Effects of damping and helical fillets on galloping vibration of stay cable installed with a rectangular lamp

Abstract

Lamps attached on the stay cable can change the stable circular cross-section of the stay cable, which is easy to cause galloping vibration. Stay cables of the Kuipu bridge in the Fuzhou City of China were equipped with damper and helical fillets. After the installation of rectangular lamps on this bridge, no obvious vibration of the stay cables was observed. This is totally different from another bridge, the Kuimen bridge in the Chongqing City of China, which has been recently reported by the authors. The purpose of this study is to clarify the effects of the helical fillets and the structural damping on galloping vibration. The features of galloping vibration of the stay cable attached to a rectangular lamp were carefully studied, and the effectiveness of damping and helical fillets in reducing the galloping vibration was evaluated. First, the force measurement wind tunnel tests on the segmental test model were carried out, and the lift and drag coefficients of the stay cable attached to the rectangular lamp were measured. The dangerous wind attack angle of galloping vibration was predicted by the galloping force coefficient. Second, 3-dimensional vibration measurement wind tunnel tests were carried out to measure the wind-induced response of the cable-lamp test models. The variation of wind-induced response of stay cable with yaw angle and wind velocity was measured for test models with and without helical fillets. In the test, the features of wind-induced vibration were studied at the damping level of 0.1%, 0.6%, 0.7%, 0.8%, and 1.0%. According to Den Hartog’s theory, the minimum coefficient of galloping force is about −9.8. The experimental results of 3-dimensional vibration measurement wind tunnel tests show that the helical fillets wrapped on the cable surface can enhance the critical wind velocity to a certain extent, but the critical wind velocity is still far lower than the design wind velocity. When the damping ratio of the stay cable is up to 1.0% (S_C = 35.1), the galloping vibration of the stay cable attached to a rectangular lamp on the Kuipu Bridge can be effectively mitigated.

Keywords

Galloping wind tunnel tests stay cables rectangular lamps damping ratio helical fillets lighting projects

Introduction

The lighting lamps installed on stay cables can change the symmetrical and stable circular cross-section of the stay cable, which is easy to cause galloping vibration. After the installation of rectangular lamps, the stay cables of the Kuimen bridge in the Chongqing City of China experienced a large amplitude of galloping vibration in 2019 (An et al., 2021). An et al. (2021) reproduced the galloping vibration of the stay cable in wind tunnel tests and found that the galloping critical wind velocity of the first mode of the longest stay cable was as low as 6.3 m/s. Li et al. (2014) studied the influence of a circular-hoop lamp on the vibration of stay cables of the Hedong Bridge in Guangzhou City of China. The results show that the critical wind velocity of galloping was only 18 m/s, which was far lower than the design wind velocity of Guangzhou. The improvement suggestions for the design scheme were put forward to effectively suppress the galloping vibration. Deng et al. (2021) studied the aerodynamic coefficients of two-dimensional stay cables with rectangular lamps by means of CFD and wind tunnel tests, and the wind attack angle range of galloping was predicted based on Den Hartog’s criterion.

Den Hartog’s theory (Den Hartog, 1932, 1956) was usually used to analyze the responses of galloping vibration and has plenty of successful examples of application in the studies of dry cables (Cheng et al., 2008a, 2008b; Macdonald and Larose, 2006, 2008a, 2008b; Matsumoto et al., 2010; Nikitas and Macdonald, 2014), iced conductors (Chabart and Lilien, 1998; Kim and Sohn, 2018) and bluff body sections (Alonso et al., 2005; Li et al., 2022; Novak, 1971, 1972; Païdoussis et al., 2011; Parkinson and Smith, 1964; Piccardo et al.,2011, 2015; Zuo et al., 2017). Some researchers (Bearman et al., 1987; Blevins, 1977, 1990; Fung, 1955) pay attention to the applicability of the quasi-steady assumption, and they believed that the reduced wind velocity should be higher than 10 (circular cylinders) or 30 (square cylinders). Under some specific circumstances, for example at the critical Reynolds number (Ma et al., 2017a, 2017b), or for low-speed galloping (Chen et al., 2017, 2021; Gao and Zhu, 2017; Mannini et al., 2014, 2018), the quasi-steady assumption might be disabled to modeling the galloping vibration. Another method of studying the features of galloping vibration is wind tunnel tests, which can accurately capture the galloping response without worrying about whether the quasi-steady assumption applies.

