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Rain–wind-induced vibration of inclined cables at limited high reduced wind velocity region
Journal of Wind Engineering and Industrial Aerodynamics 91 (2003) 1–12
Rain–wind-induced vibration of inclined cables at limited high reduced wind velocity region Masaru Matsumotoa,*, Tomomi Yagia, Mitsutaka Gotob, Seiichiro Sakaia a
Department of Global Environment Engineering, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto 606-8501, Japan b Tokyo Regional Bureau, Japan Railway Construction Public Corporation, 1-11-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-0021, Japan
Abstract Aerodynamic vibration of stayed cables, especially rain–wind-induced cable vibration, has been a rather serious problem on the design of cable-stayed bridges. This well-known phenomenon occurs at particular high reduced wind velocity regions and under raining condition. From results of wind tunnel tests, it might be explained as a vortex-induced vibration, which occurs at limited high reduced wind velocity region. Then, in this study, the effects of water rivulet and the wind turbulence on the vortex-induced vibration at high reduced wind velocity are investigated by wind tunnel tests, and the mechanism of the rain– wind-induced cable vibration is tried to be understood comprehensively. Furthermore, the role of Karman vortex in this vibration is discussed. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Rain–wind-induced vibration; Inclined cables; Vortex-induced vibration at high reduced wind velocity; Karman vortex; Water rivulet; Wind turbulence
1. Introduction Rain–wind-induced vibration of inclined cables has been a well-known phenomenon on cable-stayed bridges since the mid-1980s [1]. However, its generation mechanisms have not been entirely solved and its effective countermeasures *Corresponding author. Tel: +81-75-753-5091; fax: +81-75-761-0646. E-mail address: [email protected] (M. Matsumoto). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 3 3 1 - 8
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except additional dampers are still undeveloped. Therefore, to clarify the mechanism of this vibration is an urgent problem on the bridge engineering community. Matsumoto [2] pointed out that the axial flow behind inclined cables and the water rivulet formed on the upper side of cables play vital roles in rain–wind-induced vibration of inclined cables. The aerodynamic instabilities of inclined cables are roughly divided into galloping instability and vortex-induced oscillation. Also, the latter phenomenon can be divided into Karman vortex excitation and vortex-induced vibration at high reduced wind velocity. From observation results [3] at existent cable-stayed bridges, it is well known that rain–wind-induced vibration often occurs at the limited reduced wind velocity region around V =fD ¼ 40; 80 and so on, where V is the wind velocity, f is the vibration frequency and D is the diameter of cable. These reduced wind velocity regions are rather higher than the reciprocal of the Strouhal number of inclined cables. Then, Matsumoto [3] tried to explain the mechanism of this rain–wind-induced vibration as vortexinduced vibration at high reduced wind velocity by the interaction between Karman vortex and axial vortex. Furthermore, the strong three-dimensional characteristics of vortex shedding around the inclined circular cylinder were observed in the wind tunnel [4]. This vibration is sometimes observed under no precipitation condition [5], which is easily supposed from the results of wind tunnel tests mentioned above. However, many of the prototype cable vibrations have been observed under the raining condition. Also, the natural wind is generally a turbulent flow. Then, the effects of the water rivulet and wind turbulence on the vortex-induced vibration at high reduced wind velocity must be clarified. In this study, to understand the mechanism of this vibration, the effects of the water rivulet and wind turbulence are considered using wind tunnel tests. In this paper, the vortex-induced vibration at high reduced wind velocity is abbreviated as HSV, that is, high speed vortex-induced vibration.
2. Wind tunnel tests The wind tunnel used in this study is a room-circuit Eiffel type located at Kyoto University. The working section of the wind tunnel is 1.8 m high and 1.0 m wide, and the maximum wind velocity is 30 m/s. As an inclined cable model, a rigid circular cylinder with diameter D ¼ 54 mm, and length L ¼ 1500 mm was installed in the wind tunnel with horizontal yaw angle b=451. As shown in Fig. 1, the cylinder bored through the wall of the wind tunnel with holes of 100 or 200 mm diameter at both the upstream (left) and downstream (right) sides. These holes are called ‘‘windows’’ in this study and they were kept open to allow air to flow into the wind tunnel or out. This might prevent the axial flow from reducing its velocity. To realize the raining condition over the cable, an artificial water rivulet was installed at various angles from the stagnation point of the cable model, as shown in Fig. 2.
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Fig. 1. Top view of wind tunnel.
Fig. 2. Position and size of artificial water rivulet.
