Effectiveness of cable networks of various configurations in suppressing stay-cable vibration

Engineering Structures 31 (2009) 2851–2864

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Engineering Structures

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Effectiveness of cable networks of various configurations in suppressingstay-cable vibrationLuca Caracoglia a,∗, Delong Zuo ba Department of Civil and Environmental Engineering, Northeastern University, Boston, MA 02115, USAb Department of Civil and Environmental Engineering, Texas Tech University, Lubbock, TX 79409, USA

a r t i c l e i n f o

Article history:Received 28 August 2008Received in revised form9 July 2009Accepted 9 July 2009Available online 26 July 2009

Keywords:Cable-stayed bridgesStay-cable vibrationCable networksViscous dampersNumerical methodsFull-scale measurements

a b s t r a c t

Cross-ties have often been used as a passive mitigation system for wind- and rain-wind-inducedvibrations of stay cables on cable-stayed bridges. Recently, dampers have also been incorporated in cablenetworks formed using cross-ties in the hope of combining the mechanisms of both systems, and toachieve improvedmitigating performance. This paper presents the results from a recent study conductedto investigate the effectiveness of cross-tie and damper-cross-tie systems of various configurations. Thisinvestigation,which is based onboth analytical formulation and full-scalemeasurement, represents a steptoward the development of guidelines for the design of cable-networks for stay cable vibrationmitigation.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Stay cables have often been observed to exhibit large-amplitudevibrations under excitation from wind and especially during rain-fall. Such vibrations have been a concern for bridge designers andowners as they pose a significant threat to the safety and ser-viceability of cable-stayed bridges. Although considerable progresshas been made in identifying the nature of the vibrations, to thepresent date, the phenomenon still eludes fundamental under-standing. As a result, different excitation mechanisms have beenpostulated to be the cause of the vibrations. For example ProfessorMatsumoto [1] suggested that rain-wind-induced vibration is re-lated to interaction between along-wind Kármán vortices and axialvortices along the cylinder axis at a frequency that is much lowerthan the nominal Strouhal frequency; MacDonald and Larose [2]indicated that rain-wind-induced vibration can be related to a typeof dry-cable galloping in the critical Reynolds number range. Astudy conducted recently [3], based on full-scalemeasurement andwind tunnel tests, proposed that the prevalent rain-wind-inducedvibration is likely related to a type of dry-cable vibration due tothree-dimensional vortex shedding.While the mechanism of the vibrations remains unclear,

mitigation strategies have been developed based on the current

∗ Corresponding author. Tel.: +1 617 373 5186; fax: +1 617 373 4419.E-mail address: [email protected] (L. Caracoglia).

0141-0296/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.07.012

understanding to prevent the onset of the vibrations. One strategyuses secondary restrainers (also known as cross-ties) to connectadjacent stays to form cable networks so that the energy in a staycan be distributed to the highermodes and to the other stays in thenetwork (e.g., [4]). This strategy has been widely used in practicedue to its seemingly simplemechanismand easiness to implement,despite some concerns about its negative effect on the aestheticsof cable-stayed bridges. It has been proven to be effective in mostsituations by both analytical formulation and field observations [5].However, it has been revealed that cross-ties, as an inherent in-plane mechanism, are incapable of controlling out-of-plane cableoscillation (i.e., in the plane orthogonal to the primary cable plane).Also, cross-ties are an energy redistributionmechanism [4], and donot provide a direct source of energy dissipation. If not properlydesigned, their performance in suppressing some specific modesof vibration can be limited [6]. Due to these reasons and the factthat failures of restrainers have been occasionally observed in thefield [3], designers of stay-cablemitigation systems sometimes optfor other vibration control strategies, such as individual dampersconnected to the deck in the proximity of each stay anchorage.Traditionally, dampers and cross-ties have primarily been used

independently. In consideration of the limitations of the cross-tiesstated earlier, it appears natural to combine the energy dissipationproperty of dampers and the energy redistribution capability ofcross-ties to form a hybrid system by adding dampers to cablenetworks formed using cross-ties [7]. Example applications of thisstrategy can be found on the Normandie Bridge in France and

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2852 L. Caracoglia, D. Zuo / Engineering Structures 31 (2009) 2851–2864

Notations

The following symbols were used in this paper:

Aj,p, Bj,p complexmodal amplitudes of theYj,p eigen-function(j, p cable segment)

AjS, BjS complexmodal amplitudes of theYj,p eigen-function(j-th cable, ‘‘S’’ segment)

cDSj viscous damper coefficient (kN s/m); unit conne-cted to j-th cable

fBSL,fBSU generic cable network frequency (Hz), networks BSLand BSU

f M1r first-mode frequency of network AS1 with ‘‘locked’’dampers (i.e., rigid connectors)

f M1u first-mode frequency of network AS1 formed withcross ties only (no dampers)

f̂j (=ω01/ω0j) cable frequency ratio, j-th cableg acceleration of gravity (=9.81 m/s2)gp maximum number of connected stays for the p-th

cross-tieHj cable tension (kN), j-th cablei (=

√−1), imaginary unit

Kj,p linear spring / stiffness coefficient simulating cross-tie segment (j-th cable, p-th tie)

K̂j,p (=Kj,p[sin(ψj,p)

]2) equivalent stiffness;

non-orthogonal stay/cross-tie orientationlj,p length of the j, p cable segmentlSj length of the ‘‘S’’ segment of the j-th cableLj total length of the j-th cablemj total number of cable segments, j-th cablem̃ (=max[gp]) maximum number of connected stays

(any cross-tie)Mj,p concentratedmass at node j, p simulating the cross-

tie mass (see Fig. 1)Mj,S concentratedmass simulating the dampermass (see

Fig. 1)n total number of cablest time variablexj,p local longitudinal co-ordinate along the axis of the

j, p cable segmentxjS local longitudinal co-ordinate along the axis of the

j-th cable, ‘‘S’’ segmentyj,p(xj,p, t) transverse displacement of the j, p cable segment

(j-th cable, p-th cross-tie)yjS(xjS, t) transverse displacement of the j-th cable, ‘‘S’’

segmentYj,p

(xj,p)modal eigen-function of the j, p cable segment, asa function of xj,p

YjS(xjS) modal eigen-function of the ‘‘S’’ cable segments, asa function of xjS

α real part of the generic dimensionless networkfrequency

β imaginary part of the generic dimensionless net-work frequency

γ (=ω/ω01) dimensionless complex network fre-quency

ζDSj (=cDSj/cDS1)η modal damping ratio (a function of α and β)µj mass per unit length (kg/m), j-th cable(νj)

2(=µjHj/(µ1H1)

), mass/tension reduction parame-

ter (Eqs. (7) and (8))ξj,p (=lj,p/Lj) dimensionless length of the j, p cable

segment

ξjS (=lSj/Lj) dimensionless lenght of the j-th cable, ‘‘S’’segment

ρDS1 (=cDS1 (H1µ1)−0.5), normalized damper coefficient;reference unit (‘‘DS1’’ in Fig. 3)

ρDSj (=ρDS1ζDSj), normalized damper coefficient; unitconnected to the j-th cable

σpk,j

(=∏j−1q=k

sin ψ̂q,psinψq,p

), parameter accounting for non-

parallel orientation of the stays(σpj,j = 1.0

)χj,p

(=Mj,p/ (µ1L1)

), normalized mass coefficient, sim-

ulating the mass of a cross-tieχj,S

(=Mj,S/ (µ1L1)

