13th World Conference on Earthquake Engineering Vancouver, B.C., Canada

August 1-6, 2004 Paper No. 3068

APPLICATION OF CROSS SPECTRUM BASED MODAL IDENTIFICATION

TO OUTPUT-ONLY RECORDS OF AMBIENT VIBRATION

Kenji KANAZAWA1

SUMMARY A cross spectrum based modal identification technique, which has been developed by the authors, is examined on the applicability to ambient vibration records of buildings, and its theoretical background is also reviewed. The modal identification technique is extended from the spectral estimation scheme based on an autoregressive moving-average and moving-average (ARMAMA) model, which has following advantages: (1) Owing to cross spectrum based technique, it is not necessary to presume relationships between inputs and outputs; (2) Modal characteristics of a building can be identified accurately, even using records contaminated with local noises due to mechanical vibration or local member vibration.

INTRODUCTION Vibration tests of existing buildings have been conducted not only to investigate their dynamic behavior but to confirm the validity of their design, construction and maintenance schemes, and the results have been reflected to the improvement of structural design against large earthquakes. Of vibration tests, ambient vibration test (AVT) is to measure very small structural dynamic response driven by various kind of vibration sources in normal operation, and one can estimate the dynamic characteristics of the building in elastic range. The main advantage of AVT is that it can be conducted at low cost and data can be obtained more frequently compared to other vibration tests: e.g. force-induced test or dynamic response observation against earthquake and wind loads. In accordance with recent developments of measuring technique and equipment, the advantage of AVT is widely spreading than before. To take advantage of the measuring operation, the AVT will be hopefully applied for structural health monitoring to evaluate existing structural capability of buildings(e.g.: Rytter[1], Farrar et al. [2], Housner et al. [3]). In evaluating dynamic characteristics of the building, modal parameter identification and structural model identification are powerful tools to determine the mathematical model or the physical parameters of the building from measured records. In the earthquake engineering field, the system identification techniques based on input-output relations have so far been developed for earthquake observation records of a building under the assumption that the records at the ground floor or near the ground surface are

1 Research Engineer, Central Research Institute of Electric Power Industry, Japan.

Email: kanazawa@criepi.denken.or.jp

considered as the structural inputs to the building (e.g.: Hoshiya and Saito[4], Safak[5]). For the AVT records, however, this assumption is not always valid, because the ambient vibration of the building is excited by wind forces as well as ground motion (Izumi et al.[6]), and because the complete records of the wind forces cannot be measured in practice. Furthermore, it is difficult to measure even the inputs from the ground to the building accurately, especially when the effect of soil-structure interaction cannot be ignored (Tobita et al.[7]). Thus the system identification techniques based on input-output relations have limitation for using the AVT record. Some alternative approaches have been proposed recently, which is premised on using only output records without the use of the input records(e.g.: Peeters and De Roeck[8]). Of the output-only modal identification, one approach is to consider measured correlation functions as the free vibration signature of the building, and to employ the modal identification techniques for free vibration: e.g. Ibrahim time domain method(Ibrahim[9], Ibrahim et al.[10][11]), eigensystem realization algorithm(Juang and Pappa[12]). Another approach is to assume that the input of the building is modeled as a stochastic process (mostly white noise process), and to employ the modal identification schemes based on input-output relations: e.g. subspace-based algorithm or stochastic subspace identification (Larimore[13], Van Overschee and Moor[14]), frequency domain decomposition(Brincker et al.[15]). The authors have proposed output-only modal identification techniques which differ from the aforementioned approaches, in that they are based on cross- and auto-power spectral densities(PSDs) of linear difference models(Kanazawa et al.[16][17], Kanazawa[18]). In the proposed techniques the cross- and auto-PSDs can be modeled as an autoregressive moving-average model (ARMA model, Kanazawa et al.[16]) or an autoregressive moving-average and moving-average model (ARMAMA model, Kanazawa et al.[17]). Especially, the ARMAMA model as treated in the paper, is a new linear difference model composed of MA terms added to an ARMA model. By using the spectral analysis of the ARMAMA models, one can clearly separate correlated and uncorrelated components of the vibration records of the building[19]. Moreover, because of the advantage of the spectral analysis, the proposed output-only modal identification technique gives accurate modal parameters from the correlated components of the vibration records. In this paper, the author states the theoretical background of the output-only modal identification technique based on the cross spectrum of the ARMAMA model. And, to investigate the applicability of the modal identification technique, ambient vibration records of large coal silos are analyzed.

