statistics and identification of mode-dependent structural damping of ...

The Eighth Asia-Pacific Conference on Wind Engineering,December 10–14, 2013, Chennai, India

STATISTICS AND IDENTIFICATION OF MODE-DEPENDENT STRUCTURAL DAMPING OF CABLE-SUPPORTED BRIDGES

Yi Liu, Yaojun Ge*, Fengchan Cao, Yi Zhou, Shouqiang Wang

State Key Lab of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China, [email protected]

ABSTRACT

This paper presents general characteristics of field-measurement modal damping of 31 cable-supported bridges in China, including 6 suspension bridges and 25 cable-stayed bridges. The majority of the statistical results of bridge damping presented are analogous to most previous research results, and the new finding is the fact that the modal damping decreases with the increase of the natural frequencies. In order to study the operational variability of the dynamic properties, 1650 spanned Xihoumen Bridge, the longest box-girder suspension bridge in the world, is selected as a case study, and a classical one-stage time domain algorithm, called Data-Driven Stochastic Subspace Identification (SSI-DATA) Method, was adopted in obtaining the modal damping of the bridge by examining a whole year’s recordings through the structural health monitoring system. It seems that the measured modal damping increases with the increase of the root-mean-square values of structural accelerations.

Keywords: Modal damping, Cable-supported bridges, Field measurement, Health monitoring system, System identification

1. Introduction

Unlike the mass and stiffness properties, estimation of structural damping is a very difficult problem in structural dynamics, in which damping is the most important parameter affecting the dynamic responses of structures. The uncertainty of structural damping results in many difficulties in reliable structural design for dynamic effects [Kareem et al. (1996) and Tamura. (2012)]. The demand for a reasonable predictive model for structural damping is becoming increasingly important since the actual damping value of a bridge is still hard to be identified. It is necessary to set up a structural damping database with reliable and precise data from existing bridges with different types and conditions [Tamura et al. (1996) and Kijewski et al. (2000)]. Such a database would greatly help to develop a relatively accurate damping predictor for assessing damping ratios in design phase.

A. P. Jeary (1986) proposed a damping predictor by analyzing the damping database from the field measurement of several high-rise buildings through the Building Researcher Establishment. Davenport (1989) investigated the damping characteristics of long-span suspension bridges and cable-stayed bridges by interpreting many on-site observed data [referenced in Hiroki et al. (1997)]. Tamura (1996; 2000; 2003; 2008; 2012) introduced the Japanese database on building dynamic properties including damping ratios. Yamaguchi (1997) studied the mode dependency of damping ratios of 29 cable-stayed bridges in Japan. Fujino (2002; 2012) examined the mode dependency of damping ratios in cable-supported bridges by establishing a damping database to modify the wind resistant design. Guo (2005) analyzed the properties of the field-measured data of modal damping by using the established database, including one suspension bridge and seven cable-stayed bridges in China. However, due to the lack of on-site measured data from prototype bridges, there are only limited researches who have explored the typical characteristics of structural damping for wind-resistance design, and it is not enough to establish a high-quality database for structural damping of long-span bridges.

Proc. of the 8th Asia-Pacific Conference on Wind Engineering – Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds)Copyright c© 2013 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: 978-981-07-8011-1doi:10.3850/978-981-07-8012-8 151 719

Proc. of the 8th Asia-Pacific Conference on Wind Engineering (APCWE-VIII)

In recent years, dynamic field tests of large and flexible bridges, especially in the cases of cable-supported bridges have been conducted after the completion of construction in China, which is helpful to investigate the validity of the dynamic properties of bridges, including natural frequencies, mode shapes and modal damping ratios. Furthermore, since most recently built long-span bridges in China have been equipped with structural health monitoring (SHM) system, modal damping ratios can be identified through using SHM field measurement recordings. Of course, this is the new and may be the only effective way for us to identify the variability of dynamic properties of long-span bridges for the real-time purpose.

This paper is aimed to study the general characteristics of modal damping of cable-supported bridges by using the field-measurement results of modal damping ratios, and the measured data of 6 suspension bridges and 25 cable-stayed bridges in china have been developed into a preliminary database. On the other hand, this paper also adopted the Data-Driven Stochastic Subspace Identification (SSI-DATA) method to deal withthe one-year SHM recording data of Xihoumen Bridge to identify the bridge dynamics in time domain for examining the preliminary damping ratio database developed. 2. Statistics of Damping Ratios of Cable-supported Bridges in China

In this section the fundamental characteristics of modal damping (especially the mode dependency of structural damping in suspension and cable-stayed bridges) were studied by using the continuously updated database respectively. Table 1 and table 2 embrace the general information of the Chinese damping database.

