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Field observations of rain-wind-induced cable vibration in cable-stayed dongting lake bridge

Journal of Wind Engineering

and Industrial Aerodynamics 95(2007)303–328

Field observations of rain-wind-induced cable vibration in cable-stayed Dongting Lake Bridge

Y.Q.Ni a,?,X.Y.Wang a,1,Z.Q.Chen b ,J.M.Ko a

Department of Civil and Structural Engineering,The Hong Kong Polytechnic University,

Hung Hom,Kowloon,Hong Kong

School of Civil Engineering,Hunan University,Changsha,Hunan 410082,PR China Received 15March 2004;received in revised form 1July 2006;accepted 18July 2006

Available online 7September 2006

Abstract

The cable-stayed Dongting Lake Bridge has been observed several times to exhibit large-amplitude cable oscillation under simultaneous action of rain and wind.To investigate excitation mechanism and response characteristic of the rain-wind-induced stay vibration,a series of ?eld measurements lasting 45days have been conducted on the Dongting Lake Bridge by deploying accelerometers,anemometers and rain gauge for continuous monitoring.This paper presents the measurement results of rain and wind excitations as well as dynamic response of a typical stay in three rain-wind excitation events.The measurement data show that under speci?c combination of rain and wind,the maximum acceleration response of the cable reaches 10g and the maximum displacement response (peak-to-peak)is around 0.7m.It is revealed that the large-amplitude rain-wind-induced stay oscillation occurs in the bridge when the mean wind velocity at deck level ranges from 6to 14m/s,the wind attack angle (relative yaw angle)ranges from 101to 501,and the rainfall is light to moderate (less than 8mm/h).For the observed cable,the overall dominant mode of cable vibration during rain-wind excitations is the third mode.However,in the evolution process of this kind of vibration,the dominant mode may differ for different response segments.r 2006Elsevier Ltd.All rights reserved.

Keywords:Cable-stayed bridge;Cable vibration;Rain-wind excitation;Field measurement

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Corresponding author.Tel.:+852********;fax:+852********.

E-mail address:ceyqni@https://www.sodocs.net/doc/a8562856.html,.hk (Y.Q.Ni).

On leave from School of Civil Engineering,Hunan University of Science and Technology,Xiangtan,Hunan 411201,China.

1.Introduction

Owing to large ?exibility,relatively small mass and extremely low inherent damping,cables in cable-supported bridges are susceptible to vibration caused by various excitation mechanisms.For example,incidences of large-amplitude cable oscillation in cable-stayed bridges under speci?c combinations of rain and wind have been reported worldwide (Hikami and Shiraishi,1987;Pacheco and Fujino,1993;Matsumoto et al.,1995a ;Poston,1998;Irwin et al.,1999;Persoon and Noorlander,1999;Main et al.,2001;Chen and Tanaka,2002;Ni et al.,2002).This kind of large-amplitude vibration may cause reduced life of the cables and their connections due to fatigue and breakdown of protections against corrosion (Pacheco and Fujino,1993;Poston,1998),and invoke the risk of losing public con?dence to the bridges (Persoon and Noorlander,1999).In the past decade,the mitigation of rain-wind-induced cable vibration by means of aerodynamic,mechanical and structural means has been extensively studied.

Although a lot of research efforts have been made,the excitation mechanism of rain-wind-induced cable vibration is still an imperfectly understood phenomenon (Matsumoto et al.,1995b ;Verwiebe and Ruscheweyh,1998).The research on rain-wind excitation mechanisms has been conducted by wind tunnel testing (Hikami and Shiraishi,1987;Ohshima and Nanjo,1987;Matsumoto et al.,1990,1992;Kinoshita et al.,1991;Flamand,1995;Honda et al.,1995;Bosdogianni and Olivari,1996),analytical and numerical modeling (Yamaguchi,1990;Geurts et al.,1998;Verwiebe,1998;Ruscheweyh,1999;Wang and Xu,1999;Peil et al.,2003),and ?eld observation (Matsumoto et al.,1989,2003;Main and Jones,1999;Persoon and Noorlander,1999;Main et al.,2001;Schwarzkopf and Sedlacek,2003).The proposed excitation mechanisms include water rivulet formation on upper surface of a cable,axial ?ow in a wake of cable,low-frequency vortex shedding along the cable axis,and vortex-induced vibration at high reduced wind speed.In order to verify the validity of these proposed excitation mechanisms,understanding the excitation and response features of rain-wind-induced cable vibration actually occurring in the ?eld becomes extremely important.Although visible observations of the rain-wind-induced vibration have been reported for numerous cable-stayed bridges,in situ measurement data of this kind of cable vibration at full-scale conditions are seldom available.