THEORETICAL BACKGROUND OF CROSS SPECTRUM BASED MODAL IDENTIFICATION

The cross spectral based modal identification technique is composed of cross spectrum estimation and its application to modal identification. In the below description of the theoretical background, at first, the spectral estimation scheme using the ARMAMA model is reviewed briefly. The detail of the spectral estimation theory is found in references[17][19]. Then, after deriving the cross spectrum expression for dynamic responses in the general physical model, the author will present the theory of the modal identification technique by using relation of cross spectrum between the ARMAMA model and the physical model. Cross spectrum form of ARMAMA model[17][19] Let )(tx and )(ty be measured responses on stationary process, the ARMAMA models are defined in discrete time t as

)()(

)()(

)(

)()(

1

1

1

1

tezC

zDte

zA

zBtx x

x

x

x

x−

−

−

−+= )1(

)()(

)()(

)(

)()(

1

1

1

1

tezC

zDte

zA

zBty y

y

y

y

y

−

−

−

−

+= )2(

where )(te , )(tex and )(tey are mutually independent white noise sequences with unit variances; 1−z is

the unit-delay operator; )( 1−zAx , )( 1−zAy , )( 1−zCx and )( 1−zC y are the AR operators as

∑=

−− =n

j

jxx zjazA

0

1 )()( , ∑=

−− =n

j

jyy zjazA

0

1 )()( , ∑=

−− =n

j

jxx zjczC

0

1 )()( , ∑=

−− =n

j

jyy zjczC

0

1 )()( ; )3(

1)0( =xa , 1)0( =ya , 1)0( =xc , 1)0( =yc ; )4(

)( 1−zBx , )( 1−zBy , )( 1−zDx and )( 1−zDy are the MA operators as

∑=

−− =n

j

jxx zjbzB

0

1 )()( , ∑=

−− =n

j

jyy zjbzB

0

1 )()( , ∑=

−− =n

j

jxx zjdzD

0

1 )()( , ∑=

−− =n

j

jyy zjdzD

0

1 )()( . )5(

In equations(1) and (2), )(tx and )(ty are driven by the common input )(te , therefore, the first terms of the right-hand sides in each equation can represent correlated components between )(tx and )(ty . On the other hand, the second terms of the right-hand sides in each equation relate uncorrelated components between )(tx and )(ty . Equations (1) and (2) can be rewritten in the linear difference forms as

∑∑∑∑∑∑ −−+−−=−−j k

xxxj k

xxj k

xx kjtekajdkjtekbjckjtxkajc )()()()()()()()()( )6(

∑∑∑∑∑∑ −−+−−=−−j k

yyyj k

yyj k

yy kjtekajdkjtekbjckjtykajc )()()()()()()()()( )7(

Each model in equations (6) and (7) consists of one AR term and two MA terms, therefore, the authors proposed to call them "Auto-Regressive Moving-Average and Moving-Average (ARMAMA)" models. The cross-PSD )(tS xy between )(tx and )(ty , and the auto-PSD )(tS xx of )(tx which correspond to

equations (1) and (2) can be obtained as follows:

)()(

)()()(

1

11

−

−− =

zAzA

zBzBzS

yx

yxxy )8(

)()(

)()(

)()(

)()()(

1

1

1

11

−

−

−

−− +=

zCzC

zDzD

zAzA

zBzBzS

xx

xx

xx

xxxx )9(

It is noted that the auto-PSD of )(ty is similarly expressed in equation (9). The cross- and auto-PSD forms in equations (8) and (9) can be simplified as follows

)()(

)(

)(1

1−

−=

−

−∑

=zAzA

zlR

zSyx

n

nl

lrs

xy )10(

)()()()(

)(

)(11

2

21−−

−=

−

−∑

=zCzCzAzA

zlR

zSxxxx

n

nl

lww

xx )11(

where

∑∑= =

−+=n

k

n

jxyyxrs jklRjakalR

0 0

)()()()( , )12(

∑∑∑∑= = = =

−++−=n

k

n

j

n

k

n

hxxxxxxww hkjilRhakajciclR

0 0 0 0

)()()()()()( ; )13(

)(lRxy is the cross-correlation function of )(tx and )(ty ; )(lRxx is the auto-correlation function of )(tx .

