**1. Introduction**

One of the main elements that governs the dynamic behavior of cable-stayed bridges is their stay cables [1]. This structural system has both a high flexibility and a low damping, which makes it susceptible to su ffer both from di fferent vibratory problems [2] and exhibit significant changes in its modal properties induced by the modification of the operational and environmental conditions [3].

The vibratory problems observed in cables of cable-stayed bridges may be classified in terms of the structural elements excited during the vibration phenomenon into the following [2]: (i) localglobal vibratory problems, in which the vibrations involve the excitation of both the cables and the deck of the structure [4]; and (ii) local vibratory problems, in which only the cables of the structure are excited laterally [5]. These vibratory problems may be caused by either of the following: (i) direct excitation sources, such as road tra ffic, wind [6] or earthquake action [7], or (ii) indirect excitation sources, such as linear internal resonances, parametric excitations or dynamic bifurcations.

In this paper, we focus on the case of wind-induced vibrations of stay cables, as this is the source problem of many vibratory issues reported in the literature [2]. As wind-induced vibrations can cause di fferent structural problems on stay cables (like fatigue or comfort problems), two types of measures are normally adopted to mitigate the cable vibrations [8], consisting of either of the following: (i) modifying its natural frequency via the installation of a secondary net of cables [9]; or (ii) increasing its damping ratio via the installation of external control systems [2]. Such control systems for stay cables may be classified into three di fferent groups [8]: (i) active [10]; (ii) semi-active [11]; and (iii) passive [12].

Active control systems for stay cables focus on controlling the dynamic response of the cable via the modification of its tensional state [13]. For this purpose, some kind of actuator, following the orders of a controller, acts on the cable in order to minimize the di fference between the actual response of the cable (recorded by a sensor) and the allowable response value [14]. Although the theoretical research on the use of these devices has experienced a significant growth in recent years, their practical implementation in real cable-stayed bridges has been limited due to their high cost and the robustness problems associated with the power supply needed to guarantee their operation [2].

On the other hand, semi-active control systems focus on modifying the constitutive parameters of external damping devices deployed to control the response of the stay cable under external actions [15]. Among the di fferent semi-active devices, magnetorheological dampers have been widely studied and implemented in real cable-stayed bridges [16]. Although semi-active damping devices outperform their passive damping counterparts [17] with a lower cost than active control systems, their e fficiency is limited when they are employed under uncertainty conditions, since their performance highly depends on the control algorithm considered for the design [18].

Finally, passive control systems for stay cables focus on increasing the damping ratio of the cables via the installation of external devices, whose characteristic parameters are originally designed to mitigate the dynamic response of the structural system [19]. Due to the robustness of such passive damping devices [20], they have been installed successfully on numerous real cable-stayed bridges to reduce wind-induced vibrations [21]. Nevertheless, these devices present as main limitation, a lower flexibility to adapt the system response to the variability of both the external actions and the modification of the stay cable parameters induced by loading, when compared to the active and semiactive devices. In order to overcome this limitation, two strategies may be adopted as outlined: either (i) to install a hybrid control system [22]; or (ii) to design the passive damping device taking into account these uncertainty conditions via a robust design method [23].

Di fferent design methods have been developed for this purpose. Among the di fferent proposals, Kovacs was the first researcher to study the optimum design of viscous dampers for stay cables [24]. Subsequently, Pacheco et al. provided a universal curve which allows the representation of the modal damping of the first vibration mode of a taut cable in terms of the damping coe fficient of the viscous damper [25]. The maximum of this curve corresponds to the optimum damping ratio of the taut cable when a viscous damper is installed on it. Later, Krenk et al. obtained an analytical expression for this curve [26]. Alternatively, other authors, such as Yoneda and Maeda, proposed an analytical model of the damped cable to determine the optimum parameters of the passive damper [27]. Although the design parameters obtained following any of these approaches are similar, so that they are currently employed for the practical design of passive damping devices, they fail to take into account a key aspect: the uncertainty associated with the variation of both the external actions and the modification of the modal properties of the stay cables [28].

In order to overcome this limitation, a motion-based design method [29] under uncertainty conditions is formulated, implemented and further validated in this paper. In fact, this proposal generalizes the formulation of a well-known design method, the so-called motion-based design method under deterministic conditions [30], to the abovementioned uncertainty conditions. The proposed motion-based design method under uncertainty conditions transforms the design problem into a constraint multi-objective optimization problem. Hence, the main objective of this problem is to find the optimum values of the characteristic parameters of the passive damping device which meet the design requirements for the structure. For this purpose, a multi-objective function is defined in terms of these parameters, together with an inequality constraint aimed at guaranteeing the compliance of the design requirements. Such design requirements are defined in terms of the vibration serviceability limit state of the structure. Since this serviceability limit state is defined under stochastic conditions, the failure probability of its compliance must be limited [31] and a reliability analysis must be performed [32,33]. For practical engineering applications [34], an equivalent reliability index is usually considered instead of the probability of failure. Thus, the formulation of the inequality constraint is realized in terms of the reliability index, which cannot exceed an allowable value [35]. For the computation of the reliability index, a sampling technique, the Monte Carlo method has been considered herein [36].

