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Engineering Structures

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Wake-flow-induced vibrations of vertical hangers behind the tower of along-span suspension bridge

Wenli Chena,b,⁎, Donglai Gaoa,b, Hui Lia,b, Hui Huc

a Key Lab of Structures Dynamic Behavior and Control of Ministry of Education, Harbin Institute of Technology, Harbin 150090, Chinab Key Lab of Smart Prevention and Mitigation of Civil Engineering Disasters of Ministry of Industry and Information Technology, Harbin Institute of Technology, Harbin150090, Chinac Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA

A R T I C L E I N F O

Keywords:Tower wake flowVertical hangersVortex sheddingForced vibrationResonance

A B S T R A C T

Violent vibrations have been reported in the vertical hangers of long-span suspension bridges, especially forthose located in the vicinity of the towers. In the present study, an experimental investigation is performed tocharacterize the wake-flow-induced vibrations of vertical hangers behind the tower of a suspension bridge. Thetower column and vertical cable models are determined by using a long-span suspension bridge with a geo-metrical scale ratio of 1:10. Regular vortex shedding from the tower column model is detected in the near wakewith a Strouhal number (St) of 0.20, and the turbulence intensity of the wake flow behind the tower columnmodel is found to be quite high. Arranged at different stations behind the tower column model, the verticalcables experience violent vibrations. The vibration frequencies of the vertical cables are synchronized with thevortex shedding from the upstream tower model within a certain velocity range, during which severe cablevibrations take place. When the incoming wind speed becomes high, the cable vibrations exhibit multimodecharacteristics. It is also found that the vertical cables arranged at the rear are subject to the combined inter-ferences of the tower column model and the front cables. As a result, the vibration responses of the rear cablesare more violent than those of the front cables.

1. Introduction

Cylindrical structures are commonly used in structural and bridgeengineering. Examples of these structures are vertical hangers of sus-pension bridges, stay cables of cable-stayed bridges, and overheadpower lines. Because of relatively low stiffness and damping ratios,cable structures are susceptible to wind-induced vibrations. Vortex-in-duced vibration (VIV) and rain-wind-induced vibration (RWIV) of staycables were reported in previous studies [19,10]. The VIV of cylindricalstructures is a subject that has attracted extensive attention. Apart fromits relevance to practical engineering, it is also of great importance fromthe perspective of fundamental fluid dynamics. It is well known thatflow around a circular cylinder is characterized by flow separation andalternating vortex shedding downstream in the near wake. When thefluid velocity increases, the shedding frequency approaches the naturalfrequency of a given oscillating cylinder, and then the two frequenciessynchronize. This synchronization, which is generally referred to aslock-in [3], may occur and induce vibration of the cylinder. The VIV ofa circular cylinder was comprehensively reviewed by Williamson and

Govardhan [15], Sarpkaya [13], Gabbai and Benaroya [9], Bearman[2], and others.

For a circular cylinder immersed in the wake of another one, thewake interference from the upstream bluff body can result in com-pletely different fluid and structural behaviours in comparison with anisolated cylinder. The body-wake interaction and aerodynamic inter-ference between two closely separated circular cylinders have beenintensively studied, as reviewed by Zdravkovich [16] and Sumner [14].Zdravkovich [17] classified the flow past two tandem cylinders intothree regimes: (I) the extended-body regime, when the cylinders arearranged so close that the shear layers rolled up from the upstream oneshield the downstream one, and the gap flow between the cylinders isnearly stagnant; (II) the reattachment regime, where the shear layersare rolled up from the upstream cylinder, then reattach on the down-stream one and finally result in an insignificant gap flow; (III) the co-shedding regime, where the shear layers separate alternately in the gap,and the gap flow becomes significant in this case.

Bokaian and Geoola [4] experimentally investigated the response ofan elastically mounted circular cylinder immersed in the vicinity of an

https://doi.org/10.1016/j.engstruct.2018.05.049Received 16 August 2016; Received in revised form 13 April 2018; Accepted 14 May 2018

⁎ Corresponding author at: Key Lab of Smart Prevention and Mitigation of Civil Engineering Disasters of Ministry of Industry and Information Technology, Harbin Institute ofTechnology, Harbin 150090, China.

E-mail address: cwl_80@hit.edu.cn (W. Chen).

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0141-0296/ © 2018 Elsevier Ltd. All rights reserved.

