## 4 Numerical Analysis and Discussion
### Vibration Control of the PTMD  
The response of the transmission tower is shown in Figure 5 and the PTMD under multi-component seismic excitations is analyzed. To compare with the vibration reduction effect of the PTMD, the response of the transmission tower with the TMD is also carried out. The mass ratio between the PTMD and the transmission tower is 2%. The length of the cable is 0.08 m. The gap between the mass block and limiting device is 0.02 m. The vibration reduction ratios of the TMD and PTMD can be expressed as follows:  
\[\eta_{D}=\frac{D_{0}-D_{c}}{D_{0}}\times 100\% \tag{6}\]  
\[\eta_{A}=\frac{A_{0}-A_{c}}{A_{0}}\times 100\% \tag{7}\]  
\[\eta_{F}=\frac{F_{0}-F_{c}}{F_{0}}\times 100\% \tag{8}\]  
where, \(\eta_{D}\), \(\eta_{A}\), and \(\eta_{F}\) are the vibration reduction ratios of displacement, acceleration, and axial forces, respectively; \(D_{0}\), \(A_{0}\), and \(F_{0}\) are the maximum response of the displacement, acceleration and axial force of the transmission tower without control, respectively. \(D_{c}\), \(A_{c}\), and \(F_{c}\) are the maximum response of the displacement, acceleration, and axial force of the transmission tower with control, respectively. The responses of the transmission tower with the TMD, PTMD, and without control were subjected to the conditions of the El Centro earthquake and are shown in Figure 9. It can be seen from the displacement and acceleration time history curves at the top of the transmission tower that the PTMD can effectively reduce the response of the displacement and acceleration. Due to double the vibration control characteristics of the PTMD, the vibration reduction effect of the PTMD is better than that of the TMD, and the response of the transmission tower with PTMD is always smaller than that of the TMD. Note that the vibration control of the PTMD is stable. The PTMD can reduce the maximum axial force of the transmission tower, and the vibration reduction effect is different along the height of the transmission tower. Table 2 listed the vibration reduction ratio of the transmission tower under multi-component seismic excitations. The vibration reduction ratios of the transmission tower under different seismic excitations are different. Analyzing the vibration reduction ratio of the transmission tower under the El Centro earthquake conditions, the TMD can effectively reduce the peak value of displacements in the longitudinal and transverse directions by 21% and 26%, but the vibration reduction ratios of 29% \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**ID** & **Earthquake** & **Event Date** & **Magnitude** & **Station** \\ \hline EQ1 & Imperial Valley & 18 May 1940 & 6.9 & El Centro \\ EQ2 & Northridge & 17 January 1994 & 6.6 & La-Baldwin Hills \\ EQ3 & Kobe & 16 January 1995 & 6.9 & Oka \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Seismic records. and 44% of the transmission tower with the PTMD are larger than those of with the TMD. The RMS (root mean square) reduction ratios of the displacements of the PTMD in the longitudinal and transverse directions are 54% and 54%, greater than those of the TMD which are 36% and 12%. The vibration reduction ratios of the acceleration peak values of the PTMD in the longitudinal and transverse directions are 37% and 26%, and the RMS reduction ratios in the longitudinal and transverse directions are 52% and 36%, which are larger than those of the TMD. In terms of axial force, the maximum axial force of the transmission tower with the PTMD is reduced by 28%, but the vibration reduction ratio of the TMD is only 10%. The results are similar to the response of the transmission tower under the Northridge and Kobe earthquake conditions shown in Table 2. It can be seen from the table that the reduction ratio of the PTMD is significantly larger than that of the TMD owing to the double reduction characteristics.