128 



THEORY OF SEAKEEPING 



43 so 



Nur^te'- of Swings 



Fig. 22 Effect of forward speed on rolling of model of "Royal Sovereign" class of battleship 

 (from A. M. Robb, "Theory of Naval Architecture," 1952, p. 278) 



B/A = 2kwo 



C/A = 0)0= = {27r/T)- 



coo = ciiTular frequency of free undamped oscilla- 

 tions 



T = period of undamped oscillation. 



Damping increases the period of oscillation so that it 

 becomes 



T, = T/{1 - K-y^- (33) 



The well-known solution of ecjuation (32) for a par- 

 ticular case, in which ^ = is taken at the ma.ximum 

 initial inclination </)o; i.e., when d(i>/dt = 0, is 



4> = 4>oC~'"'' cos o>t (M) 



The plot of roll angle versus time is shown in Fig. 21, 

 taken from Robb (1952, p. 259). It appears to consist 

 of a series of nearly sinusoidal half-cycles, the ampli- 

 tudes of which diminish as shown by the exponential 

 factor in equation (34). If the rate of amplitude de- 

 crease is not too rapid, the tangency points of the en- 

 velope curve to the oscillatory curve can be taken ap- 

 proximately at cos ccqI = 1. Substituting the value of 

 too in terms of the period T and expressing the time in 

 terms of T, n — t/T, the equation of the envelope is 

 obtained as 



<p = 0oe --''"' (35) 



On the other hand, W. Froude (1874) shows that dec- 

 remental equation (31) when linearized, a = a and b 

 = 0, corresponds to the equation of the en\"elope curve 



B = /1.2/.CO0 = 4Aa/T 



(39) 



1 , 00 



n = -= log -- 

 2a <p 



(36) 



2a has been written above instead of Froude's a, as 

 Froude recorded n in half-cycles while counting in whole 

 cycles is now customary. Ecjuation (36) also can be 

 expressed as 



4> = 0uc--^" (37) 



From comparison of equations (35) and (37) it follows 

 that 



The foregoing has been presented in detail since the 

 decrement al e(iuation (31) is widely used in British litera- 

 ture, and most: of the practical data on damping of ships 

 in rolling is available as coefficients a and b. However, 

 descriptions of the relationship of these coefficients to 

 the ecjuations of motion are few and not clear. A good 

 but very brief one was given by Williams (1952). A 

 much more widely used method in general vibratory 

 problems is the "logarithmic decrement" defined as 



log {<t>l/(p2) 



(40) 



where 4>i and </>2 are amplitudes of any two succeeding 

 oscillations. Tiie logarithmic decrement is measured 

 by the slope of log 4> versus the number of cycles of oscil- 

 lation n. It is related to the nondimensional damping 

 coefficient k, and to the coefficient a of the linearized 

 decremental equation by 



2a 



(41) 



a = TTK 



(38) 



5.32 Empirical data on damping in roll. In the cur- 

 rent literature on ship motions, strong emphasis is put on 

 nonlinearitj' of damping, and reported \-alucs of the coeffi- 

 cients a and b are shown to vary widely and irregularly. 

 Their relationship to a ship form is hardly ever dis- 

 cussed. However, if the coefficient a of the linearized 

 equation is considered, an idea of the order of magnitude 

 can be established. Assuming, for instance, that the 

 linearized equation must match equation (31) at 4> 

 = 7.5 deg,-' Table 3 can be compiled as an example from 

 readily available data. 



The range of fluctuation for the values of a in each 

 group shown in Table 3 is approximately 50 per cent 

 from the mean for ships without bilge keels, and 40 per 

 cent for those with bilge keels. This fluctuation is sur- 

 prisingly small, considering the fact that neither the 

 details of ship forms nor the details of keel and bilge keel 

 constructions have been considered. The powerful effect 

 of bilge keels on rolling is brought out vividly by the 

 table and also by Figs. 21 and 22. 



The effect of the forward speed on damping in roll 



and the coefficient of damping B in equation (25) is 



^' Corresponding to the mean value for a fairly heavy rolling. 



