# A Field Investigation on Vortex-Induced Vibrations of Stay Cables in a Cable-Stayed Bridge
## 4 Correlation between Cable Response and Wind Characteristics
### Estimation Method of the Participative Modes of VIVs  
The mean wind speed profile acting on the stay cables was not uniform, because the investigated cable-stayed bridge was constructed in the atmospheric boundary layer. According to the log law, the mean wind speed profile in the atmospheric boundary layer can be defined as the following:  
\[U(z)=U(z_{0})\left(\frac{z}{z_{0}}\right)^{\alpha} \tag{2}\]  
where \(z_{0}\) is the reference height above the ground, \(z\) is the arbitrary height above the ground, \(U(z_{0})\) is the mean wind speed at the height \(z_{0}\), \(U(z)\) is the mean wind speed at the height, \(z\), and \(\alpha\) is the terrain roughness index, equal to 0.12, 0.16, 0.22, and 0.30 for A, B, C, and D terrains, respectively, according to the Chinese Standard GB 5009-2001 (National Standard of the People's Republic of China, 2001 version, see [16]). According to the wind data measured by the anemometers installed at the tower top and bridge deck, the correlation of the mean wind speeds between the tower top and deck level of the bridge was obtained, as shown in Figure 14. The mean wind speed on the tower top is nearly 1.16 times that on the bridge deck level, according to the fitting line. Figure 14: Correlation between mean wind speeds at tower top and deck level of the bridge. Figure 13: Vibration amplitude versus turbulence intensity in April. In the present study, the height of the deck level is 59.0 m above the ground, and the corresponding height of the reference height \(z_{0}\) (i.e., the deck anemometer installing height) is 65.0 m above the ground. The height of the tower top is 151.0 m above the bridge deck, and the corresponding height, \(z\), of the anemometer of the tower top is 212.1 m, with the anemometer 2.1 m above the tower top. According to Equation (2), the terrain roughness index, \(\alpha\), can be calculated as 0.125, which is close to 0.12 because the investigated bridge is constructed near the sea; therefore, the terrain of the bridge is close to a type of A terrain. Next, according to \(U(z_{0})\) by the anemometer at the deck level, the mean wind speed at an arbitrary height can be calculated by Equation (2). Actually, the wind is not perpendicularly acting on the stay cable plane, as shown in Figure 15, and the yaw angle of the stay cable plane is \(\beta\). The inclination angle of the stay cable is \(\theta\). The incoming velocity is decomposed into two components that are perpendicular to the cable spanwise direction: \(U(z)\cos\beta\) and \(U(z)\sin\beta\sin\theta\). The axial flow component \(U(z)\sin\beta\cos\theta\) is neglected in the present study. The relative velocity \(U_{0}(z)\) and the attack of angle \(\alpha_{0}\) are subsequently obtained, as follows:  
\[U_{0}(z)=U(z)\sqrt{\cos^{2}\beta+\sin^{2}\beta\sin^{2}\theta} \tag{3}\]  
\[\alpha_{0}=\arctan(\tan\beta\sin\theta) \tag{4}\]  
Because the Reynolds number varies at different heights of the wind velocity profile, the Strouhal number, \(\text{St}(z)\), can be estimated by using the following formula [17]:  
\[ \begin{array}{l}\text{St}(z)=0.1853+0.0261\times\exp\left(-0.9\times L_{R} {}^{2.3}\right)\\ L_{R}=\log\left(\text{Re}(z)/1.6\times 10^{3}\right)\\ \text{Re}(z)=\frac{U_{0}(z)D(z)}{v}\end{array}  \tag{5}\]  
where, \(D(z)\) is the cable diameter at the height of z (in this study, it is constant along the cables), \(\text{Re}(z)\) is the Reynolds number depending on z, \(v\) is the coefficient of kinematic viscosity, and \(U_{0}(z)\) is the incoming relative velocity. The vortex-shedding frequency, \(f_{\text{s}}(z)\), of a long cable can be expressed as the following:  
\[f_{\text{s}}(z)=\text{St}(z)\frac{U_{0}(z)}{D(z)} \tag{6}\]  
According to the velocities at the upper and lower ends of the stay cable, the vortex-shedding frequencies of two ends of the stay cable, \(f_{\text{SL}}\) and \(f_{\text{SU}}\), can be calculated from Equation (6).