Structural Dynamic Analysis with Generalized Damping Models E-Book

Sondipon Adhikari

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 342

Veröffentlichungsjahr: 2014

Beschreibung

Since Lord Rayleigh introduced the idea of viscous damping in his classic work "The Theory of Sound" in 1877, it has become standard practice to use this approach in dynamics, covering a wide range of applications from aerospace to civil engineering. However, in the majority of practical cases this approach is adopted more for mathematical convenience than for modeling the physics of vibration damping.

Over the past decade, extensive research has been undertaken on more general "non-viscous" damping models and vibration of non-viscously damped systems. This book, along with a related book Structural Dynamic Analysis with Generalized Damping Models: Analysis, is the first comprehensive study to cover vibration problems with general non-viscous damping. The author draws on his considerable research experience to produce a text covering: parametric senistivity of damped systems; identification of viscous damping; identification of non-viscous damping; and some tools for the quanitification of damping. The book is written from a vibration theory standpoint, with numerous worked examples which are relevant across a wide range of mechanical, aerospace and structural engineering applications.

Leseprobe

Table of Contents

Preface

Nomenclature

Chapter 1: Parametric Sensitivity of Damped Systems

1.1. Parametric sensitivity of undamped systems

1.2. Parametric sensitivity of viscously damped systems

1.3. Parametric sensitivity of non-viscously damped systems

1.4. Summary

Chapter 2: Identification of Viscous Damping

2.1. Identification of proportional viscous damping

2.2. Identification of non-proportional viscous damping

2.3. Symmetry-preserving damping identification

2.4. Direct identification of the damping matrix

2.5. Summary

Chapter 3: Identification of Non-viscous Damping

3.1. Identification of exponential non-viscous damping model

3.2. Symmetry preserving non-viscous damping identification

3.3. Direct identification of non-viscous damping

3.4. Summary

Chapter 4: Quantification of Damping

4.1. Quantification of non-proportional damping

4.2. Quantification of non-viscous damping

4.3. Summary

Bibliography

Author Index

Index

ToSonia AdhikariSunanda AdhikariandTulsi Prasad Adhikari

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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Library of Congress Control Number: 2013951215

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN: 978-1-84821-670-9

Preface

Among the various ingredients of structural dynamics, damping is one of the least understood topics. The main reason is that unlike the stiffness and inertia forces, damping forces cannot always be obtained from “first principles”. The past two decades have seen significant developments in the modeling and analysis of damping in the context of engineering dynamic systems. Developments in composite materials including nanocomposites and their applications in advanced structures, such as new generation of aircrafts and large wind turbines, have led to the need for understanding damping in a better manner. Additionally, the rise of vibration energy harvesting technology using piezoelectric and electromagnetic principles further enhanced the importance of looking at damping more rigorously. The aim of this book is to systematically present the latest developments in the modeling and analysis of damping in the context of general linear dynamic systems with multiple degrees-of-freedom. The focus has been on the mathematical and computational aspects. This book will be relevant to aerospace, mechanical and civil engineering disciplines and various sub-disciplines within them. The intended readers of this book include senior undergraduate students and graduate students doing projects or doctoral research in the field of damped vibration. Researchers, professors and practicing engineers working in the field of advanced vibration will find this book useful. This book will also be useful for researchers working in the fields of aeroelasticity and hydroelasticity, where complex eigenvalue problems routinely arise due to fluid–structure interactions.

There are some excellent books which already exist in the field of damped vibration. The book by Nashif et al. [NAS 85] covers various material damping models and their applications in the design and analysis of dynamic systems. A valuable reference on dynamic analysis of damped structures is [SUN 95]. The book by Beards [BEA 96] takes a pedagogical approach toward structural vibration of damped systems. The handbook by Jones [JON 01] focuses on viscoelastic damping and analysis of structures with such damping models. These books represented the state of the art at the time of their publications. Since these publications, significant research works have gone into the dynamics of damped systems. The aim of this book is to cover some of these latest developments. The attention is mainly limited to theoretical and computational aspects, although some references to experimental works are given.

One of the key features of this book is the consideration of general non-viscous damping and how such general models can be seamlessly integrated into the framework of conventional structural dynamic analysis. New results are illustrated by numerical examples and, wherever possible, connections are made to well-known concepts of viscously damped systems. A related title, Structural Dynamic Analysis with Generalized Damping Models: Analysis [ADH 14], is complementary to this book, and, indeed, they could have been presented together. However, for practical reasons, it has proved more convenient to present the material separately.