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identical and fixed one. Two kinds of instability were found in thedynamic tests, namely, vortex resonance and galloping. The gallopingresponse occurred only when the downstream cylinder was well sub-merged in the near wake of the upstream one. The vortex sheddingfrequency was always found to lock to the oscillation frequency. Inaddition, the vibration characteristics were observed to remain un-affected although the turbulence intensity was changed. In addition, thegalloping amplitudes were found to be sensitive to the aspect ratio ofthe cylinder models.

Brika and Laneville [5] investigated the dynamic behaviours of along flexible cable in the wake of a stationary cylinder with a similargeometry. It was found that for tandem arranged cables, the dynamicresponse of the downstream cable was nonhysteretic, the synchroni-zation onset was at higher reduced velocities, and the synchronizationregion was wider than that of an isolated cylinder. Hover and Trian-tafyllou [11] studied a cylinder placed 4.75 diameters behind a sta-tionary cylinder of the same geometrical size. An in-line configurationwas found to produce large-amplitude galloping responses, and anupward extension of the frequency lock-in range of the reduced velocitywas observed. The frequency resonance onset was found at nearly thesame reduced velocity for an isolated circular cylinder, whilst a largephase change of the lift force occurred at higher reduced velocities.

Assi et al. [1] experimentally investigated the mechanism of wake-induced vibrations (WIV) of a pair of tandem cylinders. A typical WIVresponse was found to be characterized by a build-up of amplitudepersisting to high reduced velocities. This was different from the typicalVIV response that often occurs in a limited resonance range. The re-searchers proposed that the WIV of the downstream cylinder was ex-cited by the unsteady vortex-structure interactions between the down-stream cylinder and the upstream wake: coherent vortices interferingwith the downstream cylinder could induce fluctuations in the fluidforce that were not synchronized with the cylinder motion. A phase lagbetween the cylinder motion and the fluid force was favourable andsupported the positive energy transferred from the flow to the structureto maintain the vibrations. On the other hand, if the unsteady vorticeswere removed from the wake of the upstream cylinder, the WIV wouldhardly be generated.

In the present study, a bundle of vertical hangers of a suspensionbridge suffer from wake interference from the upstream tower, and thewake-flow-forced buffeting responses are experimentally investigated.This paper is organized as follows. The engineering background andproblem are described in Section 2, the model configuration and ex-perimental details are given in Section 3, the wake flow characteristicsbehind the tower column are presented in Section 4, the dynamic re-sponses of vertical cables subject to tower wake flow are presented inSection 5, and some discussions and concluding remarks follow inSection 6.

2. Background and problem description

The long-span suspension bridge investigated in the present studyjoins Zhoushan Archipelago to Ningbo, Zhejiang Province, P.R. China,as illustrated in Fig. 1. With a main span of 1650m, this is the longestbridge in China and the second longest suspension bridge in the world(after Akashi Kaikyō bridge, main span of 1991m, Japan). The towersrise to a height of 236.5 m above the sea. They support the main cablesfrom which the twin box girders are suspended through hundreds ofvertical hangers. Each bundle of hangers is composed of four separatesteel wire ropes, as shown in Fig. 2. Instead of an isolated verticalhanger, a bundle of separated hangers is usually adopted in suspensionbridges. Examples include Akashi Kaikyō bridge in Japan and theGolden Gate bridge in the U.S.

The length of the longest vertical hangers of the long-span suspen-sion bridge, which are located in the vicinity of the bridge tower, isabout 169.7m. Because of relatively low stiffness and damping ratios,these long and flexible vertical hangers of the long-span suspension

bridge are sensitive to wind and may suffer from wind-induced vibra-tions. Violent vibrations of the vertical hangers have been observed. Insitu monitoring data and video records reveal that characteristics of theviolent vibrations occurring in the vertical hangers of the suspensionbridge can be summarized as follows: (1) violent vibrations of thevertical hangers are frequently observed in a wind speed range of14–18m/s; (2) in the vicinity of the bridge tower, vibrations of thevertical hangers are particularly violent, i.e. the longest verticalhangers; and (3) frequent collisions between vertical hangers are wit-nessed and recorded. It is noteworthy that the violent vibrations ofvertical hangers have also been reported at Akashi-Kaikyō bridge[7,12,8]. It is observed that the excessive vibrations of the downstreamropes were excited by the upstream ropes, indicating the occurrence ofwake-induced flutter. Significant vibrations were recorded especiallyduring typhoons, and the high-damping rubber dampers installed tosuppress vortex-induced vibrations of these ropes were found to bedamaged. To improve their aerodynamic characteristics, the verticalhanger ropes were wrapped with helical wires of 10mm in diameter[7].

