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Smart Structures and Systems, Vol. 26, No. 5 (2020) 575-589
DOI: https://doi.org/10.12989/sss.2020.26.5.575
Copyright © 2020 Techno-Press, Ltd.
http://www.techno-press.org/?journal=sss&subpage=7 ISSN: 1738-1584 (Print), 1738-1991 (Online)
1. Introduction
Cables are widely used in structural engineering because
of their high axial strength-to-weight ratio (Irvine 1981).
However, due to their high flexibility in the lateral direction
and low intrinsic damping, they are prone to vibrations
induced by various types of excitations (De Sá Caetano
2007, Fujino et al. 2012, Hikami and Shiraishi 1988,
Matsumoto et al. 2001). Mechanical dampers are
commonly installed on long cables to increase cable energy
dissipation capacity and thus suppress cable vibrations
(Chen et al. 2003).
The research on the cable-damper system can be traced
back to the 1980s. Carne (1981) and Kovacs (1982) were
among the first researchers to investigate the vibrations of a
taut cable with an attached damper, both focusing on the
first-mode damping ratio when the damper location is near
Corresponding author, Research Associate Professor,
E-mail: linchen@tongji.edu.cn
a Ph.D. Student, E-mail: fangdiandi@tongji.edu.cn
b Professor, E-mail: lmsun@tongji.edu.cn
one cable end. Later, Yoneda and Maeda (1989), Uno et al.
(1991) and Pacheco et al. (1993) attempted to develop
universal design formulas for a taut cable with a viscous
damper by using the complex eigenanalysis method. By
grouping the non-dimensional damper/cable parameters
properly, Pacheco et al. (1993) managed to obtain the
“universal estimation curve” which applies to the first
several cable modes for a damper at an arbitrary location
close to a cable end. Xu and Yu (1998a, b) developed an
efficient and accurate transfer matrix formulation using
complex eigenfunctions and focused on the influences of
sag and out-plane vibrations. Krenk (2000) developed an
exact analytical formulation of the frequency equation of a
taut cable with a viscous damper and obtained an
asymptotic approximation for the damping ratios of the first
few modes for damper locations near one cable end.
Subsequently, the analytical method has been extended to
consider cable sag (Krenk and Nielson 2002). Main and
Jones (2002a, b) followed the same approach and analyzed
the case of a damper located arbitrarily along the cable.
They pointed out the importance of damper-induced
frequency shifts in characterizing the response of the
system. Furthermore, to simulate realistic cable-damper
systems, more cable parameters have been included, such as
Cable vibration control with internal and external dampers:
Theoretical analysis and field test validation
Fangdian Di1a, Limin Sun1,2b and Lin Chen1
1Department of Bridge Engineering, Tongji University, 1239 Siping Road, Shanghai, China
2State key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, 1239 Siping Road, Shanghai, China
(Received October 19, 2019, Revised July 13, 2020, Accepted August 22, 2020)
Abstract. For vibration control of stay cables in cable-stayed bridges, viscous dampers are frequently used, and they are
regularly installed between the cable and the bridge deck. In practice, neoprene rubber bushings (or of other types) are also
widely installed inside the cable guide pipe, mainly for reducing the bending stresses of the cable near its anchorages. Therefore,
it is important to understand the effect of the bushings on the performance of the external damper. Besides, for long cables,
external dampers installed at a single position near a cable end can no longer provide enough damping due to the sag effect and
the limited installation distance. It is thus of interest to improve cable damping by additionally installing dampers inside the
guide pipe. This paper hence studies the combined effects of an external damper and an internal damper (which can also model
the bushings) on a stay cable. The internal damper is assumed to be a High Damping Rubber (HDR) damper, and the external
damper is considered to be a viscous damper with intrinsic stiffness, and the cable sag is also considered. Both the cases when
the two dampers are installed close to one cable end and respectively close to the two cable ends are studied. Asymptotic design
formulas are derived for both cases considering that the dampers are close to the cable ends. It is shown that when the two
dampers are placed close to different cable ends, their combined damping effects are approximately the sum of their separate
contributions, regardless of small cable sag and damper intrinsic stiffness. When the two dampers are installed close to the same
end, maximum damping that can be achieved by the external damper is generally degraded, regardless of properties of the HDR
damper. Field tests on an existing cable-stayed bridge have further validated the influence of the internal damper on the
performance of the external damper. The results suggest that the HDR is optimally placed in the guide pipe of the cable-pylon
anchorage when installing viscous dampers at one position is insufficient. When an HDR damper or the bushing has to be
installed near the external damper, their combined damping effects need to be evaluated using the presented methods.