The Dongting Lake Bridge is a three-tower prestressed concrete cable-stayed bridge recently built in Hunan,China.It has a total length of 880m,consisting of two main spans of 310m each and two side spans of 130m each.The central tower is 125.7m high and the side towers are 99.3m high each.The bridge deck is 23.4m wide with four traf?c lanes,with the clearance height of 25.0m above water level.It is supported by 222cables of size ranging from 28to 201m in length and 99to 159mm in diameter,which are coated into smooth surface with polyethylene (PE).Shortly after opening to traf?c in the end of 2000,the bridge has been observed several times to exhibit severe cable vibration under low wind speed and light-to-moderate rain.The frequent occurrence of such rain-wind-induced vibration has agitated the administrative authority and management engineers who ?nally adopted MR-based damping technology for cable vibration mitigation (Chen et al.,2004).It also provides a good test-bed for in situ measurement of excitation and response characteristics of cable vibration under the rain-wind excitation conditions and other excitation conditions.For this purpose,a typical cable of 122m long was selected for monitoring,and the MR dampers were intentionally dismantled from the cable to cater for vibration.An instrumentation system,consisting of 15accelerometers,one displacement

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Y.Q.Ni et al./J.Wind Eng.Ind.Aerodyn.95(2007)303–328305 transducer,two tri-axial anemometers,one rain gauge,and one data acquisition and processing system,has been used for continuous45-day measurements in the bridge site. This paper presents the measurement data during three typical rain-wind excitation events and analysis results of cable response at different time segments.The measurement data-based analyses on rain-wind excitation mechanism(mainly on the mechanism of vortex-induced cable vibration at high reduced wind velocity)and on wind-induced and traf?c-induced cable oscillation will be reported separately.

2.Description of measured cable and instrumentation system

Fig.1shows the elevation and plan of the Dongting Lake Bridge.The bridge axis direction is NNW201.The previous observations on the Dongting Lake Bridge indicated that rain-wind-induced cable vibration in this bridge occurred commonly under a combined action of yawed north wind and rain,and behaved as large-amplitude oscillation at all stays declining in the direction of the wind?ow and small or invisible oscillation at the cables inclining in the direction of the wind?ow.Hence,the cable A12,as shown in Fig.1,was selected for response measurement.Fig.2illustrates the cable A12after the dampers are dismantled.The main parameters of the cable A12are as follows:length L?121.9m, inclination angle a?35:21,diameter D?119mm,initial tension T?3150KN,mass per unit length m?51.8kg/m,elastic modulus E?2.0?105MPa.The?rst four in-plane modal frequencies of the cable are1.07,2.14,3.20and4.23Hz,respectively.Referring to Fig.3,the instrumentation system includes the following primary components:

One three-axis ultrasonic anemometer at the top of south side tower,2m high above the tower top(Fig.3(a)).It is situated at an elevation of102m;

One three-axis ultrasonic anemometer at deck level near by the cable A12,4m stretching out from the deck edge with a horizontal cantilever(Fig.3(b)).It is situated at an elevation of26m;

One rain gauge at deck level near by the cable A12(Fig.3(c));

Four uniaxial accelerometers on the locations of L/6and L/20from the lower anchorage for cable in-plane and out-of-plane acceleration measurement(Fig.3(d)); Eight uniaxial accelerometers on neighboring MR-damped cables A10,A11,A13and

A14for in-plane and out-of-plane acceleration measurement;

Plan

Fig.1.Dongting Lake Bridge and deployment of sensors.

Two uniaxial accelerometers at south side tower near by the upper anchorage of cables A11and A13for tower longitudinal acceleration measurement;

One uniaxial accelerometer at the deck for vertical acceleration measurement;

Fig.2.Cable A12with dampers being

dismantled.

Fig.3.Sensor installation for ?eld measurement:(a)anemometer at tower top;(b)anemometer at deck level;(c)rain gauge at deck level;and (d)accelerometers on cable A12.

