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Design of viscous dampers targeting multiple cable modes
Engineering Structures 31 (2009) 2797–2800
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Short communication
Design of viscous dampers targeting multiple cable modes F. Weber a,∗ , G. Feltrin a , M. Maślanka b , W. Fobo c , H. Distl c a
Swiss Federal Laboratories for Materials Testing and Research, Structural Engineering Research Laboratory, Überlandstrasse 129, CH-8600 Dübendorf, Switzerland
b
AGH University of Science and Technology, Department of Process Control, al., Mickiewicza 30, 30-059 Cracow, Poland
c
Maurer Söhne GmbH & Co. KG, Frankfurter Ring 193, D-80807 Munich, Germany
article
info
Article history: Received 16 June 2008 Received in revised form 26 February 2009 Accepted 23 June 2009 Available online 4 July 2009 Keywords: Cable Damper Damping ratio Viscous
abstract Rain–wind induced stay cable vibrations may occur at different cable eigenfrequencies. Therefore, external transverse dampers have to be designed for several target cable modes. The resulting modal damping ratios have to fulfil Irwin’s criterion for minimum Scruton number such that rain–wind induced vibrations can be excluded. For this situation, this paper presents a systematic and easy applicable design procedure for linear viscous dampers that respects Irwin’s criterion, minimizes the damper position and leads to almost minimum variance of the target modal damping ratios. Minimum damper position is preferable from the aesthetic point of view, and it minimizes the installation costs, reduces the damper support flexibility and thereby increases the damper efficiency. Minimum variance of the target modal damping ratios maximizes the safety against large amplitude vibrations due to the unpredictability of the predominant mode. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The common countermeasure to suppress rain–wind induced stay cable vibrations is to increase the cable damping ratio by external oil dampers positioned close to the lower cable end (Fig. 1a). According to Irwin’s criterion [1], the cable damping ratio against rain–wind induced vibrations has to be at least
ζ min >
Scmin ρair D2
, Scmin = 10 (1) m where Sc is the Scruton number, m the cable mass per unit length, ρair the air density and D the cable diameter. Since oil dampers behave like linear viscous dampers [2], oil dampers can be optimized by optimal tuning of their damper coefficient c for maximum cable damping according to [3]: cnopt ∼ =
T
(2) aωn where T is the cable tension force, a the damper position and ωn the radial frequency of the undamped mode n. The resulting cable damping ratio ζn is almost mode independent; see [3]:
ζn = µζnmax ∼ =µ
a 2l
∗
Corresponding author. Tel.: +41 44 823 45 36; fax: +41 44 823 44 55. E-mail addresses: [email protected] (F. Weber), [email protected] (G. Feltrin), [email protected] (M. Maślanka), [email protected] (W. Fobo), [email protected] (H. Distl). 0141-0296/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2009.06.020
(3)
where µ is the damper efficiency, ζnmax the theoretic maximum damping ratio and l the cable length. Assuming optimal damper tuning, the lowest position of the damper on the cable that respects Irwin’s criterion becomes 2lζ min . ζ min = ζn → amin ∼ = µ
(4)
Since a is the distance from the cable support to the damper measured along the cable (Fig. 1a), amin will be denoted as the minimum damper position in the following. The minimized damper position with constraint (1) is most desirable from the aesthetic point of view and it minimizes the damper support costs. This result can be used in (2) in order to derive a closed-form solution for the damper coefficient that combines optimal damper tuning, minimum damper position and safety against rain–wind induced cable vibrations as follows: cnopt
min(a)
=
µT 2ζ min lωn
.
(5)
Eq. (5) necessitates that the most critical mode n, hence the target mode, is known. This might be the case for the retrofit of a stay cable bridge but it is most unlikely if dampers are installed during the construction of the bridge as countermeasure against future, unknown cable vibrations. Then, dampers are to be designed for several target modes due to the unpredictability of the predominant mode, as for example suggested in [4]. In this case, the design of the damper coefficient and position may require iterations in order to meet constraint (1), which usually results in strongly varying
2798
F. Weber et al. / Engineering Structures 31 (2009) 2797–2800
b
a
Fig. 1. (a) Transverse damper on cable; (b) normalized damping ratio as a function of non-dimensional damping parameter.
b
a
Fig. 2. Damper position for required minimum damping ratio of modes 1–2 and 1–5 depending on η/n.