Keywords: stay cable; vibration control; sag effect; viscous damper; HDR damper; modal damping; parameter
optimization
575
Fangdian Di, Limin Sun and Lin Chen
the bending stiffness (Main and Jones 2007, Fujino and
Hoang 2008), cable sag (Xu and Yu 1998a, b, Yu and Xu
1998, Wang et al. 2005, Nielsen and Krenk 2003, Krenk
and Nielsen 2002, Fujino and Hoang 2008) and inclination
(Xu and Yu 1998a, Fujino and Hoang 2008, Hoang and
Fujino 2007, Sun and Huang 2008). Besides, dampers
beside an ideally viscous type have been addressed, such as
taking into account the damper intrinsic stiffness (Zhou and
Sun 2006), nonlinearity (Main and Jones 2002a, b, Chen
and Sun 2016, Sun et al. 2019b) and damper support
flexibility (Fujino and Hoang 2008, Sun and Huang 2008)
and damper mass (Duan et al. 2019a, b).
However, as cables are increasingly long in long-span
cable-stayed bridges, a single damper installation may no
longer be able to provide enough damping. Caracoglia and
Jones (2007) hence investigated the combined damping
effect of two viscous dampers on a stay cable. Their result
suggests that the use of dampers in close position cannot
enhance the maximum attainable modal damping in the
cable. In contrast, dampers at opposite ends can increase the
supplemental damping. A similar conclusion has been
arrived by Hoang and Fujino (2008). Zhou et al. (2014) had
studied a taut cable with a damper and a spring while the
position of the spring was arbitrary along the cable. Cu et
al. (2015) investigated the damping of a cable equipped
with two HDR dampers. Other methods for enhancing near-
anchorage damper performance have also been explored
recently, for example, by using a passive negative stiffness
device (Chen et al. 2015, Zhou and Li 2016). Particularly, a
viscous damper combined with pre-compressed springs was
studied by Chen et al. (2015). Chen et al. (2016) has
proposed to use both rotational and transverse dampers for
supplementing cable damping. Inerter devices have also
been combined with viscous dampers for cable vibration
control (Sun et al. 2017, Lu et al. 2017, 2019, Wang et al.
2019, Huang et al. 2019).
It is noted that in practice, rubber bushings (or other
types) are commonly installed inside steel guide pipes at
cable anchorages for reducing cable bending stress near the
ends (Nakamura et al. 1998). The bushings have been
modeled as springs in the previous studies, while they also
have some energy dissipation effects. Takano et al. (1997)
found that the stiffness of the bushing could significantly
reduce damper effectiveness. Main and Jones (2003)
analyzed a taut cable with a bushing (linear spring) and a
viscous damper (linear dashpot), and an asymptotic solution
is obtained which leads a generalized design curve for a
linear viscous damper accounting for the bushing stiffness.
When a single external damper is inadequate for cable
vibration control, it is straightforward to replace the bushing
by an HDR damper, referred to as the internal damper in the
guide pipe. Practically, this is more feasible than other
strategies such as installing dampers near both cable ends.
Indeed, the combination of a viscous damper and an HDR
damper mounted at opposite ends was studied by Hoang
and Fujino (2008), and it was found that the total damping
is approximately the sum of their separate damping effect.
However, the case of a viscous damper and an HDR damper
installed at the same end has not been studied till date.
Besides, in the previous studies concerning two dampers on
Fig. 1 A shallow cable with an internal damper and an
external damper
a cable, cable sag and damper intrinsic stiffness have not
been investigated in detail. Notably, Yoneda et al. (1995)
studied two dampers with stiffness near the same cable end
but using an approximate method based on the equivalence
of the damper parameters or the installation position.
Therefore, this study performs a comprehensive analysis of
the damping effect of a shallow cable attached with an
external damper and an internal damper. The external
damper is considered to be a viscous damper with intrinsic
stiffness, and the internal one is described using an HDR
model, which can simulate both the bushings and also the
rubber dampers. Asymptotic formulas for modal damping
ratios are derived when both the external and internal
dampers are close to cable anchorages, either on the same
end or opposite ends of a cable. Parametric analysis is then
carried out to discuss their joint effects, particularly, on
multimode cable damping. For the case when the two
dampers are installed close to the same cable end, an
experimental study has been conducted to validate the
theoretical estimation.
The rest of this paper is organized as follows. Sec. 2
presents the characteristic equation of the system using the
complex modal analysis. Sec. 3 derives asymptotic
solutions for cases when the dampers are installed
respectively close to the two cable ends and close to the
same cable end, and performs the parametric analysis. Sec.
4 introduces the experimental study and compares the
results. Sec. 5 concludes the paper.
2. Characteristic equations of a shallow cable with
both internal and external dampers
A shallow cable with an internal damper and an external
damper is shown in Fig. 1. The cable is fixed at both ends.
The cable chord length is denoted by, the cable axial
tension is indicated by , the mass per unit length is
denoted by, and the axial stiffness is denoted by.
The external damper is a viscous damper with intrinsic
stiffness, and and are the stiffness and damping
coefficients respectively; and, it is installed close to one
cable end at a distance of. The internal damper is an
HDR damper with stiffness and loss factor, and the
distance of the HDR from the nearer cable end is denoted
by. As long cables are concerned here, cable bending
stiffness is neglected. Cable inherent damping is ignored as
it is practically very small.
For describing static profile and dynamic displacements
of the cable, a coordinate system is defined for the cable-
576
Cable vibration control with internal and external dampers: Theoretical analysis and field test validation
damper system with x starting from the left cable anchorage
pointing rightwards along the cable chord, and the static
profile and the dynamic displacement are denoted by
and, respectively, with = time.
