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Journal of Wind Engineering

and Industrial Aerodynamics 89 (2001) 633647

Vortex-induced cable vibration of cable-stayed

bridges at high reduced wind velocity

Masaru Matsumotoa,*, Tomomi Yagia, Yoshinori Shigemurab,

Daisuke Tsushimac

aDepartment of Global environment Engineering, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto

606-8501, Japanb Project Engineering Department, Kawasaki Heavy Industries Ltd., 2-4-1, Hamamatsu-cho, Minato-ku,

Tokyo 105-6116, JapancDesign Department, Bridge Division, Ishikawajima-Harima Heavy Industries, Co., Ltd., 5-17, Hikarimachi,

Kure, Hiroshima 737-8515, Japan

Abstract

In this paper, mechanisms of vortex-induced vibration of inclined cables at high reduced

wind velocity region are discussed using results from a series of wind tunnel tests. As a

conclusion, this aerodynamic instability of inclined cables would occur by the fluid interaction

between Karman vortex and axial vortex. Also, the axial flow along the cable axis and the

upper water rivulet control this aerodynamic instability. Furthermore, three dimensional

properties of vortex shedding around the cable must play important roles in these mechanisms.

# 2001 Elsevier Science B.V. All rights reserved.

Keywords: Inclined cables; Vortex-induced vibration; Karman vortex; Axial vortex; Axial flow;

Water rivulet

1. Introduction

The reduction of wind-induced cable vibration is one of the major problems in

designing cable-stayed bridges. There are many kinds of wind-induced cable

vibration, such as Karman vortex excitation, galloping, rain and wind induced

vibration, vortex-induced oscillation at high wind velocity and so on. Hikami [1]

found the rain and wind induced cable vibration of inclined cables at the Meiko

*Corresponding author. Tel.: +81-75-753-5091; fax: +81-75-761-0646.

E-mail address: [email protected] (M. Matsumoto).

0167-6105/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 7 - 6 1 0 5 ( 0 1 ) 0 0 0 6 3 - 0

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Nishi Bridge, and he succeeded to reproduce the similar vibration with artificial

precipitation in a wind tunnel. Matsumoto [2] pointed out that inclined cables can

also oscillate under no precipitation condition and its instability caused by the axial

flow behind the cables. Then, Matsumoto [3] clarified that the upper water rivuletand the axial flow behind the inclined cable play important roles in the galloping

instability and also in the velocity restricted response at the high reduced wind

velocity. However, all of the generation mechanisms of wind-induced cable

vibrations are not always well explained yet. Especially, the mechanism for the

velocity restricted vibration at high reduced wind velocity, which is considered as a

sort of vortex-induced vibration, is still unknown.

From the observation results of prototype cables, this kind of vibration always

occurs at the reduced wind velocity V=fD=20, 40, 60, 80 and so on, where V is awind velocity, f is a frequency of vibration and D is a diameter of cable [4]. It should

be noted that these particular wind velocities are rather higher than the resonance

wind velocity region of Karman vortex excitation. Matsumoto [5] tried to explain the

mechanism of this phenomenon by three dimensional interaction between Karman

vortex and axial vortex along the cable. In this paper, the discussion is focused on the

generation mechanism of this vortex-induced oscillation of inclined cables at high

wind velocity, which seems to be one of the most significant problems in cable

aerodynamics of bridges at present. To find out a solution to reduce the cable

vibration aerodynamically or even mechanically, it is necessary to clarify this

generating mechanism. Then, a series of wind tunnel tests were conducted, which

were measurements of aerodynamic lift force of stationary/oscillating circularcylinders, unsteady pressure along the cylinders and flow visualization tests, as

follows.

2. Wind tunnel tests

The wind tunnel used in this study is a room-circuit type, which working section is

1.8 m height and 1.0 m width, and the maximum wind velocity is 30 m/s. A circular

cylinder as a model of cable was installed with yaw angle in the wind tunnel. The

mounting position of the cylinder can be defined as the vertical angle a and thehorizontal yaw angle b, see Fig. 1. Then, the relative yaw angle b* , which defines

relative angle between the wind direction and the cable axis, can be introduced as

follows:

b* arcsin cos a sin b: 1

However, in this study, the vertical angle was chosen as a 08. Therefore, the cable

position against wind direction can be defined by only the horizontal yaw angle b.