Finally, in order to validate the performance of the proposed method, it was applied to the robust design of three different passive damping devices (viscous, elastomeric, and friction dampers) where they are installed on the longest stay cable of the Alamillo bridge (Seville, Spain). To this end, only the effect of the rain–wind interaction phenomenon and the turbulent component of the wind action were considered. The results were compared with those obtained applying a conventional approach. This comparative study reveals that the proposed method allows the reduction of the cost of the passive damping devices while ensuring the structural reliability of the stay cable.

The manuscript is organized as follows: First, the motion-based design method under uncertainty conditions is described in detail. Next, a damper-cable interaction model under wind action, based on the finite element (FE) method, is presented. Subsequently, the performance of the proposed method is illustrated and further validated with a case-study (Alamillo bridge, Seville, Spain). In the final section, some concluding remarks are drawn to complete the paper.

#### **2. Motion-Based Design of Structures under Uncertainty Conditions**

#### *2.1. Motion-Based Design of Structures under Deterministic Conditions*

Structural optimization is a computational tool which can be used to assist engineering practitioners in the design of current structural systems [37]. Thus, this computational tool allows the optimum size, shape or topology of the structure to be found which meet the design requirements established by the designer/manufacturer/owner. Among the different structural optimization methods, the performance-based design method has been widely employed to design passive damping devices for civil engineering structures [23,30]. When the design requirements are defined in terms of the vibration serviceability limit state of the structure, the performance-based design method is denominated the motion-based design method [29]. This general design method was adapted herein for the design of passive damping devices when they are used to control the dynamic response of civil engineering structures. As assumption, all the variables, involved in this problem, are deterministic.

Thus, the motion-based design method under deterministic conditions transforms the design problem into a constrained multi-objective optimization problem. Therefore, the main objective of this problem is to find the optimum value of the characteristic parameters of the passive damping devices which guarantee an adequate serviceability structural behavior. For this purpose, a multi- objective function is minimized. The multi-objective function, f(θ), is defined in terms of the characteristic parameters, θ, of the considered passive damping devices. Additionally, the space domain is constrained including two restrictions in the optimization problem: (i) an inequality constraint, <sup>g</sup>*det*(θ); and (ii) a search domain. [<sup>θ</sup>min, <sup>θ</sup>max]. As the relation between the objective function and the design variables is nonlinear, global optimization algorithms are normally considered to solve this constrained multi-objective optimization problem. [38]. Accordingly, the motion-based design problem under deterministic conditions can be formulated as follows:

$$\begin{aligned} \text{Find } \boldsymbol{\Theta} \text{ to minimize } \mathbf{f}(\boldsymbol{\Theta})\\ \text{Subjected to } \begin{cases} & \mathbf{g}\_{dct}(\boldsymbol{\Theta}) \le 0 \\ & \boldsymbol{\Theta}\_{\text{min}} < \boldsymbol{\Theta} < \boldsymbol{\Theta}\_{\text{max}} \end{cases} \end{aligned} \tag{1}$$

where θ is the vector of the design variables; f(θ) is the multi-objective function to be minimized; θmin and θmax are the lower and upper bounds of the search domain; and <sup>g</sup>*det*(θ) is a function which defines the inequality constraint.

Therefore, the key aspect of this optimization problem is the definition of the inequality constraint. In the case of slender civil engineering structures, whose design is conditioned by their dynamic response [29], the compliance of the vibration serviceability limit state can be considered for this purpose. According to the most advanced design guidelines [6,34], the vibration serviceability limit state of a structure is met if the movement of the structure, *ds*(θ), which can be characterized by its displacement, velocity or acceleration, is lower than an allowable value, *dlim*, defined in terms of the considered comfort requirements. Thus, the inequality constraint of the abovementioned optimization problem may be expressed as follows:

$$\mathbf{g}\_{\rm det}(\boldsymbol{\Theta}) = \frac{d\_{\rm s}(\boldsymbol{\Theta})}{d\_{\rm lim}} - 1 \le 0 \tag{2}$$

Finally, as the result of this multi-objective optimization process, a set of possible solutions is obtained. This set of possible solutions is denominated the Pareto front. Accordingly, a subsequent decision-making problem must be solved, the selection of the best solution among the di fferent elements of this Pareto front. Two possible alternatives are normally considered for this purpose [23]: (i) the selection of the best-balanced solution among all the elements of the Pareto front; and (ii) the consideration of additional requirements to solve this decision-making problem. The selection between both alternatives depends on the designer's own criterion and the particular conditions of the problem.

