160 



THEORY OF SEAKEEPING 



and b, 1949) and Grim (1956), and also can be derived 

 by an extension of Haskind's (1946) work.^ 



Work oriented in this direction was presented in papers 

 by Suyehiro (1920, 1924) and Watanaljc (1938), but as 

 yet, a straightforward formulation of the problem in the 

 form of a set of conventional coupled differential ec[ua- 

 tions of motion has not been presented. 



Three-mode rolling, heaving and side-swaying motion 

 in smooth water was described by Ueno (1942) and, in 

 connection with evaluation of hydrodynamic pressures, 

 by Grim (1956). It was shown that, in this case, heav- 

 ing motions do not exert a significant influence on roll and 

 side sway. This, however, has no significance as far as 

 the influence of heaving on rolling in waves is concerned. 

 This was discussed as an isolated feature by Rankine 

 (1864a).'' Side .sway is significant in defining the total 

 amount of motion damping. In general, without a fully 

 stated mathematical formulation for rolling in waves, 

 similarities and differences between rolling in waves and 

 rolling in smooth water cannot be brought to light. 



Suyehiro (1920, 1924) and Watanabe (1938) presented 

 experimental data on ship rolling in side waves with par- 

 ticular attention to side sway and side drift. The hori- 

 zontal component of the orbital motion of a ship's center 

 of gravity in beam waves was found to \'ary from 0.71 to 

 1.24 of the movement of water particles in waves; i.e., 

 of the amount assumed by W. Froude (1861). As a re- 

 sult, the apparent center of rolling shifted over a range 

 from below the keel to above LWL. It also shifted to the 

 lee side of a ship, indicating a heaving component in the 

 motion. 



Under Fi'oude's assumption, conditions are symmet- 

 rical on the lee and windward sides of a rolling ship 

 since the passing simple gravity waves are assumed to be 

 undisturbed b}' the ship. In reality, there is a large 

 amount of ship-wave interference. The energy dissi- 

 pated in damping as a result of rolling and side-swaying 

 is taken out of the wave, reducing its amplitude. "\^isu- 

 ally, a relatively becalmed area is found on the lee side of 

 a ship. The forces exerted by waves are, therefore, 

 greater on the windward side, and a net force causing a 

 ship to drift to leeward is generated in addition to that 

 caused by the wind. 



As has been mentioned earlier, the only features of the 

 rolling problem considered in practice are the significance 

 of synchronism and the importance of damping in con- 

 trolling ship motions. Realization of the significance of 

 .synchronism leads to specification of the period of I'oll 

 which is controlled by choosing a suitable metacentric 

 height. In the past other aspects of the rolling motion 

 had only academic interest and were not visibly con- 

 nected with practical needs. They become important. 



' Investigation.s of a long ship in oljliciue waves, neglecting the 

 water flow effect,?, were made by Kriloff (1898) and Cartwright 

 and Rydill 1957. In the last reference cross couplings were not 

 considered. 



' The elTect of heaving may be important for a small ship in 

 steep beam waves. The reduction of the apparent gravity force 

 by the water acceleration at wave crests may drastically reduce 

 the righting moment. 



0.25 



0.50 



0.75 

 n 



1.00 



1.25 



1.50 



Fig. 5 Magnification factor plotted versus tuning factor in cal- 

 culated steady nonlinear rolling of a ship in regular waves (from 

 Vedeler, 1953). Here "amplitude" designates ratio of roll- 

 ing amplitude to maximum wave slope, and n = Tu/T„ where 

 To is natural period of rolling for very small amplitudes and T, is 

 wave period 



however, if coupling with other modes of motion is con- 

 sidered. This will be discus.sed later. 



2.22 Nonlinearities in rolling in side waves. 



Later studies of the rolling of ships concentrated almost 

 entirely on the effects of nonlinearities. As will be dis- 

 cussed later in connection with motions in irregular seas, 

 this does not appear to be a fruitful direction. The ex- 

 perimental work concentrated on nonlinearit^y of damp- 

 ing while the theoretical work emphasized the nonlinear- 

 it}^ of the righting moment. The first has been discussed 

 in Sections 2-5.3 and 5.32, and Fig. 2-21 shows that non- 

 linearity of damping does not affect significantly the na- 

 ture of free rolling. 



The righting arm h is connected with metacentric 

 height GM by the expression 



h = GM sin 0' 



where <i>' is the angle of inclination of a ship with respect 

 to the water surface. For simplicity, W. Froude (1861) 

 made the usual a.ssumption that for small angles of roll 



h = GM 0' 



(9) 



Froude showed that this expression (with G^I assumed 

 as constant) was true in fact for a ship form closely re- 

 sembling the battleships of his day. The rolling motion 

 resulting from the foregoing assumption is "isochro- 