Violent oscillations of the vertical hangers and their collisions raisedconcerns regarding safety and durability in the bridge engineeringcommunity. Some possible excitation mechanisms were proposed toexplain the violent oscillations of the vertical hangers, especially forthose in the vicinity of the bridge tower. Conventional vortex-inducedexcitation as a possible mechanism responsible for the violent oscilla-tions was excluded for the simple reason that the onset velocity of thelock-in range is quite low. The natural frequency of the hangers in thevicinity of the tower is about 0.45–0.52 Hz (dependent on the length),and their diameter is 0.088m. The hangers are observed to vibratemainly with first-mode frequency. If the cylindrical structures’ Strouhalnumber (St) is assumed to be 0.2, then the onset velocity of the vortex-induced vibration is estimated to be 0.2m/s. When the bridge is subjectto crosswinds, wake galloping responses of the two rear vertical hangersare likely to take place owing to the aerodynamic interference of thefront hangers. However, wake galloping cannot fully explain the violentoscillations since the spacing between the front and rear verticalhangers is about nine times the hanger diameters, at which the up-stream interference is believed to be not that substantial. Zhang and Ge[18] suggested that all vertical hangers were hung from the maincables, and that the buffeting responses of the main cables wouldtherefore be an excitation mechanism of the vibration of the verticalhangers. The researchers proposed a ‘main-cable buffeting induced re-sonance’ to explain the violent oscillations of the vertical hangers ob-served in the suspension bridge. Some would argue that it is still un-clear whether the main-cable buffeting could excite such violentvibrations in the vertical hangers.

Previously proposed mechanisms, including the wake galloping re-sponse and main-cable buffeting-induced resonance, cannot fully ex-plain this phenomenon. However, the excitation mechanism of thevertical hangers and the role of the bridge tower have not been thor-oughly clarified. Further investigations are therefore needed. As men-tioned above, violent oscillations of the vertical hangers are observed inthe vicinity of the bridge tower, so the interference effects of the towerwake cannot be neglected when the direction of the incoming wind isparallel or oblique to the bridge axis. Under this circumstance, theneighbouring vertical hangers are immersed in the wake of a bridgetower column and cannot be exempted from body-wake interference. Inthe present study, the interference effects of the bridge tower column onthe vertical hangers are experimentally investigated. An excitationmechanism, i.e. tower wake flow forced vibrations, is proposed to ex-plain the violent oscillations of the vertical hangers when the incomingwind attacks along the direction of the bridge axis.

3. Experimental details

An experiment was conducted in the larger test section of the Joint

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Laboratory of Wind Tunnel and Wave Flume, Harbin Institute ofTechnology, P. R. China. The dimensions of the test section are as fol-lows: width of 6.0m, height of 3.6m, and length of 50m. During thewind tunnel tests, the maximum wind speed is 30m/s.

3.1. Tower column model

Each tower of the long-span suspension bridge consists of two se-parated vertical columns connected by three horizontal couplingbeams, as can be seen from Fig. 1. The main cables are supported on topof the vertically erected columns. When the incoming wind is parallel tothe bridge axis, the vertical hangers are influenced by a single towercolumn. With a geometrical scale ratio of 1:10, a tower column model isused to reproduce the upstream tower interference, as shown in Fig. 3.The section of the tower column model is 0.65m×0.85m, and itsheight is 3.3 m. The solid blockage of the tower column model is

estimated to be 9.93%, which is relatively high for wind tunnel tests.However, the blockage effects on the wake flow can be neglected in thepresent study since it has enough space for the near-wake vortexshedding from the tower column model. It is noteworthy that the solidblockage of the hanger model is extremely low. The corners of thecolumn model are cut, as shown in Fig. 4. The tower column model ismade of timber plates with a thickness of 12 mm to ensure its bodyintegrity and rigidity. The tower column model is placed on a base plateelevated 0.1 m over the tunnel floor, and is fixed to the tunnel ceiling. Itshould be noted that the bridge towers in situ are not strictly stationarywhen subject to wind loads, but their vibrations are negligible incomparison with the violent oscillations of the flexible vertical hangers.Therefore, the tower column model is firmly fixed in the present study,and no vibration has ever been detected during wind tunnel tests.

