Computational methods for complex eigenproblems in finite element analysis of structural systems...

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Comput. Methods Appl. Mech. Engrg. 195 (2006) 6448–6462

Computational methods for complex eigenproblemsin finite element analysis of structural systems with

viscoelastic damping treatments

Fernando Cortes *, Marıa Jesus Elejabarrieta

Department of Mechanical Engineering, Mondragon Unibertsitatea, Loramendi 4, 20500 Mondragon, Spain

Received 3 October 2005; received in revised form 18 January 2006; accepted 19 January 2006

Abstract

In this paper efficient numerical methods to approximate the complex eigenvalues and eigenvectors in non-proportional and non-vis-cous systems are presented. These methods are specially conceived for practical engineering applications making use of the finite elementanalysis to determinate the effect that potential damping treatments have on the natural frequencies and mode shapes of structural sys-tems. Considering the solution of the undamped problem, the complex eigenpair is estimated by finite increments using the eigenvectorderivatives. For non-proportional viscous systems with low and medium damping, a simple single-step technique is presented whoserapidity and accuracy is verified by means of numerical applications. For higher damped systems an incremental approach is proposedthat keeps the accuracy without significantly increasing the computational time. For non-viscously damped systems a fast iterativemodality is suggested, which allows to approximate, in an efficient and simple way, the complex eigenpair. As numerical applications,the study of a metallic beam with free layer damping treatment is completed using finite element procedures, where the damping materialis modelized by an exponential model whose parameters are obtained from curve fitting to experimental data.� 2006 Elsevier B.V. All rights reserved.

Keywords: Complex modes; Damping treatments; Finite element analysis; Non-classical damping; Non-viscous damping

1. Introduction

Viscoelastic treatments are extended procedures for structural vibration reduction using damping materials (mono-graphs on this subject can be found in literature, e.g. Refs. [1–3]). In engineering applications, the efficacy of the dampingtreatments may be measured by means of the damped system eigenvalues. Indeed, the symmetric eigenproblem with vis-cous damping is given by the second order system [4]

s2r Mþ srCþ K

� �ur ¼ 0; ð1Þ

where M, C and K are the mass, viscous damping and stiffness n · n symmetric matrices. M is positive definite and K ispositive definite or semi-positive definite. For the general case, also called non-proportional case, the rth eigenpair, withr 6 2 · n, is given by the complex eigenvalue sr and the complex eigenvector ur, which appear in conjugate pairs,

srþn ¼ �sr ð2Þand

0045-7825/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2006.01.006

* Corresponding author. Tel.: +34 943 794700; fax: +34 943 791536.E-mail addresses: [email protected] (F. Cortes), [email protected] (M.J. Elejabarrieta).

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F. Cortes, M.J. Elejabarrieta / Comput. Methods Appl. Mech. Engrg. 195 (2006) 6448–6462 6449

urþn ¼ �ur; ð3Þ

respectively, where ð��Þ denotes the complex conjugate operator. An exception occurs when the M, C and K matrices verifythe condition presented by Caughey and O’Kelly [5],

KM�1C ¼ CM�1K; ð4Þ

then, the system possess normal modes, i.e. the eigenvectors are real and they are the same as that of the undamped system.A common technique for solving the general non-proportional complex eigenproblem consists in the transformation of

Eq. (1) into a linear system given by the state-space equations

�K 0

0 M

� �ur

_ur

� �¼ sr

C M

M 0

� �ur

_ur

� �; ð5Þ

where _ur ¼ srur. With this technique the positive definition of the matrices is no longer verified and the size of the problemis doubled, thus, the computational efforts are seriously enlarged. The eigenvalues sr are the same as those of the originalsystem and the eigenvectors zr are related with the former by

zr ¼ur

srur

� �. ð6Þ

The large-scale systems may be solved by the classic complex Lanczos’s [6] or Arnoldi’s [7] methods. An alternative is pre-sented by Leung [8], who analyses a complex subspace iteration method. More recently, some methods which work in theoriginal space and solve the quadratic eigenvalue problem have been developed, such as the complex vector iteration processpresented by Ruge [9] and the complex subspace iteration procedure by Fisher [10].

All these methods are not appropriate in systems with non-viscous damping, whose motion equation in free vibration isrepresented in time-domain [11] by

M€uðtÞ þZ t

�1Gðt � sÞ _uðsÞdsþ KuðtÞ ¼ 0; ð7Þ

where t is the time, s is the retardation time, G(t) is the damping function matrix and u(t), _uðtÞ and €uðtÞ are the displacementvector and its first and second time derivatives, respectively. Then, the eigenproblem is given by

s2r Mþ sr

eGðsrÞ þ K�

ur ¼ 0; ð8Þ

where ð~�Þ denotes the Laplace transformation. This eigenproblem becomes a non-linear matrix system due to the termeGðsrÞ, wherefrom the eigenvalues may be computed [12] by solving the m roots of the characteristic equation,

det s2r Mþ sr

eGðsrÞ þ K�

¼ 0; ð9Þ

whose order m is generally higher than 2 · n,

m ¼ 2� nþ p; p P 0; ð10Þthat is, the non-viscous character of the system introduces p extra poles. Consequently, the eigenmodes can be classified inelastic modes, related to the 2 · n complex conjugate eigenvalues, and non-viscous modes, which are introduced by the non-viscous nature of the damping mechanism. This fact is a major difference between the viscous and non-viscously dampedsystems. For stable passive systems, these p extra eigenvalues are negative real eigenvalues, thus, the associate non-viscousmodes are overcritically damped modes, i.e. they have not oscillatory behaviour. An interesting and exhaustive analysis ofa one degree of freedom non-viscous system is presented by Muller in Ref. [13].

