Application of Structural Control Systems for the Cables of Cable-Stayed Bridges: State-of-the-Art and State-of-the-Practice

Abstract

Stay cables are one of the key elements of cable-stayed bridges and are characterized by lightweight, low inherent damping, and high flexibility. They are continuously subjected to small-to large-amplitude vibrations due to various types of dynamic loads that may, in the long term, cause fatigue and fracture problems for the cable system, and may eventually compromise the safety of cable-stayed bridges. Thus, several countermeasures including surface profiling, cross-ties, and structural vibrational control systems have been used to improve the dynamic performance of stay cables. This article presents a comprehensive state-of-the-art and state-of-the-practice review of structural vibration control systems specifically designed and used for the cables in cable-stayed bridges. Generally, the stay cable dampers are classified as internal and external dampers. Consequently, important aspects of each control strategy are highlighted and various types of devices and their designs are discussed to find the best control solution for suppressing the cable vibrations.

Full text

Vol.:(0123456789)
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Archives of Computational Methods in Engineering
https://doi.org/10.1007/s11831-021-09632-4
REVIEW ARTICLE
Application ofStructural Control Systems fortheCables
ofCable‑Stayed Bridges: State‑of‑the‑Art andState‑of‑the‑Practice
AhadJavanmardi1,2 · KhaledGhaedi2,3· FuyunHuang1,4 · MuhammadUsmanHanif5· AlirezaTabrizikahou6
Received: 26 February 2021 / Accepted: 4 July 2021
© CIMNE, Barcelona, Spain 2021
Abstract
Stay cables are one of the key elements of cable-stayed bridges and are characterized by lightweight, low inherent damping,
and high flexibility. They are continuously subjected to small-to large-amplitude vibrations due to various types of dynamic
loads that may, in the long term, cause fatigue and fracture problems for the cable system, and may eventually compromise
the safety of cable-stayed bridges. Thus, several countermeasures including surface profiling, cross-ties, and structural
vibrational control systems have been used to improve the dynamic performance of stay cables. This article presents a com-
prehensive state-of-the-art and state-of-the-practice review of structural vibration control systems specifically designed and
used for the cables in cable-stayed bridges. Generally, the stay cable dampers are classified as internal and external dampers.
Consequently, important aspects of each control strategy are highlighted and various types of devices and their designs are
discussed to find the best control solution for suppressing the cable vibrations.
Nomenclature
CEC Cycle energy control
CFD Computational fluid dynamics
CVD Controlled viscous damping
D Diameter of the cable
DG Dry galloping
EHC Energy harvesting circuit
EIMD Electromagnetic inertial mass damper
EMD Electromagnetic device
EMDEH Electromagnetic damper cum energy
harvester
EMSD Electromagnetic shunt damper
EMSD-ID Electromagnetic shunt damper-inerter damper
device
FPB Friction pendulum bearing
FRP Fiber-reinforced polymer
HDR High-damping rubber
ID Inerter damper
IG Ice galloping
IMD Inertial mass damper
IVA Inerter-based vibration absorber
L Length of cable
LQG Linear–quadratic Gaussian
LQR Linear quadratic regulator
LRB Laminated rubber bearing
m Mass
MR Magneto-rheological
MSM Mode superposition method
NSD Negative stiffness damper
PSD Positive stiffness damper
P-VE Pseudo-viscoelastic
RWIV Rain-wind-induced vibration
Sc Scruton number
SMA Shape memory alloy
TET Targeted-energy-transfer
TID Tuned inerter damper
* Fuyun Huang
huangfuyun@fzu.edu.cn
Ahad Javanmardi
ahadjavanmardi@gmail.com
1 College ofCivil Engineering, University Town, Key Lab
ofFujian Province, Fuzhou University, 2 Xueyuan Road,
Fuzhou350108, China
2 Center ofResearch andDevelopment, PASOFAL
Engineering Group, 52200KualaLumpur, Malaysia
3 Civil Engineering Department, Faculty ofEngineering,
University ofMalaya, KualaLumpur, Malaysia
4 Fujian Provincial Key Laboratory onMulti-disasters
Prevention andMitigation inCivil Engineering, Fuzhou
University, Fuzhou350108, Fujian, China
5 School ofCivil andEnvironmental Engineering (SCEE),
National University ofSciences andTechnology,
IslamabadH-12, Pakistan
6 Institute ofBuilding Engineering, Poznan University
ofTechnology, Piotrowo 5, 60-965Poznan, Poland

A.Javanmardi et al.
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TMD Tuned mass damper
TMD-MR Tuned mass damper-Magnetorheological
TM-HDR Tuned mass-high damping rubber
VID Viscous inerter damper
VIMD Viscous inertial mass damper
VIV Vortex-induced vibration
VSD Viscous-shear damper
WG Wake galloping
𝜁
Damping ratio
𝜌
Density of air
1 Introduction
Cable-stayed bridges are one of the most aesthetically
appealing and popular bridges worldwide due to their advan-
tages i.e., lightweight, relatively small cross-sections,and
high structural efficiency. Meanwhile, the race for achiev-
ing a longer span cable-stayed bridge is entering a new
era with the Russky Bridge (with a main span of 1104m)
holding the current world record. On the other hand, cable-
stayed bridges are characterized by low structural damping,
complex dynamic behavior and longer natural time peri-
ods [1–3]. Their relatively small section and lighter weight
make them susceptible to oscillations under dynamic load-
ings such as wind, earthquake, pedestrian and traffic loads
[4]. Moreover, as the span length of cable-stayed bridges
increases, the flexibility of the structure and the length of
stay cables also increase, which may result in aerodynamic
stability issues.
As shown in Fig.1, stay cables are generally subjected
to direct vibration due to the wind acting along the cable
and indirect vibration through cable anchorages at the deck
and towers due to wind, traffic, and earthquake loads. Cable
vibrations are associated with several problems for the cable-
stayed bridges, i.e., (i) comfort reduction for the passing
traffic, (ii) resonance, causing the structural damage, (iii)
fatigue issues in cables, anchorages, and other components
that could reduce their service life, and (iv) potential pre-
mature failure of the corrosion protection system. Several
cable-stayed bridges such as Puente Real Bridge [5, 6],
Saint-Nazaire Bridge7, Zarate Brazo Largo Bridge [7],
Dubrovnik Bridge [8], Kap Shui Mun Bridge [9], Veterans
Memorial Bridge [10], Fred Harman Bridge [11], and Inter-
national Guadiana Bridge [12] have reportedly been dam-
aged due to dynamic loads, as shown in Fig.2.
A structural control system is defined as a system that
reduces vibrational responses of structures due to different
types of dynamic loads. Broadly speaking, structural con-
trol systems are classified as passive, active, semi-active and
hybrid control systems [13]. Passive control systems enhance
the dynamic performance of the structure by adding stiffness
or damping or a combination of both without a need for a
control algorithm or any external source of energy. Active
control systems receive the essential dynamic response
parameters of the structure under the dynamic loads from the
sensors and analyze the data using a predefined algorithm
that controls and activates externally powered actuators to
counter the external dynamic loads. The active systems
need a dedicated and uninterrupted power source for proper
operation. Semi-active control systems are similar to active
control systems. However, they require a small supply of
energy (like a battery), which is advantageous in providing
continuous protection of the structures during power failure
under extreme external excitations. Hybrid control systems
are a combination of passive, active and semi-active control
systems in order to improve the performance, efficiency, and
stability of the control system. Prior to the date, huge num-
bers of structural control systems have been proposed and
used for civil structures [14–30]. Nonetheless, the structural
vibration control system was found to be a promising solu-
tion for cable-stayed bridges in order to improve the dynamic
response and minimizing vibrations [31–38].
The design service life of cable-stayed bridges is typi-
cally over 100-year, owing to their massive construction
cost. While engineers are aiming to achieve longer main
spans, the key inevitable challenge of cable-stayed bridges
remains to be the complex dynamic behavior and cable
vibrations due to wind and aerodynamic instabilities. Thus,
several countermeasures for suppressing the cable vibra-
tions have been proposed and adopted in practice. Over the
past few decades, great progress has been made from theory
Fig. 1 Different types of
dynamic loads in the cable-
stayed bridges
Source of vibraon
Wind-induced vibraons Live Loads Earthquakes
Vortex-induced
vibraon
Wind-rain-induced
vibraon GallopingVehicle
Pedestrian
Train
Near field
Earthquake
Far field
Earthquake
Ice galloping
Dry ga lloping
Wake galloping

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
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to utilization for the structural control systems specifically
designed for stay cables. This paper presents a state-of-the-
art and state-of-the-practice review of structural vibration
controls used specifically for the cables of the cable-stayed
bridges to suppress the cable vibrations due to dynamic
loads. In this context, an overview of the vibration problems
of stay cable under dynamic loads is provided. Then, various
structural control systems for stay cables are classified and
systematically reviewed. Subsequently, a summary of stay
cable dampers and a list of cable-stayed bridges equipped
with stay cable dampers are given. Finally, concluding
remarks on suppressing vibrations of cables by means of
control systems are given.
2 Vibrations ofStay Cables
As mentioned earlier, stay cables may suffer vibrations
due to direct excitation by the wind- or indirectly through
the movements of the deck and towers (see Fig.1). As
a direct source of vibration on stay cables, the wind is
one of the most challenging aspects of the design of
cable-stayed bridges. Figure3 shows the main vibration
phenomena associated with stay cables [7, 39–42]. Vari-
ous aeroelastic phenomena in cables are motion-induced
vibration, vortex-induced vibration (VIV), buffeting, rain-
wind-induced vibration (RWIV), wake galloping (WG),
dry galloping (DG), ice galloping (IG), aerodynamic
Fig. 2 Damage to the cable system of the cable-stayed bridges due
to dynamic loads. a Guide pipe fracture failure due to wind and the
strengthening it for the Fred Harman Bridge [11] b Protective pipe
damage of a stay cable of the Dubrovnik bridge due to wind loads [8].
c Damage to the cable and anchorage of the International Guadiana
Bridge due to cable-deck dynamic interaction and heavy traffic loads
[12]