The related book, Structural Dynamic Analysis with Generalized Damping Models: Analysis [ADH 14] focuses on the analysis of linear systems with general damping models. This book, Structural Dynamic Analysis with Generalized Damping Models: Identification, deals with the identification and quantification of damping. There are ten chapters and one appendix in the two volumes combined ? covering analysis and identification of dynamic systems with viscous and non-viscous damping.

In [ADH 14] Chapter 1 gives an introduction to the various damping models. Dynamics of viscously damped systems are discussed in Chapter 2. Chapter 3 considers dynamics of non-viscously damped single-degree-of-freedom systems in detail. Chapter 4 discusses non-viscously damped multiple degree-of-freedom systems. Linear systems with general non-viscous damping are studied in Chapter 5. Chapter 6 proposes reduced computational methods for damped systems. A method to deal with general asymmetric systems is described in the appendix.

In this book, Structural Dynamic Analysis with Generalized Damping Models: Identification, Chapter 1 describes parametric sensitivity of damped systems. Chapter 2 describes the problem of identification of viscous damping. The identification of non-viscous damping is detailed in Chapter 3. Chapter 4 gives some tools for the quantification of damping.

This book is the result of the last 15 years of research and teaching in the area of damped vibration problems. Initial chapters started taking shape when I offered a course on advanced vibration at the University of Bristol. The later chapters originated from the research work with numerous colleagues, students, collaborators and mentors. I am deeply indebted to all of them for numerous stimulating scientific discussions, exchanges of ideas and, on many occasions, direct contributions toward the intellectual content of the book. I am grateful to my teachers Professor C. S. Manohar (Indian Institute of Science, Bangalore), Professor R. S. Langley (University of Cambridge) and, in particular, Professor J. Woodhouse (University of Cambridge), who was heavily involved with the works reported in Chapters 2–4 of this book. I am very thankful to my colleague Professor M. I. Friswell with whom I have a long-standing collaboration. Some joint works are directly related to the content of this book (Chapter 1 of this book in particular). I would also like to thank Professor D. J. Inman (University of Michigan) for various scientific discussions during his visits to Swansea. I am thankful to Professor A. Sarkar (Carleton University) and his doctoral student M. Khalil for joint research works. I am deeply grateful to Dr A. S. Phani (University of British Columbia) for various discussions related to damping identification and contributions toward Chapters 2 and 5 of [ADH 14] and Chapter 2 of this book. Particular thanks go to Dr N. Wagner (Intes GmbH, Stuttgart) for joint works on non-viscously damped systems and contributions in Chapter 4 of [ADH 14]. I am also grateful to Professor F. Papai for involving me in research works on damping identification. My former PhD students B. Pascual (contributed in Chapter 6 of [ADH 14]), J. L. du Bois and F. A. Diaz De la O deserve particular thanks for various contributions throughout their time with me and putting up with my busy schedules. I am grateful to Dr Y. Lei (University of Defense Technology, Changsha) for carrying out joint research with me on non-viscously damped continuous systems. I am grateful to Professor A. W. Lees (Swansea University), Professor N. Lieven, Professor F. Scarpa (University of Bristol), Professor D. J. Wagg (University of Sheffield), Professor S. Narayanan (Indian Institute of Technology (IIT) Madras), Professor G. Litak (Lublin University), E. Jacquelin (Université Lyon), Dr A. Palmeri (Loughborough University), Professor S. Bhattacharya (University of Surrey), Dr S. F. Ali (IIT Madras), Dr R. Chowdhury (IIT Roorkee), Dr P. Duffour (University College London), and Dr P. Higino, Dr G. Caprio and Dr A. Prado (Embraer Aircraft) for their intellectual contributions and discussions at different times. Besides the names mentioned here, I am also thankful to many colleagues, fellow researchers and students working in this field of research around the world, whose names cannot be listed here due to page limitations. The lack of explicit mentions by no means implies that their contributions are any lesser. The opinions presented in the book are entirely mine, and none of my colleagues, students, collaborators and mentors have any responsibility for any shortcomings.

I have been fortunate to receive grants from various companies, charities and government organizations including an Advanced Research Fellowship from UK Engineering and Physical Sciences Research Council (EPSRC), the Wolfson Research Merit Award from the Royal Society and the Philip Leverhulme Prize from the Leverhulme Trust. Without these findings, it would have been impossible to have conducted the works leading to this book. Finally, I want to thank my colleagues at the College of Engineering at Swansea University. Their support proved to be a key factor in materializing the idea of writing this book.