), normalized mass coefficient, sim-

ulating the mass of a damperψq,p relative inclination angle between cross-tie seg-

ment q, p and stay qψ̂q,p relative inclination angle between cross-tie seg-

ment q, p and stay q+ 1ω generic complex circular frequency of the cable

network (rad/s)ω01 (=π/L(H1/m1)0.5) real ‘‘native cable’’ circular fre-

quency of the reference stay (rad/s)ω0j real ‘‘native cable’’ circular frequency of the j-th

cable (rad/s)

Subscripts:

j, q stay-cable indexp cross-tie segment index

the Leonard Zakim Bunker Hill Bridge in Boston, Massachusetts,USA [8], where both viscous dampers and cross-ties are installedto suppress stay-cable vibrations [9]. Similar hybrid configurationswere also discussed in [10]. Systems with dampers installedbetween stays at internal locations were investigated in [11–13],but will not be discussed in the present paper.The performance of such hybrid damper-cross-tie systems has

also been observed to be adequate except in some specific cases [5].However, the advantages and potential disadvantages of this typeof system over the traditional cross-tie-only and damper-onlysystems remain to be assessed. Furthermore, while the design ofdampers, when they are implemented independently, has beenguided in practice by the so-called ‘‘universal damping curves’’developed by analyzing the free vibration of a damper attachedto a stay (e.g., [14]), the design of cable networks, especially thoseemploying thehybrid damper-cross-tie systems, is often simplifiedand still lacks adequate guidelines.Based on analytical formulation and interpretation of full-scale

measurement data, an investigation was conducted to assess theperformance of cable networks of both the ‘‘classical type’’, inwhich cross-ties are the only mitigation devices, and the hybrid‘‘damper-cross-tie’’ type, inwhich various numbers of dampers canbe present. This paper presents the results from this investigationand attempts to promote development of rational guidelines forthe design of cable networks.

2. General formulation

An analytical method for analyzing the linear in-plane freevibration of cable networks without external energy dissipationdevices was developed in [15]. It was subsequently extended toenable the simulation of hybrid networks with a limited numberof viscous dampers [7]. (A similar approachwas recently employedfor the analysis of vibrations on overhead transmission lines [16]).

L. Caracoglia, D. Zuo / Engineering Structures 31 (2009) 2851–2864 2853

Fig. 1. Generalized model of a cable network with multiple dampers.

In this study, the original model is generalized to include amore comprehensive set of ‘‘damper-cross-tie’’ configurations andto simulate the presence of multiple external viscous dampersconnected to the stays, either in-line with the restrainers or not.The formulation is briefly revisited in this section.As depicted in Fig. 1, in the proposed model, the stays in the

network are simulated by a set of parallel cables, interconnectedby means of cross-ties. Each jth cable (j = 1, . . . , n) in Fig. 1 isdivided by restrainers into mj segments of length lj,p. The cross-ties are modeled by linear spring elements of stiffness Kj,p, with jand p being the cable and the segment indices.Each element of the j-th cable (j = 1, . . . , n) is simulated as a

linear taut string and the free vibration is represented by the waveequation [17]. As an example the equation of motion of the p-thsegment (p = 1, . . . ,mj) is

Hj∂2yj,p/∂x2j,p = µj∂2yj,p/∂t2. (1)

In Eq. (1), yj,p(xj,p, t) is the transverse dynamic displacement ofsegment j, p between nodes Pj,p−1 and Pj,p as a function of timet and local coordinate xj,p. The parameters Lj, Hj and µj are thelength, tension and mass per unit length of the cable. If the motionof each element is in the form of yj,p(xj,p, t) = Re[Yj,p(xj,p)eiωt ],with i =

√−1 and ω being a complex circular frequency due

to the presence of the dampers, Eq. (1) are reduced to ordinarydifferential equations in terms of complex mode shapes Yj,p(xj,p)as

Hjd2Yj,p/dx2j,p + µjω2Yj,p = 0. (2)

The dampers are anchored to the deck and are simulated bydashpots. These dampers are divided into a group consisting ofunits installed in-line with a restrainer and another group thatis not oriented in-line with the cross-ties. Normalized dampercoefficients [7,14], irrespective of the group, are defined as ρDSj =cDSj(H1µ1)−0.5 = ρDS1ζDSj with j being the stay index and cDSjthe viscous damper coefficient. These coefficients are normalizedwith respect to the reference cable with j = 1 and to the damperunit ‘‘DS1’’ through parameters ζDSj = cDSj/cDS1 and ρDS1 =cDS1 (H1µ1)−0.5. If a damper is installed close to the cable anchorageand is not in-line with a restrainer, one additional ‘‘S’’ segmentof length lSj is added to the j-th cable at the deck level (Fig. 1).Expressions, similar to Eqs. (1) and (2), can be derived for the ‘‘S’’segments in terms of YjS(xjS) but are omitted for brevity.The eigen-frequency ω in Eq. (2) is complex because of the

presence of the dampers. This frequency can be normalized withrespect to the fundamental ‘‘native’’ circular frequency of anunrestrained reference stay, ω01 = π/L(H1/µ1)0.5 [17] to yieldω = γω01, with γ = α + iβ being a dimensionless complexfrequency. The solution to Eq. (2) can be expressed as

Yj,p(xj,p) = Aj,p sin(γπL−1j f̂jxj,p

)+ Bj,p cos

(γπL−1j f̂jxj,p

), (3a)

YjS(xjS) = AjS sin(γπL−1j f̂jxjS

)+ BjS cos

(γπL−1j f̂jxjS

). (3b)

Eq. (3a) are associated with the internal segments of a cable(the ‘‘non-S’’ segments), and Eq. (3b) are related to the responseof the ‘‘S’’ segments. The unknown amplitudes Aj,p, Bj,p and AjS , BjSare complex, and f̂j = ω01/ω0j is the j-th ‘‘cable frequency ratio’’.Dimensionless coefficients, such as the cable frequency ratio, areemployed to allow for derivation of the general-form solution toEq. (2). In fact, this formulation allows for the implementation of anumerical solution for large networks with several stays.Eqs. (3a) and (3b) are solved to obtain the unknown amplitudes

by means of a set of compatibility, continuity and equilibriumequations. Vanishing of displacements at cable ends (with ξj,p =lj,p/Lj and ξjS = lSj/Lj), and continuity at the nodes connectingconsecutive segments on the same stay (internal and ‘‘S’’ segments)are represented by Eqs. (4), (5a) and (5b), as follows:

Bj,1 = 0, AjS sin(γπ f̂jξjS

)+ BjS cos

(γπ f̂jξjS

)= 0,

with j = 1, . . . , n; (4)

Aj,pj sin(γπ f̂jξj,pj

)+ Bj,pj cos

(γπ f̂jξj,pj

)− Bj,pj+1 = 0,

with j = 1, . . . , n; pj = 1, . . . ,mj; (5a)

AjS sin(γπ f̂jξjS

)+ BjS cos

(γπ f̂jξjS

)− BjS = 0,

with j = 1, . . . n; (5b)

Internal continuity in the transverse direction along each cross-tie is considered in the following equation

K̂j,p(Bj+1,p+1 − Bj,p+1

)=

j∑k=1

γπH1L−11 σpk,j

{νk[Ak,p cos

(γπ f̂kξk,p

)− Bk,p sin

(γπ f̂kξk,p

)− Ak,p+1] − γπχk,pBk,p+1

}, (6)

with p = 1, . . . , m̃− 1; j = 1, . . . , (gp − 1); m̃ = max[gp].In Eq. (6) the quantity (νk)2 = (µkHk) / (µ1H1) is a mass-

tension reduction factor, which accounts for the relative reductionof mass and tension of each cable with respect to the referencestay; the parameter gp denotes the maximum number of cablesinterconnected by the p-th cross-tie. In the sameequation, the non-orthogonal orientation of stays relative to restrainers and the non-parallel orientation of the stays are respectively accounted for byequivalent stiffness K̂=j,pKj,p

[sin(ψj,p)

]2 and by σ pk,j = ∏j−1q=k

sin ψ̂q,psinψq,p

,

with ψq,p and ψ̂q,p being relative inclination angles of each cross-tie segment q, p relative to stays q and q+1, respectively (σ pj,j = 1).Inertial terms (γπχk,pBk,p+1) are introduced in Eq. (6) to accountfor the mass of the secondary system, represented by lumpedelementsMj,p at each node Pj,p (Fig. 1), internal to the network andnormalized as χj,p = Mj,p/ (µ1L1).