As the wind tunnel tests, the following measurements have been conducted: (1) The aerodynamic lift force of the stationary cable model was measured using load-cells at both ends of the model. (2) The fluctuating wind velocity in the wake of the stationary cable model was measured using a hot-wire anemometer, as shown in Fig. 3. This measurement was conducted along the cable model from the upstream side to downstream side. (3) The dynamic response and damping of the spring supported cable model were measured. The cable model was supported as one degree of freedom in the heaving direction by four springs. For all of these tests, the horizontal yaw angle of the cable model is fixed as b ¼ 451: The wind velocity of approaching flow varied from 0.5 to 10 m/s. Then,
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Fig. 3. Position of hot-wire anemometer.
Reynolds numbers for these tests are within 1800pRep36000 and belong to the subcritical range.
3. Aerodynamic lift force of stationary inclined cable model To understand the effects of water rivulet on driven force of vortex-induced vibration at high reduced wind velocity (HSV), the unsteady aerodynamic lift force on the stationary inclined cable model with the artificial water rivulet was measured. The lift force was measured using load-cell installed at both ends of the cylinder as mentioned above. Generally, the aerodynamic lift force on the inclined cable model is not uniformly distributed on the cylinder. Therefore, the lift forces measured at the upstream and at the downstream sides were analyzed separately, instead of evaluating the summing up data. The artificial water rivulet was installed on the stationary cable model with angles y ¼ 451; 601, 631, 681, 721, 751, 901, respectively. The horizontal yaw angle of the cable model is b ¼ 451; the windows (100 mm) were open and the mean wind velocity is V=4m/s under the smooth flow. The power spectrum densities (PSD) of unsteady lift force measured at the upstream side load-cell are plotted in Fig. 4. In this figure, a result of the case without the water rivulet is also plotted. There are two dominant frequency components: one is for Karman vortex excitation and the other is supposed as the driven force of vortex-induced vibration at high reduced wind velocity (HSV). These low frequency components fv correspond to the reduced wind velocity V =fvD ¼ 402180: When the water rivulet is located at y ¼ 681; 721, 751, the low frequency components of lift force are extremely large. Therefore, it seems that if the water rivulet is formed around y ¼ 721; HSV might easily occur. This result agrees with the fact that the cable vibration is usually observed under raining condition. Also, the PSD diagram of unsteady lift force measured at the downstream side is shown in Fig. 5. In this case, the low frequency components of aerodynamic lift force are never observed, but only Karman vortex components are dominant. Also, the PSD values for Karman vortex shedding at the downstream side are extremely larger than those at the upstream side. Therefore, it seems that the vortices formation at the upstream and downstream sides are totally different.
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Fig. 4. Power spectrum density of unsteady lift force on stationary cylinder with artificial rivulet (from load-cell at upstream side, with 100 mm windows, b ¼ 451; V ¼ 4 m=s; in smooth flow).
Fig. 5. Power spectrum density of unsteady lift force on stationary cylinder with artificial rivulet (from load-cell at downstream side, with 100 mm windows, b ¼ 451; V ¼ 4 m/s, in smooth flow).
4. Fluctuating wind velocity in the wake of stationary inclined cable model The fluctuating wind velocity in the wake of the stationary inclined cable model was measured along the model. The circular cylinder was mounted in the wind tunnel with horizontal yaw angle b ¼ 451: The hot-wire anemometer was set in the wake as shown in Fig. 3 and its position along the cable direction was varied from the upstream side to the downstream side. Then, the effects of water rivulet and wind turbulence on this fluctuating wind velocity were investigated. The power spectrum density (PSD) diagrams of fluctuating wind velocity in the wake along the cable direction are shown in Figs. 6 and 7. The mean wind velocity in the wind tunnel is fixed as V ¼ 6 m/s, and the window size at both the ends of model is 100 mm. The PSD diagram for a case measured under smooth flow is plotted in Fig. 6. The low frequency components, which may be supposed as axial vortex
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Fig. 6. Power spectrum density of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in smooth flow, without rivulet, with 100 mm windows, V ¼ 6:0 m/s, b ¼ 451).
Fig. 7. Power spectrum density of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in smooth flow, with rivulet y ¼ 721; with 100 mm windows, V ¼ 6:0 m/s, b ¼ 451).
components [4], are observed through span-wise direction. Especially at the upstream side, which corresponds to small X/D, theses low frequency components are extremely dominant. These frequency components correspond to reduced wind velocity around V =fvD ¼ 80; where fv is the dominating frequency of fluctuating velocity. This result is also well consistent with results from surface pressure measurements of the inclined cylinder [4]. To investigate the effects of the water rivulet, an artificial water rivulet was installed on the cylinder at y ¼ 721: The PSD diagram is shown as Fig. 7. It becomes clear that the water rivulet enhances the low frequency components at the mid-span and also the downstream side of the cable model. Therefore, the water rivulet may encourage the generation of vortex-induced vibration at high reduced wind velocity, and these results are also consistent with the results of the previous section. To discuss the turbulence effects on the fluctuating wind velocity, the wind turbulence Iu ¼ 6:5% was applied to this cable model without the rivulet. The experimental conditions are the same as the previous case, except that the mean wind velocity is V ¼ 4 m/s and window size at both ends of the cylinder is 200 mm. Also, the hot-wire anemometer was set at a very close position to the downstream side
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Fig. 8. Power spectrum density of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in smooth flow, without rivulet, with 200 mm windows, V ¼ 4:0 m/s, b ¼ 451).
