CALCULATION OF HYDRODYNAMIC FORCES BY STRIP THEORY 



345 



Table 5 Values of co-t/g for Synchronous Speed, i/6, k; and k', of the Various Ship Forms for Comparison with Hovelock's 



Submerged Spheroid 



Model L/B 



Havelock's submerst'd spheroid • 8 



Series 60 (Model 1445) 7.5 



V-bow (Model 1616) 7.5 



T-2 Tanker (Model 1444) 7.4 



Destroyer (Model 1723) ■ , . 9.4 



Trawler (Model 1699A) 5.8 



Yacht (Model 1699B) 3.5 



Lengthened trawler (Model 1699C) 10.3 



Lengthened j-acht (Model 1699D) 8.0 



" ki for the ship forms (with free surface correction ^-4) 



, , _ added moment of inertia 



moment of inertia of water displaced by a ship. 



mass of water displaced by a ship 



since they have not yet been developed for ships of nor- 

 mal form, they will be ignored in the present work; the 

 damping will be taken as computed for two-dimensional 

 flow by Grim. It is gratifying ne\-erthele,ss to have 

 measures of possible error such as Figs. 16 and 17 to 

 replace arbitrary assumptions made in the earlier paper 

 as to the applicability of the strip method of analysis. 

 The damping force for each section is 



vN(0 = [i + ^e - T'e].V(t) (38) 



where N{^), the damping force per unit vertical velocity 

 of the ship section, is given by eciuation (36). The total 

 damping force and moment of the hull are then obtained 

 by integration over the length, thus the damping force is 



fN(0[i + ^e-ve]dt (39) 



and the damping moment is 



fN{^)\z + ^d^Ve]^(lt (40) 



The contributions of these expressions to the coefficients 

 of the coupled equations of motion are designated by the 

 subscript 2 



h = fNi^d^ 



e, = E-2 = fNm rf? 



g, = -YfN(Qdk = -Vh, (41) 



B, = fNm' dk 



c,= -vfNmdt 



The integrations are carried out numerically as in all 

 cases in this paper, and over the length of the hull. 



When all the forces and moments proportional to z 

 and d and their derivatives with respect to time, which 

 are presented in the foregoing development of the po- 

 tential theory with corrections for free-surface effects, 

 are combined with the inertial and restoring forces and 

 moments the coefficients of the equations of motion be- 

 come 



(42) 



where the integrations are taken over the length of the 

 hull. 



Terms c anil G, the coefficients of the displacement in 

 heave z in the force and moment ecjuations respectively, 

 depend only on the changes in the displacement of the 

 ship. With the linearizing assumption these are evalu- 

 ated on the basis of the beam of a section. Terms cj and 

 C, the coefficients of the angular displacement 6, depend 

 on displacement changes and also on the kinematics of 

 the fluid flow resulting from the ship being at an in- 

 stantaneous angle of trim. 



It is seen that the damping-force coefficient h in heave 

 is a function of frequency of encounter Wc but is inde- 

 pendent of forward speed V per se. This appears to be 

 confirmed by the experimental work of Golovato (1956). 

 However, the damping moment coefficient B in pitch 

 and the cross-coupling coefficients e and E are composed 

 of dynamic terms proportional to V and dissipative terms 

 independent of V (except as the frequency of encounter 

 oje is a function of V). While the dynamic terms con- 

 tribute only a little to B, they make \-ery important con- 

 tributions to the cross-coupling coefficients. 



' S* represents beam. 



