## 3 Damper-Cable Interaction Model under Wind Action
### Modelling the Damper-Cable Interaction  
The analysis of the dynamic behavior of stay cables has been studied extensively over the last four decades. Thus, analytical [40], numerical [30], and experimental studies [41] have been performed for this purpose. Among the different proposals, a numerical method, the FE method was considered herein to develop a damper-cable interaction model. This method presents three main advantages when it is implemented for this particular problem [30]: (i) its easy implementation for practical civil engineering applications; (ii) it allows a direct interaction of element with different constitutive laws (cable and dampers); and (iii) it simplifies the simulation of some effects such us the nonlinear behavior of the cable [40], the sag effect [42], and the influence of the external dampers on the modal properties of the cables (locking effect) [43]. The implementation of the FE method for this particular problem is based on the numerical integration of the weak formulation of the differential equilibrium equation of a vibrating stay cable in the lateral direction. Figure 2 shows an inclined cable of length, \(L\) [m], suspended between two supports at different level which presents a sag, \(d_{c}\) [m], with respect to the axis aligned with the two supports. The application of a small displacement causes the motion of a generic point from the self-weight configuration, \(P\), to, \(P^{\prime}\),where \(u_{c}\) and \(v_{c}\) represent the component of the movement of the cable respectively in the parallel and perpendicular direction to the axis traced between the two supports. The equation, which governs the vibration of a taut cable in the lateral direction under the assumptions of linear and flexible behavior, may be expressed as follows:  
\[H\frac{\partial^{2}v_{c}}{\partial x^{2}}-\frac{\partial^{2}}{\partial x^{2}} \left(EI\frac{\partial^{2}v_{c}}{\partial x^{2}}\right)=m\frac{\partial^{2}v_ {c}}{\partial t^{2}} \tag{7}\]  
where \(v_{c}\) is the lateral displacement of the cable [m]; \(H\) is the axial force of the cable [N]; \(EI\) is the bending stiffness of the cable [Nm\({}^{2}\)] (where \(E\) is the Young's modulus [N/m\({}^{2}\)] and \(I\) is the moment of inertia of the cross-section of the cable [m\({}^{4}\)]); and \(m\) is the mass per unit length of the cable [kg/m]. According to the Equation (7), the vibration of the cable is governed by both its tensional state and its bending stiffness [44; 45]. Additional phenomena can be simulated via the selection of the adequate finite-element. A nonlinear two-node element with six degrees of freedom per node has been considered herein to simulate the cable behavior. This element allows both the nonlinear geometrical and stress-stiffness behavior of the cable to be simulated adequately [46]. In order to take into account, the initial tensional and deformational state of a stay cable during either a modal or a transient analysis, a preliminary static nonlinear analysis must be performed. In this preliminary static analysis, the equilibrium form of the cable under its self-weight and a preliminary axial force is achieved. As a result of this analysis, both the stress and the shape of the cable are updated, which is a key aspect to simulate numerically its real behavior. Subsequently, the modelling problem must focus on the simulation on the passive damping devices behavior. Three passive damping devices were considered herein (Figure 2). For these three passive damping devices, a linear constitutive law was assumed. The effect of these three passive damping devices on the cable may be simulated by an equivalent damping force. Each equivalent damping force is related to the energy that each damping device is able to dissipate, and it is opposed to the movement of the cable. Thus, each passive damping device has been modelled by a finite element whose behavior is equivalent to the corresponding damping force (Figure 2).