HYDRODYNAMIC FORCES 



127 



10 20 30 40 50 60 70 

 Seconds 



20 25 30 

 Seconds 



Fig. 2 1 Curves of declining rolling oscillations in still water for 

 a destroyer at zero speed (from A. M. Robb, "Theory of Naval 

 Architecture," 1952, p. 2 59). (.a) Full-size ship, (b) complete 

 model, (f) naked hull. Linearized envelope curves are shown 

 by solid lines 



computed. In general, / = lo + I", where /o is the mo- 

 ment of inertia of the ship itself and /" the added or hy- 

 drodynamic moment of inertia. Or, if /' is the moment 

 of inertia of the water displaced by a ship, I" = l^.J' ■ 

 The coefficient of accession to inertia in roll ^-^j- can be 

 estimated roughly by comparison with ellipsoids, or can 

 be computed by the strip theory. Generally, it is small, 

 and Manning (1942, pp. 21, 27) indicated that direct 

 use of equation (I^O) with ship's radius of gyration /; was 

 satisfactory. Grim (1956), considering low oscillation 

 frequency, listed the expressions for ccjmputation of the 

 accession to inertia fc^x for ship sections defined by F. AI. 

 Lewis' (1929) formulas. As an example, for a ship sec- 

 tion with a draft 0.4 of the beam and a section coefficient 

 of 0.93, kri is found to be 0.07. The coefficient of ac- 

 cession to inertia k^^ is increased, however, by the addi- 

 tion of appendages, particularly bilge keels. 



In the asymptotic case of a high fre(iuenc3', the rolling 

 of a rectangular ship section of draft/beam ratio 1 '2 is 

 equivalent to rotation of a submerged square prism. 

 For this case Wendel (see Figs. 4 and .5) computed /o„ = 

 0.276. The addition of bilge keels, 0.083 of the beam in 

 width, gave a computed Avi = 0.63 (by interpolation of 

 the data in Fig. 4). T. B. Abell (1916) found experi- 

 mentally the corresponding values of 0.28.5 and 0.72 for 

 /i:„. The water viscosity effects did not, therefore, 

 change significantly the values of the added masses. 



The importance of taking the effect of bilge keels into 

 account has been vividly demonstrated by these studies. 



The foregoing data refer to water of infinite depth. 

 The added mass in rolling, as well as in heaving and 

 pitching, is strongly affected by the shallowness of water. 

 Data on the effect of shallowness on rectangular ship 

 sections will be found in Koch's (1933) paper. The 

 depth of water must, therefore, be taken into account in 

 ;ill vil)rati(iii iiiid rolling experiments. 



5.3 Damping Moments. It has been established by 

 W. Froude (1861, 1872, 1874), and confirmed by others, 

 that the value of damping coefficient B for a bare hull 

 depends mostly on dissipation of energy in waves and 

 only to a small extent on viscous forces. In Baumami's 

 (1937) experiments with rolling circular cylinders, it was 

 observed that, once excited, oscillations damped out 

 very slowly. Bilge keels can cause a large dissipation of 

 energy by \'orticity, as will be shown in Section 5.33. 



The primary sovu'ce of information on damping to date 

 has been the observations made on the rate of amplitude 

 decay in rolling in smooth-water oscillation. Such ex- 

 periments were first conducted by W. Froude (1874) 

 and thereafter by many others (for instance, Ciawn, 

 1940; Williams, 1952). " Fig. 20 shows the plot of roll 

 amplitude (/> versus the number /(. of oscillations as oli- 

 tained by Froude (1874) for tuo ships. Fronde found 

 that the experimental data were closely fitted by a curve 

 of the type 



-d4>/dn = d<t> + /></>- (31) 



where the coefficients d and h were different for various 

 ships and are shown by the legend on the figure."' Also 

 shown are the best ffis using only the first (linear) or 

 the second (c[uadratic) terms with a suitable adjustment 

 of coefficients.-'- The adjusted coefficient will be desig- 

 nated by a. The reasonably good fit by a single linear 

 term should be noted. Froude and many suliseciuent 

 writers emphasized the nonlinearity of the damping 

 coefficient as shown in equation (31). 



5.31 Linear approximation. The linear approxima- 

 tion to the extinction v\w\v {d4>il>i = ac^) is particularly 

 valuable because of its direct connection with the damp- 

 ing coefficient B in the linear differential equation (25). 

 This equation can be rewritten for free oscillations in calm 

 water as 



(/) -|- 2x0)00 + 'JJo'</ 







(32) 



t-here 



'" A Ijar has been placed over the letters as a reminder that 

 they should not be confused with the coefficients n,A and b,B in 

 the differential equations of motion ( 1 and 25). The coefficient 

 B in the latter case is defined as a function of equation (31); i.e., 

 B = B{<t>) = f{drt>/d'i)- The symbols <i and b used by Froude in 

 Fig. 20 correspond to il and h used in this monograph. 



^' This jirocedure is to lie contrasted with the one used l),y 

 Golovato (l(t57a,6) wlio linearized the similarly expressed damp- 

 ing in heaving merely l)y letting b = 0, and not changing the value 

 of d. It will be understood that the word "linearized" as used 

 in this monograph implies adjustments of both coefficients so 

 as to give the best approximation to a function in the range of the 

 independent variable of greatest interest in practical jiroblems. 