In engineering applications aimed at studying the effect of damping treatments on the stationary response, the non-vis-cous modes have not practical interest because their contribution disappears over time. Therefore, under this assumption,only the elastic modes have practical interest and they are the only modes considered in this work.

The particular case in which the system possess normal modes is analysed by Adhikari [14], who states that the condi-tions for the existence of normal modes in non-viscously damped systems are

KM�1GðtÞ ¼ GðtÞM�1K; ð11ÞMK�1GðtÞ ¼ GðtÞK�1M ð12Þ

and

MGðtÞ�1K ¼ KGðtÞ�1

M. ð13Þ

6450 F. Cortes, M.J. Elejabarrieta / Comput. Methods Appl. Mech. Engrg. 195 (2006) 6448–6462

In the general case, taking into account that in practical applications the mechanical systems are represented by large scalematrix systems, e.g. using finite element procedures, neither Eq. (9) nor iterative modalities of the eigenmethods previouslyrevised are efficient in engineering applications.

Therefore, in this paper computational methods for solving the complex eigenproblem in viscoelastically damped systemsare developed and integrated in finite element algorithms for practical engineering applications. The basis of these new meth-ods is taken from a technique developed by the authors [15] for a damped system with structural damping matrix H, whichsolves the linear eigenproblem ðs2

r Mþ Kþ iHÞur ¼ 0. Indeed, these new procedures begin from the undamped eigensolu-tions and then, they approximate the complex eigenpair using the eigenvector derivatives. Thus, the methods that computethe eigenvector derivatives in symmetric damped systems are summarised first. Next the new method for systems with non-proportional viscous damping matrix is presented, and an iterative modality for non-viscously damped systems is proposedlater. Finally, numerical applications that prove the effectiveness of the presented methods are illustrated using finite elementtechniques in structural systems with damping treatments. The algorithms of the different procedures enunciated in thispaper and the finite element matrices used in the numerical applications are compiled in Appendices A and B, respectively.

2. Derivatives of eigenvalues and eigenvectors in symmetric damped systems

Fox and Kapoor’s [16] and Nelson’s [17] classical methods are used to compute the eigenderivatives in undamped sys-tems. The procedures developed until the end of the 1980s are reviewed by Murthy and Haftka [18]. More recently, Lee andJung [19] have presented an algebraic method that gives the exact solution using a simple algorithm. The study of the eig-enderivatives in symmetric non-conservative systems with non-repeated eigenvalues is more recent. The more interestingmethods are those working in the original space, in which the quadratic eigenproblem is given by Eq. (1), where the rthcomplex eigenvector ur may be normalized arbitrarily [20],

uTr ð2srMþ CÞur ¼

1

cr; ð14Þ

where cr is the normalization factor and (Æ)T denotes the transpose vector. There are several ways in which the normaliza-tion factor can be chosen. The most consistent with the traditional modal analysis practice is the one using

cr ¼1

2sr; ð15Þ

which, in undamped systems, degenerates to the familiar unit modal mass normalization

uTr Mur ¼ 1. ð16Þ

For obtaining the derivative of the rth eigenvalue, Eq. (1) is differentiated,

s0rð2srMþ CÞur þ ðs2r M0 þ srC

0 þ K0Þur þ ðs2r Mþ srCþ KÞu0r ¼ 0; ð17Þ

where (Æ) 0 represents the derivative with respect to any parameter. Premultiplication by uTr yields

s0ruTr ð2srMþ CÞur þ uT

r ðs2r M0 þ srC

0 þ K0Þur þ uTr ðs2

r Mþ srCþ KÞu0r ¼ 0; ð18Þwhere the last term is zero by virtue of Eq. (1) and of the symmetry of the matrices. Then, the derivative of the rth eigen-value becomes

s0r ¼ �uT

r ðs2r M0 þ srC

0 þ K0Þur

uTr ð2srMþ CÞur

. ð19Þ

To compute the eigenvector derivative, a vector fr can be defined as follows:

fr ¼ �s0rð2srMþ CÞur � ðs2r M0 þ srC

0 þ K0Þur ð20Þand then the eigenvector derivative u0r and fr relationship is deduced from Eqs. (17) and (20),

s2r Mþ srCþ K

� �u0r ¼ fr; ð21Þ

wherefrom u0r cannot be solved directly because the dynamic stiffness matrix ðs2r Mþ srCþ KÞ is singular. In recent years

several methods that facilitate the evaluation of the eigenvector derivatives have been developed. Adhikari [21] presentsa method based on the modal superposition of Fox and Kapoor’s technique in which the eigenvector derivative u0r yields

