## 5 Dynamic Loading Mitigation of Offshore Platform Using TMD
### Dynamic Responses and Discussion
#### 5.2.1 Aligned Typhoon and Typhoon-Wave  
Figures 10-12 systematically demonstrate the comparative responses of acceleration, velocity, and displacement in the \(X\)- and \(Y\)-axis induced by aligned typhoon-wave loads on the Deck equipped with TMD (DTMD) versus the Initial Design deck (IDD), in both the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Soil**} & \multicolumn{2}{c}{**Depth (m)**} & \multirow{2}{*}{**Effective**} & \multirow{2}{*}{**Friction**} & \multirow{2}{*}{**Undrained Shear**} \\ \cline{3-4} \cline{6-7}  & & **Top (m)** & & & **Bottom (m)** & & **Weight (kN/m\({}^{3}\))** & **Angle (\({}^{\circ}\))** & **Strength (kPa)** \\ \hline
1 & Sand & 0 & 10 & 9 & 23\({}^{\circ}\) & - \\
2 & Sand & 10 & 14.2 & 9.2 & 23.5\({}^{\circ}\) & - \\
3 & Sand & 14.2 & 21.5 & 9.5 & 26.5\({}^{\circ}\) & - \\
4 & Sand & 21.5 & 33.1 & 9.6 & 27\({}^{\circ}\) & - \\
5 & Clay & 33.1 & 51.1 & 8.8 & - & 96 \\
6 & Sand & 51.1 & 53.1 & 9.7 & 29\({}^{\circ}\) & - \\
7 & Clay & 53.1 & 65.1 & 9 & - & 134 \\
8 & Sand & 65.1 & 67.9 & 9.8 & 30\({}^{\circ}\) & - \\
9 & Clay & 67.9 & 75.5 & 9.2 & - & 163 \\
10 & Clay & 75.5 & 81.9 & 9.2 & - & 179 \\
11 & Sand & 81.9 & 82.8 & 9.9 & 30.5\({}^{\circ}\) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Soil property. Figure 9: Two excitation patterns. (**a**) Aligned typhoon-wave loads; (**b**) Misaligned typhoon-wave loads. time and frequency domains. In the \(X\)-axis direction, the maximum values of displacement, velocity, and acceleration in the time domain for the IDD are 0.17 m, 0.68 m/s, and 0.42 m/s\({}^{2}\), respectively. For the DTMD, the corresponding values are 0.15 m, 0.52 m/s, and 0.27 m/s\({}^{2}\), representing reductions of 18.18%, 13.33%, and 26.19%, respectively. In the \(Y\)-axis direction, the maximum values of displacement, velocity, and acceleration in the time domain for the IDD are 0.06 m, 0.20 m/s, and 0.19 m/s\({}^{2}\), respectively. For the DTMD, the corresponding values are 0.04 m, 0.17 m/s, and 0.13 m/s\({}^{2}\), representing reductions of 33.33%, 15%, and 31.58%, respectively. For both \(X\)- and \(Y\)-axis directions, the acceleration and velocity responses exhibit predominant spectral energy concentration in the low-frequency range, while the displacement displays two distinct peaks within the same frequency range. The implemented TMD demonstrates significant mitigation of low-frequency peaks in acceleration and velocity, with additional reduction observed in displacement peaks. To gain an in-depth understanding of the TMD's performance, the relative energy dissipation ratio is used to quantify its energy dissipation efficiency: \(E_{dis}/E_{inp}\). Where \(E_{inp}\) is the input wave energy, and \(E_{dis}\) is the energy dissipated by the TMD. Details of the formula can be referred to in [37]. Since this paper mainly focuses on the TMD's performance and the TMD Is Installed at the top, the Input energy at the top deck and the energy dissipation of the TMD are calculated. The simulation results show that the relative energy dissipation ratio is 62.3% in the \(X\)-direction and 65.7% in the \(Y\)-direction. Figure 13 illustrates the influence of the TMD on the base shear at the leg bottom under aligned typhoon-wave load conditions. In the \(X\)-axis direction, the maximum and minimum base shear values for the IDD are 4715.8 kN and \(-743.8\) kN, respectively,  
Figure 10: Deck displacement under aligned typhoon and typhoon-wave. (**a**) Time-domain simulation results in the \(X\)-axis direction; (**b**) Frequency-domain simulation results in the \(X\)-axis direction; (**c**) Time-domain simulation results in the \(Y\)-axis direction; (**d**) Frequency-domain simulation results in the \(Y\)-axis direction. while the corresponding values for the DTMD are 3880.8 kN and \(-667.0\) kN, representing reductions of 17.71% and increases of 10.33%, respectively.