## 5 Results and Discussion
### Suppression of Free Vibration  
During free vibration tests, an initial displacement is applied to the piping system, and the displacement response is recorded. Figure 7 illustrates the time domain displacement curves of the piping system under the control of TMD and PTMDs. For a clear comparison, the initial displacement of the free vibration is 20 mm for all conditions and the time from the initial displacement to decay to 10% of the initial displacement (2 mm) is defined as the control time. Figure 7a illustrates the displacement response of the piping system subject to free vibration with and without TMD control. The control times are 26.5 s (without control) and 48.8 s (with TMD control), and the corresponding damping ratios are 0.58% and 0.29%, respectively. As shown in Figure 7b,c, when VE-PTMDs are  
Figure 6: (**a**) Experimental setup and (**b**) data acquisition system. employed, the vibration attenuation is more pronounced. The control times are 1.6 s (with spring steel type VE-PTMD) and 2.85 s (with pendulum type VE-PTMD), respectively, and the corresponding system damping ratios are 10.4% and 5.6%. Figure 7d,e show the free vibration responses of the piping system with spring steel type SMA-PTMD control and with pendulum type SMA-PTMD control, respectively. The beating phenomenon is observed for both types of SMA-PTMDs, and is different from the smoothly decaying trend of vibration intensity with VE-PTMDs control. The responses of the SMA-PTMDs are similar, and only one beat cycle is observed. First, the displacements decay rapidly to zero, then slightly increase, and then decrease gradually. From visual observation, when an instantaneous displacement is applied to the piping system, the SMA-PTMD worked immediately with strong poundings. When the displacement decayed rapidly to zero, the displacement of the mass of the PTMD is observed to be large. The beating suggests the transferal of energy between the piping system and the mass of the PTMD. Following the approach to zero displacement, the impact of the mass onto the pipe induces a small increase in the displacement. Finally, successive collisions between the mass of the PTMD and the pipe attenuate the displacement of the pipe to zero. Figure 7: Experimental results of free vibration: (**a**) With tuned mass dampers (TMD); (**b**) with spring steel type VE–PTMD; (**c**) with pendulum type VE–PTMD; (**d**) with spring steel type SMA–PTMD; (**e**) with pendulum type SMA–PTMD. For comparison, the results of free vibration tests are summarized in Table 2. As seen, after installing the TMD, the control time is extended from 26.5 s to 48.8 s, which is 184% of that without control. The response with TMD is worse when compared to the case of without control because the TMD is not optimally designed [88; 89]. The addition of the mass, which has a low damping ratio, decreases the overall damping ratio of the entire assembly. Since the TMD is not optimized, the damping ratio stays at a low value. From the [PERSON] analysis [90], this is because the TMD's auxiliary damping is too small, and it is far from the optimal damping. Thus, the TMD cannot dissipate energy effectively and lead to a continual, mutual transmission of energy between the pipe and the TMD. This is the reason for the beating phenomenon and the extension of the control time. In contrast, the control effects of four PTMDs are evident. The control times are from 1.6 s to 2.9 s, which correspond to 6% to 11% of the corresponding time of the piping system without control. This shows that all four PTMDs effectively and efficiently suppress the displacement response of the piping system during the free vibration. With the TMD control, the system damping ratio is reduced from 0.58% to 0.29%; however, the damping ratios are increased significantly with PTMD control (4.0%\(\sim\)10.4%). The damping ratios of the PTMDs are 6.9-17.9 times larger than when there is no control. This shows the effectiveness of when bounding is introduced to increase damping.