# Effect of Dynamic Loading Conditions on Maximizing Energy Dissipation of Metallic Dampers
## 4 Dynamic Optimization
### Dynamic Optimization Results  
Dynamic optimization was accomplished with the three variables (\(x_{b}\), \(x_{c}\), and \(x_{up}\)). A total of nine maximum ground acceleration cases were tested, from 0.05 g to 0.25 g. A frequency of excitation, 0.5 Hz, was first applied, which was the same as that of the previous structural dynamic analysis in Section 3.4. The optimization results and improvements are shown in Fiugres 13 and 14a and Table 3. The plastic energy dissipation increased in all the cases compared to those of the initial and statically optimized case (S1). The improvements from case S1 were significant for small acceleration and decreased with an increase in the given acceleration scale (Figure 14a). It should be noted that extremely high improvement was obtained for smaller acceleration. This is because the statically optimized shape scarcely reaches plastic deformation at this level of acceleration. For comparison of a dynamically optimized case (D3) with the static optimization result (S1), the expected hysteresis loops are shown in Figure 14b under a sine function loading. The percentage displayed on the graph is the rate of change of the ALLPD values compared to the basic model before optimization. In this case, it can be considered as the area of the hysteresis loop. Under the small acceleration (Maximum 0.1 g), the energy dissipation of case S1 decreases, whereas D3 dissipates more energy. There were significantly different trends between the results under small and large acceleration conditions. The optimized values of the variable \(x_{b}\) were larger than \(x_{c}\) at high acceleration, a result that is similar to the static optimization results. On the other hand, the optimized value of \(x_{b}\) was lower than \(x_{c}\) at low acceleration. The transition of this trend coincides with the inflection point in Figure 12 from negative to positive improvement of  
Figure 12: Improvement of case S1 in energy dissipation from the initial shape analyzed with the dynamic optimization model. the dissipation capability of the statically optimized shape. The value of the variable \(x_{w}\) was independent of the change in ground acceleration scales. In addition, the effect of the number of excitation cycles on the dynamic optimization was analyzed. As mentioned earlier, in order to reduce the computational cost of the iterative operation conducted by the optimization algorithm, only one cycle of sine wave was first applied as the loading condition. However, considering that a more stationary response is required for the actual loading such as an earthquake, a new set of the dynamic optimization results was derived. In this case, the total time period of the excitation was extended to 8 s to increase the number of sine-wave cycles four times. The derived optimization variable values are shown in Figure 15b. Consequently, the changing trend of the optimized variable values due to the differences in the peak acceleration was more stably revealed, but the point at which the trend changes was maintained, the same as in  
Figure 14: The energy dissipation of the dynamically optimized dampers: (**a**) Improvement of results; (**b**) Hysteresis loops of S1 and D3. Figure 13: Dynamically optimized shapes under various acceleration conditions: (**a**) Representative stress distribution of the novel topology (Case D3); (**b**) Optimized results of all cases. the case of a single cycle. Therefore, it was found that the one-cycle condition was sufficient to see the macroscopic changes in the trend of the dynamic optimization results. Based on this result, the effect of excitation frequencies was analyzed as in the next section.