**5. Conclusions**

Stay cables are prone to vibrate under wind-induced vibrations, so that passive damping devices are usually employed to control their response. Nevertheless, the performance of these damping devices is directly a ffected by the sensitivity of the stay cables to both the variability of the external actions and the modification of the constitutive modal properties of the cables induced by the changes of the operational and environmental conditions. Accordingly, it is necessary to establish design methods which overcome these limitations and can be easily implemented for practical engineering applications.

For this purpose, a motion-based design method under uncertainty conditions was proposed and implemented herein. In this approach, the design problem is transformed into a constrained multi-objective optimization problem. Thus, the di fferent components of the multi-objective function are defined in terms of the characteristic parameters of the considered passive damping device; and an inequality constraint is additionally included to guarantee an acceptable probability of failure of the structural system. As design criterion to evaluate the probability of failure, the compliance of the vibration serviceability limit state (according to the FHWA guidelines) was considered. Therefore, the computation of the probability of failure was performed via a reliability index. In this manner, the compliance of the vibration serviceability limit state is met if the reliability index is greater than an allowable value (according to the European guidelines). A sampling technique, the Monte Carlo method, was considered to determine numerically this index.

The performance of the method was validated numerically via its implementation for the design of three di fferent passive damping devices (viscous, elastomeric, and friction dampers) when they are used to control the wind-induced vibrations of the longest stay cable of the Alamillo bridge (Seville). To this end, the e ffects of the rain-wind interaction phenomenon and the turbulent component of the wind action were considered as excitation sources. Additionally, and for comparison purposes, the passive damping devices were also designed according to a conventional method. As result of this study, a clear reduction of the values of the characteristic parameters of the dampers was obtained when the motion-based design method was applied, when compared to the results of the conventional method. Thus, the proposed method allows improvement of the design of passive damping devices for stay cables under wind-induced vibrations considering uncertainty conditions. This improvement is reflected in a reduction of both the size and the budget of the devices, which facilitates its installation. Nevertheless, despite the good performance of the proposed approach, further studies are recommended to validate experimentally the long-term behavior of passive damping devices designed according to this proposal.

**Author Contributions:** Conceptualization, J.N.-P. and J.F.J.-A.; Methodology, J.F.J.-A.; Programming, J.N.-P. and G.Q.; Software, J.N.-P. and G.Q.: Validation, J.F.J.-A. and I.M.D.; Writing—Original Draft Preparation, J.N.- P. and J.F.J.-A.; Writing—Review & Editing, J.F.J.-A.; Supervision, I.M.D. and A.S.; Funding Acquisition, I.M.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially funded by the Ministry of Science, Innovation and Universities (Government of Spain) under the Research Project SEED-SD (RTI2018-099639-B-I00). Additionally, the co-author, Javier Naranjo-Pérez, was supported by the research contract, USE-17047-G, provided by the Universidad de Sevilla.

**Conflicts of Interest:** The authors declare no conflict of interest.
