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 semi-active 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]. Different design methods have been developed for this purpose. Among the different proposals, [PERSON] was the first researcher to study the optimum design of viscous dampers for stay cables [24]. Subsequently, [PERSON] 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 coefficient 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, [PERSON] et al. obtained an analytical expression for this curve [26]. Alternatively, other authors, such as [PERSON] and [PERSON], 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.