J. Wind Eng. Ind. Aerodyn. 99 (2011) 18–26

Contents lists available at ScienceDirect

Journal of Wind Engineeringand Industrial Aerodynamics

0167-61

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jweia

Wind load identification using wind tunnel test data by inverse analysis

Jae-Seung Hwang a, Ahsan Kareem b, Hongjin Kim c,n

a School of Architecture, Chonnam National University, Gwangju 500-757, Republic of Koreab Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, IN 46556, USAc School of Architecture & Civil Engineering, Kyungpook National University, Daegu 702-701, Republic of Korea

a r t i c l e i n f o

Article history:

Received 21 January 2009

Received in revised form

20 September 2010

Accepted 22 October 2010Available online 12 November 2010

Keywords:

Modal wind load

Across-wind

Load identification

Inverse problem

Kalman filter

Wind tunnel test

Aeroelastic model

05/$ - see front matter & 2010 Elsevier Ltd. A

016/j.jweia.2010.10.004

esponding author. Tel.: +82 53 950 7539; fax

ail address: hjk@knu.ac.kr (H. Kim).

a b s t r a c t

The method for modal wind load identification from across-wind load responses using Kalman filter is

presented and verified using the wind tunnel test data. The Kalman filter is utilized for the inverse

identification from limited measured responses and the closed-form of Kalman filter gain in modal space

is derived for different types of measured response solving the Riccati equation. The wind induced

responses used for the verification are measured responses from an aeroelastic wind tunnel test of a

rectangular shaped concrete chimney. The displacement responses of the top part of the model are

measured and used for the wind load identification, but the acceleration responses obtained by numerical

differentiation of displacement are also used in order to evaluate the effect of response type on the

identification result. It is found from the identification results that the proposed method identifies the

modal across-wind load from measured responses with quite accuracy and the acceleration response

yields more accurate wind load identification than displacement response.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The wind-induced force acting on a building varies dependingon the terrain condition, topographic condition, and surroundingbuildings as well as the shape of the building. Therefore, there existlimits on actual wind load estimation based on the wind loaddesign codes. The use of wind tunnel test is often recommended inorder to achieve more reliable structural performance in the designof tall buildings. The wind tunnel test has gained wide acceptancebecause it provides not only the design wind loads for a certainstructure but also key information for establishing and improvingthe design codes.

Recently, the measurement systems with enhanced reliabilityand durability have been developed and the effectiveness of sitemonitoring has been increasing. The monitoring system has beenintroduced to civil engineering starting from long bridges and tallbuildings, and the collected data has been applied to maintenance,performance evaluation, and disaster prevention (Campbell et al.,2005; Li et al., 2003). The actual measured data from sitemonitoring system is also useful for evaluating and modifyingthe wind tunnel test result. Although the wind tunnel test is a mostrefined method for predicting design wind loads of a structurecurrently, the actual behavior of the structure may differ from thetest result due to uncertainties and limitations of test. Accordingly,

ll rights reserved.

: +82 53 950 6590.

it is required to perform the research to minimize the discrepancybased on the measured data.

The force identification from the response of a structure is one ofthe typical inverse problems. The inverse problems have beenstudied and developed in three ways depending on the magnitudeand location of forces to be identified. First, studies on the estimationof unknown magnitude of force with known location have beenperformed and an example is to identify the impulsive loads actingon wheels of an airplane during landing (Williams and Jones, 1948).The second is on identifying the location and the magnitude of forceand examples include an effort to find the location and magnitude ofan impulsive loading acting on the input/output head of computerdisk (Briggs and Tse, 1992). The third is on the estimation of movingloads and an example is to obtain the moving vehicle load acting onthe bridge slab (Yang and Yau, 1997).

Recently, the authors proposed a procedure to identify themodal loads from the structural response (Hwang et al., 2009). Forthe identification of the modal loads, the Kalman filter is employedto estimate unmeasured responses required in the process ofdetermining modal loads. The effects of the type of responseand noise on the modal load estimation were evaluated in thefrequency domain through numerical analysis of a single-degree-of-freedom (SDOF) and a multi-degree-of-freedom (MDOF) sys-tems. It was observed that the acceleration response was relativelymore stable and robust in external load identification than otherresponse components and the noise was amplified at the highfrequency range for the displacement and velocity responses whilethe noise was not amplified in the identified external load in the

J.-S. Hwang et al. / J. Wind Eng. Ind. Aerodyn. 99 (2011) 18–26 19

case of acceleration response. Further, the results showed that themodal loads can be identified accurately from limited measure-ments of acceleration response on the basis of the assumption thatthe measured response can be transformed to the modal responseby a proper orthogonal decomposition.