Fig. 2. First and second prototype networks, BSU and BSL, on the BS-line, south tower, Fred Hartman Bridge, Houston, Texas, USA.

Global force equilibrium equations at each cross-tie accountfor the potential presence of an external damper in-line with therestrainer below the lower stays (Fig. 1), as indicated below:

− iρDSpBgp,p+1 =gp∑j=1

σpj,gp

{νj

[Aj,p cos

(γπ f̂jξj,p

)− Bj,p sin

(γπ f̂jξj,p

)− Aj,p+1

]− γπχj,pBj,p+1

}, (7)

with p = 1, . . . , m̃− 1.In Eq. (7) the quantities νj, σ

pk,j and χj,p are the mass-tension,

equivalent inclination and normalized mass factors, describedabove.Finally, force equilibrium equations simulating the behavior of

a damper in the proximity of a cable anchorage at the deck leveland not in-line with a cross-tie (See unit labeled with cDSq and ‘‘S’’-segment dampers in Fig. 1) are:

− iρDSjBjS ={νj

[Aj,mj cos

(γπ f̂jξj,mj

)−Bj,mj sin

(γπ f̂jξj,mj

)− AjS

]− γπχjSBjS

}. (8)

In Eq. (8) the inertial forces related to the mass of the dampers,Mj,S (Fig. 1), are also simulated by term γπχjSBjS with χjS =Mj,S/ (µ1L1), a normalized mass parameter.FromEqs. (4)–(8), a homogeneous systemof 2r = 2

(∑mj + n

)equations can be assembled inmatrix formas SΦ = 0, inwhich thecomplexmatrix S consists of a set of transcendental expressions asa function of γ , linearly dependent on Φ ∈ C2r , i.e., the vector ofunknowns Aj,p, Bj,p and AjS , BjS . This system represents an eigen-value problem, which is numerically solved in this study for theeigen-values, γ .

3. Case study

3.1. Background information on the prototype network configura-tions and the full-scale measurement system

Three cable networks on the Fred Hartman Bridge in Houston,Texas, USA are used as prototypes in the present study. Thisbridge is a twin-deck cable-stayed type with two parallel mainspans of 381 m in length. The main spans and the four sidespans are supported by 192 stays ranging from 59 m to 198 min length. To suppress wind- and rain-wind-induced vibrations,both cross-ties and passive viscous dampers of various typeshave been installed on the stays. To study the nature of wind-and rain-wind-induced stay-cable vibrations, and to evaluate theperformance of countermeasures in mitigating these vibrations,

a full-scale measurement system was installed on the bridge inOctober 1997 and since then continuously monitored the bridgeuntil August 2005. The measurement system primarily consistedof two types of accelerometer which monitored the vibrationsof a number of stays and the oscillation of the bridge decksat multiple locations, propeller-based UVW anemometers and apropeller-vane anemometer, which measured wind speed anddirection at deck level and at the tower-top, respectively, and raingauges which measured the rainfall at the bridge site. An onsitecomputer continuously monitored the transducers. Every timepredetermined threshold in either wind speed or stay vibrationwas exceeded, the system sampled all channels for five minutesat 40 Hz. More details of the full-scale measurement system canbe found in [18].The first and second prototypes were modeled based on one

of the BS-line stay systems (south tower) on the bridge, which isschematically shown in Fig. 2. These configurations represent thestate of the cable plane between March 1999 and April 2004. Twocable networks were present on these stays: one labeled as BSU(upper network) consisting of stays BS17 to BS24 interconnectedby three restrainers, and the other labeled BSL (lower network)between stays BS13 to BS15, which were interconnected by onecross-tie. The Restrainers in network BSU are labeled as 1-BS, 2-BS and 3-BS. Stay BS16 was not mitigated by cross-ties. Undesiredvibration of this staywas controlled by a viscous damper. Locationsof the accelerometers, used to monitor cable vibrations, are alsoindicated in the figure.The third configuration represents the network consisting of

a plane of the AS-line stays, which originate from the southtower and partially support the east main span. This networkis designated the AS network in subsequent discussions. In thislarge network, stays AS13 to AS24 are connected together by threerestrainers labeled 1-AS, 2-AS and 3-AS. At a later stage viscousdampers D1 to D12 were installed on each stay from AS13 to AS24to supplement the existing cross-tie system (restrainers 1-AS to 3-AS). Fig. 3 represents the configuration of this network betweenJune 2004 and September 2005. The configuration of both thedampers and the restrainers was derived from the specificationssupplied by the designers. In particular, each damper is locatedat a distance between 1.5 and 5.0 m from the stay anchorage atdeck level. As indicated in Fig. 3, accelerometers were installed ona number of stays and on selected locations of the deck to monitorvibrations of the structure.Properties of the cables and restrainers of the three systems

were derived from design specifications. Since the numerical so-lution is extremely sensitive to the frequencies of the unrestrainedand undamped stays, whichwill be subsequently referred to as the‘‘native frequencies’’ of the stays, the cable tensions (Hj) used in the


Fig. 3. Third prototype network on the AS-line, south tower, Fred Hartman Bridge.

Table 1Main properties of stays BS13 to BS24 of the BS-line (reference stay: BS24).

Stay Mass (kg/m) Tension Hj (N) Length Lj (m) Lj/LAS24 Native freq. (Hz) f̂j (−) vj (−)

AS24 76.03 3.80E+06 197.85 1.00 0.565 1.00 1.00AS23 76.03 4.10E+06 183.06 0.93 0.634 0.89 1.04AS22 70.12 3.79E+06 168.40 0.85 0.690 0.82 0.96AS21 70.12 3.95E+06 154.08 0.78 0.770 0.73 0.98AS20 70.12 3.59E+06 139.70 0.71 0.810 0.70 0.93AS19 65.20 4.08E+06 125.78 0.64 0.994 0.57 0.96AS18 52.87 2.68E+06 112.28 0.57 1.002 0.56 0.70AS17 52.87 2.51E+06 99.38 0.50 1.096 0.52 0.68AS16 47.93 2.26E+06 87.33 0.44 1.242 0.45 0.61AS15 47.93 2.61E+06 76.55 0.39 1.523 0.37 0.66AS14 47.93 2.08E+06 67.34 0.34 1.545 0.37 0.59AS13 32.50 1.74E+06 59.52 0.30 1.944 0.29 0.44

Table 2Main properties of stays AS13 to AS24 of the AS-line (reference stay: AS24).

