# Study on Local Vibration Control of the 100 m X-BOW Polar Exploration Cruise Ship
## 3 Results
### Local Vibration Control Analysis of TMD  
In order to carry out TMD control on the local vibration of the 100 m polar expedition cruise ship, the main frequency of the excitation source should be determined first, and the frequency search range should be determined according to the main frequency of the excitation source, which is generally \(\pm\)15% of the main frequency of the excitation source. The local modes of the 100 m polar expedition cruise ship should be searched in the frequency search range, and the local modes to be controlled should be selected. The modal amplitude and modal mass of the local structure were extracted by Nastran, and the equivalent mass of the local mode was calculated according to Equations (12) and (13). Using the equivalent mass, the optimal frequency ratio of TMD was obtained from Equation (4), and the optimal damping ratio of TMD was obtained from Equation (5). The vibration reduction effect of TMD was analyzed by the finite element method. The X-Bow polar expedition cruise vessel, with a length of 100 m, exhibits a primary excitation frequency that is pivotal for its propeller blade excitation. This frequency is identified at 12.5 Hz, with a search bandwidth extending from 11.25 Hz to 13.75 Hz. Within this specified bandwidth, a comprehensive modal analysis of the ship's structural integrity was conducted. Notably, at a frequency of 13.4 Hz, pronounced local modal responses were observed at the stern section of the seventh deck, as depicted in Figure 7. A detailed examination of the data, as presented in Table 2, reveals that the measured vibration velocity at this specific location reached a peak value of 1.07 mm/s. This value surpasses all other recorded measurements within the scope of Table 2. Consequently, it is inferred that the localized structural dynamics at the 13.4 Hz mode are particularly susceptible to the resonant effects induced by the propeller's blade frequency excitation. Because this mode shape is coupled with other local mode shapes, the equivalent mode mass of this local mode shape cannot be calculated by the whole modal. Therefore, it is necessary to extract the centralized mass matrix and modal deformation of each node in the mode shape and use Equation (14) to calculate the equivalent mass of the local mode. \begin{table}
\begin{tabular}{c c c c c} \hline
**Deck** & **Area** & **Test Value (mm/s)** & **Simulation Value** & **Error** \\ \hline DECK7 & The sun deck & 1.07 & 0.83 & 22.43\% \\ \hline DECK6 & Passenger compartment & 0.046 & 0.04 & 13.04\% \\ \hline DECK5 & The restaurant & 0.31 & 0.3 & 3.23\% \\ \hline DECK4 & Passenger compartment & 0.22 & 0.18 & 18.18\% \\ \hline DECK3 & Passenger compartment & 0.11 & 0.094 & 14.55\% \\ \hline DECK2 & Laundry room & 0.09 & 0.08 & 11.11\% \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of vibration response simulation value and test value effective value. Figure 7: Local modes of the seventh sun deck. (**a**) Global map of local modes; (**b**) Enlarged view of local mode. where the second-order mode is the main control target. After loading the TMD, the control effect of the response at 13.4 Hz is more obvious, but some working conditions have a certain amplification effect on the response at 10 Hz and 18.8 Hz. The response of the frequency band from 12 Hz to 15 Hz is amplified, as shown in Figure 9, which shows that when the mass ratio is small, as the mass ratio increases, the vibration absorption effect of the TMD becomes obvious, and the response curve is flat. In order to obtain the most suitable TMD scheme for this structure from Table 3, the control effect of the TMD response at 13.4 Hz and the amplification effect at 10 Hz and 18.8 Hz need to be comprehensively evaluated. The control effect can be evaluated through Equation (29), such that  
\[\Delta L=20\text{lg}(X_{0}/X_{1}) \tag{29}\]  
where \(X_{0}\) is the vibration amplitude at 13.4 Hz when TMD is not installed, \(X_{1}\) is the vibration amplitude at 13.4 Hz after TMD is installed, and \(\Delta L\) is the control effect.