Prior to wind tunnel tests on the vibration responses of the verticalcables subject to tower wake flow, the wake characteristics behind the

Fig. 1. Glimpse of investigated long-span suspension bridge.

Fig. 2. Main cables and vertical hangers viewed from top of south tower.

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tower column were first measured by using hot-wire anemometry(54N80 Multichannel, Dantec Dynamics). By analysing the fluctuatingvelocities, the mean velocity of the wake flow, turbulence intensity, andvortex shedding frequency from the tower column model could beobtained. As shown in Fig. 4, the red1 dots denote the positions of a hot-wire anemometer placed behind the tower model. At each point, theprobe of the hot-wire anemometer is fixed at 1.2m above the windtunnel floor. To explore the possible influence of the vertical cables onthe vortex shedding process from the tower column mode, during thewind tunnel tests on the vibration responses of the vertical cablessubject to tower wake flow, the vortex shedding frequency from thetower column was simultaneously measured. In the hot-wire measure-ments, the sampling rate of the hot-wire anemometer was 1000 Hz, andthe sampling time for each case was 60 s.

3.2. Vertical hanger models

Four identical cables were employed to simulate a bundle of verticalhangers. With an external diameter of 8.8mm, each vertical cable has atotal length of 3000mm, as shown in Fig. 3. It should be mentioned thatthe diameter of the cables is also defined by a geometrical scale ratio of

1:10. Each cable model is a steel wire rope coated with a smooth PVCskin. These vertical cables are firmly supported by four independentbrakes. With the same geometrical ratio of 1:10, four vertical cables arefixed in a rectangular arrangement. The spacing in the cross-flow (CF)direction is 60mm and 30mm in the in-line (IL) direction, as illustratedin Fig. 5. The mass per unit length of the cable model was calculated tobe 0.158 kg/m. Moreover, two identical National Advisory Committeefor Aeronautics (NACA) 0012 aerofoil models (with a chord lengthc= 567.1 mm) are placed at both ends of the vertical cables to guidethe flow and eliminate the end effects, as can be seen in Fig. 3.

A bundle of four vertical cables is arranged in four positions alongthe axis, as denoted by the red dots in the grey dashed-dotted line inFig. 4, to ensure the incoming wind is parallel to the tower-cable con-figuration. First, the vertical cables are arranged 2400mm away fromthe tower column model. Based on the distance from the tower columnmodel, this case is abbreviated as a test case of D2400. When the cablesare shifted downstream to 4200mm, 6000mm, and 7800mm awayfrom the tower column model, these test cases are abbreviated as casesD4200, D6000, and D7800, respectively. It is worth noting that thespacing in the wind tunnel test was also defined by a geometrical scaleratio of 1:10. Only four distances were considered in the present studybecause the fifth bundle of vertical hangers of the long-span suspensionbridge are 96m away from the tower, and thus, the tower interferenceson this station are minimal and even negligible. To provide a baselinefor the investigation of interference effects, four vertical cables and an

Fig. 3. Experimental setup of vertical cables behind tower model (case D4200).

Fig. 4. Top view of tower column model and positions of hotwire measurements.

1 For interpretation of color in Fig. 4, the reader is referred to the web version of thisarticle.

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isolated cable were also tested without the upstream tower columnmodel.

Eight accelerometers (B&K 4507B) were employed to measure thecross-flow (CF) and in-line (IL) vibration responses of these four verticalcables subject to the tower wake flow. The accelerometers were allfixed at 1.0m over the lower end and 2.0m from the upper end, asshown in Fig. 3. The sampling rate was 1000 Hz, and the sampling timeof each case was 60 s. Free vibration tests were conducted in still air toobtain some structural parameters of the vertical cable models. Basedon the free vibrational time histories, the fundamental natural fre-quencies (fN) of the cable models were calculated to be 3.0 Hz, and theirfrequency deviations were limited to a maximum of 1% by adjusting thetension forces of the cables. In addition, the damping ratios of the cablemodels were measured as about 0.58%. It can be seen that the cablemodels’ mass and damping ratios were quite low. Two free vibrationtests (both before and after wind tunnel tests) were conducted for eachcase to ensure that no tension slack had occurred in the steel wire ropesduring the test. Since the vertical cables are flexible, rather than rigidand segmental, their higher modal frequencies can be observed in thefree vibration tests. The nth-order modal frequencies of the vibratingvertical cables were found to be multiples of the fundamental naturalfrequency and were very close to 3n Hz. It should be noted that thethird-order modal frequencies of the vertical cables could not be mea-sured owing to the position of the accelerometers (at 1/3 the height ofthe vertical cables).