A.Javanmardi et al.
1 3
interference, drag crisis phenomena, etc., [10, 43–49]. It
has also been reported that two adjacent cables having
alike configurations may experience different vibrational
modes during wind-induced vibrations [50]. Meanwhile,
the RWIV may cause coupling of in-plane and out-of-
plane vibrations of the cables [51]. Indirect excitation or
support-induced excitation of the cables is generated by
the motion of the girder and pylons through the cables’
anchorages or supports [42]. In addition, strong coupling
between local and global vibrations may occur when local
and global natural frequencies become nearly the same.
For example, the cable and anchorage system of a stay
cable of the International Guadiana Bridge was damaged
due to indirect excitation (heavy traffic loads), as shown in
Fig.2c. This damage reflects the importance of cable-deck
dynamic interaction as a combination of the global vibra-
tional mode of the deck and the local vibrational mode of
the cables [12]. Therefore, the complex characteristics of
the cable-deck interaction significantly affect the perfor-
mance of stay cable dampers, and it should be considered
during the design stage of the stay cable dampers [52].
According to U.S. Federal Highway Administration
(FHWA) [46], the maximum allowable amplitude of the
cable vibration should be on the order of the cable diam-
eter. In practice, a criterion based on Scruton number (Sc)
(also known as Irwin’s criterion [53]) is used to evaluate
the damping capability of cables. The Scruton number is
a non-dimensional number defined as:
where m is mass per length of the cable;
𝜁
is the damping
ratio;
𝜌
is the density of air and D is the diameter of the
cable. Based on Irwin’s criterion, the Sc should be equal to
or greater than 10 for smooth circular cable, and for cable
with pipe surface treatments, the Sc should be greater than
(1)
S
c=
m𝜁
𝜌
D2
5 [54]. If a cable does not meet this requirement, additional
damping should be provided to control the cable vibrations.
3 Vibrational Control Systems forStay
Cables
A few countermeasures are used to minimize the vibra-
tions of the cables of cable-stayed bridges such as cable
surface treatment, using cross-ties and vibrational control
systems for stay cables [55]. Cable surface treatment gen-
erally reduces the RWIVs [56]. There are several types
of surface treatments i.e. helical fillet, rings, and pattern
intended [57, 58]. The cross-ties are secondary cables that
transversely connect the stay cables together to form cable
networks and increase the in-plane stiffness of the cables.
Though the cross-ties are found to be effective in reduc-
ing the in-plane vibrations of the cables [59] but they are
less appealing in terms of aesthetics points of view. The
cross-ties create intermediate supports at the stay cables,
which cause an increase in the natural frequency of the stay
cables for the vertical modes [60, 61]. The cross-ties can
also increase the modal damping of the stay cables, how-
ever, the damping increment depends on the properties of the
cross-ties and their connection with cables [62, 63]. Some-
times, these countermeasures are simultaneously adopted
to achieve higher efficiency in minimizing the vibrations of
stay cables [64]. For the sake of conciseness, countermeas-
ures other than structural control systems are not further
discussed. Stay cable dampers are one of the most notable
solutions in controlling the vibrations of stay cables. Gener-
ally, stay cable dampers are two types, internal dampers and
external dampers. As shown in Fig.4, stay cable dampers
can be installed near the cable anchorage to the deck or, if
necessary, at both anchorage ends of the cable (a much less
common approach). Moreover, by increasing the damping of
cable, stay cables act as tuned mass dampers that may also
Fig. 3 Cable vibrations phenomena [42, 49]

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
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help in reducing the vibrations of deck or pylons [10, 41].
The research on the importance of stay cable vibrations and
its alleviation such as stay cable dampers dates back to the
1980s [65, 66]. The stay cable dampers not only improve
the vibrational response of the individual cables but also
provides additional damping to the entire bridge, which can
also enhance the seismic performance of the bridge [67].
Furthermore, several stay cable and vibration control device
manufacturers such as BBR VT International, Freyssinet,
Maurer AG, Structural/VSL, etc., have proposed and imple-
mented a variety of stay cable dampers. As a result, a large
number of modern cable-stayed bridges are equipped with
stay cable dampers worldwide. In the following sections,
the two types of stay cable dampers are discussed in detail.
3.1 Internal Dampers
Internal dampers (also called integrated dampers) are placed
entirely within the anchor pipe of the stay cables or inte-
grated within the cable system as shown in Fig.5. As the
name indicates, the internal dampers provide a smooth outer
shape. Nakamura etal. [68] studied the effect of high-damp-
ing rubber (HDR) dampers on the vibrational response of
stay cables. The HDR damper was effective in reducing the
RWIVs of stay cables. However, the study concluded that
for parallel stay cables, the HDR dampers were less effective
for wake galloping vibrations.
Hwang etal. [72, 73] proposed to use two types of iso-
lation systems for long stay cable that integrated entirely
inside the anchorage system of the cable. The first proposed
system consisted of an isolation system namely laminated
rubber bearing (LRB) and an internal damper, as shown in
Fig.6a. The second type of isolation system was friction
pendulum bearing (FPB) as illustrated in Fig.6b. The results
of the numerical study indicated that the proposed isolation
systems outperformed the optimal external passive damper
for the stay cable in reducing the wind-associated vibrations
of the cables. Although the concept of the proposed isolation
systems was numerically analyzed and found to be effective
but more detailed studies are needed to address the crucial
factors affecting the cable’s vibrations for the isolation sys-
tem of the cables.
Wang and Wu [74] invented the smart damper for reduc-
ing the vibrational response of the hybrid fiber-reinforced
polymer (FRP) cable. The smart damper consisted of viscoe-
lastic material integrated between two layers of the hybrid
cables and had two different configurations as illustrated in
Fig.7. The study concluded that the smart damper having
the discontinuous distribution of viscous material provided
the same optimal damping for the stay cable and hada bet-
ter static behavior as compared with the continuous con-
figuration. In addition, the smart damper performance for
reducing the out-of-plane vibrations was better than the in-
plane vibrations. Furthermore, the result of the numerical
study of the proposed smart system was validated with the
experimental test results [75] and it was confirmed that with
the increase of the vibration amplitude (in-plane and out-
of-plane vibrations), the modal damping ratio of the smart
damper-cable system also increases.
Egger etal. [76] proposed the multiple-mass-particle
impact damper that had a similar configuration as the smart
damper. As illustrated in Fig.8, the multiple-mass-particle
impact damper consisted of a stay strand inside a hollow
constrainer that was integrated within the cable. The con-
stitutive model of the damper was derived and the pro-
posed impact damper was tested on a real cable through
Fig. 4 Installation location of stay cable dampers in the cable-stayed bridges

A.Javanmardi et al.
1 3
an experimental test. The investigation results showed that
the multiple-mass-particle impact damper was effectively
provided additional damping for a wide range of vibrational
modes of the stay cable.
3.2 External Dampers
External dampers are externally mounted to the stay cables
as shown in Fig.9. In practice, external dampers are
Fig. 5 Some of the most used internal dampers. a Square friction
damper (courtesy of BBR VT International [69]), b Internal viscous
damper (courtesy of BBR VT International [69]), c Internal elasto-
meric damper (courtesy of Freyssinet [70]), d Internal hydraulic
damper (courtesy of Freyssinet [70]), e Internal radial damper (cour-
tesy of Freyssinet [70]), f Internal friction damper [71]
Fig. 6 Detailing of isolation systems proposed for the stay cable. a LRB [72] and b FPB [73]

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
1 3
usually placed near to the cable’s anchorage fora very long
cable, as they are more efficient than the internal dampers.
In this regard, Chen and Sun [77] developed an experimen-
tal framework for the laboratory scale testing of stay cable
with attached dampers such as viscous and magneto-rhe-
ological (MR) dampers. The results of the proposed setup
framework for the stay cable dampers were comparable
with those obtained by the full-scale cable tests.
One of the early investigations on mitigating the stay
cable vibrations of cable-stayed bridges due to wind load
was done by Wianecki in 1983 [65]. During the construc-
tion of the Brotonne cable-stayed bridge, a storm caused
the stay cables of the bridge to vibrate up to their third
fundamental mode. Consequently, hydraulic dampers (vis-
cous dampers) were installed between the stay cables end
and deck to suppress the vertical and horizontal vibra-
tions of the cables. Aerodynamic instabilities with large
amplitudes were also observed in the stay cables of the
Erasmus Cable-Stayed Bridge shortly after its completion
[80]. The initial investigation indicated that RWIVs were
the cause of aerodynamic instabilities of the stay cables.
Alternatively, the hydraulic damper was proposed and
installed to stabilize the stay cables.
Pacheco etal. [81] developed a simple design procedure
for designing the external viscous damper for the stay cable.
The proposed simplified procedure was based on a universal
estimation curve that relates the modal damping ratio of the
cable with the cable parameters (length, mass per unit length
and fundamental frequency), the mode number, the attached
damper, the damper size, and the damper location. The cal-
culated modal damping by this method was found to be high
enough to mitigate most of the wind-induced vibrations.
Nonetheless, the damper effectiveness may reduce without
considering the non-viscous properties of the damper and
the sag effect of the cable. Yu and Xu [82, 83] investigated
the usage of multi-pairs oil dampers for vibrational control
of the stay cables. The design formulation of the damper was
based on a hybrid method that considered several parameters
including cable inclination angle, cable sag, cable inherent
damping, damper stiffness, damper direction, and others
[84]. The dynamic in-plane and out-of-plane responses of
stay cables subjected to harmonic excitations were deter-
mined in frequency domains. It was observed that the modal
damping of the first in-plane mode of the long cables with
significant sag was much smaller than the other in-plane
and out-of-plane modes due to cable-frequency avoidance
effects. It was found that the sensitivity analysis should be
conducted in order to find the optimum damper properties
and position for practical applications. Consequently, it was
recommended to use one pair of viscous dampers at each
end of stay cables to achieve the required modal damping
ratios. Furthermore, Xu etal. [85] validated the theoretical
formula of a stay cable with oil damper through free vibra-
tion and force vibration tests. Maximum modal damping
was achieved based on the optimization of the damper size,
which significantly reduced the vibration of the cable.
Tabatabai and Mehrabi [86] developed a complex eigen-
value problem based on the governing differential equa-
tion for vibration of the stay cable equipped with a viscous
damper that includes the sag extensibility and cable bend-
ing stiffness. However, their investigation was limited to
the first vibrational mode of the cable and concluding that
Fig. 7 Smart damper configurations for the hybrid FRP cable [74]
Fig. 8 The 3D view of multiple-mass-particle impact damper in a
stay cable [76]

