## 5 Aeolian Vibration and Its Controlling  
### Equivalent Lift Force of Vortex-Induced Vibration  
Generally, the FE model of the full structure was directly created and assessed for its mechanical behavior. Compared with axial tensile stress, minor bending stress was always ignored in the flexible and long-span cable. Thus, the two-force member, such as the link element without considering bending, was appropriate for modeling overhead transmission lines. As it is similar to conventional cable-supported structures, components of the long-span CRS, including conductors, insulators, cables and connectors, were meshed by a two-node link element. The external wind force was subjected to all nodes of the above components (Figure 20). Wind load effect during VIV was simplified and equivalent to lift force, as the following expression [11]:  
\[F_{L}(t)=\frac{1}{2}\rho DV^{2}C_{L}(t) \tag{6}\]  
where \(\rho\) is air density (kg/m\({}^{3}\)), \(V\) is wind speed (m/s), \(C_{L}(t)\) is equivalent sectional lift coefficient related to time history \(t\). Figure 19: The influence of number of spans of conductor on maximum horizontal displacement of conductor (wind speed: 10 m/s). Figure 20: Typical wind lift loading mode in FE model of CRS (2 spans). (**a**) Conductor, (**b**) cable. The vortex-induced force was expressed as the function of air density, mean wind speed, the diameter of the cross-section and corresponding aerodynamic parameters [22; 23]. Assuming that vortex shedding is an essentially sinusoidal process, the sectional lift coefficient was expressed as:  
\[C_{L}(t)=\sqrt{2}C_{L^{\prime}}\sin(2\pi f_{w}t) \tag{7}\]  
where \(f_{w}\) is the frequency of vortex shedding (Hz), \(C_{L^{\prime}}\) is R.M.S. (root mean square) lift coefficient, which was approximated the calculation as below [24]:  
\[C_{L^{\prime}}=0.045+1.05\times(1-\text{Re}/1600)^{4.5} \tag{8}\]  
where Re is the Reynolds number. Experiment data showed that the above formula was available for \(260<\text{Re}\leq 1.6\times 10^{3}\). Within the above range, the Reynolds number was approximately calculated from the Strouhal number, St, \(\text{Re}\simeq(0.2139-\text{St})/4\) and \(\text{St}=0.185\) in this case [25]. Considering the balance of wind-induced vibration of the structural system, the frequency of vortex shedding (\(f_{w}\)) was equal to the frequency of structure vibration (\(f\)). Based on the Strouhal equation, the relationship between vibration frequency and wind speed was expressed as follows:  
\[f=f_{w}=V\frac{\text{St}}{D} \tag{9}\]  
Energy consumption of the Single Degree-Of-Freedom (SDOF) system with viscous damping under harmonic load (\(W\)) was expressed as below:  
\[W=\int_{o}^{T}\!\!f_{d}ydt=cA^{2}\omega^{2}\!\int_{0}^{T}\!\!\cos^{2}(\omega t -\varphi)dt=\pi cA^{2}\omega \tag{10}\]  
where \(A\) is the vibration amplitude of conductor and \(A=2y_{\text{max}}\), \(f_{d}\) is the damping force, \(c\) is the damping coefficient, \(\omega\) is the circular frequency of harmonic vibration, \(t\) is the time (s). During the VIV process, dissipating power of the conductor per length was equivalent to the energy consumption of damping forces. According to Equation (10), energy consumption was expressed as below [26]:  
\[W=P_{c}t\Delta l=P_{c}\Delta l/f=\pi cA^{2}\omega \tag{11}\]  
where \(P_{c}\) is the dissipating power of conductor per length, and the dissipating power of conductor per length was evaluated as below ([PERSON] and [PERSON] 2018):  
\[P_{c}=4\pi^{4}m^{2}EI\frac{y^{2}\text{max}f^{5}}{T^{2}} \tag{12}\]  
where \(T\) is the tensile force of conductor (N), \(m\) is the mass per unit length of conductor (kg/m) and \(EI\) is equivalence sectional flexural rigidity of conductor (N\(\cdot\)m\({}^{2}\)). Thus, damping coefficient \(c\) was obtained:  
\[c=\frac{1}{2}\pi^{2}m^{2}EI\frac{f^{3}}{T^{2}} \tag{13}\]  
where \(EI\) is sectional flexural rigidity of conductor (N\(\cdot\)m\({}^{2}\)).