Table 1: Structural Damping Ratios of 6 Suspension Bridges in China

No Bridge name Center Span Length (m) Girder material Natural

frequency Damping

ratio 1 Ningbo Qingfeng Bridge 280 S-C* Composite 0.508 0.019 2 Humen Bridge 888 S* 0.119 0.0248 3 Yichang Changjiang Highway Bridge 960 S 0.164 0.015 4 Tsing Ma Bridge 1377 S 0.069 0.0084 5 Jiangyin Changjiang Bridge 1385 S 0.0549 0.0126 6 Xihoumen Bridge 1650 S 0.054 0.021

Table 2: Structural Damping Ratios of 25 Cable-stayed Bridges in China

No Bridge name Center Span Length (m) Girder material Natural

frequency Damping

ratio 1 Dezhou Xinhe Bridge 90 RC* 0.44 0.0468 2 Shenyang Gonghe Bridge 120 RC 0.88 0.059 3 Jilin Linmenjiang Bridge 132.5 RC 0.48 0.00398 4 Fulin 4th Bridge 140 RC 0.8 0.013 5 Feilongdao Bridge 150 S-C Composite 0.85 0.0068 6 Nanchang Bayi Bridge 160 RC 0.46 0.0189 7 Jiujiang Bridge 160 RC 0.4297 0.07399 8 Jianyi Bridge 165 RC 0.45 0.0122 9 Fengtai Huaihe Bridge 224 RC 0.525 0.089 10 Ningbo Waitan Bridge 225 S 0.55 0.0187 11 Yibin Zhongba Jinshajing Bridge 252 RC 0.391 0.02 12 Tianjin Yonghe Bridge 260 RC 0.288 0.0318 13 Luzhou Taian Changjiang Bridge 270 RC 0.352 0.014

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14 Dongyin Yellow River Bridge 288 S 0.45 0.014 15 Badong Changjiang Bridge 388 RC 0.33 0.0151 16 Wuhan Changjiang 2nd Bridge 400 RC 0.27 0.0152 17 Shanghai Nanpu Bridge 423 S-C Composite 0.36 0.0127 18 Hongkong Kap Shui Mun Bridge 430 S-C Composite 0.39 0.028 19 Coastal Highway Liaohe Bridge 436 S 0.37 0.0556 20 Hongkong Tingkau Bridge 475 S-C Composite 0.165 0.0141 21 Anqing Changjiang Highway Bridge 510 S 0.273 0.007 22 Fujian Qingzhou Minjiang Bridge 605 S-C Composite 0.226 0.007 23 Shanghai Changjiang Tunnel Bridge 730 S 0.23 0.028 24 Hubei Edong Changjiang Bridge 926 S-C Composite 0.18 0.0073 25 Jiangsu Sutong Bridge 1088 S 0.082 0.0164 Notes: RC-- Reinforced Concrete; S—steel; S-C -- Steel-Concrete

In view of the field measurements which are not only for the wind resistant design but

also for the consideration of vehicle-induced dynamic responses, the modal parameters of medium span bridges have been also accumulated and included in this database. It is noteworthy that among the several testing methods for measuring the structural damping of bridges, most of the bridges we amassed were based on the ambient vibration measurements. Moreover, to speak of the usage of the bridges, highway bridges account for 99%.

2.1 Relation between Structural Damping and Natural Frequency

The relationship between structural damping and natural frequency has been explored in many prior studies, and the results given by varies pundits (Davenport (1989); Yamaguchi (1997); Fujino (2002; 2012); Guo (2005))are analogical, namely, there is a poor correlation between the modal damping and the natural frequency in cable-stayed bridges as against the legible relation in suspension bridges. As a validation, fig.1 shows the relation between the modal damping and the corresponding natural frequency of both cable-stayed bridge and suspension bridge.