A.Javanmardi et al.
1 3
viscous dampers not only provide sufficient modal damp-
ing for the first mode of the cable but could also improve
the modal damping of the higher modes. In addition, the
cable sag effect was found to be insignificant on the result-
ing cable damping ratios, while the bending stiffness had a
great influence. To overcome the design limitation of the
viscous damper for a single mode of stay cables, Wang etal.
[87] proposed an optimal design procedure for the viscous
dampers for controlling the multi-mode vibrations of the
cable. The design algorithm of the viscous damper was
based on the optimal Linear–Quadratic Gaussian (LQG)
control theory. Consequently, the results of the numerical
analysis showed that the proposed method could effectively
provide a high level of damping for the cables over a broad
range of vibrational frequencies. Fujino and Hoang [88]
derived a formula for designing a viscous damper for the
stay cable by assuming a finite support stiffness of the cable.
In the design formula, several parameters such as the sag
and flexural rigidity of the cable were considered. Weber
etal. [89] proposed a design procedure for the linear vis-
cous damper to increase multi-mode damping of the stay
cable with the minimum damper distance from the cable
support. The proposed design procedure was only depend-
ent on the cable properties (length, mass per unit length,
tension force) and the damper position without the need
for numerical iterations. Due to the unpredictability of the
predominant mode of the stay cables, this method assures
the safety of the cables during the large amplitude vibra-
tions. Cheng etal. [90] proposed an energy-based method
for evaluating the damping ratio of a cable with an attached
viscous damper. The proposed energy-based method had
no limitation on the damper location and was found to
be advantageous to determine the overall damping of the
cable-damper system during the preliminary design stage.
Furthermore, Zhou etal. [91] performed a stochastic analy-
sis on the coupling effect of the in-plane and out-of-plane
vibrations of stay cable with attached viscous dampers. The
results of the parametric study indicated that the in-plane
and out-of-plane displacements of the cable-damper system
were independent of the loading direction when the vis-
cous coefficient of the damper and the excitation level were
constant. However, when the damper size was constant, the
Fig. 9 Some of the most used external dampers. a Viscous damper, b MR damper (courtesy of Maurer AG [78]), c MR damper (courtesy of
Maurer AG [78]), d HDR damper [79]

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
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in-plane and out-of-plane displacements of the cable-damper
system were increasing with the increase of the excitation
level. Javanbakht etal. [92] developed a control-oriented
model for simulating the dynamic behavior of a stay cable
with an attached damper. The proposed model was based
on the mode superposition method (MSM) with enhanced
shape functions and account different boundary conditions
(hinged-hinged and fixed–fixed boundary conditions), bend-
ing stiffness and sag effect of the cable. The proposed model
required lesser numbers of modes to satisfy the convergence
criterion due to the presence of the static correction term.
Moreover, the proposed model performed better in terms
of the convergence rate and numerical accuracy over the
other methods, which shows its potential for control design
applications of the stay cable with external dampers. Sun
etal. [93] improved the reduced-order model for the stay
cable with attached viscous damper. The improved model
accounted for the sag-extensibility parameter, which is also
known as Irvine parameter that includes the sag and stiff-
ness effects. Numerical studies was performed on cables
having different sag-extensibility parameters with one and
two viscous dampers having different nonlinearities. The
cable vibrational modes that were affected by the sag param-
eters, only considered for the numerical analysis. The results
showed that for the small sag-extensibility parameter, the
nonlinear damper had higher modal damping as compared
to the linear damper because the nonlinearity induced the
energy bleeding effect. Nonetheless, for a certain range of
the sag-extensibility parameter, the maximum modal damp-
ing ratio of the linear dampers was greater than the nonlinear
dampers. When the sag-extensibility parameter was large,
the nonlinear and linear damper had almost the same modal
damping. Moreover, different damping mechanism was
observed for symmetric modes when two similar dampers
were symmetrically installed at the cable ends as compared
to the summation of the optimal damping obtained by each
damper alone. Aforementioned difference was depended on
the sag-extensibility of the cable and the nonlinearity of the
damper.
Main and Jones [94] performed a field test on linear
viscous dampers mounted on stay cables of Fred Hartman
Bridge in Houston, USA. The dampers were tuned and
optimized for the first vibrational mode of stay cables. The
analysis of the field test confirmed the effectiveness of the
linear damper on mitigating the stay cable vibrations due
to the RWIVs. However, it was highlighted that the linear
damper performance was mode-dependent, therefore, it was
suggested to use the nonlinear damper for suppressing the
stay cable vibrations. In addition, it was pointed out that the
damping coefficient of the linear viscous damper should be
optimum in order to maximize mitigating the in-plane vibra-
tions of the cable due to RWIV [95]. Furthermore, Main
and Jones [96, 97] used an analytical approach based on the
complex eigenvalue problem for the free vibration of stay
cable with the external linear and nonlinear dampers. The
investigation of linear and nonlinear dampers for the stay
cables concluded that the nonlinear dampers could achieve
optimal performance for a wide range of modes of the cables
as compared to linear dampers. A similar conclusion was
also obtained from the energy equivalence approach [98] and
multi-harmonic balance method [99], the nonlinear dampers
are more advantageous in supplying modal damping for a
wide range of vibrational modes of the stay cables as com-
pared to linear dampers. Danhui etal. [100] also found that
the damping coefficient and cable tension forces have sub-
stantial effects on the modal damping of the cable-damper
system and these parameters should be used as design vari-
ables during the design stages. The advantage of the non-
linear viscous damper over the linear viscous damper is that
the nonlinear damper provides the maximum damping at
different vibration amplitudes for each vibration mode of the
cable while the linear damper can only provide maximum
damping for one particular mode of the cable [101]. Unlike
the linear dampers, the nonlinear dampers can transfer the
induced energy from the lower modes of the cable to higher
modes [99]. Due to this unique feature, the nonlinear damp-
ers can be placed closer to the cable anchorage and tuned to
the lower vibrational modes of the cables.
Caracoglia and Jones [102] investigated the effective-
ness of utilizing two viscous dampers at each end of the
cable to suppress the in-plane vibrations of the stay cable.
The analytical analysis of the universal estimation curve
was extended to a practical universal curve for two viscous
dampers attached to a stay cable. When the dampers were
spaced closely to each other on the cable, the stiffness of
the cable-damper system increased, while the damping
ratio decreased. Nonetheless, it was recommended to use
the dampers at both ends of the stay cable to have a higher
damping level for the system. Hoang and Fujino [103] also
studied the combined effect of two viscous dampers attached
to a cable. Two cases i.e., (i) one damper at each end of the
cable and (ii) two dampers at the same end of the cable
were considered for studying the combined effect. It was
concluded that the maximum combined damping effect of
the two dampers was achieved when they were attached to
each end of the cable. When the dampers were located at
the same end of the cable, the modal damping was mostly
from the outer damper located at a longer distance from
the cable end. In other words, utilizing two dampers on the
same end of the cable has no advantages over the use of a
single damper in increasing the modal damping of the stay
cable. Fournier and Cheng [67] studied the effect of damper
and support stiffness on the efficiency of the linear viscous
damper in suppressing the cable vibrations. The results of
the experimental and numerical study revealed that the effi-
ciency of the damper reduced when the damper stiffness

A.Javanmardi et al.
1 3
increased or the support stiffness decreased. The damper
support stiffness had a more profound impact on the effi-
ciency of the damper, and it was suggested to install the
damper closer to the mid-span of the cable. Even so, if the
damper size exceeds a certain capacity, the damper stiff-
ness and damper support stiffness may have adverse effects
on the damper efficiency. Furthermore, in order to have a
higher efficiency of the linear viscous damper, it is better to
have stiff support and keep the damper stiffness as low as
possible. The results of the experimental test also confirmed
that the damper stiffness should be minimized to achieve
higher modal damping for the cables [101]. Meanwhile, the
nonlinear force can dissipate a larger amount of energy for
the higher-mode oscillations. Additionally, the rotational
restrainer effect has an adverse effect on the damping ratio
of the viscous damper attached to the cable [104]. Therefore,
the rotational restrainer effect should be accounted for in the
design of the damper to supply adequate modal damping to
the cable. Nguyen and Macdonald [105] studied the gallop-
ing instability effect on a stay cable with two orthogonally
attached viscous dampers. It was pointed out the conven-
tional galloping analysis may overestimate the critical wind
speed for the galloping effect. Moreover, the complexity of
the mode shapes should be accounted for, as it shows the
cable instability during the oscillations.
Low frequency and large amplitude oscillations due to
RWIVs, severely damaged the guide pipes of the anchorage
of a cable of Fred Hartman Bridge (see Fig.1)[11]. Subse-
quently, the investigative team recommended implement-
ing the hydraulic dampers as one of the countermeasures
to suppress the wind-induced vibrations of the stay cables.
Thus, two prototype hydraulic dampers were installed on
two stay cables of the bridge. The prototype hydraulic
dampers were optimized for damping in the first mode of
the cables. After three years of monitoring the prototype
dampers, the bridge experienced over 35,000 events of wind-
induced vibrations. The results of monitoring confirmed
the positive performance of the damper in suppressing the
stay cable vibrations. However, the prototype dampers had
hydraulic oil leakage, and the out-of-plane vibrations of the
stay cable causeddamge due to the rotation of the damper at
its supports as shown in Fig.10a. Consequently, the support
configurations of the final damper design were changed as
shown in Fig.10b, c to avoid any damage to the dampers
Fig. 10 a Damage to the prototype damper due to out-of-plane vibration of the cable, b final hydraulic damper design, and c hydraulic dampers
installed on the stay cables of the Fred Hartman cable-stayed bridge [11]

