## 8 Performance Comparison among TMD, TMDI, and Pendulum PTMD
### Performance Comparison  
The pendulum PTMD considerably reduced the displacement and acceleration of the top floor compared to TMD and TMDI (Figure 16). However, displacement responses are reduced more than acceleration responses. The TMDI and TMD were less effective in  
Figure 15: El-Centro earthquake excitation: (**a**) acceleration time history, and (**b**) acceleration spectrum. controlling the top floor's acceleration, as shown in Figure 16b, although they can reduce the displacement responses, as shown in Figure 16a. Similar to the performance against VF excitations described in Section 6, TMDI is less effective than equivalent TMD under earthquake excitation. Pendulum PTMD reduces the magnitude of excessive vibration and the duration of unwanted vibration, as observed in Figure 16. Responses of the structure under an earthquake in the frequency domain are shown in Figure 17. A significant contribution to displacement response comes from the first mode, but in the case of acceleration, both the first and the second modes contribute. TMD does not reduce higher modes, TMDI has slightly better performance on higher modes, and pendulum PTMD has more success in attenuating higher modes than the other two devices. It should be noticed that both axes in Figure 17 are on a logarithmic scale. Floorwise STD displacements and STD accelerations are presented in Figure 18a,b, respectively. It is observed that pendulum PTMD reduced the displacement at the 42 nd floor by 52.9% (STD) and 20% (peak). However, TMD and TMDI decreased the same response by 37.2% (STD), 11.9% (peak), 10.9% (STD), and 4.1% (peak). Although acceleration responses are reduced less in comparison to displacement responses, pendulum PTMD reduces top floor acceleration by 20.5% (STD) and 10.6% (peak), which is much higher compared to the reduction brought by TMD and TMDI. Interestingly, the reduction in peak acceleration by TMDI (6.5%) is slightly higher than the TMD (2.6%). This is because TMDI has enhanced control performance than TMD for higher modes, which can be observed from a closer look at Figure 18. This well-known fact supports the result that TMD cannot reduce peak acceleration, which is most often a significant cause of damage and destruction in high-rise buildings during seismic events. Nevertheless, pendulum PTMD performs better than TMD and TMDI among all modes, including the first mode. Figure 19 presents the floor-wise peak displacement and acceleration of the building. The floor displacement increases linearly over the height of the building, but the acceleration increases non-linearly. This observation shows that higher modes contribute to the acceleration response, and the first mode is dominant in the displacement response. Figure 16: Uncontrolled and controlled displacement and acceleration responses of the 42 nd floor, in the time domain, under El-Centro earthquake excitation: (**a**) displacement response and (**b**) acceleration response. Figure 17: Response spectra of the 42 nd story under El-Centro earthquake: (**a**) displacement and (**b**) acceleration. Figure 18: Floorwise responses under El-Centro earthquake: (**a**) STD displacement, (**b**) STD acceleration. Figure 20 shows the shear force on each floor induced by earthquake excitation and the performance comparison of each control device. STD shear at the top is reduced by 22.2% using pendulum PTMD, TMD reduces it by 14%, and TMDI reduces the same response by 11.5%. The performance of both TMD and TMDI is close to reducing both STD and peak shear at the top floor, as described in Figure 20. Considerable reduction in base shear is accomplished using pendulum PTMD over both TMD and TMDI. Nearly 56% reduction in STD base shear is achieved using pendulum PTMD, whereas TMDI and TMD reduce the same response by 33.7% and 47%, respectively. However, a smaller reduction in maximum base shear is obtained compared to STD base shear, as shown in Figure 21. The inter-story drift ratio is a significant performance criterion for the high-rise building under earthquakes.