The static profile of a shallow cable can be accurately
approximated by the parabolic function (Irvine 1981)
(1)
The sag at cable mid-span is denoted by , and is the
gravitational acceleration. Cable inclination can be
considered in the definition of the non-dimensional sag
parameter (Irvine 1981).
The cable is divided by the two dampers into three cable
elements, and the connection points are denoted by with
index . Where and correspond to the
left and right anchorages of the cable, and and
correspond to the left and right damper installation points
respectively. An element-wise coordinate system is defined
with the horizontal axis starting from pointing
towards , as shown in Fig. 1. A typical element is a
segment with a length of between the points and
, and its vertical dynamic displacement is denoted by
. The cable vibration induces an additional
horizontal tension . Thus, the equation of each cable
segment is governed by the following partial differential
equation (Irvine and Caughey 1974) as
(2)
The dynamic cable tension can be obtained from the
elastic elongation of the cable. Denote the initial length of
the cable element as and the element length after elastic
elongation as . The increment of the cable tension along
the chord is , and thus the elasticity relation of
the cable can be written as
(3)
where denotes the cable dynamic displacement
along the chord and the last relation follows from the
motion of a point from its initial position
. Including the boundary conditions of
and at the cable anchorages, multiplying Eq. (3) by
and then integrating the left- and right-hand
sides over the cable span, one obtains
(4)
where
Considering free in-plane vibrations of the cable,
solutions to Eq. (2) are expressed as
(5)
where is the complex circular frequency, ,
and denote the complex vibration amplitudes of the
corresponding time-dependent variables. Substituting Eq.
(5) into Eq. (2), one finds
(6)
where
is the wavenumber.
Similarly, Eq. (4) becomes
(7)
The transverse vibrations of a general cable element at
its left and right ends are denoted by and
respectively. In free vibrations of the cable, one finds
and . The
solutions of can then be obtained, as
(8)
Noting that
(9)
Eq. (7) becomes
(10)
where the Irvine parameter is defined as
Eq. (10) is multiplied by, leading to
(11)
From Eq. (8), one finds the following expressions
577
Fangdian Di, Limin Sun and Lin Chen
(12a)
(12b)
The internal force equilibrium at the damper location is
expressed as
=0
(13)
where is the force-deformation relation
of the external damper expressed in the frequency domain,
and 1+i represents the force-deformation
relation of the HDR damper in the frequency domain
(Fujino and Hoang 2008). Note that subscripts 1 and 2
correspond to the viscous damper and HDR damper,
respectively.
Finally, collecting Eqs. (11) and (13) and after
simplification, three equations are obtained as
0
(14a)
(14b)
1+i
(14c)
Eq. (14) can be written in a matrix form as
(15)
where is a zero vector and
1+i
The characteristic equation of the system is given by the
determinant of the coefficient matrix in Eq. (15). After
simplification, the following equation is obtained
+2+2+4=0
(16)
where
For limiting cases related to particular values of and
, the solutions to Eq. (16) are discussed as follows.
1) When =0, Eq. (16) simply becomes ,
whose solutions correspond to the eigenfrequencies of a
shallow cable without any attachments.
2) When =0 or =0, only one damper is attached
to the cable. For example, let and =0, Eq. (16)
reduces to+2, corresponding to the characteristic
equation of a shallow cable with a viscous damper, as
discussed by Krenk and Nielsen (2002) where the damper
stiffness was absent though.
3) When , Eq. (16) is reduced to=0,
which gives solutions to the “locked-damper” modes of the
cable, i.e., no vertical displacement at the two nodes,
and . Two groups of solutions are obtained when the
factors in the left-hand side of =0 are respectively equal
to zero. These are real oscillatory modes characterized by
the three segments of the cable oscillating independently
(Sun et al. 2019a).
4) Other limiting cases include that either or is
infinite, leading to a system similar to a single shallow
cable with a damper (Krenk and Nielsen 2002, Fujino and
Hoang 2008), which is hence not discussed any more.
3. Modal damping of a cable with both internal and
external dampers
Providing cable parameters, damper properties and
578
Cable vibration control with internal and external dampers: Theoretical analysis and field test validation
damper locations, the wavenumber can be solved from Eq.
(16) numerically for modes of concern. Here, the argument
principle method (Chen et al. 2016) is used when necessary.
For normal cases when dampers are close to a cable end,
asymptotic formulas of the complex wavenumber are
derived for lower modes. The derivation is based on a small
perturbation on the real wavenumber of a cable without any
attachments. Recall that the wavenumber of a shallow
cable, denoted by with superscript 0 indicating no
damper attached and the mode index, is given by
(17)
(18)
The complex wavenumber can be assumed a small
perturbation from as . Once the
wavenumber is obtained, the modal damping is determined
afterwards by
(19)
3.1 Dampers at the opposite ends
3.1.1 Nearly antisymmetric mods
Eq. (16) can be written as
The asymptotic result is obtained from Eq. (20) by
assuming in the right-hand side. The wavenumber
equation then reduces to
The asymptotic expression valid for and
is further obtained by introducing the approximation that
0.5, and using the one-term Taylor expansion
of the sine and cosine terms on the right-hand side. The
wavenumber increment is then explicitly expressed as
1+
(22)
It is convenient to introduce the non-dimensional
damper parameters as
,
where is the dimensionless viscous coefficient, is the
dimensionless damper stiffness, and
is the
dimensionless spring coefficient of the HDR damper.