Then, the top view of the wind tunnel is shown in Fig. 2, and the yaw angle b and the

distance X from the side wall are defined as in Fig. 2. Also, the diameter D of rigid

cable model used in this study is 50 mm or 54 mm.In this series of wind tunnel tests, the total aerodynamic lift force of the cable

model and the pressure distribution along the cable axis were measured under

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various conditions. The windows on both the wall of wind tunnel were opened to

support the cable model at the both ends of cylinder, which diameter is 200 mm.

These windows can be closed to control the end flow condition by installing so called

end plates. In some cases, the artificial water rivulet was attached on the cable model

as shown in Fig. 3, and the position of the rivulet is defined by the angle y from the

stagnation point. On the cable model for measuring surface pressure, 48 pressuretaps are installed in a single line along the cable axis. Then, the position of the

pressure hole is defined by y and X as mentioned above.

Fig. 2. Top view of wind tunnel.

Fig. 3. Position of artificial water rivulet.

Fig. 1. Attitude of inclined cable.

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Also, flow visualization tests at the wake of the cable model were done using

smoke wire technique.

3. Unsteady aerodynamic lift force of stationary cylinder

The rigid circular cylinder was mounted in the wind tunnel with yaw angle b=08,

and the total aerodynamic lift force was measured in the stationary situation. For the

basic case, the power spectrum distribution of the lift force at the wind velocity

V=3 m/s in smooth flow is shown in Fig. 4. There are two dominating frequency in

the figure, the higher one corresponds to Strouhal number 0.2 and the lower one

seems to correspond to the reduced wind velocity V=fD=20, where the vortex-

induced oscillation may occur.The result of the case with yaw angle b=458 is shown in Fig. 5. And the case with

additional artificial axial flow using a vacuum cleaner is also shown in Fig. 6. In both

the cases, the lower frequency components in the above basic case move to further

lower frequency region, which corresponds to the reduced wind speed V=fD=40.This is also the occurring reduced wind velocity of this vortex-induced oscillation.

Therefore, the axial flow along the cable axis may have some contributions to the

generation of the vortex-induced oscillation at higher wind velocity. Furthermore,

the result of the case attaching end plates is shown in Fig. 7. In this case, the air flow

cannot be supplied from the outside of the wind tunnel to the cable direction. From

the results, the lower frequency components of the lift force seem to be totallyvanished. On the other hand the component of Karman vortex shedding is

remarkably enhanced. Then, it becomes clear that the axial flow has an important

role in this phenomenon of higher wind velocity region.

Fig. 4. Power spectrum density of lift force (without end plates, V=3.0 m/s, b=08, in smooth flow).


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To investigate the effects of the turbulence of natural wind, the measurement of

the lift force under the turbulent flow, where the intensity Iu=12.7%, was conducted

in the same way as previous cases. However, the lower frequency components, which

seems to be the generation source of the vortex-induced oscillation in high wind

speed, never tends to be enhanced, see Fig. 8.

4. Unsteady aerodynamic lift force of oscillating cylinder

The total aerodynamic lift force of forced vibrating cylinder was measured.

However, due to the restriction of wind tunnel facility for this experiment, the cable

Fig. 5. Power spectrum density of lift force (without end plates, V=3.0 m/s, b=458, in smooth flow).

Fig. 6. Power spectrum density of lift force (with artificial axial flow, V=3.0 m/s, b=08, in smooth flow).


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model was mounted in the wind tunnel with horizontal yaw angle b=08, instead of

b=458. To realize the similar flow condition to the inclined state of cable, the

artificial axial flow was applied in some cases. Also, the cable model was forced to

vibrate in vertical direction by the frequency f0=2 Hz and the amplitude 2Z=D=0.4.Then, Scanlans aerodynamic derivative H*1 [6] is introduced in this study, which can

be calculated as following expression:

H*1 2L cos j

prD2Zf0; 2

Fig. 7. Power spectrum density of lift force (with end plates, V=3.0 m/s, b=08, in smooth flow).