Stay Mass (kg/m) Tension Hj (N) Length Lj (m) Lj/LBS24 Native freq. (Hz) f̂j (−) vj (−)

BS24 76.03 3.73E+06 197.85 1.00 0.560 1.00 1.00BS23 76.03 4.29E+06 183.06 0.93 0.648 0.86 1.07BS22 70.12 3.79E+06 168.40 0.85 0.690 0.81 0.97BS21 70.12 2.60E+06 154.08 0.78 0.625 0.90 0.80BS20 70.12 3.59E+06 139.70 0.71 0.810 0.69 0.94BS19 65.20 3.20E+06 125.78 0.64 0.881 0.64 0.86BS18 52.87 2.68E+06 112.28 0.57 1.002 0.56 0.71BS17 52.87 2.39E+06 99.38 0.50 1.071 0.52 0.67BS16 47.93 2.34E+06 87.33 0.44 1.265 0.44 0.63BS15 47.93 1.90E+06 76.55 0.39 1.300 0.43 0.57BS14 47.93 1.60E+06 67.34 0.34 1.355 0.41 0.52BS13 32.50 1.65E+06 59.52 0.30 1.893 0.30 0.43

simulation were derived from the actual frequency values of thestays, which were estimated based on the unrestrained vibrationsrecorded by the full-scale measurement system. The main prop-erties of the AS-line and BS-line stays are summarized in Tables 1and 2. For each viscous damper in the AS network (j = 1, . . . , 12),the normalized damper coefficient ρDSj = ρDS1ζDSj is expressed interms of the coefficient ρDS1 (ρDS1 = 1.72 according to design) ofdamper ‘‘DS1’’ installed on stay AS13 (Fig. 3). In all the simulations,the ratio ζDSj = cDSj/cDS1 is selected as a constant derived fromthe design values. The damper coefficients of the dampers not in-line with a restrainer (cDSj) range between 29 kN× s/m (DS1) and73 kN×s/m (DS12). Dampers D5 and D9 are in-line with Restrain-ers 2-AS and 3-AS, respectively. The damper properties are sum-marized in Table 3.The full-scalemeasurement systemon the FredHartman Bridge

recorded a number of types of stay cable vibration, including theclassical low-amplitude Kármán-vortex-induced vibration, the so-called rain-wind-induced vibration and a class of large-amplitude

dry-cable vibrations over a range of high reduced velocity. Itwas observed that due to the three-dimensional nature of thewind-stay environment, some vibrations hadmultiplemodal com-ponents and that some vibration in the individual modes hadsignificant components in both the in-plane and out-of-plane di-rections [3,18]. In addition to the vibrations induced by the directexcitation of wind or the combination of wind and rain, full-scaledata suggest that wind-induced oscillation of the bridge deck canalso lead to large-amplitude vibration of the stays due to interac-tion between the decks and the stays, when the frequency of thedeck oscillation and that of a stay mode are close and the phe-nomenon of frequency curve veering occurs [19].

3.2. Cable networks formed with cross-ties only

The effectiveness of cable networks formedwith cross-ties onlywas investigated using both numerical simulations and full-scalerecords for networks BSL, BSU and AS in the absence of dampers.Preliminary results were discussed in [5,6].


Table 3Main properties of the AS-line dampers – current design values (reference damper: DS1).

Damper ID Stay lSja (m) Damper coeff. cDSj (kN s/m) ρDSj (−) ζDSj (−)

DS12 AS24 4.85 72.97 4.29 2.50DS11 AS23 4.48 72.97 4.29 2.50DS10 AS22 4.21 58.38 3.43 2.00DS9b AS21 3.87 58.38 3.43 2.00DS8 AS20 3.54 58.38 3.43 2.00DS7 AS19 3.14 58.38 3.43 2.00DS6 AS18 2.74 43.78 2.58 1.50DS5b AS17 2.32 43.78 2.58 1.50DS4 AS16 1.86 43.78 2.58 1.50DS3 AS15 1.43 43.78 2.58 1.50DS2 AS14 1.71 29.19 1.72 1.00DS1 (Ref.) AS13 1.43 29.19 1.72 1.00a Distance from cable anchorage at the deck level.b D5S, DS9 are installed in-line with Restrainers 2-AS and 3-AS.

a

b

Fig. 4. Mode-frequency evolution curves of theBSU (BS17 toBS24) andBSL (BS13 toBS15) networks of the BS-line. (a) NET_BSU ( ) upper network frequencies arecompared to the native frequencies of individual stays: BS24 (�), BS23 (4), BS22( ), BS21 ( ), BS20, ( ), BS19 ( ), BS18 (+), BS17 ( ), BS16 (×); (b) NET_BSL( ) lower network compared to native frequencies of stays BS15 ( ), BS14 ( ),BS13 (�).

Numerical simulations were based on the configurations de-scribed in Section 3.1. In particular, observations were also basedon the results of previous studies [15] conducted to analyze theperformance of in-plane cable networks by varying the numberand the location of restrainers.Fig. 4 depicts the mode-frequency evolution chart of the BSU

(Fig. 4(a)) and BSL (Fig. 4(b)) networks in the frequency rangebetween 0.5 Hz and 5.0 Hz. In each figure, the frequencies ofthe two networks are compared with the native frequencies ofthe stays that are used to form the network, including bothfundamental and higher-mode frequencies. In Fig. 4(a), those of

Fig. 5. Mode-frequency evolution curves of the AS network (AS-line); originalconfiguration with no dampers ( ) and system with ‘‘locked dampers’’ ( ).Native cable frequencies of individual stays: AS24 (�), AS23 (4), AS22 ( ),AS21 ( ), AS20, ( ), AS19 ( ), AS18 (+), AS17 ( ), AS16 (×), AS15 ( ), AS14 ( ),AS13 (�).

BS16 are also shown for completeness. As anticipated, the networkfrequencies are in general higher than the native fundamentalfrequencies of the longest individual cables, such as 0.560 Hz,which is the fundamental frequency of BS24 in Fig. 4(a). In the samefigure, a large plateau composed of many localized modes above2.0 Hz is present due to the existence of a large number of cablesegments. For the BSL network in Fig. 4(b), the overall frequencyincrement in comparison with the fundamental native frequencyof BS15 is significant, with the location of the modal plateau beingabove 2.6 Hz.For the large AS network with twelve cables and three re-

strainers, the mode-frequency evolution is shown in Fig. 5. Thisconfiguration is represented by a diamond marker in the figure. Inthis case, the first-mode frequency of the network was predictedat about 1 Hz, while the fundamental native frequencies of thelongest stay of the AS linewas estimated to be 0.565Hz (AS24). Thefirst modal plateau coincides with a frequency range above 2.0 Hz,which is similar to the modal plateau of BSU, but the number ofmodal solutions has increased (Note the scale difference in thehorizontal axis compared to Fig. 4). Simulations were extended to‘‘plateau 2’’ in the figure. Fig. 5 also depicts a second configuration(circular marker), representing the behavior of the AS networkwith ‘‘locked dampers’’. This case will be discussed in Section 3.3.The shapes of two example modes of the BSU network, whose

frequencies were shown in Fig. 4(a), are depicted in Fig. 6 (modalscales only indicative). An important feature of cable networks isassociated with the distinction between the fundamental globalmodes with contribution from all cable segments and a large


a

b

Fig. 6. Examples of modal solutions for the BSU cable network of the HartmanBridge. (a) Global mode BSU-NM01 (0.89 Hz), (b) localized mode BSU-NM04(1.93 Hz) (α = 1.00 is equivalent to 0.560 Hz).