The above derivation from equations (8) and (9) to (10) and (11) is similar to the auto-PSD form of ARMA model proposed by Kinkel et al.[20] and Kaveh[21]. In equations (10) through (13), the PSDs are expressed without the use of the MA coefficients. By calculating the AR coefficients and the correlation functions from the measured responses )(tx and )(ty , one can estimate the cross- and auto-PSDs given by equations (10) through (13). Here, the AR coefficients can be calculated by using the Yule-Walker equations below:

1);()()(1

+≥−−=−∑=

nllRlkRjan

kxyxyx )14(

1);()()(1

+≥−=−∑=

nllRklRjan

kxyxyy )15(

and

12;)()()()()(01 0

+≥−−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−+ ∑∑ ∑

== =nllkRkajclkjRka

n

kxxxx

n

j

n

kxxx )16(

The AR coefficients )(kax and )(ka y can be calculated by sufficient sets of equations (14) and (15),

respectively. After the AR coefficients )(kax are obtained, the AR coefficients )(kcx can be also calculated by a sufficient set of equation (16). In the below examples, the author solved the sets of equations (14) through (16) by the singular value decomposition[23].

Cross spectrum form of MDOF structures[16] The dynamic behavior of a multi-degree-of-freedom(MDOF) structure consisting of n masses connected through springs and dampers is described by following matrix differential equation:

)}({)}(]{[)}(]{[)}(]{[ tftxKtxCtxM =++ &&& )17( where ][M , ][C and ][K are the mass, damping and stiffness matrices; )}({ tx and )}({ tf are the displacement and external force vectors. Here, the eigenvectors matrix of the left-hand side in equation

(17) is denoted by ][ kiφ , and let us assume the proportional damping; namely, ]][[][ Tkiki C φφ is a

diagonal matrix. Then equation (17) can be rewritten as the independent set of equations on modal coordinates:

nim

tttt

i

iiiiiii ,...,2,1;

)()()(2)( 2 =

Γ=++ ξωξωςξ &&& )18(

where iξ is the i -th modal displacement; iς and iω are the modal damping factor and the modal circular

frequency associated with iξ ; im is the modal mass of the i -th mode; iΓ is the i -th modal force given by

∑=Γk

kkii tft )()( φ )19(

Let )(thi be the i -th impulse acceleration response in equation (18), two acceleration responses )(txl&&

and )(txk&& at the k -th and l -th nodes of the MDOF structure become

∑ ∫ −Γ=i

iii

lil dth

mtx ηηηφ

)()()(&& and ∑ ∫ −Γ=i

iii

kik dth

mtx ηηηφ

)()()(&& . )20(

By taking the expectation of the product of )(txl&& and )(txk&& given by equation (20), and by applying the

Wienner-Khinchin theorem, the cross-PSD )(sSkl can be obtained for the two acceleration responses:

)()()()( * sSsHsHmm

sS jijii j j

kj

i

lilk ΓΓ⋅⋅=∑∑

φφ )21(

where )(sH j is the pulse transfer function associated with )(th j , which is given by

22

2

2)(

jjjj

ss

ssH

ωως ++= ; )22(

)(sS jiΓΓ is the cross-PSD of the modal forces iΓ and jΓ ; s denotes the Laplace parameter; * denotes the

complex conjugate operator. Here, the poles of )(sH j can be shown as

21ˆ jjjjj is ςωως −+−= and 2* 1ˆ jjjjj is ςωως −−−= )23(

Assuming that both )(sS jiΓΓ and *)(sH i )(sH j have no common poles, then the cross-PSDs in

equations (21) can be rewritten in the below partial fraction form:

)(ˆ

ˆ

ˆ

ˆ

ˆˆ)( 0*

*

*

*

sBss

B

mss

B

mss

B

mss

B

msS lk

j j

lj

j

kj

j

lj

j

kj

j

kj

j

lj

j

kj

j

ljlk +

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−+

−+

++

+=∑

φφφφ )24(

where

∑ −=ΓΓ+×=i

ssijijj

i

kikj

jsSsHsHss

mB

ˆ

* )()()()ˆ(φ

, ∑ =ΓΓ−×=i

ssjijij

i

lilj

jsSsHsHss

mB

ˆ

* )()()()ˆ(ˆ φ )25(

and )(0 sBlk is the remainder function. As shown in equation (24), the cross PSD of the acceleration responses of the MDOF mechanical structure can be expressed as fractional functions in term of its eigenvalues js . By letting kl = in

equation (24), the auto-PSD can be also obtained. In the above derivation, the author derived the cross-PSD of the relative acceleration responses of the structure with proportional damping. In the case of absolute acceleration responses of a structure with non-proportional damping, the cross-PSD can be expressed as the similar form in equation (24)[17]. Modal identification technique To relate the cross-PSD of the ARMAMA model to that of physical model, equation (10) is translated into the Laplace plane:

