## 5 Analyses and Discussion of the Whipping Factor
### Discussion on the Vertical Longitudinal Bending Component  
From Section 5.1, it was found that the vertical longitudinal bending component is dominant in elastic vibration. Therefore, it is important to investigate the correlation between whipping in the vertical longitudinal bending component and the sea state and ship operational conditions. Figure 8a shows the correlation between the whipping factor and the mean wave period, Figure 8b presents the correlation between the whipping factor and ship speed, and Figure 8c shows the correlation between the whipping factor and the encounter wave period. In each figure, the data with different relative wave directions are distinguished with different marker colors, with linear regression results shown with solid straight lines. As shown in Figure 8a, there is a slight downward trend in the head, beam, and following waves. The slope of the approximately straight line is smaller in the case of beam waves than in the other cases. In the head and following seas, the wave period corresponding to the wave length equal to the ship length causes a greater hull girder wave-induced bending moment. In the case of this ship, it is approximately 14 s. Therefore, we can observe the tendency of the whipping factor to decrease toward a wave period of 14 s because of the larger wave response component. Under beam seas, such effects cannot be expected, and this may explain the smaller slope of the regression curve for the beam sea. In Figure 8c, the plot points shift to the left in the head waves and shift to the right in the following waves compared with those in Figure 8a. By plotting the data for the encounter wave period, it was more apparent that the whipping factor tended to decrease when the encounter wave period increased. Next, looking at Figure 8b, in the head wave, the whipping factor increased as the ship speed increased, and the correlation was positive, with a correlation coefficient of 0.452. In addition, in the case of the beam and following waves, the approximately straight line is level or descends, and the whipping factor also decreases as the ship speed increases. The correlation coefficients were 0.149 and \(-\)0.085, respectively. It is considered that this occurs because the impact from the seawater applied to the bow  
Figure 7: (**a**) Cumulative frequency distribution of whipping factor (H\({}_{1/3}>\) = 3.0 m), (**b**) exceedance probability distribution of longitudinal stress (H\({}_{1/3}>\) = 3.0 m). becomes stronger as the ship speed increases in the head sea, but, in contrast, it becomes milder in the following sea. In the vertical bending component, the whipping factor was found to exceed 2.0 in some cases. To determine the mechanism to induce such a large whipping factor, we made a close-up observation of those data. Table 5 includes the sea state in which the whipping factor of 2.199 was calculated. As presented here, the ship navigated in head seas with a forward speed of 19 kt. Figure 9 shows the longitudinal bending stress time series, showing occasionally large whipping responses. The Weibull distribution plot of the peak values is shown in Figure 10, where we can observe the large whipping factor. For a close-up observation of the time-series response when the large whipping was observed, Figure 11 shows the time zone in which the stress was maximum in the time-series stress data shown in Figure 9. The vibrational component is also plotted in the figure. The vibrational component is defined as the high-frequency response within 0.3-2.0 Hz, which is obtained by subtracting the wave component from the raw data. The figure shows that at approximately 310 s, whipping occurred and the amplitude of the vibrational component increased due to the impact force of the waves. Thereafter, it attenuated until 325 s, following which, the vibrational response suddenly increased again at approximately 328 s, presumably due to the impact of a large wave, which excited the bow part accidentally in the phase to accelerate the existing whipping response.