number of localized modes at higher frequencies (modal plateau).Fig. 6(a) shows a typical example of a global mode (the first mode,BSU-NM01, at 0.89Hz), and Fig. 6(b) depicts a localizedmode, i.e., ahigher mode in which only a portion of the network is activelyinvolved. Specifically, in Fig. 6(b), the mode BSU-NM04 with afrequency of 1.93 Hz is dominated by the vibration of stay BS21.The change in the modal frequencies and shapes of the cables

when they are interconnected through the cross-ties results inenergy redistribution in the system when external excitationis applied. The effectiveness of this mechanism is assessed byinterpreting the vibrations of the stays recorded by the full-scale measurement system. Due to the fact that the stays wereinstrumented over different periods of time, this paper will usestays that are associated with the largest datasets to interpretthe effectiveness or ineffectiveness of the mitigation systems. Aswill be seen subsequently, a number of these stays (stays AS1and AN24) are not necessarily in the cable networks subjectedto analytical study. However, since they representatively revealthe performance of the mitigation systems, and since they areindeed in similar networks, it is reasonable to use these stays forillustration purposes.Full-scale data suggest that the cross-ties were generally

effective in preventing the onset of various types of stay cablevibrations. As an example, Fig. 7(a) shows the 14-s root-mean-square (RMS) in-plane and lateral (out-of-plane) displacementsof stay AS1 recorded from October 1997 to September 1998,

a

b

Fig. 7. One-minute RMS displacements of stay AS1 (a) before and (b) afterinstallation of cross-ties.

before the cross-ties were added, and Fig. 7(b) shows the RMSdisplacements of the same stay fromMay 1999 to December 2003,when AS1 was interconnected to a number of adjacent stays withcross-ties. The accelerometer was installed 6 meters above thedeck level, at a distance equal to 7% of the total stay length.measured from the cable anchorage.The RMS displacements were computed based on the dis-

placement time histories, which were obtained by numericallyintegrating the acceleration of the stays recorded by the ac-celerometers. Although the vibrations were not presented againstthe corresponding wind data, since the duration of the monitoringof the restrained vibration is considerably long, the effectivenessof the cross-ties is apparent. In this case, vibration suppression inthe lateral direction is due to the fact that, since wind- and rain-wind-induced vibrations are aeroelastic, a mitigation mechanismin either direction is capable of suppressing vibration componentsin both directions, unless the oscillation primarily occurs in theunmitigated direction.However, the full-scale data also revealed a number of li-

mitations of cross-ties as a mitigation strategy for stay cablevibrations. Fig. 8 shows the vibrations of stay AN24 (which wasin a cable network similar to the AS network shown in Fig. 3) inthe in-plane and lateral directions in its native modes betweenApril, 1999 and December, 2002, when the stay was connected toadjacent stays using cross-ties. The accelerometer that monitoredthe vibrationswas located 6mabove the deck level, or at a distanceequal to 8% of total stay length from the cable anchorage.The modal displacements were obtained by decomposing the

displacement time histories using a sixth order Butterworthfilter. This figure suggests that while the cross-ties successfullysuppressed vibrations in many of the lower native modes of thestay, they appeared ineffective in mitigating vibrations in the


Fig. 8. Mean modal vibration amplitude of stay AN24 (central bridge span, northtower) after cross-tie installation.

fourth (2.25 Hz) and the eighth (4.52 Hz) native modes. Evidenceof vibration in native modes other than numbers 4 and 8 wasnegligible, although some of these modes were indeed observed atvery low amplitudes. The ineffectiveness of the cross-ties for thisnetwork is due to the fact theywere evenly spaced and tied toAN24at locations very close to the nodal points of the fourth nativemodeso that modes 4, 8, 12 etc. of this stay remain as in-plane nativemodes of the cable network, i.e., they are essentially not restrainedby the cross-ties and, as a result energy in these modes cannot beeffectively redistributed to the other modes of stay AN24 or theadjacent stays.

Another often overlooked limitation of cross-ties is, as outlinedabove, that they are essentially a mechanism in the in-planedirection so that their effectiveness is marginal in the lateraldirection. Fig. 9 shows the acceleration time histories and thecorresponding power spectral density functions for an examplerecord for stay AS20 in the AS network. Two types of frequencycomponent are present in the in-plane response (acc-z) of the stay:three native staymodes (thirdmode: 2.49 Hz, sixthmode: 4.98 Hz,seventh mode: 5.81 Hz) and a number of insignificant localizedcable networkmodes (represented by the small peaks in the powerspectrum of the in-plane acceleration over the frequency rangeof approximately 4.5 Hz to 6.5 Hz) due to the effect of the cross-ties. The fact that the native modes are still present in the in-planeresponse is because the connections between the cross-ties and thestays are not perfectly tight in this case. In the lateral-direction,however, only frequency components in the native stay modes(third mode: 2.49 Hz; fifth mode: 4.15 Hz; sixth mode: 4.98 Hz)can be detected.Such limitation of the cross-ties in the lateral direction can also

be seen in the statistics of recorded vibrations. As an example,Fig. 10(a) shows the one-minute RMS displacement of stay AS20during the time period between March 1999 and June 2004,while it was in the AS network shown in Fig. 3. Cluster A inthe graph represents quasi-static vibrations of the stay due todeck oscillation [19], and cluster B represents rain-wind-inducedvibrations associated with wind approaching in a direction veryclose to the projection of the cable axis in the horizontal plane. Thequasi-static vibrations are not in either the nativemodes of stays orthe localized or globalmodes of the cable network. The frequenciesof these vibrations are instead the same as the frequencies of the

Fig. 9. Time histories and the corresponding power spectra of an example record showing responses in the in-plane and lateral directions (AS20).


deck oscillation. This type of vibration is not a matter of primaryconcern. The rain-wind-induced vibrations represented by clusterB are in the third and sixth native modes (2.4 Hz and 4.8 Hz,respectively) of the stay. These vibrations in the lateral directionare problematic since the restraint by the cross-ties is limited inthis direction, as stated above. Also, Fig. 10(a) indicates that lateralvibrations did not occur often for stay AS20. This is due to thefact that at the Fred Hartman Bridge, simultaneous occurrence ofrain and wind approaching in directions close to the bridge axiswere not often observed. For other bridges and for stays locatedin an environment where wind directions are close to the axialorientation of the stays, this ineffectiveness of cross-ties can bemore problematic.Finally, this limitation of the cross-ties has alsomanifested itself

in their inability in suppressing deck-induced stay-cable vibrationin the out-of-plane direction. Fig. 10(b) shows the vibration locusof stay BS24 (BSU network in Fig. 4) during an event when the deckwas oscillating in its third vertical mode at about 0.57 Hz [19],which is close to the fundamental frequency of this stay (andTable 1). Although the deck oscillationwas in the vertical direction,it can be seen that the vibration of stay BS24 was more significantin the lateral direction than in the in-plane direction. This lateralvibration of stay BS24 is believed to be induced by deck-inducedquasi-static oscillation of the adjacent stays (BS17 to BS23) in theBSU network in Fig. 2, whose vertical planes are not coincidentwith that of BS24 because of the three-dimensional layout ofthe cables on this bridge. When the oscillation of the deck doeshave a lateral component, such as in the case of oscillation in thetorsional modes, the ineffectiveness of the cross-ties can be morepronounced.