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One vibration transducer at the deck for vertical displacement measurement;

One data acquisition and processing system in the bridge site.

This system was settled in situ in mid-March 2003.Continuous ?eld measurements were conducted from 24March to 11May 2003.3.Correlation of wind velocities at different heights

Wind velocity and direction were measured by two three-axis ultrasonic anemometers at the tower top and deck level,respectively.Fig.4shows a sample of wind velocity records and 3-min mean wind velocities at the two heights.It is seen that the characteristic of wind velocity is basically identical at the tower top and at deck level.A statistical analysis indicates that the ratio of mean wind velocities at the tower top and at deck level is around 1.55.The correlation of wind velocities at the two heights can be expressed approximately as

U ez 1T?U ez 2Tz 1

z 2 a ,(1)

where z 1and z 2denote height from the ground;U (z 1)and U (z 2)are wind velocity at the heights z 1and z 2,respectively;a is an exponential coef?cient.The wind fetch around the bridge is almost uniform.With the measurement data,a is estimated by curve ?tting to be 0.2723.In the following,only the wind velocity and direction data measured at deck level are presented and analyzed.

The orientation of the anemometers was calibrated along the bridge axis,namely,when wind direction has an angle y with respect to the bridge axis (refer to Fig.1),the direction output of the anemometers is also y .Considering the inclination angle of stay cable and orientation angle between stay axis and bridge axis in plan view,the wind direction will be represented by the relative yaw angle b n de?ned as (Matsumoto et al.,1995b ;Main et al.,2001)

b n ?sin à1esin b cos a T,

(2)

0510152025W i n d v e l o c i t y (m /s )

:00

0510*******-m i n u t e s m e a n w i n d v e l o c i t y (m /s )

Time (1 April 2003)

:00

17:3018

:00

:30

:00

:30

:0021:30

:00

17:3

018:00

:30

19:0

:30

20:30

21:0

:30

(a)

(b)

Fig.4.Measured wind velocities at tower top and deck level:(a)records from anemometers;and (b)3-min mean wind velocities.

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where b is the horizontal yaw angle and a the inclination angle of stay cable.The relative

yaw angle b n de?nes relative angle between the wind direction and the cable axis as shown in Fig.5.It is equal to zero when the wind direction is perpendicular to the cable axis and equal to 90o when the wind direction is parallel to the cable axis.A positive b n corresponds to the cable declining in the direction of the wind ?ow,while a negative b n corresponds to the cable inclining in the direction of the wind ?ow.4.Measurement results under rain-wind excitations

Three typical rain-wind excitation events were observed on 1–2April,18–19April and 28–29April 2003,respectively.In the ?rst rain-wind excitation event,large-amplitude stay cable vibration occurred with intermission from 4:00pm of 1April to 3:00am of 2April under a combined action of wind and rain.Fig.6shows the time history of in-plane and out-of-plane acceleration responses of the cable A12at L /6location during the ?rst rain-wind excitation event.In this event,the maximum cable acceleration response approached to 10g.Fig.7illustrates the corresponding wind velocity,wind direction and rainfall,respectively.By comparing Figs.6and 7,it is observed that when the rain stopped occasionally (19:10–20:10and 21:50–22:20of 1April)or the wind direction changed dramatically (0:05–0:25of 2April),the cable ceased to vibrate or reduced its response suddenly.The cable vibration amplitude varied signi?cantly with the weather condition and large-amplitude oscillation could not keep a stationary level for a long time.Fig.8shows the power spectral density of cable in-plane and out-of-plane acceleration responses.It is evident that the dominant mode of both in-plane and out-of-plane responses for the cable in this rain-wind excitation event is the third mode.The second and fourth modes also participate in the cable vibration,but the fundamental mode (?rst mode)has no signi?cant contribution to the rain-wind-excited cable response.

In the second rain-wind excitation event,large-amplitude stay cable vibration occurred discontinuously from 8:00am of 18April to 7:00am of 19April under simultaneous wind and rain.Fig.9shows a time segment of in-plane and out-of-plane acceleration responses of the cable A12at L /6location during the second rain-wind excitation event.The largest cable acceleration response was found about 10g.Fig.10illustrates the corresponding wind velocity,wind direction and rainfall,respectively.It is seen that when the wind velocity was low (much less than 10m/s)or the wind direction deviated considerably,the

Fig.5.De?nition of yaw and inclination angles.

