Computer-Aided Civil and Infrastructure Engineering 23 (2008) 373–388 Modal Identification Study of Vincent Thomas Bridge Using Simulated Wind-Induced Ambient Vibration Data Xianfei He, Babak Moaveni, Joel P. Conte ∗ & Ahmed Elgamal Department of Structural Engineering, University of California at San Diego, USA & Sami F. Masri Department of Civil and Environmental Engineering, University of Southern California, USA Abstract: In this article, wind-induced vibration re- sponse of Vincent Thomas Bridge, a suspension bridge located in San Pedro near Los Angeles, California, is sim- ulated using a detailed three-dimensional finite element model of the bridge and a state-of-the-art stochastic wind excitation model. Based on the simulated wind-induced vibration data, the modal parameters (natural frequen- cies, damping ratios, and mode shapes) of the bridge are identified using the data-driven stochastic subspace iden- tification method. The identified modal parameters are verified by the computed eigenproperties of the bridge model. Finally, effects of measurement noise on the sys- tem identification results are studied by adding zero-mean Gaussian white noise processes to the simulated response data. Statistical properties of the identified modal parame- ters are investigated under an increasing level of measure- ment noise. The framework presented in this article will allow us to investigate the effects of various realistic dam- age scenarios in long-span cable-supported (suspension and cable-stayed) bridges on changes in modal identifica- tion results. Such studies are required to develop robust and reliable vibration-based structural health monitoring methods for this type of bridge, which is a long-term re- search objective of the authors. ∗ To whom correspondence should be addressed. E-mail: jpconte@ ucsd.edu. 1 INTRODUCTION Vibration-based structural health monitoring has been the subject of significant research in structural engineer- ing in recent years. The basic premise of vibration-based structural health monitoring is that changes in structural characteristics such as mass, stiffness, and energy dis- sipation mechanisms influence the vibration response characteristics of structures. Therefore, changes in dy- namic features such as modal parameters and quantities derived thereof are often used as damage indicators in structural damage identification and health monitoring. Salawu (1997) presented a review on the use of natural frequency changes for damage detection. It is, however, challenging if not impossible to localize the detected damage (e.g., to obtain spatial information on the dam- age) from changes in natural frequencies only. Pandey et al. (1991) introduced the concept of mode shape cur- vature for damage localization. In their study, both a cantilever and a simply supported beam model were used to demonstrate the effectiveness of using changes in modal curvature as a damage indicator to detect and localize damage. As another mode shape based dam- age indicator, Pandey and Biswas (1994) proposed the use of changes in the dynamically measured flexibility matrix to detect and localize damage. They showed that C 2008 Computer-Aided Civil and Infrastructure Engineering. Published by Blackwell Publishing, 350 Main Street, Malden, MA 02148, USA, and 9600 Garsington Road, Oxford OX4 2DQ, UK.
Transcript
Computer-Aided Civil and Infrastructure Engineering 23 (2008) 373–388
Modal Identification Study of Vincent Thomas BridgeUsing Simulated Wind-Induced Ambient
Vibration Data
Xianfei He, Babak Moaveni, Joel P. Conte∗ & Ahmed Elgamal
Department of Structural Engineering, University of California at San Diego, USA
&
Sami F. Masri
Department of Civil and Environmental Engineering, University of Southern California, USA
Abstract: In this article, wind-induced vibration re-sponse of Vincent Thomas Bridge, a suspension bridgelocated in San Pedro near Los Angeles, California, is sim-ulated using a detailed three-dimensional finite elementmodel of the bridge and a state-of-the-art stochastic windexcitation model. Based on the simulated wind-inducedvibration data, the modal parameters (natural frequen-cies, damping ratios, and mode shapes) of the bridge areidentified using the data-driven stochastic subspace iden-tification method. The identified modal parameters areverified by the computed eigenproperties of the bridgemodel. Finally, effects of measurement noise on the sys-tem identification results are studied by adding zero-meanGaussian white noise processes to the simulated responsedata. Statistical properties of the identified modal parame-ters are investigated under an increasing level of measure-ment noise. The framework presented in this article willallow us to investigate the effects of various realistic dam-age scenarios in long-span cable-supported (suspensionand cable-stayed) bridges on changes in modal identifica-tion results. Such studies are required to develop robustand reliable vibration-based structural health monitoringmethods for this type of bridge, which is a long-term re-search objective of the authors.∗To whom correspondence should be addressed. E-mail: [email protected].
1 INTRODUCTION
Vibration-based structural health monitoring has beenthe subject of significant research in structural engineer-ing in recent years. The basic premise of vibration-basedstructural health monitoring is that changes in structuralcharacteristics such as mass, stiffness, and energy dis-sipation mechanisms influence the vibration responsecharacteristics of structures. Therefore, changes in dy-namic features such as modal parameters and quantitiesderived thereof are often used as damage indicators instructural damage identification and health monitoring.Salawu (1997) presented a review on the use of naturalfrequency changes for damage detection. It is, however,challenging if not impossible to localize the detecteddamage (e.g., to obtain spatial information on the dam-age) from changes in natural frequencies only. Pandeyet al. (1991) introduced the concept of mode shape cur-vature for damage localization. In their study, both acantilever and a simply supported beam model wereused to demonstrate the effectiveness of using changesin modal curvature as a damage indicator to detect andlocalize damage. As another mode shape based dam-age indicator, Pandey and Biswas (1994) proposed theuse of changes in the dynamically measured flexibilitymatrix to detect and localize damage. They showed that
the flexibility matrix of a structure can be easily and ac-curately estimated from a few low-frequency vibrationmodes of the structure. Methods based on changes inidentified modal parameters to detect and localize dam-age in structures have also been further developed forthe purpose of damage quantification (i.e., estimation ofthe extent of damage). Among these methods are strain-energy based methods (Shi et al., 2002), the direct stiff-ness calculation method (Maeck and De Roeck, 1999),and sensitivity-based finite element (FE) model updat-ing methods (Friswell and Mottershead, 1995; Teughelsand De Roeck, 2004). A comprehensive literature surveyon vibration-based structural health monitoring meth-ods can be found in a number of recent publications(Doebling et al., 1996; Farrar and Jauregui, 1998; Sohnet al., 2003).
