HYDRODYNAMIC FORCES 



129 



Roulis en eau 

 calme au point fixe 

 Ms 00226 



Modele avec quiUes 

 de roulis 



Peri ode 



4,0 



T=0.40 



Ts= 4.395(1) 

 des poids +ournan+s 



4.5 



S.O 



Fig. 23 Effect of forward speed on steady rolling of a ship 

 model (from Brard, 1949). Rolling in smooth water was in- 

 duced by harmonically oscillating moment approximately 

 equal to moment produced by waves of height/length ratio of 

 0.0226 at zero speed. Speed parameter y = 27rK/.gT where V 

 is speed, T period of rolling. T, = 4.395 is natural period 

 for small angles of roll 



Table 3 Coefficient a of the Linearized Decremental 

 Equation for Several Ships at Zero Forward Speed 



0.020 



0.082 



1 W. Froude(1874). 



2 Robb (1952, p. 271). 

 (1940). 



The data from W. Froude and Gawn 



appears to be considerable. This 

 by Robb (1952, pp. 275 and 278). 



is discussed briefly 

 Fig. 22, taken from 



K()l)l), shows a curve of declining angles for a model of a 

 Royal tiiwercign chiss battleship tested in a towing tank 

 at different speeds. Fig. 23 shows the effect of the for- 

 ward speed according to Brard (1949). 



5.33 Theoretical knowledge of damping in roll. 

 Theoretical studies to date are very few and their appli- 

 cability uncertain. The work of Ursell (19-18a, 6) can 

 be mentioned. Cartwright and Rydiil (1957) quoted an 

 e.xpression for damping from an inipublished work of 

 Ursell (1946). By use of the strip theory they calcu- 

 lated the cfiefficient k = 0.088 for the Discouerij II as 

 compared with the observed k = 0.0675. The discrep- 

 ancy is in fact much larger since the theoretical work of 

 Ursell was based on smooth elliptical sections, while 

 Table 3 shows that the experimental value corresponds 

 (although on the low side) to ships with usual appen- 

 dages. The value based on Ursell in this case is over 

 four times the value obtained from rolling of ships with- 

 out bilge keels. 



From observations at low speeds at sea, Williams 

 (1952) obtained the corresponding average value of 

 a = 0.275, i.e., k = 0.088 for HMS Cumberland. Cart- 

 wright and Rydiil (1957) have found k = 0.0675 for the 

 research shi]3 Discovery II. 



Ursell (1949rt) presented an analytical derivation of the 

 wave amplitude caused by rolling of noncircular cylin- 

 ders on the water surface. He applied this derivation to 

 cylinders described by a certain transformation which 

 ga\-e an approximately rectangular section with large 

 rotmding of corners. The wave amplitude at infinity 

 was found to be, to the first order, 



0.mk-4>{b + d){b + l.05d)\b - \.2Qd\ 



(42) 



where 



/,■ = oi-/g 



<p = amplitude of rolling 



b = half-beam 



(/ = draft 



II denotes absolute value 



An interesting featiu'e of this expression is that wave 

 amplitude (and therefore damping) vanishes when b 

 = 1.26c/ or roughl.y d = O.-iOB, where B is the beam. 

 At this d/B ratio the damping can be obtained only 

 from bilge keels. 



When d = 0.5B, the amplitude is 



40 



k-ct>b' 



(43) 



In a previous work, Ursell (1948) showed that for a 

 thin vertical plate rolling about the line of intersection 

 with the water surface the wave amplitude is 



3 



fc>rf' 



(44) 



It was mentioned that the foregoing derivations are 

 valid when the product kb is small. An exact definition 

 of smallness was not given, but generally the cciuations 