u0r ¼ �cr

2uT

r ð2srM0 þ C0Þur

�ur �

X2n

k¼1k 6¼r

ck

uTk s2

r M0 þ srC0 þ K0

� �ur

sr � skuk

� . ð22Þ


This method is relatively simple to implement but for obtaining the exact solution the complete modal base is required.Friswell and Adhikari [22] extend Nelson’s method to non-proportionally damped systems with complex modes, in whichthe eigenvector derivative u0r is decomposed in a particular solution br and another homogeneous drur,

u0r ¼ br þ drur; ð23Þ

where the scalar dr is given by

dr ¼ �cr uTr ð2srMþ CÞbr þ uT

r s0rMþ srM0 þ 1

2C0

� ur

� ð24Þ

and the br vector is computed by a pivoting procedure on Eq. (21). The advantage of this method is that the exact derivativeof an eigenvector is computed using only the corresponding eigenvalue and eigenvector. Based on Ref. [19], Lee et al. [23]and Choi et al. [24] solve a (n + 1) matrix system by means of a numerically stable technique that preserves the band andsymmetry of the matrices.

3. Proposed method for non-proportional viscous systems

3.1. Introduction

The proposed method begins from the undamped eigensolution given by

ð�kr;0Mþ KÞur;0 ¼ 0; ð25Þwhere kr,0 and ur,0 are the undamped eigenpair, in which sr;0 ¼ i

ffiffiffiffiffiffiffikr;0

p, and i ¼

ffiffiffiffiffiffiffi�1p

is the imaginary operator. By consid-ering the damping of the system, the viscous matrix varies from zero to C, while the M and K matrices remain constant.Thus, the eigenvector will be modified as follows:

ur ¼ ur;0 þ Dur; ð26Þwhere Dur is the complex eigenvector variation. The present method computes an approximation of the eigenvector incre-ment Dur by taking the derivatives of Eqs. (19) and (21) as linear finite increments, with C = 0, DM = 0, DK = 0 andDC = C, that yields

Dsr ¼ �1

2

uTr;0Cur;0

uTr;0Mur;0

ð27Þ

and

ðs2r;0Mþ KÞDur ¼ �Dsrð2sr;0Mþ CÞur;0; ð28Þ

respectively. Different modalities may be used to compute the finite increment Dur from Eq. (28): the modal AFK (Adhikaribased on Fox and Kapoor), FAN (Friswell and Adhikari based on Nelson) and the algebraic LKJ (Lee, Kim and Jung)and CJKL (Choi, Jo, Kim and Lee). When the approximate eigenvector is known, Eq. (1) may be premultiplied by uH

r ,

uHr s2

r Mþ srCþ K� �

ur ¼ 0; ð29Þ

where (Æ)H denotes the complex conjugate transpose or Hermitian operator (see, e.g., Ref. [4]), and the eigenvalue sr is up-dated as

sr ¼ xr �fr þ iffiffiffiffiffiffiffiffiffiffiffiffiffi1� f2

r

q� ; ð30Þ

in which

x2r ¼

uHr Kur

uHr Mur

¼ kr

mrð31Þ

and

fr ¼1

2xr

uHr Cur

uHr Mur

¼ cr

2xrmr; ð32Þ

where mr, kr, xr, cr and fr describe the mass, stiffness, undamped natural frequency, damping and damping ratio modalparameters, respectively, although the meaning is vaguely different to that used in other systems, such as in proportionaldamped systems. The study will be completed in three stages:


1. In the first stage it will be assumed that the eigenvectors of the damped system are real and the same as those of theundamped, ur = ur,0; it will be proved that this approach is valid for small damping.

2. In the second, the increment Dur will be computed from Eq. (28) by a single-step technique. The numerical applicationsof Section 5 will prove that this approximation is applicable in systems with medium damping.

3. In the last stage, Dur is computed by multiple steps by means of an incremental technique applicable in systems withhigher damping.

3.2. First approach: the eigenvectors remain constant

In this first approach, it is supposed that the eigenvectors of the system are not influenced by the damping, consequently,they are real and the same as those of the undamped, thus, the increment of the eigenvector is null,

Dur ¼ 0. ð33ÞTherefore, the approximation of the rth eigenvalue is given by

sr ¼ xr;0 �fr þ iffiffiffiffiffiffiffiffiffiffiffiffiffi1� f2

r

q� ; ð34Þ

where xr,0 is the rth undamped eigenfrequency and fr is computed by

fr ¼1

2

uTr;0Cur;0

uTr;0Kur;0

. ð35Þ

This result corresponds to the Basile hypothesis for small damping [25], employed in the uncoupled mode superpositionmethod (UMSM) [26] for lightly damped systems, where fr < 0.01. When the damping cannot be supposed small, this ap-proach is not accurate enough.

3.3. Second stage: the single-step method

An improvement is proposed by considering that the eigenvectors vary due to the damping introduced in the system asEq. (26) indicates, and next, the eigenvalues are updated by Eqs. (30)–(32). The present method may be implemented indifferent modalities in function of the procedure used for the computation of the eigenvector increment Dur from Eq.(28): AFK, FAN, LKJ and CJKL. The numerical applications of Section 5 confirm that this method using the AFKand FAN varieties are the most efficient in full and truncated problems, respectively, which are summarized in TablesA.1 and A.2 of Appendix A, respectively.