To develop a robust and reliable structural healthmonitoring methodology, it is essential to investigate theeffects of realistic damage scenarios on structural modalproperties. Because it is inconvenient or impossible tostudy the changes in structural modal parameters causedby various damage scenarios and damage levels throughactual tests on a real structure during its service life, dy-namic response simulation of the structure based on awell calibrated and validated FE model thereof providesan essential tool in structural health monitoring research.In this article, a simulation platform is presented to sim-ulate the wind-induced (ambient) vibration response ofVincent Thomas Bridge (VTB) using a detailed three-dimensional (3D) FE model of the bridge and a state-of-the-art stochastic wind excitation model. The VTB is asuspension bridge that crosses over the main channel ofLos Angeles Harbor in San Pedro, California. The bridgewas constructed in the early 1960s with an overall lengthof approximately 1850 m, comprising the main span of457 m and 154 m spans on either side. Generally, traf-fic, wind, microtremors, and their combinations are themain sources of ambient excitation for bridges. This ar-ticle focuses on realistic simulation of the wind-inducedresponse of VTB and system identification of the bridgebased on its simulated wind response data.
Wind loads, including self-excited (caused by the in-teraction between wind and structural motion) and buf-feting forces (caused by the fluctuating wind velocityfield), are dependent on the geometric configuration ofthe bridge deck section, the reduced frequency of thebridge, and the incoming wind velocity fluctuations. Inthe simulation, the self-excited forces are representedin the time domain by means of convolution integralsinvolving aerodynamic impulse functions and structuralmotions. To simulate properly the stochastic character-istics of buffeting forces, the longitudinal (along-winddirection) and vertical spatially discrete wind velocityfields along the bridge axis are simulated as two inde-
pendent stochastic vector processes according to theirprescribed power spectral density matrices. The spectraof the longitudinal and vertical wind velocity fields areassumed to remain constant along the bridge axis andthe coherence function of the wind velocity fluctuationsat two different positions along the bridge is taken as themodel proposed by Davenport (1968).
In the second part of the article, the dynamic prop-erties of the bridge are identified using the data-driven stochastic subspace identification method (VanOverschee and De Moor, 1996) based on the low-amplitude simulated wind-induced response of VTB.The system identification results are verified by the com-puted eigenproperties of the bridge FE model, whichallows us to assess the performance of the above output-only system identification method when applied to wind-excited long-span suspension bridges. To study the ef-fects of measurement noise on the system identificationresults, zero-mean Gaussian white noise processes areadded to the simulated output signals. Statistical prop-erties (bias and coefficient of variation) of the identifiedmodal parameters are investigated under increasing lev-els of measurement noise.
The framework presented in this article will allow usto investigate systematically the effects of various re-alistic damage scenarios in long-span cable-supportedbridges on changes in modal identification results ob-tained from ambient vibration data. Such studies are re-quired to develop robust and reliable vibration-basedstructural health monitoring methods for this type ofbridge, which is a long-term research objective of theauthors.
2 AERODYNAMIC FORCES
2.1 Self-excited forces
The differential equations of motion of a bridge sub-jected to aerodynamic forces with respect to the staticequilibrium position can be expressed as
M x(t) + C x(t) + K x(t) = F(t) = Fse(t) + Fb(t) (1)
where x(t), x(t), and x(t) = nodal displacement, velocity,and acceleration response vectors, respectively; M, C,and K = structural mass, damping, and stiffness matrices,respectively; F = nodal load vector, and the subscripts seand b denote the self-excited and buffeting aerodynamicforce components, respectively.
For harmonic structural motion, the self-excited forcessuch as lift Lse, drag Dse, and pitching moment Mse (seeFigure 1) per unit span of the bridge are typically ex-pressed as (Scanlan, 1978a; Simiu and Scanlan, 1996;Chen et al., 2000a, 2000b)
Modal identification study of Vincent Thomas Bridge 375
H
U+
w(t)
y
z
x
ph
se b
se
se b
b
L + L
M + M
D + D
B
u(t)
Fig. 1. Aerodynamic forces on bridge deck section.
Lse(t) = 12ρU2 B
[KH∗
1hU
+ KH∗2
Bα
U+ K2 H∗
3 α
+ K2 H∗4
hB
+ KH∗5
pU
+ K2 H∗6
pB
](2a)
Dse(t) = 12ρU2 B
[KP∗
1pU
+ KP∗2
Bα
U+ K2 P∗
3 α
+ K2 P∗4
pB
+ KP∗5
hU
+ K2 P∗6
hB
](2b)
Mse(t) = 12ρU2 B2
[KA∗
1hU
+ KA∗2
Bα
U+ K2 A∗
3α
+K2 A∗4
hB
+ KA∗5
pU
+ K2 A∗6
pB
](2c)
where ρ = air density; U = mean wind velocity; B =bridge deck width; ω = circular frequency of vibration;K = ωB/U = reduced frequency;H∗
i , A∗i and P∗
i (i =1, . . . , 6) = flutter derivatives; and h, p, and α = verti-cal, lateral, and torsional displacement, respectively. Itshould be noted that the formulation of the self-excitedforces in Equation (2) cannot be applied directly intime domain simulation, as the flutter derivatives arefrequency dependent. For arbitrary structural motion,the self-excited forces per unit span can be expressed interms of impulse response functions as (Lin and Yang,1983; Chen et al., 2000a, 2000b)
Lse(t) = 12ρU2
[∫ t
−∞fLh(t − τ )h(τ ) dτ
+∫ t
−∞fLp(t − τ )p(τ ) dτ
+∫ t
−∞fLα(t − τ )α(τ ) dτ
](3a)
Dse(t) = 12ρU2
[∫ t
−∞fDh(t − τ )h(τ ) dτ
+∫ t
−∞fDp(t − τ )p(τ ) dτ
+∫ t
−∞fDα(t − τ )α(τ ) dτ
](3b)
Mse(t) = 12ρU2
[∫ t
−∞fMh(t − τ )h(τ ) dτ
+∫ t
−∞fMp(t − τ )p(τ ) dτ
+∫ t
−∞fMα(t − τ )α(τ ) dτ
](3c)
where the f Xy(· · ·) terms in the integrands are the im-pulse response functions of the self-excited forces, whichare associated with the aerodynamic force componentindicated by the subscript. To evaluate the self-excitedforces in the time domain based on Equation (3), it is nec-essary to determine the aerodynamic impulse responsefunctions based on the experimentally determined flut-ter derivatives. For this purpose, the self-excited forcesare taken to the frequency domain via Fourier transfor-mation of Equations (2) and (3) as⎡⎢⎢⎣
F [Lse(t)]
F [Dse(t)]
F [Mse(t)]
⎤⎥⎥⎦
= 12ρU2
⎡⎢⎢⎣
K2 (iH∗1 + H∗
4 ) K2(iH∗
5 + H∗6
)K2 B (iH∗
2 + H∗3 )
K2(iP∗
5 + P∗6
)K2
(i P∗
1 + P∗4
)K2 B (iP∗
2 + P∗3 )
K2 B (iA∗1 + A∗
4) K2 B(iA∗
5 + A∗6
)K2 B2 (iA∗
2 + A∗3)
⎤⎥⎥⎦
·
⎡⎢⎢⎣
F [h(t)]
F [p(t)]
F [α(t)]
⎤⎥⎥⎦
(4)
⎡⎢⎢⎣
F [Lse(t)]
F [Dse(t)]
F [Mse(t)]
⎤⎥⎥⎦
= 12ρU2
⎡⎢⎢⎣
F [ fLh(t)] F [ fLp(t)] F [ fLα(t)]
F [ fDh(t)] F [ fDp(t)] F [ fDα(t)]
F [ fMh(t)] F [ fMp(t)] F [ fMα(t)]
⎤⎥⎥⎦
·
⎡⎢⎢⎣
F [h(t)]
F [p(t)]
F [α(t)]
⎤⎥⎥⎦ (5)