In this study, the load identification method is further applied tothe modal wind load using a wind tunnel test data. The windinduced responses are the measured data from the aeroelastic windtunnel test of a rectangular shaped chimney, which is designed tosimulate the sway motions in the directions of strong and weakaxes in their first modes. The displacement responses of the toppart of the model are measured and used for the wind loadidentification, but the accelerations obtained by numericallydifferentiating displacements are also used in order to evaluatethe effect of response type on the identification result.

The structural responses used for identification of wind loadscan be displacement, velocity, and acceleration depending on themeasurement system, but always involves noise regardless of theresponse type. Further, the exact dynamic properties of the targetstructure are required for identifying the wind loads from theresponses through the inverse analysis, but different dynamicproperties to exact ones can be used due to modeling errors.Accordingly, the main objective of this study is to quantitativelyevaluate the effect of the limitation and errors in the inverseidentification of wind loads from measured data, and to verify theidentification method using the aeroelastic wind tunnel test data.The method presented in this paper can be further applied toevaluate the accuracy of wind tunnel test by comparing theidentified wind loads acting on a building using actually measuredresponses to the wind loads obtained from the test.

In the following sections, the analytical procedure is formulatedfor SDOF systems based on the assumption that the dynamicbehavior of the chimney is governed by the first mode response.The procedure, however, can be generalized for MDOF systemsusing the mode superposition as presented in Hwang et al. (2009).This assumption makes possible to derive the closed form ofKalman filter gain in modal space for different types of measuredresponse solving.

The external load estimation technique presented in this papercan be extended to the higher modes by decomposing themeasured experimental response of a structure into modalresponses using the mode decomposition methods such as properorthogonal decomposition (POD) method and independent com-ponent analysis (ICA) method. The modal loads of higher modes canbe estimated from the corresponding decomposed modalresponses. In this paper, however, the SDOF system and the firstmodal wind load are only considered because the measuredresponses from a wind tunnel test are mainly governed by thefirst mode response.

2. Modal wind load identification from structural responses

In this section, a procedure for identification of modal windloads from structural responses is briefly introduced. The Kalmanfilter is used for the inverse identification from limited measuredresponses (Liu et al., 2000; Hwang et al., 2009).

2.1. Wind load identification using Kalman filter

If the entire responses of a structure are known, the wind loadsacting on the structure can be computed as

f ¼Mn €uþCn _uþKnu ð1Þ

where f is an n�1 wind load vector, u is an n�1 displacementvector, and Mn, Cn, and Kn are n�n mass, damping, and stiffness

matrices, respectively. Since the measurement of entire responsesfor all degrees of freedom is not possible in practice, the estimationof responses from a few measured responses is required.

In order to estimate the unmeasured responses, the Kalmanfilter is utilized (Grewal and Andrews, 1993). For the estimation ofstructural responses using the Kalman filter, it is more advanta-geous to define the equations in state space. Transforming theequation of motion in Eq. (1) into a state space form yields

_z ¼AzþBf ð2aÞ

y¼ CzþDf þe ð2bÞ

where z is a 2n�1 state vector, y is a p�1 measured responsevector, A, B, C, and D are 2n�2n, 2n�n, p�2n, and p�n systemmatrices, respectively, and e is a p�1 noise vector where p is thenumber of measured response.

Using the Kalman filter and limited measured responses, y, theentire responses of a structure can be estimated as

_z¼ ðA�GCÞzþGy ð3Þ

where G is the Kalman filter gain and z is estimated states. Theestimated states from the measured response is then expressed as

z¼ ½u _u�T

ð4Þ

where u is an estimated displacement vector from measuredresponses y and is denoted differently to the actual displacementvector.