Fig. 9. Power spectrum density of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in turbulent flow Iu=6.5%, without rivulet, with 200 mm windows, V ¼ 4:0 m/s, b ¼ 451).
wall, which was not measured in the previous case. The PSD diagram measured under smooth flow is plotted in Fig. 8 and the low turbulent case (Iu ¼ 6:5%) is shown in Fig. 9. The smooth flow case shows very similar characteristics to the previous case in Fig. 6, where the difference of experimental conditions are just the wind velocity and the window size. However, it becomes clear that the 6.5% turbulence flow extremely enhances the low frequency components of fluctuating wind velocity in the wake, especially at the upstream and the downstream sides. Therefore, it can be said that the wind turbulence may enhance remarkably the instability of this vortex-induced vibration at high reduced wind velocity. Furthermore, the wavelet analyses for both the cases were done, and the results for 5 s at the position X =D ¼ 3:0 are shown in Figs. 10 and 11. Morlet’s function was used as the mother function in the analyses and the wavelets are expressed by absolute values in these plots. From these plots, it is also clear that the low frequency components around V =fvD ¼ 40 exist strongly in the case of turbulent flow than in the smooth flow case. In the PSD analyses shown in Fig. 9, the low frequency components are just dominant. However, using these wavelet analyses, it becomes
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Fig. 10. Wavelet analyses of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in smooth flow, without rivulet, with 200 mm windows, V ¼ 4:0 m/s, b ¼ 451; X =D ¼ 3:0).
Fig. 11. Wavelet analyses of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in turbulent flow Iu ¼ 6:5%; without rivulet, with 200 mm windows, V ¼ 4:0 m/s, b ¼ 451; X =D ¼ 3:0).
clear that Karman vortices also exist unsteadily. This must be a very significant fact to realize the role of Karman vortex in this aerodynamic vibration.
5. Response of spring supported inclined cable model According to the above-mentioned results, the effects of water rivulet and wind turbulence might be essential factors of the vortex-induced vibration at high reduced
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wind velocity, which is usually observed as the rain–wind-induced vibration in many cable-stayed bridges. Then, these effects on the dynamic response of rigid inclined cylinder were examined by free vibration tests. The cable model was supported by four springs and allowed to move in the heaving direction. Also, its rolling motion was restricted by piano wires. The horizontal yaw angle of the model was b ¼ 451 and the diameter of windows on the wind tunnel wall was 200 mm at both the ends of the model. The cases discussed in this study are (a) in smooth flow without rivulet, (b) in smooth flow with rivulet at y ¼ 721; (c) in turbulent flow (Iu ¼ 6:5%) without rivulet and (d) in turbulent flow (Iu ¼ 6:5%) with rivulet at y ¼ 721: The velocity–amplitude–damping (V –A–d) diagrams for these 4 cases are plotted in Figs. 12–15. In these figures, 2A=D denotes the non-dimensional double amplitude of the heaving response, where A; D are the amplitude and diameter of the model, respectively. The natural frequencies of the system are f ¼ 1:6621:69 Hz, the masses per length are m ¼ 0:48920:526 kg/m and the logarithmic decrements at 2A=10 mm are d=0.0043–0.0050. Then, Scruton numbers defined by Sc ¼ 2md=ðrD2 Þ are about 1.23–1.40, where r is air density. Bold lines in these plots denote zero damping curves, which shows the amplitude of response. In the plot of Fig. 12, which is the case under smooth flow and without the artificial rivulet, a galloping oscillation occurred around reduced wind velocity V =fD ¼ 70 and also Karman vortex-induced vibration was observed. However, a tendency of vortex-induced vibration in high reduced wind velocity (HSV) was never observed. Then, the rivulet was installed on the cable model at y ¼ 721 under smooth flow condition, and the results are shown in Fig. 13. The onset wind velocity of galloping becomes lower than the previous case (a); however the absolute values of negative damping are smaller than case (a). Furthermore, a faint tendency of HSV can be seen around V =fD ¼ 40250: Therefore, it might be considered that the onset velocity of galloping appears to be lower due to the effects of this vortex-induced vibration.
Fig. 12. Velocity–amplitude–damping diagrams of spring supported inclined cable model (in smooth flow, without rivulet, with 200 mm windows, b ¼ 451).