3.4. Third stage: the incremental approach

For highly damped systems, an improvement of the previous method is proposed by an incremental procedure. The sim-plest way to proceed is to divide the total variation of the viscous matrix DC = C in qmax equal increments DCq, according to

DCq ¼1

qmax

C. ð36Þ

Tables A.3 and A.4 of Appendix A show the algorithms for this incremental method using the AFK and FAN modalitiesfor full and truncated problems, respectively.

4. Proposed method for non-viscously damped systems

When the properties of the viscoelastic materials are frequency dependent, the eigensystem is written as Eq. (8) indicates.The derivatives of the eigensolutions of this kind of systems are presented by Adhikari in Ref. [27]. For that, Eq. (8) isdifferentiated,

s0rð2srMþ eGðsrÞÞur þ s2r M0 þ sr

eG0ðsrÞ þ K0�

ur þ ðs2r Mþ sr

eGðsrÞ þ KÞu0r ¼ 0; ð37Þ

where the operator (Æ) 0 represents the total derivative. In effect, Adhikari decomposes the total derivative eG 0ðsrÞ in partialderivatives,

o½eGðsrÞ�oa

¼ osr

oaoeGðsÞ

os

��s¼sr

þ oeGðsÞoa

��s¼sr

; ð38Þ

Fig. 1. Iterative scheme for variable damping on frequency.


where a represents the design parameter. But the proposed method uses the total increment of the damping matrices, there-fore total derivatives are used in this paper and then, the incremental form of Eq. (37), with eGðsrÞ ¼ 0, DM = 0, DK = 0

and DeGðsrÞ ¼ eGðsrÞ, becomes

srð2DsrMþ eGðsrÞÞur þ ðs2r Mþ KÞDur ¼ 0; ð39Þ

that is equivalent to the viscous one if the following assessment is effectuated,

C�r ¼ eGðsrÞ; ð40Þwhere the matrix C�r is now complex, i.e., it is not already purely viscous. Therefore, the real and imaginary components ofthe C�r matrix induce dissipative and elastic forces, i.e. proportional to the velocity and to the displacement, respectively.The latter induce variations on the storage modulus of the material in function of the frequency, as is identified in exper-imentations (see e.g. Refs. [1–3] for details). Even if C�r is complex, the procedure for viscous systems of Section 4 can beimplemented into the iterative scheme shown in Fig. 1 (although the mathematical operations are more laborious).

The procedure begins by solving the problem without damping. When the damping matrix is evaluated at the computedeigenfrequency, the method is applied in the way that is indicated in the algorithms of Tables A.5 and A.6 of Appendix Afor calculating the new complex eigenpair. The iterative error e may be defined as

e ¼ maxReðsr;j � sr;j�1Þ

Reðsr;j�1Þ;Imðsr;j � sr;j�1Þ

Imðsr;j�1Þ

� ; ð41Þ

in this way, the eigenvalues converge in the imaginary and real part as well (even if the real part normally converges moreslowly). This approach solves only once the undamped eigenproblem, and then the iterations are effectuated on the incre-ments, so the convergence criterion is rapidly achieved.

5. Numerical applications

5.1. Problem definition

The effect of the partial coverage on a free layer damping cantilever beam is studied by means of the finite elementmethod. The dimensions of the metallic base layer are 200 mm of length, 1 mm of thickness and 10 mm of width. Thoseof the damping layer are 100 mm, 1.59 mm and 10 mm, respectively (see Fig. 2).

Fig. 2. Metallic beam with free layer damping treatment.


The properties of the materials are taken from the experimental characterisation effectuated by the authors [28] on AISIT 316 L stainless steel laminated sheet and on 1/1600 thickness Soundown Vibration Damping Tile material [29]. Indeed,Young’s modulus and density of the elastic material are Ee = 176.2 · 109 Pa and qe = 7782 kg/m3, respectively, and thatof the damping material is qd = 1423 kg/m3. Poisson’s coefficient me = md = 0.3 is chosen for both materials, typical for iso-tropic elastic materials. The measurements of the storage modulus E(x) and loss factor g(x) of the viscoelastic material areshown in Table 1. In next Section these experimental data will be fitted to the exponential damping model [30] for param-eter identification.

Rectangular finite elements under plane stress assumption are used (see e.g. Refs. [31,32] for details). The stiffness matrixK is computed by full integration and the consistent mass matrix M is employed. For the damping material, the viscous C

and the non-viscous eGðsÞ matrices are proportional to the stiffness matrix due to the fact that the latter is originated byforces proportional to the strain e and the former to the strain rate _e. The details about the formulation of these matricesare shown in Appendix B.

Only one finite element is chosen along thickness of each material and equal length is taken for all elements along span.Systems of different size are studied, from 52 to 252 non-null degrees of freedom (DOF). The largest model is shown inFig. 3.

The computations are carried out in double arithmetic precision (16 digits) on a personal computer using Matlab [33]under windows environment. The numerical applications are divided in two groups: the first proves the rapidity and accu-racy of the new method for the single-step and incremental approaches using a viscous or Kelvin–Voigt damping model.For systems with variable damping on frequency, the efficiency of the iterative modality is verified in the second numericalapplication.