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
1 3
due to out-of-plane vibrations of the cable. In addition, the
modal analysis of the bridge indicated that 95% of the cables
had a dominating response greater than the third vibrational
mode.
Chen etal. [106] configured lateral and rotational viscous
dampers for controlling the cable vibrations. As shown in
Fig.11, the proposed stay cable damper was a combination
of a central viscous damper that provides lateral damping
and two eccentric viscous dampers providing rotational
damping, which were arranged parallelly. The numeri-
cal analysis showed that by combining the lateral damper
with rotational dampers, the optimal damping of the cable
increased up to 30% and 15% as compared with a single
lateral damper for the clamped and pinned support configu-
rations, respectively.
Jensen etal. [107] proposed to use a tuned mass damper
(TMD) between the closely spaced stay cables of the bridge
as shown in Fig.12a. A band-limited white noise function
was used as the excitation input and only the linear behavior
of the cables was considered for the numerical study. The
results showed that the TMD caused a 14% reduction in
the vibrational response of the two closely spaced cables.
Hijmissen etal. [108] found that apart from the TMD param-
eters (the TMD mass, spring, and damping constants), the
bending stiffness of the cable with an attached TMD greatly
influences the modal damping of the first tuned mode of the
cable-damper system, while for the higher modes its effect
is insignificant. As illustrated in Fig.12b, Sun etal. [109]
also proposed a similar approach for suppressing adjacent
cables’ vibrations by using a tuned inerter damper (TID).
The approximate solution of the damper was derived for the
optimal tuning and damping ratio of the TID. The TID was
found to be effective in maximizing the damping ratio of
the lower bound of all targeted modes of the cables. On the
contrary, the TID had an insignificant damping ratio increase
for the in-phase modes of the twin cable system.
Hovenring Bridge (Netherland) is a ring-shaped cable-
stayed bridge, designed for bicycles. Wind-induced vibra-
tions were observed on several stay cables of the bridge
shortly after its construction in 2011 [110]. An on-site inves-
tigation and numerical analysis were conducted to find the
modal parameters of the bridge. It was found that the cables
were excited due to vortex-induced vibrations, thereafter, a
type of TMD called Stockbridge-type damper was installed
on the cables as a mitigation measure. The Stockbridge-
type damper is a multiple degrees-of-freedom TMD that is
usually used for electric power transmission lines. Conse-
quently, the Stockbridge-type dampers were installed on the
Fig. 11 The schematic view the lateral and rotational dampers
installed on a stay cable [106]
Fig. 12 Analytical models for suppressing the vibrations of the adjacent cables by using a TMD [107] and b TID [109]

A.Javanmardi et al.
1 3
stay cables of the bridge as showing in Fig.13 and found to
be effective in suppressing the cables’ vibrations.
Fujino and Hoang [88] derived an approximate formula
for the HDR damper externally attached to a stay cable. The
damping force of the HDR damper is independent of the
frequency due to the hysteretic characteristics of the rubber
material. The cable sag and flexural rigidity of the cable
were included in the proposed design formula and the sup-
port stiffness was assumed to be finite. Cu and Han [111]
studied the effect of different parameters of HDR damper
including, spring factor, material loss factor, and the damper
location on dynamic behaviors of the stay cable. It was sug-
gested that the HDR dampers are more suitable for short
stay cables or can be combined with other dampers to have
higher efficiency. Le [112] investigated the boundary condi-
tion effect on the performance of the HDR damper attached
to a cable. The results showed that the efficiency of the HDR
damper increases when the rotational restrain of the cable
has a finite stiffness. Moreover, the modal damping was
found to be independent of the boundary condition and the
vibrational modes.
Ni etal. [113] developed a neuro-control algorithm for
the semi-active MR damper to control the vibrations of stay
cables. The effectiveness of the proposed neural network
control methods was tested numerically and experimentally
on a scaled cable model with an attached MR damper to
its lower anchorage. The study concluded that the proposed
control algorithm for the semi-active MR damper could pro-
gressively control the vibrational response of the stay cables.
Johnson etal. [114] performed a comparative study between
the semi-active damper and passive damper for mitigating
the vibrations of the stay cables. Several parameters such as
cable sag, inclination angle, and longitudinal flexibility of
the cables were considered. The results indicated that the
semi-active damper was 30–40% more efficient than the pas-
sive damper in reducing the vibrational response of the stay
cables. Nevertheless, the damping force of the MR damper
without current (in the passive mode) increases with the
increase of the excitation frequency [115]. The MR damper
having a large current could provide a similar damping force
for a wide range of frequencies for the stay cables. Weber
etal. [116] performed experimental studies to evaluate the
effectiveness of the MR damper in reducing the vibrational
response of the stay cables. Several constant currents of
the MR damper were considered for two different control
strategies, namely, maximum damping of one mode and
maximum damping of several modes. Results of the tests
indicated that although the MR damper reduced the in-plane
vibrations of the stay cable,but it was less effective for con-
trolling the out-of-plane vibrations of the cable, especially
at higher frequencies. Therefore, it was suggested to tune the
MR damper in the damper plane. The optimal current level
of the MR damper could provide maximum modal damp-
ing to a target mode. However, the damper current had an
insignificant effect on the modal damping when the aim was
to reduce multi-mode vibration of the cable.
Dongting Lake Bridge experienced severe cable oscil-
lations due to RWIV after the construction [117]. Alterna-
tively, it was suggested to equip the stay cables with semi-
active MR dampers. Initially, a couple of semi-active MR
dampers were installed on a few cables, and field measure-
ment and real-time monitoring were conducted for 45days.
After confirming the effectivenessof the proposed control
system at the end of the test trial, a total 156 of twin-damp-
ers were installed on the stay cables of the Dongting Lake
Bridge, as shown in Fig.14. A decade later after the instal-
lation of the MR dampers on the Dongting Lake Bridge,
several dampers were randomly selected and their service
condition was evaluated [118]. It was found that 80% of the
selected dampers could supply a considerable damping force
to the cables after 10years of service. The two most crucial
problems associated with the MR dampers were found to
be the precipitation of the iron particle and the leakage of
the MR fluid. Therefore, it was suggested to increase the
durability of the MR dampers and protect them from harsh
environments. In addition, it was also recommended to posi-
tion the piston end in the upward direction to avoid leakage
due to gravitational force.
Duan etal. [119] developed a state-derivative feedback
control for the semi-active MR damper to suppress the vibra-
tion of the stay cables. The flowchart of the proposed semi-
active control system is shown in Fig.15. The cable accel-
eration was measured by a single accelerometer attached to
near the lower end of the cable. The cable was excited by
the sweeping sine function and the sinusoidal step relaxation
function. With the real-time data collection, the control algo-
rithm was mostly commanded the appropriate dissipative
Fig. 13 Stockbridge-type dampers installed on the stay cables of the
Hovenring Bridge [110]

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
1 3
force for the MR damper to control the stay cable vibrations
(see Fig.16). In conclusion, high equivalent damping of the
stay cable was achieved through the proposed semi-active
control system.
Christenson etal. [120] investigated the use of an MR
shear mode damper with a clipped optimal H2/ LQG control
method in reducing the vibrational response of the stay
cable. As shown in Fig.17, the MR shear mode damper
was made of a foam paddle saturated with MR fluid passing
through a pair of parallel steel plates with an attached coil
of copper wire at one end of the plates. When the MR fluid-
saturated paddle moves in the magnetic field, it generates
the damping force. The H2/LQG control algorithm used the
output feedback from the force and displacement sensors to
determine the appropriate control force for the damper. The
Fig. 14 Semi-active MR dampers installed on the Dongting Lake Bridge
Fig. 15 Semi-active state-derivative control in reciprocal state space framework for the MR damper of a stay cable
Fig. 16 Stay cable acceleration response with and without a semi-
active MR damper under sweeping sine function excitations [119]
Fig. 17 The MR shear mode damper [120]