Fig. 8. Power spectrum density of lift force (without end plates, V=5.0m/s, b=458, in turbulent flow

Iu=12.7%).


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where L is amplitude of the lift force, j is phase difference of the lift force from the

motion velocity of cable model, and r is air density. The positive value of H*1denotes aerodynamically unstable situation.

First of all, the result of the basic case measured in smooth flow is shown in Fig. 9.Around reduced wind speed V=fD=30, H*1 has positive value, and this seems tocorrespond to the lower frequency components in Fig. 4. It should be noticed that

the unstable region appears around V=fD=30 instead of V=fD=40, because thiscable model is not inclined. To investigate the effect of the water rivulet, the artificial

rivulet was installed on the cable surface, where y=638 and 728, and the results are

shown in Figs. 10 and 11, respectively. When the rivulet is set on y=638, H*1becomes positive at nearly V=fD=20 and around 70. Also, the result of the casey=728 suggests the cable might become unstable around V=fD=20, 40, 60, 100.Then, it becomes clear that the water rivulet seems to enhance the aerodynamic

instabilities. In the next case, the artificial axial flow was added using same method as

mentioned before and the result is shown in Fig. 12. From this result, there is a

possibility to generate the velocity restricted vibration around V=fD=40. However,according to the previous study [3], the artificial axial flow induces galloping

instability in the higher wind velocity region. The main difference of the both

experiments is the applying condition of artificial axial flow, which means the axial

Fig. 9. Aerodynamic derivative H1 (b=08, in smooth flow).

Fig. 10. Aerodynamic derivative H1 (with artificial rivulet y=638, b=08, in smooth flow).


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flow in this study was rather weaker than in the previous study. Therefore, it seems

that the type of aerodynamic instability, which means the velocity restricted type or

galloping, depends on the condition of axial flow.

Next, the aerodynamic derivative H*1 were measured in turbulent flow instead of

smooth flow, where the intensity of turbulence is Iu=12.7%. Fig. 13 shows a resultin turbulent flow without any attachments. In this case, H*1 shows negative values

through the all velocity range, which means the cable is aerodynamically stable.

Fig. 11. Aerodynamic derivative H1 (with artificial rivulet y=728, b=08, in smooth flow).

Fig. 12. Aerodynamic derivative H1 (with artificial axial flow, b=08, in smooth flow).

Fig. 13. Aerodynamic derivative H1 (b=08, in turbulent flow Iu=12.7%).


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Figs. 14 and 15 show results of the case with the artificial rivulet at y=638 and 728,

respectively. When y=638, H*1 has positive value around V=fD=20, 60, 80. Also,H*1 becomes positive at under V=fD=20 and around V=fD=120. These positive

values of H*1 are much higher than those measured in smooth flow. Therefore, theturbulence of approaching flow can stimulate the aerodynamic instability of velocity

restricted responses. Furthermore, applying the artificial axial flow under the

condition of turbulent flow, H*1 shows extremely unstable situation between

V=fD=20 and 60, see Fig. 16.

Fig. 14. Aerodynamic derivative H1 (with artificial rivulet y=638, b=08, in turbulent flow Iu=12.7%).

Fig. 15. Aerodynamic derivative H1 (with artificial rivulet y=728, b=08, in turbulent flow Iu=12.7%).

Fig. 16. Aerodynamic derivative H1 (with artificial axial flow, b=08, in turbulent flow Iu=12.7%).


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Then, it becomes clear that the axial flow should be one of the main factors to

control the instability of not only galloping but also vortex-induced vibration at high

wind velocity.

5. Karman vortex and axial vortex

Using the same data of the unsteady lift force of Fig. 5, where the stationary

cylinder was mounted with b=458, the wavelet analysis was tried, to observe the

variation of the lift force against time. The result is plotted in Fig. 17. The lift force

induced by Karman vortex is intermittently amplified, and also the low frequency

component is detected which corresponds to the reduced wind velocity about

V=fD=40. To understand the relationship between the amplified Karman vortexcomponent and the lower frequency component, the flow visualization tests were

conducted.

Using the smoke wire method, existence of so-called axial vortex is confirmed in

the wake of yawed cylinder, see Fig. 18. This axial vortex has different vortex axis

from Karman vortexs one, which means this vortex axis has some vertical angle.