Due to the large slenderness ratio and low damping, stay cable is sensitively prone to various types of wind-induced vibrations, such as dry galloping (DG) (Cheng et al., 2008a, 2008b; Matsumoto et al., 2010), ice galloping (IG) (Foti et al., 2017; Li et al., 2016a), vortex-induced vibration (VIV) (Chen et al., 2015; Li et al., 2021; Liu et al., 2021) and rain-wind-induced vibration (RWIV) (Hikami and Shiraishi, 1988; Jing et al. 2017; Li et al., 2016b). Installing a damper at the end of the stay cable is a successful countermeasure to suppress the wind-induced vibrations of the stay cable. The results of wind tunnel tests show that the higher-order VIV of stay cables can be effectively suppressed if the damping ratio is up to 0.48% (Liu et al., 2021). The damping ratio of 0.5% is enough to suppress the RWIV of the stay cable (Gu and Du, 2005; Li et al., 2016b). However, Hua et al. (2020) found that increasing the damping ratio to 3.2% cannot effectively suppress the galloping of the main cable of a suspension bridge during construction. An et al. (2021) studied the galloping of stay cables with rectangular lamps and found that the critical wind velocity of galloping vibration is still far lower than the design wind velocity when the damping ratio is up to 0.6%. The minimum damping required to suppress wind-induced vibrations varies for different cross sections and different types of vibrations. Determining the minimum damping to suppress galloping vibration is of great importance for the design of the damper.

The early applications of helical fillets on stay cables were the Normandy Bridge (Flamand, 1995) and the Øresund Bridge (Larose and Smitt, 1999), which were used to suppress the RWIV of the stay cable. After 30 years of practice, the effectiveness of helical fillets in mitigating the RWIV has been proved. Furthermore, the helical fillet attached to the stay cable is also useful to reduce the vortex-induced vibration (Liu et al., 2021). However, some researchers found that the helical fillet on the stay cable has an adverse effect on the DG (Christiansen et al., 2018a, 2018b). Until now, the effectiveness of the helical fillet on the galloping vibration of stay cable has been seldom studied.

In the May of 2019, a large amplitude of cable vibration of the Kuimen bridge was observed after installing rectangular lamps on the stay cables. An et al. (2021) successfully reproduced the vibration of the stay cables by using wind tunnel tests and found the mechanism of the cable vibration is galloping. In the August of 2020, similar rectangular lamps as the Kuimen bridge were also installed on the stay cables of the Kuipu bridge. But no obvious vibration of the stay cables of this bridge was observed. Different from the Kuimen bridge, there are helical fillets on the surface and dampers at the end of the stay cables of the Kuipu bridge. These might be the main reason for the different vibrations of the two bridges.

The purpose of this study is to clarify the effects of the helical fillets and the structural damping on galloping vibration. The features of galloping vibration of the stay cable attached to a rectangular lamp were carefully studied, and the effectiveness of damping and helical fillets in reducing the galloping vibration was evaluated. First, the force measurement wind tunnel tests on the segmental test model were carried out, and the lift and drag coefficients of the stay cable attached to the rectangular lamp and helical fillet were measured. The dangerous wind attack angle of galloping vibration was predicted by the galloping force coefficient. Second, 3-dimensional vibration measurement wind tunnel tests were carried out to measure the wind-induced response of the cable-lamp test models. The variation of wind-induced response of stay cable with yaw angle and wind velocity was measured for test models with and without helical fillets. In the test, the features of wind-induced vibration were studied at the damping level of 0.1% (S_C = 3.51), 0.6% (S_C = 21.06), 0.7% (S_C = 24.57), 0.8% (S_C = 28.08), and 1.0% (S_C = 35.1).