With an increment of 0.4m/s, the inflow velocities investigated inthe wind tunnel tests were in the range of 7.9–16.3 m/s. It should bementioned that the incoming airflow was smooth and the impacts ofatmospheric boundary layer (ABL) were neglected in the present study.

4. Wake characteristics behind the tower column

The characteristics of the wake flow behind the tower column modelwere first measured by using a hot-wire anemometer, as shown inFig. 4. The power spectra and dominant frequencies of the vortexshedding from the upstream tower column model at different inflowvelocities were identified through fast Fourier transforms (FFTs). Fig. 6illustrates the measured vortex shedding frequencies for cases D2400,D4200, and D6000 under different inflow velocities. The fitted linerepresents a Strouhal number (St) of 0.20. It is worth noting that whenthe hot-wire anemometer was placed 7.8 m behind the tower columnmodel, i.e. for the case of D7800, no regular vortex shedding frequencywas detected at the five monitoring positions. This result suggests thatthe vortex shedding is substantially weakened at these positions owingto the longer distance from the tower column model.

The mean velocities at different locations behind the tower columnmodel are illustrated in Fig. 7. It can be seen that the mean velocities ofthe wake flow witness a notable decrease, especially in the case ofD2400, owing to solid blockage of the upstream tower column model.In addition, for cases D4200, D6000, and 7800, the mean velocitydistribution was found to be quite similar. Fig. 8 illustrates the varia-tions in the turbulence intensity at different stations behind the towercolumn model. For the test case of D2400, the turbulent intensity ismeasured at higher than 30%, indicating a highly turbulent flow in thetower model wake. In addition, it is found that a longer distance fromthe upstream interfering model contributes to a lower turbulence in-tensity level. When the measurement position is 7.8 m away from thetower column model, the turbulence intensity is measured at approxi-mately 12.8%. In comparison with the incoming turbulence intensity,the interfering tower column model acts as an upstream turbulencegenerator. It should be mentioned that both the mean velocities andturbulence intensity are measured at locations along the axis at whichthe vertical cables are arranged for different test cases.

5. Dynamic responses of vertical cables subject to tower wakeflow

First, some important nomenclatures should be clarified: f is thedominating frequency of the vibrating cable; fV denotes the frequency

Fig. 6. Vortex shedding frequency from tower against velocity of incomingairflow.

Fig. 7. Mean value of velocities at different stations behind tower columnmodel.

Fig. 5. Schematic of bundle of vertical cables in wake.

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at which the vortices shed from the upstream interfering tower columnmodel, as presented in Fig. 6; and fN is the natural frequency of thecable that vibrates in the still air, i.e. fN=3.0 Hz. It is worth noting thatduring the wind tunnel tests, vibrations of the vertical cables are mostly

dominated by the first modal frequency, so f takes the value of the firstmodal frequency of the vibrating cables in the present study.

5.1. Wake flow forced vibration responses

The cross-flow (CF) and in-line (IL) vibration responses of the ver-tical cables were measured by using eight accelerometers for differenttest cases. The root-mean-square (RMS) values of the resultant com-bined accelerations were calculated and are presented in Fig. 9. Whenthe vertical cables were exempted from the tower column interference,their vibration responses were found to be similar to that of an isolatedcable. When the vertical cables are subject to tower wake flow, it can beseen from the figure that the vibration responses become much moreviolent owing to the upstream interfering tower column model. Itshould be mentioned that the wake flow velocity measured behind thetower column model experiences a decrease owing to the solid blockageeffect, as can be seen in Fig. 7.

Two front cables (see cable 1 and cable 2 in Fig. 5) exhibit simila-rities in vibration responses, and so do the rear two cables (see cable 3and cable 4 in Fig. 5). In addition, the vibration responses of the rearcables are found to be larger than the front cables. Fig. 10 compares thevibration response of one front cable (cable 1) and a rear one (cable 3)without tower column interference. It is shown that the rear cablesuffers interference from the upstream one, and its vibration is ampli-fied. Therefore, the rear cables in Fig. 9 suffered from the combined

Fig. 8. Variations of turbulence intensity at different stations behind towercolumn model.

Fig. 9. RMS of accelerations of vertical cables: (a) cable 1, (b) cable 2, (c) cable 3, and (d) cable 4.