Furthermore, define the following group parameter
. Eq. (22) thus becomes
1+
(23)
The damping ratio can then be extracted from the
complex wavenumber using Eq. (19), leading to
(24)
Further simplification of Eq. (24) gives
(25)
Eq. (25) is the asymptotic modal damping of the cable
with an internal damper and an external damper when they
are respectively close to the two cable ends. The expression
includes two terms, corresponding to the contribution of
each of the two dampers. The maximum modal damping
ratio in the cable is achieved when each damper is
respectively under the optimal condition, that is
opt 1+ opt
1+
(20)
(21)
579
Fangdian Di, Limin Sun and Lin Chen
0.5
1+
+0.5
1+1+
When the loss factor is zero, the HDR damper is
reduced to a spring with stiffness , and the second term of
Eq. (25) equals zero. When the spring factor of the HDR
damper is infinite, the second term of Eq. (25) also equals
zero. In those special cases, the HDR damper has no
contribution to the cable modal damping.
3.1.2 Nearly symmetric modes
Eq. (16) can be rewritten as
(26)
The left- and right-hand sides of Eq. (26) are divided
by
, leading to
Noting that
Eq. (27) can then rewritten as
For and , the following approximations
can be introduced
(30)
Meanwhile, the left-hand side of Eq. (29) is linearized
around the undamped wavenumber, i.e.,
(31)
With these manipulations, the expression for the
complex wavenumber increment is obtained from Eq. (29)
as follows
(27)
=1+
=1+
1+
(28)
(29)
580
Cable vibration control with internal and external dampers: Theoretical analysis and field test validation
(32)
Further using the non-dimensional parameters, Eq. (32)
is written as
Correspondingly, the damping ratio is obtained as
Define the reduction factor owing to the cable sag effect
as Krenk and Nielsen (2002)
with the cable element index . As and
. The reduction factor is not sensitive to the damper
location (as will be shown in Fig. 2). Hence, further using
the following approximation.
One can obtain a simpler expression as
(35)
The preceding asymptotic expression for modal
damping has two parts, respectively representing the
contribution from the two dampers when installed
separately. For symmetric modes, the right-hand side of Eq.
(35) includes the reduction factor due to the influence of the
cable sag, as derived by Krenk and Nielsen (2002). The
maximum modal damping ratio in the cable is achieved
when each of the dampers is respectively tuned to the
optimal condition, that is
opt 1+, opt
1+ and
0.5
1+
+0.5
1+1+
3.1.3 Parametric analysis
The influence of cable sag: Note that in the cable-stayed
bridges the value ofis normally less than 3 (Tabatabai
and Mehrabi 2000). For example, for the longest
cables of the Sutong bridge. The longest cable of Dubai
creek tower (under construction) is of more than 700 m, and
approximately . In this context, the range of the
Irvine parameter is considered to be for
parametric analysis. The reduction factorfor the first
four vibration modes () are plotted in Fig. 2 for
. For the antisymmetric modes (), the
Irvine parameterdoes not affect (= 1). For the
symmetric modes (), the influence of sag must
be considered, especially for the first symmetric mode
(Krenk and Nielsen 2002). It is known that the cable sag
modifies the mode shape of symmetric cable modes.
Particularly, relative cable motion near cable ends is
reduced, such that less energy can be dissipated when the
damper is installed near the cable end. This influence
becomes more pronounced as the Irvine parameter increases
(in the range of) and as the damper is installed
closer to the cable end, as showed in Fig. 2.
Analysis of the parameters of dampers: Fig. 3 shows the
cable damping curves when the external damper with
intrinsic stiffness is installed at
and
the internal HDR damper with different loss factors and
optimal stiffness is installed at
but close to the
other end, for the first and second cable modes. Eqs. (25)
and (35) show that the total modal damping effect is
approximately the sum of the contributions from the
dampers when installed separately, as also seen in Fig. 3.
Therefore, when considering cable sag, the damping
effect of two dampers installed respectively close to the two
(33)
(34)
581
Fangdian Di, Limin Sun and Lin Chen
Fig. 2 Variation of the reduction factor due to the
influence of cable sag
(a) n = 1
(b) n = 2
Fig. 3 Modal damping versus the viscous coefficient of
the external damper of a cable with both internal
and external dampers
cable ends can still be appreciated by analyzing the cable
with each of the dampers separately. Fig. 3 also suggests a
feasible solution to supplement additional damping when
the damping provided by the external damper near the
cable-deck anchorage is insufficient due to the sag effect
and the damper stiffness, i.e., by installing an HDR damper
near the cable-pylon anchorage.