A.Javanmardi et al.
1 3
results of the experimental and numerical analyses showed
that the semi-active MR shear mode damper reduced the
in-plane vibrations of the cable by 50%. Moreover, the semi-
active control strategy vibrational response was 20% better
than the optimal passive control of the same damper.
Zhou etal. [121] performed a comparative study between
the performance of the MR damper and viscous damper for
controlling the in-plane and out-of-plane vibrations of a stay
cable through three-dimensional finite difference method.
When two or more vibration modes of the cable were
excited, the MR damper performed better in controlling the
cable vibrations as compared to the viscous damper. Other-
wise, if only a single mode of the cable was excited and the
viscous damper was tuned for that mode, both the dampers
had a similar performance. Ying etal. [122] developed a
semi-active control method for parametrically excited insta-
bility of stay cables based on the linear quadratic regulator
(LQR) control method for the MR dampers. The effect of
several parameters including the control force bound, time
delay, and state observation errors on the instability of the
cable were analyzed for controlling the transverse vibra-
tions (out-of-plane vibrations) of the cable. In conclusion,
the MR damper with the optimal control configuration was
effective in controlling the parametrically excited instabil-
ity of stay cables, especially for the minimum parameter-
excitation amplitude and unstable regions. Johnson etal.
[123] showed that the semi-active MR damper locating at
2% length of the cable from the cable end, had a 63% better
performance than the optimal viscous damper at the same
location. As mentioned earlier, the longer cable is, the higher
sensitivity to oscillations during the wind; therefore, the MR
dampers (with a similar configuration as shown in Fig.14)
were installed for the longest stay cables of Shandong Bin-
zhou Yellow River Highway Bridge [124]. For this bridge,
the semi-active control was developed based on the state
feedback of acceleration response from the accelerometers
located at several locations of the cable. The optimal damp-
ing ratio of the cable with the MR damper having the semi-
active control algorithm was substantially higher than the
passive MR damper configuration due to the pseudo negative
stiffness characteristic of the semi-active control. Li etal.
[125] proposed active and semi-active control systems with
negative stiffness for vibrational control of the cable. In the
numerical analysis, a pseudo-viscoelastic (P-VE) damper
was replaced with the active and semi-active control sys-
tems because the P-VE damper had similar characteristics
and behaviors. The LQR control algorithm was used for the
semi-active and active control of the MR dampers. Subse-
quently, it was concluded that the active and semi-active
control system with negative stiffness could provide a larger
damping ratio and perform better than the passive dampers
in reducing the vibrations of the cable due to enhancement
of the cable displacement. The RWIV was observed on the
longest stays of the Alamillo cable-stayed bridge [126]. The
observed vibration caused discomfort for the passing pedes-
trians. Several countermeasures were proposed including
using MR dampers for the longest cables. The MR damp-
ers were installed at about 3% of the cable length from
the cables’ anchorage such that they were located below
the pedestrian handrail. The field measurement after the
installation of the MR dampers concluded that the damper
increased the damping of the cables and thereafter no RWIV
was observed.
Two control algorithms i.e., modulated homogeneous
friction algorithm and the balance logic algorithm were
proposed by Zhou etal. [127] for the semi-active control
of MR damper attached to a cable. The performance of the
semi-active MR damper with these control algorithms was
compared to an optimal viscous damper tuned for the first
in-plane vibrational mode of the cable. The numerical study
indicated that the MR damper with the modulated homo-
geneous friction control algorithm and the viscous damper
almost had a similar performance in reducing the vibrational
response of the cable. Meanwhile, in some cases, the balance
logic control algorithm outperformed the modulated homo-
geneous friction control algorithm in mitigating the vibra-
tion of the cable with the MR damper. Weber and Distl [128]
also developed two control algorithms i.e., cycle energy con-
trol (CEC) and controlled viscous damping (CVD) for the
semi-active control of MR damper in suppressing the stay
cable vibrations. As shown in Fig.18, the proposed control
algorithms were based on the energy-equivalent adaptive
cable damping systems for the real-time controlling of the
MR dampers. The CVD control algorithm takes the opera-
tional temperature of the MR damper into account. The full-
scale field testing of the MR dampers with the two proposed
control systems on the stay cables of the Sutong Bridge
(China) and the Eiland Bridge (Netherlands) confirmed the
independency of the cable damping from the amplitudes and
frequencies of the vibrations for both algorithms. The CEC
and CVD semi-active controls of the MR damper are highly
practical as stay cables may vibrate at different amplitudes
and frequencies based on the wind condition. The CVD sys-
tem was also implemented for the stay cables of the Russky
Bridge (Russia). The CVD system found to be worthy by
computing the actual temperature of the MR damper. This
assures the force tracking that is independent of the actual
damper temperature, which is very important for bridges
with extremely harsh environmental conditions.
Zhao etal. [129] developed a new type of MR damper
with a new optimal equivalent control algorithm based on
the LQR control algorithm. As Fig.19 illustrates, the pro-
posed damper consisted of a single rod MR damper having a
spring-floating compensation device. In order to evaluate the
damper performance in reducing the in-plane vibration of a
cable, a numerical analysis was performed on a stay cable

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
1 3
with two semi-active MR dampers. The performance of the
semi-active control in reducing the vibrational response of
the cable was better than the optimal passive control. Mean-
while, the new optimal equivalent control algorithm with
minimum feedbacks requirements had comparable perfor-
mance to thatof the LQR control.
Electromagnetic damper cum energy harvester (EMDEH)
was proposed by Shen and Zhu [130] for the application of
the stay cable vibrations. The EMDEH was made of an elec-
tromagnetic device (EMD) connected to an energy harvest-
ing circuit (EHC). As illustrated in Fig.20, the application of
the EMDEH for a stay cable was numerically studied in the
MATLAB/Simulink environment. The EMDEH was found
to be efficient in reducing the cable vibrations that was as
good as an optimally viscous damper. Moreover, it was also
found from the simulation that EMDEH had the capability
of regenerating power for small electronic devices such as
wireless sensors. Afterward, Cai and Zhu [131] improved the
performance of the EMDEH by redesigning the EHC unit.
In the redesigned EHC unit, a buck-boost converter having
a low-power microcontroller unit was used to regulate the
duty cycle based on the feed-forward signal. The improved
EMDEH provided stable damping to the cable and compara-
tively enhanced the cable vibrations.
Lu etal. [132] proposed the viscous inertial mass
damper (VIMD) as a negative stiffness device for reduc-
ing the stay cable vibrations. As demonstrated in Fig.21,
the VIMD consisted of a viscous damping section and
an inertial mass section. The inertial mass section was
made of a threaded rod and a ball nut that transfer the
Fig. 18 Control flowchart of the MR damper based on a the CEC algorithm and b the CVD algorithm [128]
Fig. 19 Schematic detailing of
a new MR damper proposed by
Zhao etal. [129]

A.Javanmardi et al.
1 3
linear movement of the threaded rod to rotate the ball
nut and the mass tube. As a result, the magnified inertial
force was changed to a linear motion. The viscous damp-
ing section was made of a piston rod connected to the
threaded rod, viscous fluid material and a piston head to
produce the damping force. By installing the VIMD at
the vicinity of the cable anchorage, the modal damping
of the cable was substantially increased for the first four
vibrational modes. In addition, the performance of the
VIMD in reducing the cable vibration was outperformed
the optimal viscous damper. Wang etal. [133] studied the
cable sag effect on the performance of an inertial mass
damper (IMD) attached to a stay cable. It was found that
the IMD was able to alleviate the sag adverse effect on the
vibrational response of the stay cable. The sag increased
the vibrational frequency of the cable, while it decreased
the optimal damping coefficients and the inertial mass of
the IMD for the nearly symmetrical vibrational modes.
However, the sag had a less significant effect on the asym-
metric modes.
Shi etal. [134] investigated the dynamic behavior of
a cable with a passive negative stiffness damper (NSD)
through analytical and numerical approaches. The NSD
was modeled as a combination of a viscous damper with a
negative stiffness spring. The results of the analysis showed
that with the increase of the negative stiffness of the NSD,
its damping ratio also increases. Even though the NSD
increased the damping ratio of the cable but it also reduced
the stiffness of the cable that may cause instability in the
cable system. Therefore, the stiffness of the NSD should be
calculated accurately for the stay cable. Lastly, it was found
that the NSD was more effective than the conventional vis-
cous dampers in reducing the cable vibrations. Following
this, Shi etal. [135] performed an experimental test on a
stay cable with a new type of NSD called magnetic NSD to
validate the numerical results. As demonstrated in Fig.22a,
the magnetic NSD consisted of a movable magnet placed
between two fixed magnets on a shaft, which were covered
by a conductive pipe [136]. The experimental results proved
that the provided modal damping for the cable by the mag-
netic NSD was four times higher than the optimal viscous
damper. It was also pointed out that the flexural rigidity of
stay cables and boundary conditions should be taken into
account in order to have an accurate analysis and results.
Shi etal. [137] also compared the performance of the pas-
sive NSD with the LQR and output feedback active control
in suppressing the cable vibrations. The comparative study
demonstrated that the NSD could supply a similar high
Fig. 20 The model of the stay cable with EMDEH system in MATLAB/Simulink [130]
Fig. 21 The detailing of VIMD [132]

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
1 3
damping level as the LQR active control to the stay cable.
In addition, the output feedback control with two feedback
variables had comparable performance to that of the NSD. It
was concluded that the NSD could provide sufficient damp-
ing to the cable as a completely passive control system and
may even be more practical as it need not any algorithm
or external source of energy. Zhou and Li [138] proposed
another type of NSD for mitigating the stay cable vibrations.
As shown in Fig.22b, the NSD consisted of an oil damper
and two compressed springs. The NSD characteristics and
behaviors were experimentally and numerically studied.
The results of the investigation showed that the proposed
NSD effectively reduced the single-mode and multi-mode
vibrations of the cable and it is more effective in providing
modal damping to the cable than the oil damper. Javanbakht
etal. [139] research also showed the performance superior-
ity of the passive NSD in reducing the cable oscillations over
passive positive stiffness damper (PSD). Similar to other
researchers’ findings for NSDs, the stability of the system
was found to be crucial for the NSD in suppressing the cable
vibrations. Therefore, a criterion was defined for determin-
ing the stiffness of the NSD based on the length, the bending
stiffness and tension force of the cable as well as the support
stiffness and the installation location of the damper. A new
design procedure aiming to optimize NSD for wind-induced
multi-mode vibrational control of the stay cable was pro-
posed by Javanbakht etal. [140]. The proposed NSD opti-
mization approach could effectively reduce a single mode
as well as multi-mode vibrations of the stay cable equipped
with NSD. Additionally, the performance of the proposed
approach found to be much higher than the PSD, zero stiff-
ness damper and even the LQR active control with a similar
peak control force at the installation position. Further, Javan-
bakht etal. [141] showed that the flexible support could
increase the damping ratio of the cable-NSD system, lead-
ing to reduce the size and negative stiffness of the NSD and
eventually minimize the cost and maintenance of the device.
Furthermore, Dong and Cheng [142] investigated the effect
of the support stiffness on the performance of the cable-
NSD system through laboratory experiments. It was found
that the impact of the support stiffness on the efficiency of
the NSD depends on the stiffness of the NSD. Accordingly,
flexible support is beneficial for the cable-NSD system with
strong negative damper stiffness, whereas rigid support is
more suitable for the cable-NSD system with weak negative
damper stiffness.
Chen etal. [143] performed a unified analysis on the
NSD and different types of inerter-based vibration absorb-
ers (IVAs) i.e., (i) IMD, (ii) TID, (iii) a viscous damper in
parallel with TID, and (iv) a viscous damper in series with
an inerter, aiming to reduce the multi-mode vibrations of
the stay cable. The analytical study indicated that the NSD
enhanced the modal damping of multiple modes of the cable.
In addition, the TID had superior efficiency among the other
IVAs for controlling the multi-mode vibrations of the stay
cable. To attain the best performance, the TID frequency
should be tuned to the lowest frequency of the target modes
of the cable and have a relatively large inertance. Gao etal.
[144] developed an optimum design procedure for the vis-
cous inerter damper (VID) based on output feedback control
Fig. 22 Details of two types of NSDs proposed by a Shi etal. [135] and b Zhou and Li [138] for the stay cables