Last, but by no means least, I wish to thank my wife Sonia and my parents for their constant support, encouragement and putting up with my ever-increasing long periods of “non-engagement” with them.

Sondipon ADHIKARIOctober 2013

Nomenclature

y

j

modal coordinates (in Chapter 3, [ADH 14])

forcing function in the Laplace domain

displacement in the Laplace domain

matrix containing

matrix containing

Φ

matrix containing the eigenvectors

ϕ

j

vector of initial velocities

non-viscous proportional damping functions (of a matrix)

Y

k

a matrix of internal eigenvectors

y

k

j

j

th eigenvector corresponding to the

k

th the internal variable

PSD

power spectral density

0

a vector of zeros

Lagrangian (in Chapter 3, [ADH 14])

δ

(

t

)

Dirac-delta function

δ

jk

Kroneker-delta function

Γ(•)

gamma function

γ

Lagrange multiplier (in Chapter 3, [ADH 14])

(•)*

complex conjugate of (•)

(•)

T

matrix transpose

(•)

–1

matrix inverse

(•)

–

T

matrix inverse transpose

(•)

H

Hermitian transpose of (•)

(•)

e

elastic modes

(•)

nv

non-viscous modes

derivative with respect to time

space of complex numbers

space of real numbers

⊥

orthogonal to

Laplace transform operator

inverse Laplace transform operator

det(•)

determinant of (•)

diag [•]

a diagonal matrix

for all

imaginary part of (•)

∈

belongs to

∉

does not belong to

⊗

Kronecker product

Laplace transform of (•)

real part of (•)

vec

vector operation of a matrix

O

(•)

in the order of

ADF

anelastic displacement field model

adj(•)

adjoint matrix of (•)

GHM

Golla, Hughes and McTavish model

MDOF

multiple-degree-of-freedom

SDOF

single-degree-of-freedom

Chapter 1

Parametric Sensitivity of Damped Systems

Changes of the eigenvalues and eigenvectors of a linear vibrating system due to changes in system parameters are of wide practical interest. Motivation for this kind of study arises, on the one hand, from the need to come up with effective structural designs without performing repeated dynamic analysis, and, on the other hand, from the desire to visualize the changes in the dynamic response with respect to system parameters. Furthermore, this kind of sensitivity analysis of eigenvalues and eigenvectors has an important role to play in the area of fault detection of structures and modal updating methods. Sensitivity of eigenvalues and eigenvectors is useful in the study of bladed disks of turbomachinery where blade masses and stiffness are nearly the same, or deliberately somewhat altered (mistuned), and one investigates the modal sensitivities due to this slight alteration. Eigensolution derivatives also constitute a central role in the analysis of stochastically perturbed dynamical systems. Possibly, the earliest work on the sensitivity of the eigenvalues was carried out by Rayleigh [RAY 77]. In his classic monograph, he derived the changes in natural frequencies due to small changes in system parameters. Fox and Kapoor [FOX 68] have given exact expressions for the sensitivity of eigenvalues and eigenvectors with respect to any design variables. Their results were obtained in terms of changes in the system property matrices and the eigensolutions of the structure in its current state, and have been used extensively in a wide range of application areas of structural dynamics. Nelson [NEL 76] proposed an efficient method to calculate an eigenvector derivative, which requires only the eigenvalue and eigenvector under consideration. A comprehensive review of research on this kind of sensitivity analysis can be obtained in Adelman and Haftka [ADE 86]. A brief review of some of the existing methods for calculating sensitivity of the eigenvalues and eigenvectors is given in section 1.6 (Chapter 1, [ADH 14]).

The aim of this chapter is to consider parametric sensitivity of the eigensolutions of damped systems. We first start with undamped systems in section 1.1. Parametric sensitivity of viscously damped systems is discussed in section 1.2. In section 1.3, we discuss the sensitivity of eigensolutions of general non-viscously damped systems. In section 1.4, a summary of the techniques introduced in this chapter is provided.