damping ratios and does not minimize the damper position. Therefore, this paper describes a systematic design procedure for linear viscous dampers that have to dampen several target modes with respect to Irwin’s criterion (1), minimize the damper position and generate almost minimum variance of the resulting modal damping ratios, which is most desirable due to the unpredictability of the predominant mode. 2. Proposed solution For a cable of length l with a viscous damper at a distance a close to one end, the normalized damping ratio ζn /(a/l) of mode n may be approximated as derived in [3] as follows:
ζn ∼ η n = 1 + ηn2 (a/l) cl T
a
ωn . l
ηi ηj = . 2 1 + ηi 1 + ηj2
(8)
Assuming linear cable behaviour, i.e. ωn = nω1 , Eq. (7) leads to (6)
i
ηi = ηj . j
where ηn is the non-dimensional damping parameter
ηn =
the non-dimensional damping parameter divided by mode number n, then the damping curves do not coincide; see Figs. 2a and b for the cases of target modes 1–2 and 1–5. It can be observed from these figures that the region above the bold envelope curve describes all possible damper designs that meet constraint (1). The envelope minimum characterizes the damper design for minimum damper position which still fulfils Irwin’s criterion. The design of the damper for this point is derived by equating the modal damping ratios of the lowest target mode i and the highest target mode j; thus
(7)
Plotting ζn /(a/l) against ηn leads to congruent normalized damping curves showing maximum damping at ηn = 1; see [5] (Fig. 1b). However, if the relative damper position divided by the minimum required modal damping ratio ζ min is plotted against
(9)
Putting (9) into (8) yields √ a third-order √ polynomial in ηi . The three solutions are 0, − (i/j) and + (i/j). The solution 0 describes the situation where the damper coefficient is 0, which is equivalent to the cable without a damper (Fig. 1a), whereas the negative solution does not have a physical meaning. The positive min(a) solution yields the non-dimensional damping parameter ηi−j that produces equal damping ratios of the highest and lowest
F. Weber et al. / Engineering Structures 31 (2009) 2797–2800
a
2799
b
Fig. 3. (a) Target modal damping ratios and (b) corresponding minimum damper positions.
modes of the target mode range i–j and minimizes the damper position:
s min(a) i−j
η
=
i j
.
(10)
The minimum damper position that guarantees at least ζ min to all target modes is derived by putting the result of Eq. (10) into Eq. (6), replacing ζn by ζ min and taking the damper efficiency µ into account.
a min l
=
i−j
ζ min (i + j) . √ µ ij
(11)
Putting the obtained expressions for the non-dimensional damping parameter (10) and the minimum damper position (11) in Eq. (7) and taking advantage of ωi = iω1 for linear cable behaviour yields the desired damper coefficient as follows: opt
ci−j
min(a)
=
µT . ζ min lω1 (i + j)
(12)
This damper coefficient produces equal damping ratios of the lowest and highest modes of the target mode range, minimizes the damper position and fulfils Irwin’s criterion. By combining Eq. (9) and (10), the non-dimensional damping parameter for the minimized damper position, target mode range i − j and mode n becomes
n n (a) (a) ηimin = ηimin = √ . −j −j n
i
ij
(13)
Putting the results of Eqs. (11) and (13) into Eq. (6), the resulting damping ratio of mode n for the minimized damper position and target modes i − j becomes
ζ min n(i + j) (a) ζimin = . −j n µ(n2 + ij)
(14)
As expected, Eq. (14) leads to ζ min for n = i and n = j, respectively, whereas the modes between the lowest and highest target modes have higher damping ratios. This is visible in Fig. 2b, where the curves of modes 2–4 could generate ζ min even with smaller damper position and therefore generate more damping than ζ min for damper position minimized for target modes 1–5. Fig. 3a depicts the same behaviour for the examples of target modes 1–2, 1–3, 1–4 and 1–5 and Fig. 3b the corresponding damper positions. As mentioned above, the minimum damper position
