iterative method for exponential damping identification

Computer-Aided Civil and Infrastructure Engineering 00 (2014) 1–15

Iterative Method for Exponential DampingIdentification

Yuhua Pan

School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China and China Academy of BuildingResearch, Beijing 100013, China

&

Yuanfeng Wang*

School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China

Abstract: Exponential damping is a new potentialdamping model for dynamic analysis of systems. This ar-ticle outlines a complex mode procedure for identifyingthe exponential damping model and discusses its appli-cability and limitations. A new iterative method for re-laxation factor is proposed, without using the full set ofmodal data, which is simple, direct, and compatible withconventional modal testing procedures. Combined withthe frequency response functions (FRFs)-based modelupdating method, the finite element model with such non-viscous damping is updated. The proposed method andseveral related issues are illustrated by numerical exam-ples. It is shown that the finite element model updatedmethod for the systems with exponential damping canpredict accurately not only the natural frequencies butalso the FRFs of the systems.

1 INTRODUCTION

Characterization of damping in a vibrating system isan important area of research in structural dynamics(Lazan, 1968; Nashif et al., 1985; Osinski, 1998). Duringthe last hundred years, various assumptions of damp-ing models (Biot, 1958; Makris, 1997; Woodhouse, 1998;Gounaris et al., 2007; Nakamura, 2007; Franchetti andModena, 2009; Genta and Amati, 2010) have been pre-sented for damping phenomena and each one has itsown dynamic analysis method and application scope.

*To whom correspondence should be addressed. E-mail: [email protected].

Due to the complexity of damping mechanism, viscousdamping model is widely used to model the energy dissi-pation in a vibrating system for its theoretical simplicity.However, this model assumes that the damping forceis dependent on the instantaneous generalized velocityand implies that energy dissipation per cycle is linearlyproportional to the frequency of the system. It cannottruly represent the complicated damping character inactual structures (Miklestad, 1952; Crandall, 1970; Bert,1973). Therefore, further study of other damping mod-els, which can better describe the damping character ofa vibrating system and its corresponding dynamic anal-ysis methods, would be a significant subject in engineer-ing and other related areas.

Damping models in which the dissipative forcesdepend on any variable other than the instantaneousgeneralized velocities are called nonviscous dampingmodels (Wagner and Adhikari, 2003). One type of non-viscous damping models is based on the fact that thedissipative forces depend on the time history of mo-tion, which are represented by convolution integrals be-tween velocities and decaying kernel functions. Withthis model, the damping force of a single-degree-of-freedom (SDOF) is expressed as

fd(t) =∫ t

0G(t − τ )x(τ )dτ (1)

where G(t) is the damping kernel function.Among a wide variety of mathematical expressions,

the exponential damping model seems to be a promising

C© 2014 Computer-Aided Civil and Infrastructure Engineering.DOI: 10.1111/mice.12077

2 Pan & Wang

candidate in which the damping function is describedby

G(t) =kmax∑k=1

ckμke−μk t , for t ≥ 0,k = 1,2, . . . , kmax (2)

where c is the damping coefficient; μ is the relaxationfactor; and kmax denotes the number of different expo-nential models employed to describe the damping be-havior of the system. In the limiting case when μk→�,k = 1, 2, . . . , kmax, the equation of motion reduces to thatof a viscously damped system with an equivalent viscousdamping coefficient as

c =kmax∑k=1

ck (3)

Thus, the exponential damping model is a furthergeneralization of the classical viscous damping model.

Due to the enhanced damping model, traditionalmethods for the viscous damping model cannot be di-rectly applied on such nonviscously damped systems.As a result, in recent years, some researchers have con-ducted their investigations in this field, for example,Adhikari (2002), and Cortes and Elejabarrieta (2006)presented a modal analysis method and a numericalmethod, respectively, to treat the complex eigenval-ues and eigenvectors in nonviscously damped linear dy-namic systems. Other dynamic analysis methods pro-posed specially for such damping model can be foundin recent references (Wagner and Adhikari, 2003; Ad-hikari and Wagner, 2004; Cortes et al., 2009; Pan andWang, 2013).

For the purpose of accurate simulation of dy-namic systems, a variety of new frequency-domainidentification methods have been developed by re-searchers specifically for damping identification (Phaniand Woodhouse, 2007; Prandina et al., 2009). Thesemethods can be classified into two groups: modal meth-ods (Lancaster, 1961; Ibrahim, 1983; Minas and Inman,1990; Adhikari and Woodhouse, 2001a; Adhikari andPhani, 2009; Phani and Woodhouse, 2009) and matrixmethods (Fritzen, 1986; Chen et al., 1996; Lee and Kim,2001; Ozgen and Kim, 2007). The former, based onthe modal parameters deduced from the frequency re-sponse functions (FRFs), can be called indirect method.The latter, based directly on the measured FRF ma-trix to identify damping matrix, can be called directmethod. However, the existing damping identificationmethods are generally restricted to the viscous damping.There are still few research literatures on the nonvis-cous damping identification. Adhikari and Woodhouse(2001b) made a first attempt to extend the complexmode analysis method to the exponential nonviscous

damping systems. This method belongs to the indirectmethod, in which higher precision identification param-eters of full set of modes are required to identify damp-ing matrices. In addition, the study on relaxation factoris also limited, which is a focus of the authors’ research.

In the past two decades, growing interest has beenattracted by researchers on structural system identifi-cation (Jiang and Adeli, 2005; Adeli and Jiang, 2006;Jiang et al., 2007; Sirca and Adeli, 2012), structuralhealth monitoring (Ching et al., 2006; Park et al.,2007), and damage identification (Jiang and Adeli, 2007;Jafarkhani and Masri, 2011; Raich and Liszkai, 2012;Xiang and Liang, 2012), with the aim to develop a math-ematical model, monitor the performance of structures,and assess the conditions of systems. In these fields, fi-nite element model (FEM) updating methods providean effective way to obtain an accurate numerical model,which is very important for parameter identification,damage detection, and condition assessment of engi-neering structures (Zhou et al., 2013). These methodscan also be classified into two groups: indirect method(Imregun and Visser, 1991; Mottershed, 1993; Moaveniet al., 2009) that is based on the modal parametersdeduced from the FRFs and direct method (Lin andEwins, 1994; Kwon and Lin, 2004; Lin and Zhu, 2006;Asma and Bouazzouni, 2007; Arora et al., 2009a, b;Esfandiar et al., 2010; Garcia-Palencia and Santini-Bell,2013) that is based directly on the measured FRF matri-ces. The latter has an advantage that cumulative errorsfrom modal parameters identification can be avoided.