With respect to fig.1 group 1 (a1&b1), the modal damping decreases with the increase of the natural frequency, and the bridges with smaller center span length seem to have higher natural frequency and damping ratio. However, unlike suspension bridges the tendency of the modal damping with natural frequency can be approximately expressed as Eq. (1) (c and b are constants regard to the structure properties, especially, the value of b is around -1) [Davenport et al. (1989)], the data are more scattered in fig.a1 regarding the different bridges. Therefore, we cannot simply give a specific equation to describe such a tendency. The regression value of b regarding the selected suspension bridges is shown in fig.1 (b1).

bc fξ = ⋅ (1)

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0.0 0.6 1.2 1.8 2.40.00

0.02

0.04

0.06

Dam

ping

ratio

Natural frequency (Hz)

Sutong Bridge Shanghai Changjiang Tunnel Bridge Qingzhou Minjiang Bridge Hongkong Tingkau Bridge Badong Changjiang Bridge

a1

0.0 0.6 1.2 1.8

0.00

0.01

0.02

0.03 Xihoumen Bridge Jiangyin Bridge Yichang Bridge Humen Bridge Ningbo Qingfeng Bridge

Dam

ping

Rat

io

Natural Frequency (Hz)

bX=-0.99 bJ=-0.43bY=-1.06 bH=-1.41bN=-1.13

b1

0.0 0.2 0.4 0.6 0.8 1.00.00

0.03

0.06

0.09

0.12

Dam

ping

ratio

Natural frequency (Hz)

a2

Vertical bending mode Lateral bending mode Torsional mode

0.0 0.2 0.4 0.6

0.00

0.01

0.02

0.03 Vertical bending mode Lateral bending mode Tosional mode

Dam

ping

ratio

Natrual frequency (Hz)

b2

Fig.1. Relation between modal damping and natural frequency a1&a2, cable-stayed bridge; b1&b2, suspension bridge

As to fig.1 group 2 (a2&b2), the different mode of the damping ratio of different types

of bridge shows a significant difference. In fig.1 (a2), the damping ratio of all the three modes slightly decreases when the natural frequency is less than 0.28Hz, and then shows an upward trend, but declines when the natural frequency surpasses 0.5Hz. With less dispersed data, the pattern of the damping ratio versus natural frequency of suspension bridge is more obvious as can be seen from the fig.1 (b2). Apart from the lateral bending mode, the tendency of the damping ratio for both vertical and torsional mode is similar. The data plunge remarkably when the natural frequency is under 0.7Hz, and then go up slowly with the increase of the natural frequency. Besides, it is quite apparent that the damping ratio of torsional mode is lower than the vertical and the lateral bending mode, since the natural frequency of vertical bending mode and lateral bending mode is usually lower than the torsional mode.

2.2 Relation between Structural Damping and Span Length

It has been found that for either cable-stayed bridges or suspension bridges, the lowest natural frequency decreases monotonically with the increase of the span length [Yamaguchi et al. (1997) and Fujino. (2002)] however, the modal damping performs differently as against the natural frequency. As shown in fig.2 (a), the tendency of the damping ratio is quite the contrary to the previous discussion of the damping-frequency relation in cable-stayed bridges. In hopes of illustrating this phenomenon, it is worthwhile noticing that the bridges under about 250m are almost the reinforced concrete bridges with single tower and the bridges over that span length are twin towers bridges with varies girder materials and structure types. As it were, it is reasonable to indicate that with the increase of the span length, the stiffness of the bridges falls, thereby the natural frequency of the bridges descends, and in turn, the modal damping ascends. Unluckily, it still remains cloudy (including the trend shown in fig.1 (a2)) that why the upward tendency of damping ratio ends at that particular span length, and the

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value drops until the span length of the bridge reaches 600m and then rises mildly afterwards. It is of great possibility that this phenomenon is associated with the type of the structure and the material the bridges possess.

0 200 400 600 800 1000 12000.00

0.03

0.06

0.09

0.12

Dam

ping

ratio

Center span length (m)

a

Vertical bending mode Lateral bending mode Torsional mode

0 450 900 1350 18000.00

0.01

0.02

0.03 Vertical bending mode Lateral bending mode Tosional mode

Dam

ping

ratio

Center span length (m)

b

Fig.2. Relation between modal damping and center span length a, cable-stayed bridge; b, suspension bridge

As the tendency depicted in fig.2 (a) is connected to the fig.1 (a2), so the fig.2 (b) is

linked to fig.1 (b2). As indicated by fig.1 (b), the modal damping has a fair correlation with the center span length, and the threshold is around 1400m. Once the center span length exceeds that threshold, the modal damping stops declining and begins to grow with the increase of the span length. Whereas the real mechanism of this phenomenon maintains unknown, we still need further study.