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Fig. 13. Velocity–amplitude–damping diagram of spring supported inclined cable model (in smooth flow, with rivulet y ¼ 721; with 200 mm windows, b ¼ 451).
Fig. 14. Velocity–amplitude–damping diagram of spring supported inclined cable model (in turbulent flow Iu=6.5%, without rivulet, with 200 mm windows, b ¼ 451).
In the turbulence case (Iu ¼ 6:5%), which is shown in Fig. 14, the galloping oscillation and Karman vortex-induced oscillation seem to be stabilized, however around V =fD ¼ 40250 and 80 there are tendency of this velocity-limited oscillation (HSV). In Fig. 15, a result of the case with rivulet under turbulence flow shows the response of vortex-induced vibration at V =fD ¼ 40: From the V –A–d diagram, velocity-limited instabilities can also be found around V =fD ¼ 60: Consequently, this vortex-induced vibration guides the galloping onset velocity to the lower velocity region. Therefore, from these results of free vibration tests it becomes clear that the water rivulet induces the velocity-limited instability, and a suitable combination of the rivulet and the wind turbulence might be significant factors for the aerodynamic instability.
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Fig. 15. Velocity–amplitude–damping diagram of spring supported inclined cable model (in turbulent flow Iu=6.5%, with rivulet y ¼ 721; with 200 mm windows, b ¼ 451).
There is still an unknown fact in this study. The results of full-scale measurements and elastic cable experiments in the wind tunnel usually show the velocity-limited response [3]; however, the rigid body experiments like these also show divergent vibration, which seems to be galloping. It seems that the strong interferences of vortices around the body in galloping instability might exist. To clear this problem, further investigations are needed.
6. Mechanism of inclined cable aerodynamic vibration The water rivulet on the upper surface of the inclined cable must be a key of the mechanism of rain–wind-induced vibration, which is considered as the vortexinduced vibration at high reduced velocity (HSV) in this study. The observational results at full-scale cable-stayed bridges indicate that this velocity-limited cable vibration often occurs under the precipitation condition. Also, the cable attitude often has a negative slope along the wind direction, which means wind comes from the bridge tower side with a certain horizontal yaw angle [3]. The position of the water rivulet must be determined by a wind velocity, a wind direction and a material of cable jacket. Then, if the water rivulet is formed at a certain position, which corresponds to around y¼ 6812751 in this study, the instability of this vortexinduced vibration is enhanced and cable vibration might occur. Also, the natural wind at the site of the bridges may have some turbulence, and this fact well agrees with a result of this study, in which the wind turbulence may enhance the instability of this cable vibration. The pressure field around and along a long inclined cable must be investigated further to understand the complicated mechanism of this vibration. Especially, the role of Karman vortex in this vibration must be considered. Then, the effects of the water rivulet and the wind turbulence should be clarified in detail. Also, the intensity
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of turbulence might be one of the parameters to control this vibration. Furthermore, the interaction between galloping instability and the Karman vortex should be explained.
7. Conclusions It becomes clear that the water rivulet on the upper surface of the cable and the wind turbulence play vital roles in the mechanism of the vortex-induced vibration at high reduced velocity. This fact well agrees with the full-scale observational data of rain–wind-induced vibration. Then, the rain–wind-induced vibration might be explained as the vortex-induced vibration at high reduced velocity. Also, the unsteady shedding of Karman vortices must be a significant factor to understand this phenomenon. Furthermore, the aerodynamic interaction between galloping instability and the Karman vortex should be clarified.
Acknowledgements This work was supported in part by a Grant-in-Aid for Scientific Research (A) from Japan Society for the Promotion of Science.
References [1] Y. Hikami, Rain vibrations of cables in cable-stayed bridge, J. Wind Eng. 27 (1986) 17–28 (in Japanese). [2] M. Matsumoto, N. Shiraishi, H. Shirato, Rain–wind-induced vibration of cables of cable-stayed bridges, J. Wind Eng. Ind. Aerodyn. 41–44 (1992) 2011–2022. [3] M. Matsumoto, Observed behavior of prototype cable vibration and its generation mechanism, in: A. Larsen, S. Esdahl (Eds.), Bridge Aerodynamics, Balkema, Rotterdam, 1998, pp. 189–211. [4] M. Matsumoto, T. Yagi, Y. Shigemura, D. Tsushima, Vortex-induced cable vibration of cable-stayed bridges at high reduced wind velocity, J. Wind Eng. Ind. Aerodyn. 89 (2001) 633–647. [5] J.A. Main, N.P. Jones, Full-scale measurements of stay cable vibration, Wind Engineering into the 21st Century, Proceedings of the Tenth International Conference on Wind Engineering, Copenhagen, Denmark, 1999, pp. 963–970.