In this article, the application of complex mode analy-sis theory on the exponential damping model is outlinedfirst and its applicability and limitation are discussed.And then, an effective iterative method for the relax-ation factor is proposed. Finally, combined with the re-cently proposed FRF-based model updating method,the FEM with such nonviscous damping is updated. Nu-merical examples are provided to illustrate the validityof the proposed approach.

2 BACKGROUND OF EXPONENTIAL DAMPINGIDENTIFICATION

2.1 Approximate expression of FRF

Considering an exponentially damped system, a FRFHef (ω), which is between the applied force at fth degreeof freedom (DOF) and the response at eth DOF, is de-fined as (Adhikari, 2000)

Hef (ω) =N∑

k=1

[Rkef

iω − sk+

R∗kef

iω − s∗k

]+

m∑j=2N+1

R jef

iω − s j

(4)

Iterative method for exponential damping identification 3

where Rkef and R∗kef

are the efth element of the residuematrices corresponding to the pole sk and s∗

k , respec-tively. The relationship between residues Rkef and modeshapes is

Rkef = γk zek zfk (5)

where zek is the eth element of kth mode shape. Thepoles sk are related to the natural frequencies λk by

sk = iλk (6)

The first part of the right-hand side of Equation (2.1)corresponds to elastic modes and the second part corre-sponds to nonviscous modes aroused by the exponen-tial damping model. The elastic modes are generallycomplex as damping is nonproportional in nature. Forlightly damped systems, the kth complex natural fre-quencies corresponding to the elastic modes can be ex-pressed as

λk = ωk + iωkξk (7)

where ξk and ωk are the kth damping ratio and un-damped natural frequency, respectively.

Compared with the elastic modes, the nonviscousmodes may be quite small. The second part of the right-hand side of Equation (2.1) may be neglected in actualvibration test. In this case, the FRF of exponentiallydamped systems can be represented in an approximateway as

Hef (ω) ≈N∑

k=1

[Rkef

iω − sk+

R∗kef

iω − s∗k

](8)

which is the same with viscously damped systems. Thus,the traditional vibration modal testing procedure basedon the viscous damping theory can also be suitable forthis kind of exponentially damped systems.

2.2 Fitting of FRFs

From Equation (8), it is observed that Hef (ω) is a linearfunction of the residues in the numerator, whereas it is anonlinear function of the poles located in the denomina-tor. The linear parameters of residues can be obtainedby a linear least-square approach and the nonlinear pa-rameters of poles can be obtained by a nonlinear op-timization approach. In a practical vibration test, mostoften only measuring one row or line of the FRF matrixis sufficient to determine these parameters. For exam-ple, in a hammer test, the response measurement pointis kept fixed while the excitation point varies accordingto a selected grid on a structure. Thus, in Equation (8),e that is fixed can be omitted for brevity. Suppose thenumber of hammer points (i.e., DOF of the structure) is

N and the number of modes retained in the study is m,from Equation (8), we obtain

H f (ω) =N∑

k=1

[ψ1k (ω)Akf + ψ2k (ω)A∗kf ] (9)

for f = 1, 2, . . . , N, where

ψ1k (ω) = − (iω)r

ω − λk, ψ2k (ω) = (iω)r

ω + λ∗k

(10)

Akf = iRkef (11)

and r = 0, 1, and 2 corresponds to displacement, veloc-ity, and acceleration transfer function, respectively.

Combining Equations (11) and (5), one has

Akf = iγ k zek zfk (12)

When γ k = 1/(2iλk), the standard-mass-normalized or-thogonal modes are obtained by substituting this valueof γ k into Equation (12)

Akf = zek zfk

2λk(13)

As mentioned above, e that is the response measure-ment point is fixed. For f = e, from Equation (13), thereis

zek =√

2λk Akf , for k= 1, 2, . . . , m (14)

Substituting zek from the above into Equation (13),the mode shapes are obtained as

zfk = 2λk Akf

zek=

√2λk

Akf√Ake

(15)

for k = 1, 2, . . . , m, and f = 1, 2, . . . , N.The detailed procedure of linear–nonlinear optimiza-

tion approach for modal parameter identification can beseen in the reference Adhikari (2000).

2.3 Fitting of damping coefficient matrix

Consider an exponentially damped free vibration sys-tem, its dynamic equation can be expressed as

Mx(t) +∫ t

0G(t − τ )x(τ )dτ + Kx(t) = 0 (16)

where M and K ∈ RN×N are the mass and stiffness ma-

trices of the system, respectively; G(t) is the dampingkernel function matrix. The exponential damping is de-fined as

4 Pan & Wang

G(t) =Cμe−μt (17)

where C ∈ RN×N is the damping coefficient matrix of the

system. The Laplace transformation of Equation (16) isobtained as

s2Mz + sL(s)z + Kz = 0 (18)

where L(s)= L[G(t)] ∈ CN×N ; L[•] denotes the Laplace

transformation; and s j ∈ C and z j ∈ CN are the jth com-

plex natural frequency and complex mode, respectively.For lightly damped vibration systems, based on the first-order perturbation theory proposed by Woodhouse(1998), the jth complex natural frequency and modeshape can be approximately expressed as

λ j ≈ ±ω j+iL′j j (ω j )/2 (19)

z j ≈ x j+iN∑

k=1k = j

ω j L′k j (ω j )