#### *2.2. Motion-Based Design of Structures under Stochastic Conditions*

In order to generalize the implementation of the motion-based design method to scenarios with stochastic conditions, it is necessary to consider during the design process the uncertainty associated with the variability of both the external actions and the modal properties of the structure. For this purpose, two types of methods are normally employed [33]: (i) probabilistic methods; and (ii) fuzzy logic methods. Between these two methods, a probabilistic approach was considered herein because engineering practitioners are more used to dealing with probability concepts than with fuzzy logic problems. Concretely, a structural reliability method [39] was adapted herein to deal with the aforementioned uncertainty. According to this method, the vibration serviceability limit state can be expressed as a probabilistic density function, <sup>g</sup>*unc*(θ), which is defined in terms of the capacity of the structure, *Cs*, and the demand of the external actions, *Da*(θ) (where both terms are random variables characterized by their probability density function). Thus, the vibration serviceability limit state can be defined as follows:

$$\mathcal{g}\_{\text{unc}}(\boldsymbol{\theta}) = \begin{cases} \mathbb{C}\_{\text{s}} - D\_{\text{d}}(\boldsymbol{\theta}) & \text{if } \mathcal{g}\_{\text{unc}}(\boldsymbol{\theta}) \text{ is assumed normally distributed} \\\ \frac{\mathbb{C}\_{\text{s}}}{D\_{\text{d}}(\boldsymbol{\theta})} & \text{if } \mathcal{g}\_{\text{unc}}(\boldsymbol{\theta}) \text{ is assumed log -- normally distributed} \end{cases} \tag{3}$$

The above relation (Equation (3)) allows the computation of the probability of failure of the structural system, *pf*(θ), to the vibration serviceability limit state. This probability of failure, *pf*(θ), may be determined as follows:

$$p\_f(\boldsymbol{\Theta}) = \begin{cases} \text{Prob}[g\_{\text{unc}}(\boldsymbol{\Theta}) < 0] & \text{if } g\_{\text{unc}}(\boldsymbol{\Theta}) \text{ is assumed normally distributed} \\ \text{Prob}[g\_{\text{unc}}(\boldsymbol{\Theta}) < 1] & \text{if } g\_{\text{unc}}(\boldsymbol{\Theta}) \text{is assumed log -- normally distributed} \end{cases} \tag{4}$$

On the other hand, as it is shown in Figure 1, it is possible to characterize the probability of failure, *pf*(θ)**,** via an equivalent index, the so-called reliability index, β*s*(θ).

**Figure 1.** Probability density function of the vibration serviceability limit state, <sup>g</sup>*unc*(θ): (**a**) <sup>g</sup>*unc*(θ) follows a normal distribution; and (**b**) <sup>g</sup>*unc*(θ) follows a log-normal distribution.

The relation between the probability of failure, *pf*(θ), and the reliability index, β*s*(θ), may be expressed as follows:

$$p\_f(\boldsymbol{\Theta}) = \begin{cases} \mathcal{F}\_{\mathbb{S}\_{\text{aux}}}(0) = \boldsymbol{\Phi}(\frac{\mu\_{\text{grav}}(\boldsymbol{\Theta})}{\sigma\_{\text{grav}}(\boldsymbol{\Theta})}) = \boldsymbol{\Phi}(-\boldsymbol{\beta}\_s(\boldsymbol{\Theta})) & \text{normalized} \\ \mathcal{F}\_{\mathbb{S}\_{\text{aux}}}(1) = \boldsymbol{\Phi}\left(\frac{\ln \mu\_{\text{C}\_s}/\mu\_{\text{D}\_x}(\boldsymbol{\Theta})}{\sqrt{\sigma\_{\text{lic}\_s}^2 + \sigma\_{\text{linD}\_x(\boldsymbol{\Theta})}^2}}\right) = \boldsymbol{\Phi}(-\boldsymbol{\beta}\_s(\boldsymbol{\Theta})) & \text{log -- normally distributed} \end{cases} \tag{5}$$

where <sup>F</sup>*gunc* is the cumulative probability distribution function of <sup>g</sup>*unc*(θ); μ*gunc* (θ) and <sup>σ</sup>*gunc* (θ) are respectively the mean and standard deviation of <sup>g</sup>*unc*(θ); Φ is the standard normal cumulative distribution function; μ*Cs* and μ*Da* (θ) are respectively the mean of the probabilistic distribution function of *Cs* and *Da*(θ); and <sup>σ</sup>*lnCs* and <sup>σ</sup>*lnDa*(θ) are respectively the standard deviation of the lognormal distribution of *Cs* and *Da*(θ).