376 He et al.
where F[ . . . ] = Fourier transform operator. Thus, therelationship between aerodynamic impulse responsefunctions and flutter derivatives can be obtained by com-paring Equations (4) and (5) term by term:
F [ fLh(t)] = K2(iH∗1 + H∗
4
)(6a)
F [ fLp(t)] = K2(iH∗5 + H∗
6
)(6b)
F [ fLα(t)] = K2 B(iH∗
2 + H∗3
)(6c)
F [ fDh(t)] = K2(iP∗5 + P∗
6
)(7a)
F [ fDp(t)] = K2(iP∗1 + P∗
4
)(7b)
F [ fDα(t)] = K2 B(iP∗
2 + P∗3
)(7c)
F [ fMh(t)] = K2 B(iA∗
1 + A∗4
)(8a)
F [ fMp(t)] = K2 B(i A∗
5 + A∗6
)(8b)
F [ fMα(t)] = K2 B2(i A∗2 + A∗
3
)(8c)
Using the above equations, the self-excited forces canbe used in time domain analysis based on Equation (3)once the flutter derivatives are obtained from wind-tunnel experiments. The experimental flutter derivativesin the above equations are usually obtained at a discreteset of reduced frequencies K(ωk). Then, the rationalfunction approximation method known as Roger’s ap-proximation is used to estimate the aerodynamic forcecoefficients defined in Equations (6)–(8), also known asaerodynamic transfer functions, as continuous functionsof the reduced frequency K (Roger, 1977; Chen et al.,2000a; Lazzari et al., 2004). For example, let
F [ fLh(t)] = K2(iH∗
1 + H∗4
)=
[CLh,1 + iCLh,2
Bω
U+ CLh,3
(iBω
U
)2
+n∑
k=4
CLh,kiω
dLh,kUB
+ iω
⎤⎥⎦ (9)
where CLh,i and dLh,k (dLh,k ≥ 0; i = 1, . . . , n and k =4, . . . , n) = frequency-independent coefficients. The firstand second terms on the right-hand side of Equation (9)represent the noncirculatory static-aerodynamics andthe aerodynamic damping, respectively; the third termdenotes the additional aerodynamic mass, which is usu-ally negligible; and the rational terms represent the un-steady components which lag the velocity term and allowan approximation of the time delays through the posi-tive values of parameters dLh,k. The value of n indicates
the level of accuracy in this approximation. Based onEquation (9), the following relations can be derived:
H∗1 (v) = v
2πCLh,2 +
n∑k=4
v3
2πd2Lh,kv
2 + 8π3CLh,k dLh,k
(10a)
H∗4 (v) = v2
4π2CLh,1 − CLh,3 +
n∑k=4
v2
d2Lh,kv
2 + 4π2CLh,k
(10b)
where v = 2π/K = reduced velocity. Therefore, thefrequency-independent coefficients CLh,i and dLh,k canbe determined through least squares fitting of flutterderivatives obtained experimentally at discrete reducedfrequencies/velocities. The above rational function rep-resentation of the aerodynamic transfer function for theself-excited lift force component induced by the verticalstructural motion (see Equation (9)) can be extendedinto the Laplace domain by introducing the Laplace pa-rameter s = iω. Then, the self-excited lift force compo-nent induced by vertical structural motion can be derivedby substituting the inverse Laplace transformation ofF [f Lh(t)](iω=s) into the corresponding component inEquation (3) as
Lse,h(t)
= 12ρU2
[CLh,1h(t) + CLh,2
BU
h(t)
+ CLh,3B2
U2h(t) +
n∑k=4
CLh,k
∫ t
−∞e− dLh,kU
B (t−τ )h(τ ) dτ
]
(11)
The self-excited lift force components induced by lat-eral and torsional structural motions can be obtainedsimilarly. Thus, the total self-excited lift force can be ex-pressed as
Lse(t)
= Lse,h(t) + Lse,p(t) + Lse,α(t)
= 12ρU2
[CLh,1h(t) + CLh,2
BU
h(t) + CLh,3B2
U2h(t)
+n∑
k=4
CLh,k
∫ t
−∞e− dLh,kU
B (t−τ )h(τ ) dτ
]
+ 12ρU2
[CLp,1 p(t) + CLp,2
BU
p(t) + CLp,3B2
U2p(t)
+n∑
k=4
CLp,k
∫ t
−∞e− dLp,kU
B (t−τ ) p(τ ) dτ
]
Modal identification study of Vincent Thomas Bridge 377
+ 12ρU2 B
[CLα,1α(t) + CLα,2
BU
α(t) + CLα,3B2
U2α(t)
+n∑
k=4
CLα,k
∫ t
−∞e− dLα,kU
B (t−τ )α(τ ) dτ
](12)
Formulations for the self-excited drag force Dse(t) andself-excited moment Mse(t) can be derived similarly.These derivations are not shown here for the sake ofbrevity.