(ω2j − ω2

k )xk (20)

where λ j = −is j ; ω j is the jth undamped natural fre-quency; L′

kl(ω j ) =xTk L(ω j )xl is the jth complex mode

damping; and x j ∈ RN is the jth real mode shape, which

satisfy orthogonal relations with M and K as

xTk Mxl = δkl, xT

k Kxl = ω2jδkl (21)

for f = 1, 2, . . . , N.From L′

kl(ω j ) =xTk L(ω j )xl and L(s)= L[G(t)] ∈

CN×N , one obtains

L′kl(ω j ) = μC ′

kl

(μ+iω j )= μ2C ′

kl

(μ2+ω2j )

− iμω j C ′

kl

(μ2+ω2j )

(22)

where C ′kl = xT

k Cxl . Substituting Equation (22) intoEquations (19) and (20), then

λ j ≈ ω j+C ′

j j

2μω j

(μ2+ω2j )

+iC ′

j j

2μ2

(μ2+ω2j )

(23)

z j ≈ x j+N∑

k=1k = j

μω j

(μ2+ω2j )

ω j C ′k j

(ω2j − ω2

k )xk

+ iN∑

k=1k = j

μ2

(μ2+ω2j )

ω j C ′k j

(ω2j − ω2

k )xk

(24)

Write

z j = u j+iv j (25)

where u j and v j ∈ RN are both real. For lightly damped

vibration systems, neglecting the high-order terms inEquation (23), the jth undamped nature frequency isobtained as

ω j ≈ Re(λ j ) (26)

where Re(•) denotes to take the real part. Likewise, ne-glecting the high-order terms in Equation (24), that isu j ≈ x j . Thus, the real and complex parts of complexmode shape vectors have the following approximaterelationship:

v j ≈m∑

k=1Ak j uk, where

Ak j = μ2

(μ2+ω2j )

ω j C ′k j

(ω2j − ω2

k )

(27)

To minimize the error, a Galerkin approach can beadopted to obtain the Ak j . The undamped mode shapesul , for l = 1, 2, . . . , m, are taken as “weighting func-tions.” The error from representing vj by the series sumEquation (27) can be expressed as

ε j = v j −m∑

k=1

Ak j uk (28)

Using the Galerkin method on ε j ∈ R, one obtains

uTl ε j = uT

l

(v j −

m∑k=1

Ak j uk

)= 0 (29)

The above equations can be combined in matrix form

WA = D (30)

where W = UTU ∈ Rm×m and D = UTV ∈ R

m×m with

U = [u1,u2, · · · um] ∈ RN×m,

V = [v1,v2, · · · vm] ∈ RN×m

(31)

Then, the coefficient matrix can be obtained by

A = W−1D = (UTU)−1UTV (32)

From Equation (27), the off-diagonal elements of themodal damping matrix can be derived from

C ′k j = (μ2+ω2

j )

μ2

(ω2j − ω2

k )

ω jAk j (33)

for k, j = 1, 2, . . . , m; k � j and from Equation (23),the diagonal elements of the modal damping matrixare

C ′j j = 2Im(λ j )

(μ2+ω2j )

μ2, for j= 1, 2, . . . , m (34)

where Im(•) denotes to take the imaginary part. Theabove two equations together completely define themodal damping matrix C′ ∈ R

N×m . If U ∈ RN×N is the

complete undamped modal matrix, then the dampingmatrices in the modal coordinates and original coor-dinates are related by C′ = UTCU. Thus, given C′,


um

uk uk uk uk

umum

( )g t ( )g t ( )g tthN

Fig. 1. A 9-DOF damped system.

the damping matrix in the original coordinates can beeasily obtained by the inverse transformation as C =UT−1

C′U−1. For the case when the full modal matrix isnot available, that is, U ∈ R

N×m is irreversible, a pseu-doinverse is required to obtain the damping matrix inthe original coordinates. The damping matrix in theoriginal coordinates is then given by

C = [(UTU)−1UT]TC′[(UTU)−1UT] (35)

Note that the above method is also suitable forthe identification of viscous damping matrix. When μ

→�, Equations (27), (33), and (34) are correspondinglytransformed into

Ak j = ω j C ′k j

(ω2j − ω2

k )(36)

C ′k j = (ω2

j − ω2k )

ω jAk j and C ′

j j = 2Im(λ j ) (37)

3 NUMERICAL RESULTS OF DAMPINGMATRIX IDENTIFICATION

To verify the validity and to discuss the applicable scopeof the above method, a 9-DOF system with exponen-tial damping shown in Figure 1 is studied for numericalsimulation. The relaxation factor μ can be expressed innondimensional form by

μ = 1/(γ Tmin) (38)

where Tmin is the period of the highest undamped nat-ural frequency and θ = γ Tmin is the so-called charac-teristic time constant. When γ is small compared withunity, the damping behavior can be expected to be es-sentially viscous, but when γ is of order unity or big-ger, nonviscous effects should become significant. Bothproportional and nonproportional damping coefficient

matrices are considered for identification. The mass andthe stiffness matrices of the system are

K = ku

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 −1−1 2 −1

. . .. . .

. . .−1 2 −1

. . .. . . −1−1 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

M = muI

(39)

where ku = 4 × 105 N/m, mu = 1 kg, and I is the 9 orderunit matrix.

3.1 Identification of exponential damping model

In the simulation, the damping kernel function isG(t) = Cg(t) = Cμe−μt with the damping coefficientmatrix C = cI and c = 25 Ns/m, which correspondsto the mass proportional damping. The Gaussian whitenoise is applied on each DOF, respectively, to get theresponses of the first DOF of the system by using theFFT method proposed by Pan and Wang (2013). Then,identify the damping coefficient matrices from the FRFH1 f (ω) by using the above method.

When γ = 0.002, from Equation (38), if μ→�, theexponential damping model should show near-viscousbehavior. Use both viscous damping and exponentialdamping to identify the damping coefficient matrix, re-spectively. The results calculated by using the completeset of nine modes are shown in Figure 2.

