4.3.2. Elastomeric Damper

Subsequently, the motion-based design of the elastomeric damper under uncertainty conditions may be addressed. The design problem of this elastomeric damper may be defined as follows:

$$\begin{aligned} \text{find } \boldsymbol{\Theta} = \begin{bmatrix} c\_{d,\boldsymbol{\varepsilon}'} \ k\_{d,\boldsymbol{\varepsilon}} \end{bmatrix} \text{to minimize } \mathbf{f}(\boldsymbol{\Theta}) = \begin{bmatrix} f\_{1\prime} \ f\_{2} \end{bmatrix} = \begin{bmatrix} c\_{d,\boldsymbol{\varepsilon}'} \ k\_{d,\boldsymbol{\varepsilon}} \end{bmatrix} \\ \text{subject to} \begin{cases} c\_{\min} < c\_{d,\boldsymbol{\varepsilon}} < c\_{\max} \\ k\_{\min} < k\_{d,\boldsymbol{\varepsilon}} < k\_{\max} \\ \beta\_{\boldsymbol{s}}(\boldsymbol{\Theta}) \ge \beta\_{\boldsymbol{t}} = 1.35 \end{bmatrix} \end{aligned} \tag{22}$$

As result of the design process, the parameters of the elastomeric damper (*cd*,*<sup>e</sup>* and *kd*,*<sup>e</sup>*) were obtained. The best solution among all the elements of the Pareto front was *cd*,*<sup>e</sup>* = 1.22 × 10<sup>5</sup> sN/m and *kd*,*<sup>e</sup>* = 1.30 × 10<sup>5</sup> N/m. The reliability index associated with this solution is, β*s*(θ) = 1.49, which met the design requirements. Figure 7 shows the maximum displacement at the mid-span of the cable damped by the elastomeric damper for the di fferent elements of the sample.

**Figure 7.** Maximum displacement at the mid-span of the stay cable with the elastomeric damper for the different elements of the sample.