2.2 Buffeting forces
The buffeting forces per unit span of the bridge are com-monly expressed as (Scanlan, 1978b; Simiu and Scanlan,1996; Chen et al., 2000a, 2000b)
Lb(t) = −12ρU2 B
{CL χLbu
2u(t)U
+ [C′L + CD] χLbw
w(t)U
}(13a)
Db(t) = 12ρU2 B
{CD χDbu
2u(t)U
+ C′DχDbw
w(t)U
}(13b)
Mb(t) = 12ρU2 B2
{CM χMbu
2u(t)U
+ C′MχMbw
w(t)U
}(13c)
where CL, CD, CM = mean lift, drag, and moment coef-ficients, respectively; C′
L = dCL/dα, C′D = dCD/dα and
C′M = dCM/dα (α = angle of attack of the wind); u(t) and
w(t) = wind velocity fluctuations in the longitudinal andvertical directions, respectively; χLbu, χLbw, χDbu, χDbw,χMbu, χMbw = aerodynamic admittance transfer func-tions between wind velocity fluctuations and buffetingforces. Their squared magnitudes are known as aerody-namic admittance functions. Similar to the self-excitedforces in Equation (2), the buffeting forces in Equation(13) cannot be used directly in time domain analysis, be-cause the aerodynamic admittance transfer functions arefrequency dependent. From Equation (13), the buffetingforce spectra can be derived as
SLb(ω) = 14ρ2U2 B2
{4C2
L|χLbu|2 Suu(ω)
+ [C′L + CD]2|χLbw|2Sww(ω)
}(14a)
SDb(ω) = 14ρ2U2 B2
{4C2
D|χDbu|2 Suu(ω)
+ C′2D|χDbw |2 Sww(ω)
}(14b)
SMb(ω) = 14ρ2U2 B4
{4C2
M|χMbu|2 Suu(ω)
+ C′2M|χMbw |2 Sww(ω)
}(14c)
where Suu(ω), Sww(ω) = power spectral density functionsof longitudinal and vertical wind velocity fluctuations,respectively. It should be noted that the statistical cor-relation between longitudinal and vertical wind velocityfluctuations is neglected in deriving Equation (14). Theaerodynamic admittance functions are assumed to beidentical and approximated by the Liepmann function(Liepmann, 1952) as
|χr (ω)|2 = 1/ (
1 + πωBU
)(15)
where the subscript r refers to Lbu, Lbw, Dbu, Dbw,Mbu, and Mbw. Equivalent wind power spectral densityfunctions are defined as
Sequu(ω) = Suu(ω)/(1 + πωB/U) (16a)
Seqww(ω) = Sww(ω)/(1 + πωB/U) (16b)
Thus, the buffeting forces can be obtained from theequivalent wind velocity fluctuations simulated using theequivalent wind power spectral density functions as
Lb(t) = −12ρU2 B
{CL
2ueq(t)U
+ [C′L + CD]
weq(t)U
}
(17a)
Db(t) = 12ρU2 B
{CD
2ueq(t)U
+ C′D
weq(t)U
}(17b)
Mb(t) = 12ρU2 B2
{CM
2ueq(t)U
+ C′M
weq(t)U
}(17c)
2.3 Rational function approximationof flutter derivatives
This section illustrates the rational function representa-tion of self-excited force coefficients (see Equations (6)–(8)) through least squares fitting of flutter derivativesdetermined experimentally at discrete reduced frequen-cies/velocities (see Equation (10)).
Because the aerodynamic coefficients of VTB are notavailable, the flutter derivatives H∗
1 to H∗4 and A∗
1 to A∗4,
determined experimentally for the William Preston LaneBridge (WPLB) (J. D. Raggett, Personal communica-tion, 2004) located in Baltimore, Maryland, are used forVTB, which is similar in size and design. The remain-ing dimensionless aerodynamic coefficients are taken as(J. D. Raggett, Personal communication, 2004): H∗
5 =H∗
6 = 0, A∗5 = A∗
6 = 0 and P∗2 = P∗
3 = P∗4 = P∗
5 = P∗6 = 0;
CL = 0, CM = 0, CD = 0.162 (CD is estimated as 2.5 timesthe projected frontal area per unit length of the bridgedeck normalized by the deck’s width); C′
L∼= K(H∗2
1 +H∗2
4 )1/2 = 1.415, C′M
∼= K(A∗21 + A∗2
4 )1/2 = 0.238 and C′D =
0; and P∗1 = −2CD/K.
378 He et al.
0 2 4 6 8 10 12-4
-3
-2
-1
0
Reduced velocity, 2πU/Bω
0 2 4 6 8 10 12-0.1
0
0.1
0.2
0.3
0.4
Reduced velocity, 2πU/Bω
0 2 4 6 8-1
0
1
2
3
4
5
Reduced velocity, 2πU/Bω
0 2 4 6 8
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Reduced velocity, 2πU/Bω
A2*[Exp. data]
A3*[Exp. data]
A2*[Fitted curve]
A3*[Fitted curve]
H2*[Exp. data]
H3*[Exp. data]
H2*[Fitted curve]
H3*[Fitted curve]
A1*[Exp. data]
A4*[Exp. data]
A1*[Fitted curve]
A4*[Fitted curve]
H1*[Exp. data]
H4*[Exp. data]
H1*[Fitted curve]
H4*[Fitted curve]
Fig. 2. Rational function approximations of flutter derivatives for William Preston Lane Bridge (J. D. Raggett, Personalcommunication, 2004).
In this study, two lag terms (i.e., n = 5) are usedin the rational representations (see Equation (9)). Fig-ure 2 shows a comparison of the flutter derivatives esti-mated from the rational function representations (e.g.,see Equation (10)) and those measured in wind tunneltests for WPLB. The excellent agreement obtained in-dicates that the self-excited forces on the bridge decksection can be approximated by the rational functionsconsidered with very good accuracy.
3 SIMULATION OF WIND-INDUCED RESPONSEOF VINCENT THOMAS BRIDGE
A detailed 3D FE model of VTB (see Figure 3) de-veloped in the structural analysis software ADINA(ADINA R&D Inc., 2002) was used in this study for
Fig. 3. Three-dimensional finite element model of Vincent Thomas Bridge.
the simulation of the wind-induced response of VTB.This FE model is composed of 3D linear elastic (tension-only) truss elements to represent the main suspension ca-bles and suspender cables, 3D linear elastic membraneand shell elements to model the reinforced-concretebridge deck, stringers supporting the deck on the floortrusses, and beam-column elements to model the stiff-ening trusses, the lateral braces between the stiffeningtrusses and the tower shafts. The floor trusses were mod-eled with 3D elastic beam-column and truss elements.This FE model consists of approximately 8,900 nodesand 9,400 elements, resulting in approximately 22,000 de-grees of freedom (DOFs). The natural frequencies of thelow frequency vibration modes computed from this FEmodel of VTB are given in Table 1 where they are com-pared with (1) the corresponding natural frequencies
Modal identification study of Vincent Thomas Bridge 379
Table 1System identification results
Abdel-Ghaffar et al. (1992)
Natural frequency [Hz] Damping ratio (%) Identified freq. [Hz] Computed freq. [Hz]
Mode no. Identified Computed Identified Specified MAC Ambient Earthquake 2D Model 3D Model
previously identified from actual ambient vibration dataand earthquake records (Abdel-Ghaffar et al., 1992),and (2) the corresponding natural frequencies com-puted from other validated FE models of VTB (Abdel-Ghaffar et al., 1992). This comparison shows that theFE model of VTB used in this study captures reasonablywell the dominant low-frequency vibration modes of thisbridge.