Figs. 4-5 show the damping of a shallow cable
respectively attached with an external damper and an
internal damper. Consider that the dampers are installed
at
and the cable sag is . Fig. 4(a)
shows the cable damping curves of the antisymmetric
modes when the viscous damper with different values of the
intrinsic stiffness, i.e., . Fig. 4(b)
(a) Nearly antisymmetric vibrations
(b) Nearly symmetric vibrations
Fig. 4 Modal damping of a cable with a single viscous
damper (
) both internal and
external dampers
(a) Nearly antisymmetric vibrations
(b) Nearly symmetric vibrations
Fig. 5 Modal damping of a cable with a single HDR
damper (
)
shows the cable damping curves of the symmetric modes
0 5 10 15 20
0.0
0.3
0.6
0.9
1.2
lpL=0.02
lpL=0.04
Mode 1
Mode 3
Mode 2,4
lpL=0.01
Rp
0 1 2 3 4
0.000
0.004
0.008
0.012
n
=1
=0.5
=0.1
=0
13
2
2
0.02
0.10
11
3
l L l L
k
K
==
=
=+
=
0 1 2 3 4
0.000
0.005
0.010
0.015
n
=1
=0.5
=0.1
=0
13
2
2
0.02
0.10
11
3
l L l L
k
K
==
=
=+
=
0 1 2 3 4
0.000
0.003
0.006
0.009
0.012
1.00k=
0.50k=
0.10k=
n
( )
1
22
1+
nl
L
k
+
2,4,n=
0k=
0 1 2 3 4
0.000
0.003
0.006
0.009
0.012
0.10k=
n=3,5,7
n=1
n
( )
11
22
1+
nlR
L
k
+
1,3,n=
0 1 2 3 4
0.000
0.001
0.002
0.003
0.004
0.005
=0.5
=1
=0.3
n
( )
3
222 2,4,
1+ +
nl
Kn
L
KK
= ，
K
0 1 2 3 4
0.000
0.001
0.002
0.003
0.004
0.005
=1n=3,5,7
n=1
n
( )
33
222
1+ +
nl
KR
L
KK
K
1,3,n=
582
Cable vibration control with internal and external dampers: Theoretical analysis and field test validation
when the intrinsic stiffness parameter of the viscous damper
is . Fig. 5(a) shows the cable damping curves of
the antisymmetric modes when the loss factor of the HDR
damper varies, i.e., . Fig. 5(b) shows the
cable damping curves of the symmetric modes when the
loss factor of the HDR damper is.
From Fig. 4, it is seen that the stiffness of the viscous
damper has a great influence on the damping effect of the
viscous damper. The reduction effects of damper stiffness
and cable sag should be considered in the design of viscous
damper, especially for the first symmetric mode vibration.
Fig. 5 indicates that the loss factor of HDR damper
determines the maximum damping achievable. With the
development of high-damping rubber materials, the loss
factor of HDR damper can reach a value larger than 1.0.
Hence, the HDR damper can be used for vibration reduction
of medium length cables. Besides, it is a feasible
countermeasure to improve cable damping by installing an
HDR damper inside the guide pipe of the cable-pylon
anchorage, when the external damper between the cable and
the bridge deck is insufficient.
3.2 Dampers at the same end
3.2.1 Nearly antisymmetric modes
When the viscous damper and the HDR damper are
installed at the same end, the HDR damper is closer to the
cable end, namely and . The distance
between the external damper and the nearer cable anchorage
is denoted by . Eq. (16) can be
reduced to the following form using the similar
approximations as used previously, as
Further using the approximations for and
leads to
1+
(37)
For the dampers at the same end, the following non-
dimensional parameters are defined as
,
Note the differences in the definitions of these non-
dimensional parameters as compared to those in the
previous case. Eq. (37) thus becomes
1+
(38)
Finally, the damping ratio is given as
Fig. 6 Comparison of the damping curves obtained via
different methods
3.2.2 Nearly symmetric modes
Like the simplification from Eq. (26) to Eq. (30), and
considering and , Eq. (16) can be
transformed into the following form
(40)
Using the non-dimensional damper parameters, Eq. (40)
becomes
(41)
Recall the definition of the reduction factor and in this
case for the external damper define
Then, the damping ratio is expressed as
020 40 60 80 100
0.000
0.006
0.012
0.018
Yoneda-1
Yoneda-2
Main
Present
Exact
23
2
3879kN, 278m
0
6
.02, 0.01
kN m, 0 06 ,6 =97.
HL
l L l L
Kk
==
==
==
c
(36)
(39)
583
Fangdian Di, Limin Sun and Lin Chen
3.2.3 Parametric analysis
The case of a spring and a viscous damper: When the
loss factor is zero, the HDR damper is a spring with a
stiffness , Eq. (39) can be rewritten into the following
form
For a small value of ,
. Eq. (42) can further be rewritten into the following
form
In previous studies, the bushing has been considered as
a spring (Main and Jones 2003, Takano et al. 1997). Yoneda
et al. (1995) proposed two equivalent methods (see
Appendix A) for analyzing the effect of the bushing on the
damper performance. Main and Jones (2003) derived an
explicit asymptotic expression for the effective damper
location (see Appendix B). Fig. 6 displays the comparison
of these methods to the present formula, where a cable has a
length of , the cable tension is ,
the cable sag parameter , a spring with stiffness
is installed at
, and a
viscous damper with the intrinsic stiffness is
installed at
. The numerical solution of Eq.