3.3. Cable networks with cross-ties and dampers—numerical simula-tion of network modes

A parametric study was conducted to assess the performanceof the hybrid ‘‘damper-cross-tie’’ system with multiple dampers(DS1 to DS12, Table 3), as shown in Fig. 3. To limit the numberof independent variables associated with the dampers in the ASnetwork, the normalized damper coefficient of ‘‘DS1’’ in Table 3ρDS1 was allowed to vary within a suitable interval, while therelative damper coefficient ratios of other units ζDSj = cDSj/cDS1were assumed to be equal to the design values indicated in thetable.In a cable networked restrained by a ‘‘damper-cross-tie’’

system, the frequencies and eigen-modes are influenced by thenormalized damper coefficients of each unit (i.e., ρDSj in Table 3).The evolution of the frequency solution in the complex plane forthe AS network, as a function of ρDS1, is represented in terms ofthe dimensionless frequency α and damping ratio η (0 ≤ η ≤ 1),computed as η =

{(β/α)2/

[1+ (β/α)2

]0.5}. The solutions arecharacterized by two limiting real-frequency cases: ρDS1 = 0i.e., undamped solution without dampers and ρDS1 → +∞, i.e., alldampers are locked and can be replaced by a rigid link to the deck.For intermediate values of ρDS1, complex eigen-modes are usuallyunderdamped with η ≤ 1. As in the case of a stay attached with aviscous damper [14], other solutions are also possible dependingon the damper coefficients: over-damped modes with α = 0,i.e., non-oscillatory rapid decay (β > 0 but finite), and criticalmodeswith frequency α 6= 0 and (η→ 1)β →+∞.Real frequencies corresponding to undamped and locked

damper configuration are shown in Fig. 5 in Section 3.2. In partic-ular, Fig. 5 confirms that, as anticipated, locking the dampers tothe deck has an effect exclusively on the global modes of the ASnetwork, labeled as M1 to M3 (18% increment for the fundamentalmode fM1r = 1.18 Hz in comparison with fM1u = 1.01 Hz with

a

b

Fig. 10. Examples of cross-tie inadequacy in suppressing lateral vibration:(a) Displacement amplitude of stay AS20 after installation of cross-ties (ASnetwork); (b) vibration locus of stays BS24 under the excitation of deck oscillation(BSU network).

no dampers), while no significant effect in terms of frequency in-crease is evident for the first group of localized modes (e.g., modesM4 to M25 of the AS network), since modal plateau 1 in Fig. 5 ispractically coincident in the two cases. For higher global modes,e.g., mode M30, some difference is observed.Frequency-damping trajectories associated with the funda-

mental global modes M1 and M2 of the AS network with twelvedampers are shown in Fig. 11(a) in the range between 0.5 Hz and2.0 Hz as a function of ρDS1; some of the higher-frequency local-ized solutions around 2 Hz (modes M5 to M7 of the AS network)are presented in Fig. 11(b). In these figures, real frequencies asso-ciated with the limiting cases are indicated as undamped (u) andlocked dampers (r), alongwith the direction of increasing damping(ρDS1).Fundamental network modes of the AS network (modes M1

andM2) are underdamped. Each trajectory originates from the ‘‘u’’nodes and terminates at the ‘‘r ’’ solutions as ρD1 → ∞, and ischaracterized by a general frequency increment. From Fig. 11(a),it can be concluded that the current design (ρDS1 = 1.72, node‘‘a1’’) does not correspond to optimal damping for the fundamentalmodes, i.e., the local maximum on each (α, η) trajectory. Eventhough the damping ratio is significant (about 3.5% for mode M1and 6.0% for mode M2), higher damping for these network modesmight be achieved if larger dampers were employed.In contrast, modal behavior for the higher and localized modes

is quite complicated due to the presence of a large number ofdampers. This is especially evident for modes M6 and M7 inFig. 11(b) (Note the scale difference in the vertical axis). Analysis ofthe numerical simulations also suggests the potential presence of


Table 4Performance of the AS damper-cross-tie system — global modes M1, M2.

Case # Damper configuration (‘‘P’’ present; ‘‘–’’ absent) Maximum dampingnode (Mode M1)

Maximum dampingnode (Mode M2)

DS1 DS2 DS3 DS4 DS5 DS6 DS7 DS8 DS9 DS10 DS11 DS12 α η ρDS1 α η ρDS1

1 – P P P P P P P P P P P 1.935 0.084 7.6 3.031 0.124 6.72 – – P P P P P P P P P P 1.935 0.084 7.6 3.029 0.125 6.73 – – – P P P P P P P P P 1.933 0.084 7.5 3.015 0.128 6.54 – – – – P P P P P P P P 1.933 0.084 7.5 3.015 0.128 6.55 – – – – – P P P P P P P 1.817 0.016 5.4 2.841 0.067 3.56 – – – – – – P P P P P P 1.815 0.015 5.3 2.835 0.066 3.47 – – – – – – – P P P P P 1.812 0.013 5.4 2.833 0.065 3.48 – – – – – – – – P P P P 1.812 0.013 5.4 2.833 0.065 3.49 – – – – – – – – – P P P 1.792 0.002 5.4 2.686 0.001 3.510 – – – – – – – – – – P P 1.791 0.001 4.6 2.686 0.001 3.011 – – – – – – – – – – – P 1.790 0.001 4.6 2.685 0.000 3.012 – – – – P – – – P – – – 1.928 0.083 7.8 3.022 0.128 6.9

Note: α, normalized frequency; η, damping ratio; ρDS1 , reference damper coefficient.

Table 5Performance of the AS damper-cross-tie system - localized modes M5, M7.

Case # Damper configuration (‘‘P’’ present; ‘‘–’’ absent) Maximum damping node(Mode M5)

Maximum dampingnode (Mode M7)

DS1 DS2 DS3 DS4 DS5 DS6 DS7 DS8 DS9 DS10 DS11 DS12 α η ρDS1 α η ρDS1

1 – P P P P P P P P P P P 3.8085 0.00130 14.0 3.8943 0.00175 19.02 – – P P P P P P P P P P 3.8085 0.00130 14.0 3.8943 0.00176 19.03 – – – P P P P P P P P P 3.8084 0.00131 14.0 3.8943 0.00176 19.04 – – – – P P P P P P P P 3.8084 0.00131 14.0 3.8943 0.00176 19.05 – – – – – P P P P P P P 3.8034 0.00029 16.0* 3.8915 0.00048 3.46 – – – – – – P P P P P P 3.8032 0.00029 16.0* 3.8915 0.00047 3.47 – – – – – – – P P P P P 3.8029 0.00032 16.0* 3.8914 0.00044 3.48 – – – – – – – – P P P P 3.8029 0.00032 16.0* 3.8914 0.00044 3.49 – – – – – – – – – P P P 3.8039 0.00000 2.3 3.8927 0.00001 2.210 – – – – – – – – – – P P 3.8039 0.00000 2.0 3.8927 0.00000 1.911 – – – – – – – – – – – P 3.8039 0.00000 2.0 3.8927 0.00000 2.012 – – – – P – – – P – – – 3.8082 0.00131 14.0 3.8941 0.00177 19.0

Note: α, normalized frequency; η, damping ratio; ρDS1 , reference damper coefficient.* Maximum damping corresponds to the upper limit of ρDS1 .

critically damped modes for this network in this range of frequen-cies. For such modes, frequencies (α) and normalized ‘‘critical’’damping coefficients (ρDS1) are usually computed by numericallyscanning the non-trivial solutions for large β , corresponding to thecondition η→ 1, i.e., critical damping. Details are described in [7].As a general tendency, no beneficial effects are observed in

terms of achievable maximum damping for these local modes(with maximum η estimated at about 0.2% for modes M5 andM7 from the figure). It must be noted that these maxima areconsiderably lower than the values that could be achieved if cross-ties were not installed and the damper size was optimized basedon the ‘‘universal curve’’ for individual cable/damper [14].Fig. 12 depicts an example of complex eigen-function of mode

