158 



THEORY OF SEAKEEPING 



fected. The phases were not discussed, and, indeed, such 

 a discussion would have been meaningless without tak- 

 ing heave-pitch coupling into account . 



2.14 Significant characteristics of ship motions; sig- 

 nificance of phase relationships. It appears that motions 

 of a ship can reasonaltly be predicted either by model 

 tests or by calculations. The resultant cjuantitative data 

 must be interpreted, however, so as to obtain a somewhat 

 intangible description of the "seagoing fjualities" or "sea- 

 kindliness" of a ship. This description must evidently 

 be connected with the nature of a ship's service. The 

 amplitude of motion is important in certain special 

 cases; the minimum motion is desired for an aircraft 

 carrier in order to facilitate airplane landing, and it may 

 be desired for naval ships in order to provide a more stal)le 

 gun and missile platform. In commercial ships the am- 

 plitudes of motion do not appear to be important by 

 themselves, and other phenomena connected with mo- 

 tions may be more decisive in determining the seakind- 

 liness of a ship. Accelerations, which are proportional 

 to the square of the frequency as well as to the amplitude, 

 are more important for passenger and transport ships. 

 Geller (1940) and Shaw (1954) proved that there is a 

 direct connection between accelerations and sea-sick- 

 ness. On fishing trawlers accelerations impose hardships 

 on the crew at work (Mockel, 1953). On modern cargo 

 ships, on the other hand, crew accommodations are not far 

 from amidships and accelerations in pitch and heave are 

 of minor imp(5rtance in regard to the crew con\-enience. 

 These accelerations are significant, however, in the de- 

 sign of the structures supporting cargoes in the No. 1 

 hold. The critical conditions limiting the sea speed of 

 cargo ships appear to be shipping water and slamming, 

 the latter occurring mostly in light ship conditions. 

 These factors probably also limit the operational speed of 

 naval ships. 



Shipping of water and slamming are affected as much 

 by phase relationships as by amplitudes of motions. As 

 can be seen in Figs. 1, 2 and 3 the phase lag continu- 

 ously increases with increase of a ship's speed, and so 

 with increase in the frequency of wave encounter. In the 

 ca.se of a simple harmonic oscillator, i.e., equation 2-(l), 

 the phase is near zero at low freciuency, is 90 deg at 

 synchronism, and asymptotically approaches 180 deg at 

 high frequency. A similar pattern occurs in coupled 

 motion, except that the synchronous speeds in pitching 

 and hea\-ing are different and phase angles are somewhat 

 modified. The important fact brought out by this 

 phase-angle behavior is that a ship tends to follow wave 

 motion at a speed well below synchronism with the result 

 that slamming and shipping of water do not occur. At a 

 speed above the synchronous one a ship may have the 

 same amplitude of motion, but because of a large phase 

 angle tlie descending bow is likely to impinge on the 

 flank of an oncoming wave. Thus, conditions are favor- 

 able for slamming and shipping of water. This has al- 

 ready been discussed in Section 2-7 and made e\'ident in 

 Figs. 2-33 and 34. 



2.2 Rolling, Heaving, and Side Sway. Motion in the 



210 



D 



a. 



0.4 0.8 1.2 1.6 2.0 2,4 

 Ship Speed Vm M/Sec. 

 I I I I L 



3.2 



0.4 



0.8 1.2 I.G 



F=v/-/r 



2.0 



2.4 



Fig. 4 Phase lags heave to pitch, hp-n-, and pitch to wave, 

 (Sir-p (from Akita and Ochi, 195 5) 



three modes of rolling, heaving and side sway, with no 

 pitching, yawing, and surging, is approximated by a 

 ship in side .=;wells without forward speed. Were each 

 mode of the motion independent, it would be represented 

 by equation 2-(25). The motions, however, affect each 

 other and therefore are represented by a system of three 

 coupled equations of nine terms each on the left-hand 

 sides, similar in structure to the pair of heave-pitch 

 equations (2). In the present case, predominating parts 

 of the heaving and side-sway motions are caused by the 

 orbital motion of water in waves which is essentially of 

 equal magnitude in the \-erticai and horizontal direc- 

 tions. It is not definitely known whether or not one of 

 these motions can be neglected, thus simplifjang this 

 problem. The coupling of motions has usually been dis- 

 regarded in the literature on ship rolling in waves ex- 

 cept in the case of roll stabilization. In the latter case 

 (Chadwick and Klotter, 1955a and 6), heaving motion 

 was neglected and coupled rolling and side-sway e(|ua- 

 tions were treated. 



W. Froude (18G1), who first developed a theory of ship 

 rolling, is largely responsible for the lack of an explicit 

 treatment of coupling. Fronde's simple and elegant 

 solution satisfied his immediate needs, and, in fact, ap- 

 pears to have provided all the rolling information of prac- 