(16) is regarded as the exact solution. The comparison
shows that the asymptotic expression in the present study
gives the most accurate estimate, the results obtained by
using the two methods proposed by Yoneda et al. (1995) are
almost the same, and the result from Main and Jones (2003)
is more conservative.
To derive the asymptotic expression for the optimal
damping parameter of the viscous damper, from Eq. (43) or
(44), one finds
(45)
by letting
. Substituting into Eqs. (39)
and (42) or Eqs. (43) and (44), leads to the maximum modal
damping.
Fig. 7 shows the cable damping curves when the internal
damper (spring) with different stiffness is installed at
and the external damper with the intrinsic
stiffness is installed at
, for the first
and second cable modes. Fig. 7 shows the apparent
influence of the internal damper on the effectiveness of
external damper for cable vibration mitigation. The internal
damper (spring) reduces the maximum attainable damping
ratio in each mode and increases the corresponding optimal
value of the external damper coefficient, as implied by Eq.
(45).
The case of an HDR damper and a viscous damper: This
section further analyzes the dependence of the modal
damping on the loss factor of the HDR damper for
. Fig. 8 shows the damping curves along with the
parameter . It is seen that a nonzero loss factor would
further reduce the maximum attainable damping ratio of the
(a) n = 1
(b) n = 2
Fig. 7 Dependence of modal damping on spring
stiffness and the damper coefficient parameter
0 1 2 3 4
0.000
0.004
0.008
0.012
n
0.5K=
1K=
10K=
2
3
2
0.02
0.01
0.10
=0
3
lL
lL
k
=
=
=
=
0K=
0 1 2 3 4
0.000
0.005
0.010
0.015
n
0.5K=
1K=
10K=
2
3
2
0.02
0.01
0.10
=0
3
lL
lL
k
=
=
=
=
0K=
(42)
(43)
(44)
584
Cable vibration control with internal and external dampers: Theoretical analysis and field test validation
(a) n = 1
(b) n = 2
Fig. 8 Dependence of modal damping on the loss factor
of the HDR
(a) n = 1
(b) n = 2
Fig. 9 Dependence of modal damping on the position of
the HDR damper
cable. The negative effect is nevertheless very small and
negligible, as displayed by the closeup view in Fig. 8.
Fig. 9 shows the dependence of modal damping on the
(a) Design of the viscous damper targeting multiple cable modes
(b) Multimode damping ratios of a cable with a single external
damper and with both external and internal dampers
Fig. 10 Multimode damping effects of the dampers on
the cable
HDR installation position. When the damping coefficient of
the viscous damper is small (the left side of the intersection
of damping curves), the HDR damper plays a major role in
providing damping, and a larger installation position of the
HDR damper leads to increase in modal damping. If the
damping coefficient of the viscous damper is already large
(the right side of the intersection of damping curves), the
presence of the HDR damper harms the performance of the
viscous damper, and this effect becomes more pronounced
when the HDR is closer to the viscous damper.
In practical design, the parameters of an external damper
are determined according to the damping curves of the first
several cable modes (Weber et al. 2009). For example, as
shown in Fig. 10(a), the first five modes are considered to
optimize the damping coefficient of a viscous damper with
intrinsic stiffness installed at
, where the Irvine parameter of the cable is set as
, the optimal damping coefficient is found to be
. With these damper parameters
determined, Fig. 10(b) plots the modal damping ratios of
the first 20 modes of the cable. To appreciate the influence
of the internal damper on the performance of the external
damper, an internal damper with stiffness and two
different loss factors (0.2 and 1.0 respectively) is considered
being installed at
Correspondingly, the
damping ratios of the cable with the two dampers are shown
in Fig. 10(b). Note that the approximate solution can only
guarantee the accuracy of cable modal damping ratios for
the first several modes (Krenk 2000). The results in Fig.