1.1. Parametric sensitivity of undamped systems

The eigenvalue problem of undamped or proportionally damped systems can be expressed by

[1.1]

1.1.1. Sensitivity of the eigenvalues

We rewrite the eigenvalue equation as

[1.2]

[1.3]

The functional dependence of p is removed for notational convenience. Differentiating the eigenvalue equation [1.2] with respect to the element p of the parameter vector we have

[1.4]

Premultiplying by , we have

[1.5]

Using the identity in [1.3], we have

[1.6]

[1.7]

1.1.2. Sensitivity of the eigenvectors

Different methods have been developed to calculate the derivatives of the eigenvectors. One way to express the derivative of an eigenvector is by a linear combination of all the eigenvectors

[1.8]

[1.9]

Premultiplying by , we have

[1.10]

[1.11]

Using these, we have

[1.12]

From this, we obtain

[1.13]

To obtain the jth term αjj, we differentiate the mass orthogonality relationship in [1.11] as

[1.14]

Considering the symmetry of the mass matrix and using the expansion of the eigenvector derivative, we have

[1.15]

Utilizing the othonormality of the mode shapes, we have

[1.16]

The complete eigenvector derivative is therefore given by

[1.17]

From equation [1.17], it can be observed that when two eigenvalues are close, the modal sensitivity will be higher as the denominator of the right-hand term will be very small. Unlike the derivative of the eigenvalues given in [1.7], the derivative of an eigenvector requires all the other eigensolutions. This can be computationally demanding for large systems. The method proposed by Nelson [NEL 76] can address this problem. We will discuss Nelson’s method in the context of damped systems in the following sections.

1.2. Parametric sensitivity of viscously damped systems

The analytical method in the preceding section is for undamped systems. For damped systems, unless the system is proportionally damped (see section 2.4, Chapter 2 of [ADH 14]), the mode shapes of the system will not coincide with the undamped mode shapes. In the presence of general non-proportional viscous damping, the equation of motion in the modal coordinates will be coupled through the off-diagonal terms of the modal damping matrix, and the mode shapes and natural frequencies of the structure will, in general, be complex. The solution procedures for such non-proportionally damped systems follow mainly two routes: the state-space method and approximate methods in the configuration space, as discussed in Chapters 2 and 3 [ADH 14]. The state-space method (see [NEW 89, GÉR 97], for example) although exact in nature, requires significant numerical effort for obtaining the eigensolutions as the size of the problem doubles. Moreover, this method also lacks some of the intuitive simplicity of traditional modal analysis. For these reasons, there has been considerable research effort in analyzing non-proportionally damped structures in the configuration space. Most of these methods either seek an optimal decoupling of the equation of motion or simply neglect the off-diagonal terms of the modal damping matrix. It may be noted that following such methodologies, the mode shapes of the structure will still be real. The accuracy of these methods, other than the light damping assumption, depends upon various factors, for example frequency separation between the modes and driving frequency (see [PAR 92a, GAW 97] and the references therein for discussions on these topics). A convenient way to avoid the problems that arise due to the use of real normal modes is to incorporate complex modes in the analysis. Apart from the mathematical consistency, by conducting experimental modal analysis, we also often identify complex modes: as Sestieri and Ibrahim [SES 94] have put it “… it is ironic that the real modes are in fact not real at all, in that in practice they do not exist, while complex modes are those practically identifiable from experimental tests. This implies that real modes are pure abstraction, in contrast with complex modes that are, therefore, the only reality!” But surprisingly, most of the current application areas of structural dynamics, which utilize the eigensolution derivatives, e.g. modal updating, damage detection, design optimization and stochastic finite element methods, do not use complex modes in the analysis but rely on the real undamped modes only. This is partly because of the problem of considering an appropriate damping model in the structure and partly because of the unavailability of complex eigensolution sensitivities. Although, there have been considerable research efforts toward damping models, sensitivity of complex eigenvalues and eigenvectors with respect to system parameters appears to have received less attention.

In this section, we determine the sensitivity of complex natural frequencies and mode shapes with respect to some set of design variables in non-proportionally damped discrete linear systems. It is assumed that the system does not possess repeated eigenvalues. In section 2.5 (Chapter 2, [ADH 14]), the mathematical background on linear multiple-degree-of-freedom discrete systems needed for further derivations has already been discussed. Sensitivity of complex eigenvalues is derived in section 1.2.1 in terms of complex modes, natural frequencies and changes in the system property matrices. The approach taken here avoids the use of state-space formulation. In section 1.2.2, sensitivity of complex eigenvectors is derived. The derivation method uses state-space representation of equation of motion for intermediate calculations and then relates the eigenvector sensitivities to the complex eigenvectors of the second-order system and to the changes in the system property matrices. In section 1.2.2.3, a two degree-of-freedom system that shows the “curve-veering” phenomenon has been considered to illustrate the application of the expression for rates of changes of complex eigenvalues and eigenvectors. The results are carefully analyzed and compared with presently available sensitivity expressions of undamped real modes.