is a desirable optimization criterion because it minimizes the damper support costs and hardly affects the original bridge design. Moreover, small damper support dimensions lower the support flexibility and thereby increase the damper efficiency, as shown for example by [6]. In contrast, in order to maximize the structural safety, the damper should be optimally tuned to the predominant mode. In general, this mode is unknown. Therefore, it would be advantageous to design the damper for equal damping ratios of the target modes. Since this is not possible for viscous dampers for more than two target modes, see Eq. (8), the best feasible solution is minimum variance of the target modal damping ratio. In the following, this solution is investigated and compared to the results of the minimum damper position. The variance and standard deviation, respectively, of the target modes 1–3, 1–4 and 1–5 are therefore calculated as a function of η. Figs. 4a and b show the results for target modes 1–3 and 1–4, respectively. The circle denotes the values of η and standard deviation that correspond to the minimum damper position and fulfil constraint (1). Going to the left or right on this curve for constant damper position means that constraint (1) is either not fulfilled for the lowest or highest mode. By increasing the damper position, similar standard deviation curves can be derived. On each of these curves the point is determined where the standard deviation is minimized and constraint (1) is fulfilled. Connecting all these points yields the trajectories in bold style in Figs. 4a and b. Hence, the trajectory minimum – indicated by the large cross – represents the damper design for minimum standard deviation of the target modes which also respect Irwin’s criterion. It is seen from Figs. 3 and 4 that only the case of target modes 1–3 requires a slightly different damper tuning if the goal is minimum standard deviation of the target modal damping ratios in contrast to minimum damper position. To reach the minimized standard deviation in this case, the damper position has to be increased by approximately 0.9% of its minimum value (Fig. 3b). The resulting standard deviation then becomes approximately 0.6% smaller than for the minimum damper position. 3. Conclusions A design procedure for linear viscous dampers for damping of several target cable modes has been presented. The method minimizes the damper position with the constraint that all target modes show at least the minimum required damping ratio which is thought to be sufficient to avoid rain–wind induced stay cable vibrations. The minimum damper position minimizes the damper
2800
F. Weber et al. / Engineering Structures 31 (2009) 2797–2800
a
b
Fig. 4. Standard deviation of damping ratios: (a) for target modes 1–3 and (b) for target modes 1–4.
support costs, only marginally affects the bridge aesthetics and enables large damper efficiency due to minimal support flexibility. Several target modes are specified due to the unpredictability of the predominant cable mode for a bridge under construction. The method is suitable for real applications since it does not need iterations and only requires the cable properties tension force, length, mass per unit length and damper position, which are known already during the design phase of the bridge. The method produces equal damping ratios of the lowest and highest target modes whereas the damping ratios of the other target modes are higher. Due to the unpredictability of the predominant cable mode, it would preferable to design linear viscous dampers in a way that all target modes show equal damping ratios. However, this is impossible for linear viscous dampers due to their modal dependence. Therefore, minimum variance of the target modal damping ratios is the best that can be achieved. The damper designs for minimum damper position and minimum variance of the target modal damping ratios are then compared for the cases of target mode ranges 1–2, 1–3, 1–4 and 1–5. It turns out that the method for minimum damper position also minimizes the variance of the target modal damping ratios in the cases of damping modes 1–2, 1–4 and 1–5. Only in the case of damping modes 1–3 do the results of the two methods slightly differ, by less than 0.9%. Thus,
the method for minimum damper position is also preferable from the structural safety point of view. Acknowledgements The authors gratefully acknowledge the financial support of the Swiss Federal Laboratories for Materials Testing and Research (EMPA), Dübendorf, Switzerland, and the technical support of the industrial partner Maurer Söhne GmbH & Co. KG, Munich, Germany. References [1] Irwin PA. Wind vibrations of cables on cable-stayed bridges. In: Building to last structures congress: Proceedings of the 15th structures congress. ASCE, Reston; 1997. p. 383-7. [2] Weber F, Fobo W, Distl H. Damping of several single mode vibrations with linear viscous dampers. In: Proc 7th int symp on cable dynamics. 2007. p. 409–14. [3] Krenk S. Vibrations of a taut cable with an external damper. J Appl Mech 2000; 67(4):772–6. [4] Wang XY, Ni YQ, Ko JM, Chen ZQ. Optimal design of viscous dampers for multimode vibration control of bridge cables. J Eng Struct 2005;27(5):792–800. [5] Kumarasena S, Jones NP, Irwin P, Taylor P. Wind induced vibrations of stay cables. Interim final report. Report RDT 05-004. February 2005. [6] Zhou Y, Sun L. Complex modal analysis of a taut cable with a three-element maxwell damper. In: Proc 6th int symp on cable dynamics; 2005. p. 397–404.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Short communication