Since the reflected damping property of the system isclose to the viscous one, the fitted matrices obtained byboth damping models identify the damping in the sys-tem very well. The identified results of diagonal damp-ing values, obtained by both damping models, are veryclose to the true damping value c = 25 Ns/m, be-cause the identified damping values on the position ofthe biggest error for viscous damping and exponentialdamping are 24.99 and 25.01, respectively, and the rela-tive errors of them are both only 0.04% compared withthe true value. So, the correctness and validity of theabove identification methods for both viscous dampingand exponential damping are proved. It is also illus-trated that the exponential damping model is suitablefor the damping identification of the viscously dampedsystems.

When γ is larger, the damping model shows obvi-ous nonviscous behavior. Figure 3 shows the resultsof both the fitted viscous damping matrix and the ex-ponential damping matrix. With the increase of γ ,for the case γ = 0.2, the result of fitting the viscous

6 Pan & Wang

24

68

24

68

0

10

20

30

j-th DOF(a)k-th DOF

Fitte

dvi

scou

sda m

pin gCjk

(Ns /

m)

24

68

24

68

0

10

20

30

j-th DOF(b)k-th DOF

Fitte

dex

pone

ntia

ldam

pingCjk

(Ns/

m)

Fig. 2. Fitted damping coefficient matrix for γ = 0.002.(a) Viscous damping. (b) Exponential damping.

model cannot represent the true damping behavior,while the result of fitting the exponential model clearlydemonstrates the accuracy of the method that is iden-tical to the true value, see Figure 3. The average ofthe main diagonal damping values identified by vis-cous damping model is only 15.19, whose relative errorreaches to 39.24%, while the the average one identifiedby exponential damping model is 25.03, which is veryclose to the true value. It further shows that the applica-tion of different damping models has a great influenceon identifying the system damping. When the systemshows obvious nonviscous damping behavior, the expo-nential damping model can also identify the dampingmatrices accurately.

3.2 Identification of nonproportional damping matrix

Considering the identification of nonproportionaldamping matrix, the simulated system is identical to thatshown in Figure 1 with the same damping kernel func-

24

68

24

68

0

10

20

30

j-th DOF(a)k-th DOF

Fitte

dvi

scou

sda m

pingCjk

(Ns /

m)

24

68

24

68

0

10

20

30

j-th DOF(b)k-th DOF

Fitte

dex

pone

ntia

ldam

pingCjk

(Ns/

m)

Fig. 3. Fitted damping coefficient matrices for γ = 0.2.(a) Viscous damping. (b) Exponential damping.

tion. The damping coefficient matrix is still a diagonalmatrix with the damping elements c1 = 25 Ns/m be-tween the 4th and 6th DOFs, and c2 = 15 Ns/m in otherDOFs. The fitting results calculated by using the com-plete set of nine modes for the case γ = 0.2 are shownin Figure 4.

When using the viscous damping model, the identi-fied damping value of the 5th DOF on the main diag-onal is 15.42, whose relative error reaches to 38.22%compared with the true value 25, and the identified val-ues of the 1st and 8th DOFs are 8.63 and 9.26 with therelative errors 42.47% and 38.27% compared with thetrue value 15, respectively. When using the exponentialdamping model, the average of the 4th–6th DOFs iden-tified results on the main diagonal is 25.01, and the av-erage of other DOFs is 15.01. They are both close to thetrue values. It is shown that when the nonproportionallydamped system obviously reflects nonviscous behavior,the result of the fitted viscous damping matrix has alarger deviation from the truth, while the result of the


24

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24

68

0

10

20

30

j-th DOF(a)k-th DOF

Fitte

dvi

scou

sda m

pin gCjk

(Ns /

m)

24

68

24

68

0

10

20

30

j-th DOF(b)k-th DOF

Fitte

dex

pone

ntia

l dam

pingCjk

(Ns/

m)

Fig. 4. Fitted damping coefficient matrices for γ = 0.2. (a)Viscous damping. (b) Exponential damping.

fitted exponential damping matrix can still predict theright value and spatial distribution of nonproportionaldamping.

3.3 Identification of Rayleigh damping matrices

Consider a commonly used Rayleigh damping, whichhas first two-order damping ratios ξ 1 = ξ 2 = 0.01. Thedamping matrix can be established by linear combina-tion of mass and stiffness matrices. Here, we only pro-vide the fitted results of the exponential damping modelin Figure 5, which are almost the same with the viscousone under the case γ = 0.002.

It is shown that for the Rayleigh type damping, theresults obtained by whether fitting the viscous or the ex-ponential damping model can represent the real damp-ing when the completed set of nine modes is used, seeFigure 5a. However, the quality of the fitted dampingmatrices by both damping models deteriorates as thenumber of modes used to fit the damping matrices is re-

24

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24

68

-20

-10

0

10

20

30

j-th DOF(a)k-th DOF

Fitte

dex

pone

n ti a

l dam

pin g

Cjk

(Ns/

m)

24

68

24

68

-20

-10

0

10

20

30

j-th DOF(b)k-th DOF

Fitte

dex

pone

ntia

l dam

ping

Cjk

(Ns/

m)

Fig. 5. Fitted exponential damping coefficient matrices, γ =0.002. (a) Completed set of nine modes and (b) first five

modes.

duced. When using the first five modes only, the spatialdistributions of identified damping matrix show reason-able approximation to the real behavior, while their val-ues have larger deviations from the true value, as shownin Figure 5b.

3.4 Discussion

Through the above numerical simulations, we can getthe following important conclusions.

1. The complex mode analysis method and its damp-ing identification procedure, which are based onthe nonproportional damping theory, are suitablefor the proportional damping (or Rayleigh damp-ing). It shows that the proportional damping is thespecial condition of the nonproportional damping.

2. The above proposed method can also be applied tothe exponential damping model, which is a moregeneral damping model than viscous one.