0121

*

*

*

*

)()()()( xy

n

mj yj

j

xj

jm

j yj

j

yj

j

xj

j

xj

jxy S

sssssssssssssS +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

−−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

−+

−−+

−−= ∑∑

+==

γβγγββ )26(

where xjs and yjs are the poles in the Laplace plane, which are calculated by solving 0)( 1 =−zAx and

0)( 1 =−zAy , and by translating the solutions from the z-plane into the Laplace plane; jβ and jγ are the

residues associated with the poles xjs− and yjs given by

1

11)(

−=

−− −

=

xjzz

xjxyj z

zzzSβ and

yjzz

yjxyj z

zzzS

=

− −= )( 1γ )27(

in which xjz and yjz are the solutions in 0)( 1 =−zAx and 0)( 1 =−zAy , respectively; 0xyS denotes the

constant term. In equation (26), the first summation denotes m pairs of complex conjugate components, that is, the first summation can be associated with vibration characteristics, because the poles of each modal characteristics appear as a pair of complex conjugates. Let us relate the first summation in equation (26) to

the cross-PSD form of the physical model in equation (24). According to equation (24), even when any responses on the physical model are selected as the two records, the cross-PSDs always have common

poles corresponding to the eigenvalues such as js− , *ˆ js− , js and *ˆ js . On the other hand, in equation

(26) the first summation of the cross-PSDs cannot have common poles even when )(tx and )(ty are selected to the responses in the building. To overcome the discrepancy, it is assumed that the AR

operators )( 1−zAx and )( 1−zAy are replaced by the same AR operators )( 1−zA in the cross-PSD

estimation, and also assumed that the AR operator )( 1−zA is common among all cross-PSDs of the responses of the objective building. These assumptions must be reasonable because the AR operators for the correlated components are represented as characteristics of eigenvalues (e.g.: natural frequency, damping factor) of a mechanical structure, and also because the eigenvalues are common at all the nodal

points of the structure. By introducing these assumptions, let js be the poles in 0)( 1 =−zA , the cross-

PSD of the ARMAMA model can be rewritten as

0121

*

*

*

*

)()()()( xy

n

mj j

j

j

jm

j j

j

j

j

j

j

j

jxy S

sssssssssssssS +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

−−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

−+

−−+

−−= ∑∑

+==

γβγγββ )28(

Similarly the auto-PSD in equation (11) can be rearranged the expression as

)()()()(

)( 0121

*

*

*

*

sSssssssssssss

sS xx

n

mj j

j

j

jm

j j

j

j

j

j

j

j

jxx +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

′+

−−

′+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

′+

−

′+

−−

′+

−−

′= ∑∑

+==

γβγγββ )29(

where )(0 sS xx is the remainder function of s , which contains the components for the poles in

0)( 1 =−zCx . Here, let us suppose that )(tx is fixed to a response at the node l as a reference record and )(ty is a response at the node k ( nk ,...,2,1= ), then n -1 cross-PSDs and an auto-PSD can be obtained as the forms in equations (28) and (29). By equating equations (28) and (29) with equation (24), the below relation of the residues can be found:

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

nj

j

j

j

lj

lnj

lj

lj

m

B

φ

φφ

γ

γγ

MM

2

12

1

ˆ )30(

where lkjγ denotes the residue for the j th mode in the PSD between the node l and k .

As shown in equation (30) the j -th residue vectors are proportional to the j -th mode vectors in physical

model, therefore, the modal vectors in physical model can be estimated from the ratios of the residue lkjγ .

It should be noted that such modal vectors are independent of external forces acting to the objective building. Thus the modal properties can be determined without the use of input records. The natural

circular frequency and damping factor associated with the modal vectors }{ lkjγ can be derived from using

equation (23):