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-10

-8-6-4-20246 8

10I n -p l a n e a c c e l e r a t i o n (g )

-10

-8-6-4-202468

10O u t -o f -p l a n e a c c e l e r a t i o n (g )

Time (18:30 1 April to 1:30 2 April 2003)

(a)(b)0:

18:3019:3020:3021:3022:3023:30

0:3018:3019:3020:3021:3022:3023:30

Fig.6.Time history of cable response in ?rst rain-wind excitation event:(a)in-plane acceleration;and (b)out-of-plane acceleration.

W i n d v e l o c i t y (m /s )

-10

01020304050607080

90W i n d d i r e c t i o n β* (d e g r e e )

Time (18:30 1 April to 1:30 2 April 2003)

18:3019:3020:3021:3022:3023:30

3018:3019:3020:3021:3022:3023:30

1:30

0:3

23:

1:3

Time (18:30 1 April to 1:30 2 April 2003)

(a)(b)0

12345

6R a i n f a l l (m m /h )

(c)Fig.7.Wind velocity,wind direction and rainfall in ?rst rain-wind excitation event:(a)wind velocity at deck level;(b)wind direction (b *);and (c)rainfall.

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cable exhibited very small response.It is worth mentioning that even in the case of small amount of rainfall,large cable vibration might be generated with proper wind velocity and wind direction.Fig.11shows the power spectral density of cable in-plane and out-of-plane acceleration responses.It is indicated again that the dominant mode of both in-plane and out-of-plane responses for this cable under rain-wind excitation is the third mode.

In the third rain-wind excitation event,large-amplitude stay cable vibration occurred from 11:00pm of 28April to 3:00am of 29April with combined wind and rain.Fig.12shows a time segment of in-plane and out-of-plane acceleration responses of the cable A12at L /6location during the third rain-wind excitation event.The maximum cable acceleration response exceeded 10g.Fig.13illustrates the corresponding wind velocity,wind direction and rainfall,respectively.It is evident that when the wind velocity was lower than a certain value or the wind direction deviated beyond a certain range,the cable ceased to vibrate.

-15-50510

I n -p l a n e a c c e l e r a t i o n (g )

-50510

15O u t -o f -p l a n e a c c e l e r a t i o n (g )

Time (18 April 2003)

10:4010:1011:1011:4012:10

-10-10-1510:10

10:40Time (18 April 2003)

11:1011:4012:10

(a)

(b)

Fig.9.Time history of cable response in second rain-wind excitation event:(a)in-plane acceleration;and (b)out-of-plane acceleration.

345678

910

24681012

14P S D o f i n -p l a n e a c c e l e r a t i o n (g 2/H z )

345678

910

3500P S D 0f o u t -o f -p l a n e a c c e l e r a t i o n (g 2/H z )

Frequency (Hz)

400030002500200015001000500(a)

(b)

Fig.8.Power spectral density of cable response in ?rst rain-wind excitation event:(a)in-plane acceleration;and (b)out-of-plane acceleration.

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Fig.14shows the power spectral density of cable in-plane and out-of-plane accelerations.The dominant mode of cable in-plane and out-of-plane responses in this event is the third mode again.It is,therefore,concluded that rain-wind-induced vibrations

024681012141618

20W i n d v e l o c i t y (m /s )

-30

90W i n d d i r e c t i o n β* (d e g r e e )

R a i n f a l l (m m /h )

12:10

Time (18 April 2003)

11:4011:1010:4010:10

Time (18 April 2003)

10:10

10:4011:1011:4012:10

Time (10:10 ~12:10, 18 April 2003)

10:10

10:4011:1011:4012:10

(a)(b)(c)

Fig.10.Wind velocity,wind direction and rainfall in second rain-wind excitation event:(a)wind velocity at deck level;(b)wind direction (b *);and (c)rainfall.

345678

910

0P S D o f i n -p l a n e a c c e l e r a t i o n (g 2/H z )

345678910

0500

1000

1500

P S D o f o u t -o f -p l a n e a c c e l e r a t i o n (g 2/H z )

8000600040002000

Frequency (Hz)

(a)(b)

Fig.11.Power spectral density of cable response in second rain-wind excitation event:(a)in-plane acceleration;and (b)out-of-plane acceleration.