8 Pan & Wang

However, the above method for damping matrixidentification has obvious limitations. First, this methodis needed to obtain each order mode shape, frequency,and damping ratio by fitting the FRFs. Then, secondaryidentification of damping matrix is made according tothe identified modal parameters. This requires that allthe identified mode frequencies, damping ratios, and es-pecially shapes must have higher reliability and accu-racy, to identify accurate damping matrix. In the actualvibration tests, the error caused by the identificationprocedure of modal parameters through FRF in somecases may be even larger than that caused by the theo-retical model inaccuracy. The influence of mode shapeerror on identified damping matrix is more obvious. Thesecondary identification of damping matrix by such ex-perimental modal parameters allows the problem of er-ror accumulation to exist, leading to the identified re-sults not being credible. Second, from Section 3.3, it canobtain the exact identification results only in the casethat the complete set of modes is used. The identifi-cation error will be larger when only the first few or-der modes are used. It is often difficult to obtain accu-rate high-order modes in vibration tests, which limits theapplication of this method. Third, there may not be areliable or available method for identifying relaxationfactor μ at present, which requires further study. Thus,based on existing researches, a new iterative method forrelaxation factor μ in the exponential damping model isproposed. Combined with the FRF-based model updat-ing method, numerical simulation of a cantilever beamstructure is provided as an illustrated example.

4 FRF-BASED MODEL UPDATING METHOD

4.1 Iterative method for relaxation factor

For the Rayleigh damping, it is assumed that the damp-ing matrix is proportional to the mass and stiffness ma-trices as

C = a0M + a1K (40)

where a0 and a1 are the mass-proportional and stiffness-proportinal damping coefficients, respectively. Accord-ing to the orthogonality of mode, from Equation (21),expression for the jth mass-normalized modal dampingcan be obtained as

C j = xTj Cx j = a0xT

j Mx j + a1xTj Kx j = a0 + a1ω

2j (41)

Then, from Equation (41), the relationship betweendamping ratio and frequency is

ξ j = C j

2ω j= a0

2ω j+ a1ω j

2(42)

For exponential damping, assume that the dampingcoefficient matrices of kernel function have the formcorresponding to the proportional damping

G(t) =C0μ0e−μ0t + C1μ1e−μ1t (43)

where C0 = a0M and C1 = a1K; μ0 and μ1 are the re-laxation factors related to mass and stiffness, respec-tively. Based on the first-order perturbation method,the Laplace transform of G(t) is obtained by

L′kl(ω j ) = μ0C ′

0,kl

(μ0 + iω j )+ μ1C ′

1,kl

(μ1 + iω j )

= μ20C ′

0,kl

(μ20 + ω2

j )+ μ2

1C ′1,kl

(μ21 + ω2

j )

− i

(μ0ω j C ′

0,kl

(μ20 + ω2

j )+ μ1ω j C ′

1,kl

(μ21 + ω2

j )

)(44)

where

C ′0,kl = a0xT

k Mxl = a0δkl and

C ′1,kl = a1xT

k Kxl = a1ω2jδkl

(45)

for k, j = 1, 2, . . . , N. Substituting Equation (45) intoEquation (19), approximate expression for the jth com-plex natural frequency can be obtained as

λ j ≈ ω j + a0

2μ0ω j

(μ20 + ω2

j )+ a1ω

2j

2μ1ω j

(μ21 + ω2

j )

+i

(a0

2μ2

0

(μ20 + ω2

j )+ a1ω

2j

2μ2

1

(μ21 + ω2

j )

) (46)

Comparing the imaginary parts of Equation (7) andEquation (46), for the exponential damping, the rela-tionship between damping ratio and frequency can beexpressed as

ξex, j = a0

2ω j

μ20

(μ20 + ω2

j )+ a1ω j

2μ2

1

(μ21 + ω2

j )

= a0

2ω j

1(1 + (ω j/μ0)2)

+ a1ω j

21

(1 + (ω j/μ1)2)

(47)

From Equation (47), when μ0, μ1→�, the expo-nential proportional damping reduces to a commonRayleigh viscous damping.

It is noted that the damping part proportional to themass denotes an external damping force of a vibrationstructure, such as air damping, which is more close tothe fluid viscous damping force, while the damping partproportional to the stiffness comes from internal mate-rial damping, which may have the nonviscous behav-ior. Thus, considering μ0→� may be more realistic,


then only the relaxation factor μ1 is needed to be deter-mined. Neglecting μ0 in the first term of Equation (47),one obtains

ξex, j = a0

2ω j+ a1ω j

21

(1 + (ω j/μ1)2)(48)

From Equation (48), the nonviscous behavior of ex-ponential damping is mainly controlled by the ratio ofω j/μ1. All order damping ratios of exponential dampingare smaller than those of viscous damping. It is foundthat the greater μ1 is the value of ω j/μ1 is more closeto 0, and the exponential damping is more close to theviscous one. When μ1 is a constant, with the increaseof ω j , value of ω j/μ1 is also increasing, correspond-ingly, the damping ratio ξex, j decreases more obviously.It illustrates that the nonviscous behavior of exponen-tial damping performs more significantly in higher or-der modes. This characteristic may provide a train ofthought for the identification of relaxation factor μ1.

Introducing parameter γ 1, which denotes the highestorder frequency to relaxation factor ratio, that is

γ1 = ωmax

2πμ1= 1

μ1Tmin(49)

So,

μ1 = (1/γ1Tmin) (50)

It is apparent that exponential damping is more closeto viscous damping when μ1 is smaller. From the aboveanalysis, we can conclude that for lower order modes,nonviscous behavior of exponential damping model isnot significant, whereas for higher order modes, thedamping ratios of exponential damping will be dif-ferent from those of viscous damping. Thus, first as-sume that γ 1 = 0.002, from Equation (50), the corre-sponding relaxation factor μ1 is obtained. Then, usingEquation (48), the first two-order damping ratios canbe calculated with the established set of equations asfollows:(

ξex,1

ξex,2

)= 1

2

(1/ω1 ω1/(1 + (ω1/μ1)2)

1/ω2 ω2/(1 + (ω2/μ1)2)

)(a0

a1

)(51)