A.Javanmardi et al.
1 3
for mitigation of the multi-mode vibrations of the cable. In
this approach, the velocity of the cable at the damper loca-
tion was used as the feedback input. Higher modal damping
for the intermediate modes of the cable was achieved by
the proposed method as compared to other methods, while
the modal damping of the first and the higher modes of the
cable was slightly lesser. Finally, the VID with the proposed
method had a substantially better performance than the vis-
cous damper in reducing the multi-mode vibrations of the
cable.
Shi and Zhu [145] proposed an inerter damper for the
stay cables. The inerter damper was a type of NSD damper
with a unique feature. Unlike other typical NSDs, the force
of the inerter damper is directly proportional to the accel-
erations, therefore, its stiffness changes with the vibrational
frequency, which eventually avoids any structural instability.
The inerter damper added substantial damping to a tuned
mode of the cable that was higher than the optimal viscous
damper. However, for the rest of the vibrational modes of
the cable, the added damping was relatively limited as if
the cable was locked at the damper location. In conclusion,
the inerter damper could provide the optimal damping for a
specific target mode of the stay cable. Li etal. [146] devel-
oped another inerter-based damper called electromagnetic
inertial mass damper (EIMD), which had negative stiffness.
The EIMD primarily consisted of an inerter that provides
inertance and an electromagnetic damper provides electro-
magnetic damping. Figure23 illustrates the detailing of the
EIMD. The numerical model of the cable-EIMD system was
developed based on the complex eigenvalue analysis and
compared with the experimental results. When the EIMD
was installed at 2.5% length of the cable, the first modal
damping ratio of the cable was substantially higher than the
similar cable with the optimal viscous damper. For the opti-
mal design of EIMD, the inertance and damping coefficient
should have optimal values. It was also found that the EIMD
is also effective in suppressing the vibration of the higher
modes. Li etal. [147] used a combination of electromagnetic
shunt damper (EMSD) and inerter damper (ID) to control
the vibration of the cables. The direct current was used in the
shunt circuit of the EMSD-ID device to control the electro-
magnetic force generated by the motor. A full-scale experi-
ment showed that the EMSD-ID provided a substantially
large modal damping for the fundamental mode of the cable.
Li etal. [148] studied the vibrational response of a stay
cable equipped with a single shape memory alloy (SMA)
damper. The analytical analysis results indicated that the
SMA damper was able to control the vibrations dominated
by the first few modes of the cable. It was also found that the
optimal location of the SMA damper should be about 0.2%
of the cable length in order to suppress the oscillations of
the cable over broadband frequency excitations. Zuo etal.
[149] proposed a new type of SMA damper for controlling
the cable vibrations, as shown in Fig.24. The damper opti-
mization procedure was also proposed and its performance
was compared with an active control system based on the
LQR algorithm. Two SMA dampers were located at 0.2%
of the cable length from its end to control the in-plane and
out-of-plane vibrations. The results of the numerical analysis
showed that the optimized SMA dampers had almost similar
Fig. 23 Detailing of the EIMD [146]
Fig. 24 The SMA damper detailing proposed by Zuo etal. [149]

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
1 3
performance as the active control system. Furthermore, Zou
and Li [150] confirmed the effectiveness and application
of the same SMA damper on a stay cable of a cable-stayed
bridge through numerical and experimental studies. Nev-
ertheless, the SMA damper could effectively reduce cable
vibrations at temperatures between −25 and 50 °C [151].
Dieng etal. [152] showed the maximum in-plane displace-
ment reduction can be achieved when the SMA damper is
placed nearby the maximum vibration amplitude. How-
ever, this location may not be always a practical solution
for cables of the cable-stayed bridges. The SMA damper
also reduced the oscillation time and vibration amplitude of
the cable. Mekki and Auricchio [153] evaluated the perfor-
mance of the SMA damper in controlling the vibrations of
stay cables and comparing it with the TMD performance. It
was found that the SMA damper performance depends on
the length, cross-sectional area and location of the damper
on the stay cable. The comparative study also indicated that
the SMA performance was better than an optimal TMD in
controlling the harmonic and the high free vibrations of the
stay cable. Furthermore, the fatigue life of the SMA damper
for a stay cable was found to be 4 million cycles at the strain
near 1% [154].
Takano etal. [155] proposed an alternative vibrational
countermeasure for the stay cable of the Tsurumi Tsubasa
Bridge. As shown in Fig.25, a combination of an HDR
damper and an oil damper (viscous damper) was installed
on the stay cables. The vibration test results confirmed the
effectiveness of the proposed control system. Furthermore,
no wind-induced vibrations have been observed for the
stay cables since the installation of the proposed counter-
measure. Zhou etal. [156] studied the combined effect of a
viscous damper with a spring in reducing the vibration of
the cables as an alternative to the cable-cross-tie-damper
system. The results indicated that the combination of the
damper and spring was effective if both were utilized near
the cable anchorage. When the damper and spring were
placed at each end of the cable, the effect of the spring on
the cable damping was insignificant. For the case where the
damper was installed near to cable anchorage and the spring
was far from the cable anchorage, the spring considerably
changed the maximum damping ratio corresponded to the
optimum damper constant. For this case, a specific location
of the cable was found for the spring that could effectively
add more damping and stiffness to the specific mode of the
cable. The fundamental cable frequency increased when
the damper and spring were installed in parallel due to the
spring stiffness. Another significant effect of adopting spring
on the cable was reducing the free length of the cable and
alternatively changing the regime of the damper location.
Nonetheless, more profound studies should be conducted
to conclude the practical application of the damper-spring
system in suppressing the cable vibrations. Zhou etal. [157]
also investigated the effect of the cross-tie on the damp-
ing of a cable-damper system. Two identical parallel cables
with attached dampers nearby the cables’ anchorage and a
cross-tie were considered for the numerical study. It was
concluded that when the cross-tie is used with the cable-
damper system, its effect should be considered as the cross-
tie changes the frequency of the system. Moreover, the
cross-tie could substantially increase the damping of the
lower vibrational modes of two cables with attached damp-
ers near the anchorage [157, 158]. Ahamd etal. [158, 159]
showed that the cross-tie should be placed far away from
Fig. 25 Schematic detailing of a combined (internal and external dampers) stay cable control system for the Tsurumi Tsubasa Bridge [155]

A.Javanmardi et al.
1 3
the dampers in order for the dampers to be more effective
in the cable-cross-tie-damper system. Furthermore, it was
suggested to increase the cross-tie flexibility to maximize the
attainable damping ratio of the cable-cross-tie-damper sys-
tem. However, this might decrease the in-plane frequency of
the system that may cause more localized vibrations within
the system. Zhou etal. [160] introduced a concentrated mass
to the cable-damper system to improve the efficiency of the
viscous damper. The results of the numerical study promoted
that combing a mass with the viscous damper-cable system
caused the cable to behave as if it was attached to a semi-
active damper with negative stiffness properties. The maxi-
mum achievable modal damping of the cable-damper-mass
system was obtained when the damper and mass were closed
to the cable anchorage and the coefficient of the nondimen-
sional mass was lesser than the critical value. It was also
noted that the mass location should be designed properly,
as the inappropriate installation of the mass may reduce
the damping of the damper for some particular vibrational
modes of the cable. Zhou etal. [161] examined the effect
of connecting two cables by a linear viscous damper as an
alternative to the cross-tie. The effect of several parameters
for two stay cables having harp arrangement and intercon-
nected with a viscous damper was addressed. The analytical
study demonstrated that the interconnected viscous damper
considerably increased the multi-mode modal damping
of the two neighboring cables and slightly increased their
vibrational frequencies. Liu etal. [162] studied the conse-
quence of the integration of the viscoelastic dampers within
the cross-ties on reducing the vortex-induced vibration of
the stay cables. For this purpose, a wind tunnel test was
performed on three stay cables that were interconnected with
cross-ties, while viscoelastic dampers were installed within
the connection of the cross-ties. The test results proved that
the viscoelastic dampers within the cross-ties were able to
enhance the stiffness and damping of the cables that eventu-
ally reduced the vortex-induced vibration of the stay cables.
Jiang etal. [163] proposed a new spring-damper restrain-
ing system to be placed at the end of the stay cable. As
shown in Fig.26, the restraining system consisted of a vis-
cous damper and an elastic spring horizontally attached to
the end of the movable support of the cable. In other words,
the proposed restraining system made one side of the cable
movable. Although the results of the numerical and experi-
mental studies showed that the proposed system had an over-
all 2% damping ratio in the transverse direction and was
effective in reducing the in-plane vibrations of the cables,
but the performance of the system depends on the cable axial
movement at the flexible support. Moreover, the proposed
system needs a more thorough investigation to be a practical
solution for the mitigation of the cable vibrations, as of some
other concerns i.e., out-of-plane vibrations, fatigue, etc., may
compromise the performance of the proposed system as well
as the safety of the stay cable.
Cai etal. [164, 165] proposed the tuned mass damper-
magneto-rheological (TMD-MR) damper to mitigate the
stay cable vibrations. As shown in Fig.27a, the TMD-MR
damper consisted of a TMD and an MR damper. The TMD-
MR damper can be implemented at any location of the stay
cable, which advantageous over other external dampers that
have to be installed near the lower end of the cable. Several
parameters such as cable inclination, cable geometry-elastic-
ity parameter, damper position, mass ratio, frequency ratio,
damper damping ratio, and tuning mode were considered
Fig. 26 The spring-damper restraint system for controlling the cable
vibration [163]
Fig. 27 Detailing of a TMD-MR damper [164, 166], and b movable TMD-MR damper [167]