2.3 Damping Values in High Modes

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08 2nd mode 3rd mode Eq.2: 2nd mode Eq.2: 3rd mode

Dam

ping

ratio

in 2

nd

and

3rd

mod

es

Damping ratio in 1st mode

a0.000 0.008 0.016 0.024 0.032

0.000

0.008

0.016

0.024

2nd mode 3rd mode Eq.2: 2nd mode Eq.2: 3rd mode

Dam

ping

ratio

in 2

nd a

nd 3

rd m

odes

Damping ratio in 1st mode

b

Fig. 3 Damping database and application of Eq. (2) a, cable-stayed bridge; b, suspension bridge

Fig.3 delineates an intriguing relationship between the fundamental modes and the

higher modes of damping. Anteriorly, Kareem (1996) presumed damping to be proportional to stiffness which increases in higher modes, and examined the applicability of the equation--put forward by Yokoo and Akiyama (1972) for the ratio of damping values in the higher modes of vibration to the damping in terms of the associated frequency—based on the data base studied by Tamura et al. (1994). Interestingly, depart from the previous espoused outlooks, the results depicted in fig.3 are consistent with the assumption made by Saul and Jayachandran that lower damping in higher modes in comparison with the first mode which may have resulted in overemphasizing the relative importance of higher modes [quoted by

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Kareem et al. (1996)]. In this regard, we proposed a modified expression in Eq. (2) which is akin to the one that Yokoo and Akiyama (1972) assumed

'

1 1

1 1n nfCf

ξξ

= + − (2)

Where 'C is a constant with the same value approximately equal to 0.38 given by Yokoo and Akiyama (1972). In addition, by analyzing the data available in the database, it can be noted that based on the data selected 68% (cable-stayed bridges) of the cases the damping in the 2nd mode is higher than the damping in the 1st mode, the corresponding value for the 3rd mode was found to be 84% (cable-stayed bridges). Whilst the average value of the ratio of the 2nd to the 1st mode damping is 0.98 and for the ratio of the 3rd to the 1st is 0.72. As for the case of suspension bridges, 83.3% of the cases the 2nd mode damping ratio is greater than the 1st mode, which is same as the 3rd mode, and the average value of the ratio of 2nd mode to 1st mode damping is 0.76, the 3rd to the 1st is 0.55.

In effect, to clarify the causation of this contradiction between bridges and buildings, we have to recall the correlational analysis of the modal damping and natural frequency depicted in fig.1 group 1 (a1&b1). To appear consistent, the modal damping in 3rd mode is inferior to the 2nd mode. No wonder the results in fig.3 are inconsistent with the forgone conclusion made by Kareem (1996). However, the value of the constant 'C which can be utilized effectively in both bridges and buildings remains a mystery.

To sum up the previous discussion, within a particular bridge, it is rational to rate that the damping ratio decreases with the increases of the natural frequency, irrespective of the greater scatter the cable-stayed bridges possess. The correlations between the damping ratio and natural frequency as well as span length regarding different bridges are still elliptical, even though the relation between natural frequencies with span length is explicit as has been pointed out by many researchers. Herewith, the key point of this problem is to figure out how to pinpoint the inner connection between the damping ratio and natural frequency in respect to different bridges.

3. Identification of Modal Damping Ratios of Xihoumen Bridge based on SHM

3.1 Xihoumen Bridge and its SHM System

Submerged in marine environment and located in the strong wind area where includes typhoon in summer and monsoon in winter, Xihoumen Bridge is the main part of the connection project—Zhoushan Island-Mainland Connection Project—which links Zhoushan archipelago and Ningbo city in Zhejiang province around China East Sea. The bridge has two continuous span suspended by main cables (578m+1650m), and a four-lane expressway on the twin box steel girder which is 36m wide and 3.51m high. The whole bridge was completed and opened for traffic in December 25, 2009, and created a world record for box-girder suspension bridge. The layout of the bridge is shown in fig.4.