jj s=ω )31(

jj sargcos−=ς )32(

APPLICATIONS TO AMBIENT VIBRATION RECORDS

Specimens of the silos The silos are constructed in a coal-fuel power station, which can store the coal up to 70,000 tons. Figure 1 shows an overview of the silos, in which the AVTs were conducted at the two silos. As shown in the figure, the two silos are called Silo A and Silo B, respectively. The silos are reinforced concrete (RC) cylindrical shell structures with the wall height of 65 meters and inner diameter of 46 meters. The thickness of the RC wall changes linearly with height, which is 1.3 meters at the bottom, and is 0.5 meters at the top. Ambient vibration test To evaluate changes of modal properties with the amount of the stored coal, the AVTs of the two silos were conducted in different coal storage states. For Silo A, two AVTs were conducted i.e., in the empty and full-stored state. For Silo B, three AVTs were conducted in the empty state, in the one-third storage state and in the full-stored state. Figure 2 shows the deployment of three-component accelerometers. The sensors for Silo A were set on the bottom and top floors at every 30 degrees azimuthally along with the circumference of the shell wall. In addition to the arrangement similar to Silo A, for Silo B accelerometers were set on the east-side wall of the RC shell along a flight of stairs. In both cases, because the measuring points were more than the sensors available, the ambient vibration records of all measuring points were obtained by two successive measurements with replacing several accelerometers. In detail, two measurements were conducted in the south-side and the north-side for Silo A, or in the east-side and the west-side for Silo B, respectively, and the record of each measurement was obtained for one hour with the sampling time 200 Hz. Test results and discussion The modal properties of the two silos would be identified by applying the proposed cross spectrum based modal identification to the ambient vibration records. Figure 3 shows the results of the mode shapes on the roof and the associated natural frequencies and damping factors for Silo A in the full-stored state. As shown in the figure, six modal properties were obtained and some natural frequencies of them are closely spaced. The 1st and 2nd modes shown in Fig. 3(a) and (b) are the translational ones on the NS and EW directions, where the shell axial deformation is dominant along the height of the silo. The 3rd and 4th modes shown in Fig. 3(c) and (d) are both the 2nd oval shapes where the radial deformation is dominant on the roof, whereas both axes of the oval shape cross at the angle of 45 degree. The 5th mode shown in Fig. 3(e) is the 3rd oval shape. And the 6th mode shown in Fig. 3(f) is the translational one in NS direction, which is similar to the 1st mode shape, however, the motions of inner coal will be in opposite phase. As observed above, the proposed modal identification technique gives accurate model property estimates even when the closely spaced modes exist. Figure 4 also shows 3D spatial mode shapes of the 1st and 3rd modes, where the displayed amplitudes at the basement are magnified 20 times. As shown in the figures, the motions of the basement can be clearly captured with the deformations of soil-structure dynamic interaction effect phenomena.

Since the mode shapes can be clearly distinguished from one another in above ways, one can discuss the dynamic characteristics in more detail. Figure 5 is one of the example, which shows relationships between eigenvalues and the amount of stored coal. As shown in the figure, the natural frequency in every mode shape decreases with the increase of the amount of the stored coal, whereas the tendencies of decreasing ratios are clearly different from each other. Furthermore, the value of damping factors in the translational mode increase with the increase of the amount of the stored coal, but those in oval mode decrease.

CONCLUSION

The proposed cross spectrum based modal identification technique have been shown to estimate modal properties of a building from the output-only records such as those by ambient vibration test (AVT). The proposed technique has three major advantages as follows: 1. Owing to cross spectrum based technique, it is not necessary to presume relationships between

inputs and outputs before modal identification. 2. Dynamic characteristics of a building can be identified accurately, even using data contaminated

with local noises such as those by mechanical vibration or local members vibration. 3. Each modal property can be well identified even when the close-spaced modes exist. Especially, the first advantage of treating unknown inputs is effective to analyze the ambient vibration records. This effectiveness is proved throughout the examinations on the AVTs of the two large coal silos. Furthermore, conventional modal identification techniques using structural output only introduce the assumption that the structural inputs are white noise or wide-band process sequences: e.g., [9]-[15]. On the other hand, the proposed technique requires the assumption that all poles (eigenvalues) in the vibration model and the inputs (e.g.: external forces or ground accelerations) are different from each other. This assumption is less restrictive. The author also believes the proposed technique is effective to earthquake response records of mechanical structures, where the input-output based modal identification have been often employed. Expansion of the applicability to the earthquake engineering field will be one of the future work.

ACKNOWLEDGMENTS The authors are grateful to Mr. Katsuhiko Miyazumi of Shikoku Electric Power Co. Inc. for measuring assistance of the ambient vibration test of the large coal silos. And the author gratefully thanks Dr. Kazuta Hirata of CRIEPI for discussing the draft of the paper.

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Figure 1 Overview of coal silos. Figure 2 Sensor locations.

Figure 3 Results of the mode shapes on the roof of Silo A in the full-stored state.

(a) the 1st mode corresponded with Fig.4(a) (b) the 3rd mode corresponded with Fig.4(c)

Figure 4 The mode shapes of Silo A in the full-stored state.

Figure 5 Dependency of eigen-properties upon the amount of stored coal, determined by the proposed technique.