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0:20-50510

I n -p l a n e a c c e l e r a t i o n (g )

0510

15O u t -o f -p l a n e a c c e l e r a t i o n (g )

-10-15Time (29 April 2003)

0:501:201:50

0:20

Time (29 April 2003)

0:501:201:50

-10-15-5(a)

(b)

Fig.12.Time history of cable response in third rain-wind excitation event:(a)in-plane acceleration;and (b)out-of-plane acceleration.

02468101214161820W i n d v e l o c i t y (m /s )

012345678910R a i n f a l l (m m /h )

0:20

Time (29 April 2003)

0:501:201:50

0:20

1:50

Time (29 April 2003)

1:200:500:20

Time (29 April 2003)

0:501:201:50

(a)

(b)

(c)Fig.13.Wind velocity,wind direction and rainfall in third rain-wind excitation event:(a)wind velocity at deck level;(b)wind direction (b *);and (c)rainfall.

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910

P S D o f i n -p l a n e a c c e l e r a t i o n (g 2/H z )

910

0500100015002000

2500P S D o f o u t -o f -p l a n e a c c e l e r a t i o n (g 2/H z )

90006000

3000

Frequency (Hz)

(a)

(b)

Fig.14.Power spectral density of cable response in third rain-wind excitation event:(a)in-plane acceleration;and (b)out-of-plane acceleration.

3691215

012345R M S i n -p l a n e a c c e l e r a t i o n (g )

3 6 9 1215

1 min mean wind velocity (m/s)

(a)

(b)

Fig.15.Cable RMS response versus 1-min mean wind velocity:(a)in-plane acceleration;and (b)out-of-plane acceleration.

-90

01234501234R M S i n -p l a n e a c c e l e r a t i o n (g )

1 min mean wind direction β* (degree)

-60

-30

-90

-60

-30

(a)

(b)

Fig.16.Cable RMS response versus 1-min mean wind direction (b *):(a)in-plane acceleration;and (b)out-of-plane acceleration.

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occurring in the cable A12are dominated by the third mode.However,it is worth mentioning that for different cables (even adjacent cables with similar con?gurations)in a cable-stayed bridge,their dominant modes in rain-wind-induced vibrations may differ with each other (Ni et al.,2002).

A statistical analysis has been conducted with the measurement data from all three rain-wind excitation events.For the convenience of comparison with measurement data obtained by other investigators (Main et al.,2001;Main and Jones,2001),the recorded cable acceleration signals were divided at 1-min intervals and the root-mean-square (RMS)response of the cable for each interval is obtained against the corresponding 1-min mean wind velocity,1-min mean wind direction and amount of rainfall,respectively.Figs.15–17plot the cable RMS response versus mean wind velocity,mean wind direction (b n )and rainfall,respectively.It is evident from the ?gures that the critical mean wind velocity for

010

12345R M S i n -p l a n e a c c e l e r a t i o n (g )

Rainfall (mm/h)

Rain (mm/h)

(a)

(b)

Fig.17.Cable RMS response versus rainfall:(a)in-plane acceleration;and (b)out-of-plane acceleration.

Fig.18.Cable RMS in-plane acceleration versus reduced wind velocity.

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rain-wind-induced cable vibrations is 6–14m/s and the critical mean wind direction (relative yaw angle b n )ranges from 101to 501.The rain-wind-induced cable vibrations with large amplitude occurred in light to moderate rain (less than 8mm/h).During the 45-day ?eld measurements,no large-amplitude cable vibration was observed in the case of heavy rain as wind velocity was often reduced to 1–2m/s when heavy rain came down.Fig.18illustrates RMS in-plane acceleration response versus reduced wind velocity (V /fD )where V is the wind velocity,f the vibration frequency and D the cable diameter.It is seen that the large-amplitude responses correspond to the reduced wind velocity ranging from 15to 35.Fig.19shows the relation of RMS in-plane acceleration to RMS out-of-plane acceleration of the cable under rain-wind excitations.It is found that the in-plane acceleration response amplitude is approximately two times the out-of-plane acceleration response amplitude.Table 1provides the measured turbulence intensities at the tower top and the deck level during the three rain-wind excitation events,where I u indicates the turbulence intensity in

51-m i n u t e R M S o f i n -p l a n e a c c e l e r a t i o n (g

)

1-minute RMS of out-of-plane acceleration (g)

0.5 1.5 2.5

Fig.19.Cable RMS in-plane acceleration versus RMS out-of-plane acceleration.