If we assume that the measured first two-order damp-ing ratios, which related to frequencies ω1 and ω2, re-spectively, have the relationship as ξ1 = ξex,1 and ξ2 =ξex,2, from Equation (51), the proportional coefficientsof exponential damping can be obtained by

(a0

a1

)= 2

(1/ω1 ω1/(1 + (ω1/μ1)2)

1/ω2 ω2/(1 + (ω2/μ1)2)

)−1 (ξ1

ξ2

)(52)

In that way, for relatively higher order mode damp-ing ratios ξex, j , 3� j � N, its value calculated by Equa-tion (48) will be different from the measured ξ j . Substi-tuting ξex, j = ξ j into Equation (48), there is

Cμ21 = B (53)

with

C j = a0 + a1ω2j − 2ξ jω j and B j = 2ξ jω

3j − a0ω

2j (54)

for 3 � j � N, where C = [C3, C4, · · · C j ]T and B =[B3, B4, · · · B j ]T are column vectors, composed of C j

and B j , respectively. For j > 3, Equation (53) is anoverdetermined set of equations, which can be solvedby the pseudoinverse way. A new relaxation factor isthen calculated by

μ21= [CTC]−1CTB (55)

Define the relaxation factor error as ε = |μ1 − μ1|. Ifthe error is small enough to meet certain error range,say ε � 10−10, then the estimated μ1 is true value. Oth-erwise, recompute by substituting μ1 = μ1 into Equa-tion (52).

In summary, the procedure can be described by thefollowing steps:

1. Assume the first guess of the relaxation factorγ 1 (or μ1). Assume that the system is viscouslydamped, that is, γ 1 = 0.002.

2. Calculate the coefficients a0 and a1 of exponentialproportional damping from Equation (52).

3. Obtain a new value of relaxation factor μ1 fromEquation (55), and its error ε = |μ1 − μ1|.

4. If ε � 10−10, then the estimated μ1 is true value.Otherwise, set the final value of μ1 as the currentvalue μ1 = μ1, and go back to step 2.

4.2 Model updating for exponentially damped systems

FRF-based FEM updating method was proposed in re-cent years, which has a distinct advantage that it avoidsthe step of modal parameter identification; thus, errorsfrom modal parameter extraction can be eliminated.

In the development of FRF-based model updatingmethod, the following identities relating dynamic stiff-ness matrix Z and receptance FRF matrix H for the ana-lytical model as well as the actual structure, respectively,can be written

ZaHa = I (56)

ZeHe = I (57)

10 Pan & Wang

where superscripts a and e denote analytical (likean FE model) and experimental models, respectively.Expressing Ze as

Ze = Za + �Z (58)

Substituting Equations (56) and (58) into Equa-tion (57), the following matrix equation is obtained:

�ZHe = Za(Ha − He) (59)

Premultiplying the above equation by Ha and usingZaHa = HaZa = I gives

Ha�ZHe = Ha − He (60)

If only the jth column of experimentally measuredFRF matrix He

j is available, then the above equation isreduced to

Ha�ZHej = Ha

j − Hej (61)

With this method, physical-variables-based updatingparameter formulation is used. Linearizing �Z with re-spect to being the vector of the physical variables asso-ciated with individual or group of finite elements gives

�Z =Np∑

k=1

∂Za

∂pk�pk (62)

where Np is the total number of updating parameters.Substituting �Z into Equation (61), one obtains

Np∑k=1

{Ha ∂Za

∂pkHe

j

}�pk = �H j (63)

If the measured FRFs have enough frequency pointsn, Equation (63) can constitute a set of overdeterminedalgebraic equations as

S�p = �H (64)

where

�H =

⎡⎢⎢⎢⎣

Haj (ω1) − He

j (ω1)

...

Haj (ω1) − He

j (ω1)

⎤⎥⎥⎥⎦ (65)

S =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

Ha(ω1)∂Za

∂p1He

j (ω1) · · · Ha(ω1)∂Za

∂pNp

Hej (ω1)

.... . .

...

Ha(ωn)∂Za

∂p1He

j (ωn) · · · Ha(ωn)∂Za

∂pNp

Hej (ωn)

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(66)

10 9 8 7 6 5 4 3 2 1

Fig. 6. Finite element model of the cantilever beam.

Separate real part and imaginary part of Equa-tion (64) {

Re(S)�pm = Re(�H)

Im(S)�pd = Im(�H)(67)

where �pm denotes the vector of updated parametersrelated to mass and stiffness matrices, and �pd de-notes the vector of damping matrix updated parame-ters. These parameters can be obtained by the pseu-doinverse approach similar to the procedure discussedin Section 2.3.

The dynamic stiffness matrix Za of an exponentiallydamped system can be expressed as

Za(ω) = −ω2M + iω(C0 + C1μ1

μ1 + iω) + K (68)

5 NUMERICAL EXAMPLE

Consider a reinforced concrete cantilever beam as anexample to illustrate the feasibility of the proposedmethod. The beam’s geometric dimension is L × B× H = 2 m × 0.2 m × 0.2 m and its elastic mod-ulus and density are E = 3.0 × 1010 N/m2 and ρ =2.5 × 103 kg/m3, respectively. Bernoulli beam elementmodel is used. Using the accurate dynamic iterativeconvergence method proposed by Friswell et al. (1995,1998) to eliminate the rotational DOF, 10 × 10 massand stiffness matrices that match the measured testingnumber of DOF are obtained, see Figure 6. Table 1shows the condensation results for 10 undamped naturalfrequencies.