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interference of the upstream tower column model and the cables ar-ranged in front. As a result, their dynamic responses were greatly am-plified.

Moreover, for the cables arranged at 6.0 m and 7.8 m away from thetower column model, i.e. for the test cases of D6000 and D7800, their

vibration responses were not as severe in comparison with the test casesof D2400 and D4200. As was revealed in Section 4, the vortex sheddingis weakened, and the wake flow velocity gradually recovers at furtherdownstream positions.

When the inflow velocity is low and the frequency of the upstreamvortex shedding is below the natural frequency of the cables, the cablevibration responses slowly grow with the wind velocity. Owing to theturbulent wake flow, the vertical cables exhibit buffeting-like re-sponses. As the inflow velocity increases to 9.9m/s, the frequency ofthe upstream vortex shedding approaches the natural frequency of thecables. Then, their frequencies synchronize, and frequency resonancestend to develop. In the velocity range of 9.9–13.9m/s, the cable vi-bration responses are found to be quite violent. In addition, frequentcollisions between the front and rear cables can be observed within thevelocity range of 9.9–13.9 m/s during the wind tunnel tests. Within thisvelocity range, the vibration responses of the vertical cables are re-ferred to as tower wake flow forced buffeting. When the inflow velocitycontinues to increase, the frequency of the upstream vortex sheddingincreases correspondingly. As a result, the frequency synchronization isweakened, and the vibration responses experience a gradual decrease.However, the downward tendency dose not last for long, since buffetingresponses occur. As a consequence, the RMS values of the resultantaccelerations are found to climb again.

5.2. Frequency detuning

The frequency ratios of f/fN under different test conditions are il-lustrated in Fig. 11. In the wind tunnel tests for the case of D2400, thevertical cable models are found to vibrate with a frequency larger thantheir natural frequency, i.e. f/fN > 1, owing to the increased axialtension force of the cable. As the inflow velocity continues to increase,

Fig. 11. f/fN ratios against velocity of incoming airflow: (a) D2400, (b) D4200, (c) D6000, and (d) D7800.

Fig. 10. Vibration responses of one front cable and one rear cable withouttower model interference.

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the cable vibrating frequency ratio (f/fN) experiences a steady rise [seeFig. 11(a)]. When the velocity of the incoming airflow reaches 13.1m/s, the vibration frequencies of the cables (f) are about 1.39 times thenatural frequency (fN). However, when the velocity of the incomingflow increases from 13.9 m/s to 14.3 m/s, the vertical cables witness asudden and remarkable drop in vibration frequencies, as can be ob-served in Fig. 11(a). For the test case of D4200, the frequency detuningexhibits a good similarity to the case of D2400. Moreover, the fre-quency ratios fV/f are calculated to investigate the correlation betweenthe dynamic behaviours of the vertical cables and the vortex sheddingfrom the upstream tower column model, as shown in Fig. 12. The fV/fratios show a similar upward tendency with an increase of inflow ve-locity for the test cases of D2400, D4200, and D6000. For the cases ofD2400, the fV/f ratios are found to be close to 1.0 in a wide velocityrange of 9.9–13.9 m/s.

As was revealed by Fig. 6, the frequency of the vortices shed fromthe tower column model grows monotonously with the inflow velocity.Therefore, the vibration frequencies of the vertical cables are mainlydominated by the upstream vortex shedding within this velocity range.When the cable models are placed 7.8 m downstream behind the towercolumn model, the fV/f ratios exhibit a nearly linear increase, and thedomination is barely observed. It is worth noting that the fV values inFig. 12(d) are obtained by calculation according to the Strouhal number(fV=USt/D) rather than direct hot-wire measurements. As mentionedabove, the hot-wire measurements failed to identify regular vortexshedding at this location. Therefore, it can be concluded that for thecase of D7800, the influence of upstream vortex shedding is remarkablydiminished but did not completely disappear.

It should also be observed from Fig. 12 that the resonance mainlytakes place when fV/f=0.95–1.05. The natural frequency of the

vertical hangers immersed in the wake of the bridge tower is in therange of 0.45–0.52 Hz. Then, the upstream vortex shedding frequencywhen resonance occurs is estimated to be 0.40–0.54 Hz. The presentstudy also indicates that the Strouhal number of the bridge towercolumn suffering from parallel airflow is 0.20. As a result, the resonancewind speed is estimated to be U= fV D/St=(13.0–17.6) m/s, which isroughly consistent with field observations.