1.2.1. Sensitivity of the eigenvalues

The equation of motion for free vibration of a linear damped discrete system with N degrees of freedom can be written as

[1.18]

where M, C and are mass, damping and stiffness matrices, is the vector of the generalized coordinates and denotes time. The eigenvalue problem associated with equation [1.18] is given by

[1.19]

Complex modes and frequencies can be exactly obtained by the state-space (first-order) formalisms. Transforming equation [1.18] into state-space form, we obtain

[1.20]

where is the system matrix and is the response vector in the state space given by

[1.21]

In the above equation, is the null matrix and is the identity matrix. The eigenvalue problem associated with the above equation is now in terms of an asymmetric matrix and can be expressed as

[1.22]

where sj is the jth eigenvalue and is the jth right eigenvector that is related to the eigenvector of the second-order system as

[1.23]

The left eigenvector associated with sj is defined by the equation

[1.24]

where (•)T denotes matrix transpose. For distinct eigenvalues, it is easy to show that the right and left eigenvectors satisfy an orthogonality relationship, that is

[1.25]

and we may also normalize the eigenvectors so that

[1.26]

The above two equations imply that the dynamic system defined by equation [1.20] possesses a set of biorthonormal eigenvectors. As a special case, when all eigenvalues are distinct, this set forms a complete set. Henceforth in our discussion, it will be assumed that all the system eigenvalues are distinct.

Suppose the structural system matrices appearing in [1.18] is a function of a parameter p. This parameter can be an element of a larger parameter vector. This can denote a material property (such as Young’s modulus) or a geometric parameter (such as thickness). We wish to find the sensitivity of the eigenvalues and eigenvectors with respect to this general parameter. We aim to derive expressions of derivative of eigenvalues and eigenvectors with respect to p without going into the state space.

For the jth set, equation [1.19] can be rewritten as

[1.27]

where the regular matrix pencil is

[1.28]

[1.29]

Differentiating the above equation with respect to pj, we obtain

[1.30]

where stands for , and can be obtained by differentiating equation [1.28] as

[1.31]

Now taking the transpose of equation [1.27] and using the symmetry property of Fj, it can shown that the first and third terms of the equation [1.30] are zero. Therefore, we have

[1.32]

Substituting from equation [1.31] into the above equation, we obtain

[1.33]

From this, we have

[1.34]

[1.35]

This is exactly the well-known relationship derived by Fox and Kapoor [FOX 68] for the undamped eigenvalue problem. Thus, equation [1.34] can be viewed as a generalization of the familiar expression of the sensitivity of undamped eigenvalues to the damped case. Following observations may be noted from this result:

– The derivative of a given eigenvalue requires the knowledge of only the corresponding eigenvalue and eigenvector under consideration, and thus a complete solution of the eigenproblem, or from the experimental point of view, eigensolution determination for all the modes is not required.

– Changes in mass and/or stiffness introduce more change in the real part of the eigenvalues whereas changes in the damping introduce more change in the imaginary part.

Since is complex in equation [1.34], it can be effectively used to determine the sensitivity of the modal damping factors with respect to the system parameters. For small damping, the modal damping factor for the jth mode can be expressed in terms of complex frequencies as denoting real and imaginary parts, respectively. As a result, the derivative can be evaluated from

[1.36]

This expression may turn out to be useful since we often directly measure the damping factors from experiment.

1.2.2. Sensitivity of the eigenvectors

1.2.2.1. Modal approach

We use the state-space eigenvectors to calculate the derivative of the eigenvectors in the configuration space. Since zj is the first N rows of ϕj (see equation [1.23]), we first try to derive and subsequently obtain using their relationships.

Differentiating [1.22] with respect to pj, we obtain

[1.37]

Since it has been assumed that A has distinct eigenvalues, the right eigenvectors, ϕj, form a complete set of vectors. Therefore, we can expand as

[1.38]

[1.39]

Using the orthogonality relationship of left and right eigenvectors from the above equation, we obtain

[1.40]

The ajk as expressed above is not very useful since it is in terms of the left and right eigenvectors of the first-order system. In order to obtain a relationship with the eigenvectors of second-order system, we assume

[1.41]

where . Substituting ψj in equation [1.24] and taking transpose, we obtain

[1.42]

Elimination of ψ1j from the above two equation yields

[1.43]

By comparison of this equation with equation [1.19], it can be seen that the vector M–1ψ2j. is parallel to zj; that is, there exists a non-zero βj ∈ such that

[1.44]

Now substituting ψ1j, ψ2j and using the definition of ϕj from equation [1.23] into the normalization condition [1.26], the scalar constant βj can be obtained as

[1.45]

Using ψ2j from equation [1.44] into the second equation of [1.42], we obtain

[1.46]

The above equation along with the definition of ϕj in [1.23] completely relates the left and right eigenvectors of the first-order system to the eigenvectors of the second-order system.