Outline of the Kuipu bridge and the lamp installation

The Kuipu Bridge, which is located in Fuzhou City, Fujian Province, China, is taken as the background of this study. This bridge is a cable-stayed bridge with two pylons and two cable planes symmetrically arranged. There are 136 stay cables in total. The elevation of the Kuipu Bridge is shown in Figure 1, together with the cable numbers. In order to suppress the RWIV response of the stay cable, a helical fillet with a diameter of 5 mm and a pitch of 6-8D (D is the diameter of the stay cable) is arranged. The diameter of stay cables varies from 124 mm to 162 mm. The range of the inclination angle of stay cables is from 35° to 73°. The length of stay cables varies from 48.7 m to 142.5 m. The detailed parameters of stay cables are shown in Table 1. In the August of 2020, a lighting project was launched to install lamps on the Kuipu Bridge, including the lamps on stay cables. The lamps and their electrical wires were accommodated by a rectangular aluminum alloy box along the cable axis. The lamps had a circular shape with a diameter of 50 mm, and they were uniformly installed on the facade of the rectangular box with five lamps per meter, as shown in Figure 2. Here, we name the system including the rectangular box, electrical wires, and lamps as “rectangular lamp”. The cross-section of the rectangular lamp is 64 mm by 48 mm, as shown in Figure 3. There is a small radian at the corner of the rectangle. The arc length is 2.36 mm, and the radius of the arc is 1.5 mm.

Figure 1.

Elevation of the Kuipu Bridge (unit: m).

Table 1.

Detailed information of the stay cables of the Kuipu Bridge.

Number	Diameter D (mm)	Inclination angle θ (°)	Mass (kg/m)	Length (m)	Frequency f (Hz)	S _c
S1	124	72.76	63.70	48.65	2.65	3.43
S2	121	66.25	60.00	54.11	2.29	3.40
S3	121	60.79	60.00	59.70	2.10	3.40
S4	121	56.21	60.00	65.55	1.97	3.40
S5	121	52.37	60.00	71.65	1.85	3.40
S6	124	49.14	63.70	77.99	1.68	3.44
S7	124	46.39	63.70	84.55	1.56	3.44
S8	124	44.03	63.70	91.22	1.44	3.44
S9	133	42.03	71.70	98.02	1.30	3.36
S10	133	40.04	71.70	104.60	1.25	3.36
S11	135	38.31	76.90	111.29	1.16	3.50
S12	135	37.47	76.90	119.06	1.11	3.50
S13	147	36.09	90.40	125.86	0.98	3.47
S14	147	36.18	90.40	130.41	1.00	3.47
S15	147	35.81	90.40	134.22	1.00	3.47
S16	162	35.68	111.20	138.37	0.93	3.52
S17	162	35.51	111.20	142.46	0.92	3.52

Figure 2.

Photos of stay cable attached with a rectangular lamp.

Figure 3.

Dimensions of rectangular lamp for prototype stay cables (unit: mm).

Scruton number is defined as follows

S_{C} = \frac{m ζ}{ρ D^{2}}

(1)

where m is the mass per unit length of the cylinder; ζ is the structural damping; ρ is the air density and D is the diameter of the cylinder. The values of the Scruton number are also listed in Table 1. It can be found that all values of the stay cables on the Kuipu bridge are as low as 3.50 if the damping ratio is 0.1%.

The fundamental frequency of Cable S17 is the smallest one among all the cables. This indicates that the galloping critical wind velocity of Cable S17 should be the lowest one. Furthermore, it appears that the inclination angle has little effect on galloping critical wind velocity according to An et al. (2021). Therefore, Cable S17 was taken as the background of this study.

Outline of wind tunnel tests

Both force and vibration measurement wind tunnel tests were conducted in the high-velocity test section of the HD-2 Boundary Layer Wind Tunnel (HD-2BLWT) at Hunan University, Changsha, China. The HD-2BLWT is a closed-circuit-type wind tunnel. The dimension of the test section is 3 m (width) × 2.5 m (height) ×17 m (length) and the wind velocity can be varied continuously up to 58 m/s. The longitudinal turbulence intensity in the bare tunnel is less than 0.5% for U >2 m/s.