In the wake of the completion of the bridge, a sophisticated structural health monitoring (SHM) system is designed and installed on this bridge, and sensors for monitoring of wind and wind-induced vibration are incorporated in the SHM system. Six three-dimensional ultrasonic anemometers were assembled at the south and the north quarter points as well as the middle span section of the deck respectively, and each section has two anemometers on the east and west sides of the bridge deck. For this type of anemometer, the highest sampling frequency is 32Hz, the measurement wind speed range is 0~65m/s, and the accuracy is 1.5% rms. Besides, 12 servo accelerometers sampled at 100Hz were set up to measure the lateral, vertical and torsional acceleration of the bridge deck. The known locations of the anemometers and accelerometers were roughly presented in fig.4.

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Fig. 4 Overall dimensions of the Xihoumen Bridge and instrumentation layout. AC10, AC13, accelerometers on the left side of the bridge deck; AC12, AC15, accelerometers on the right side of the bridge deck; UA1, UA3, anemometers on the left side of the bridge deck;UA2,UA4, anemometers on the right side of the bridge deck;

3.2 Data-driven Stochastic Subspace Identification

As it has been often pointed out, the accuracy of structural damping identified through structure health monitoring when the bridge is in-service is untrustworthy. Except for the different ambient conditions, the non-stationary loads and the relatively small amplitudes of response as well as the viscous damping hypothesis which is against with the nonlinearities in structural systems have all served to cloud the issue, and the veracity of full-scale data is categorically has to do with the appropriate use of output-only system identification methods which varied in precision and efficiency [Tamura. (2012)]. Out of consideration for the bias and variability of the damping ratios, in this study, SSI-DATA method [Peeters et al. (1999)] which has a competitive edge in system identification was applied to an 1-year (2009/12/01~2010/11/12) acceleration response (Particularly, the vortex-induced vibration of the measured data was excluded) gathered from Xihoumen Bridge. A brief review of this method is presented next.

In practice, the measurements are evaluated at only l sensor locations and are available at discrete time instants as well, thus the sample time and noise is always influencing the data. The discrete–time state-space representation of a linear time-invariant system of order n is defined as

1+ = + +k k k kx Ax Bu w (3a) υ= + +k k k ky Cx Du (3b)

Where ×∈ n nA R is the state transition matrix, which completely characterizes the dynamics of the system through its eigenproperties; ×∈ n mB R is the discrete input matrix; ×∈ l nC R is the output matrix that specifies how the inner states are transformed into the measured system response/output; ×∈ l mD R is the direct transmission matrix; 1×∈ n

kx R denotes the state vector; 1×∈ l

ky R represents the measured system response at discrete time ( )= Δt k t along l DOFs; 1×∈ m

ku R is the load vector describing the m inputs in time; 1×∈ nkw R is the process noise due

to external disturbances, modeling inaccuracies and unknown input excitation; 1υ ×∈ lk R is the

measurement noise due to sensor accuracies and also unknown input excitation. Both noise terms kw and υk are immeasurable vector signals which can be assumed to be zero-mean, white vector sequences with the following covariance matrix:

( )υ δυ

=i T Tj j ijT

i

w Q SE w

S R (4)

Where E is the expected operator; δij is the Kronecker delta; Q, R, S are the process and measurement noise auto/cross-covariance matrices.

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In practical applications, the load vector input function ku is often unknown /unmeasured (especially in the ambient vibration testing) and only the response of structure is measured. In this case, the discrete-time state-space model in Equation (3) is extended to the following stochastic version without ku terms:

1+ = +k k kx Ax w (5a) υ= +k k ky Cx (5b)

Compared to the two-stage time-domain system identification methods such as Natural Excitation Technique (NExT) [James et al. (1993)] combined with the Eigensystem Realization Algorithm (ERA) [Juang and Pappa, (1985)], SSI-DATA does not require any pre-processing of data to calculate auto/cross-correlation functions or auto/cross-spectra of output data. Moreover, robust numerical techniques such as QR factorization, singular value decomposition (SVD) and least squares are utilized in this method. As the frequencies and damping ratios of the structure can be determined from the eigenvalue of the state-space matrix A, the principle procedure of SSI-DATA is to extract matrix A as well as the output matrix C from the output-only data, the main steps can be epitomized as follows [He et al. (2008)]: (1) Form an output Hankel matrix and partition it into “past” and “future” output sub-matrix; (2) Calculate the orthogonal projection matrix of the row space of the “future” output sub-matrix into the row space of the “past” output matrix using QR factorization; (3) Obtain the system observability matrix and Kalman filter state estimate via SVD of the projection matrix; (4) Using the available Kalman filter state estimate, extract the discrete-time system state-space matrices as a least squares solution.