Rain–wind-induced vibration of inclined cables at limited high reduced wind velocity region Masaru Matsumotoa,*, Tomomi Yagia, Mitsutaka Gotob, Seiichiro Sakaia a
Department of Global Environment Engineering, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto 606-8501, Japan b Tokyo Regional Bureau, Japan Railway Construction Public Corporation, 1-11-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-0021, Japan
Abstract Aerodynamic vibration of stayed cables, especially rain–wind-induced cable vibration, has been a rather serious problem on the design of cable-stayed bridges. This well-known phenomenon occurs at particular high reduced wind velocity regions and under raining condition. From results of wind tunnel tests, it might be explained as a vortex-induced vibration, which occurs at limited high reduced wind velocity region. Then, in this study, the effects of water rivulet and the wind turbulence on the vortex-induced vibration at high reduced wind velocity are investigated by wind tunnel tests, and the mechanism of the rain– wind-induced cable vibration is tried to be understood comprehensively. Furthermore, the role of Karman vortex in this vibration is discussed. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Rain–wind-induced vibration; Inclined cables; Vortex-induced vibration at high reduced wind velocity; Karman vortex; Water rivulet; Wind turbulence
1. Introduction Rain–wind-induced vibration of inclined cables has been a well-known phenomenon on cable-stayed bridges since the mid-1980s [1]. However, its generation mechanisms have not been entirely solved and its effective countermeasures *Corresponding author. Tel: +81-75-753-5091; fax: +81-75-761-0646. E-mail address: [email protected] (M. Matsumoto). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 3 3 1 - 8
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except additional dampers are still undeveloped. Therefore, to clarify the mechanism of this vibration is an urgent problem on the bridge engineering community. Matsumoto [2] pointed out that the axial flow behind inclined cables and the water rivulet formed on the upper side of cables play vital roles in rain–wind-induced vibration of inclined cables. The aerodynamic instabilities of inclined cables are roughly divided into galloping instability and vortex-induced oscillation. Also, the latter phenomenon can be divided into Karman vortex excitation and vortex-induced vibration at high reduced wind velocity. From observation results [3] at existent cable-stayed bridges, it is well known that rain–wind-induced vibration often occurs at the limited reduced wind velocity region around V =fD ¼ 40; 80 and so on, where V is the wind velocity, f is the vibration frequency and D is the diameter of cable. These reduced wind velocity regions are rather higher than the reciprocal of the Strouhal number of inclined cables. Then, Matsumoto [3] tried to explain the mechanism of this rain–wind-induced vibration as vortexinduced vibration at high reduced wind velocity by the interaction between Karman vortex and axial vortex. Furthermore, the strong three-dimensional characteristics of vortex shedding around the inclined circular cylinder were observed in the wind tunnel [4]. This vibration is sometimes observed under no precipitation condition [5], which is easily supposed from the results of wind tunnel tests mentioned above. However, many of the prototype cable vibrations have been observed under the raining condition. Also, the natural wind is generally a turbulent flow. Then, the effects of the water rivulet and wind turbulence on the vortex-induced vibration at high reduced wind velocity must be clarified. In this study, to understand the mechanism of this vibration, the effects of the water rivulet and wind turbulence are considered using wind tunnel tests. In this paper, the vortex-induced vibration at high reduced wind velocity is abbreviated as HSV, that is, high speed vortex-induced vibration.
2. Wind tunnel tests The wind tunnel used in this study is a room-circuit Eiffel type located at Kyoto University. The working section of the wind tunnel is 1.8 m high and 1.0 m wide, and the maximum wind velocity is 30 m/s. As an inclined cable model, a rigid circular cylinder with diameter D ¼ 54 mm, and length L ¼ 1500 mm was installed in the wind tunnel with horizontal yaw angle b=451. As shown in Fig. 1, the cylinder bored through the wall of the wind tunnel with holes of 100 or 200 mm diameter at both the upstream (left) and downstream (right) sides. These holes are called ‘‘windows’’ in this study and they were kept open to allow air to flow into the wind tunnel or out. This might prevent the axial flow from reducing its velocity. To realize the raining condition over the cable, an artificial water rivulet was installed at various angles from the stagnation point of the cable model, as shown in Fig. 2.
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Fig. 1. Top view of wind tunnel.
Fig. 2. Position and size of artificial water rivulet.
As the wind tunnel tests, the following measurements have been conducted: (1) The aerodynamic lift force of the stationary cable model was measured using load-cells at both ends of the model. (2) The fluctuating wind velocity in the wake of the stationary cable model was measured using a hot-wire anemometer, as shown in Fig. 3. This measurement was conducted along the cable model from the upstream side to downstream side. (3) The dynamic response and damping of the spring supported cable model were measured. The cable model was supported as one degree of freedom in the heaving direction by four springs. For all of these tests, the horizontal yaw angle of the cable model is fixed as b ¼ 451: The wind velocity of approaching flow varied from 0.5 to 10 m/s. Then,
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Fig. 3. Position of hot-wire anemometer.