0 1 2 3 4
0.000
0.003
0.006
0.009
0.012
0.8 1.0 1.2 1.4
0.0080
0.0082
0.0084
0.0086
0.0088
0.0090
n
2
3
2
0.02
0.01
0.10
0.5
3
lL
lL
k
K
=
=
=
=
=
0
=
1
=
0 1 2 3 4
0.000
0.004
0.008
0.012
0.016
0.8 1.0 1.2 1.4
0.0116
0.0118
0.0120
0.0122
0.0124
2
3
2
0.02
0.01
0.10
0.5
3
lL
lL
k
K
=
=
=
=
=
n
0
=
1
=
0 1 2 3 4
0.000
0.003
0.006
0.009
0.012
( )
23
2
0.03
0.10
1
0.5
3
l l L
k
K
+=
=
=
=
=
n
3
0
=
0.03
lL
0 1 2 3 4
0.000
0.004
0.008
0.012
0.016
( )
23
2
0.03
0.10
1
0.5
3
l l L
k
K
+=
=
=
=
=
n
3
0
=
0.03
lL
0 1 2 3 4
0.000
0.006
0.012
0.018 Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
( )
00
1n
3
2
2
0
0.03
3 =0.10
lL
lL
k
=
=
=,
2 4 6 8 10 12 14 16 18 20
0.000
0.004
0.008
0.012
0.016
Modes
External Damper
External Damper & Internal Damper(
=)
External Damper & Internal Damper(
=)
3
2
2
0.01
0.02
0.10
0.5
3
lL
lL
k
K
=
=
=
=
=
585
Fangdian Di, Limin Sun and Lin Chen
10(b) are computed numerically. As shown in Fig. 10(b),
considering multimode vibration control of the cable, the
internal damper impairs the effect of the external damper
for lower modes. For higher modes, the influence of the
internal damper system damping gradually decreases.
Comparing the impacts of the internal damper with two
different loss factors (0.2 and 1.0 respectively) on the
external damper, the negative effect of an internal damper
with a larger loss factor on the external damper is more
significant except for the first mode. The internal damper
stiffness could significantly reduce the damping achieved
by the external damper, which has been interpreted as a
reduction of the effective distance between the external
damper and the cable end or an increase in intrinsic stiffness
of the external damper (Yoneda et al. 1995, Main and Jones
2003). The energy dissipation capacity of the internal
damper can be interpreted as increasing damping coefficient
of the external damper. As such, after the installation of the
internal damper, the equivalent coefficient of the external
damper is larger than the optimal one as indicated by the
dashed line in Fig. 10(a). As seen from the damping curves
in Fig. 10(a), increasing damping coefficient of the external
damper decreases cable model damping ratio except for the
first mode. It is also seen in Fig. 10(b) that the damping
ratio decreases (except for the first mode) with increasing
loss factor of the HDR damper.
3.3 Discussion and remarks
The theoretical analysis shows that the total damping
effect provided by the dampers installed at opposite ends is
about the sum of separate contributions of the dampers,
which extend the conclusion in the previous studies
concerning taut cables and ideally viscous dampers
(Caracoglia and Jones 2007, Hoang and Fujino 2008). For
multimode cable vibration control, the parameters of the
external damper can be optimized according to the first
several cable modes (Weber et al. 2009). For the internal
damper (HDR damper), the loss factor should be as large as
possible, and the stiffness of damper should be optimized
according to the loss factor as opt 1+
. A larger
installation distance is essential to improve the damping
effect, and the length of the guide pipe should be designed
accordingly. For long cables, the sag effect and the damper
inherent stiffness impair the damper control performance on
the lower modes of the cable vibration. Installation of
internal dampers near the bridge tower side can improve the
damping in such cases.
To reduce cable bending stresses near the anchorages,
the neoprene rubber bushings, are desired to be installed on
stay cables inside the steel guide pipes. From the theoretical
analysis of this study, the additional transverse stiffness of
the bushings could significantly reduce the external damper
effectiveness, and improving the energy dissipation capacity
of the internal dampers is insignificant for multimode
vibration control of the cable. Therefore, in the premise of
fulfilling the design requirements for reducing the bending
stress, the stiffness of internal damper should be as small as
possible, and the installation distance should be as far away
from the external damper as possible. Compared with the
installation of dampers at the opposite ends, the energy
dissipation of the internal damper is not so important for
cable vibration control in the lower vibration modes. But for
some higher vibration modes, when the external damper is
located on the node of the vibration modes, the internal
HDR damper with greater loss factor can help reduce the
vibrations.
4. Experimental study
For validation of the theoretical analysis in section 3.2,
experiments have been carried out on a cable of the Sutong
Bridge in China. The length of the tested cable is
L =454.1 m, the mass per unit length is m
=77.65 kg/m,
the axial tension is H
=5099 kN, and the Irvine parameter
is . A viscous damper is installed about 10.689 m
away from the cable anchorage point. The damping
coefficient of the viscous damper is about c
=90 kN./m
(the equivalent coefficient for in-plane vibrations of the two
viscous dampers shown in Fig. 11), which is optimized for
the first eleven cable modes. The intrinsic stiffness of the
Fig. 11 The experiment setup
586
Cable vibration control with internal and external dampers: Theoretical analysis and field test validation
(a) Experimental results
(b) Theoretical results
Fig. 12 Comparison of the measured and theoretical
damping ratios
viscous damper is ignored based on laboratory test results.
The internal damper is an HDR damper installed at the end
of the steel cable guide pipe, about 3.987 m away from the
cable anchorage point, the loss factor of HDR damper is
about , and the stiffness of HDR damper is about
K =2000 kN/m.