Test models and test cases for force measurement

The geometric scale ratio of the force measurement test model is 1:1. The length and diameter of the cable model are respectively 1200 mm and 162 mm, and the pitch of the helical fillets is 7.4D (D is the diameter of the cable). The cable model was installed vertically at the center of the turntable. The High-frequency force balance (HFFB) used to measure the aerodynamic forces were connected to the bottom of the cable model. To eliminate the end effect, a circular end plate with a diameter of 500 mm was arranged at the upper end of the model. The circular end plate was fixed on a steel pipe protruding from the roof of the wind tunnel. The gap between the end plate and the upper-end face of the test model is about 3 mm. Figure 4 shows the photo of the test model installed in the test section.

Figure 4.

Photo of the test model installed in the test section.

The relationship between the flow and the test model was defined by wind attack angle α, as shown in Figure 5. The wind attack angle α of the incoming wind can be changed by rotating the turntable. Uniform flow with a wind velocity of 9 m/s was adopted in the force measurement wind tunnel tests. The aerodynamic forces were measured in the range of α = 0°–360° with an interval of 2°. The sampling frequency of the HFFB was 1000 Hz and the sampling time was 30 s.

Figure 5.

Definition of wind attack angle α.

Only the average values of the aerodynamic forces were considered in this study. The mean forces directly measured by the HFFB are F_X and F_Y, which are along the X direction and Y direction of the balance (body axis). The relationship between the mean forces F_D and F_L in the wind axis and the mean forces directly measured by the balance in the body axis is

F_{D} = F_{X} \cos α + F_{Y} \sin α

(2)

F_{L} = F_{Y} \cos α - F_{X} \sin α

(3)

and the mean drag and lift coefficients, C_D and C_L, can be obtained from

C_{D} = \frac{F_{D}}{0.5 ρ U^{2} D L}

(4)

C_{L} = \frac{F_{L}}{0.5 ρ U^{2} DL}

(5)

in which, U is the mean wind velocity of the approaching flow; D and L are the diameter and the length of the test model, respectively; ρ is the density of air. In consideration of the effect of the boundary layer near the bottom floor, the aerodynamic coefficients are numerically modified. It is worth noting that the clockwise rotation of the flow around the model is positive in this paper, which means that the upward movement of the model is positive, contrary to the model definition in some articles (An et al., 2021; Deng et al., 2021). The coefficient of galloping force is defined as follows

C_{g} = C_{D} - \frac{d C_{L}}{d α}

(6)

According to Den Hartog’s criterion (Den Hartog, 1932), the galloping maybe take place if the galloping force coefficient C_g < 0.

Test models and test cases for vibration measurement

Elastic suspension of rigid segment model was used in vibration test. The main structural parameters such as frequency, mass, damping, etc., should be carefully simulated in the test model. In the test, three dimensionless parameters, inertia parameter(m/ρD²), elastic parameter (U/f D), and damping ratio were simulated. A geometric scale ratio of less than 1:1 was adopted in the design of the segmental test model, which has been discussed in An et al. (2021). The target stay cable was Cable S17 with a diameter of 162 mm, a mass per unit length of 111.2 kg/m, and a fundamental frequency of 0.92 Hz. The geometric scale was 1:3.24, and the diameter of the cable model was 50 mm. The detailed similarity scales of the test model were given in Table 2. In Table 2, the number of S_C is calculated with a damping of 0.1%. The length of the model was 1500 mm. Two models, which were with and without the helical fillets, were designed in the vibration tests to study the effects of the helical fillets. The parameters of the two models were the same except for the helical fillets. The models wrapped with helical fillets were consistent with the prototype of the stay cable on the Kuipu Bridge. The diameter of the helical fillets of the test model was 1.5 mm. The two vibration measurement models were shown in Figure 6.

Table 2.

Similarity scales of the elastic test models.

