# Shared Tuned Mass Dampers for Mitigation of Seismic Pounding
## 3 Numerical Study
### Effectiveness of STMDs
The optimal STMD, therefore, does not provide any clear advantage over the TMD installed on building A alone. The mean (over 462 ground motions) ratios between the bounding distance of the AC controlled and SC controlled structures to that of the uncontrolled structure are 0.79 and 0.76, respectively. Controlling building A alone is, on average, about 3% less effective than using an STMD. The stiffnesses of AC and SC schemes are \(8.96\times 10^{5}\) N/m and \(9.68\times 10^{5}\) N/m, respectively. Their damping coefficients are \(4.73\times 10^{4}\) kg/s and \(2.27\times 10^{4}\) kg/s. This shows that the SC scheme results in similar performance like the AC scheme, but with a much smaller dashpot. This might be, in theory, an advantage of sharing the device. However, there might be practical difficulties in properly connecting the dashpot between the two buildings, and suitability of the scheme needs to be judged on a case by case basis. To illustrate the effectiveness of AC and SC schemes in reducing the minimum separation distance between the two buildings, reduction ratio, which is defined as minimum separation distance of the controlled building divided by that of the uncontrolled building is shown in Figure 8. The smaller this ratio, the more effective is the control scheme. Effectiveness is investigated against different ground motion and structural response parameters. Only those ground motions with PGA larger than 0.2 g are used in the figure. Here, g represents acceleration due to gravity. Ground motions weaker than this are not considered as they were found to cause a low risk of bounding. Figure 8c shows two different clusters of data, one below \(\sim\)25 s, and the other to its right. The cluster to the right has very few data points, and any apparent trend in this cluster is not reliable. When duration is less than 25 s, the Pearson correlation coefficient between reduction ratio and duration is \(-\)0.44 with a p-value of 0.0011, which shows that the correlation is statistically significant, and not by random chance. Figure 8b shows that the effectiveness of the TMDs is independent of PGA. There seems to be a weak negative correlation between the bounding distance of the uncontrolled structure and the effectiveness of TMD schemes. The Pearson correlation coefficient was found to be \(-\)0.17 with a large p-value of 0.21, which means that any apparent trend in the data is statistically insignificant. The same observation can be made in Figure 8d, which shows that the effectiveness of TMD schemes is independent of spectral displacement at the fundamental period of the uncontrolled structure. In all these cases, there is no significant difference between the SC and AC schemes, which means that sharing the TMD does not seem to provide additional benefit in preventing bounding. We next compare the effectiveness of the SC scheme to that of the AC scheme for different mass ratios of the TMD. Mass of the TMD is taken as 1%, 2%, and 3% of the total mass of building A. The optimal solutions for these mass ratios were found with frequency ratios of 1.02, 1.04, and 1.04 and damping ratios of 7%, 11%, and 13%. The effectiveness of the devices is quantified by the mean ratios of bounding distance of SC and AC schemes to that of uncontrolled structure, denoted by \(r_{SC}\) and \(r_{AC}\) respectively. A value of this ratio less than 1 means the control systems, on average, reduce bounding distance. These ratios are shown in Figure 9. As the mass ratio increases, the performance of SC is better than that of AC. However, even for a mass ratio of 3%, which corresponds to 24% of floor mass, the SC scheme is only about 6% better than the AC scheme. We repeated the analysis with different frequencies of the two buildings and arrived at similar results. Figure 7: Root displacement of building A (**left**) and minimum separation distance to avoid bounding (**right**) when subjected to 462 earthquake ground motions. The abscissa and ordinate correspond to uncontrolled and controlled structures, respectively.