5.3. Vibration responses of the vertical cables in the case of D4200

For different cases in the wind tunnel tests, the vibration responsesof the vertical cables subject to upstream tower column interferenceexhibit similarities. In this section, the dynamic responses of the ver-tical cables for the test case of D4200 are presented and discussed toreveal some insight into the wake-flow-induced cable vibrations.

A time-frequency analysis of the vibration response was conductedto investigate the change in participating frequency mode with time.Three typical velocities of the incoming airflow were chosen and ana-lysed: 9.2m/s, 11.9m/s, and 16.3m/s. The time-frequency analysis [6]was carried out by using a wavelet transform. The wavelet adoptedherein was the Gabor wavelet, which is a Gaussian modulated by acomplex exponential defined as

⎜ ⎟= ⎛⎝

− + ⎞⎠

ψ tσ π

tσ

πjf t( ) 1( )

exp2

22 1/4

2

2 0 (1)

where f0 is the central frequency, and σ determines the frequencybandwidth. Fig. 13 shows the time-frequency analysis for the CF vi-brations of cable 2 and cable 4 at a wind speed of 9.2 m/s. For cable 2,which is arranged in front, the first modal frequency (f1= 3.20 Hz)dominates the time history and the second modal frequency

Fig. 12. fV/f ratios against velocity of incoming airflow: (a) D2400, (b) D4200, (c) D6000, and (d) D7800.

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(f2= 6.37 Hz) appears for a limited period only. Thus, cable 2 experi-ences nearly single-mode buffeting. It is worth noting that the firstmodal frequency of the vibrating cables becomes 3.20 Hz, which ishigher than the natural frequency (fN=3.0 Hz) measured in still air.For cable 4 arranged at the rear, the first modal frequency(f1= 3.20 Hz) also dominates the vibration, while the second(f2= 6.28 Hz) and fourth (f4= 12.75 Hz) modal frequencies jointly andintermittently appear in the time history.

In Section 3.2, it was suggested that the third-order modal fre-quency of the vertical cables could not be measured owing to the po-sition of the accelerometers. Therefore, it can be concluded that be-cause of the combined interference of the tower column model and thecable arranged in front, cable 4 at the rear experiences multimodebuffeting responses. With respect to the IL vibration of cable 2, thesecond modal frequency dominates (f2= 6.40 Hz) the vibration timehistory, as shown in Fig. 14. In addition, the first (f1= 3.43 Hz) andfourth modal (f4= 13.5 Hz) frequencies with smaller amplitudes arenoticeable in the vibration time history of cable 2. The bottom panel ofFig. 14 shows that the IL vibration of cable 4 is also mainly dominatedby the second modal frequency (f2= 6.40 Hz), which is similar to thefront cable 2.

The participating modes of cable 4 are more complex, and somemuch higher modal frequencies are found in the time history. This in-dicates a multimode buffeting response in the in-line direction. In ad-dition, the vibration responses of the rear cables are found to be largerthan those of the front cables. When the high-speed wake flow passesthe front cable, whose diameter is rather small, some high-frequencyand small-scale vortices can be generated and transported downstream.Because the rear cables suffer from the combined interference of theupstream tower column model and the cables arranged in front, theirdynamic responses are further amplified.

The frequency analysis in Section 5.2 revealed that when the velo-city of the incoming airflow is 11.9 m/s, the wake flow forced buffeting

of vertical cables takes place. Fig. 15 plots the time-frequency analysisfor the CF vibrations of cable 2 and cable 4. It can be seen that the time-frequency behaviours of the cable vibrations in the cross-flow directionare dominated by the first modal frequency (f1= 3.75 Hz) in the timehistories of both cables. Likewise, higher modal frequencies are foundto participate in the CF vibration of cable 4 at the rear. The time-fre-quency analysis for the IL vibrations demonstrates that the first modalfrequencies (f1= 3.69 Hz) synchronously dominate in the time historiesof cable 2 and cable 4, as shown in Fig. 16. Figs. 11 and 12 reveal that atan inflow velocity of 11.9 m/s, the vibration frequency of the cablesexperiences a lock-in onto the vortex shedding frequency from theupstream tower column model. As a result, the cables vibrate with theirfirst modal frequency in both the CF and IL directions. It should also benoted that for cable 4, the participation of higher modal frequencies islimited, indicating a strong frequency lock-in by the upstream towercolumn model.