If the estimated states converge to the actual states, Eq. (2a) canbe re-written replacing actual states by estimated states as

_z¼ AzþBf ð5Þ

Using the pseudo-inverse in Eq. (5), the wind loads can be obtainedfrom the estimated states as

f ¼ Bþ ð _z�AzÞ ð6Þ

where B+ is the pseudo-inverse of the matrix B. From Eq. (3), thewind loads can also be obtained as

f ¼ BþGðy�CzÞ ð7Þ

It is noted from Eq. (7) that the wind loads can be identified directlyfrom limited measured responses, y, and estimated states, z, byutilization of the Kalman filter.

2.2. Modal wind load identification

For a structure such as chimney whose dynamic behavior isgoverned by the first mode response, the procedure describedabove can be simplified to identify the first modal wind load fromthe first mode response. If the effect of higher modes on thedynamic behavior is considerably large, each modal wind load canbe identified from the higher modal responses that can be obtainedfrom the structural responses utilizing mode decompositionmethods such as POD method (Liang et al., 2002).

The governing equation in modal space is

Mni €Z iþCni _ZiþKniZi ¼fTi f ¼ Fi ð8Þ

where Mni, Cni, and Kni are the modal mass, damping, and stiffness ofthe i-th mode, respectively, and Zi, fi, and Fi are the modaldisplacement, mode vector, and modal wind load of the i-th mode,respectively. In state-space form, the equation of motion for the i-thmode is given by

_wi ¼_Z i

€Z i

" #¼

0 1

�M�1ni Kni �M�1

ni Cni

" #Zi

_Z i

" #þ

0

M�1ni

" #Fi ð9aÞ

vi ¼ ½�M�1ni Kni�M�1

ni Cni�wiþM�1ni FiþBi ð9bÞ

J.-S. Hwang et al. / J. Wind Eng. Ind. Aerodyn. 99 (2011) 18–2620

where wi is a 2�1 state composed of modal displacement andvelocity of the i-th mode, vi is the modal response of the i-th modeobtained from the modal decomposition, and zi is the noise. In theabove state-space equation, the case of acceleration feedbackwhere modal acceleration is used as a measured response is onlypresented.

Simplifying Eq. (9) yields

_wi ¼ AiwiþBiFi ð10aÞ

vi ¼ C iwiþDiFþBi ð10bÞ

where Ai, Bi, Ci, and Di are 2�2, 2�1, 1�2, and 1�1 modal systemmatrices, respectively. Similarly to Eqs. (6) and (7), the modal windload can be identified from the estimated modal states _wi and wi as

Fi ¼ Bþi ð_wi�AiwiÞ ð11aÞ

Fi ¼ Bþi Giðvi�CiwiÞ ð11bÞ

where Bþi is the pseudo-inverse of the matrix Bi and Gi is the i-thmode Kalman filter gain. The variable wi in Eq. (11) is the estimatevariable of Kalman filter defined in modal space and can becalculated as

_wi ¼ ðAi�GiC iÞwiþGini ð12aÞ

vi ¼ C iwi ð12bÞ

The Kalman filter gain, Gi, in Eq. (11) can be obtained in closed-formanalogous to that of SDOF system. System matrices Ai and Bi inEq. (9a) can be re-written as

Ai ¼0 1

�o2i �2xioi

" #¼

0 1

�ki �di

" #ð13aÞ

Bi ¼0

1

� �ð13bÞ

where oi and xi are the modal frequency and damping ratio,respectively. ki and di in Eq. (13) are modal stiffness and dampingcoefficient, respectively, which are introduced to derive the Kalmanfilter gain in closed form. Unlike Eq (9a), the inverse term of modalmass is omitted in Eq. (13b) for the simplification of derivation, butthe modal wind load will be calculated by multiplying modal massduring the identification process. System matrices Ci and Di, whichare presented in Table 1, differ depending on the measuringresponse used for identification. The modal mass in matrix Di ofacceleration feedback in Table 1 is also omitted for simplification.

Using matrices Ci and Di given in Table 1, the optimal Kalmanfilter gain can be obtained solving the following Riccati equationgiven as

AiPiþPiATi �ðCiPþDiQ 1BT

i ÞTðDiQ 1DT

i þQ 2Þ�1ðCiPiþDiQ 1BT

i Þ

þBTi Q 1Bi ¼ 0 ð14Þ

where Pi is a Riccati matrix, which is a solution of the Riccatiequation, Q1 and Q2 are the covariance matrices of the external loadand noise, respectively, and 0 is a zero matrix. Using the solution ofthe Riccati equation, the optimal Kalman filter gain is obtained as

Gi ¼ ðBiQ 1DTi þPiC

Ti ÞðDiQ 1DT

i þQ 2Þ�1

ð15Þ

Table 1System matrices Ci and Di for different response types.