5.2. Parameter identification

The non-viscous character of the damping material is modelized by means of the exponential model [30], whose dampingfunction in Laplace domain is given by

Table 1Experimental data of the storage modulus and loss factor of the damping material

x (rad/s) E (109 Pa) g

75.40 1.00 0.8376.03 1.10 0.78117.5 0.89 0.96119.4 1.03 0.88147.0 0.97 0.92147.7 1.01 0.91164.6 0.96 0.95165.9 1.05 0.89203.6 1.04 0.99204.2 1.07 0.921924 1.32 0.801925 1.30 0.792154 1.48 0.822176 1.58 0.83

Fig. 3. Finite element discretization with 252 DOF.


eGðsÞ ¼ c1þ st0

; ð42Þ

where c is the viscous parameter and t0 is the relaxation time. The equivalent dynamical complex modulus E*(x) (see Refs.[1–3] for details) in frequency domain is consequently given by

E�ðxÞ ¼ Ed þct0

ixt0

1þ ixt0

; ð43Þ

where Ed is the relaxed modulus of the damping material. The curve fitting was effectuated by means of Nelder–Mead [34]minimisation method, which is implemented in the fminsearch function of Matlab. The function that was minimised is

Xlmax

l

jljE�ðxlÞ � E�l j; ð44Þ

where l is the index of summation, lmax is the total number of experimental data, j Æ j represents the modulus of a complexnumber, jl is a weighting factor, E*(xl) is the value of the fitting curve at the frequency xl, given by Eq. (43), andE�l ¼ Elð1þ iglÞ is the lth measured complex modulus at the same frequency xl, which are gathered in Table 1. The weight-ing factor jl may be arbitrarily chosen and in this application is taken as a function of the frequency, given by

jl ¼xl

xmax

; ð45Þ

where xmax is the maximum considered frequency (in that study was 400 Hz or 2513 rad/s), with the purpose of giving big-ger weigh to the higher frequencies for the reason that the measuring method does not provide accurate results at the lowestfrequencies. The fitted values of the three parameters of the exponential model are shown in Table 2, and the comparisonbetween the experimental data and the fitted curves may be seen in Fig. 4.

From this figure it should be noted that the exponential model is able to reproduce the increasing behaviour of the stor-age modulus and the peak of the loss factor, but the accuracy of this model is not under study in this work.

5.3. Numerical applications for non-proportional viscous damping

In this section numerical applications for non-proportional viscous damping are presented, in which the eigenproblem

s2r ðMe þMdÞ þ srCþ ðKe þ KdÞ

� �ur ¼ 0 ð46Þ

must be solved, where the (Æ)e and (Æ)d subscripts are related to the elastic and damping materials, respectively. The rapidityand accuracy of the proposed method will be compared with those of the state-space QZ factorisation [35] implemented inthe eig(A, B, ‘qz’) function of Matlab. The modulus Ed and the viscous coefficient c are taken from Table 2 for constructingthe elastic Kd and the viscous C matrices of the finite elements of the viscoelastic material (see Appendix B for details).

The first numerical example evaluates the rapidity of the present method employing the single-step approach in functionof the size of the system. Five systems of different size are studied, from 52 to 252 non-null degrees of freedom (DOF).

Table 2Parameters of the exponential model

Ed (106 Pa) c (103 Pa s) t0 (10�6 s)

572.4 878.6 347.2

Fig. 4. Experimental data and fitted curves for (a) storage modulus E and (b) loss factor g.

Table 3CPU time in function of the DOF of the system for different methods

DOF 52 102 152 202 252

QZ 0.125 0.625 2.391 5.749 11.51T0 0.016 0.047 0.078 0.094 0.172

Present

AFKFANLKJCJKL

8>><>>:

0.063 0.109 0.203 0.329 0.5320.094 0.750 2.343 5.859 11.700.126 0.985 3.765 10.09 21.950.125 0.796 2.953 8.030 16.78


Table 3 shows the CPU time, in seconds, invested to solve all complex modes. The time needed by the QZ factorisationis shown in the first row. In the second one the time T0 is presented, which represents the necessary time for solving theundamped problem. The rest of the rows contain the time needed by the proposed method using different modalities, inwhich the T0 time and that of the normalization of the complex eigenvectors as Eq. (15) indicates, are included.

From Table 3 it can be pointed out that the proposed method using the AFK derivatives requires lower computationaltime than the exact QZ transformation, and the differences are more evident as the size of the system becomes larger.

When only some modes are required, neither QZ transformation nor AFK modality of the proposed method are effi-cient for solving truncated systems. The methods that only need the actual eigenpair are the most efficient; among them, theFAN derivatives show the lowest computational time. Consequently, in the next numerical applications in which truncatedproblems will be solved, the FAN derivatives will only be considered.

To prove the accuracy of the proposed method, the four first modes of the exact solution of the 252 DOF system pro-vided by the QZ factorisation is compared with those of the one-step and incremental FAN modalities. The modal prop-erties fr and xr, given by Eqs. (31) and (32), respectively, are shown in Table 4 for the three methods.

From this table it can be noted that the damping ratio fr is the parameter that presents the most important differences,the errors on xr are practically unappreciated. These inaccuracies are bigger as the damping ratio fr is higher. For the lowermodes whose damping ratio is lesser than 0.02, the errors provided by the proposed method are practically unappreciated.These errors begin to be remarkable from damping ratios around 0.04, but it can be seen how the iterative modality withtwo increments reduces the errors of the fourth mode from 54% to 3.1% without considerably increasing the computationaltime, because the more important computational efforts are done by solving the undamped eigensolutions.

