344 



THEORY OF SEAKEEPING 



i.e 



10 



14 



16 18 



20 



Fig. 



16 Ratios of three-dimensional to two-dimensional cal- 

 culations of damping coefficients for submerged spheroid of 

 L/B = 8 (Fig. 1 of Havelock, 1956. Subscript H indicates 

 heaving, P pitching, and 5' strip method) 



10 



12 



^L/g 



16 18 



20 



Fig. 17 Ratios of three-dimensional to two-dimensional cal- 

 culations of damping coefficients for thin "Michell" ship of 

 B/H = 2 (Fig. 2 of Vossers, 1956. Subscript H indicates 

 heaving, F pitching, and S strip method) 



Holstein (1936, 1937o and 19376) and Havelock (1942) 

 represented the heaving body by .a di.stribution of 

 harmonically pulsating sources along the bottom. In 

 that case the amplitude ratio A is given by 



A 



-hf 



sin {hy) 



(37) 



where A'o = top^/g, y is the half-beam 5/2, and / = S/B, 

 the mean draft of a ship section. This theoretical result 

 was confirmed by Holstein 's own experiments; the re- 

 sults of the many experiments appear consistent, yet 

 some doubt may be felt because of the smallness of the 

 test tank. The theory is approximate and acceptance of 

 it necessarily hinges on agreement with experimental 

 results. 



A more nearly exact theory was developed by Ursell 

 (1949, 1953, 1954) for a heaving semicylinder, and by 

 Grim (1953) for a number of analytically defined sections 

 closely approaching practical ship .sections. In the ca.se 

 of a semi-circular section Grim's results agree with Ur- 

 sell's. In their theory the damping force is calculated as 

 a boundary-value problem. Unfortunatelj' no experi- 

 mental verification has been provided. 



Both theories, that of Holstein and Havelock which 

 does not satisfy the boundary condition on the surface of 

 the body and that of Grim which fulfills all the boundary 

 conditions with a good degree of accuracy, give approxi- 

 mately the same results at the frequency for synchronism 

 for a normal ship, but very different results at higher fre- 

 quencies. The Holstcin-Havelock values of .4, equation 

 (37), were used in the 1955 paper where the results of the 

 calculations generally indicated overdamping. In recent 

 work Grim's ^-values, which are presented in the form of 

 charts, have been substituted. This has given a better 

 correlation of the calculated and experimental ampli- 

 tudes of the motions of the ship models. 



The experiments of Golovato (1956) have been men- 

 tioned already in connection with coefficients of added 

 mass. In those experiments the coefficient h of damping 

 force was also measured. The experimentally measured 

 h was found to be lower than the coefficients computed 

 by the different methods of Havelock and Grim, that 



computed by Grim's method being closer to the measured 

 value. Another comparison by Golovato of computed 

 and measured coefficients using the experimental data of 

 Haskind and Riman (1946) gave similar results. 



The computed coefficients are ba.sed on the strip 

 method of analysis which assumes two-dimen.sional fluid 

 flow, while Golovato's and Haskind and Riman's tests 

 were made with ship models, i.e., in three-dimensional 

 flow. The relation of damping in three-dimensional flow 

 to damping in two-dimensional flow can be estimated on 

 the basis of the work of Havelock (1956) and of Vossers 

 (1956). Havelock calculated the damping coefficient by 

 two methods, three-dimensional and strip, for a sub- 

 merged spheroid with length-beam ratio of 8, a fair value 

 for comparison with ship models. Vossers made similar 

 computations from Haskind 's theoretical results (1946) 

 for a "thin ship" in the sense of the approximations in- 

 troduced by Michell (1898) in his theory of wave re- 

 sistance of ships. The results of both are reproduced here 

 in Figs. 16, 17 as ratios of the coefficients of damping 

 in heave or pitch by the two methods.'' It should be 

 remembered that accurate evaluation of damping is most 

 important in the vicinity of synchronism. The values of 

 the frequency parameter co-L/g (or (x^-L/g in the figures) 

 at synchronism are gi\'en in the last two columns of 

 Table 5 for the eight models to which the computational 

 method outlined here has been applied. (Also included 

 in Table 5 are the parameters L/B, k-i and k' for com- 

 parison with Havelock's submerged spheroid.) 



Omitting from consideration the original trawler and 

 yacht, the lowest values of the parameter w-L/g are 10.4 

 in heave for the T-2 tanker and 11.7 in pitch for the 

 Series 60 hull, while for the models with displacement- 

 length ratio of 60 co-L/g is above 17.6. At the.se values 

 Figs. 16 and 17 indicate negligible corrections for damp- 

 ing in heave and small corrections for damping in pitch 

 (0 to 15 per cent based on Havelock's computations and 

 10 to 20 per cent from Vossers'). Since the corrections 

 for thi'ee-dimensional effect are negligible or small and 



" The subscript S in the figures denotes "strip" method. 



