# Investigation on Ship Hydroelastic Vibrational Responses in Waves
## 4 Analysis of Numerical and Experimental Results
### Hydroelastic Responses in Irregular Waves
#### 4.4.4 Probability Distribution of \(M_{bv}\)  
It is of great significance to investigate the asymmetric characteristics of \(M_{BV}\)s in hogging and sagging bending states of the ship especially in severe seas. The comparisons of probability of exceedance of \(M_{BV}\)s amidships are shown in Figure 29. As can be observed, the numerical results show the same tendency as the experimental counterparts for all the cases, although with some degree of departure because of the limitation of sample size and other reasons. The sagging \(M_{BV}\)s are usually larger than the hogging \(M_{BV}\)s due to the high frequency slamming \(M_{BV}\)s, especially at large \(M_{BV}\) regime. For example, the sagging \(M_{BV}\)s are almost twice the values of the hogging \(M_{BV}\)s at the probability of exceedance of \(10^{-3}\) for irregular wave condition 1. In addition, the nonlinearity of ship hull geometry also contributes to the asymmetric characteristics of the total \(M_{BV}\)s, as has been analyzed in the regular wave conditions.  
To further investigate the relationship between the low frequency \(M_{BV}\)s and the total \(M_{BV}\)s, the corresponding scatter diagrams are drawn in Figure 30, where the hogging and sagging \(M_{BV}\)s for both low frequency \(M_{BV}\)s (obtained by Fourier filtering mentioned above) and total \(M_{BV}\)s in each wave encountering period are counted and presented. The points in first and third quadrants denote hogging and sagging \(M_{BV}\)s, respectively. The diagonal that separates both the first and third quadrants into two equal parts denotes that the low frequency \(M_{BV}\)s are equal to the total \(M_{BV}\)s, meaning that the high frequency slamming \(M_{BV}\)s are zero. As illustrated in Figure 30, the high frequency slamming \(M_{BV}\)s have more effects on the sagging \(M_{BV}\)s than the hogging counterparts. The nonlinear behavior is more pronounced for experimental results than numerical results, especially for the higher speed condition, which can be explained by the fact that only weakly nonlinear effects are considered in the present hydroelasticity theory.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multicolumn{2}{c}{**Condition 1 (V = 5 km, H\({}_{123}\) = 18.5 m, Tz = 12.5 s)**} & \multicolumn{2}{c}{**Condition 2 (V = 18 km, H\({}_{123}\) = 9.185 m, Tz = 11.5 s)**} \\ \cline{2-7}  & **Experiment** & **Calculation** & **Error** & **Experiment** & **Calculation** & **Error** \\ \hline Peak significant value (M\(\mathrm{Nm}\)) & 2519.82 & 2912.30 & 15.58\% & 1604.38 & 1792.53 & 11.73\% \\ \hline Trough significant value (M\(\mathrm{Nm}\)) & \(-\)3896.74 & \(-\)4235.16 & 8.68\% & \(-\)1880.67 & \(-\)2140.40 & 13.81\% \\ \hline Mean value (M\(\mathrm{Nm}\)) & \(-\)287.73 & \(-\)454.16 & \(---\) & \(-\)37.25 & \(-\)132.46 & \(---\) \\ \hline Standard deviation (M\(\mathrm{Nm}\)) & 2247.94 & 2595.24 & 15.45\% & 1216.78 & 1466.81 & 20.55\% \\ \hline \hline \end{tabular}
\end{table}
Table 8: Statistical parameters of total \(M_{BV}\)s at amidship.  
Figure 29: Probability of exceedance of total \(M_{BV}\) responses in irregular waves. (**a**) Irregular wave condition 1; (**b**) Irregular wave condition 2.