Field tests have been performed to measure the damping
of the cable with only the external dampers and with both
viscous and HDR dampers, respectively. The cable was
excited to vibrate in a target mode, and then the vibration
was left to decay freely after its amplitude reached a certain
value. Modal damping of this mode was then computed
from the free-decaying cable response. Fig. 11 demonstrates
the testing scheme. Accelerometers were placed on the
cable about 13 m above the bridge deck to measure cable
accelerations during vibrations. For vibration excitation, an
exciter was attached to the cable also about 13 m above the
bridge deck. Steel wires were also used for manual
excitations when the exciter was inadequate for exciting
vibrations in lower-modes. Fig. 11 shows photos of the
installed sensors and the exciter.
Fig. 12(a) illustrates the results of the field tests. For
almost all modes, the cable modal damping decreases with
the installation of the internal damper. Fig. 12(b) shows the
corresponding theoretical results. Note that the measured
cable inherent damping (about 0.05%) is added in the
theoretical damping ratio. From Fig. 12(a), it is seen that the
damping effect of the external damper (viscous damper) is
reduced in the presence of the internal damper (HDR
damper), which qualitatively agree with the theoretical
results as shown in Fig. 12(b). However, quantitative
discrepancies between the measured and the estimated
damping ratios are observable for all the modes. The
measured damping in mode 2 is particularly larger than the
theoretical prediction mainly because of the unsuccessful
vibration excitation (Chen et al. 2020). For the other modes,
the theoretical damping ratios are larger than the respective
measurements, which is attributed to the efficiency loss
probably induced by the damper joints and nonlinear
behaviors. The cable damping ratio of the fundamental
mode was not measured due to the difficulty in exciting the
cable vibrations in low frequencies. Hence, the damping
ratio of the first cable mode is not shown in Fig. 12.
5. Conclusions
This paper presents a comprehensive theoretical analysis
of a shallow cable with an external damper and an internal
damper together with an experimental study for validation.
The external damper is considered to be a viscous damper
with intrinsic stiffness, and the internal damper is regarded
as an HDR damper. The sag of the cable is considered.
Asymptotic expressions are derived for modal damping
ratio in cases of the dampers close to either the same cable
end or the opposite cable ends. The following conclusions
can be drawn.
• Considering intrinsic stiffness of the external damper
and cable sag effect, when the dampers are installed at
opposite ends of the cable, the total damping effect is still
asymptotically the sum of the contributions from each of
the dampers when installed separately. The maximum
modal damping ratio of the cable is achieved when each
damper is respectively tuned to the optimal condition. An
internal damper installed near the bridge tower can
compensate for efficiency loss of the external damper due
to the damper stiffness and the sag effect. This strategy can
be a feasible countermeasure for long cable vibration
mitigation in practice.
• When the dampers are installed near the same cable
end, the presence of the internal damper decreases the
maximum attainable damping provided by the external
damper for lower cable modes. Specifically, the transverse
stiffness introduced by the internal damper significantly
reduce the effectiveness of the external damper. The closer
the internal damper is to the external damper, the greater the
reduction effect is. The loss factor of the internal damper
has pretty limited influences on the performance of the
external damper for multimode cable vibration control.
Therefore, the influence of the internal damper (or rubber
bushings) on the effectiveness of external cable dampers
should be evaluated in the damper design, using the
accurate asymptotic expressions or the numerical method
provided in this study.
• The results of the field tests are found to be
qualitatively consistent with the theoretical results. The
modal damping of the cable, for almost all tested modes,
decrease after the installation of the internal damper.
Note that the paper mainly focuses on the in-plane
vibrations of the cable. However, theoretical development
in this study also applies to cable out-plane vibrations,
1 2 3 4 5 6 7 8 9 10 11 12 13
0.000
0.005
0.010
0.015
0.020
External Damper
External & Internal Dampers
Modes
1 2 3 4 5 6 7 8 9 10 11 12 13
0.000
0.005
0.010
0.015
0.020
External Damper
External & Internal Dampers
Modes
587
Fangdian Di, Limin Sun and Lin Chen
where the effect of cable sag is absent.
Acknowledgments
This study was partly supported by the National Natural
Science Foundation of China (Grant nos. 51978506,
51608390), which is gratefully acknowledged.
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HJ
Appendix A
For the case of a spring and a viscous damper installed
near the same cable support, Yoneda et al. (1995) proposed
two simplified methods based on equivalence for estimating
the combined damping effect. They are referred to as
Yoneda 1 and Yoneda 2 respectively in the following. In the
first method (Yoneda 1), the influence of the spring has been
interpreted as reducing the effective distance between the
viscous damper and the cable end. The second method
considers an increment of the intrinsic stiffness of the
viscous damper equivalently introduced by the spring.
Yoneda 1: The modal damping ratio is given as
(46)
where
(47)
Note that is equivalent installation length of damper,
and
(48)
with
(49)
Yoneda 2: The cable modal damping ratio is computed
by
(50)
where is the equivalent stiffness parameter of damper,
defined as
(51)
Appendix B
Main and Jones (2003) had given an explicit asymptotic
expression for the effective damper location, and using this
effective damper location, the universal curve of is
generalized to account for the influence of the bushing
(spring). The effective length in Eq. (46) is derived as
(52)
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