Given that each complex eigenvalue pair of the A matrix corresponds to one vibration mode of the structure, once the system state-space matrices are determined, the modal parameters of the N=n/2 vibration modes can be given by

2 2ln( ) / / 2 ln( ) / 2λ π λ π= Δ =i i i sf t f (6)

2Re(ln( )) / / 2ξ λ π= −i i s if f (7) Where λi is the ith eigenvalue of matrix A , 2 1λ −i and 2λ i (i=1, 2, 3,…, N)are complex conjugate pairs; Δt and sf are the sampling time and sampling frequency respectively.

.3.3 Identification Results of Modal Damping Ratios

Seeing that the influence of traffic cannot be removed, a common thread runs through this problem is to select the measured response data artificially so as to minimize the traffic-induced effect. Therefore, by dividing the data into hourly segments (a representative segment of measured vertical acceleration of bridge deck is shown in fig.5) and then examining the tens of thousands of time histories of the acceleration response, the data mainly from 23:00:00~04:59:59 were sorted out, because during this period of time, the traffic was low and the bridge ambient vibrations were driven mainly by wind (this conclusion can be manifested by the following study). Since the bridge is submerged in the ocean environment, the wind orthogonal to the bridge is relatively strong, and it can be seen from both fig.6 and fig.7. In fig.6, it is conspicuous that during the measured period, the most frequent winds came from the northwest, with the wind direction nearly perpendicular to the longitudinal bridge axis. The relation between the vertical acceleration of the bridge deck (AC12) and the 10-min mean wind speed demonstrate that the acceleration of the deck increases with the increase of the wind velocity.

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Fig. 5 Time history of a 1-hour vertical acceleration of bridge deck from AC12

225o

0o

30o

45o

60o

90o

120o

150o

180o210o

240o

270o

300o

330o

510

1520

25m/s

Longitudinal bridge axis

Fig.6 Wind rose diagram during the

measured period (2009/12/01- 2010/11/12) from UA2

0 5 10 15 20 250

1

2

3

4

5

6

RM

S of

acc

eler

atio

n (c

m/s

2 )10-min mean wind speed (m/s)

Fig.7 Wind velocity and RMS acceleration at north quarter point of

the span from AC12&UA2

In applying SSI-DATA, the data from channel AC12, AC13, and AC15 were adopted. To identify the modal parameter of the lower vibration modes (in the case of large and flexible bridges, the natural frequency of the predominant modes is less than 1Hz), the filtered measured data were first down sampled to 10Hz (the Nyquist frequency 5=Nyqf Hz ) so as to improve the computational efficiency of SSI-DATA. It is common knowledge that the choice of the parameter i and j for Hankel matrix are pivotal, they have a distinct influence on the accuracy of the identified results and the computational effort. In practical applications, toensure the uniform estimate of the system matrix, parameter j must be greater than i, i.e. j > 20i [Xiaoxiang et al. (2009)]. As for parameter i, it is strongly correlated with the sampling frequency and the fundamental frequency of the structure, and the empirical value is

02≥ si f f ( sf is the sampling frequency, 0f is the structural basic frequency) [Reynders et al. (2008)]. For this reason, the dimension of block for the Hankel matrix regards to this study is 400 35600× . Additionally, due to the scant of enough accelerometer sensors, it’s impossible to extract the mode shape of the bridge through system identification. Hence, by comparing the identified modal frequency with the previous system identification studies of the same bridge based on ambient vibration test pre-operation, the real mode was elected.