Finally, it is important to remark the increasing character of the modal damping ratio fr in function of the frequency. Ineffect, this fact is due to the viscous (or Kelvin–Voigt) model employed for the characterisation of the viscoelastic material,that assumes that its damping capacity increases with the frequency (see e.g. [1]). This fact is not reflected in experimentalmeasurements, justifying the use of non-viscous damping models, which consider the dependence on frequency of thedamping.

5.4. Numerical application for non-viscous damping

In this section the non-viscous character of the viscoelastic material will be considered and the non-linear eigenproblem

s2r ðMe þMdÞ þ sr

eGðsrÞ þ ðKe þ KdÞ�

ur ¼ 0 ð47Þ

will be solved making use of the method presented in Section 4. The relaxed modulus Ed and the parameters c and t0 of thedamping matrix eGðsÞ (see Appendix B for details) are taken from Table 2. The 252 DOF system is under study. First of allthe damping is not considered and the first order linear system

s2r;0ðMe þMdÞ þ ðKe þ KdÞ

� ur;0 ¼ 0 ð48Þ

Table 4Modal properties of the first four modes computed by QZ and FAN modalities

QZ FAN 1 step FAN 2 increments

x1 (103 s�1) 0.3259 0.3259 0.3259f1 0.00475 0.00475 0.00475x2 (103 s�1) 1.9314 1.9314 1.9314f2 0.01682 0.01686 0.01684x3 (103 s�1) 5.4009 5.4017 5.4010f3 0.04005 0.04154 0.04018x4 (103 s�1) 10.705 10.724 10.704f4 0.08069 0.12452 0.08308

Table 5Modal properties of the first four modes of the undamped and non-viscous systems

xr,0 (103 s�1) xr (103 s�1) fr

Mode 1 0.3259 0.3259 0.0047Mode 2 1.9302 1.9310 0.0113Mode 3 5.3890 5.3918 0.0085Mode 4 10.535 10.548 0.0052


is solved. In order to consider the non-viscous behaviour, the algorithm presented in Section 4 is utilised making use ofFAN derivatives to compute only four modes. The results of the undamped and damped eigenvalues are shown in Table5, when the convergence tolerance Tol = 0.001 was chosen.

From this table it should be proved that the damping ratio fr increases up to the second mode, and then diminishes. Thisfact is due to the dependence of the damping capacity on frequency, see Fig. 3, in which the curve of the loss factor presentsa maximum around 200 Hz, or 1250 rad/s. This frequency is comprised between the first and the second modes of the struc-ture, which explains the maximum damping ratio achieved at the second one.

Regarding the accuracy of the results, in can be signalled that the maximum damping ratio is not bigger than 0.02, thus,from Table 4 it can be estimated that the accuracy is acceptable in this practical application.

6. Concluding remarks

Computational procedures that allow to evaluate in a simple way the effect of the damping treatments in structural sys-tems were presented in this paper. The proposed methods were specially conceived to be integrated in finite element pro-cedures for engineering applications, in which the effect of damping on natural frequencies and mode shapes is under study.Numerical examples have been presented for a partial coverage in a metallic beam, wherefrom it should be concluded that:

1. For non-proportional damping, the numerical applications have verified the efficacy of the proposed method using AFKand FAN derivatives in full and truncated problems, respectively.

2. For non-viscous damping, the classical complex eigenmethods are not satisfactory due to the fact that the damping var-ies with frequency, and the methods that solve the characteristic equation are not adequate in large scale systems. Theproposed iterative modality has been applied in an engineering application in which the complex eigensolutions havebeen computed in a simple way.

Acknowledgements

The authors would like to thank the reviewers for the suggestions of some improvements for the paper and ModestoMateos from Mondragon Unibertsitatea for the help provided in English grammar.

Appendix A. Algorithms

See Tables A.1–A.6.

Appendix B. Finite element matrices

In this appendix the finite element formulation of the matrices that were employed in this paper are presented (see theReferences for details on finite element monographs). Four node bilinear isoparametric rectangular finite elements ofdimensions a · b (see Fig. B.1(a)) are considered, under plane stress hypothesis.

Table A.1Algorithm for single-step method using AFK derivatives

1. Solve the undamped problem s2r;0Mþ K

� ur;0 ¼ 0 with uT

r;0Mur;0 ¼ 12. Compute fr :¼ �sr,0Cur,0

3. Compute Dur :¼ � 1

4sr;0uT

r;0Cur;0

� ur;0 �

X2n

k¼1k 6¼r

sr;0

2sk;0

uTk;0frur;0

sr;0 � sk;0uk;0

4. Compute ur :¼ ur,0 + Dur

5. Compute sr using Eqs. (30)–(32)6. Normalize ur using Eq. (15)

Table A.4Algorithm for incremental method using FAN derivatives



r;0Mur;0 ¼ 12. Compute DC :¼ 1

qmaxC

3. For q from 1 to qmax do

4. Compute Dsr;q�1 :¼ � 12 uT

r;q�1DCur;q�1

5. Compute fr;q :¼ �sr;q�1 2Dsr;q�1Mþ 1þ q�1sr;q�1

� DC

� ur;q�1

6. Compute Dr;q :¼ s2r;q�1Mþ sr;q�1ðq� 1ÞDCþ K

7. Find the k position of the element with largest absolute value in ur,q�1 vector8. Construct Dr;q by zeroing out the k row and column of Dr,q and setting the k diagonal element to 19. Construct �fr;q by zeroing out the kth element of fr,q

