# Effect of Dynamic Loading Conditions on Maximizing Energy Dissipation of Metallic Dampers
## 4 Dynamic Optimization  
### An Analytical Model for Dynamic Optimization  
The prototype bridge model presented in the section above was created to illustrate the possible tendency of the damper's behavior under dynamic loading. Many more aspects of the model need to be reconsidered, such as its precise dimensions and properties, in order to represent a real structural environment for a damper. However, it is impossible to utilize a realistic structural model in the optimization process, since the computational cost will be excessively demanding. Thus, an analytical model is proposed in this section for the dynamic optimization process. This model is based on a concept similar to that of the preceding model. The overall configuration of this analytical model is explained in Figure 10.  
Both ends of the damper are only allowed to move in one lateral direction with fixed movements. This dynamic model contains two additional components--the concentrated mass and the spring set. These components are also applied to each end of the member, representing the complete structure on which the damper will be installed. Several sets of values for the stiffness and concentrated mass that could be matched to a specific structure \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Structural Model Analysis**} & \multirow{2}{*}{**Initial**} & **Optimized** & **Initial** & **Optimized** \\  & & **(Case S1)** & **(Case S1)** \\ \cline{2-4}  & **Dynamic Analysis (Max. 0.1 g)** & **Dynamic Analysis (Max. 0.2 g)** \\ \hline ALLPD [kN\(\cdot\)m] & 4.82 & 4.61 & 18.91 & 21.27 \\ Rate of change & & 4.42\% & +12.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Energy dissipation of each optimized shape with a structure under dynamic loading. Figure 9: Prototype bridge dimensions and boundary conditions. can be considered. The stiffness of spring set 1 and spring set 2 is calculated from the expected stiffness of the prototype column, pier 1 and pier 2 in Figure 9. The point mass of the upper end is derived by considering the mass of the superstructure. The other mass at the lower end originated from the mass of the upper portion of the pier where the damper was installed. Therefore, for the bridge structures, the concentrated mass at the upper end is larger than the other mass at the lower end. Here, a spring stiffness of 1840.78 kN/m, and 50 and 4 tons of concentrated mass, were applied to match the previously presented prototype structure. The analysis result of the simplified model with the parameter assumptions was compared with that of the target structure, i.e., the prototype bridge. The comparison was conducted utilizing the initial geometry of the damper. The responses of both cases, which were relative lateral displacements between the upper and lower ends of the damper, were well matched (Figure 11). An evaluation of the optimized dampers was conducted again, using the dynamic analytical model. As before, the energy dissipation capacities of the initial shape and the  
Figure 11: Specifications of the control points. Relative lateral displacement between the upper and lower ends of the damper under sine-function loading with maximum 0.2 g acceleration. Figure 10: Simplified model for dynamic optimization: (**a**) Model configuration; (**b**) Deflection of the model under the applied excitation. two optimized shapes were compared, with two different maximum ground accelerations. A tendency to closely match the previous structural analysis was confirmed. The optimized shapes showed less energy dissipation under small ground accelerations. The scale of dissipation decreased to a smaller degree compared to the total structural analysis. This was because the overall displacement of the analytical model was slightly smaller due to a minor intrinsic error in the model. However, the utilization of this model is still valid for design purposes, since it has conservative assumptions. Moreover, a detailed analysis was carried out regarding specific statically optimized cases.