CHAPTER 2 Hydrodynamic Forces 



1 Introduction 



Chapter 2 deals with definition and evaluation of hy- 

 drodynamic forces acting on the hull of an oscillating 

 ship in waves. The oscillating motion of a ship will be 

 discussed in detail in Chapter o. However, the forces 

 and motions are so closely interconnected that a com- 

 plete separation of these two subjects is not possible, and 

 a certain minimum information on motion has to l)e in- 

 cluded in Chapter 2 as well. 



The exposition given in Chapter 2, as indeed in all sub- 

 secjuent chapters, follows the policy outlined in the Intro- 

 duction to the Alonograph (pages W and \"). An attempt 

 has been made at a critical summarj^ of the existing state 

 of the art. It is expected that the reader will be stimu- 

 lated to further research by the realization of the scope 

 and the shortcomings of present knowledge of the hydro- 

 dynamic forces acting on a ship oscillating in wa\'es. A 

 summary of suggestions for research will be pro\-ided at 

 the end of the chapter. 



Because of the close relationship between the subject 

 matter of Chapters 2 and o, the bibliography for both 

 is placed at the end of Chapter )i. The I'eader is asked 

 to refer to it whenever the reference gi\-es only the year 

 of publication, thus "Davidson and Schiff (194(1)." 

 When a reference is made to other chapters it is j3 receded 

 by the chapter number; thus, "Pierson (1-1957)." 



1.1 Forces Acting on a Body Oscillating in a Fluid. 

 A continuously changing pattern of water \'elocities rela- 

 ti\-e to the hull is created when a ship oscillates in waves. 

 By x'irtue of the Bernoulli theorem, these water velocities 

 and their rates of change cau.se changes of the water 

 pressure on the hull. These pressures, acting in various 

 directions, always normal to elements of the hull surface, 

 can be resolved along three axes, x, .(/, and z, and the com- 

 ponents can be integrated over the entire area of the hull 

 so as to gi\'e the total resultant force in each of the.se di- 

 rections. The force components also can be multiplied 

 by the distance to the center of gravity of a ship, and 

 integrated to gi\'e the total moment about each axis. 

 It has been found that once the detailed derivation has 

 been carried out, the actual evaluation of the forces often 

 can be accomplished l)y a much simpler procedure in 

 terms of the body volume. 



The actual mechanism of a ship oscillating along 

 three axes — surging, side sway and heave, and rotating 

 about three axes, rolling, pitching and yawing — can be 

 complicated. Xevertheless, the basic concepts and ter- 

 minology are defined in the same way as for a simple 

 harmonic oscillator. A can buoy in heaving motion in 



low, long wa^-es is a good example of a simple forced 

 oscillation. Its motion is described by a linear differ- 

 ential c(|uation of the second order 



as + 65 -|- cz = F,) cos wt 



(1) 



Here the first term az denotes the forces connected 

 with the acceleration d'-z/dt-, and the coefficient a is a 

 mass. It is in reality the ma.ss m of the buoy itself plus 

 a certain imaginary water mass rn^, the acceleration of 

 which gi\'es a heax'ing force ecjual to the vertical result- 

 ant of all fluid pressures due to actvial acceleration of 

 water particles in many directions. This imaginary 

 mass is known as "added mass" or "hydrodynamic mass" 

 and the coefficient a is written as m + m^ = mil + k\), 

 where k, is the "coefficient of accession to inertia" in the 

 \'ertical plane. The total mass represented by the coeffi- 

 cient a is known as "apparent mass" or "\-irtual mass." 



The second term bi denotes the force proportional to 

 the instantaneous vertical velocity dz/dt. The coeffi- 

 cient h is known as "damping coefficient" for a reason to 

 be discussed shortly. In most cases it is assumed to be 

 constant. In reality it is often not constant and in ap- 

 plication to ship rolling, for instance, it often has been 

 taken as depending on velocity s(|uared i- as well as on 

 i. However, a satisfactory' description of many forms 

 of oscillation in nature is given bv the linear form of 

 Ecjuation ( 1 ) . 



The term cz is the force proportional to displacement, 

 and is usually known as the "restoring force," while the 

 coefficient c is often referred to as a "spring constant." 

 This is a force exerted per unit of displacement z. In the 

 present example of a can buoy, the constant c is the 

 hea\ing force caused by a change of draft of one unit; 

 i.e., 1 ft in the foot-pound .system. 



The term Fo cos wt on the right-hand side of equation 

 (1) is the "exciting force." In the present simple ex- 

 ample, Fo is the amplitude of the buoyant force due to 

 wave height. In a ship's case it also will depend on 

 water velocities. 



In the forced motion with harmonic exciting force, 

 e(|uation (1), the motion, after sufficient time, is also a 

 simjile harmonic so that body position at any instant is 



^0 cos (co/ -f- () 



(2) 



where oj is the circular fre(|uencv, and t is the "phase 

 lag angle." Term zo is the amplitude of motion defined 

 in its relationship to the amplitude of exciting force 

 Foby 



zo = F„[{c - aco-)-^ + 6V-1-"- (3) 



106 