Response type Matrix Ci Matrix Di

Displacement feedback [10] [0]

Velocity feedback [01] [0]

Acceleration feedback ½�o2i �2xoi� [1]

Due to the fact that the covariance matrices of the external load andnoise are not known a priori, these are assumed to be

Q 1 ¼ E½eTi ei� ¼ I ð16aÞ

Q 2 ¼ E½BTi Bi� ¼ g ð16bÞ

where I is an identity matrix, ei ¼wi�wi, and g is a factor less thanone. The Riccati matrix Pi has the following form:

Pi ¼p1 p2

p2 p3

" #ð17Þ

Substituting Eqs. (16) and (17) and system matrices definedabove for the case of acceleration feedback into Eq. (14) yields

2p2 p3þpk

p3þpk 2pdÞ

" #�

1

1þgp2

k pkð1þpdÞ

pkð1þpdÞ ð1þpdÞ2

" #

þ0 0

0 1

� �¼

0 0

0 0

� �ð18Þ

where

pk ¼�p1ki�p2di ð19aÞ

pd ¼�p2ki�p3di ð19bÞ

From Eq. (18), three independent equations can be obtained andsolving those equations yields

p1 ¼d2

i

k2i

þgðgÞki

!p3þ

di

k2i

hðgÞ ð20aÞ

p2 ¼p2

3

2gð20bÞ

p3 ¼�digþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdigÞ2�2kighðgÞ

qki

ð20cÞ

where

gðgÞ ¼

ffiffiffiffiffiffiffiffiffiffigþ1

g

sð21aÞ

hðgÞ ¼ g�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðgþ1Þ

pð21bÞ

Substituting the above Riccati matrix into Eq. (15) yields theKalman filter gain Gi as

Gi ¼

p3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

gð1þgÞ

s

�g

1þgþ

ffiffiffiffiffiffiffiffiffiffig

1þg

r266664

377775 ð22Þ

Similarly to the case of acceleration feedback, the closed-formsof the Riccati matrix and Kalman filter gain for the case ofdisplacement feedback are obtained as

p1 ¼ x ð23aÞ

p2 ¼p2

1

2gð23bÞ

p3 ¼ kip1þdip2þp1p2

g ð23cÞ

Gi ¼

p1

ki

p2

g

2664

3775 ð23dÞ

Fig. 2. Plan and elevation of the aeroelastic model. (a) Plan and coordinate system

and (b) elevation.

J.-S. Hwang et al. / J. Wind Eng. Ind. Aerodyn. 99 (2011) 18–26 21

where x is the solution of the fourth-order polynomial given as

x4þ3digx3þ4g2ðd2i þk2

i Þx2þ8kidig3x�4g3 ¼ 0 ð24Þ

The closed-forms of the Riccati matrix and Kalman filter gain forthe case of velocity feedback can also be similarly obtained as

p1 ¼p3

kið25aÞ

p2 ¼ 0 ð25bÞ

p3 ¼�digþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þðdigÞ2

qð25cÞ

Gi ¼

p2

ki

p3

g

2664

3775 ð25dÞ

3. Verification of modal wind load identification

For the verification of modal wind load identification from mea-sured responses, the method is applied to the wind tunnel test dataperformed for a rectangular shaped concrete chimney (Fig. 1). Theheight of the chimney is 210 m. Table 2 presents the dynamicproperties of the chimney and those of aeroelastic model, which aredetermined applying the appropriate scaling law. The aeroelasticmodel used for the wind tunnel test was designed to simulate thesway motions in the directions of strong and weak axes in theirfirst modes.

The wind tunnel test of the aeroelastic model was conducted atthe Eiffel type atmospheric boundary layer wind tunnel of theHyundai Institute of Construction and Technology (Jang et al.,1997). The total length of the wind tunnel is 53 m and thedimension of the measuring part is 4.5 m (width)�2.5 m(height)�25 m (length).

Fig. 1. Wind tunnel test.

Table 2Properties of the aeroelastic model.

