## 5 Conclusions  
In the present research, a ship's behaviour in a rolling motion was analysed and mathematically modelled. Firstly, it was modelled in zero-speed conditions, from a theoretical point of view, when the ship is subjected to a single and external force as a result of its own restoring moment. For some given initial assumptions, and after considering different studied cases, it was found that after the first 300 s, the angle of the rolling motion is practically nil, regardless of the ship's loading condition. Then, it was demonstrated that, contrary to what may be considered, in the long-time domain, the initial mass distribution does not influence the time to recover the upright position. A very simple mathematical relationship that shows the time needed to reach the maximum rolling angle was found, the results of which were corroborated with the corresponding angular velocities. Afterwards, once the full spectrum of sea state conditions was modelled, the mathematical models of the ship's rolling motion under the influence of beam waves with resistance, in a long-time domain and at zero-speed conditions, were obtained. In this case, relevant results were found. After the first 150-200 s and onwards, it was shown that the coupling of roll frequencies with waves' periods was periodically repeated with the same amplitude, regardless of the ship's loading condition. What is more, once this period of time elapsed, the ship rolls according to the waves' frequency, i.e., the ship's loading conditions are not relevant. This is an aspect essential to the ship's overall safety and its behaviour at sea. For instance, the cargo lashing design should consider the transversal accelerations sustained by the ship during the sea route, not the roll angle amplitude. For this reason, the results here obtained can be considered useful for ship operators given that it allows them to know, well in advance, from what moment there will be no sudden increases in acceleration which cause the lashing to break. Furthermore, in the case of passenger ships, it can be known ahead of time how to reach a stable condition as smoothly as possible for the passengers' comfort. This condition implies the worst stability (i.e., a higher Td), but it is remarkable to know that, after a certain period of time, the ship will roll according to the waves' frequencies. However, as it was also graphically shown, in the first moments under the influence of waves, the behaviour of each ship depends on the relationships between Td and Tw. Lastly, a mathematical model which allows for the calculation of the time to reach the coupling frequencies for a given rolling angle was obtained. The results of these two first studies are valid considering certain assumptions such as the ship at zero speed under the influence of beam waves, and linear factors during the rolling motion. Finally, the ship's rolling motion while sailing in waves coming from any constant direction and the ship sailing at non-zero speed was studied. In these universal conditions, the encounter period (Te) replaces the wave's period (Tw) as being the most influential in the ship's behaviour. For that reason, as the shipmaster has influence in the mentioned Te to modify navigational parameters such as the heading and ship's speed, a guide for shipmasters in the form of a 3D map and polar diagrams is proposed to improve the safety of navigation by enabling the selection of adequate values for the ship's speed and heading angle, depending on the sea conditions found. In particular, changing the heading angle and/or ship's speed will help to reach, until the coupling process is finished, an adequate rolling angle and, moreover, change the time to adjust to these relatively new sea conditions. It even indicates the ship speeds and \(\alpha\) angles to avoid in order to stay as far away as possible from synchronism conditions.