Design of viscous dampers targeting multiple cable modes F. Weber a,∗ , G. Feltrin a , M. Maślanka b , W. Fobo c , H. Distl c a
Swiss Federal Laboratories for Materials Testing and Research, Structural Engineering Research Laboratory, Überlandstrasse 129, CH-8600 Dübendorf, Switzerland
b
AGH University of Science and Technology, Department of Process Control, al., Mickiewicza 30, 30-059 Cracow, Poland
c
Maurer Söhne GmbH & Co. KG, Frankfurter Ring 193, D-80807 Munich, Germany
article
info
Article history: Received 16 June 2008 Received in revised form 26 February 2009 Accepted 23 June 2009 Available online 4 July 2009 Keywords: Cable Damper Damping ratio Viscous
abstract Rain–wind induced stay cable vibrations may occur at different cable eigenfrequencies. Therefore, external transverse dampers have to be designed for several target cable modes. The resulting modal damping ratios have to fulfil Irwin’s criterion for minimum Scruton number such that rain–wind induced vibrations can be excluded. For this situation, this paper presents a systematic and easy applicable design procedure for linear viscous dampers that respects Irwin’s criterion, minimizes the damper position and leads to almost minimum variance of the target modal damping ratios. Minimum damper position is preferable from the aesthetic point of view, and it minimizes the installation costs, reduces the damper support flexibility and thereby increases the damper efficiency. Minimum variance of the target modal damping ratios maximizes the safety against large amplitude vibrations due to the unpredictability of the predominant mode. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The common countermeasure to suppress rain–wind induced stay cable vibrations is to increase the cable damping ratio by external oil dampers positioned close to the lower cable end (Fig. 1a). According to Irwin’s criterion [1], the cable damping ratio against rain–wind induced vibrations has to be at least
ζ min >
Scmin ρair D2
, Scmin = 10 (1) m where Sc is the Scruton number, m the cable mass per unit length, ρair the air density and D the cable diameter. Since oil dampers behave like linear viscous dampers [2], oil dampers can be optimized by optimal tuning of their damper coefficient c for maximum cable damping according to [3]: cnopt ∼ =
T
(2) aωn where T is the cable tension force, a the damper position and ωn the radial frequency of the undamped mode n. The resulting cable damping ratio ζn is almost mode independent; see [3]:
ζn = µζnmax ∼ =µ
a 2l
∗
Corresponding author. Tel.: +41 44 823 45 36; fax: +41 44 823 44 55. E-mail addresses: [email protected] (F. Weber), [email protected] (G. Feltrin), [email protected] (M. Maślanka), [email protected] (W. Fobo), [email protected] (H. Distl). 0141-0296/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2009.06.020
(3)
where µ is the damper efficiency, ζnmax the theoretic maximum damping ratio and l the cable length. Assuming optimal damper tuning, the lowest position of the damper on the cable that respects Irwin’s criterion becomes 2lζ min . ζ min = ζn → amin ∼ = µ
(4)
Since a is the distance from the cable support to the damper measured along the cable (Fig. 1a), amin will be denoted as the minimum damper position in the following. The minimized damper position with constraint (1) is most desirable from the aesthetic point of view and it minimizes the damper support costs. This result can be used in (2) in order to derive a closed-form solution for the damper coefficient that combines optimal damper tuning, minimum damper position and safety against rain–wind induced cable vibrations as follows: cnopt
min(a)
=
µT 2ζ min lωn
.
(5)
Eq. (5) necessitates that the most critical mode n, hence the target mode, is known. This might be the case for the retrofit of a stay cable bridge but it is most unlikely if dampers are installed during the construction of the bridge as countermeasure against future, unknown cable vibrations. Then, dampers are to be designed for several target modes due to the unpredictability of the predominant mode, as for example suggested in [4]. In this case, the design of the damper coefficient and position may require iterations in order to meet constraint (1), which usually results in strongly varying
2798
F. Weber et al. / Engineering Structures 31 (2009) 2797–2800
b
a
Fig. 1. (a) Transverse damper on cable; (b) normalized damping ratio as a function of non-dimensional damping parameter.
b
a
Fig. 2. Damper position for required minimum damping ratio of modes 1–2 and 1–5 depending on η/n.