Property Original structure Scaling factor Scaled model

Height 210 m 1/200 1050 mm

Mass 21,744.9 ton (1/200)3 2718.1 g

Natural frequencies X-direction: 0.352 Hz 32.16 11.32 Hz

Y-direction: 0.245 Hz 39.71 9.73 Hz

Damping ratio 1% in each direction 1 1%

Since the shapes of the chimney and the site were symmetricalrespect to the X-axis, the tests were performed for 7 wind directionsat 301 intervals for the 01–1801 azimuth range. The wind speed wasincreased from 0.5 to 22.5 m/s with an interval of 0.5 m/s duringthe test. The measured response used in this study is the responseat the wind speed of 15.0 m/s.

The wind tunnel test was performed for the actual design andconstruction of a chimney in a typical manner using a displacementsensor only. The top displacement of the model was measuredusing an optical displacement sensor in order to minimize theeffect of sensor mass. A camera in the optical displacement sensorread the movement of a light-emitting-diode (LED) target installedon the top of the model and transformed the displacement into thevoltage signal. The measured response used in this study is thedisplacement response in the across-wind direction (Y-direction)when the wind blows in X-direction where the coordinate systemsand reference axes are defined in Fig. 2.

3.1. Numerical verification

The wind load acting on the structure is not readily obtainablefrom the wind tunnel test. Therefore, the numerical simulationusing the same structural parameters is performed for the numer-ical verification of the modal wind load identification. Through thenumerical verification, the effects of measured response type, noiseand modeling error on the identification results are investigated.The dynamic analysis is performed in the Y-direction and theacross-wind load is considered in the simulation.

The properties of the structure, noise and modal wind load usedfor the numerical simulation are summarized in Table 3. Based onthe assumption that the dynamic behavior of the chimney isgoverned by the first mode response, the modal mass, frequencies,and damping ratios of the first mode only are presented in the table.

Table 3Structural properties and applied wind load and noise.

Structure Mass 21,744.9 ton

Damping ratio 1% in X- and Y-directions

Natural frequency 0.352 Hz in X-direction

0.245 Hz in Y-direction

Noise in the

measurement

Type High-pass filtered white

noise

Filter bandwidth Greater than 30 Hz

Filter transfer

function

s2/s2+267s+35,531

Modal wind load Type Low-pass filtered white

noise

Filter bandwidth Less than 10 Hz

Filter transfer

function

3947.8/s2+88.9s+3947.8

Fig. 3. Correlogram between actual and identified wind loads for different

measured response types. (a) Displacement feedback, correlation coefficient¼0.81,

(b) velocity feedback, correlation coefficient¼0.97 and (c) Acceleration feedback,

correlation coefficient¼1.0.

J.-S. Hwang et al. / J. Wind Eng. Ind. Aerodyn. 99 (2011) 18–2622

The fundamental modal mass of 21,744.9 ton is obtained using thefundamental mode shape vector normalized in such a way that thetop of chimney is one. A low-pass filtered white noise is used for amodal wind load and a high-pass filtered white noise is used for anoise in the measurement. The corresponding transfer functions forthe filters are presented in Table 3.

The process to numerically verify the method for the identifica-tion of modal wind load from measured responses is summarizedas follows:

Step 1: Modal wind load and noise are generated using the givenlow and high-pass filters, respectively. The generated wind load isdenoted as the actual wind load.

Step 2: Modal responses subjected to the generated modal windload are obtained through dynamic analysis, and one response isselected for the inverse problem among displacement, velocity, andacceleration responses.

Step 3: The generated noise in Step 1 is added to the responsechosen in Step 2 and the modal wind load is identified using themethod presented in this study. The identified modal wind load isdenoted as estimated wind load. The noise level and the uncertaintyof dynamic properties such as modal damping and frequency errorsare considered in this step.

Step 4: Verification of the identification method is carried out bycomparing the estimated wind load to actual wind load in time andfrequency domains.

The numerical simulation of 600 s is performed with time stepsof 0.02 s. In Fig. 3, the correlograms between identified wind loadand actual one are presented for different types of measuredresponses. The results are obtained for each measurementresponses without adding noises. It can be seen that the correlationcoefficient of the acceleration feedback is a unit value, indicatingthat the modal wind load is identified accurately from accelerationresponses. The accuracy of identification decreases as the velocityand displacement responses are used.

