154 



THEORY OF SEAKEEPING 



500 



a> 

 (P 



300 g^ 

 o 



cn 



o 



•loo";; 



if) 



D 



Of 

 500 



CO 



300 fe-. 



Ci 



cn 

 



w 



500 



soo 



300 tn 



100 



SL 



2 4 6 



Model Speed, Ft/Sec 



2 4 G 



Model Speed, Ft/Sec 



Fig. 1 Motions of a destroyer model in waves 1.0, 1.2 5, 

 1.5 and 2.0 model lengths by 1.43 in. high (from Korvin- 

 Kroukovsky and Jacobs, 1957). Circles indicate experimental 

 data — open for heave, closed for pitch — and curves show calcu- 

 lated motions 



amplitude.s of motion, expre.ssing both the amplitude 

 and the phase hxg. Expressed mathematically 



where Z and 6 are complex immbers given l)y the solution 

 of equations (2) in (4). Z,, and du are real numbers ex- 

 pressing the amplitudes of heaving and pitching mo- 

 tions, respectively. The — 6 and — e indicate lag of the 

 maximum of ship motion behind the maximum of the 

 wave. 



Equatifjus (2) are general and represent the dynamics 

 of a two-mode coupled oscillation of a body. Hydro- 

 dynamics is involved in evaluating the coefficients of 

 these equations. The various theories of ship motions 



500 



400 f^ 



w 

 a> 



300 ^ 



200 



100 



500 



Fig. 2 Motions of a 5-ft model of Series 60-0.60 block coef- 

 ficient in waves of 1.0 and 1.5 model lengths by 1.25 in. high 

 (from Korvin-Kroukovsky and Jacobs, 1957). Circles indi- 

 cate Davidson Laboratory experimental data — open for heave, 

 solid for pitch — and curves show calculated motions. Tri- 

 angles are Taylor Model Basin experimental data, and exes Uni- 

 versity of California data 



differ by the methods used in evaluating the coefficients 

 and by the approximations im'olved in these methods. 



Kriioff (1896, 1898), Hazen and Xims (1940), Wein- 

 blum and St. Denis (1950), and St. Denis (1951) neg- 

 lected the cross coupling between heaving and pitching; 

 i.e., assumed d = c = g = D = E = G = 0. Kriioff as- 

 sumed the coefficients a and .4 as equal to the ship mass tn 

 and moment of inertia /, respectively. Weinblum and 

 St. Denis included added water masses which were esti- 

 mated by comparison with ellipsoids. They estimated 

 the damping coefficients b and B on the basis of Has- 

 kind's (194(3) derivations. St. Denis evaluated the co- 

 efficients by the strip theory using F. M. Lewis' (1929) 

 material for the added masses in coefficients a and -4 

 (neglecting surface effects), and using Ha\'elock's (1942) 

 method for evaluating the coefficients b and B. All of 

 the aforementioned authors used the Froude-Kriloft' 

 hypothesis (see Chapter 2, Section 3.14) in evaluating 

 wa\'e forces Fo and moments il/o. 



The coupled ecjuations (2) were usetl by Haskind 

 (1946), Hanaoka (1957) and Korvin-Kroukovsky and 

 Jacobs (1957). A brief outline of the principles used by 

 Haskind and by Hanaoka in evaluating the coefficients 

 was given in Section 2-6. The mathematical complexity 

 limits the applicability of these methods to ships of 

 idealized form expressible by simple mathematical for- 

 mulae. 



Korvin-Kroukovsky and Lewis (1955), and Korvin- 

 Kroukovsky and Jacobs (1957) used a strip method in 

 which the coefficients of eciuations (2) were expressed by 

 the integration of sectional properties. The details of this 

 procedure will be found in Appendix C. In this method 



