## 5 Results and Discussion
### Suppression of Forced Vibration  
In forced vibration experiments, a high-power motor with a rotating arm is used to generate a quasi-harmonic excitation. The rotating arm of the motor is connected to the middle of the pipe via a bungee cable, and the bungee cable passes through a pulley to keep the excitation horizontal. The excitations of different frequencies are achieved by adjusting the voltage of the motor and changing the rotational speed of the arm. These kinds of quasi-harmonic excitations have also been used in other pipe tests [38; 42; 66]. The frequency response curves of the piping system are plotted by recording the steady-state displacement amplitudes over different excitation frequencies of forced vibration. Figure 8 describes the frequency responses of the piping system under the conditions of the piping system without control, with the TMD control, with the spring steel type VE-PTMD control, with the pendulum type VE-PTMD control, with the spring steel type SMA-PTMD control, and with the pendulum type SMA-PTMD control. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Conditions**} & \multicolumn{2}{c}{**Time (20 mm to 2 mm)**} & \multicolumn{1}{c}{**Damping Ratio**} \\ \hline Without Control & & 26.5 s & 0.58\% \\ With TMD Control & & 48.8 s & 0.29\% \\  & & Spring Steel & 1.6 s & 10.4\% \\ With PTMD Control & & Pendulum & 2.85 s & 5.6\% \\  & & Spring Steel & 2.5 s & 4.9\% \\  & & Pendulum & 2.9 s & 4.0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The results of free vibration. The resonant frequency of the piping system is measured to be 2.57 Hz, and the maximum displacement of the system at the resonant condition is 18.90 mm. After the addition of TMD, the frequency response curve of the piping system has two peaks, which are on each side of the original resonant frequency, as shown in the green line of Figure 8. The new system's resonant frequencies are 2.12 Hz and 3.02 Hz, and the corresponding magnitudes of vibration are 16.98 mm and 12.86 mm. The blue line and red line are frequency responses of the piping system with the spring steel type VE-PTMD control and with the pendulum type VE-PTMD control. Both curves have only one peak, the resonant frequencies are both 2.46 Hz, and the corresponding magnitudes of vibration are 9.72 mm and 9.24 mm, respectively. The magenta line and the cyan line represent the frequency responses of the piping system with the spring steel type SMA-PTMD control and with the pendulum type SMA-PTMD control. Similar to the case with the TMD control, both the curves have two magnitude peaks. The peak frequencies of the response with the spring steel type SMA-PTMD are 2.38 Hz and 2.83 Hz, and the corresponding magnitudes are 7.67 mm and 8.39 mm, respectively, as shown by the magenta line in Figure 8. Two peaks of the frequency response with the pendulum type SMA-PTMD control (the cyan line) are 2.36 Hz and 2.75 Hz, and the corresponding magnitudes are 5.93 mm and 7.26 mm. At the resonant frequency of the piping system (2.57 Hz), the control effect of the spring steel type SMA-PTMD (3.91 mm) is slightly better than that of the pendulum type SMA-PTMD (5.0 mm). In order to understand the effectiveness of SMA-PTMDs at the resonant frequency (2.57 Hz), a lock-release experiment of the PTMD mass is conducted. First, the mass of the PTMDs is locked, and the PTMD and the piping system become one system. At the resonant frequency excitation, the whole system reaches the maximum displacement response. At this point, the constraint of the mass is released and the PTMD starts working. Figure 9 shows the displacement responses of the piping system before and after the mass of the PTMDs is released. After the mass of the PTMDs is released, the mass collides with the pipe, and the displacement of the pipe is rapidly reduced. After this transient process, the displacement of the pipe tends to be stable.