damping ratios and does not minimize the damper position. Therefore, this paper describes a systematic design procedure for linear viscous dampers that have to dampen several target modes with respect to Irwin’s criterion (1), minimize the damper position and generate almost minimum variance of the resulting modal damping ratios, which is most desirable due to the unpredictability of the predominant mode. 2. Proposed solution For a cable of length l with a viscous damper at a distance a close to one end, the normalized damping ratio ζn /(a/l) of mode n may be approximated as derived in [3] as follows:
ζn ∼ η n = 1 + ηn2 (a/l) cl T
a
ωn . l
ηi ηj = . 2 1 + ηi 1 + ηj2
(8)
Assuming linear cable behaviour, i.e. ωn = nω1 , Eq. (7) leads to (6)
i
ηi = ηj . j
where ηn is the non-dimensional damping parameter
ηn =
the non-dimensional damping parameter divided by mode number n, then the damping curves do not coincide; see Figs. 2a and b for the cases of target modes 1–2 and 1–5. It can be observed from these figures that the region above the bold envelope curve describes all possible damper designs that meet constraint (1). The envelope minimum characterizes the damper design for minimum damper position which still fulfils Irwin’s criterion. The design of the damper for this point is derived by equating the modal damping ratios of the lowest target mode i and the highest target mode j; thus
(7)
Plotting ζn /(a/l) against ηn leads to congruent normalized damping curves showing maximum damping at ηn = 1; see [5] (Fig. 1b). However, if the relative damper position divided by the minimum required modal damping ratio ζ min is plotted against
(9)
Putting (9) into (8) yields √ a third-order √ polynomial in ηi . The three solutions are 0, − (i/j) and + (i/j). The solution 0 describes the situation where the damper coefficient is 0, which is equivalent to the cable without a damper (Fig. 1a), whereas the negative solution does not have a physical meaning. The positive min(a) solution yields the non-dimensional damping parameter ηi−j that produces equal damping ratios of the highest and lowest
F. Weber et al. / Engineering Structures 31 (2009) 2797–2800
a
2799
b
Fig. 3. (a) Target modal damping ratios and (b) corresponding minimum damper positions.
modes of the target mode range i–j and minimizes the damper position:
s min(a) i−j
η
=
i j
.
(10)
The minimum damper position that guarantees at least ζ min to all target modes is derived by putting the result of Eq. (10) into Eq. (6), replacing ζn by ζ min and taking the damper efficiency µ into account.
a min l
=
i−j
ζ min (i + j) . √ µ ij
(11)
Putting the obtained expressions for the non-dimensional damping parameter (10) and the minimum damper position (11) in Eq. (7) and taking advantage of ωi = iω1 for linear cable behaviour yields the desired damper coefficient as follows: opt
ci−j
min(a)
=
µT . ζ min lω1 (i + j)
(12)
This damper coefficient produces equal damping ratios of the lowest and highest modes of the target mode range, minimizes the damper position and fulfils Irwin’s criterion. By combining Eq. (9) and (10), the non-dimensional damping parameter for the minimized damper position, target mode range i − j and mode n becomes
n n (a) (a) ηimin = ηimin = √ . −j −j n
i
ij
(13)
Putting the results of Eqs. (11) and (13) into Eq. (6), the resulting damping ratio of mode n for the minimized damper position and target modes i − j becomes
ζ min n(i + j) (a) ζimin = . −j n µ(n2 + ij)
(14)
As expected, Eq. (14) leads to ζ min for n = i and n = j, respectively, whereas the modes between the lowest and highest target modes have higher damping ratios. This is visible in Fig. 2b, where the curves of modes 2–4 could generate ζ min even with smaller damper position and therefore generate more damping than ζ min for damper position minimized for target modes 1–5. Fig. 3a depicts the same behaviour for the examples of target modes 1–2, 1–3, 1–4 and 1–5 and Fig. 3b the corresponding damper positions. As mentioned above, the minimum damper position
is a desirable optimization criterion because it minimizes the damper support costs and hardly affects the original bridge design. Moreover, small damper support dimensions lower the support flexibility and thereby increase the damper efficiency, as shown for example by [6]. In contrast, in order to maximize the structural safety, the damper should be optimally tuned to the predominant mode. In general, this mode is unknown. Therefore, it would be advantageous to design the damper for equal damping ratios of the target modes. Since this is not possible for viscous dampers for more than two target modes, see Eq. (8), the best feasible solution is minimum variance of the target modal damping ratio. In the following, this solution is investigated and