M1 of the AS network, coincident with the optimal solution forρDS1 = 7.60, i.e., node ‘‘a2’’ in Fig. 11(a). Real and imaginary parts ofthemode shape are shown in Fig. 12(a) and Fig. 12(b), respectively.This solution clearly differs from the current design (node ‘‘a1’’in Fig. 11(a)), but requires dampers with viscous coefficients fourtimes larger than the actual values. From the study of the realand imaginary parts, it can be seen that the energy dissipationmechanism for M1 is mainly associated with the viscous forcetransferred to damper DS5 (note the ‘‘kink’’ in the mode shape),which is in-line with restrainer 2-AS.Since simultaneous optimization of more modes (e.g., modes

M1 to M3) is difficult to achieve in practice because of the largenumber of dampers in theASnetwork system, other solutionsweretested starting from thedesign configurationwith three restrainersand all dampers in Fig. 3. For a given damper configuration andmode, an (α, η) pair corresponding to the maximum achievabledamping level ηwas determined as a function of the reference ρDS1by subsequently removing the damper units (one at a time). Table 4

summarizes the results of this investigation for the fundamentalglobal modes M1 (symmetric) and M2 (asymmetric). Twelve caseswere considered. The (α, η) pair of case 1 corresponds to optimalnode ‘‘a2’’ in Fig. 11(a). Itwas concluded that dampersDS5 andDS9,simulated as in-line with restrainers 2-AS and 3-AS, are mainlyresponsible for the performance of mode M1 (see case 12 withfrequency close to 1 Hz, η = 8% and ρDS1 = 7.60), while thepresence of other damper units has limited effects (cases 1 to 4).Drastic performance reduction is observed if either DS5 or DS9 isremoved (cases 5 to 11). A similar conclusion can bemade formodeM2 (1.7 Hz) with η = 12% and ρD1 = 6.9 (case 12).

3.4. Cable networks with cross-ties and dampers — analysis of theperformance from simulations and full-scale records

Table 5 shows the results of the parametric investigation on theperformance of the AS network modes M5 and M7 (above 2 Hz).Case 1 corresponds to the gray circular marker nodes in Fig. 11(b).It is revealed that, since the contribution to the mode shapes isprimarily associated with the vibration of internal cable segments(‘‘non-S’’ segments), the presence of the dampers has little effect(η < 0.2% in most cases) and the performance of such modes ismainly linked to effects of the cross-ties. Numerical simulationswere restricted to ρDS1 ≤ 16.0 for M5 and ρDS1 ≤ 30.0 forM7, to avoid the presence of large dampers of limited practicaluse. In contrast to initial observations, it can be noted that themodal behavior of higher-frequency localized modes may still besensitive, to some extent, to the addition or removal of selecteddamper units.Fig. 13 shows the partially unexpected evolution of the localized

mode M7; the cases simulated are the same as those reported in


a

b

Fig. 11. Frequency-damping trajectories of the AS ‘‘damper-cross-tie’’ systemwithtwelve dampers. (a) M1 and M2 first global modes, (b) M5 to M7 localized modes(real modes, u: undamped, r: locked dampers ρDS1 →∞). Arrows show increasingvalues of ρDS1 (α = 1.0 corresponds to 0.565 Hz).

a

b

Fig. 12. Complex eigen-function of mode M1 for ρDS1 = 7.60 (node ‘‘a2’’ inFig. 11(a)) and corresponding to normalized eigen-frequency γ = 1.94 + 0.16i.(a) Real part and (b) imaginary part of the mode shape.

Fig. 13. Simulated evolution of mode M7 of the AS damper-cross-tie system (α =3.9 is equivalent to 2.2 Hz). Cases 1 to 12 are described in Table 5.

Table 5. Case 1 in this figure coincides with the trajectory of M7 inFig. 11(b). The maximum damping value attainable is influencedby the damper configuration (cases 1 through 12). Moreover, thefigure confirms that the solution trajectories for highermodesmaydrastically change as dampers are progressively removed from theoriginal configuration (case 1). As an example, cases 1 through4 significantly differ from case 5. In the former cases the modalfrequency α initially decreases and subsequently increases as ρDS1is increased, resulting in an unusual trajectory. In contrast, in thelatter case, a monotonic decrement in the frequency is alwaysobserved.Two example mode shapes for higher network modes of the

localized type, i.e., M5 and M7, are shown in Figs. 14 and 15. Realand imaginary parts of the eigen-function are separately indicated.Each figure shows the mode shape associated with the eigen-frequency γ corresponding to the maximum damping nodes forthe damper configuration labeled as ‘‘case 4’’ in Table 5 and shownin Fig. 13. These correspond to ρDS1 = 16.00 for M5 (Fig. 14) andρDS1 = 19.00 forM7 (Fig. 15). Case 4was also selected since Table 5suggested that the optimal node for this configuration could beconsidered efficient in comparison with other cases in the sametable. In case 4 a relatively large damping value η is achieved fora relatively low ρDS1, i.e., for smaller dampers. The amplitudesemployed in the plots are only indicative and are selected tohighlight the typical localized nature of these modes. It can beseen in these figures thatmost vibration occurs in the intermediateportions of the network, away from the cable segments wheremost dampers are located. The placement of dampers on everystay in the proximity of the cable end is not beneficial, except forthe location of D5, which is modeled as attached to AS17 in-linewith restrainer 2-AS, also indicated in Figs. 14 and 15. This damperprovides ‘‘some’’ contribution to the damping, especially forM7; infact, the complex part of themode shape of the AS17 cable segmentclose to D5 is not negligible in Fig. 15(b).Table 5, Figs. 14 and 15 reveal the complexity in the behavior of

a hybrid cable network with multiple dampers attached betweena stay and a deck. However, they also suggest that, if a damperis optimized for the first native mode of a stay, its performancemay become inadequate for the higher modes of a network, suchas M5 to M7 in the AS network. However, even though in thecurrent practice the optimization of a damper attached to a stayis often based on the first native mode, vibration at frequenciescorresponding to M5 to M7 cannot be excluded. If frequenciescorresponding tomodesM5 toM7were of concern to the designer,Table 5 confirms that this hybrid system would not be as effective


a

b

Fig. 14. Complex eigen-function of mode M5 corresponding to the maximumdamping node for ‘‘case 4’’ damper configuration in Table 5 (ρDS1 = 14.00, γ =3.808+ 0.005i). (a) Real part and (b) imaginary part of the mode shape.

as the damper-to-stay case, optimally designed for the frequencyrange of M5 to M7.Full-scale measurement data recorded after the installation of

the dampers on the AS network confirm the numerical resultsdescribed in Section 3.2. In particular, the damper-cross-tie systemhas generally been effective in suppressing the vibrations, unlessthe oscillations were primarily in the lateral direction. A summaryof the recorded vibrations of AS22 from June 2004 to September2005, when it was mitigated by the damper-cross-tie system isshown in Fig. 16. It can be seen that during the 15 months,the stay only exhibited a limited number of events of moderate-amplitude vibrations. The vibrations in the first native mode ofthe stay, which were considerably two-dimensional, are identifiedto be due to oscillation of the bridge deck in its first torsionalmode, whose frequency is estimated to be 0.684 Hz [20] andclose to the fundamental frequency of stay AS22. The vibrationsin the fourth (2.76 Hz) and the seventh (4.85 Hz) modes areidentified to be rain-wind-induced vibration. As stated earlier, theexistence of low-amplitude native modes were due to the factthat the connections between the cross-ties and the stays werenot perfectly tight. Fig. 16 also suggests that the damper-cross-tiesystem, which consists of dampers and cross-ties in the in-planedirection of the stays, was also ineffective in mitigating vibrationsin the lateral direction. According to the full-scale data it appearsthat no improved performance could be clearly attributed to theaddition of the dampers as compared to the full-scale response ofthe AS network without dampers, as presented in Section 3.2.Fig. 17 shows another example of acceleration time histories

and the corresponding power spectra for stay AS22 while beingmitigated by cross-ties and dampers. The accelerations are

a

b

Fig. 15. Complex eigen-function of mode M7 corresponding to the maximumdamping node for ‘‘case 4’’ damper configuration in Table 5 (ρDS1 = 19.00, γ =3.894+ 0.007i). (a) Real part and (b) imaginary part of the mode shape.

