## 5 Results and Discussion
### Suppression of Forced Vibration
In the entire frequency domain, the maximum displacement responses of the piping system with the SMA-PTMDs controls occur at 2.83 Hz (spring steel type) and 2.75 Hz (pendulum type). Figure 10 compares the maximum displacement response without control and with the SMA-PTMDs control. The maximum displacement response with the pendulum type SMA-PTMD control (7.26 mm) is slightly better than that of with the spring steel type SMA-PTMD control (8.39 mm). From a practical perspective, this small difference suggests that in this case, both types of SMA-PTMDs essentially have a similar performance. A minor tuning of either type may cause the performance to become even closer. This close difference was also observed in the prior study with VE-PTMDs [45], suggesting that these two designs have a nearly equal performance in  
Figure 8: Experimental results of forced vibration. vibration suppression, and the user is free to choose a design that may better suit their engineering requirements (e.g., available materials). For comparison, the results are summarized in Table 3. With the TMD control, 97% of displacement is reduced at the resonant frequency (2.57 Hz). However, the reduction comes at the cost of the formation of two new peak regions surrounding the original natural frequency. All four PTMDs, especially the SMA-PTMDs, considerably reduce the displacement responses across the entire frequency domain. From small to large, the amount of vibration reduction can be ordered as: 49% (spring steel type VE-PTMD), 51% (pendulum type VE-PTMD), 56% (spring steel type SMA-PTMD), and 62% (pendulum type SMA-PTMD). At the resonant frequency (2.57 Hz), the displacements of the piping system are 0.54 mm (TMD), 8.99 mm (spring steel type VE-PTMD), 7.89 mm (pendulum type VE-PTMD), 3.91 mm (spring type SMA-PTMD), and 5.00 mm (pendulum type SMA-PTMD). These experimental results demonstrate that the proposed SMA-PTMDs are more effective than VE-PTMDs given the same configuration (i.e., piping system). Figure 10: Comparison of resonant vibration with SMA–PTMDs: (**a**) Spring steel type; (**b**) pendulum type. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & & & **Displacement** & **Displacement** \\
**Conditions** & & **Resonant Frequency and Corresponding** & **Reduction Ratio** & **and Reduction** \\  & & **Maximum Displacement** & **Across the Entire** & **Ratio at Resonant** \\  & & & **Frequency Domain** & **Frequency** \\ \hline \multicolumn{2}{c}{Without Control} & 2.57 Hz & 18.90 mm & - & - \\ \multicolumn{2}{c}{With TMD Control} & 2.12 Hz, 3.02 Hz & 16.98 mm, 12.86 mm & 10\% & 0.54 mm/97\% \\ \multicolumn{2}{c}{VE} & Spring Steel & 2.46 Hz & 9.72 mm & 49\% & 8.99 mm/52\% \\ \multicolumn{2}{c}{With PTMDs} & Pendulum & 2.46 Hz & 9.24 mm & 51\% & 7.98 mm/58\% \\ \multicolumn{2}{c}{control} & Spring Steel & 2.38 Hz, 2.83 Hz & 7.67 mm, 8.39 mm & 56\% & 3.91 mm/79\% \\ \multicolumn{2}{c}{} & Pendulum & 2.36 Hz, 2.75 Hz & 5.93 mm, 7.26 mm & 62\% & 5.00 mm/74\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The results of forced vibration. Figure 9: Experimental results of forced vibration when the mass of the SMA–PTMD is released mid-vibration: (**a**) Spring steel type; (**b**) pendulum type.