HYDRODYNAMIC FORCES 



135 



Fig. 29 Variation of correction coefficient for damping in heave in short waves, k:(L/X), 

 with ratio ship length/wave length, with waterplane fullness coefficient, a, as parameter (from 



Haskind, 1946) 



Damping in pilcliing wa.s gi\-en liy Haskind as 



B = B.K2-.Kg 



Damping in long waves, B, is gi\en by liim as 



B = 



1 



j.:\ 



(59) 



(60) 



4' g 



where J „ is the moment of inertia of the water plane aiiont 

 midship (i.e., about the ^/-axis). \'alnes of the coefficient 

 Kg are given in l''ig. oO. Haskind also gave the curves of 

 variation of damping with the relati\'e length of the 

 parallel section 7. 



These Haskind computations are \alid for puiv heav- 

 ing and pure pitching at zero speed. It appears that 

 within the freciuency range of practical interest, damping 

 depends but little on the speed of advance per se, hut 

 primarily on the fre(|uency of wave encounter.-" In a 

 coupled pitching and heavhig motion, as distinct from 

 pure heaving and pitching, the apparent damping coeffi- 

 cient includes additional terms dependent on both speed 

 of ath'ance and added masses. These terms occurred 

 in Ilaskind's ecjuations of coupled motion, and were dis- 

 cussed in Ha\'elock's 1955 work where he refers to them 

 as "dynamic" coupling terms. These terms also were 

 discussed by Kor\-in-Ivroukovsky and Jacobs (1957) and 

 in Appendix C 



6.5 Concluding Remarks. It appears that the work of 

 Haskind is the only comprehensive one covering the sub- 

 ject of hydrodynamic forces resulting from ship oscilla- 

 tions in wa\'es, although the wave-exciting forces are not 

 evaluated. The only comparable work (and apparently 

 further extended) is that of Hanaoka. but this is not yet 



L 

 \ 



Fig. 30 Variation of correction factor, kii, for damping in pitch 

 in short waves (from Haskind, 1946) 



adeciuately available in English. In actual applicati(Hi 

 Hanaoka, as well as Haskind, used the idealized parabolic 

 type of ship to which Michell's assumptions reasonably 

 apply. Under.standing the basic derivations of Haskind 

 requires a thorough knowledge of mathematical physics, 

 but his final results are expressed in relati^'ely simple 

 form. 



Agreement of calculated and ex]5erimental values of 

 added masses and damping forces in the region of syn- 

 chronism is much ]X)orer in this advanced treatment'-' 



'^ See Section ;i.24. 



Subject to MichcH's as.sumptions. 