Assume that the damping matrices are proportionalto the mass and stiffness matrix, respectively, that isC0 = a0M and C1 = a1K. For the first two-order modes,the damping ratios are 3% and 1.5%. The real damp-ing matrices of the beam can be obtained through theproportional coefficients a0 and a1 that are calculatedby Equation (52) for μ1→�. Assume that the damp-ing kernel function of the beam has the form of Equa-tion (43) where the relaxation factor μ0→� and μ1

awaits identification. The Gaussian white noise is ap-plied on each DOF, respectively, to get the responsesof the 1st DOF of the system by using the FFT methodproposed by Pan and Wang (2013), then 10 FRFscan be calculated for the model system, treated like


Table 1Condensation of 10 undamped natural frequencies

Modes 1 2 3 4 5 6 7 8 9 10

FEM (Hz) 27.98 175.35 491.10 963.02 1594.4 2388.7 3351.4 4481.0 5805.5 7217.0Convergence (Hz) 27.98 175.35 491.10 963.02 1594.4 2388.7 3351.4 4481.0 5805.5 7217.0

Table 2Calculated values of γ 1

Original γ 1 0.02 0.05 0.1 0.2 0.3 0.4 0.5 1 2 3 5 7

First three modes for γ 1 0.156 0.163 0.184 0.252 0.338 0.431 0.527 1.028 2.051 3.075 5.121 7.172Relative error (%) 680 226 84 26 12.7 7.75 5.4 2.8 2.6 2.5 2.4 2.5First four modes for γ 1 0.16 0.155 0.133 0.124 0.257 0.374 0.486 1.024 2.061 3.082 5.11 7.159Relative error (%) 700 210 33 −38 −14.3 −6.5 −2.8 2.4 3.1 2.7 2.2 2.3First five modes for γ 1 0.16 0.155 0.133 0.124 0.257 0.374 0.486 1.024 2.061 3.082 5.11 7.159Relative error (%) 700 210 33 −38 −14.3 −6.5 −2.8 2.4 3.1 2.7 2.2 2.3

Table 3Influence of γ 1 on the damping ratios

γ 1 0.02 0.05 0.1 0.2 0.3 0.4 0.5 1 2 3 5 7

ξ ex,3 3.10 3.10 3.09 3.08 3.05 3.01 2.97 2.64 1.86 1.27 0.69 0.46ξ 3 3.09 3.09 3.09 3.07 3.05 3.02 2.98 2.65 1.83 1.24 0.67 0.45Relative error (%) −0.3 −0.3 0.0 −0.3 0.0 0.3 0.3 0.4 −1.6 −2.4 −2.9 −2.2

ξ ex,4 5.84 5.83 5.80 5.68 5.50 5.26 4.98 3.46 1.59 0.87 0.39 0.24ξ 4 5.94 5.95 5.93 5.83 5.66 5.41 5.10 3.42 1.52 0.83 0.38 0.24Relative error (%) 1.8 2.0 2.3 2.7 2.9 2.8 2.5 −1.1 −4.6 −4.4 −3.1 −2.1

ξ ex,5 9.57 9.53 9.40 8.90 8.17 7.33 6.48 3.31 1.14 0.57 0.24 0.15ξ 5 10.30 10.32 10.25 9.75 8.87 7.80 6.72 3.12 1.06 0.54 0.24 0.15Relative error (%) 7.6 8.2 9.0 9.6 8.6 6.4 3.7 −5.7 −7.0 −5.4 −3.3 −2.2

Note: 1. ξex is the theoretical damping ratio of exponential damping model; ξ is the measured damping ratio.2. The third-, fourth-, and fifth-order theoretical damping ratios of viscous damping model are: ξv,3 = 3.10%, ξv,4 = 5.84%, and ξv,5 = 9.58%.

experimental data and used for identifying damping ra-tios by the procedure described in Section 2.

From the above discussion, with the γ 1 increasing, thecantilever beam may exhibit different damping proper-ties. When the value of γ 1 is smaller, the damping ofthe beam is close to viscous damping, otherwise, whenthe value of γ 1 is larger, the beam will exhibit nonvis-cous damping properties. For different original γ 1, iter-ative calculations are conducted by using the first three,four, and five modes, respectively, to study the calcula-tion accuracy and applicable conditions of the proposedmethod. Iterative calculation values of γ 1 are shown inTable 2. The measured damping ratios and the theoret-ical damping ratios calculated by two different dampingmodels, for different original values of γ 1, are shown inTable 3.

For the case of using the first three-order modes, theresults are drawn in Figure 7. As seen from Figure 7,

-100

0

100

200

300

400

500

600

700

0

1

2

3

4

5

6

7

8

1

Original value Iterative value

20.4 7530.50.30.20.10.050.02 1

Itera

tive

valu

e (%

)Relative error (%)

Fig. 7. Iterative calculation of γ 1 values for first three modes.

with the increase of γ 1, the relative error of the itera-tive calculation results obviously displays an exponen-tial trend of decrease. It is found that if γ 1 � 0.3, the

12 Pan & Wang

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050Viscous damping v,3

Exponential damping ex,3

Measured 3

20.4 7530.50.30.20.10.050.02 1

Fig. 8. Comparison between theoretical and measured valuesof the third-order damping ratios in different γ 1.

iterative method proposed in the article can estimatethe relaxation factor μ1(or γ 1) more accurately. Whenγ 1 = 0.3, the relative error of the iterative calculation re-sults is 12.7% and the relative error is at around 2.5%,for γ 1 in 1–7 range, see Table 2. However, with thesmaller γ 1, the relative error is larger, which is 26 for γ 1

= 0.2 and even reaches 680% for γ 1 = 0.02. This is prob-ably because when γ 1 is smaller, the nonviscous damp-ing behavior of the structure is not significant and thetheoretical values of the third-order damping ratio forboth damping models, ξ ex,3 and ξ v,3, are both nearly thesame with the measured ξ 3 = 3.09 as seen in Figure 8and Table 3. The result of μ2

1 calculated by the rightside of Equation (55) is not accurate enough, appearingeven negative. Thus, the iterative calculation with thethird-order damping ratios cannot obtain the true μ1.To ensure that square root calculation of Equation (55)does not appear imaginary, we take the absolute valueof the right side of Equation (55), for uniform. When γ 1

continues to reduce to 0.02, from the above discussion,the exponential damping model has been getting closeto viscous damping model, and the identification resultsobtained by using the two kinds of damping models arebasically no different from each other. The structure isnow considered to perform a viscous damping behavior,and a viscous damping model can be directly applied fordynamic response analysis.