Fig. 8 depicts a two-stage upward trend relation between the measured damping ratios in the first four mode and the rms of vertical accelerations (AC12), which means the system is non-proportionally damped, and the source of the non-proportionality is primarily due to friction and aerodynamic force [Nagayama (2005)]. Despite the acceptable variations of damping ratios, there seem to be clear trend between damping ratios and acceleration amplitude. From a to d (in fig.8), the two-stage upward tendency of damping ratios versus acceleration rms can be fitted with parabolas. By drawing the tangents of the fitted curves to portray the different performance of the two-stage damping trend, one can notice that the slopes of the tangents (L1&L2) of both stages are declining, at that, with the increase of the acceleration, the damping ratios are close to a certain value, especially in the 4th mode. Before explaining this phenomenon thoroughly, first, it is better to construe the relation between the

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wind speed and the measured damping ratios shown in fig.9 and the correlation between natural frequency and acceleration exhibited in fig.10. By contrast, the tendency of the measured damping ratios in 1st mode with 1-hour mean wind speed is parallel to the trend shown in fig.8 (a), coincidentally, this similarity happens to the rest of the modes (even though have not been shown in this paper), which means that as the essential source of the ambient excitation, the bridge vibration was mainly driven by wind. What’s more, since the bridge was subjected to considerably strong wind (the maximum wind velocity during the measured time is nearly 25m/s), the aerodynamic damping effect is eminent. Therefore, the measured damping ratios herein are total damping ratios which comprise structural damping and aerodynamic damping.

0 1 2 3 4 50.00

0.01

0.02

0.03

0.04 Damping ratio Fitted curve

Mea

sure

d ve

rtica

l dam

ping

ratio

in 1

st m

ode

RMS of vertical acceleration (cm/s2)

A

L

L1

L2

Adj. R-Square=0.43201Mean Square=0.09154 a

0 1 2 3 4 50.00

0.01

0.02

0.03


Mea

sure

d ve

rtica

l dam

ping

ratio

in 2

nd m

ode


Adj. R-Squar=0.30314Mean Square=0.10506

A

L

L1

L2

a

0 1 2 3 4 50.00

0.01

0.02


Mea

sure

d ve

rtica

l dam

ping

ratio

in 3

rd m

ode


Adj. R-Square=0.54484Mean Square=0.04024

A

L

L1

L2

c0 1 2 3 4 5

0.000

0.006

0.012

0.018


Mea

sure

d ve

rtica

l dam

ping

ratio

in 4

th m

ode


Adj. R-Square=-0.00331Mean Square=0.02362

A

L

L1

L2

d

Fig. 8 Amplitude dependency of damping ratio

By appreciating fig.9 penetratingly, it must be emphasized that after subtracting the

aerodynamic damping acquired from the wind tunnel experiment conducted with a sectional model, the two-stage tendency of damping ratios vs. wind speed changed dramatically. As shown in fig.9 (b), the mean value of the damping ratios reduced, but only the first stage upward trend reserved, the second stage was leveled off. It is because the contribution of the aerodynamic damping is relatively small at low wind conditions, which can be neglected. Nevertheless, the wind tunnel result suggests that the aerodynamic damping ratio increases as the wind speed grows, therefore, the measured total damping ratio dropped drastically at the higher wind speed. It should be pointed out that the overall profile of damping ratios in fig. 9 (b) shows a great consistency with the results presented by Nagayama (2005) and Siringoringo (2006) even if their outcomes were shown in damping- acceleration relation. Besides, the average value of damping ratio without the aerodynamic damping is around 6 . In respect to the negative damping ratios in fig.9 (b), they might be errors lie both in the measurement during the wind tunnel test and the system identification.

The tendency of natural frequency with vertical acceleration bears many resemblances to the precedent researches made by Nagayama (2005) and Siringoringo (2006) of Hakucho Bridge in Japan, that is to say, the natural frequency decrease as the rms acceleration (or wind speed) increases within a low acceleration range, but the downward tendency does not sustain

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for long, the natural frequency shows a tad increase as the acceleration goes up (the wind tunnel test suggests that the natural frequency of vertical bending mode increases as the wind speed increases due to aerodynamic effects [Siringoringo et al. (2006)]). However, the variation of the measured natural frequency from its mean value is very low, which is about 2.1%, and when the vertical coordinate narrowed down, this variation can be ignored. As shown in fig.10 (b), the natural frequencies of the bridge are nearly independent of the rms acceleration (this is the same with the wind speed), i.e., the aerodynamic force has only a slight impact on the structural modal frequencies [Li et al. (2011)].