In simulating the wind-induced response of VTB, theaerodynamic forces are assumed to act along the bridgedeck only. The aerodynamic parameters are assumed tobe invariant along the bridge axis and the variation ofthe aerodynamic characteristics due to static rotationof the bridge deck (under gravity loads and aerostaticforces due to mean wind velocity) is neglected for sim-plicity. The aerodynamic parameters introduced in Sec-tion 2.3 are used in the simulation. The buffeting forcesare simulated based on the simulated equivalent windvelocity fluctuation time histories according to Equation(17). The statistical correlation between longitudinal andvertical wind velocity fluctuations is ignored so that thespatially discretized wind velocity field is simulated asthe combination of two independent stochastic vectorprocesses. The simulation of the wind velocity fluctu-ations, ueq(t) and weq(t), is performed using the spec-tral representation method in conjunction with the fastFourier transform technique (Deodatis, 1996; Cao et al.,2000). The wind spectra for the longitudinal and verti-cal wind velocity fields are taken as Kaimal’s spectrum(Kaimal et al., 1972) and Panofsky’s spectrum (Lumleyand Panofsky, 1964), respectively, defined as
Fig. 4. Locations of spatially discretized aerodynamic forces (“aerodynamic” nodes) along the bridge deck.
Suu(ω, z) = 200zu2∗
4πU(z)[
1 + 50ωz2πU(z)
]5/3(18a)
Sww(ω, z) = 3.36zu2∗
4πU(z)
[1 + 10
(ωz
2πU(z)
)5/3] (18b)
where u∗ = kU(z)/ln(z/z0) = shear velocity of the windflow in m/s; z0 = roughness length in m; k ≈ 0.4; andU(z) = mean wind velocity in m/s at height z above theground level. In this study, z0 = 0.07 m, and the heightof the bridge deck above the ground is z = 60 m. Themean wind velocity, U, is taken as 10 m/s (36 km/h) tosimulate wind-induced ambient vibrations under com-mon low wind intensity. The coherence function of thewind velocity fluctuations at two different locations ofabscissas x1 and x2 along the bridge deck is taken as(Davenport, 1968; Cao et al., 2000)
Cohr (x1, x2, ω) = exp(
−λω|x1 − x2|2πU(z)
), ω > 0 (19)
where subscript r = u or w, λ = 10 for the longitudi-nal wind velocity fluctuation and λ = 8 for the verticalwind velocity fluctuation. The aerodynamic forces arediscretized at 27 locations along the bridge axis as shownin Figure 4, implying that the longitudinal and verticalstochastic wind velocity fields are discretized into twoindependent vector processes of 27 components each.Two-hour long wind velocity records are simulated with
380 He et al.
0 1000 2000 3000 4000 5000 6000 7000-4
-2
0
2
4
Time [s]
ueq(t
) [m
/s]
0 1000 2000 3000 4000 5000 6000 7000-1.5
-1
-0.5
0
0.5
1
1.5
2
Time [s]
weq
(t)
[m/s
]
Fig. 5. Simulated longitudinal, ueq(t), and vertical, weq(t),wind velocity fluctuations (U = 10 m/s).
a sampling time of t = 0.25 s. As illustration, Figure 5shows a sample of the simulated longitudinal and verti-cal wind velocity fluctuation at the center point of themain span. The estimated (from simulated time histo-ries) equivalent power spectral density function of thelongitudinal wind velocity fluctuation is compared in Fig-ure 6 to the theoretical wind spectrum (see Equations(16a) and (18a)) used to simulate the stochastic windvelocity field. Similarly, Figure 7 shows a comparison
Fig. 7. Comparison of simulated and exact coherencefunction of the longitudinal wind velocity fluctuation.
between exact and estimated (from simulated time histo-ries) coherence functions of the longitudinal wind veloc-ity fluctuations at two stations located 28.4 m apart. FromFigures 6 and 7, it is observed that the simulated wind ve-locity field along the bridge follows closely the assumedtheoretical wind spectrum and coherence function.
It is assumed that the self-excited forces per unit spanat different locations along an “aerodynamic” elementof length L (corresponding to the tributary length ofthe “aerodynamic” node located at the center of the“aerodynamic” element) are fully correlated, while therandom fluctuation of the buffeting forces per unit spanalong an “aerodynamic” element is accounted for. It isassumed that the buffeting force components induced bythe longitudinal, u, and vertical, w, wind velocity fluctua-tions are uncorrelated, because the statistical correlationbetween u and w is neglected. Based on the above as-sumptions, for example, the self-excited and buffetinglift force components acting at an “aerodynamic” nodewith tributary length L can be expressed as (Chen et al.,2000a)
Lese(t) = Lc
se(t)L (20a)
Leb(t) = L
∫ t
0[hLbu(t − τ )Lc
bu(τ ) + hLbw(t − τ )Lcbw(τ )] dτ
(20b)
where the superscript c indicates the center location ofthe “aerodynamic” element e; hLbu(t) and hLbw(t) areimpulse response functions, the Fourier transforms ofwhich, HLbu(ω) and HLbw(ω) satisfy the following rela-tion:
|HLbr (ω)|2 = 1L2
∫ L
0
∫ L
0cohLbr (x1, x2, ω) dx1 dx2 (21)
in which r = u or w; and cohLbr(x1, x2, ω) denotes the co-herence function of the lift buffeting force componentsper unit span at two different positions x1 and x2 along
Modal identification study of Vincent Thomas Bridge 381
0 50 100 150 200 250-4
-3
-2
-1
0
1
2
3x 10
-3
Time[s]
Dis
plac
emen
t [m
]
Fig. 8. Simulated vertical displacement response of VTB atcenter of main span.
the “aerodynamic” element, which is assumed to be thesame as that for the wind velocity fluctuations (see Equa-tion (19)). The drag and moment buffeting force compo-nents can be expressed in a similar way. The transferfunctions HLbr (ω) (r = u, w) in Equation (21) are alsoapproximated using rational functions for the purposeof time domain analysis, for example,
HLbr (ω) =
⎡⎢⎣C1 +
n∑k=2
Ckiω
dkUB
+ iω
⎤⎥⎦ (22)
where the coefficients Ck and dk are determined throughleast squares fitting.