Parameters	Test model	Prototype	Scales
D (mm)	50	162	1:3.24
f (Hz)	1.8	0.92	1.96:1
U (m/s)	3.3–31.8	5–53	1:1.65
m (kg/m)	10.59	111.2	1:3.24²
ζ	0.1%	0.1%	1:1
Sc	3.51	3.52	1:1

Figure 6.

Models with and without helical fillets.

A 3-dimensional test model of stay cable was adopted to carry out the vibration measurement wind tunnel tests. The orientation of the 3-dimensional stay cable can be defined by three angles φ, β, and θ, as indicated in Figure 7. φ represents the spatial relationship between the cable and the rectangular lamp, and a sketch map of φ = 0° is also shown in Figure 7, in which U₀ is the component of the approaching wind velocity U perpendicular to the cable plane.

Figure 7.

Definition of inclination angle θ, yaw angle β and position of lamp to cable φ.

θ is the inclination angle of the stay cable, β is the yaw angle of the cable plane relative to the approaching flow, and ω is the cable-wind relative angle, as shown in Figure 7. The yaw angle (β) of 0° is defined as the freestream wind perpendicular to the cable plane. φ represents the spatial relationship between the cable and the rectangular lamp, and a sketch map of φ = 0° is also shown in Figure 7. The incoming wind velocity is parallel to the OX axis. The freestream wind velocity (U) has a component that is normal to the cable axis but still aligned in the cable-wind plane (U_N = Usinω).

The cable-wind angle (ω) and the angle between the vertical cable plane and the wind component normal to the cable axis (α_N) are calculated using equations (7) and (8).

\cos ω = \sin β \cos θ

(7)

\cot α_{N} = \tan β \sin θ

(8)

It should be noted that the in-plane vibration direction refers to the direction perpendicular to the cable axis in the cable plane, and the out-of-plane vibration direction refers to the direction perpendicular to the cable plane.

To conveniently adjust the yaw angle β and the inclination angle θ in the vibration measurement wind tunnel tests, a support system was specially designed, as shown in Figure 8. The inclination angle θ of the stay cable can be changed by changing the fixed points on the four columns. The support system was installed on the turntable of the high-velocity test section of the HD-2BLWT, and the yaw angle β can be conveniently adjusted by the turntable. A photo of the test model installed on the support system in the wind tunnel is given in Figure 8.

Figure 8.

Photo of the test model installed on the support system in the wind tunnel.

The stiffness of the elastic test model was provided by eight springs connected to the two ends of the test model. The natural frequency and damping ratio of the elastic test model were identified by the method of free vibration decay. The damping ratio of the test model can be increased by pasting adhesive tapes on the springs. Five levels of damping ratio, including 0.1% (S_C = 3.51), 0.6% (S_C = 21.06), 0.7% (S_C = 24.57), 0.8% (S_C = 28.08), and 1.0% (S_C = 35.1), were obtained, and the acceleration time histories and the corresponding Fast Fourier Transforms (FFT) are shown in Figure 9.

Figure 9.

Time histories and its corresponding FFTs of the decrement displacement for different levels of structural damping: (a) ζ = 0.1%; (b) ζ = 0.6%; (c) ζ = 0.7%; (d) ζ = 0.8%; (e) ζ = 1.0%.

There are four types of test cases in the vibration measurement wind tunnel tests. All test cases are carried out in a uniform flow. In test case 1, the inclination angle of the test model with helical fillets was θ = 36° and the position angle φ = 0° the yaw angle β was changed from 0° to 360° with an interval of 2°. The test wind velocity in test case1 was always 10 m/s, and the damping ratio of the test model was 0.1%. The test parameters of test cases 1 and 2 were the same as each other, except for no helical fillets on the test model of test case 2. The dangerous range of yaw angle β could be found in test cases 1 and 2. In test cases 3 and 4, the yaw angle was fixed in the dangerous yaw angle β (330°–334°) to obtain the effects of wind velocity and damping ratio. The wind velocity in test cases 3 and 4 varied from 5.0 to 32.0 m/s and the damping ratios were 0.1%–1.0%. The detailed information on the four types of test cases was shown in Table 3.

