Vertical plane response of a ship on irregular seas

This work presents the study of the sea response on the vertical plane for a 40-m length overall coastal patrol vessel (CPV-40) of Colombia’s National Navy. The condition studied is at full load and at patrol and maximum speeds. First, we analyzed the response for regular waves with incidence angles of 90o, 120o, 150o, and 180o, obtaining the response for heave and pitch motions through the SCORES program that uses the method called “girdle theory”. From the response, it was verifiedthat the amplitudeof the response is proportional to the amplitudeof the incident wave. Thereafter, the calculation of the vertical acceleration was made for four ship positions: bow, stern, cabin, and bridge, with the greatest vertical accelerations on the bow. Finally,the study was conducted for irregular sea considering as measuring parameters the vertical acceleration on the bridge and the pitching angle for the patrol operation, from the NATO document, compliance was verified of the study parameters at RMS level. It was found that for this condition the vessel complies and, hence, the personnel will not be affected in the performance of their functions.

The coastal patrol vessel CPV-40 is a design carried out and constructed by German shipyard, FASSMER, for the Colombian National Navy and it is designed to comply coastguard functions in the Caribbean Sea and the Pacific Ocean of the Republic of Colombia.Mainly border control and marine safety operations are among the functions performed by the vessel; additionally,the ship is in capacity of performing search and rescue operations, environmental control, and humanitarian aid, among others.It has a 15day autonomy and capacity to transport 24 crew members, along with a 2,000-nautical mile reach at 12 knots under 245-ton displacement conditions and zero seat.
The ship has the following operational profile for 180 days of annualwork: -Patrol speed from 7 to 9 knots: 55% -Cruising speed from 12 to 15 knots: 35% -Maximum speed from 19 to 21 knots: 10% The following shows the study of the response on regular waves from the implementation of data files of the SCORES program, which permits finding the response to motions generated by irregular seasand regular waves.Values were obtained for frequency of encounter, heave and pitch amplitude, for two speed conditions (12 and 19 knots), and design draft of 1.86 m.

Heave
Figs. 3 and 4 show that the tendency in each of the extremes is: for small ratios of λ/L, the amplitudes tend to be null, except for incident waves at 90º; asthis ratio takes on values toward the far right, the response increases until reaching its maximum value.
The maximum amplitude at a speed of 12 knots is equal to the amplitudeof the incident wave.Upon increasing the speed to 19 knots, amplitudes are produced above the amplitude of the incident wave; this is noted from a wavelength 1.5 times the length overall.If the incidence angle of the wave train is 90°, increased response does not occur with increased speed.
For both speeds, the wave train incidence angle only affects for wavelengths below 1.5 times the length overall; for higher wavelength values, the heave response remains constant in the maximum value.
With higher speed, higher heave amplitude for incidence angles 120º, 150º, and 180º.

Pitching
Figs. 5 and 6 show that the tendency in each of the extremesis: at small wavelength/ship ratios the amplitude tends to be null; as the ratio takes on values toward the far right, the amplitude shows a negative slope.
When the incidence angle is 90° the amplitude is null, as expected, given that the wave trains on the side do not produce pitching but balance because the ship's length overall is aligned with the wave crest (front of the wave).The amplitude increases as the incidence angle increases.As the vessel's course is changed toward 90º, responses decrease.
Within an intermediate range of λ/L, the maximum amplitude value occurs, presenting the resonance phenomenon.The highest resonance responses occur when the waves come from the bow, that is, at 180°.
For vertical acceleration, four points on the ship were analyzed: bow, stern, bridge, and cabin.The motion equationfor the vertical plane is given by: From the data given by the SCORES program, values are takenfor phases and amplitudes in heave and pitch with which calculation is made of accelerations for each point and these are seen in Figs. 10 and 11.
From Figs. 10 and 11, it may be noted that the highest accelerations occur in the bow; these are given for wavelength/ship ratios between 1.2 and 1.5 when speedis 12 knots and between 1.4 and 1.5 for speedof 19 knots.The lowest accelerations occur in the cabin, which is a point to stern closet o the mid section.
Continuing with the study, heave and pitch motions were considered in irregular seas, establishing as measuring parameters acceleration on the bridge and the pitching angle.NATO [1],in its publication "Common Procedures for seakeeping in the ship design Process", established recommendations of parameter acceptance limits according to the mission for military ships.This study took as acceptance limits those recommended in said publication for personnel performance; these are detailed in the following table : This work was limited to analyzing vertical acceleration on the bridge and pitching angle measured in terms of RMS.
The condition of the ship for the analysis is at full load and at patrol speed of 12 knots, with 180° incident waves with respect to ship motion (sea by (1) • Significant wave height (Hs): 1.88 m • Period (Tp): 7.6 s.
• Wind speedBft: 10 knots From the results on regular waves, we calculated the spectrum of the parameter response from the following equation: RAOS 2 of the parameters analyzed are presented in the following figures: The response spectra were obtained from equation (2); these are presented in the following: Variance and RMS of the response are obtained from the following equation: The results for the responses and the recommended limit values in terms of RMS are presented by the following table : The representation of the spectrum of the irregular sea for the state of this study was developed from the Pierson-Moskowitz formula, which is presented in Fig. 14. ( (3) From the previous table, it can be noted that for the condition at full load, navigating at a speed of 12 knots and at sea state of 4, crew performance will not be affected.
Compliance of the recommendations also implies analysis of other events of occurrence, like: immersion of the stern, slamming, and others.These can be obtained from the moments of the response spectrum from the following equation: and amplitude, the most significant in heave amplitude is wavelength, and for the pitching amplitude it is ship speed.
• The incidence angle of the wave trains is not significant for heave; only its interaction with speed and wavelength.
• The values of maximum acceleration appear in the bow and the minimum values in the cabin, which is the nearest point to the mid section.
• In the condition at full load, navigating at a speed of 12 knots, and sea state of 4 crew performance will not be affected.
From the present study, the following may be concluded: • The response amplitude is proportional to the amplitude of the incident wave.
• Of the factors studied in this work, ship speed, incidence angle of the wave trains, wavelength Moments Mo, M2, and M4 were obtained and are represented in the following tables: (5) Figs. 1 and 2 show the lateral view of the general disposition, along with the CPV-40 shape lines, respectively.

Table 3 .
Moments M o , M 2 , and M 4 for vertical aceleration