compared to the results of the minimum damper position. The variance and standard deviation, respectively, of the target modes 1–3, 1–4 and 1–5 are therefore calculated as a function of η. Figs. 4a and b show the results for target modes 1–3 and 1–4, respectively. The circle denotes the values of η and standard deviation that correspond to the minimum damper position and fulfil constraint (1). Going to the left or right on this curve for constant damper position means that constraint (1) is either not fulfilled for the lowest or highest mode. By increasing the damper position, similar standard deviation curves can be derived. On each of these curves the point is determined where the standard deviation is minimized and constraint (1) is fulfilled. Connecting all these points yields the trajectories in bold style in Figs. 4a and b. Hence, the trajectory minimum – indicated by the large cross – represents the damper design for minimum standard deviation of the target modes which also respect Irwin’s criterion. It is seen from Figs. 3 and 4 that only the case of target modes 1–3 requires a slightly different damper tuning if the goal is minimum standard deviation of the target modal damping ratios in contrast to minimum damper position. To reach the minimized standard deviation in this case, the damper position has to be increased by approximately 0.9% of its minimum value (Fig. 3b). The resulting standard deviation then becomes approximately 0.6% smaller than for the minimum damper position. 3. Conclusions A design procedure for linear viscous dampers for damping of several target cable modes has been presented. The method minimizes the damper position with the constraint that all target modes show at least the minimum required damping ratio which is thought to be sufficient to avoid rain–wind induced stay cable vibrations. The minimum damper position minimizes the damper
2800
F. Weber et al. / Engineering Structures 31 (2009) 2797–2800
a
b
Fig. 4. Standard deviation of damping ratios: (a) for target modes 1–3 and (b) for target modes 1–4.
support costs, only marginally affects the bridge aesthetics and enables large damper efficiency due to minimal support flexibility. Several target modes are specified due to the unpredictability of the predominant cable mode for a bridge under construction. The method is suitable for real applications since it does not need iterations and only requires the cable properties tension force, length, mass per unit length and damper position, which are known already during the design phase of the bridge. The method produces equal damping ratios of the lowest and highest target modes whereas the damping ratios of the other target modes are higher. Due to the unpredictability of the predominant cable mode, it would preferable to design linear viscous dampers in a way that all target modes show equal damping ratios. However, this is impossible for linear viscous dampers due to their modal dependence. Therefore, minimum variance of the target modal damping ratios is the best that can be achieved. The damper designs for minimum damper position and minimum variance of the target modal damping ratios are then compared for the cases of target mode ranges 1–2, 1–3, 1–4 and 1–5. It turns out that the method for minimum damper position also minimizes the variance of the target modal damping ratios in the cases of damping modes 1–2, 1–4 and 1–5. Only in the case of damping modes 1–3 do the results of the two methods slightly differ, by less than 0.9%. Thus,
the method for minimum damper position is also preferable from the structural safety point of view. Acknowledgements The authors gratefully acknowledge the financial support of the Swiss Federal Laboratories for Materials Testing and Research (EMPA), Dübendorf, Switzerland, and the technical support of the industrial partner Maurer Söhne GmbH & Co. KG, Munich, Germany. References [1] Irwin PA. Wind vibrations of cables on cable-stayed bridges. In: Building to last structures congress: Proceedings of the 15th structures congress. ASCE, Reston; 1997. p. 383-7. [2] Weber F, Fobo W, Distl H. Damping of several single mode vibrations with linear viscous dampers. In: Proc 7th int symp on cable dynamics. 2007. p. 409–14. [3] Krenk S. Vibrations of a taut cable with an external damper. J Appl Mech 2000; 67(4):772–6. [4] Wang XY, Ni YQ, Ko JM, Chen ZQ. Optimal design of viscous dampers for multimode vibration control of bridge cables. J Eng Struct 2005;27(5):792–800. [5] Kumarasena S, Jones NP, Irwin P, Taylor P. Wind induced vibrations of stay cables. Interim final report. Report RDT 05-004. February 2005. [6] Zhou Y, Sun L. Complex modal analysis of a taut cable with a three-element maxwell damper. In: Proc 6th int symp on cable dynamics; 2005. p. 397–404.