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
1 3
for the parametric study. The results of the theoretical study
showed thatthe TMD-MR damper is a promising solution to
increase the modal damping of the stay cable. However, the
cable-damper system mostly was tuned for a single mode,
and multiple TMDs were not recommended as the dominat-
ing mode may be unclear for the damper design. In addition,
the experimental results showed 20–30% cable vibrations
reduction after the implementation of the TMD-MR damper
[166]. Besides, when the cable vibrational frequency was
not reaching the cable modal frequency, the MR component
of the damper was providing additional damping to the stay
cable. Finally, it was suggested to set the TMD-MR damper
in a passive mode for small vibrations. Salari etal. [167]
developed a movable damping system based on the TMD-
MR damper. The proposed damper was made of the TMD-
MR damper attached to a moving component as illustrated in
Fig.27b. The moving component had an electric engine con-
nected to the pulleys that can move along the cable. A three-
dimensional model of a cable with the movable semi-active
TMD-MR damper was created and several control strategies,
namely, passive-on, continuous sky-hook, on–off sky-hook,
and Fuzzy control algorithms were adopted for the control
of the cable vibrations. Furthermore, an innovative locating
algorithm was used to find the optimal location of the TMD-
MR damper along the cable. It was found that the proposed
damper had an insignificant influence on the cable tension.
Overall, the proposed damper with different semi-active con-
trol algorithms reduced the vibrational displacement of the
cable, however, the Fuzzy control strategy outperformed the
other employed strategies. To overcome the limitations of
the viscous damper and TMD in reducing the cable vibra-
tions, Cu and Han [168] proposed a hybrid control system.
The proposed hybrid system was a combination of these
two dampers. A comparative numerical study was performed
between four cable-damper configurations, i) cable with a
single viscous damper, ii) cable with two viscous dampers,
iii) cable with a TMD, and iv) cable with a TMD and a vis-
cous damper. It was found that the TMD was more effective
in reducing the vibration of a single target mode of cable
than the viscous damper. Nevertheless, the viscous damper
was more effective than the TMD in providing the damp-
ing for more target modes of the cable. The combination
of the viscous damper and TMD found to be more effec-
tive and efficient in reducing the vibrations of the cable as
compared with the sole implementation of these dampers.
Cu etal. [169] also proposed another hybrid damper called,
tuned mass-high damping rubber (TM-HDR) damper for
the application of stay cables. As the name indicates, the
TM-HDR damper was made of a tuned mass attached to an
HDR damper. In other words, the viscous damper and linear
spring of TMD were replaced by the HDR damper. When the
damper was installed at the center of the cable, it was inef-
fective for the even modes, even if the damper was tuned for
these modes. This was due to the zero displacement charac-
teristics of this location for even modes. It was found that the
TM-HDR damper was only effective for a single tuned mode
of the cable and its effect for other modes was insignificant.
The adjustable fluid damper with SMA actuators for
increasing the damping ratio of the stay cable was proposed
by Xu and Zhou [170]. Instead of having a fixed number of
orifices in the piston head of the fluid damper, the SMA actua-
tors were installed in the adjustable fluid damper as shown in
Fig.28. By utilizing this configuration, the numbers of orifices
were adjusted to control the damper parameters in order to
have an optimal design. The advantages of the proposed SMA
actuators were i) they act as small mechanical valves to open
and close an orifice inside the damper piston, and ii) within
a working temperature range the mechanical valves lock at
the designed position. The numerical results indicated that the
proposed damper provided adequate modal damping for the
Fig. 28 Piston head of the adjustable fluid damper with SMA actuator, a detailing, and b prototype [170]

A.Javanmardi et al.
1 3
first two modes of the stay cables. Moreover, the results of
the case study for the proposed damper in a long-span cable-
stayed bridge concluded that the damper location could be
very closed to the cable anchorage.
Weber etal. [171] studied the optimal tuning of the Cou-
lomb friction damper (as a semi-active damper) to achieve the
maximum damping for the first vibrational mode of the cable.
It was shown that the friction damper should be tuned propor-
tionally to cable amplitude at the damper location, which is
determined by amplitude feedback in real-time. In addition,
it was concluded that this damper should be tuned differently
from the linear viscous damper. Izzie etal. [172] investigated
the application of the targeted-energy-transfer (TET) device
for mitigating the vibrational response of the stay cables. The
TET device was a passive control system having linear damp-
ing and a cubic elastic restoring effect. The universal design
curves for the TET device were proposed and numerically
evaluated. It was found the amplitude-dependent damping
curve of the TET device was pretty wide and the device could
provide high damping to a wide range of the cable frequen-
cies. Besides, though the TET device was placed at a close
distance to the anchorage, but it had a satisfactory performance
even for a very long stay cable. A piecewise linear absorber
for control of the vertical vibrations of the stay cables was
proposed by Weiss etal. [173]. The piecewise linear absorber
had a nonsmooth restoring forcing function and a mass rela-
tively smaller than the modal mass of the stay cable. When the
absorber was tuned for certain initial conditions and oscilla-
tion range, the piecewise linear absorber behaved highly non-
linear through the nonlinear interactions and bifurcations of
the energies of the absorber with the cable. These nonlinear
interactions and bifurcations effectively control the vertical
vibrations of the cable under the gravity effects. Chen etal.
[174] invented the viscous-shear damper (VSD) for the multi-
mode vibrational control of the stay cable. The VSD was made
of movable shearing plates that were partially submerged in a
viscous medium inside of a casing. The behavior of the VSD
was studied experimentally in the laboratory and its analytical
formulas were derived. Thereafter, the VSDs were installed on
the cables of the Sutong Bridge to evaluate its performance
in reducing the cables’ vibrations. The results of the study
indicated that as the deformation frequency of the damper
decreased, the damping coefficient also decreased. Moreover,
the VSD effectively provided modal damping for most of the
in-plane vibrational modes (within 3Hz frequency) of the
cable, which was significantly higher than an optimal viscous
damper.
4 Summary andPractical Examples
oftheStay Cable Dampers
Table1 summarizes the stay cable dampers discussed in
Sect.3. As the table demonstrated much less research has
been conducted on internal dampers due to their limita-
tions. On the other hand, the majority of discussed stay
cable dampers were external dampers as they can achieve
higher efficiency in controlling the cables’ vibrations.
The viscous damper and MR damper are among the most
popular external dampers. The MR damper can work in
the passive or semi-active mode, which is one of its main
advantages over other stay cable dampers. Since the last
decade, a considerable amount of studies on the applica-
tion of NSDs for the stay cables have been carried out,
however, so far no practical implementation of NSDs in
the cable-stayed bridges is reported. Lastly, other types of
external dampers were found to be promising in reducing
the cable’s vibrations, nevertheless, their application was
limited to the research only.
Table2 represents a survey of cable-stayed bridges
equipped with stay cable dampers around the world. It can
also be seen from this table that internal dampers are typi-
cally passive devices. The majority of the bridges had been
equipped with hydraulic (viscous) dampers as external
dampers in the 1980s and 1990s. However, in recent dec-
ades with the advancement of technology, the MR damper
becomes more popular owing to its better performance in
controlling multi-mode vibrations of the cables. Neverthe-
less, for the short stay cables, the internal dampers are the
most common practical option while for the longer stay
cables; the external dampers are mostly used due to their
higher efficiency. Lastly, the combination of internal and
external dampers is rarely used as this configuration has
no advantage over using the external damper.
5 Concluding Remarks
This paper reviewed the state-of-the-art and the-state-of-
practice of the vibrational control systems used for the
stay cables of cable-stayed bridges. The cables of the
cable-stayed bridges are continuously subjecting to vari-
ous types of dynamic loads since the installation of them
on the bridge. Excessive cable vibrations not only cause
damage to the cable system but also compromise the over-
all safety of the cable-stayed bridges. The wind is one of
the most crucial parameters in the design of cable-stayed
bridges and its effect should be studied in order to find
effective countermeasures to control unfavorable vibra-
tions. Despite all the advancement of the computational

