, it can be verified that the errors committed by the approximate single-step technique with respect to the exact solutions, obtained by the IRAM method, are less than 0.2% when the loss factor of the material damping is lower than 1; this is a considerable improvement on other approximate methods. For higher damped systems an improvement to the previous approach is proposed by an incremental technique that keeps the accuracy without significantly increasing the computational time. The complex eigenproblem for materials whose mechanical properties are dependent on frequency is solved by a fast iterative approach, whose validity is proved using a four-parameter fractional derivative model.