Table 3.

Test cases in the vibration measurement wind tunnel tests.

Test cases	Models	θ (°)	β (°)	ζ (%)	U (m/s)
1	Model with helical fillets	36	0–360, Δβ = 2	0.1	10.0
2	Model without helical fillets	36	0–360, Δβ = 2	0.1	10.0
3	Model with helical fillets	36	334	0.1, 0.7, 1.0	5.0–32.0
4	Model without helical fillets	36	334	0.1, 0.6, 0.8, 1.0	5.0–32.0

Experimental results for force measurement

Figure 10 presents the variation of aerodynamic coefficients with wind attack angle for the test model and the comparison of aerodynamic forces obtained from different articles. The cross-section dimensions in different articles are also shown in Figure 10. Although the specific sizes are different, the cross-section forms are basically the same, and the comparison results are convincing. It can be found from Figure 10 that the maximum drag coefficient is 1.4 and the maximum lift coefficient is 0.6. An et al. (2021) found that the maximum drag coefficient of stay cable without helical fillets is 2.6 and the maximum lift coefficient is 1.38. Deng et al. (2021) calculated that the maximum drag coefficient of stay cable without helical fillets is about 3.0 and the maximum lift coefficient is about 1.3 as shown in Figure 10. It is enough to indicate that the helical fillets can reduce the drag and lift coefficients of the stay cable attached with rectangular lamp. It should be noted that there are obvious sudden decreases in the mean lift force. Figure 11 presents the variation of the galloping force coefficient C_g with wind attack angle α. Table 4 shows the range of wind attack angle for the galloping force coefficient C_g < 0, together with the minimum value of C_g and the corresponding wind attack angle. It can be found in Figure 11 and Table.4 that the maximum negative galloping force coefficient appears at α = 344°and α = 16°, and the maximum negative values of C_g are −9.8 and 9.7, respectively. This indicates the possibility of galloping vibration maybe take place near these wind attack angles. Compared with the results of stay cable without helical fillets obtained by An et al. (2021), the maximum negative value of C_g listed in Table 4 is a little smaller than those without helical fillets (−12.9). This shows that helical fillets can play a minor role in mitigating the galloping vibration of the stay cable attached to a rectangular lamp.

Figure 10.

Variation of mean drag and lift coefficients with wind attack angle.

Figure 11.

Variation of galloping force coefficients with wind attack angle.

Table 4.

Minimum galloping force coefficient and its corresponding wind attack angle.

Wind attack angles for C_g < 0 (°)	Minimum C_g	α for minimum C_g
14, 16, 26, 30–34, 40, 66–70, 206, 286, 290, 294, 312, 320–322, 342–346	−9.8 (−9.7)	344° (16°)

Experimental results for vibration measurement

Effect of yaw angle

Figures 12 and 13 present the variations of the amplitude of the inclined test model with and without the helical fillets in the in-plane and out-of-plane directions for U = 10 m/s, ζ = 0.1%, and θ = 36°, which are respectively corresponding to test cases 1 and 2 as given in Table 3. It can be found from Figure 12 that the dangerous yaw angle of the galloping vibration of the test model with the helical fillets is within the range from 332° to 336°. However, the dangerous yaw angle is within 26°–30° and 332°–336° for the test model without the helical fillets. This may be due to the helical fillets, where the critical wind velocity is increased at a specific yaw angle of 26–30°. It also can be found in Figures 12 and 13 that both in-plane and out-of-plane amplitudes are significant under dangerous yaw angles. The largest vibration amplitude occurs under the yaw angle of 334°. This indicates that the critical wind velocity under this yaw angle might be the lowest one. Therefore, the variation of galloping vibration displacement with reduced wind velocity was measured near the yaw angle of 334°, which is corresponding to test cases 3 and 4 as listed in Table 3.

Figure 12.

Variation of the amplitude of the test model with helical fillets with the yaw angle in the in-plane and out-of-plane directions.