The derivative of the system matrix A can be expressed as

[1.47]

from which after some simplifications, the numerator of the right-hand side of equation [1.40] can be obtained as

[1.48]

Since commute in product. Using this property and also from [1.19] noting that , we finally obtain

[1.49]

This equation relates the ajk with the complex modes of the second-order system.

To obtain ajj, we begin with differentiation of the normalization condition [1.26] with respect to p and obtain the relationship

[1.50]

Substitution of ψj from equation [1.46] further leads to

[1.51]

where can be derived from equation [1.46] as

[1.52]

Since Pj is a symmetric matrix, equation [1.51] can be rearranged as

[1.53]

Note that the term within the bracket is (see equation [1.46]). Using the assumed expansion of from [1.40], this equation reads

[1.54]

The left-hand side of the above equation can be further simplified

[1.55]

Finally, using the orthogonality property of left and right eigenvectors, from equation [1.54], we obtain

[1.56]

In the above equation, ajj is expressed in terms of the complex modes of the second-order system. Now recalling the definition of ϕj in [1.23], from the first N rows of equation [1.38], we can write

[1.57]

We know that for any real symmetric system, first-order eigenvalues and eigenvectors appear in complex conjugate pairs. Using usual definition of natural frequency, that is skλk and consequently , where (•)* denotes complex conjugate, the above equation can be rewritten in a more convenient form as

[1.58]

where

This result is a generalization of the known expression of the sensitivity of real undamped eigenvectors to complex eigenvectors. The following observations can be made from this result:

– Unlike the eigenvalue derivative, the derivative of a given complex eigenvector requires the knowledge of all the other complex eigenvalues and eigenvectors.

– The sensitivity depends very much on the modes whose frequency is close to that of the considered mode.

– Like eigenvalue derivative, changes in mass and/or stiffness introduce more changes in the real part of the eigenvector whereas changes in damping introduce more changes in the imaginary part.

From equation [1.58], it is easy to see that in the undamped limit C → 0, and consequently and also with usual mass normalization of the undamped modes reduces the above equation exactly to the corresponding well-known expression derived by Fox and Kapoor [FOX 68] for derivative of undamped modes.

1.2.2.2. Nelson’s method

The method outlined in the previous section obtained the eigenvector derivative as a linear combination of all the eigenvectors. For large-scale structures, with many degrees of freedom, obtaining all the eigenvectors is a computationally expensive task. Nelson [NEL 76] introduced the approach, extended here, where only the eigenvector of interest was required. Lee et al. [LEE 99a] calculated the eigenvector derivatives of self-adjoint systems using a similar approach to Nelson. This section extends Nelson’s method to non-proportionally damped systems with complex modes. This method has the great advantage that only the eigenvector of interest is required.

The eigenvectors are not unique, in the sense that any scalar (complex) multiple of an eigenvector is also an eigenvector. As a result, their derivatives are also not unique. It is necessary to normalize the eigenvector for further mathematical derivations. There are numerous ways of introducing a normalization to ensure uniqueness. For undamped systems, mass normalization is the most common. A useful normalization for damped systems that follows from equation [2.211] (Chapter 2, [ADH 14]) is

[1.59]

Differentiating the equation governing the eigenvalues [1.19] with respect to the parameter p, gives

[1.60]

Rewriting this, we see that the eigenvector derivative satisfies

[1.61]

where the vector hj consists of the first two terms in equation [1.60], and all these quantities are now known. Equation [1.61] cannot be solved to obtain the eigenvector derivative because the matrix is singular. For distinct eigenvalues, this matrix has a null space of dimension 1. Following Nelson’s approach, the eigenvector derivative is written as

[1.62]

where vj and dj have to be determined. These quantities are not unique since any multiple of the eigenvector may be added to vj. A convenient choice is to identify the element of maximum magnitude in uj and make the corresponding element in vj equal to zero. Although other elements of vj could be set to zero, this choice is most likely to produce a numerically well-conditioned problem. Substituting equation [1.62] into equation [1.61], gives

[1.63]

This may be solved, including the constraint on the zero element of vj