In this manner, the use of the reliability index, β*s*(θ), allows the computation of the vibration serviceability limit state under uncertainty conditions to be simplified. Hence, this design requirement is met if the reliability index, β*s*(θ), is greater than the allowable reliability index, β*<sup>t</sup>*, established by the designer/manufacturer/owner of the structure. In order to evaluate this inequality constraints, the reliability index, β*s*(θ), is usually computed via sampling techniques and the recommended values of the allowable reliability index, β*<sup>t</sup>*, can be found in literature [39]. In this study, Monte Carlo simulations [36] were considered in order to evaluate numerically the reliability index, β*s*(θ), and the value proposed by the European guidelines [34] was considered for the allowable reliability index, β*<sup>t</sup>*.

Finally, the motion-based design method under uncertainty conditions may be formulated as follows:

$$\begin{array}{c} \text{Find } \Theta \text{Minimize } \mathbf{f}(\boldsymbol{\Theta})\\ \text{Subjected to } \begin{cases} \text{g}\_{\text{unc}}(\boldsymbol{\Theta}) = \frac{\beta\_{\text{l}}}{\beta\_{\text{s}}(\boldsymbol{\Theta})} - 1 \le 0 \\\ \boldsymbol{\Theta}\_{\text{min}} < \boldsymbol{\Theta} < \boldsymbol{\Theta}\_{\text{max}} \end{cases} \end{array} \tag{6}$$

According to this, one of the main virtues of the motion-based design method is highlighted. The method allows the deterministic and stochastic design problems to be dealt with using a similar formulation. Only the inequality constraint must be modified to adapt the formulation to the particular conditions of each problem. This virtue facilitates the implementation of this method for the robust design of passive damping devices when they are used to control the dynamic response of slender civil engineering structures.

#### **3. Damper-Cable Interaction Model under Wind Action**

The damper-cable interaction model, considered herein to evaluate the dynamic response of a stay cable damped by different passive control systems under wind action, is described in detail in this section. First, the interaction model based on the FE method is introduced. Later, the method employed to simulate the wind action is presented.

#### *3.1. 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, *dc* [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'*,where *uc* and *vc* 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:

$$H\frac{\partial^2 \upsilon\_c}{\partial \mathbf{x}^2} - \frac{\partial^2}{\partial \mathbf{x}^2} \left( EI \frac{\partial^2 \upsilon\_c}{\partial \mathbf{x}^2} \right) = m \frac{\partial^2 \upsilon\_c}{\partial \mathbf{t}^2} \tag{7}$$

where *vc* 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 [Nm2] (where *E* is the Young's modulus [N/m2] and *I* is the moment of inertia of the cross-section of the cable [m4]); 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.

**Figure 2.** Damper-cable interaction model considered and mechanical model of each passive damper (viscous, elastomeric, and friction).

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). 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, *cd*,*<sup>v</sup>* [sN/m], and the relative velocity, .*vr*(*t*) [m/s], between the two extremes of the damper (.*vr*(*t*) = .*vd*,*<sup>A</sup>*(*t*) − .*vd*,*<sup>B</sup>*(*t*), where .*vd*,*<sup>A</sup>*(*t*) is the velocity of the extreme of the damper in contact with the cable and .*vd*,*<sup>B</sup>*(*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, *Fd*,*<sup>v</sup>*(*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, *cd*,*<sup>e</sup>* [sN/m], and the relative velocity, . *vr*(*t*) [m/s]; and (ii) an elastic component which is expressed in terms of a stiffness coefficient, *kd*,*<sup>e</sup>* [N/m], and the relative displacement between the two extremes, *vr*(*t*) [m] (*vr*(*t*) = *vd*,*<sup>A</sup>*(*t*) − *vd*,*<sup>B</sup>*(*t*), where *vd*,*<sup>A</sup>*(*t*) is the displacement of the extreme of the damper in contact with the cable and *vd*,*<sup>B</sup>*(*t*) is the

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]:

$$F\_{d, \varepsilon}(t) = c\_{d, \varepsilon} \dot{v}\_r(t) + k\_{d, \varepsilon} v\_r(t) \tag{9}$$