Table 1

Measured and design values of turbulence intensity Time

Location

Measured value Design value I u

I v I w I u I v I w April 1–2

16:51–21:36Deck (20p z p 30)0.09160.08550.07780.130.110.0716:51–21:36Tower (100p z p 150)0.06550.05280.07470.100.090.0522:10–02:48Deck (20p z p 30)0.09380.09880.08240.130.110.0722:10–02:48Tower (100p z p 150)0.07440.07670.07920.100.090.05April 18–19

7:31–9:15Deck (20p z p 30)0.09480.09120.06810.130.110.079:17–12:29Deck (20p z p 30)0.10600.08740.06550.130.110.0712:30–11:57Deck (20p z p 30)0.11780.09450.06010.130.110.07April 28–29

21:32–03:22

Deck (20p z p 30)

0.1119

0.0909

0.0561

0.13

0.11

0.07

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the longitudinal direction,I v the turbulence intensity in the lateral direction,and I w the turbulence intensity in the vertical direction.For comparison,the corresponding design values are also listed.

5.Analysis of response segments

As afore illustrated,the cable vibration response during a rain-wind excitation event does not perform as monotone increasing and then decreasing.Instead,it consists of numerous segments each of which accommodates rising and descending of the oscillation.For example,the recorded cable response in the ?rst rain-wind excitation event includes more than 20segments.In order to understand response properties of the cable at different segments,an analysis is conducted on three typical response segments obtained from the ?rst rain-wind excitation event.5.1.Response segment 1

Fig.20(a)illustrates time histories of the 1600-s response segment 1starting at 6:40pm of 1April 2003,while Fig.20(b)shows the 3-s records corresponding to large-amplitude quasi-stationary response within this segment.The maximum in-plane acceleration of the cable in this segment is about 1.5g and the corresponding wind velocity recorded at deck level is around 7–8m/s.Fig.21shows the power spectral density of in-plane and out-of-plane acceleration responses at different time intervals (each interval is 200s).It is evident that the dominant frequency of this response segment is 1.07Hz,which corresponds to the ?rst modal frequency of the cable.It is also observed from the spectral diagrams that the cable vibration is ?rst tuned to a dominant frequency,and then its amplitude increases rapidly with the effect of wind and rain.The acceleration response is mainly attributed to the ?rst three modes.The response segment 1is one of very few cases where the ?rst mode is the dominant mode of cable vibration.

Based on the measured acceleration response data,displacement response modal components of the cable in its quasi-steady state are estimated.Assume that the acceleration response in quasi-steady state can be approximated in the form of n components as

a ?

X n i ?1

A i sin eo i t tf i T,

(3)

where a is the cable acceleration response;A i the i th modal component of acceleration

response;o i the angular frequency of the i th mode;and f i denotes the phase angle of the i th mode.

Because rain-wind-induced cable vibration is dominated by only ?rst few modes,the acceleration response a can be well approximated with a combination of the ?rst 5–10modes.With the known modal frequencies o i ,the parameters A i and f i in Eq.(3)can be obtained by curve ?tting to the measured data.Then modal components of the displacement response are estimated from the identi?ed acceleration modal components.Fig.22shows a comparison of the measured in-plane acceleration response and the response reproduced by the identi?ed ?rst ?ve modal components for the 3-s quasi-

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stationary response.A good agreement between the measured and simulated responses is observed.