The dynamic response of a suspension bridge dependson the deformed bridge configuration and stress state un-der gravity loads. Therefore, first a geometric nonlinearstatic analysis is performed for the bridge under grav-ity loads only, using an incremental-iterative solutionprocedure. The aerodynamic wind forces (with buffet-ing forces based on the fluctuating wind velocity fieldsu(t) and w(t) and self-excited forces computed based onthe displacement field of the bridge relative to its staticequilibrium position under gravity loads only) are thenapplied with initial conditions given by the bridge stateunder gravity loads only. The dynamic equations of mo-tion of the bridge under aerodynamic wind loads arelinearized (geometrically) about the displacement andstress fields corresponding to gravity loads. Finally, theselinearized equations of motion are solved using the con-stant average acceleration version of the Newmark timestepping method with parameters δ = 0.5 and α = 0.25.As an illustration, Figure 8 shows the simulated verticaldynamic response of the bridge at the center of the mainspan.
Data-driven stochastic subspace identification (SSI-DATA) is one of the most advanced state-of-the-artoutput-only system identification methods, which hasalready been successfully applied for modal parameteridentification of long-span bridges based on ambient vi-bration data. The SSI-DATA algorithm extracts a linearstate-space model of the system considered using output-only measurement data directly (Van Overschee and DeMoor, 1996; Peeters and De Roeck, 2001). Comparedto two-stage time-domain system identification methodssuch as covariance-driven stochastic subspace identifi-cation (SSI-COV) (Van Overschee and De Moor, 1996)and the natural excitation technique (NExT) (Jameset al., 1993) combined with the eigensystem realiza-tion algorithm (ERA) (Juang and Pappa, 1985), SSI-DATA does not require any pre-processing of the data tocalculate auto/cross-correlation functions or auto/cross-spectra of output data (i.e., SSI-DATA is a one-stagesystem identification method). In addition, robust nu-merical techniques such as QR factorization, singularvalue decomposition (SVD), and least squares are in-volved in this method. A brief review of this method ispresented next.
The discrete-time state-space representation of a lin-ear time-invariant system of order n is defined as
z(k + 1) = Az(k) + Bu(k) (23a)
x(k) = Cz(k) + Du(k) (23b)
where A ∈ Rn×n, B ∈ R
n×l ,C ∈ Rm×n, D ∈ R
m×l = statespace matrices in discrete form; z(k) ∈ R
n = state vector;u(k) ∈ R
l = load vector (vector of loading functions);and x(k) ∈ R
m, a column vector of size m ( = number ofmeasured/output channels) which represents the mea-sured system response at discrete time t = k(t) alongm DOFs. In practical applications, the load vector in-put function u is often unknown/unmeasured and onlythe response of the structure is measured. In this case,the discrete-time state-space model in Equation (23) isextended to the following stochastic version:
z(k + 1) = Az(k) + w(k) (24a)
x(k) = Cz(k) + v(k) (24b)
where state matrices A and C are the same as in Equa-tions (23): A = state transition matrix, which completelycharacterizes the dynamics of the system through itseigenproperties, and C = output matrix that specifieshow the inner states are transformed into the measured
382 He et al.
system response/output; w(k) ∈ Rn = process noise
due to external disturbances, modeling inaccuracies (i.e.,missing high-frequency dynamics) and unknown inputexcitation (undistinguishable from the external distur-bances); and v(k) ∈ R
m = measurement noise due tosensor inaccuracies and also unknown input excitation(feed-through term). Both noise terms w(k) and v(k) areassumed to be zero-mean, white vector sequences withthe following covariance matrix:
E
⎡⎣
⎛⎝ w(i)
v(i)
⎞⎠ (
w( j)T v( j)T)⎤⎦ =
⎡⎣ Q S
ST R
⎤⎦ δi j (25)
where E[ . . . ] denotes the mathematical expectation op-erator; δi j = Kronecker delta; and Q, R, S = process andmeasurement noise auto/cross-covariance matrices.
The SSI-DATA procedure of extracting the state-space matrices A and C from output-only data can besummarized as follows: (1) Form an output Hankel ma-trix and partition it into “past” and “future” output sub-matrices. (2) Calculate the orthogonal projection matrixof the row space of the “future” output submatrix intothe row space of the “past” output submatrix using QRfactorization. (3) Obtain the system observability ma-trix and Kalman filter state estimate via SVD of theprojection matrix. (4) Using the available Kalman fil-ter state estimate, extract the discrete-time system state-space matrices as a least squares solution. To increase thecomputational efficiency of the system identification pro-cedure, only the reference “past” outputs (outputs fromthe “reference” sensors) instead of all “past” outputs areused to form the output Hankel matrix (Peeters and DeRoeck, 1999). Once the system state-space matrices aredetermined, the modal parameters (natural frequenciesand damping ratios) of the N = n/2 vibration modes canbe obtained as
where λi = i th eigenvalue of matrix A and t = sam-pling time. It should be noted that λ2i−1 and λ2i (i = 1,2, . . . , N) are complex conjugate pairs, each pair corre-sponding to a vibration mode, that is, the natural fre-quency and damping ratio obtained from λ2i−1 are thesame as those obtained from λ2i . The vibration modeshapes are obtained as
φφi = C · T2i−1 (27)
Fig. 9. Virtual array of accelerometers along the bridge deck.
where Ti denotes the i th eigenvector of matrix A. Simi-larly, T2i−1 and T2i (i = 1, 2, . . . , N) are complex conju-gate pairs of eigenvectors, each pair corresponding to avibration mode.
4.2 System identification results
A simulated array of 42 channels (21 along each sideof the bridge deck) of vertical acceleration responseof the bridge subjected to wind excitation is used forsystem identification. The simulated wind aerodynamicforces correspond to a mean wind velocity U = 10 m/s(36 km/h). Figure 9 shows the virtual accelerometer ar-ray along the bridge deck. The simulated accelerationrecords used in the identification process are 250-s longwith a sampling rate of 20 Hz, corresponding to 5,000samples per record/channel and a Nyquist frequencyf Nyq = 10 Hz. In applying SSI-DATA, these simulatedacceleration data were first low-pass filtered using a fi-nite impulse response (FIR) filter of order 512 with acut-off frequency at 1.0 Hz. Then, an output Hankel ma-trix is formed including 100 block rows based on theselow-pass filtered vibration data. The 15 channels on theeast side of the main span are used as reference “past”output channels.
The identified natural frequencies and damping ratiosare reported in Table 1 together with the correspondingcomputed natural frequencies and specified damping ra-tios of the VTB FE model used in this study. Table 1also provides modal parameters obtained from previ-ous system identification studies of VTB based on actualambient vibration data and earthquake records (Abdel-Ghaffar et al., 1992). It is observed that there is anexcellent agreement between the identified natural fre-quencies based on the simulated wind-induced responsedata and those computed from the bridge FE model. Thedifference between identified and analytically predictednatural frequency is largest for the first mode, whichcould be because the FE model of VTB used here hasthree very closely spaced modes with natural frequen-cies of 0.182, 0.226, and 0.231 Hz. The vibration modeat 0.182 Hz could not be accurately identified based onthe simulated accelerometer data. The damping ratiosidentified based on the simulated wind-induced responsedata and the damping ratios specified in the FE model ofVTB are in a good agreement considering that the esti-mation uncertainty of damping ratios is inherently largerthan that of natural frequencies.
