CALCULATION OF HYDRODYNAMIC FORCES BY STRIP THEORY 



343 



Cl 



Bi 



Q'A) 



r/, = pf (Sk,h)^ '/t 



-2rpy(^'A,A,)^ rfs'- Tp J ^^^ .^= rf,^ 



/A = pf(SUu)t d^ 

 ,, ,. CdSkJu,,, 



Dynamic Terms in z and ^ 



Attention should be called to the fact that the velocity- 

 dependent terms in z and d of equations (31), (32) and 

 (33) do not involve dissipation of energy, but only the 

 transfer of energy from one mode to another. This was 

 demonstrated by Haskind (194G) and by Havelock 

 (1955), who refers to these terms as "dynamic coupling." 



In previous studies of oscillations, velocity-dependent 

 terms appear only in the role of energy dissipation either 

 by viscosity or by wave making in the case of ships. 

 Following these earlier studies it was assumed in the 1955 

 paper that the velocity-dependent terms in the develop- 

 ment of the potential theory merely implied tlamping 

 and could be replaced by damping terms determined on 

 the basis of energy dissipation by wa\'es as a quid pro quo. 

 since the initial statement of the problem did not provide 

 for inclusion of energy-dissipation terms. Later, exami- 

 nation of the work of Fay (1957) suggested that this was 

 an error of judgment which could be responsible for the 

 poor correlation between calculated and experimental 

 phase relationships reported in the earlier paper and also 

 for the poor results obtained when applying the methods 

 of that reference to the calculation of l)cn(ling moments. 



A study of Haskind (194:0) and Havelock (1955) con- 

 firmed this and therefore the terms in i and 6, (iii), (iv) 

 and (v) of equation (31) have been reinstated. They 

 yield a heaving force due to pitching velocity d and a 

 pitching moment due to heave velocity i equal to 



and 



[-2pf(Skju)d^ - pft,d(Sk,k,)]ve 



■ pfid{Sk,k,)]Vi 



(35) 



making, Havelock gives the dynamic coupling terms as 



— pMVd for the heaving force 



and 



-fr/jl/Fi for the pitching moment 



where M is the mass of displaced water. In general 

 p y^ q, and each is given by a fairly complicated ex- 

 pression in terms of ellipsoidal co-ordinates and associated 

 Legendre functions of the second kind. For a long 

 spheroid Havelock finds that p = g = (1 -|- ki)/2, or 

 0.515 for a length-diameter ratio of 8 which is the fineness 

 ratio of the usual ship form. Haskind (1946) had found 

 that p = q for a thin or "Michell" ship symmetrical fore- 

 and-aft. 



For a prolate spheroid (S is a function of ^ alone and 



ftSd^ = -fL^dS = M/p 



Therefore expression (35) of the present development 

 becomes 



and 



-kikiMVe 



+kikiMVz 



where the bar indicates the value for the entire body. 



Thus p = q = kiki. For a circular section k2 = 1 and, 

 at the oscillating freciuency in the vicinity of synchronism 

 for most ship forms, kt is of the order of 0.75. It ap- 

 pears from the application to the models in this paper 

 that the damping in heave is reduced and in pitch is in- 

 creased by the addition of the dynamic coupling terms. 



Dissipative Damping 



It was mentioned previously that the free water surface 

 was not taken into account in the basic derivation of the 

 present paper and that a correctton for it must be intro- 

 duced independently. The effect of the free surface on 

 the \irtual mass has been allowed for approximately by 

 the use of the coefficient /m derived by Ursell for a semi- 

 cylinder (1954, and discussion, Korvin-Kroukovsky, 

 19556). (Grim, 1953, also has calculated this effect for 

 some ship-like forms but his material is not extensive 

 enough for general application.) The other well-known 

 effect of the free surface is the dissipation of energy in the 

 formation of waves which propagate away from the ship 

 in all directions. In the "strip" method of analysis it is 

 assumed that waves from each length segment d^ propa- 

 gate laterally. If the ratio of the amplitude of these 

 waves to the amplitude of the heaving motion of a ship 

 section is designed by A, the damping force per unit 

 vertical velocity v of the ship segment is expressed as 

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



N(0 



pg 



242/^^3 



(30) 



For the case of a half-immersed spheroid inider the 

 condition of a free water surface but neglecting wave 



where ui, is the frequency of the waves radiated by the 

 ship and is equal to the frequency of wave encounter. 