To further discuss the applicability of the proposedmethod, the following results are shown in Figures 9 and10, respectively, when the first four-order and five-ordermodes are used for analysis.

From Figures 9 and 10, it can be found that when thefirst four-order and five-order modes are selected for it-eration analyses, with the increase of γ 1, the calculationresults have the similar change trends with those ob-tained by only taking the first three-order modes. When

-100

0

100

200

300

400

500

600

700

0

1

2

3

4

5

6

7

8


20.4 7530.50.30.20.10.050.02 1

Rel

ativ

e er

ror

(%)

Relative error (%)

Fig. 9. Calculated γ 1 by using first four modes.

-100

0

100

200

300

400

500

600

700

800

900

0

1

2

3

4

5

6

7

8

9

10


20.4 7530.50.30.20.10.050.02 1

Rel

ativ

e er

ror (

%)

Relative error (%)

Fig. 10. Calculated γ 1 by using five modes.

γ 1 is in the range of 1–7, the error is smaller, at around2%, as seen in Table 2. However, compared with the re-sults by only using the first three modes, the accuracyof the results by using higher order modes is not ob-viously improved, but reduced in some identified val-ues of γ 1. This is because the iterative calculation er-ror of γ 1(or μ1) mainly comes from the identificationerrors of damping ratios. From Table 3, the identifieddamping ratios obtained by the method depicted in Sec-tion 2 have certain errors compared with the theoret-ical values of the exponential damping. These errorsare even bigger in identifying higher order damping ra-tios. For example, when γ 1 = 1, the errors of identifiedfourth and fifth damping ratios are −1.1% and −5.7%,respectively, while the error of the third damping ratiois only 0.4%. Therefore, when the structure performs astronger nonviscous damping behavior, the first three-order modes are appropriate for the iteration calcula-tion to obtain more accurate value of γ 1 (or μ1).

It might be thought that a useful check on the accu-racy of the proposed method could be made by com-paring the “measured” and updated FRFs, depicted inSection 4.2. The updated parameters are considered,respectively, as: density ρ and elastic modulus E for


0 100 200 300 400 500 600 700-150

-120

-90

-60

-30

0

30A

mpl

itude

(dB

)

Frequency (Hz)(a)

0 100 200 300 400 500 600 700-150

-120

-90

-60

-30

0

30

Am

plitu

de (d

B)

Frequency (Hz)(b)

(c)0 100 200 300 400 500 600 700

-150

-120

-90

-60

-30

0

30

Am

plitu

de (d

B)

Frequency (Hz)

Fig. 11. Amplitude plots of typical FRFs: (a) node 1; (b)node 3; and (c) node 7; initial analytical FRF;

simulated experimental FRF; updated FRF using viscousdamping model; updated FRF using exponentialdamping model. (See color figure in online version.)

mass matrix and stiffness matrix, and the proportion co-efficients a0 and a1 for damping matrices. Assume thatthe established FEM has errors ρa = 0.8ρ and Ea =1.2E, that is, Ma = 0.8M and Ka = 1.2K. Figure 11 showsthe typical experimental and updated FRFs for part ofthe nodes of the beam under the case γ 1 = 3.5.

It is observed from Figure 11 that the presented FEMupdating method in Section 4.2 for exponential dampingmodel has very good convergence to the real value withhigh accuracy. Thus, the feasibility of the method is vali-dated. The updated density ρ = 2.503 × 103 kg/m3, elas-tic modulus E = 3.002 × 1010 N/m2, and correspondingestimated γ1 = 3.590 are closer to the true value. How-ever, there are errors on the results of updated FRFsfor the viscous damping model, especially larger in the

24

68

10

24

68

100

50

100

150

200

j-th DOF(a)k-th DOF

24

68

10

24

68

10-2

-1

0

1

2

x 105

j-th DOF(b)k-th DOF

Fig. 12. Updated exponential damping coefficient matrices(Ns/m). (a) Mass proportional C0. (b) Stiffness

proportional C1.

third-order mode. It shows that when the structure per-forms more apparent nonviscous damping behavior (forγ 1 = 3.5), viscous damping model cannot reflect realdamping characteristics of the structure.

The updated exponential damping coefficient matri-ces are shown in Figure 12. Both the updated mass andstiffness proportional damping matrices are consider-ably close to true values, because the maximum relativeerrors are only −0.03% and 1.18%, respectively.

6 CONCLUSIONS

A comprehensive work of damping parameter identifi-cation of exponentially damped linear systems has beenconducted. The application of the complex mode anal-ysis theory on the nonviscous damping model is dis-cussed first and its feasibility and limitation are exam-ined. It is found that the complex mode analysis method

14 Pan & Wang

and its damping identification procedure are not onlysuitable for the viscously damped systems but also forthe exponentially damped systems, whether the damp-ing matrices of systems are proportional or not. It canget the exact identification results only in the case thatthe complete set of modes is used. The identification er-ror will be larger when only the first few order modesare used. In addition, there might not be a reliableor available method for identifying relaxation factor atpresent.

A new effective iterative method for relaxation factoris proposed. The method is simple, direct, and compat-ible with conventional modal testing procedures. Themode natural frequencies and damping ratios are used,but the method does not require the full set of modaldata. When a structure performs a stronger nonvis-cous damping behavior, such as γ is bigger, the firstthree-order modes are appropriate for the iteration cal-culation to obtain more accurate value of relaxationfactor. Otherwise, when γ is smaller, the damping be-havior of structures is close to viscous, and a viscousdamping model can be directly applied for dynamic re-sponse analysis.

Finally, combined with the FRF-based modelupdating method, the FEM with such nonviscousdamping is updated. Suitable numerical examples areprovided to illustrate the proposed approach. Thepresented FEM updated method for the systems withexponential damping can predict accurately not onlynatural frequencies but also the FRFs of the systems.

ACKNOWLEDGMENTS

The financial support provided by the National Nat-ural Science Foundation of China with grant number(51278038) is gratefully acknowledged.

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iterative method for exponential damping identification

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