Reynolds numbers for these tests are within 1800pRep36000 and belong to the subcritical range.
3. Aerodynamic lift force of stationary inclined cable model To understand the effects of water rivulet on driven force of vortex-induced vibration at high reduced wind velocity (HSV), the unsteady aerodynamic lift force on the stationary inclined cable model with the artificial water rivulet was measured. The lift force was measured using load-cell installed at both ends of the cylinder as mentioned above. Generally, the aerodynamic lift force on the inclined cable model is not uniformly distributed on the cylinder. Therefore, the lift forces measured at the upstream and at the downstream sides were analyzed separately, instead of evaluating the summing up data. The artificial water rivulet was installed on the stationary cable model with angles y ¼ 451; 601, 631, 681, 721, 751, 901, respectively. The horizontal yaw angle of the cable model is b ¼ 451; the windows (100 mm) were open and the mean wind velocity is V=4m/s under the smooth flow. The power spectrum densities (PSD) of unsteady lift force measured at the upstream side load-cell are plotted in Fig. 4. In this figure, a result of the case without the water rivulet is also plotted. There are two dominant frequency components: one is for Karman vortex excitation and the other is supposed as the driven force of vortex-induced vibration at high reduced wind velocity (HSV). These low frequency components fv correspond to the reduced wind velocity V =fvD ¼ 402180: When the water rivulet is located at y ¼ 681; 721, 751, the low frequency components of lift force are extremely large. Therefore, it seems that if the water rivulet is formed around y ¼ 721; HSV might easily occur. This result agrees with the fact that the cable vibration is usually observed under raining condition. Also, the PSD diagram of unsteady lift force measured at the downstream side is shown in Fig. 5. In this case, the low frequency components of aerodynamic lift force are never observed, but only Karman vortex components are dominant. Also, the PSD values for Karman vortex shedding at the downstream side are extremely larger than those at the upstream side. Therefore, it seems that the vortices formation at the upstream and downstream sides are totally different.
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Fig. 4. Power spectrum density of unsteady lift force on stationary cylinder with artificial rivulet (from load-cell at upstream side, with 100 mm windows, b ¼ 451; V ¼ 4 m=s; in smooth flow).
Fig. 5. Power spectrum density of unsteady lift force on stationary cylinder with artificial rivulet (from load-cell at downstream side, with 100 mm windows, b ¼ 451; V ¼ 4 m/s, in smooth flow).
4. Fluctuating wind velocity in the wake of stationary inclined cable model The fluctuating wind velocity in the wake of the stationary inclined cable model was measured along the model. The circular cylinder was mounted in the wind tunnel with horizontal yaw angle b ¼ 451: The hot-wire anemometer was set in the wake as shown in Fig. 3 and its position along the cable direction was varied from the upstream side to the downstream side. Then, the effects of water rivulet and wind turbulence on this fluctuating wind velocity were investigated. The power spectrum density (PSD) diagrams of fluctuating wind velocity in the wake along the cable direction are shown in Figs. 6 and 7. The mean wind velocity in the wind tunnel is fixed as V ¼ 6 m/s, and the window size at both the ends of model is 100 mm. The PSD diagram for a case measured under smooth flow is plotted in Fig. 6. The low frequency components, which may be supposed as axial vortex
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Fig. 6. Power spectrum density of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in smooth flow, without rivulet, with 100 mm windows, V ¼ 6:0 m/s, b ¼ 451).
Fig. 7. Power spectrum density of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in smooth flow, with rivulet y ¼ 721; with 100 mm windows, V ¼ 6:0 m/s, b ¼ 451).
components [4], are observed through span-wise direction. Especially at the upstream side, which corresponds to small X/D, theses low frequency components are extremely dominant. These frequency components correspond to reduced wind velocity around V =fvD ¼ 80; where fv is the dominating frequency of fluctuating velocity. This result is also well consistent with results from surface pressure measurements of the inclined cylinder [4]. To investigate the effects of the water rivulet, an artificial water rivulet was installed on the cylinder at y ¼ 721: The PSD diagram is shown as Fig. 7. It becomes clear that the water rivulet enhances the low frequency components at the mid-span and also the downstream side of the cable model. Therefore, the water rivulet may encourage the generation of vortex-induced vibration at high reduced wind velocity, and these results are also consistent with the results of the previous section. To discuss the turbulence effects on the fluctuating wind velocity, the wind turbulence Iu ¼ 6:5% was applied to this cable model without the rivulet. The experimental conditions are the same as the previous case, except that the mean wind velocity is V ¼ 4 m/s and window size at both ends of the cylinder is 200 mm. Also, the hot-wire anemometer was set at a very close position to the downstream side
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Fig. 8. Power spectrum density of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in smooth flow, without rivulet, with 200 mm windows, V ¼ 4:0 m/s, b ¼ 451).
