# PTMD Control on a Benchmark TV Tower under Earthquake and Wind Load Excitations
## 6 Vibration Control with PTMD  
### Design of the PTMD  
In this study, the PTMD is assumed to be effective in both directions, which means the impact may happen in any horizontal direction. Thus, bounding force can always be decomposed to the components along the \(x\) and \(y\) directions in every impact. The PTMD is first designed as an optimal TMD whose stiffness is first tuned to match a certain natural frequency of the host structure. The first 10 natural frequencies obtained from modal analysis results are listed in Table 2. The first and the second modes of the structure appear to be bending modes along the long and short axes. The PTMD here is assumed to be tuned as 0.11 Hz, the same as the first natural frequency of the structure. The thickness of the viscoelastic layer was set to 0.6 m during the simulation. As shown in Figure 1, the impact will occur as soon as the mass contacts the surface of the limitation device; and by assuming the viscoelastic material can be fully compressed, the maximum allowance for the deformation of the viscoelastic layer will also be 0.6 m. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mode** & **1** & **2** & **3** & **4** & **5** \\ \hline Frequency (Hz) & 0.110 & 0.159 & 0.347 & 0.368 & 0.399 \\ Mode & 6 & 7 & 8 & 9 & 10 \\ Frequency (Hz) & 0.460 & 0.485 & 0.738 & 0.902 & 0.997 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural frequencies of the reduced-order model. Figure 6: Benchmark model of Canton Tower. Usually, for this kind of high-rise structure, the optimal position to install a TMD device is on top of the structure [47]. However, considering the mast is made of lightweight steel and has a relatively small section, the local safety cannot be guaranteed if too much weight is connected. Additionally, it will be difficult to install at a certain height. Therefore, in this study, the top of the main tower is selected to be the optimal position of the PTMD. The corresponding nodal number is 27 in the reduced order model. For TMD and PTMD controls, the control effectiveness is how effective the PTMD is in controlling vibration. Theoretically, for a TMD device, the weight of the mass has to reach a certain mass ratio for obvious control effectiveness. However, for real complex civil structures, it is difficult to reach the ideal weight due to the carrying capacity of local elements around the connection. Examples of the world's largest TMD systems include the Taipei 101 building with a weight of 660 t, the Shinjuku Nomura building at 1400 t (two units), and the largest at the Shinjuku Mitsui building at 1800 t (six units) [48]. The TMD is only designed and installed for wind-induced vibration control purposes. In this study, after trial calculation, the weight of the mass used in the PTMD device is chosen to be varied from 100 t to 500 t which, when compared to the first order modal mass, \(2.17\times 10^{7}\) kg, the mass ratio is between 0.005 and 0.023. The stiffness of the connection and the optimal damping ratio is calculated as [49]:  
\[k = m_{0}(2\pi f_{1})^{2} \tag{20}\] \[\zeta_{opt} = \sqrt{\frac{3\mu_{t}}{8\cdot(1+\mu_{t})}} \tag{21}\]  
where \(f_{1}\) is the desired frequency to be tuned, usually set to be the first natural frequency of the structure, \(m_{0}\) is the weight of the mass attached to the structure, and \(\mu_{t}=m_{0}/m_{1}\) denotes the mass ratio of the mass to the generalized mass of the tuned mode. The aforementioned bounding model is adopted for the simulation of the bounding force. The impact is considered to happen as soon as the relative displacement of the mass and the connected node is less than the selected gap. For the proposed PTMD device, the preset gap value significantly affects the control effectiveness. If the gap is too large, bounding can hardly happen or too small a bounding force is generated due to the small relative velocity between the mass and the limitation.