HYDRODYNAMIC FORCES 



109 



+ x,|,V,u,c 



Fig. 1 Sketch illustrating notation used in connection with strip 

 theory (from Korvin-Kroukovsky, 195 5c) 



to them. This vertical disphxcement at any instant and 

 at any distance .r is designated r). Imagine two control 

 planes spaced dx apart at a distance ,i: from the origin, 

 and assume that the ship and water with orbital ^'eloci- 

 ties of wave motion penetrate these contrf)! surfaces. 

 Assume that the perturbation \'elocities due to the pres- 

 ence of the body are confined to the two-dimensional 

 flow between control planes; i.e., neglect the fore-and-aft 

 components of the perturbation velocities due to the 

 body, as in the 'slender body theory' of aerodynamics. 

 This form of analysis, also known as the 'strip method' or 

 'cross flow hypothesis' is thus an approximate one in the 

 sense that a certain degree of interaction between adja- 

 cent sections is neglected." 



In analyzing ship motions it is generally necessary to 

 stipulate two systems of axes, one fixed in space and one 

 fixed in the body. Thus, considering hea\-iug and pitch- 

 ing of a ship, Korvin-Kroukovskj' and Jacobs (.1957) ■* 

 stipulated an .r, y, z-system fiuxed in space (with the x-y- 

 plane in the undisturbed water surface) and a $, ij, i'-sys- 

 tem fixed in the ship. The location in the ship of the 

 origin of the latter system is arbitrary, Init the mathe- 

 matical work is simplified considerably if the origin is 

 placed at the center of gravity. A primary step in the 

 strip method of analysis is to evaluate the hydrodynamic 

 forces caused by the relative ship-wa\'e \'ertical motion at 

 a ship section located at a distance ^ from the origin. 

 The vertical velocity of this section is the summation of 

 the velocity components in heaving s and in pitching 

 ^d. When a ship is at a small angle of trim d, the draft 

 of ship sections, passing through the water slice dx, in- 

 creases with time. This gives an added vertical velocity 

 component dV. 



After the forces acting on individual ship sections are 

 evaluated, they are integrated over the ship length. 



The integral forms used to obtain various coefficients 

 for the eciua.tions of motion are given in Appendix C 

 and are discussed in Chapter 3. Use of the sectional 

 forces in computations of the hull bending moments are 

 discussed in Chapter 5, by Jacobs (5-1958) and by Dalzell 

 (5-1959). 



The forces produced by water pressures on ship sec- 

 tions can be classified l)y their nature as inertial, damp- 

 ing, and displacement. They also can be classified by 

 their cause as i-esulting from a ship's oscillation in smooth 

 water or from wave action on a restrained ship. Kriloff 

 (1896, 1898), considering only displacement forces, 

 demonstrated that the total force acting on a ship in 

 waves can be considered as the sum of these two com- 

 ponents. Korvin-Krouko-\'sky and Jacobs (1957) dem- 

 onstrated that this subdivision of forces also holds 

 (within linear theory) when the pressures are generated 

 by water acceleration. This is the direct conseciuence of 

 the linear superposition of velocity potentials defining 

 various water flows. It can be added here that the forces 

 in\'olved in pitching and heaving are caused primarily 

 by potential flows, and water \'iscosity does not appear 

 to be important in this connection. 



The sectional inertial forces will be discussed in the fol- 

 lowing Section 3.1 and the damping forces" in Section 3.2. 

 The action of dis]5lacement forces in a ship oscillating in 

 smooth water is obvious and needs no discussion. The 

 displacement effect caused by waves will be brought out 

 in the consideration of inertial forces, since the wave 

 elevations are inseparal)ly connected with water accel- 

 erations. 



3.1 Inertial Forces Acting on a Body Oscillating in 

 Smooth Water: 3.11 Conformal transformations. 

 The work on added mass in vertical oscillations most 

 often referred to is that of F. M. Lewis (1929). Lewis 

 assumed that water flow around a circular cylinder 

 floating half immersed on the water surface is identical 

 with that around a deeply immersed cylinder. Simple 

 expressions for the latter are a\-ailable in standard text- 

 books (Chapter 1, References C, D, and F). The added 

 mass of a cylinder is found from these expressions to be 

 ec[ual to the mass of water displaced by it. The coeffi- 

 cient of accession to inertia is, therefore, unity. Lewis 

 devised a conformal transformation by means of which 

 a circle is transformed into ship-like sections of various 

 beam/draft ratios and sectional coefficients. Water 

 flows corresponding to these sections were derived and 

 coefficients of accession to inertia were determined. In 

 addition to Lewis' (1929) original work, the procedure 

 was described (with ^-arious extensions) by Prohaska 

 (1947), Wendel (1950), and Landweber and de Macagno 

 (1957). The resultant relationships were also given by 

 Grim (1956). « 



The original half-immersed circle of radius r is defined 

 in complex form 



' The mathematical part of this reference is included in this 

 monograph as Appendi.\ C. 



' It will be .shown in Section 3.2 that damping forces are also 

 of inertial origin. 



« .\n independent evaluation of added masses was also made by 

 J. Lockwool Taylor (1930(;). 



