HYDRODYNAMIC FORCES 



123 



1.0 

 0.8 

 0.6 

 0.4 

 0.2 



10 



14 



le 



20 



Fig. 17 Ratios of damping coefficients for heaving and for 

 pitching (from Havelock, 1956). Damping coefficients are 

 denoted by E with subscripts H and P for heaving and pitch- 

 ing by three-dimensional theory and HS and PS, respectively, 

 by strip theory 



Fig. 18 Ratios of damping coefficients for heaving and for 

 pitching (from Vossers, 1956). Damping coefficients are 

 denoted by A' with subscripts H and P for heaving and pitch- 

 ing by three-dimensional theory and HS and PS, respec- 

 tively, by strip theory 



the elements of ship length were taken to lie cdiiical, so 

 that in passage of a body through control planes the sec- 

 tional area was taken as varial)ie and as a function of 

 time, A = A{t). This feature permitted the forward 

 speed of a body to be taken correctly into accovnit, and 

 also made the results applicable to bodies of relati\-ely 

 small length-to-diameter ratio (the length-diameter ratio 

 of 5 was used in the foregoing \-erilication). The calcu- 

 lations confirmed the applicability of the strip theory as 

 far as the inertial body characteristics and the exciting 

 forces caused by waves are concerned. The damping 

 forces were not considered. 



Havelock (1956) calculated damping forces by three- 

 dimensional theory for a spheroid of length-diameter 

 ratio of 8 (representative of usual ship proportions) and 

 compared the results with the strip theory. Fig. 17 

 shows the ratio of three-dimensional to two-ilimensional 

 damping (added subscript s) \-ersus the fiXMiuency param- 

 eter (u>~L/g). A'ossers (1956) made similar calculations, 

 generally following Haskind's (194G) methods for a sur- 

 face ship which was a,ssumed to be "thin" in the sense of 

 using the assumptions formulated by Michell (1898) in 

 the theory of ship wave resistance. The results are 

 shown in Fig. 18. 



It has been shown by Kor\-in-Kroukovsky and Jacobs 

 (1957) that in commercial .ships synchronism of pitching 

 or heaving oscillations occurs at a frequency parameter 

 over 10, and in slender fast ships, such as destroyers, at 

 about 18. At the.se frequency values the correction 

 coefficient for heaving motion is unity, while for pitch- 

 ing it is between 1.1 and 1.2. It has been shown in the 

 preceding section that application of these corrections 

 would increase the discrepancy between calculated and 

 measured damping on the Series 60 model. Further in- 

 vestigation of this question is therefore needed before 

 these corrections are put into practical use. 



3.24 Speed effect on damping. Figs. 12 and 1:5 con- 

 tain curves of experimental damping coellicients in 

 heaving and pitching for se\-erai \-alues of the Frouile 



number. The small ditferences between these curves are 

 probably within the experimental accuracy. In Fig. 7 

 the.se dift'erences due to speed were neglected and the 

 heave or pitch damping was represented by a single 

 curve for all model speeds. From a practical point of 

 view, the speed effect can be neglected. A more care- 

 ful examination of the figures reveals a rather weak 

 trend; i.e., at any gi\'en frequency the damping de- 

 creases slowlj^ with increase of speed. A similar but 

 more pronounced trend can be obser\-ed in Fig. 14 for 

 the parabolic ship model within the practical range of 

 abscissae from 0.9 to 1.5. The word "practical" refers 

 to a commercial ship operating in head seas, in which sea 

 condition the heaving and pitching behavior is most 

 critical. 



Most of the information found in the literature on the 

 effect of speed on damping may be misleading'^ because 

 of emphasis on frecjuency ranges outside the practical 

 interest, or because of incompleteness of the conditions 

 yielding the theoretical or experimental conclusions. In 

 a scientific examination of any physical phenomenon it is, 

 of course, often advisable to go beyond the practical 

 limits but in presenting the results of such broad investi- 

 gations one should indicate the region of practical appli- 

 cability. Failure to separate the speed efl'ect in damp- 

 ing from the fre(|uency effect, and failure to consider the 

 coupling between heave and pitch in a free-model experi- 

 ment are the shortcomings most frequently encountered. 

 A free-floating model may be excited in pitching oscilla- 

 tion only. Because of the cross couplings, however, 

 heaving oscillations also develop. The transfer of a 

 part of the exciting energy to heaving motion simulates 

 an exaggerated damping in pitch. 



.V pronounced dependence of the damping upon speed 

 apparently occurs at a low frequency of oscillation in 

 smooth water. In wave-excited oscillations such a low 



' Tlie Gerritsma data in Fig?. 7, 12, .aiiil 13 aro exot'iitcd. 