When the inflow velocity reaches 16.3 m/s, the time-frequencyanalysis results are illustrated in Figs. 17 and 18. For cable 2, the secondmodal frequency (f2= 7.07 Hz) dominates the vibrations in the CF di-rection, but in the IL direction, the cable vibration is jointly dominatedby the second (f2= 7.17 Hz) and fourth (f2= 14.68 Hz) modal fre-quencies. Fig. 7 reveals that the vortex shedding frequency from theupstream tower column model is lower than 5.0 Hz at this inflow ve-locity, so the second-order wake flow forced buffeting of the cablemodels has not been fully developed. While the CF vibration of cable 4is dominated by the second modal frequency (f2= 7.07 Hz), its IL vi-bration exhibits typical buffeting responses. Because of the combinedinterference of the tower column model and the cables arranged infront, the IL vibration of cable 4 is no longer simply dominated by somelower-order modal frequency. Instead, multiple modal frequencies arefound to participate actively.

It should be mentioned that in the present study, acceleration sig-nals were employed to reveal the vibration characteristics of the

Fig. 13. Time-frequency analysis of CF vibrations: cable 2 (top panel) and cable4 (bottom panel) at inflow velocity of 9.2 m/s.

Fig. 14. Time-frequency analysis of IL vibrations: cable 2 (top panel) and cable4 (bottom panel) at inflow velocity of 9.2 m/s.

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Fig. 17. Time-frequency analysis of CF vibrations: cable 2 (top panel) and cable4 (bottom panel) at inflow velocity of 16.3 m/s.

Fig. 18. Time-frequency analysis of IL vibrations: cable 2 (top panel) and cable4 (bottom panel) at inflow velocity of 16.3 m/s.

Fig. 15. Time-frequency analysis of CF vibrations: cable 2 (top panel) and cable4 (bottom panel) at inflow velocity of 11.9 m/s.

Fig. 16. Time-frequency analysis of IL vibrations: cable 2 (top panel) and cable4 (bottom panel) at inflow velocity of 11.9 m/s.

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Fig. 19. Trajectories partitioned into modes at 9.2 m/s (cable 2, D4200).

Fig. 20. Trajectories partitioned into modes at 11.9 m/s (cable 2, D4200).

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vertical cables; this could lead to an exaggeration of higher-frequencycomponents. If displacement is adopted to evaluate the vibration, thedomination of higher-mode frequencies will probably be weakenedsubstantially, and the first-mode vibrations will become more profound.

Figs. 19–21 illustrate the motion trajectories partitioned intodominant modes of cable 2 for the test case of D4200 at three differentwind speeds. It can be seen that the cable experienced complex vibra-tions. At an incoming wind speed of 9.2m/s, the first four vibrationmodes were found to participate actively, and the first mode wasdominant. In the resonance region, the first-mode cross-flow vibrationwas the most significant. When the incoming wind speed reaches13.6 m/s, higher-mode vibrations were observed. It should be notedthat the third mode is always difficult to measure owing to the presentlocation of the accelerometers.

6. Concluding remarks

In the present study, wind tunnel experiments were conducted toinvestigate the vibrations of vertical hangers behind the tower of asuspension bridge. The following conclusions were obtained.

Regular vortex shedding from the tower column model was identi-fied in the near wake with a Strouhal number (St) of 0.20. In addition, itwas noted that the upstream tower column model acts as a turbulencegenerator, and the turbulence intensity of the near wake flow behindthe tower column model was found to be quite high.

Owing to the upstream interference and the turbulent wake flow,the cable vibration responses are greatly amplified. When the frequencyof vortex shedding from the upstream tower model approaches the vi-bration frequency of the cable within a certain velocity range, vibrationresonances tend to develop, and collisions between vertical cables areobserved. Within the resonance velocity range, the vibration fre-quencies are found to be larger than their natural frequency owing tothe substantially increased tension forces.

When the incoming wind speed becomes high, the cable vibrationsexhibit multimode characteristics. It is observed that with a largerspacing from the tower column model, the cable models suffer fromfewer upstream interference effects. It can also be noted that the ver-tical cables arranged at the rear are subject to the combined inter-ference of the tower column model and the cables in front. Therefore,the vibration responses of the rear cables are more violent than those ofthe front cables.

Acknowledgements

This research work was funded by the National Natural ScienceFoundation of China (NSFC) through Grants 51378153, 51578188, and51722805, and the Fundamental Research Funds for the CentralUniversities (HIT. BRETIII. 201512 and HIT. BRETIV201803).

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