Figure 13.

Variation of the amplitude of the test model without helical fillets with the yaw angle in the in-plane and out-of-plane directions.

When the most dangerous yaw angle β is 334°, it can be calculated from equation (8) that α_N is 106°. With the definition in Figure 5, the wind attack angle is 344°. According to the force test results, the most dangerous wind attack angle is 344°. This shows that Den Hartog’s criterion successfully predicts the most dangerous angle of galloping. It is worth noting that there was no vibration in the vibration measurement test near the wind attack angle of 16° predicted by Den Hartog’s criterion. This may be because that Den Hartog’s criterion is a necessary condition for galloping to occur, not a sufficient condition. It may also be because the test wind speed is not high enough. It may also be caused by the different aerodynamic forces of the 2-dimensional model and the 3-dimensional model, which requires further research.

Effect of helical fillets

Figure 14 shows the variation of the amplitude of the inclined test model with and without helical fillets in the in-plane and out-of-plane directions with the wind velocity, which are respectively corresponding to test cases 3 and 4 as given in Table 3. It can be found from Figure 14 that the test model without helical fillets starts to vibrate at the reduced wind velocity of 80.5 (11.9 m/s of the real bridge). It seems that the vibration amplitude increases linearly with the increase of the reduced wind velocity. The galloping critical wind velocity of the model with helical fillets is 134.4 (20.0 m/s of the real bridge), which is significantly higher than that without the helical fillets. It appears that the helical fillets on the surface of the stay cable can significantly increase the galloping critical wind velocity.

Figure 14.

Variation of the amplitude of the test models with the reduced wind velocity.

Effect of damping

Figure 15 presents the variation of the amplitude of the test model with and without the helical fillets with the reduced wind velocity for different levels of damping ratios when yaw angle β = 334°. It can be found from Figure 15 that the galloping critical wind velocity of the test model with and without the helical fillets are both higher than 340 (50.6 m/s of the real bridge) when the damping ratio is up to 1.0%, and the corresponding S_C is 35.1. This indicates that the galloping vibration of the Cable S17 of the Kuipu Bridge can be successfully mitigated by increasing the damping ratio up to 1.0%.

Figure 15.

Variation of the amplitude of the test models with the reduced wind velocity at different damping (β = 334°) (a) With helical fillets (b) Without helical fillets.

Concluding remarks

Taking the Kuipu Bridge in the Fuzhou City of China as the background, the galloping vibration, and its mitigation of the stay cable attached to the rectangular lamp was studied in detail. First, the aerodynamic coefficients were measured by wind tunnel tests. The maximum drag coefficient is 1.4 and the maximum lift coefficient is 0.6. It indicates that the helical fillets can reduce the drag and lift coefficients of the stay cable attached to a rectangular lamp. The possibility of galloping is predicted by using the galloping force coefficient. According to Den Hartog’s theory, the most unfavorable wind attack angle is 344° and the corresponding coefficient of galloping force is as low as −9.8. Second, the effects of yaw angle, damping, and helical fillets on galloping were studied through a three-dimensional vibration measurement wind tunnel test. When the yaw angle is equal to 334°, the critical wind velocity is the lowest. When the damping ratio is 0.1% (S_C = 3.51), the critical wind velocity of the model without helical fillets is as low as 11.9 m/s at the yaw angle of 334°. The galloping critical wind velocity of the model with helical fillets is 20.0 m/s, which is higher than that without the helical fillets. It appears that the helical fillets on the surface of the stay cable can increase the galloping critical wind velocity, but the critical wind velocity is far from the design wind velocity. The damping ratio of 1% (S_C = 35.1) can effectively control the in-plane and out-of-plane vibration of stay cables with and without helical fillets.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project is jointly supported by the National Key Research and Development Program of China (Grant No.2017YFC0703600 and No.2017YFC0703604), the National Natural Science Foundation of China (Grant No.51578234) and Postgraduate Scientific Research Innovation Project of Hunan Province (Grant No.CX20210411).

Data availability

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

ORCID iD

Shouying Li

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