0 5 10 15 20 250.00

0.01

0.02

0.03

0.04 Measured damping ratio Wind tunnel test of sectional model

Mea

sure

d ve

rtica

l dam

ping

ratio

in1s

t mod

e

1-hour mean wind speed (m/s)

a0 5 10 15 20 25

0.00

0.01

0.02

0.03

0.04 Damping ratio without

aerodynamic damping Fitted curve

Mea

sure

d ve

rtica

l dm

apin

g ra

tio in

1st

mod

e1-hour mean wind speed (m/s)

by=0

Fig. 9 Relation between wind speed and damping ratio

and their comparison with wind tunnel test

0 1 2 3 4 50.093

0.094

0.095

0.096

0.097

Mea

sure

d na

tura

l fre

quen

cy

in 1

st m

ode


a0 1 2 3 4 5

0.075

0.100

0.125

0.150

0.175

0.200

0.225

0.250

Mea

sure

d na

tura

l fre

quen

cy


b

Fig. 10 Relation between acceleration root mean square and natural frequency

Compare the results shown in fig.8-10 (a), the cause of discrepancies in natural

frequency and damping ratio appear to be multifaceted. First, for a particular mode, within the lower rms acceleration or wind speed range, namely, the first stage upward tendency of the damping ratios in fig.8 and fig.9, is mainly due to the effect of stick-slip behavior of coulomb friction elements [Siringoringo (2006)]. In other words, a contact surface between structure members (e.g., bearings, extensive devices, sub-structures) does not move in the very low amplitude range, but begins to slip at a particular amplitude, and with the increase of the amplitude, the number of slipping contact surface ascends, thus the structure losing its stiffness, so the damping due to friction increases as the natural frequency decreases [Tamura. (2012)]. On the other side, the equilibrium point of the first stage varied from mode to mode, as can be seen from fig.8, the crossover point A and the transformation of the slop of tangent L1 can help to interpret the case. According to fig.8, the abscissa value of point A increases with the increase of mode, and the sparse degree of the damping ratio in the first four modes are changed not one whit, they move from higher amplitude to the lower amplitude, and the absolute value of the damping ratios sinks with the increase of the mode which conforms with the results presented in fig.1 (b1). All these can be illuminated as the higher the mode, the higher acceleration is needed to engender this mode, then the more easily the stuck friction elements of the bridge slip, the more slipping contact surface generated, so the suffering area

729


of the acceleration extend pronouncedly, the intersection point A moves eastward, the slop of tangent L1 descends and the value of the damping ratio falls as the mode rises.

On the second stratum of the interpretation, the upward tendency of the measured damping ratio vis-à-vis the second stage implies the presence of an aerodynamic force (has been proved by fig.9). When the wind speed is low, the influence of the aerodynamic force is small, as the wind velocity goes up, the dependencies of the modal frequency and damping ratio on the aerodynamic force become dominant. In light of the linear model of aerodynamic damping proposed by R.H. Scanlan, with the increase of the natural frequency, the reduced wind speed decreases, then the vertical aerodynamic derivative drops, which renders the aerodynamic damping falls, that’s why the second stage upward trend of the damping ratios in fig.8 slows down as the increase of the mode. Besides, granted that the viewpoint of the variability of the damping ratio discussed above is quite true, the errors lie in the measurements and the evaluation technique cannot be discounted, and in all likelihood, the two-stage upward trend of the damping ratio vs. acceleration or wind speed is caused by low signal-to-noise ratio when the amplitude of the bridge is small.

4. Conclusions

With the readily available field measured damping data in the literature on vibration studies of cable-supported bridges in China, the mode-dependent structural damping was discussed in detail. Although the inner mechanism of the structural damping maintains unknown, the characteristics of modal damping can be boiled down to a central idea which hasn’t been explicitly pointed out before, namely, the modal damping decreases as the natural frequency increases which is different from the buildings. Additionally, in the implementation of data-driven stochastic subspace identification method, a series of 1-year measured data through structural health monitoring were examined in this paper. The net results indicate that the system is non-proportionally damped and the measured modal damping increases with the increase of the rms acceleration or wind speed, the friction and aerodynamic forces both play a telling part in the non-proportionality.

3 Acknowledgements

The work described in this paper is partially supported by the NSFC under the Grant 91215302 and by the MOST under the 973 Program Grant 2013CB036301. References Tamura, Y., Suda, K., and Sasaki, A. (2000), “Damping in buildings for wind resistant design” Proc.

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