After modal components of the displacement response at the measurement point are obtained by the above method,the displacement response at arbitrary location of the cable in quasi-steady state can be estimated by modal combination with the aid of theoretical mode shapes.Fig.23illustrates the cable in-plane displacement response and its modal components at measurement point (L /6)and at mid-span for the 3-s quasi-stationary response.When the cable vibration is dominated by the ?rst mode,the maximum displacement response is expected to occur at mid-span.Therefore,from the plotted mid-span response curve we may evaluate the cable maximum displacement response for segment 1.It is seen from Fig.23that the ?rst mode contributes the overwhelming majority of displacement response.The cable displacement response amplitudes (peak-to-peak)are estimated to be 0.34m at L /6location and 0.66m at mid-span,which correspond to 2.9times and 5.5times diameter of the cable.Fig.24plots the locus diagrams of acceleration and displacement responses at L /6location.The displacement locus is

050

100150200T i F r H z )

P S D o f i n -p l a n e a c c e l e r a t i o n (g 2/H z )

0510152025

30T i F r z )

P S D o f o u t -o f -p l a n e a c c e l e r a t i o n (g 2/H z )

(a)(b)

Fig.21.Power spectral density of cable acceleration response (segment 1):(a)in-plane acceleration;and (b)out-of-plane acceleration.

8001200

-2-101

A c c e l e r a t i o n (g )

Time (s)

400

1600

800

Time (s)

801

802803

(a)(b)

Fig.20.Time history of cable acceleration response (segment 1):(a)signals in 1600s;and (b)signals in 3s.

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approximately a single ellipse,while the acceleration locus is much more complicated due to the exaggerated contribution of higher modes.

Due to the negative aerodynamic damping generated by combined wind and rain,the cable oscillation can be accelerated to very large amplitude within a few minutes.The net damping ratio of the cable in a mode can be expressed as the sum of mechanical damping ratio and aerodynamic damping ratio (Main and Jones,2001):

x net ?x mech tx aero ,

(4)

where x mech and x aero are the mechanical damping ratio and the aerodynamic damping ratio,respectively.Under speci?c combinations of wind and rain,x aero becomes possibly negative damping.As long as the negative aerodynamic damping exceeds the mechanical damping,the oscillation amplitude increases dramatically.The mechanical damping of the

00.1

I n -p l a n e d i p l a c e m e n t a t L /6 (m )

0.2

-0.1-0.2

800

801

Time (s)

802803

0.1

I n -p l a n e d i p l a c e m e n t a t L /6 (m )

0.2

-0.1

-0.2

800

801

Time (s)

802803

Mode 1 component Mode 2 component

Mode 3 component Mode 4 component Mode 5 component

Total displacement (a)(b)

Fig.23.Estimated cable displacement response (segment 1):(a)at L /6location;and (b)at L /2location.

801

802803

-15-10

-5

510

I n p l a n e a c c e l e r a t i o n (g )

800

Time (s)

https://www.sodocs.net/doc/a8562856.html,parison between measured and ?tted in-plane acceleration (segment 1).

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cable A12has been measured by ambient vibration and forced vibration tests (Ko et al.,2002).Table 2shows the identi?ed modal damping ratios for the ?rst ten in-plane vibration modes of the cable.

It is very dif?cult to accurately evaluate aerodynamic damping based on acceleration response measurement during rain-wind excitation.In this study,net negative damping ratio in a mode is roughly estimated from the identi?ed dominant modal displacement components at different instances.For response segment 1,the ?rst modal displacement component for each 3-s response is estimated at intervals of 50s.As shown in Fig.25,the displacement response amplitude increases continuously from 400to 850s.By ?tting the identi?ed displacement amplitudes with an exponentially increasing function,the net negative damping ratio for the ?rst mode is estimated to be 0.0548%.5.2.Response segment 2

Fig.26(a)illustrates time histories of the 1600-s response segment 2starting at 8:10pm of 1April 2003,while Fig.26(b)shows the 3-s records corresponding to large-amplitude quasi-stationary response within this segment.The cable largest in-plane acceleration response in this segment is nearly 2g and the corresponding wind velocity recorded at deck level is about 8m/s.Fig.27shows the power spectral density of in-plane and out-of-plane acceleration responses at different time intervals.It is found that the dominant frequency of this response segment is 2.14Hz,which corresponds to the second modal frequency of the cable.Similarly,it is observed that the cable vibration is ?rst tuned to a dominant frequency,and then increases dramatically in the dominant mode.Other modes have

0 1

-10

1 I n -p l a n e a c c e l e r a t i o n (g )

0 0

I n -p l a n e d i s p l a c e m e n t (m )

Out-of-plane acceleration (g)

0.5Out-of-plane displacement (m)

0.08

0.04-0.04-0.08

-0.1

0.1

0.2

-0.20.51.5-0.5-1.5-1

-0.5 (a)

(b)Fig.24.Vibration locus of cable response at L /6location (segment 1):(a)acceleration;and (b)displacement.