Finally, the e ffect 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 coe fficient, *cd*, *f* [sN/m], and the relative velocity, . *vr*(*t*) [m/s]; (ii) an elastic component which is expressed in terms of a sti ffness coe fficient, *kd*, *f* [N/m], and the relative displacement, *vr*(*t*) [m], and (iii) a friction component defined in terms of a static friction force, *ff* [N] (where, *ff* = μ·*N*, being μ the friction coe fficient [−] and *N* the normal force [N]) and a symbolic function, sgn. *vr*(*t*) (which returns −1, 0, and 1 in case . *vr*(*t*) < 0, . *vr*(*t*) = 0 and . *vr*(*t*) > 0, respectively). The equivalent damping force of this damper may be expressed as [50]:

$$F\_{d,f}(t) = c\_{d,f} \dot{v}\_r(t) + k\_{d,f} v\_r(t) + f\_f \cdot \text{sgn}(\dot{v}\_r(t)) \tag{10}$$

These damping devices are usually located at a certain distance, *xc* [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.

#### *3.2. Modelling the Wind Action*

Subsequently, the e ffect of the wind-induced forces was simulated numerically. The wind simulation was carried out under the assumption that the cable is a cylinder immersed in a turbulent flow [2]. Hence, the wind flow is composed of three components: (i) a mean wind velocity, U [m/s]; (ii) a fluctuating longitudinal velocity, *u*(*t*) [m/s]; and (iii) a fluctuating transversal velocity, *v*(*t*) [m/s].

The wind forces can be decomposed into a mean and a fluctuating component assuming the following hypothesis: (i) a quasi-steady behavior of the wind-induced forces; and (ii) small components of the turbulence with respect to the mean wind velocity, U [51]. The expression of these two components can be expressed as follows (assuming a linearized approximation [52]):

$$F\_D(t) = F\_D + f\_{Du}(t) + f\_{Dv}(t) \tag{11}$$

$$F\_L(t) = F\_L + f\_{L\nu}(t) + f\_{L\nu}(t) \tag{12}$$

where *FD*(*t*) is the drag force [N]; *FL*(*t*) is the lift force [N]; *FD* is the mean wind drag force; *FL* is the mean wind lift force; *fDu*(*t*) is the drag force induced by the longitudinal component of the wind; *fLu*(*t*) is the lift force induced by the longitudinal component of wind; *fDv*(*t*) is the drag force induced by the transversal component of wind; and *fLv*(*t*) is the lift force induced by the transversal component of the wind. These magnitudes can be determined using the following relationships [2]:

$$F\_D = 0.5\rho \mathcal{U}^2 D \mathcal{C}\_D \tag{13}$$

$$f\text{p}u(t) = \rho \mathcal{U}u(t)D\text{C}u\tag{14}$$

$$f\_{\rm Dv}(t) = 0.5 \rho \mathcal{U} v(t) D \left( \mathbf{C}'\_{\rm D} - \mathbf{C}\_{\rm L} \right) \tag{15}$$

$$F\_L = 0.5 \rho \mathcal{U} \mathcal{U}^2 \mathcal{D} \mathcal{C}\_L \tag{16}$$

$$f\_{\rm Lu}(\mathbf{t}) = \rho \mathcal{U} \mathbf{u}(\mathbf{t}) \mathcal{D} \mathbf{C}\_{\rm L} \tag{17}$$

$$f\_{Lv}(t) = 0.5\rho \mathcal{U}v(t)D\left(\mathbb{C}\_L - \mathbb{C}\_D'\right) \tag{18}$$

where ρ is the density of the air [kg/m3]; *D* is the outer diameter of the cable [m]; *CD* is the drag coefficient [−]; and *CL* the lift coefficient [−]. The coefficients *C D* and *C L* are the derivative of *CD* and *CL*, respectively, with respect to the angle α neighboring β (Figure 3). As the section of the cable is assumed to be circular in this study, these derivatives are therefore null because of the symmetry, and hence these two coefficients can be neglected.

**Figure 3.** Reference coordinate system, drag force component, lift force component, and wind velocity components.

Finally, in order to determine the wind forces it is necessary to generate simulations of wind velocities. For this purpose, the wave superposition spectral-based method was considered [8]. This method allows the numerical determination of a series of wind velocities via the superposition of trigonometric functions. On the one hand, the amplitude of these functions is obtained in terms of a coherence function, which considers the spatial variability of the wind velocity, and the power spectral density function of the turbulent wind velocity. On the other hand, the phase of the trigonometric functions is generated randomly. The coherence function is defined using the relationship proposes by Davenport [53]. The power spectral density function proposed by the European guidelines [54] was considered herein.