The comparison of actual and estimated winds loads in time andfrequency domains are depicted in Fig. 4. For a clear comparison, thetime histories of wind load are presented in 2–5 s and the wind loadspectra are presented in 0–25 Hz only. It can be clearly seen that theestimated wind load using acceleration response matches well to theactual wind load while the estimated wind load using displacementresponse is smaller than that using acceleration response. Infrequency domain, the spectral value of wind load estimated usingdisplacement response is smaller than that using accelerationresponse in the entire frequency except near natural frequency.

The effect of noise in measurement on the identification ispresented in Figs. 5 and 6. Fig. 5 presents the correlograms betweenidentified wind load and the actual one for different magnitudes ofnoise. Since the acceleration feedback yields the most accurate

result, the acceleration response is used for the inverse identifica-tion. The maximum values of noise are scaled to be 10%, 20%, and 30%of the peak acceleration. Fig. 6 shows the comparison of wind loadspectra when the noise level is 30%. It is observed that the wind loadspectrum of estimated wind load differs to that of the actual one onlyin the frequency range greater than 20 Hz, where the effect of noise

2 2.5 3 3.5 4 4.5 5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 106

Time (sec)

Win

d lo

ad (N

)

Actual wind loadDisp. feedbackAcc. feedback

0 5 10 15 20 250

1

2

3

4

5

6x 1012

Frequency (Hz)

Win

d lo

ad s

pect

ra

Actual wind loadDisp. feedbackAcc. feedback

Fig. 4. Comparison of actual and identified winds loads for time and frequency

domains. (a) Time history and (b) wind load spectra.

Fig. 5. Correlograms between actual and identified wind loads for different

magnitudes of noise. (a) Noise level¼10%, correlation coefficient¼0.98, (b) noise

level¼20%, correlation coefficient¼0.92 and (c) noise level¼30%, correlation

coefficient¼0.85.

J.-S. Hwang et al. / J. Wind Eng. Ind. Aerodyn. 99 (2011) 18–26 23

is large. Therefore, the wind load can be estimated more accurately ifthe noise is removed from the response using a low-pass filter.

The effect of uncertainties in dynamic modeling is investigatedand the results are presented in Figs. 7 and 8. Generally, the naturalfrequency and damping ratio can be obtained from the vibration testusing the system identification method. However, these dynamicproperties are often evaluated with uncertainty. In Fig. 7, theestimated wind load is compared to the actual wind load whenthe error in the natural frequency modeling is +5%. The accelerationresponse is used for the inverse identification and no noise is addedto the response. From the correlogram in Fig. 7(a), it can be observedthat the estimated wind load matches well to the actual one. Infrequency domain, however, the wind load spectrum of the esti-mated wind load is close to that of actual wind except near thenatural frequency of the structure (0.245 Hz). The similar observa-tion can also be found when the error in the natural frequencymodeling is �5%, and thereby the results are not presented here.

Fig. 8 shows the correlogram between the actual and theestimated wind loads when the damping ratio is set to be 2.0%while the correct damping ratio is 1% as described in Table 3. As canbe seen in Fig. 8, the estimated wind load is very close to the actualone, yielding the correlation factor of almost one. The wind loadspectrum of the estimated wind load is also very similar to that ofactual wind load, and thereby the results are omitted.

3.2. Verification of modal wind load identification using wind

tunnel test

The verification of wind load identification method from thestructural responses obtained by the wind tunnel test is performedthrough the following process. Because the actual wind loadapplied to the chimney is not known unlike the numerical

Fig. 6. Comparison of wind load spectra when the noise level is 30%.

Fig. 7. Comparison of the estimated and actual wind loads when the error in the

natural frequency modeling is +5%. (a) Correlogram, correlation coefficient¼0.98

and (b) wind load spectra.

Fig. 8. Correlogram when the damping ratio is set to be the twice of correct damping

ratio; correlation coefficient¼0.99.

Fig. 9. Estimated wind loads from different response types. (a) Time histories of

modal wind load and (b) wind load spectra.

J.-S. Hwang et al. / J. Wind Eng. Ind. Aerodyn. 99 (2011) 18–2624

verification, it is not possible to directly compare the estimatedwind load to the actual one. Therefore, the method is indirectlyverified comparing the measured responses to estimated ones,which are numerically obtained applying the identified modalwind load.