Table 2

Measured mechanical damping for in-plane vibration modes Mode no.

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th Damping ratio (%)

0.178

0.157

0.122

0.097

0.108

0.104

0.076

0.080

0.079

0.086

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319

contributions to the cable vibration only when the vibration amplitude is considerably large.The response is mainly attributed to the ?rst ?ve modes.

The displacement modal components for response segment 2are obtained by the same method as aforementioned.When the cable vibration is dominated by the second mode,the maximum displacement response should be at the location of quarter-span.Fig.28shows the cable in-plane displacement response and its modal components at measurement point (L /6)and at the location of L /4away from the lower end for the 3-s quasi-stationary period.It is seen from Fig.28that although the second mode is dominant,the ?rst mode also contributes signi?cantly to the cable response.The peak-to-peak displacement response amplitudes are 0.20m at L /6location and 0.25m at L /4location,which correspond to 1.7times and 2.0times diameter of the cable,respectively.Fig.29plots the locus diagrams of acceleration and displacement responses at L /6location.In this case,

400450550600650700750800850

0.04D i s p l a c e m e n t a m p l i t u d e o f m o d e 1 (m )

0.20.180.160.140.120.10.080.060.02

500Time (s)

Fig.25.Estimation of net damping ratio of ?rst mode (segment 1).

080012001600-3-2-10123A c c e l e r a t i o n (g )

120012011203

400Time (s)

Time (s)

1202

(a)

(b)

Fig.26.Time history of cable acceleration response (segment 2):(a)signals in 1600s;and (b)signals in 3s.

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0200

400600T i m

e F (H z )

P S D o f i n -p l a n e a c c e l e r a t i o n (g 2/H z )

050100150200T i m

F y (H z )

P S D o f o u t -o f -p l a n e a c c e l e r a t i o n (g 2/H z )

(a)

(b)

Fig.27.Power spectral density of cable acceleration response (segment 2):(a)in-plane acceleration;and (b)out-of-plane acceleration.

1200

-0.0500.2I n -p l a n e d i s p l a c e m e n t a t L /6 (m )

1203

0.150.10.05-0.1-0.15

Time (s)

12011202

1203

1200

Time (s)

12011202

(a)

(b)Fig.28.Estimated cable displacement response (segment 2):(a)at L /6location;and (b)at L /4location.

-10 1 -2-10 11.520

0I n -p l a n e d i s p l a c e m e n t (m )

I n -p l a n e a c c e l e r a t i o n (g )

Out-of-plane acceleration (g)

Out-of-plane displacement (m)

0.150.10.05-0.05-0.1

-0.05

0.05

0.5 -0.5-1.5

1.5

-1.5-0.50.5(a)(b)

Fig.29.Vibration locus of cable response at L /6location (segment 2):(a)acceleration;and (b)displacement.

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both the displacement and acceleration loci are twin-ellipse formation because of simultaneous participation of the ?rst and second modal components.

The net negative damping ratio for the second mode is roughly estimated from the identi?ed dominant modal displacement components.Fig.30plots the identi?ed second modal components from 850to 1150s at intervals of 50s.By ?tting the identi?ed displacement amplitudes with an exponential function,the net negative damping ratio for the second mode is estimated to be 0.016%.5.3.Response segment 3

Fig.31(a)illustrates time histories of the 3600-s response segment 3starting at 10:20pm of 1April 2003,while Fig.31(b)shows the 3-s records corresponding to large-amplitude

1250850

95010501150

0D i s p l a c e m e n t a m p l i t u d e o f m o d e 2 (m )

0.080.06

0.04

0.02

850

9501050Time (s)

1150Fig.30.Estimation of net damping ratio of second mode (segment 2).

400

800

05100510A c c e l e r a t i o n (g )

443

Time (s)-10-10

-5-10-5Time (s)

442

441

440

(a)

(b)Fig.31.Time history of cable acceleration response (segment 3):(a)signals in 3600s;and (b)signals in 3s.

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Field observations of rain-wind-induced cable vibration in cable-stayed dongting lake bridge

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