Fig. 9. Power spectrum density of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in turbulent flow Iu=6.5%, without rivulet, with 200 mm windows, V ¼ 4:0 m/s, b ¼ 451).
wall, which was not measured in the previous case. The PSD diagram measured under smooth flow is plotted in Fig. 8 and the low turbulent case (Iu ¼ 6:5%) is shown in Fig. 9. The smooth flow case shows very similar characteristics to the previous case in Fig. 6, where the difference of experimental conditions are just the wind velocity and the window size. However, it becomes clear that the 6.5% turbulence flow extremely enhances the low frequency components of fluctuating wind velocity in the wake, especially at the upstream and the downstream sides. Therefore, it can be said that the wind turbulence may enhance remarkably the instability of this vortex-induced vibration at high reduced wind velocity. Furthermore, the wavelet analyses for both the cases were done, and the results for 5 s at the position X =D ¼ 3:0 are shown in Figs. 10 and 11. Morlet’s function was used as the mother function in the analyses and the wavelets are expressed by absolute values in these plots. From these plots, it is also clear that the low frequency components around V =fvD ¼ 40 exist strongly in the case of turbulent flow than in the smooth flow case. In the PSD analyses shown in Fig. 9, the low frequency components are just dominant. However, using these wavelet analyses, it becomes
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Fig. 10. Wavelet analyses of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in smooth flow, without rivulet, with 200 mm windows, V ¼ 4:0 m/s, b ¼ 451; X =D ¼ 3:0).
Fig. 11. Wavelet analyses of fluctuating wind velocity in the wake of stationary inclined circular cylinder, where fv denotes the dominating frequency of fluctuating wind velocity (in turbulent flow Iu ¼ 6:5%; without rivulet, with 200 mm windows, V ¼ 4:0 m/s, b ¼ 451; X =D ¼ 3:0).
clear that Karman vortices also exist unsteadily. This must be a very significant fact to realize the role of Karman vortex in this aerodynamic vibration.
5. Response of spring supported inclined cable model According to the above-mentioned results, the effects of water rivulet and wind turbulence might be essential factors of the vortex-induced vibration at high reduced
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wind velocity, which is usually observed as the rain–wind-induced vibration in many cable-stayed bridges. Then, these effects on the dynamic response of rigid inclined cylinder were examined by free vibration tests. The cable model was supported by four springs and allowed to move in the heaving direction. Also, its rolling motion was restricted by piano wires. The horizontal yaw angle of the model was b ¼ 451 and the diameter of windows on the wind tunnel wall was 200 mm at both the ends of the model. The cases discussed in this study are (a) in smooth flow without rivulet, (b) in smooth flow with rivulet at y ¼ 721; (c) in turbulent flow (Iu ¼ 6:5%) without rivulet and (d) in turbulent flow (Iu ¼ 6:5%) with rivulet at y ¼ 721: The velocity–amplitude–damping (V –A–d) diagrams for these 4 cases are plotted in Figs. 12–15. In these figures, 2A=D denotes the non-dimensional double amplitude of the heaving response, where A; D are the amplitude and diameter of the model, respectively. The natural frequencies of the system are f ¼ 1:6621:69 Hz, the masses per length are m ¼ 0:48920:526 kg/m and the logarithmic decrements at 2A=10 mm are d=0.0043–0.0050. Then, Scruton numbers defined by Sc ¼ 2md=ðrD2 Þ are about 1.23–1.40, where r is air density. Bold lines in these plots denote zero damping curves, which shows the amplitude of response. In the plot of Fig. 12, which is the case under smooth flow and without the artificial rivulet, a galloping oscillation occurred around reduced wind velocity V =fD ¼ 70 and also Karman vortex-induced vibration was observed. However, a tendency of vortex-induced vibration in high reduced wind velocity (HSV) was never observed. Then, the rivulet was installed on the cable model at y ¼ 721 under smooth flow condition, and the results are shown in Fig. 13. The onset wind velocity of galloping becomes lower than the previous case (a); however the absolute values of negative damping are smaller than case (a). Furthermore, a faint tendency of HSV can be seen around V =fD ¼ 40250: Therefore, it might be considered that the onset velocity of galloping appears to be lower due to the effects of this vortex-induced vibration.
Fig. 12. Velocity–amplitude–damping diagrams of spring supported inclined cable model (in smooth flow, without rivulet, with 200 mm windows, b ¼ 451).
