## 5 Dynamic Loading Mitigation of Offshore Platform Using TMD
### Dynamic Responses and Discussion
#### 5.2.3 Experiment on Frequency Detuning Sensitivity of TMD  
To assess the sensitivity of TMDs to their own frequency detuning, this study conducted supplementary numerical analyses focusing on the impacts of mass or stiffness variations caused by disturbances on TMD performance. Building upon the previous experimental results, the misaligned typhoon and typhoon-wave condition was selected as the analysis backdrop, as it poses challenges to TMD performance and demonstrated favorable control effects in the original study. The research concentrated on the frequency offset in the deck's \(Y\)-direction, as well as displacement and acceleration response metrics. Four disturbance scenarios were designed in this chapter, where the TMD's mass was either reduced or increased by 10% of its original value. Similarly, the TMD's stiffness was either decreased or increased by 10% of its original value, while keeping other variables constant. These four parameter-altered scenarios were individually compared against the undisturbed TMD scenario, and the results are presented in Figures 18 and 19. To better demonstrate the sensitivity of TMD to frequency detuning, this paper processes the simulation data. First, the frequency offset of TMD, denoted as \(\Delta f_{off}\), is introduced (Equation (26)). \[\Delta f_{off}=\frac{f_{new}-f_{0}}{f_{0}}\times 100\% \tag{26}\]  
where \(f_{0}\) is the original TMD frequency, i.e., the value \(f_{opt}\) calculated by Equation (6), and \(f_{new}\) is the disturbed frequency. \(f_{new}\) can be calculated by Equation (27). Figure 17: Base shear forces under misaligned 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. \[f_{new}=\frac{1}{2\pi}\sqrt{\frac{K_{TMD}}{M_{TMD}}} \tag{27}\]  
In addition, two vibration reduction performance offsets are introduced: the displacement suppression offset \(\Delta d_{off}\) and the acceleration suppression offset \(\Delta a_{off}\) (Equations (28) and (29)). \[\Delta d_{off}=\frac{d_{new}-d_{0}}{d_{0}}\times 100\% \tag{28}\]  
\[\Delta a_{off}=\frac{a_{new}-a_{0}}{a_{0}}\times 100\% \tag{29}\]  
where \(d_{0}\) is the structural displacement suppressed by the TMD in Section 5.2.2, and \(d_{new}\) is the structural displacement suppressed by the TMD after perturbation. \(a_{0}\) is the structural acceleration value suppressed by the TMD in Section 5.2.2, and \(a_{new}\) is the structural acceleration value suppressed by the TMD after disturbance. The specific calculation results are presented in Table 3. It can be found that the perturbation direction is consistent with the theoretical expectation: \(\Delta f_{off}=\propto 1/\sqrt{M_{TMD}}\), \(\Delta f_{off}=\propto\sqrt{K_{TMD}}\). Both the reduction in mass and the increase in stiffness result in a positive frequency offset; however, the vibration reduction effect exhibits asymmetric attenuation. The sensitivity of mass perturbation is higher than that of stiffness perturbation. The displacement attenuation amount (\(-8.60\%\)) for a \(10\%\) reduction in stiffness is \(1.96\) times that for a \(10\%\) reduction in mass (\(-4.38\%\)). The acceleration attenuation amount (\(9.38\%\)) for a \(10\%\) reduction in stiffness is \(1.73\) times that for a \(10\%\) reduction in mass (\(-5.40\%\)). Figure 18: The time-domain simulation results of deck acceleration in the \(Y\)-axis direction under misaligned typhoon and typhoon-wave loads. (**a**) The comparison between \(90\%\) M\({}_{TMD}\) and \(100\%\) M\({}_{TMD}\); (**b**) The comparison between \(110\%\) M\({}_{TMD}\) and \(100\%\) M\({}_{TMD}\); (**c**) The comparison between \(90\%\)\(K_{TMD}\) and \(100\%\)\(K_{TMD}\); (**d**) The comparison between \(110\%\)\(K_{TMD}\) and \(100\%\)\(K_{TMD}\).