10. Solve br,q from the linear equation system Dr;qbr;q ¼ �fr;q

11. Compute dr;q :¼ � 1

2sr;q�1uT

r;q�1 ð2sr;q�1Mþ ðq� 1ÞDCÞbq þ Dsr;q�1Mþ 1

2DC

� ur;q�1

� 12. Compute Dur,q�1 :¼ br + drur,q�1

13. Compute ur,q :¼ ur,q�1 + Dur,q�1

14. Compute sr,q using Eqs. (30)–(32) in which cr;q :¼ uHr;qðqDCÞur;q

15. Normalize ur,q using Eq. (15)

Table A.2Algorithm for single-step method using FAN derivatives



r;0Mur;0 ¼ 12. Compute Dsr;0 :¼ � 1

2 uTr;0Cur;0

3. Compute fr :¼ �sr,0(2Dsr,0M + C)ur,0

4. Compute Dr :¼ s2r;0Mþ K

5. Find the k position of the element with largest absolute value in ur,0 vector6. Construct Dr by zeroing out the k row and column of Dr and setting the k diagonal element to 17. Construct �fr by zeroing out the kth element of fr

8. Solve br from the linear equation system Drbr ¼ �fr

9. Compute dr :¼ �uTr;0Mbr � 1

4sruT

r;0ð2Dsr;0Mþ CÞur;0

10. Compute Dur,0 :¼ br + drur,0

11. Compute ur :¼ ur,0 + Dur,0

12. Compute sr using Eqs. (30)–(32)13. Normalize ur using Eq. (15)

Table A.5Algorithm for iterative method using AFK derivatives



r;0Mur;0 ¼ 1

2. Initialize j :¼ 1, define the convergence tolerance Tol and the maximum number of iterations jmax

3. Repeat4. Compute C�r;j :¼ eGðsr;j�1Þ5. Compute fr;j :¼ �sr;0C�r;jur;0

6. Compute Dur;j�1 :¼ � 1

4sr;0uT

r;0C�r;jur;0

� ur;0 �

X2n

k¼1k 6¼r

sr;0

2sk;0

uTk;0fr;jur;0

sr;0 � sk;0uk;0

7. Compute ur,j :¼ ur,0 + Dur,j�1

8. Compute sr,j using Eqs. (30)–(32)9. Normalize ur,j using Eq. (15)10. Compute the error er,j and increase j :¼ j + 1Until er,j 6 Tol or j > jmax

Table A.3Algorithm for incremental method using AFK derivatives



r;0Mur;0 ¼ 12. Compute DC :¼ 1

qmaxC

3. For q from 1 to qmax do4. Compute fr,q :¼ �sr,q�1DCur,q�1

5. Compute Dur;q�1 :¼ � 1

4sr;q�1uT

r;q�1DCur;q�1

� ur;q�1 �

X2n

k¼1k 6¼r

sr;q�1

2sk;q�1

uTk;q�1fr;qur;q�1

sr;q�1 � sk;q�1uk;q�1

6. Compute ur,q :¼ ur,q�1 + Dur,q�1

7. Compute sr,q using Eqs. (30)–(32) in which cr;q :¼ uHr;qðqDCÞur;q

8. Normalize ur,q using Eq. (15)


Table A.6Algorithm for iterative method using FAN derivatives



r;0Mur;0 ¼ 12. Compute Dr :¼ s2

r;0Mþ K

3. Find the k position of the element with largest absolute value in ur,0 vector4. Construct Dr by zeroing out the k row and column of Dr and setting the k diagonal element to 15. Initialize j :¼ 1, define the convergence tolerance Tol and the maximum number of iterations jmax

6. Repeat7. Compute C�r;j :¼ eGðsr;j�1Þ8. Compute Dsr;j�1 :¼ � 1

2 uTr;0C�r;jur;0

9. Compute fr;j :¼ �sr;0 2Dsr;j�1Mþ C�r;j

� ur;0

10. Construct �fr;j by zeroing out the kth element of fr,j

11. Solve br,j from the linear equation system Drbr;j ¼ �fr;j

12. Compute dr;j :¼ �uTr;0Mbr;j � 1

4sr;0uT

r;0 2Dsr;j�1Mþ C�r;j

� ur;0

13. Compute Dur,j�1 :¼ br,j + dr,jur,0

14. Compute ur,j :¼ ur,0 + Dur,j�1

15. Compute sr,j using Eqs. (30)–(32)16. Normalize ur,j using Eq. (15)17. Compute the error er,j and increase j :¼ j + 1Until er,j 6 Tol or j > jmax


