122 



THEORY OF SEAKEEPING 



A/L 



Fig. 1 5 Maximum heaving-force coefficient for submerged 

 spheroid of L D = 5 in varying wave lengths. Solid curves 

 indicate Havelock's (1954) solution, circles Korvin-Krou- 

 kovsky's strip-method solution. C. = (heaving force) /V? 

 (from Korvin-Kroukovsky, 195 5i) 



X/L 



Fig. 16 Maximum pitching-moment coefficient for submerged 



spheroid (see caption Fig. 15). C,,, = (pitching moment)/ 



Lvfl (from Korvin-Kroukovsky, 1955b) 



method call^ for tests and analyses of special ship forms 

 designed with the specific purpose of isolating various 

 theoretical features. Under this program only a few 

 tests on normal ship forms are needed. These will indi- 

 cate the target for the investigation and will pn)\'iile the 

 final check on the ultimately synthesized information. 

 Under this analytical and experimental program, tests 

 of mathematically defined ship forms will play an impor- 

 tant part. The research program should include in- 

 vestigations of prismatic bodies (two-dimensional flow) 

 as well as three-dimensional forms with \'arious types of 

 sections and different distributions of sectional types and 

 areas along the ship length. This subject will be dis- 

 cussed further in Section (i. 



It appears that inclination of ship sides at the water- 

 line is one qualitati\'ely evident factor which affects 

 added masses and affects damping characteristics to a 

 greater degree. The physical conditions of this prf)blem 

 were described in Section 3.12. Theoretical and experi- 

 mental investigations are recommended. These can 

 start with an analysis of the damping of the 60-deg wedge 

 of Dimpker's (1934) experiments. 



3.22 Nonlinear effects in evaluating mean damping. 

 Attention should be called to the fact that definitions of 

 the added masses and damping forces in the foregoing ex- 

 position have been based on the assumption of linear 

 differential equations of motion with constant coeffi- 

 cients. In reality, the coefficients are not constant. 

 Even in the case of a wall-sided ship, such as the model 

 used by Golovato, the damping will vary with the in- 

 stantaneous mean draft of the ship sections; i.e., the fac- 

 tor/ in the exponent of equation (22). The analy.ses of 

 Golo\'ato and Gerritsma gi\'e a certain mean ^'alue of the 

 damping to be used in a linear theory. This mean \-alue, 

 however, may be affected by the nonlinearity in sectional 

 properties. 



The ultimate amplitude ratio A is the result of inter- 

 ference of many wa\'e systems which ha\'e their origin at 

 different elements of the body area. Such an interfer- 



ence can be very sensitive to the varying conditions at 

 the generating elements. Not only the ratio A for two- 

 dimensional sections but three-dimensional corrections 

 as well can be affected. Havelock's and Vosser's three- 

 dimensional corrections, discussed in Section 3.23, are 

 based on infinitely small displacements and cannot bring 

 out the effects discussed here. 



The author believes that it is important to inaugurate 

 research on evaluating the mean damping characteristics 

 of ship sections (two-dimensional flow) taking into ac- 

 count the draft and beam \'ariations during an oscillation 

 cycle. The oscillations themselves can .still be assumed 

 to be harmonic.'*' 



A similar research is suggested for three-dimensional 

 corrections, taking into account the cycle variation of 

 conditions at the ends of a pitching ship. 



3.23 Three-dimensional effects. It has been men- 

 tioned that the strip theory is approximate in that cer- 

 tain interaction Ijetween adjacent sections is neglected. 

 This was realized by F. M. Lewis (1929) and J. Lockwood 

 Taylor (1930) who developed corrective procedures for 

 their field of interest, namely, ship vibrations with a cer- 

 tain defined number of nodal points. These procedures 

 do not appear to be usable for a ship in waves. 



In order to verify the accuracy of the strip method of 

 calculations this method has been applied to a sul^merged 

 spheroid mo\'ing under waves (Korvin-Kroukovsky, 

 19.556), and compared with results f)l)tained by more 

 precise methods used by Haveloek (1954) and Cummins 

 (1954). These more precise methods cannot be applied 

 to a normal ship form in waves but can be used for ellip- 

 soids. The results are shown in Figs. 15 and 16. The 

 agreement appears to be satisfactory for all practical pur- 

 poses. Attention should be called to the fact that in the 

 strip method as used by Korvin-Kroukovsky (19556, c) 



'* This assumption is justified by the fact that shij) motions are 

 represented mathematical!}' bj' the double integral of forces with 

 respect to time. The motions, therefore, reflect primarily the 

 mean conditions and are not verj' sensitive to instantaneous force 

 changes. 



