###### Abstract  
Shipmasters must make several quick decisions with respect to the ship's speed and heading when new sea conditions approach. The implications for ship stability and risky situations are known but there are no guides on how they should be addressed in this limited period of time. In the present paper, and from three points of view, the ship's rolling motion in the long-term domain is analysed. Firstly, the ship's behaviour after the influence of a single and external force was studied. Secondly, the influence of successive regular beam seas, with resistance and at zero-speed conditions, was analysed. Finally, the influence of wave direction on a ship sailing at non-zero speed was investigated. Results showed that once five minutes elapse, the rolling angle tends to be null regardless of the ship's loading condition and that after a certain period of time, a coupling of the ship's rolling frequency with the waves' period and angle amplitude occurs. This circumstance was noted after three minutes for all of the ship loading conditions. Finally, novel guides for shipmasters in the form of 3D maps and polar diagrams were proposed to improve the ship's behaviour-altering navigational parameters (heading and ship's speed) when sailing in changing weather conditions. Therefore, for the three approaches, the relevant results and novel mathematical relations of linear factors were obtained which can be considered useful and applicable by the ship operators of most fishing and merchant fleets (regardless of their sizes) when they are operating under normal loading conditions. shipmaster's guide; ship's behaviour; rolling motion; coupling; different sea conditions; safe navigation +
Footnote †: journal: Applied  
0000-0002-4000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-00000-0000-00000-  
instruments, equipment, and cargo lashing system. This fatigue can trigger the failure of the lashing system and put the ship's safety at risk, with negative consequences on the vessel, crewmembers, and cargo claims. In fact, many interesting research works are focused on the problems caused by the ship's rolling motion in certain equipment installed on board [9]. However, many authors agree that the study of the rolling motion is a difficult task due to the complexity of the involved problems. Some of them are the presence of external forces, which affect the ship's stability and cause great difficulty in the mathematical model proposed [10]. For that reason, traditionally mixed techniques using empirical formulas together with the most advanced experimental processes and Computational Fluid Dynamics (CFD) techniques have been used, both in damaged and intact ship conditions [6; 11; 12]. To conclude, the rolling angles with large amplitudes and excessive accelerations experienced by ships should be a cause of concern with respect to ensuring the safety of sea transportation. Moreover, excessive rolling and losses of stability are one of the most important causes of capsizing [13; 14; 15]. For that reason, researchers such as [PERSON] et al. [16] carried out interesting studies of this motion in beam waves. With a combination of parameters such as rolling angles, wave height, ship speed, and draught, they established the cause of a ship's capsizing. In addition, their research focused on capsizing, providing a ship survivability probabilistic diagram. However, in our opinion, it is not necessary to reach a capsizing condition in order to have a non-safety situation on board. Other authors [17] also worked on roll motion modelling, generating correlated stochastic processes and simulating the ship capsizing as a function of the intensity of the wave force. They concluded that under weak wave conditions, the ship finds a stationary state after a long period of time and without reaching a capsizing condition. Capsizing occurs when the ship is sailing in strong wave conditions. However, we consider that ship capsizing does not always occur in bad sea conditions.