## 6 Results
### Reduction of Oscillations Caused by Sea Waves  
The effect of wave forces was simulated using an external force, as defined in Equation (9), with amplitude \(A\) and angular frequency \(w\). The application of this external force at the edge of the barge generates a moment, as described in Equation (10). In the experiment, the amplitude \(A\) was set to 90,000 kN, with \(w\) initially set to \(2\pi\times 0.08\) rad/s for the first 100 s, then increased to \(2\pi\times f2\) rad/s for the subsequent 100 s. Both the tower and the platform were initially at rest. Three different experiments are executed using different values of \(f2\); these are \(f2=0.75*0.54\) Hz, \(f2=0.54\) Hz and \(f2=1.25*0.54\) Hz. The frequency of the second main vibration mode of the tower is 0.54 Hz; therefore, these experiments excite this vibration mode. Figure 11: TTD and platform pitch angle for optimized passive TMD (dotted red line) and optimized amplitude-based semi-active TMD (green line). Figures 12 and 13 compare the TTD and platform pitch angle obtained with the proposed control strategies. During the first 100 s, the system response is similar for both the passive and the frequency-based semi-active controls. This confirms that the semi-active control successfully detects the current oscillation frequency (0.08 Hz) and assigns the optimal spring stiffness and damping coefficient parameters, which coincide with those of the optimized passive TMD. As a result, both signals appear overlapped in the figure during this period. Only the optimized passive TMD results are included in the figure to avoid clutter, as a comparison with other passive TMD configurations was already presented in Figure 10.  
Figure 12: TTD under induced external frequency. Comparison of semi-active and passive strategies. **(a)**\(\Omega=0.75\times 0.54\) Hz, **(b)**\(\Omega=0.54\) Hz, **(c)**\(\Omega=1.25\times 0.54\) Hz. After 100 s, the frequency of the external force changes to 0.54 Hz, exciting the second main vibration mode of the tower and altering the system response. The tower oscillation amplitude is reduced more effectively by the frequency-based semi-active control (red line) compared to the optimized passive TMD (dashed blue line) and the amplitude-based semi-active control. This demonstrates that the proposed control strategy not only accurately detects the oscillation frequency but also dynamically adjusts the TMD parameters (spring stiffness and damping coefficient) accordingly. These results validate the effectiveness of the proposed control in mitigating wind turbine vibrations. Figure 13: Platform pitch angle under induced external frequency. Comparison of semi-active and passive strategies. (**a**) \(\Omega\) = 0.75 \(\times\) 0.54 Hz, (**b**) \(\Omega\) = 0.54 Hz, (**c**) \(\Omega\) = 1.25 \(\times\) 0.54 Hz. Table 5 compares the suppression rate of the passive and semiactive strategies. The suppression rate of the frequency-based semi-active control is higher than that of the other strategies. The highest difference appears when the frequency matches the main vibration mode; in this case, the suppression rate of the frequency-based semi-active control is 24.59%, which is significantly higher than that of the optimized passive TMD (2.57%). Figure 14 presents a comparison of the power spectral density (PSD) of the TTD for the different TMD strategies when f2 = 0.54 Hz. As expected, two prominent peaks are observed: one at 0.08 Hz and another at 0.54 Hz. The strategy that most effectively dampens the low-frequency peak (0.08 Hz) is the optimized passive TMD. This can be attributed to the fact that the passive TMD was specifically optimized during a free decay test to minimize its standard deviation. In the free decay scenario, the low-frequency oscillation contributes most significantly to the standard deviation, which is why the optimization process targets this frequency. This also explains why the optimized passive TMD performs poorly at the higher frequency (0.54 Hz), resulting in the largest peak at that frequency.