# Effect of Dynamic Loading Conditions on Maximizing Energy Dissipation of Metallic Dampers
## 3 Dynamic Analysis of Static Optimization
### Static Optimization Results  
The optimized shapes were derived under four different stroke conditions. The optimization history curves of each case are illustrated in Figure 6a. The overall optimized results of the three variables are provided in Figure 6b and Table 1. The cases are named S1-S4, corresponding to the four stroke conditions. The optimization stroke conditions of each case are specified in the table, and the e volume and increased stiffness of the damper in the elastic region of the optimized dampers are also presented. The results of the dissipated energy and the improvement of each case were declared with the stroke conditions of S4, i.e., nine cycles with an increase in strokes. This unification of the loading conditions was applied to determine the relative evaluation of the results. Figure 5: Simplified static loading scheme for optimization: (**a**) Three cases with a uniform scale of strokes; (**b**) Additional case with a stroke increase. Figure 7 compares the stress distribution of the final optimized model (S1) at the edge, with the initial shape of the damper. This example results from the final 50 mm stroke-optimized model. The plotted graph shows the von Mises stress at each element located at the edge of the model. The dotted line indicates the stress distribution at the edge of the model with the initial geometry, and the solid line denotes the optimized geometry. The solid line of the optimized model proves that the stresses are well distributed, and more elements are in a plastic state than they were in the initial model. The number of elements at the edge that reached the plastic state showed an increase of approximately +180%. This result of the optimization process eventually increased the hysteresis loop area (Figure 8). \begin{table}
\begin{tabular}{c c c c c c} \hline
**Cases** & **Initial** & **S1** & **S2** & **S3** & **S4** \\ \hline Optimized Stroke size & & 25 mm & 50 mm & 100 mm & Max. 100 mm \({}^{*}\) \\ \hline \(x_{b}\) [mm] & 80 & 123.37 & 113.87 & 123.23 & 126.61 \\ \hline \(x_{c}\) [mm] & 80 & 81.34 & 84.60 & 81.42 & 80.10 \\ \hline \(x_{av}\) [mm] & 80 & 40.00 & 40.00 & 40.00 & 40.00 \\ \hline Volume [m\({}^{3}\)] & 3.840 & 3.840 & 3.839 & 3.840 & 3.836 \\ \hline Elastic stiffness [MN/m] & 2.314 & 3.705 & 3.730 & 3.709 & 3.672 \\ \hline ALLPD [kN\(\cdot\)m] & 53.76 & 93.99 & 91.01 & 93.91 & 93.64 \\ \hline Improvement [\%] & & +74.9 & +69.3 & +74.7 & +74.2 \\ \hline \end{tabular} \({}^{*}\) Case S4: The case with stroke increase.  
\end{table} 
Table 1: Shape variable values, energy dissipation, and improvement of each optimized model with a fixed thickness. Figure 6: Static optimization results: (**a**) Optimization history curves of cases S1–S4; (**b**) Optimized shapes with static analysis (S1–S4). Figure 8: Hysteresis loops of the optimized shapes: (**a**) Case S1; (**b**) Case S2; (**c**) Case S3; and (**d**) Case S4. Figure 7: Plastic strain distribution (initial and S1) and stress distribution at the edge (S1). All the optimized damper shapes showed significant improvement in energy dissipation, with every case demonstrating 70-75% improvement. Through the shape optimization, the width of the top and bottom ends became larger and that of the middle became smaller. This result of the optimization process increased the number of elements in a plastic state and eventually enhanced the hysteresis loop area (Figure 8). The hourglass shape was predictable and was already derived by a number of other researchers [1; 13; 29]. However, variety in the optimization results with respect to changes in the loading conditions was scarcely observed. The results were very slightly diversified only in the case of S2, which was under a repetitive 9-cycle 50 mm stroke. Other loading conditions, which were 9-cycle 25 mm (S1), 9-cycle 100 mm (S3), and 9-cycle with a stroke increase (S4), led to similar shapes. There were only vague correlations between the geometrical changes of the optimized results and the increases in stroke size.