From the next flow visualization tests shown in Fig. 19, it becomes clear that the

axial vortex is shed once three Karman vortex sheddings from the upper and lower

side of the cable. Then, the image figure of the interaction between Karman vortex

and axial vortex is shown in Fig. 20. The Karman vortex along the main flow is

enhanced every third vortex shedding, because of the interaction between Karman

vortex and axial vortex, when the shedding frequency of the axial vortex is 1=3 of theshedding frequency of Karman vortex. This interpretation coincides with the results

of Shirakashi [7], in which he pointed out the existence of vortex shedding with 1=3

Fig. 17. Unsteady aerodynamic lift force of stationary yawed cable (b=458).


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times frequency of conventional Karman vortex. Then, this frequency of the

amplified vortex corresponds to the reduced wind velocity about V=fD=20. If

Karman vortex is amplified every fifth or sixth of its vortex shedding, thenthe vortex-induced oscillation might occur at V=fD=40, and this can explain theexistence of the low frequency component of the unsteady lift force in Fig. 5. In the

Fig. 18. Visualized axial vortex in the wake of inclined cable (without end plate, b=458, V=0.5 m/s, in

smooth flow).

Fig. 19. Visualized axial vortex and intermittently enhanced Karman vortex in the wake of inclined cable

(without end plate, b=458, in smooth flow).


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same manner, the occurrence of the vortex-induced vibration in V=fD=60, 80 can beexplained.

6. Unsteady pressure distribution along cable axis

As mentioned in the previous section, the generation mechanism of this vortex-induced oscillation can be explained by interaction between axial vortex and

Karman vortex. However, this interaction may have strong three dimensional

characteristics along the cable axis. Therefore, the unsteady surface pressure

distributions along the cable direction were measured to observe their three

dimensional characteristics. The experimental conditions are the yaw angle b=458,

without end plates, the wind velocity V=4 m/s, in smooth flow. The pressure taps

are distributed along the cable axis at y=1358 from the stagnation point, and the

positions are denoted by X=D, see Fig. 2. Then, the power spectrum density ofunsteady pressure on the yawed stationary cylinder is shown in Fig. 21. At the all

measured positions on the model, the very low frequency components about 1.5 Hzare observed, which are considered as the axial vortex. Especially around X=D=2.5where is the upstream side, these low frequency components are extremely

dominated.

This position is almost same place as where the axial vortex was observed in

Fig. 18. Also around X=D=6, the low frequency components are dominated. On theother hand, at the pressure holes between X=D=14 and 18 where is the downstreamside, the frequency components of Karman vortex shedding are dominated, and the

Strouhal number is about 0.15. Therefore, it becomes clear that the characteristics of

unsteady pressure along the cable axis are not uniform and these properties may be

determined by the end conditions of cable, the yaw angle b and so on.For the next case, the yawed cable model was forced to vibrate in vertical direction

by the frequency f=2 Hz and the amplitude 2Z=D=0.4, and the unsteady pressure

Fig. 20. Illustration of axial vortex and Karman vortex around yawed cable.


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was measured as same as the stationary case. The reduced wind velocity is set as

V=fD=40. Using band-pass filter, the synchronized frequency component of theunsteady pressure at the each point was extracted, and then its standard deviation

along the cable axis is plotted as shown in Fig. 22. The standard deviation has

relatively large value around X=D=2 and 6 where are considered as same place asthe strong source of the axial vortex in the previous discussion. Also, taking account

of the phase difference between the displacement and the pressure, these places in the

upstream side of the cable have great contributions to oscillating the cable.

These three dimensional characteristics of flow conditions around the inclined

cable must be rather important factor to understand the complex generating

mechanism of the vortex-induced vibration at high wind speed. However, these three

dimensional properties are rather sensitive to the experimental conditions, especiallyto the end condition of the cable model. Therefore, to clarify these characteristics,

further studies are really needed.

Fig. 21. Power spectrum density of pressure on stationary cylinder (without end plates, b=458, pressure

hole ofy=1358, V=4 m/s, in smooth flow).

