4.3.3. Friction Damper

Finally, the motion-based design of the friction damper under uncertainty conditions was performed. The design problem of this friction damper may be formulated as follows:

$$\begin{aligned} \text{find } \boldsymbol{\Theta} = \begin{bmatrix} c\_{d,\varepsilon} \ k\_{d,\varepsilon'} \ f\_f \end{bmatrix} \text{to minimize } \mathbf{f}(\boldsymbol{\Theta}) = \begin{bmatrix} f\_l \ f\_{2} \ f\_{3} \end{bmatrix} = \begin{bmatrix} c\_{d,\varepsilon'} \ k\_{d,\varepsilon'} \ f\_f \end{bmatrix} \\ \text{subject to} \begin{cases} c\_{\min} < c\_{d,f} < c\_{\max} \\ k\_{\min} < k\_{d,f} < k\_{\max} \\ f\_{f\_{\min}} < k\_f < f\_{f\_{\max}} \\ \beta\_s(\boldsymbol{\Theta}) \ge \beta\_t = 1.35 \end{bmatrix} \end{aligned} \tag{23}$$

After the design process, the optimum value of the damping coefficient, stiffness coefficient, and friction force which characterize the friction damper were obtained. The optimum solution was *cd*, *f* = 1.24 × 10<sup>5</sup> sN/m, *kd*, *f* = 6.74 × 10<sup>4</sup> N/m and *ff* = 2.95 × 10<sup>4</sup> N. The reliability index associated with this solution was β*s*(θ) = 1.55, which met the design requirements. Figure 8 shows the maximum displacement at the mid-span of the cable damped by the friction damper for the different elements of the sample.

**Figure 8.** Maximum displacement at the mid-span of the stay cable damped by the friction damper for the different elements of the sample.

#### *4.4. Discussion of Results*

Finally, the performance of the proposed method was validated comparing the abovementioned results with the ones provided by a conventional one, the optimum damping coefficient of the Pacheco's universal curve [25]. This optimum value for a viscous damper can be determined using the following relationship:

$$c\_{opt} = 0.10 \frac{mL\omega\_1}{\frac{x\_c}{L}}\,,\tag{24}$$

where ω1 = 2π *f*1 is the fundamental angular natural frequency of the stay cable [rad/s] and *xc* is the distance between the anchorage of the cable and the point where the damper is implemented (Figure 2). As in the remaining cases, the viscous damper is located at the point, *xc* = 0.03*L*, with respect to the lower anchorage. The optimum damping coe fficient, according to this conventional method for the viscous damper was *copt* = 1.64 × 10<sup>3</sup> sN/m.

Thus, two main conclusions may be obtained via the comparison of the abovementioned results: (i) the motion-based design method under uncertainty conditions allows reduction of the characteristic parameter of the viscous damper by about 35% with respect to the conventional method; and (ii) for this case-study, the viscous damper appears to be the best choice to control the dynamic response of the longest cable of the Alamillo bridge, as a minimum value of the damping coe fficient was obtained for this passive damper. The proposed method allows a better adjustment to the design requirements of the problem, reducing, as consequence, the size and the cost of the passive damping devices. Hence, the performance of the motion-based design method, for this particular problem, has been validated.