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
1 3
methods such as nonlinear time-history analysis and com-
putational fluid dynamics (CFD) analysis for studying the
wind effects on the cable-stayed bridges, wind-tunnel test
with all its setbacks and limitations found to be a more
reliable tool for analyzing the aeroelastic instability, espe-
cially for long-span bridges [190]. To avoid instabilities,
dampers had been used for reducing vibrations of the stay
cables since the 1980s. The stay cable dampers are classi-
fied as internal and external dampers. Noteworthy that the
best location of a damping system for a stay cable is within
the cable center [191]; however, it is almost impractical to
implement most of the dampers at this location. Therefore,
the location of stay cable dampers is typically restricted
to the anchorage ends of the cables due to practical and
aesthetic reasons. Based on the comprehensive review, the
following remarks can be made:
I. In regards to damper classification although internal
dampers such as elastomeric dampers are among the
very first dampers used for the stay cables, yet they
are still being adopted in the new bridges especially
for short cables owing to their esthetical smooth outer
shape. Meanwhile, due to higher efficiency, external
dampers like hydraulic damper and MR damper are
widely used for relatively long cables.
II. In regards to the type of popular devices for stay
cables
• The optimal damping ratio of the passive cable
dampers (except NSDs) can achieve mostly for one
mode of the stay cables, while for the semi-active
dampers the optimal damping ratio can be achieved
for several modes of the cable through the control
algorithm. In light of active and semi-active dampers
for the stay cables, accelerometers are the necessary
sensors to deliver real-time vibrational data to the
control unit and adjust the required damping force
of the dampers to suppress the vibrations.
Table 1 Stay cable dampers summary
No Device Abbreviation Type
1 High-damping rubber [68, 88, 111, 112] HDR Internal damper/external damper
2 Laminated rubber bearing [72] LRB Internal damper
3 Friction pendulum bearing [73] FPB Internal damper
4 Smart damper [74, 75] – Internal damper
5 Multiple-mass-particle impact damper [76] – Internal damper
6 Viscous damper [11, 65, 67, 80–106, 156–163, 170] – Internal damper/external damper
7 Tuned mass damper [107, 108] TMD External damper
8 Tuned inerter damper [109] TID External damper
9 Stockbridge-type damper [110] – External damper
10 Magneto-rheological Damper [77, 113–119, 121–129] MR External damper
11 MR shear mode damper [120] – External damper
12 Electromagnetic damper cum energy harvester [130, 131] EMDEH External damper
13 Viscous inertial mass damper [132] VIMD External damper
14 Inertial mass damper [133] IMD External damper
15 Negative Stiffness Damper [125, 134–143, 145] NSD External damper
16 Viscous inerter damper [144] VID External damper
17 Electromagnetic inertial mass damper [146] EIMD External damper
18 Electromagnetic shunt damper and inerter damper [147] EMSD-ID External damper
19 Shape memory alloy [148–154] SMA External damper
20 HDR and viscous damper [155] – Combination of internal and
external dampers
21 Tuned mass damper-magneto-rheological [164–166] TMD-MR External damper
22 Movable TMD-MR damper [167] – External damper
23 Viscous damper with TMD [168] – External damper
24 Tuned mass-high damping rubber [169] TM-HDR External damper
25 Coulomb friction damper [171] – External damper
26 Targeted-energy-transfer device [172] TET External damper
27 Piecewise linear absorber [173] – External damper
28 Viscous-shear damper [174]VSD External damper

A.Javanmardi et al.
1 3
Table 2 Summary of cable-stayed bridges with stay cable dampers
No Name—Construction year Overall length
(main span) (m)
Country Control device Description
1 Brotonne Bridge-1977 [65] 1278.40 (697.5) France Hydraulic damper External damper
2 Rande Bridge-1981 (2015*) [175] 704.58 (400.14) Spain Hydraulic damper External damper
3 Luling Bridge-1983 [176] 678.2 (372.5) USA HDR dampers Internal dampers (at both ends of
the cables) installed in 2012
4 Aratsu Bridge-1988 [177] 345 (185) Japan Hydraulic dumpers External damper
5 Alamillo Bridge-1992 [126] 200 Spain MR damper External damper
6 Tsurumi Tsubasa Bridge-1994
[155]
1021 (510) Japan HDR damper + Hydraulic damper Combination of Internal and Exter-
nal dampers
7 Puente Real Bridge-1994 [5, 6] 265 (136) Spain Friction dampers Internal damper
8 Fred Hartman Bridge-1995 [11] 754 (381) USA Hydraulic damper External damper
9 Erasmus Bridge-1996 [80] 802 (285) Netherlands Hydraulic damper External damper
10 Kap Shui Mun Bridge-1997 [9] 750 (430) China Friction damper Internal damper
11 Meiko Grand Bridges-1998 [178] 1170 (590) Japan HDR damper and Viscous shear-
type damper
Internal and External dampers at
different cables
12 Vasco Da Gama Bridge-1998 [70] 829 (420) Portugal Elastomeric dampers Internal damper
13 Uddevalla Bridge-2000 [179, 180] 772 (414) Sweden Friction damper Internal damper
14 Dongting Lake Bridge-2000 [117] 880(620) China MR dampers External damper
15 Second Nanjing Bridge-2001
[148]
1238 (628) China Elastomeric dampers Internal damper
16 Dubrovnik Bridge-2002 [8] 518 (304.5) Croatia MR dampers External damper
17 Eilandbrug Kampen Bridge-2003
[78, 128]
412 (150) Netherlands MR damper External damper
18 Leonard P. Zakim Bunker Hill
MemorialBridge-2003 [181]
436 (227) USA Elastomeric Damper Internal damper
19 Shangdong Binzhou Yellow Riv-
erHighway Bridge-2004 [124]
768 (600) China MR damper External damper
20 Millau Viaduct Bridge–2004 [70] 2460 France Elastomeric dampers Internal damper
21 Veterans Glass City Skyway
Bridge–2007 [182]
374 USA MR damper External damper
22 Kanchanapisek Bridge-2007 [183] 941 (500) Thailand Radial damper Internal damper
23 Plock Bridge–2007 [70] 615 (375) Poland Elastomeric dampers Internal damper
24 Serebryany Bor Bridge–2007 [70] 1460 (409.5) Russia Elastomeric dampers Internal damper
25 Sutong Bridge–2008 [128, 184,
185]
2088 (1088) China Hydraulic damper and MR
damper
External damper
26 Megyeri Bridge-2008 [70] 1862 (300) Hungary Elastomeric dampers Internal damper
27 Niederrheinbrücke Bridge–2009
[186]
773 (376) Germany Elastomeric dampers Internal damper
28 Ponte Del Mare-2009 [187] 147.6 Italy Elastomeric dampers Internal damper
29 Wuhan Tianxingzhou Changjiang
RiverBridge–2009 [188]
1092 (504) China Lever mass damper External damper
30 Christopher S. Bond Bridge-2010
[71]
523 USA Elastomeric dampers Internal damper
31 E’dong Yangtze River
Bridge-2010 [188]
1486 (936) China Viscous-shear damper External damper
32 Jingyue Changjiang River High-
wayBridge–2010 [188]
1444 (816) China Lever mass damper External damper
33 Hovenring Bridge-2011 [110] 72** Netherlands TMD External damper
34 Rao II Bridge–2011 [70] 230 (120) Vietnam Elastomeric dampers Internal damper
35 John James Audubon Bridge-2011
[71]
860 (482) USA Friction dampers Internal damper
36 Awa Shirasagi Ohashi
Bridge-2012 [79]
505 (260) Japan HDR dampers External damper

Application ofStructural Control Systems fortheCables ofCable-Stayed Bridges:…
1 3
• The viscous damper is one of the most commonly
used dampers for the stay cables. The viscous
damper can have a linear or a nonlinear property.
However, the nonlinear damper has superior per-
formance in dissipating more energy and providing
modal damping for multiple modes of the cables.
Moreover, the nonlinear damper is more efficient
when the damper needs to be installed at a closer
distance to the cable anchorage.
• The MR-damper is found to be a very promising
solution for vibrational control of the stay cables.
The MR damper can provide substantial modal
damping to a single-mode or multi-mode of cable.
The MR damper is especially effective for long stay
cables subjecting to a wide range of vibrational
frequencies. Several semi-active control algorithms
have been proposed for control of the semi-active
MR damper and so far no comparative study has
been performed between them, yet their efficiency
has been proven in reducing the cable vibrations.
Noteworthy, the temperature variation may affect
the performance of the MR damper. Therefore, the
control algorithm for the semi-active MR damper
should account for the operating temperature of the
damper, especially for harsh environmental condi-
tions.
• In recent years, the NSD has received great atten-
tion in suppressing the cable vibrations. The NSD
can supply modal damping to a single-mode or
multi-mode of the cable. As a matter of the fact,
the NSD as a passive control system can provide
comparable modal damping to the cable as the
active control system. However, the NSD may
cause instability in the cable system; therefore, an
accurate calculation of the negative stiffness is the
essential key to overcome this drawback.
• The SMA damper can efficiently mitigate the vibra-
tions of the cables over broadband excitations.
The SMA damper can work effectively in a very
harsh environment (at a temperature between −25
to + 50º C) while its life span is relatively longer
than other dampers. However, the SMA dampers
are comparatively costlier than other practical
dampers.
III. In regards to damper design for the stay cable: a
number of parameters of the cable including the
length, mass per unit length, inclination angle, ten-
sion force, sag-extensibility parameter, flexural
rigidity, inherent damping, fundamental frequency,
and mode number should be considered during the
selection and design of the damper. Meanwhile,
parameters of the damper i.e., the type, size, stiffness,
location, direction, boundary conditions, and other
relevant parameters should be used for the design
of the damper. In addition, other conditions such as
environmental conditions (temperature), support stiff-
ness, and the presence of the cross-ties also should
be taken into account for the design of dampers. The
connection configuration of the damper with cable
is also an important aspect of the external damper.
When the damper is designed to suppress the in-plane
vibrations of the cable, the support configuration of
the damper should allow the out-of-plane rotations to
avoid any damage or failure in the damper.
Acknowledgements The discussion expressed in this paper are those
of the authors and do not represent the above-mentioned companies.
Funding The authors greatly acknowledge the support of the China
Postdoctoral Science Foundation (2020M682074) for this research
work.
*The bridge was expanded in 2015
** Deck outer diameter of the roundabout flyover
Table 2 (continued)
No Name—Construction year Overall length
(main span) (m)
Country Control device Description
37 Russky Bridge-2012 [128] 1885.53 (1104) Russia (i) Hydraulic dampers (ii) MR
damper
External dampers (i) for short and
(ii) long cables, respectively
38 Xiangshangang Bridge–2012
[188]
1376 (688) China Lever mass damper External damper
39 Second Jiujiang Bridge-2013
[188]
1405 (818) China Lever mass damper External damper
40 Yalu River Bridge–2014 [188] 3026 (636) China Lever mass damper External damper
41 Bouregreg Bridge-2016 [189] 742 (376) Morocco (i) Hydraulic dampers and (ii)
Radial dampers
Internal damper for (i) short cables
and (ii) long cables, respectively

A.Javanmardi et al.
1 3
Data availability All data, models, and code generated or used during
the study appear in the submitted article.
Declarations
Conflict of interest On behalf of all authors, the corresponding author
states that there is no conflict of interest.
Ethical approval This article does not contain any studies with human
participants or animals performed by any of the authors.
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