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Fig. 13. Velocity–amplitude–damping diagram of spring supported inclined cable model (in smooth flow, with rivulet y ¼ 721; with 200 mm windows, b ¼ 451).
Fig. 14. Velocity–amplitude–damping diagram of spring supported inclined cable model (in turbulent flow Iu=6.5%, without rivulet, with 200 mm windows, b ¼ 451).
In the turbulence case (Iu ¼ 6:5%), which is shown in Fig. 14, the galloping oscillation and Karman vortex-induced oscillation seem to be stabilized, however around V =fD ¼ 40250 and 80 there are tendency of this velocity-limited oscillation (HSV). In Fig. 15, a result of the case with rivulet under turbulence flow shows the response of vortex-induced vibration at V =fD ¼ 40: From the V –A–d diagram, velocity-limited instabilities can also be found around V =fD ¼ 60: Consequently, this vortex-induced vibration guides the galloping onset velocity to the lower velocity region. Therefore, from these results of free vibration tests it becomes clear that the water rivulet induces the velocity-limited instability, and a suitable combination of the rivulet and the wind turbulence might be significant factors for the aerodynamic instability.
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Fig. 15. Velocity–amplitude–damping diagram of spring supported inclined cable model (in turbulent flow Iu=6.5%, with rivulet y ¼ 721; with 200 mm windows, b ¼ 451).
There is still an unknown fact in this study. The results of full-scale measurements and elastic cable experiments in the wind tunnel usually show the velocity-limited response [3]; however, the rigid body experiments like these also show divergent vibration, which seems to be galloping. It seems that the strong interferences of vortices around the body in galloping instability might exist. To clear this problem, further investigations are needed.
6. Mechanism of inclined cable aerodynamic vibration The water rivulet on the upper surface of the inclined cable must be a key of the mechanism of rain–wind-induced vibration, which is considered as the vortexinduced vibration at high reduced velocity (HSV) in this study. The observational results at full-scale cable-stayed bridges indicate that this velocity-limited cable vibration often occurs under the precipitation condition. Also, the cable attitude often has a negative slope along the wind direction, which means wind comes from the bridge tower side with a certain horizontal yaw angle [3]. The position of the water rivulet must be determined by a wind velocity, a wind direction and a material of cable jacket. Then, if the water rivulet is formed at a certain position, which corresponds to around y¼ 6812751 in this study, the instability of this vortexinduced vibration is enhanced and cable vibration might occur. Also, the natural wind at the site of the bridges may have some turbulence, and this fact well agrees with a result of this study, in which the wind turbulence may enhance the instability of this cable vibration. The pressure field around and along a long inclined cable must be investigated further to understand the complicated mechanism of this vibration. Especially, the role of Karman vortex in this vibration must be considered. Then, the effects of the water rivulet and the wind turbulence should be clarified in detail. Also, the intensity
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of turbulence might be one of the parameters to control this vibration. Furthermore, the interaction between galloping instability and the Karman vortex should be explained.
7. Conclusions It becomes clear that the water rivulet on the upper surface of the cable and the wind turbulence play vital roles in the mechanism of the vortex-induced vibration at high reduced velocity. This fact well agrees with the full-scale observational data of rain–wind-induced vibration. Then, the rain–wind-induced vibration might be explained as the vortex-induced vibration at high reduced velocity. Also, the unsteady shedding of Karman vortices must be a significant factor to understand this phenomenon. Furthermore, the aerodynamic interaction between galloping instability and the Karman vortex should be clarified.
Acknowledgements This work was supported in part by a Grant-in-Aid for Scientific Research (A) from Japan Society for the Promotion of Science.
References [1] Y. Hikami, Rain vibrations of cables in cable-stayed bridge, J. Wind Eng. 27 (1986) 17–28 (in Japanese). [2] M. Matsumoto, N. Shiraishi, H. Shirato, Rain–wind-induced vibration of cables of cable-stayed bridges, J. Wind Eng. Ind. Aerodyn. 41–44 (1992) 2011–2022. [3] M. Matsumoto, Observed behavior of prototype cable vibration and its generation mechanism, in: A. Larsen, S. Esdahl (Eds.), Bridge Aerodynamics, Balkema, Rotterdam, 1998, pp. 189–211. [4] M. Matsumoto, T. Yagi, Y. Shigemura, D. Tsushima, Vortex-induced cable vibration of cable-stayed bridges at high reduced wind velocity, J. Wind Eng. Ind. Aerodyn. 89 (2001) 633–647. [5] J.A. Main, N.P. Jones, Full-scale measurements of stay cable vibration, Wind Engineering into the 21st Century, Proceedings of the Tenth International Conference on Wind Engineering, Copenhagen, Denmark, 1999, pp. 963–970.