Fig. 22. Standard deviation of synchronized frequency component of unsteady pressure on forced

vibrating cylinder (without end plates, b=458, pressure hole of y=1358, V=4 m/s, in smooth flow,

Z=0.01 m, f=2 Hz, V=fD=40).


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7. Effects of upper water rivulet

This three dimensional interaction between Karman vortex and axial vortex,

which is mentioned above, might be able to explain the mechanism of the cableaerodynamic vibrations at high reduced wind velocity. However, many of these

vibrations have been observed under precipitation condition, and then the effect of

water rivulet on this vortex-induced vibration must be clarified. Therefore, the

unsteady aerodynamic lift forces on stationary yawed cable with an upper artificial

water rivulet were measured in the wind tunnel in the same way as Section 3. The

aerodynamic lift forces were measured at both the ends of the cable model, and they

were not summed up to evaluate the total lift force, but analyzed independently.

These experiments were carried out under following conditions; the yaw angle of

Fig. 23. Power spectrum density of unsteady lift force at upstream side of stationary cylinder with

artificial rivulet (without end plates, b=458, V=4 m/s, in smooth flow).

Fig. 24. Power spectrum density of unsteady lift force at downstream side of stationary cylinder with

artificial rivulet (without end plates, b=458, V=4 m/s, in smooth flow).


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cable model is b=458, the wind velocity is 4 m/s, under the smooth flow. Then, the

position of upper rivulet was varied as y=458, 608, 638, 688, 728, 758 and 908. The

power spectrum densities of the lift force at the upstream side and at the downstream

side are shown in Figs. 23 and 24, respectively. Generally, the low frequencycomponents of the aerodynamic lift force which must be the driven force of vortex-

induced vibration at the high reduced velocity are remarkable at the upstream side

and this result agrees with previous discussions. From Fig. 23, it becomes clear that

these low frequency components are extremely enhanced when the artificial rivulet is

set at y=688, 728 and 758. Therefore, the upper water rivulet may enhance the

instability of this vortex-induced vibration at the high reduced velocity.

8. Conclusions

The existences of the axial flow and the upper water rivulet on the yawed cable

have important roles on the occurrence of the vortex-induced vibration at high wind

velocity. Furthermore, the combination of the axial flow, the water rivulet and the

turbulence flow may enhance this aerodynamic instability.

The mechanism of this vibration would be the fluid interactions between Karman

vortex and axial vortex. Then, this result can easily explain the occurrence of

vibration at multiples of reduced wind velocity 20 or 40.

Also, the three dimensional characteristics of vortex shedding are observed and

this seems to strongly depend on the end conditions and the yaw angle of the cable.For further understanding of the generation mechanisms of real cables, the

comparison of the end condition between the cable model in the wind tunnel and

the real cables seems to be rather important.

References

[1] Y. Hikami, Rain vibrations of cables in cable-stayed bridge, J. Wind Eng. 27 (1986) 1728 (In

Japanese).

[2] M. Matsumoto, C.W. Knisely, N. Shiraishi, M. Kitazawa, T. Saito, Inclined-cable aerodynamics,

Structural Design, Analysis & Testings Proceedings, Structural Congress, ASCE, San Francisco, USA,

1989.

[3] M. Matsumoto, N. Shiraishi, H. Shirato, Rain-wind induced vibration of cables of cable-stayed

bridges, J. Wind Eng. Ind. Aerodyn. 4144 (1992) 20112022.

[4] M. Matsumoto, Y. Shigemura, Y. Daito, T. Kanamura, High speed vortex shedding vibration of

inclined cables, Proceedings of International Seminar on Cable Dynamics, Tokyo, Japan, 1997,

pp. 2735.

[5] M. Matsumoto, Observed behavior of prototype cable vibration and its generation mechanism, in:

A. Larsen, S. Esdahl (Eds.), Bridge Aerodynamics, Balkema, Rotterdam, 1998, pp. 189211.

[6] R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivatives, J. Eng. Mech. Div. ASCE, 97

EM6 (1971) 17171737.

[7] M. Shirakashi, Y. Ishida, S. Wakiya, Higher velocity resonance of circular cylinder in crossflow,

J. Fluids Eng. ASME, 107 (1985) 392396.


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