J.-S. Hwang et al. / J. Wind Eng. Ind. Aerodyn. 99 (2011) 18–26 25

Step 1: The displacement responses of the top of the model aremeasured from the wind tunnel test, and acceleration responsesare calculated by numerically differentiating displacements.During the wind tunnel test, the displacement response is onlymeasured, and thereby the acceleration is calculated from thedisplacement. The noise in the measured response is removedusing a low-pass filter.

Step 2: The modal wind load is identified from the responsesusing the method presented in this study. The only displacementand acceleration responses are used in this study for thesimplification.

Step 3: A dynamic analysis is performed applying the identifiedwind load, and the resulting responses are compared to themeasured ones.

Fig. 9 presents the estimated wind loads from displacement andacceleration responses in time and frequency domains. Similar tothe results of numerical verification, the modal wind load identifiedfrom displacement response is smaller than that identified fromacceleration response. Comparison of the measured responses toestimated ones, which are numerically obtained through a dynamicanalysis applying the identified modal wind load, is presented inFig. 10. In Fig. 10(a), acceleration time histories are shown in19–21 s for better comparison. It is observed that the acceleration

Fig. 10. Comparison of estimated and measured accelerations. (a) Acceleration time

histories and (b) acceleration power spectra.

feedback yields closer results to the measured responses comparedto displacement feedback. The same observation can be made fromthe frequency domain comparison presented in Fig. 10(b).

Fig. 11 shows the correlogram between the estimated and themeasured acceleration for different measured response types. Thecorrelogram in Fig. 11(a) is obtained using the acceleration feed-back and the resulting correlation factor is 0.92. Fig. 11(b) isobtained using the displacement feedback and the resultingcorrelation factor is 0.77. From the results, it is noted that theidentification from the acceleration response yields more accuratewind load identification than displacement response.

In addition, the measured displacement response is comparedin Fig. 12 to the numerically calculated ones using the identifiedwind loads as inputs. The comparison is made in time domain inFig. 12(a) and in frequency domain in Fig. 12(b). Since the unfilteredmeasured displacement time history is almost identical to thefiltered one, it is not included in Fig. 12(a). It can be seen fromFig. 12(a) that the use of acceleration yields a closer displacementresponse to the measured one. It can be also noticed from Fig. 12(b)that the result using the acceleration response as the primaryresponse parameter matches well to the unfiltered original mea-sured displacement in the most frequency range except the highfrequency range where the effect of noise is large. Especially, the

Fig. 11. Correlograms between estimated and measured acceleration for different

measured response types. (a) Acceleration feedback, correlation factor¼0.92 and

(b) displacement feedback, correlation factor¼0.77.

Fig. 12. Comparison of estimated and measured displacements.

J.-S. Hwang et al. / J. Wind Eng. Ind. Aerodyn. 99 (2011) 18–2626

acceleration feedback yields the almost identical displacementresult near the first natural frequency in which the structuralvibration concentrates.

4. Conclusion

The modal wind load identification from structural responsesusing Kalman filter is presented. The Kalman filter is utilized for theidentification from limited measured responses. The closed-form of

Kalman filter gain in modal space is derived for different types ofmeasured response solving the Riccati equation.

The method is applied to an aeroelastic model of chimney andverified using numerically generated wind load and wind tunneltest responses. The results of numerical verification indicate thatthe modal wind load identified from acceleration response matchesthe most to the actual wind load. The accuracy of identification isaffected by the level of noise in the measured response, but the useof low-pass filter for the noise removal can improve the results. Theeffect of uncertainties in dynamic modeling is investigated and it isfound that the discrepancy in frequency affects the identificationresult more than the damping ratio.

From the verification using wind tunnel test responses, it isfound that the acceleration response yields more accurate windload identification than displacement response. It is also shownthat the acceleration obtained by numerically differentiatingmeasured displacement can also be used for reliable results whenthe acceleration response is not measured. The method is verifiedcomparing the measured responses to estimated ones, which arenumerically obtained applying the identified modal wind load. Theresults indicate that the proposed method identifies the modalwind load from measured responses with quite accuracy.

Acknowledgement

This work was supported by the Grant of the Korean Ministry ofEducation, Science and Technology (The Regional Core ResearchProgram/Biohousing Research Institute).

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