## 4 Application Example
### Motion-Based Design of Passive Damping Devices under Uncertainty Conditions  
Three different passive damping devices were considered for this study: (i) viscous damper; (ii) elastomeric damper; and (iii) friction damper. The FE method was employed to simulate the behavior of these damping devices. The software Ansys [57] was employed for this purpose. Figure 2 depicts the mechanical models, which simulate the behavior of each damper. For each passive damper, the following model was considered: (i) the viscous damper was modelled by a 1D element (COMBIN14) whose characteristic parameter was the damping coefficient, \(c_{d,r}\) [sN/m]; the elastomeric damper was also modelled by a 1D element (COMBIN14) whose characteristic parameters were the damping coefficient, \(c_{d,r}\) [sN/m], and the stiffness coefficient, \(k_{d,r}\). [N/m]; and (iii) the frictioin damper was modelled by a 1D element (COMBIN40) whose characteristic parameters were the damping coefficient, \(c_{d,f}\) [sN/m], the stiffness coefficient, \(k_{d,f}\) [N/m], and the friction force, \(f_{f}\) [N]. Consequently, the different dampers were implemented in the numerical model and designed according to the motion-based design method under uncertainty conditions. The three dampers were installed at a length of \(x_{c}=0.03L\) according to the recommendations of Ref. [2]. The damper-cable interaction model is shown in Figure 2.  
A search domain, \([\theta_{\min},\ \theta_{\max}]\), for the characteristic parameters of the dampers was included in the optimization problem to ensure the physical meaning of the solutions obtained. The search domain was defined as follows: (i) the lower bound of the search domain, \(\theta_{\min}\), was defined as \(\theta_{\min}=\left[c_{min},k_{min},\ f_{f_{min}}\right]\) (where \(c_{min}\) is the minimum value of the damping coefficient; \(k_{min}\) is the minimum value of the stiffness coefficient, and \(f_{f_{min}}\) is the minimum value of the friction force); and (ii) the upper bound of the search domain, \(\theta_{\max}\), was defined as \(\theta_{\max}=\left[c_{max},k_{max},\ f_{f_{max}}\right]\) (where \(c_{max}\) is the maximum value of the damping coefficient; \(k_{max}\) is the maximum value of the stiffness coefficient, and \(f_{f_{max}}\) is the maximum value of the friction force). The lower, \(c_{min}\), and upper, \(c_{max}\), bounds of the damping coefficient were determined considering both the requirement of the Scruton number [6] and the optimum damping coefficient of the Pacheco's universal curve [25]. According to this, the following bounds were established: (i) \(c_{min}=4.8\times 10^{4}\) sN/m; and (ii) \(c_{max}=1.64\times 10^{5}\) sN/m. This search range guarantees that any solution of this design problem avoids the occurrence of the rain-wind interaction phenomenon. The search domain of the stiffness coefficient and the friction force were based on the results of previous research [2]. According to these results, the following bounds were established: (i) for the stiffness coefficient, \(k_{min}=5\times 10^{4}\) N/m and \(k_{max}=5\times 10^{5}\) N/m; and (ii) for the friction force, \(f_{f_{min}}=1\times 10^{4}\) N and \(f_{f_{max}}=4\times 10^{4}\) N.  
In order to avoid falling into a local minimum, a global computational algorithm was considered for this optimization problem. Among the different computational algorithms, genetic algorithms were considered herein [65] for its simplicity and great efficiency to solve structural optimization problems. Genetic algorithms are nature-inspired computational algorithms based on [PERSON]'s natural selection theory. According to this, each possible value of the characteristic parameters of the damper is identified as a chromosome. Subsequently, each set of characteristic parameters is grouped into an individual (parameter vector). Later, the value of this parameter vector is improved via an iterative process where the value of the objective function is optimized.