B.1. Interpolation functions

The displacement vector uP of any point P is written in function of its components u and v

uP ¼ u v½ �T; ðB:1Þ

that are interpolated in function of the ith nodal displacements ui and vi. The isoparametric formulation is applied, thatmeans that the same shape functions used to define the geometry of the element are employed to interpolate the displace-ments. If the natural or intrinsic coordinate system (n,g) is utilised, which defines the parent element shown in Fig. B.1(b),the interpolation functions Ni are given by

Ni ¼1

4ð1þ ninÞð1þ gigÞ; ðB:2Þ

where ni and gi are the coordinates of the ith node. Thus, the approximation of the displacement uP of the P point yields

uP ¼ Nu; ðB:3Þwhere N is the matrix of the interpolation functions given by

N ¼N 1 0 N 2 0 N 3 0 N 4 0

0 N 1 0 N 2 0 N 3 0 N 4

� �ðB:4Þ

and u represents the nodal displacement vector

u ¼ u1 v1 u2 v2 u3 v3 u4 v4½ �T. ðB:5Þ

Fig. B.1. Four node rectangular finite element: (a) original and (b) parent elements.


B.2. Stiffness matrix

The stiffness matrix K is given by the integral

K ¼ tZ Z

ABTDB dA; ðB:6Þ

where t is the thickness of the element, A represents the surface and D is the elasticity matrix which relates the elastic stressre with the strain e,

re ¼ De; ðB:7Þthat, under plane stress assumption, is given by

D ¼ E1� m2

1 m 0

m 1 0

0 01� m

2

2664

3775; ðB:8Þ

where E and m are the elasticity modulus and Poisson’s coefficient, respectively. The B matrix is given by the strain–displacement e–u relationship

e ¼ Bu; ðB:9Þgiven by

B ¼

o

ox0

0o

oyo

oyo

ox

26666664

37777775

N. ðB:10Þ

Numerical integration is generally used to compute Eq. (B.6), but here, due to the rectangular shape of the elements, theexact integral can be easily computed (even if the stiffness of the system is overestimated), which is shown in Table B.1.

B.3. Mass matrix

The consistent mass matrix may be computed from the kinetics energy of the system,

M ¼ tZ Z

ANTqNdA; ðB:11Þ

where q is the density of the material. The full integration of this integral is shown in Table B.1.

B.4. Damping matrix

The damping matrix eGðsÞ of the non-viscous material can be obtained from the dissipation energy, that gives

eGðsÞ ¼ tZ Z

ABTVðsÞBdA; ðB:12Þ

where V(s) represents the matrix that relates the dissipative stress rd with the strain rate _e,

rd ¼ VðsÞ_e. ðB:13ÞThe plane stress hypothesis implies that the V(s) matrix may be written as

VðsÞ ¼eGðsÞ

1� m2

1 m 0

m 1 0

0 01� m

2

2664

3775; ðB:14Þ

due the similitude of Eqs. (B.7) and (B.13). eGðsÞ is the damping function in Laplace domain, that in this paper the expo-nential model

eGðsÞ ¼ c1þ st0

ðB:15Þ

Table B.1Finite element matrices

K ¼ Et24abð1� m2Þ

4a2ð1� mÞ þ 8b2 3abð1þ mÞ 2a2ð1� mÞ � 8b2 �3abð1� 3mÞ �2a2ð1� mÞ � 4b2 �3abð1þ mÞ �4a2ð1� mÞ þ 4b2 3abð1� 3mÞ4b2ð1� mÞ þ 8a2 3abð1� 3mÞ �4b2ð1� mÞ þ 4a2 �3abð1þ mÞ �2b2ð1� mÞ � 4a2 �3abð1� 3mÞ 2b2ð1� mÞ � 8a2

4a2ð1� mÞ þ 8b2 �3abð1þ mÞ �4a2ð1� mÞ þ 4b2 �3abð1� 3mÞ �2a2ð1� mÞ � 4b2 3abð1þ mÞ4b2ð1� mÞ þ 8a2 3abð1� 3mÞ 2b2ð1� mÞ � 8a2 3abð1þ mÞ �2b2ð1� mÞ � 4a2

4a2ð1� mÞ þ 8b2 3abð1þ mÞ 2a2ð1� mÞ � 8b2 �3abð1� 3mÞ4b2ð1� mÞ þ 8a2 3abð1� 3mÞ �4b2ð1� mÞ þ 4a2

4a2ð1� mÞ þ 8b2 �3abð1þ mÞ4b2ð1� mÞ þ 8a2

266666666664

377777777775

M ¼ qabt36

4 0 2 0 1 0 2 04 0 2 0 1 0 2

4 0 2 0 1 04 0 2 0 1

4 0 2 04 0 2

4 04

266666666664

377777777775

C ¼ cE

K eGðsÞ ¼ eGðsÞE

K

F.

Co

rtes,M

.J.

Eleja

ba

rrieta/

Co

mp

ut.

Meth

od

sA

pp

l.M

ech.

En

grg

.1

95

(2

00

6)

64

48

–6

46

26461


was used, in which c is the viscous coefficient and t0 is the relaxation time. The similitude between Eqs. (B.6) and (B.12),and between Eqs. (B.8) and (B.14), involves that the stiffness and damping matrices are proportional, as is indicated inTable B.1. The particular case in which the relaxation time tends to zero, t0! 0, implies that the damping matrix eGðsÞdegenerates in the viscous C, as is illustrated in Table B.1.

References

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