Modal identification study of Vincent Thomas Bridge 383
90
270
180
f1 = 0.214 Hz
90
270
f2 = 0.226 Hz
90
270
f3 = 0.357 Hz
90
270
f4 = 0.455 Hz
90
270
f5 = 0.514 Hz
90
270
f6 = 0.576 Hz
90
270
f7 = 0.687 Hz
90
270
0
f8 = 0.815 Hz
Fig. 10. Polar plot representation of vibration mode shapes identified using SSI-DATA.
It is important to mention that the identified nat-ural frequencies and damping ratios also include thecontribution of the aerodynamic stiffness and dampinginduced by the wind–structure interaction, which de-pend on the reduced wind velocity (2πU/Bω), the ge-ometrical configuration of the bridge section and theapproach wind flow. Due to wind–structure interaction,the identified modal parameters (especially the damp-ing ratios) of long-span cable-supported bridges underwind loading can vary significantly with wind velocitywhen approaching the flutter onset velocity. In a hybridexperimental-analytical flutter analysis of the JianyinBridge, a suspension bridge with a main span of 1,385m and a streamlined closed box steel girder (36.8-mwide and 3-m high), modal aerodynamic damping ra-tios of some vibration modes reached values of approxi-mately up to 10% (from corresponding structure modaldamping ratios of 0.5%) at the flutter onset wind ve-locity (Ding et al., 2002). More studies about effects ofwind–structure interaction on dynamic characteristics oflong-span cable-supported bridges can be found in Mat-sumoto et al. (1996), Miyata et al. (1997), Larsen (1998),
-400-200
0200
400
-100
10-1
01
f1 = 0.214 Hz
Width [m
]-400
-2000
200400
-100
10-1
01
f2 = 0.226 Hz
-400-200
0200
400
-100
10-1
01
f3 = 0.357 Hz
-400-200
0200
400
-100
10-1
01
f4 = 0.455 Hz
-400-200
0200
400
-100
10-1
01
f5 = 0.514 Hz
Length [m]
Width [m
]-400
-2000
200400
-100
10-1
01
f6 = 0.576 Hz
Length [m] -400-200
0200
400
-100
10-1
01
f7 = 0.687 Hz
Length [m] -400-200
0200
400
-100
10-1
01
f8 = 0.815 Hz
Length [m]
Fig. 11. 3D representation of normalized vibration mode shapes identified using SSI-DATA.
Chen et al. (2000a, 2000b, 2001), Matsumoto et al. (2002),and Chen and Kareem (2003).
The vibration mode shapes the identified using state-space model based system identification methods suchas SSI-DATA are in general complex valued. Figure 10represents in polar plots (i.e., rotating vectors in the com-plex plane) the mode shapes of VTB identified using SSI-DATA based on the simulated (wind-induced) ambientvibration data. These polar plots have the advantage toshow directly the extent of the nonproportional damp-ing characteristics of a vibration mode. If all complexvalued components of a mode shape vector are collinear(i.e., in phase or 180 degrees out of phase), this vibra-tion mode is said to be classically (or proportionally)damped. On the other hand, the more these mode shapecomponents are scattered in the complex plane, the morethe vibration mode is nonclassically (or nonproportion-ally) damped. However, measurement noise, estimationerrors, modeling errors, and aerodynamic damping canalso cause a truly classically damped mode to be identi-fied as nonclassically damped. Figure 10 shows that mostof the vibration modes (modes #1, 4–8) identified in this
384 He et al.
-400-200
0200
400
-100
10-1
01
f1 = 0.226 Hz
Width [m
]-400
-2000
200400
-100
10-1
01
f2 = 0.231 Hz
-400-200
0200
400
-100
10-1
01
f3 = 0.364 Hz
-400-200
0200
400
-100
10-1
01
f4 = 0.459 Hz
-400-200
0200
400
-100
10-1
01
f5 = 0.511 Hz
Length [m]
Width [m
]
-400-200
0200
400
-100
10-1
01
f6 = 0.571 Hz
Length [m] -400-200
0200
400
-100
10-1
01
f7 = 0.684 Hz
Length [m] -400-200
0200
400
-100
10-1
01
f8 = 0.823 Hz
Length [m]
Fig. 12. 3D representation of vibration mode shapes computed from the finite element model of VTB.
study are either perfectly or nearly classically damped.A 3D representation of the normalized mode shapes ofthe bridge deck for these identified vibration modes isgiven in Figure 11. Normalization was performed by pro-jecting all mode shape components onto their principalaxis (in the complex plane) and then scaling this pro-jected mode shape vector for a unit value of its largestcomponent. The identified space-discrete mode shapeswere interpolated between the virtual sensor locationsusing cubic splines along both sides of the bridge deckand straight lines across the deck.
The modal assurance criterion (MAC) (Allemang andBrown, 1982) is used to compare the identified and com-puted (“exact”) vibration mode shapes. The MAC value,bounded between 0 and 1, measures the degree of corre-lation between corresponding identified and computedmode shapes as
where ∗ denotes the complex conjugate transpose. AMAC value of 0 indicates that the corresponding iden-tified and computed mode shapes are completely un-correlated, while a MAC value of 1 indicates perfectcorrelation between them. The MAC values for all pairsof identified and computed mode shapes are also givenin Table 1. For all vibration modes identified, except forthe second one, there is a very good-to-excellent agree-ment between corresponding identified and computedmode shapes. The high degree of nonclassical dampingidentified for the second mode (see Figure 10) could bethe reason behind the low MAC value obtained for this
mode. Such high degree of nonclassical damping couldbe true/physical (possibly due to wind–structure interac-tion) or could be caused by estimation and/or modelingerrors. Three-dimensional representations of the modeshapes computed from the FE model of VTB are shownin Figure 12, which can be compared directly with theiridentified counterparts in Figure 11.