Fig. 16. Mean vibration amplitude of stay AS22 from full-scale records whilemitigated by the damper-cross-tie system.

used herein to illustrate the components of oscillation at highfrequencies of interest about the seventh native mode of the cable.It can be seen that while the amplitude of the lateral acce-

leration was quite large, that of the in-plane acceleration was verysmall. It also can be observed in the power spectrum of the in-plane vibration that, although the oscillation in this direction wasdominated by the seventh native mode of stay AS22 at 4.84 Hz,it had frequency components that are not the native naturalfrequencies of the stay, such as the component at 5.12 Hz. Thesenon-native frequencies are believed to be those of some of thelocalized in-plane network modes of modal plateau 2 in Fig. 5. Inthe case of the lateral vibration, as expected, the seventh native


Fig. 17. Time histories and power spectra of a second example record showing seventh-mode vibration of stay AS22 while mitigated by cross-ties and dampers.

mode of stay AS22 is visible only while localized network modesare absent. This record confirms that when a damper-cross-tiesystem is oriented in the in-plane direction of the stays, it isalso inadequate in mitigating vibrations primarily in the lateraldirection.

4. Conclusions and recommendations

This study investigated the performance of cross-tie anddamper-cross-tie systems in mitigating wind- and rain-wind-induced vibrations. The combination of parametric numericalsimulations and interpretation of full-scale stay-cable vibrationdata were used. The objective of the parametric studies conductedon a set of prototype networks was two-fold: to identify thein-plane frequencies and mode shapes of cable networks withcross-ties only, and to investigate the performance of hybridconfigurations, in which multiple dampers can be attached tothe stays. The objective for interpreting the full-scale recordswas three-fold: analysis of the dynamic response of real cross-tie systems with or without dampers, interpretation of the resultsof the numerical simulations along with potential improvementin the dynamic modeling and experimental verification of theeffectiveness of cross-tie systems in suppressing stay vibration.In general, both the simulation results and the full-scale data

showed that cross-tie systems without the addition of damperscan be effective in mitigating stay-cable vibrations. However thesesystems are not effective in suppressing vibrations primarily in thelateral direction of stay cables forwhich the ‘‘nativemodes’’ are not

significantly affectedby themitigation systems, since this directionis orthogonal to the primary network orientation.Numerical analyses indicated that localized modes at higher

frequencies can potentially still be a concern since the combinationof cross-ties and dampers only marginally improves energydissipation capability in these modes, for which the dampers areusually not optimized. Adding a damper to every stay in a cablenetwork formed using cross-ties may be unnecessary since theenergy dissipation capabilities, mainly evident in the fundamentalmodes of the network, can be achieved using a reduced number ofoptimally-designed dampers.Analysis of the full-scale records revealed that the monitored

stays did not generally exhibit significant large-amplitude vibra-tions in either the global or localized network modes. However,traceable frequency components representing the localized net-work modes were identified. Also, previous work on cross-ties onthe same bridge [15] had confirmed the existence of some of theglobal modes, even though the measured vibration amplitude wasvery low.The interpretation of the numerical results and the full-scale

records provided the motivation for the preliminary developmentof guidelines for the design of such systems against stay cablevibrations. These considerations can be summarized as follows:

• Cross-ties are inherently an energy redistribution mechanismbut not an energy dissipation mechanism. Quantification oftheir effectiveness is currently difficult because it still relies onthe assessment of frequency shifts of the stays.


• Cross-ties alone can be effective in mitigating large amplitudeoscillations in the lower (global) modes of a cable network.This is primarily due to the large increment in modal mass thatresults when the stays are linked together;• In addition to the global modes, a cable network exhibits alarge number of localized vibration modes. If these modes areexcited, they are more difficult to control since they can occuron individual stays between the anchor points of the cross-ties.• A related issue is that an evenly-spaced configuration of cross-ties is not a good practice. It must be noted that whenrestrainers are sometimes introduced to reduce sag effects inlong stays, they can be placed in an evenly-spaced pattern [21].However, in the case of free-vibration dynamics higher modesof the native cables are not suppressed, even in the plane of thesystem.• Oscillation of the stays cannot be excluded in the directionorthogonal to the plane of the cross-tie system because of thelimited performance of the restrainers in this lateral direction.This inadequacy of the systems can be a concern, sincelateral vibration may create large stresses at either restrainer-stay or damper-stay connection, which can potentially inducelocalized damage.• Adding dampers in the in-plane direction of a cross-tie systemcan induce non-negligible damping levels (otherwise absent) inthe lower modes of the network in this plane, and results inan improvement of the control. It appears, however, that thebenefits can be redundant since both analytical studies and fieldobservations suggest that cross-ties by themselves are alreadyan effective system for these modes. If dampers are added toeither increase or complement the performance of cross-ties,the study suggests that it is not necessary to attach a damper toevery stay.• Hybrid damper-cross-tie systems are complex and must be an-alyzed as networks with attached dampers. The use of sim-plified simulation methods, such a simple dynamic analysis ofeach stay, cannot fully characterize the response of the system.Adding cross-ties to a stay configuration modifies the free-vibration characteristics completely and can render the dam-pers ineffective, since the free-vibration dynamics of the sys-tem is fundamentally altered.• The behavior of the network in the higher modes primarily de-pends on the geometric properties of the cross-ties and theirrelative position along each stay. Therefore, its performancecan be mainly improved by a better design of the secondaryrestrainers.• Simulations confirm that the optimization of more networkmodes at the same time for damper-cross-ties system is difficultto achieve in practice because of the inherent complexityassociated with the large number of independent parametersto be determined (i.e., damper coefficients and location of theunits).• Since cross-ties are inherently a mechanism in the in-planedirection, adding damper in this direction cannot overcomethis limitation of cable networks. It appears that addition ofdampers in the orthogonal direction of a cable network can bean attractive strategy. This can be especially true for bridgeswith exceptionally long stays, since the effectiveness of thedamper-only strategy will be limited by the clamp length ofthe dampers. Both the dynamic modeling of such orthogonal

cross-damper-system and the optimization of cross-tie-multi-damper configurations, however, require future study.

Acknowledgments

Professor Nicholas P. Jones, Dean of the G.W.C. Whiting Schoolof Engineering at Johns Hopkins University, is gratefully ac-knowledged for valuable discussion during the preparation of themanuscript. The late Mr. William Jack Spangler, Senior InstrumentDesigner, Johns Hopkins University, is gratefully acknowledged forhis active contributions to the design of the experimental instru-mentation for the original project. Mr. Randall W. Poston, WDPConsulting Engineers, Austin, Texas, is acknowledged for provid-ing the technical data on the cable dampers. The full-scale in-vestigation presented in this study was sponsored by NationalScience Foundation (Grant No. 0305903) and the Texas Depart-ment of Transportation of the United States.

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Effectiveness of cable networks of various configurations in suppressing stay-cable vibration

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