## 3 Damper-Cable Interaction Model under Wind Action
### Modelling the Damper-Cable Interaction
This assumption has two advantages: (i) the relative movements between the damper and the cable, which govern the behavior of the damper, were obtained straight; and (ii) the effect of the dampers on the modal properties of the structure was taken into account directly. First, the effect of a viscous damper is equivalent to a damping force which is proportional to a damping coefficient, \(c_{d,v}\) [sN/m], and the relative velocity, \(\dot{v}_{r}(t)\) [m/s], between the two extremes of the damper (\(\dot{v}_{r}(t)=\dot{v}_{dA}(t)-\dot{v}_{dB}(t)\), where \(\dot{v}_{dA}(t)\) is the velocity of the extreme of the damper in contact with the cable and \(\dot{v}_{dB}(t)\) is the velocity of the extreme of the damper in contact with the deck, as it is illustrated in Figure 2. The viscous damping force of this damper, \(F_{d,v}(t)\), may be expressed as [47]:  
\[F_{d,v}(t)=c_{d,v}\dot{v}_{r}(t) \tag{8}\]  
Second, the effect of the elastomeric damper may be simulated via the Kelvin-Voigt model. The equivalent viscoelastic damping force is characterized by two components: (i) a viscous damping component which is expressed in terms of a damping coefficient, \(c_{d,e}\) [sN/m], and the relative velocity, \(\dot{v}_{r}(t)\) [m/s]; and (ii) an elastic component which is expressed in terms of a stiffness coefficient, \(k_{d,e}\) [N/m], and the relative displacement between the two extremes, \(v_{r}(t)\) [m] (\(v_{r}(t)=v_{d,A}(t)-v_{d,B}(t)\), where \(v_{d,A}(t)\) is the displacement of the extreme of the damper in contact with the cable and \(v_{d,B}(t)\) is the  
Figure 2: Damper-cable interaction model considered and mechanical model of each passive damper (viscous, elastomeric, and friction). displacement of the extreme of the damper in contact with the deck, as it is illustrated in Figure 2. The viscoelastic damping force of this damper may be defined as [48; 49]:  
\[F_{d,e}(t)=c_{d,e}\dot{v}_{r}(t)+k_{d,e}v_{r}(t) \tag{9}\]  
Finally, the effect of the friction damper may be mimicked via the extended Kelvin-Voigt model. The definition of the equivalent damping force involves three components: (i) a viscous damping component which is expressed in terms of a damping coefficient, \(c_{d,f}\) [sN/m], and the relative velocity, \(\dot{v}_{r}(t)\) [m/s]; (ii) an elastic component which is expressed in terms of a stiffness coefficient, \(k_{d,f}\) [N/m], and the relative displacement, \(v_{r}(t)\) [m], and (iii) a friction component defined in terms of a static friction force, \(f_{f}\) [N] (where, \(f_{f}=\mu\cdot N\), being \(\mu\) the friction coefficient [\(-\)] and \(N\) the normal force [N]) and a symbolic function, \(\text{sgn}\Big{(}\dot{v}_{r}(t)\Big{)}\) (which returns \(-1\), \(0\), and \(1\) in case \(\dot{v}_{r}(t)<0\), \(\dot{v}_{r}(t)=0\) and \(\dot{v}_{r}(t)>0\), respectively). The equivalent damping force of this damper may be expressed as [50]:  
\[F_{d,f}(t)=c_{d,f}\dot{v}_{r}(t)+k_{d,f}v_{r}(t)+f_{f}\cdot\text{sgn}\Big{(} \dot{v}_{r}(t)\Big{)} \tag{10}\]  
These damping devices are usually located at a certain distance, \(x_{c}\) [m], of the lower anchorage of the stay cable (Figure 2) due to constructive limitations. Nevertheless, due to their mechanical characteristics, they can have influence on both the damping and the natural frequencies (locking effect) of the stay cable.