To study the effects of measurement noise on thesystem identification results, zero-mean Gaussian whitenoise processes are added to the simulated bridge vi-bration response data. Statistical properties (mean andstandard deviation) of the estimated modal parametersare investigated for increasing levels of measurementnoise. For this purpose, a set of 100 identifications was
Table 2Mean and coefficient of variation (COV) of the identified
natural frequencies normalized to their counterpartsidentified based on noise-free data at different measurement
Modal identification study of Vincent Thomas Bridge 385
Table 3Mean and coefficient of variation (COV) of the identifieddamping ratios normalized to their counterparts identifiedbased on noise-free data at different measurement noise
performed at each of three different measurement noiselevels (2%, 5%, and 10%). For a given output chan-nel, the noise level is defined as the ratio (in percent)of the root mean square (RMS) of the added noise pro-cess to the RMS of the simulated acceleration response.The added measurement noise processes are assumed
0 2 5 100.998
0.9985
0.999
0.9995
1
1.0005Mode 1
f nois
e / f no
ise-
free
0 2 5 100.998
0.999
1
1.001
1.002Mode 2
0 2 5 100.998
0.999
1
1.001
1.002Mode 3
0 2 5 100.998
0.9985
0.999
0.9995
1
1.0005Mode 4
0 2 5 100.9997
0.9998
0.9999
1
1.0001
1.0002
1.0003Mode 5
Noise level [%]
f nois
e / f no
ise-
free
0 2 5 100.998
0.9985
0.999
0.9995
1
1.0005Mode 6
Noise level [%]
0 2 5 100.9997
0.9998
0.9999
1
1.0001
1.0002
1.0003Mode 7
Noise level [%]
0 2 5 100.998
0.999
1
1.001
1.002Mode 8
Noise level [%]
mean
mean +/- std
Fig. 13. Statistics (mean, mean ± one standard deviation) over 100 trials of the identified natural frequencies normalized to theircounterparts identified based on noise-free data at different measurement noise levels.
statistically independent across the output channels andover the 100 realizations considered. The statistics (meanand coefficient of variation) over 100 trials of the identi-fied natural frequencies and damping ratios normalizedto their counterparts identified based on simulated noise-free data are reported in Tables 2 and 3, respectively,for the three measurement noise levels considered. Thecoefficient of variation of a random variable is definedas the ratio of its standard deviation to its (absolute)expected value (mean). In addition, mean and mean ±one standard deviation of the identified natural frequen-cies and damping ratios normalized to their counterpartsidentified based on simulated noise-free data are rep-resented graphically in Figures 13 and 14, respectively.Based on the results reported in Tables 2 and 3 and plot-ted in Figures 13 and 14, it is observed that both thebias and coefficient of variation of the identified natu-ral frequencies and damping ratios introduced by themeasurement noise increase with increasing noise levelas expected. However, bias and coefficient of variationdue to measurement noise remain very small (negligi-ble) for the identified natural frequencies (see Table 2and Figure 13). Although they are significantly larger forthe identified damping ratios (see Table 3 and Figure 14),they remain relatively small as it is well known that the
386 He et al.
0 2 5 100.9
0.95
1
1.05Mode 1
ξ nois
e / ξ no
ise-
free
0 2 5 100.9
0.95
1
1.05Mode 2
0 2 5 100.9
1
1.1
1.2
1.3
1.4
1.5Mode 3
0 2 5 100.98
0.99
1
1.01
1.02
1.03
1.04
1.05Mode 4
0 2 5 100.99
1
1.01
1.02
1.03Mode 5
Noise level [%]
ξ nois
e / ξ no
ise-
free
0 2 5 100.9
1
1.1
1.2
1.3Mode 6
Noise level [%]
0 2 5 100.99
1
1.01
1.02
1.03Mode 7
Noise level [%]
0 2 5 100.9
1
1.1
1.2
1.3
1.4
1.5Mode 8
Noise level [%]
mean
mean +/- std
Fig. 14. Statistics (mean, mean ± one standard deviation) over 100 trials of the identified damping ratios normalized to theircounterparts identified based on noise-free data at different noise levels.
estimation uncertainty of damping ratios is inherentlylarger than that of natural frequencies. It was also foundthat the measurement noise considered has very small(negligible) effects on the identified mode shapes.
5 CONCLUSIONS
Wind-induced ambient vibration of Vincent ThomasBridge (VTB), a long-span suspension bridge locatedin San Pedro near Los Angeles, California, is simu-lated using a detailed three-dimensional FE model ofthe bridge and a state-of-the-art stochastic wind exci-tation model including both buffeting and self-excitedforces. Based on these simulated ambient vibration data,modal parameters of the low-frequency vertical vibra-tion modes of VTB are identified using data-drivenstochastic subspace identification (SSI-DATA), a state-of-the-art output-only system identification method. Theidentified modal parameters are in good agreement withthe computed (“exact”) modal parameters obtained di-rectly from the FE model of VTB, which themselves arein good agreement with the corresponding modal pa-rameters of VTB identified by other researchers using
actual ambient vibration data. This system identificationstudy also provides the opportunity to investigate the ac-curacy of the modal identification results obtained usingSSI-DATA in the case of a large and complex (virtual)structural problem for which the “exact” modal param-eters (modal parameters of the FE model of VTB) areknown, which is usually not the case when system identi-fication methods are applied directly to real-world struc-tures and data.
The effect of measurement noise on the identifiedmodal parameters is investigated. Measurement noise issimulated by adding statistically independent zero-meanGaussian white noise processes to the finite element sim-ulated bridge response to wind excitation along a set ofdegrees of freedom (virtual output channels). The sta-tistical properties (mean and coefficient of variation)of the identified modal parameters are investigated un-der an increasing measurement noise level. Both biasand coefficient of variation of the identified natural fre-quencies and damping ratios introduced by the measure-ment noise increase with increasing noise level as ex-pected. However, bias and coefficient of variation dueto measurement noise remain very small (negligible)for the identified natural frequencies. Although they are
Modal identification study of Vincent Thomas Bridge 387
significantly larger for the identified damping ratios, theyremain relatively small as it is well known that the esti-mation uncertainty of damping ratios is inherently largerthan that of natural frequencies.
The methodology and study presented in this articleprovide a validated framework for studying the effects ofrealistic damage scenarios in long-span cable-supported(suspension and cable-stayed) bridges (e.g., corrosion-induced losses in stiffness and strength of main cablesand suspenders at different locations along the bridge)on modal identification results. These effects representthe basis for developing robust and reliable vibration-based structural health monitoring systems for long-spancable-supported bridges.
ACKNOWLEDGMENTS
This study was supported by the National Science Foun-dation under ITR Grant No. 0205720. The authors wishto express their thanks to Dr. Jon D. Raggett, Presidentof West Wind Laboratory, Inc. and Dr. Xinzhong Chenat Texas Tech University for providing the aerodynamicparameters needed for the wind response simulation andfor very useful suggestions and insightful discussions re-garding the stochastic wind excitation model. Any opin-ions, findings, and conclusions or recommendations ex-pressed in this article are those of the authors and do notnecessarily reflect those of the sponsor.
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