## 3 Results
### Ship's Rolling Motion in Calm Waters in a Resistant Environment  
In this sub-section, the ship's behaviour with different loading conditions (Td) in a linear time domain under the influence of a single external force is presented with the ship in the idle condition.  
By analysing the \(\text{e}^{-\ \lambda\ \cdot\ \text{t}}\) factor from Equation (28), it can be predicted that when time (t) is infinite, the rolling angle amplitude tends to be zero. However, when the rolling angles are represented graphically over time, as shown in Figure 2, it is seen that in all cases of the ship's loading condition (Td = 8 s, 10 s, 12 s, and 14 s), the rolling angle tends to be null in the same time range, around 300 s. The only difference between the different studied cases is the frequency of the rolling angles approaching zero. This can be considered a very relevant result which, to some extent, contradicts what is traditionally assumed, that is, the greater the stability (lower Td), the less time to recover the upright position and vice versa. What is more, regardless of the ship's loading conditions, the elapsed time for it to recover the upright condition is the same, in all cases.  
Furthermore, as can be observed from Figure 2, in the four represented cases, after the same period of time has elapsed, the roll amplitude is practically the same. This is another important issue because the supposed advantage of greater stability (Td = 8 s, Figure 2a) does not translate into a lower rolling angle amplitude. Therefore, from this point of view, it would be better to have less stability (Td = 14 s, Figure 2d) because the comfort of crew members and/or passengers would be much better. In addition, equipment fatigue would be reduced.  
In order to verify that the only difference in the ship's behaviour between the different studied cases is the frequency of rolling angles, which tend to be zero, the equation of angular velocity, corresponding to the initial restoring moment (\(\theta^{\prime}\)RM), was obtained. The angular velocity is calculated by the following equation:  
\[\theta^{\prime}_{\text{RM}}=\ \theta_{\text{M}}\cdot\ \text{e}^{-\ \lambda\ \cdot\ t}\cdot[\lambda\cdot\cos(\omega_{1}\cdot\ t)\ -\ \omega_{1}\cdot\sin(\omega_{1}\cdot\ t)]\ -\ \theta_{\text{M}}\cdot\lambda\cdot\ e^{-\ \lambda\ \cdot\ t}\cdot\ \Big{[}\cos(\omega_{1}\cdot\ t)\ +\ \frac{\lambda}{\omega_{1}}\cdot\sin(\omega_{1}\cdot\ t)\Big{]}, \tag{42}\]  
Figure 2: Rolling angle and the corresponding angular velocity for different loading conditions. (a) Td = 8 s, (b) Td = 10 s, (c) Td = 12 s, and (d) Td = 14 s.  
Figure 2 shows, on each of the rolling angle graphs for each loading condition, the corresponding angular velocity as per Equation (42). From them, it can be deduced that the frequency, understood as the velocity in harmonic motion, corresponds with the previously mentioned relevant results.  
Once it was graphically demonstrated in Figure 2 that after the first 300 s the ship's rolling motion with any loading condition is practically nil, the exponential limit curve valid for any ship's loading condition, which represents the maximum rolling angle, was plotted as shown in Figure 3. The obtained relevant result can be expressed by the term \(\theta_{\mathrm{M}}\cdot\mathrm{e}^{-\lambda\cdot\mathrm{t}}\), and it can be applicable to any type of vessel.  
According to this mathematical term, represented in the red colour in Figure 3, it is possible to calculate the time to reach a certain rolling angle proposed by the ship operator. Therefore, it is a useful tool available for ship operators to know, for example, the time available to change the ship's loading condition, improving the ship's behaviour.