HYDRODYNAMIC FORCES 



107 



It is of interest to establish what work is done by an 

 osciUatiiig body on the fluid per cyele of oscillation. Ou 

 the basis of etiuation (2) : 



£ = Jo COS [cot -j- t) 



dz = — co2o sin ((j)t + i)(U 

 z = —wZo sin (o)t -\- t) 

 S = —w-Zq COS (wt + e) 



and the woi'k done, in the period T, by the acceleration 

 forces 



a I z dz — Oio'^'zo- 



(4) 



X 



X I sin {cot + e) cos [ut + c)dt = (5) 

 Jo 



by damping forces proportional to z 



c)lll 



= hooWT/2 (()) 



b \ zdz = bu-Zi," I sin- (co^ + 

 Jo Jo 



by restoring forces, 



'T 



c I z dz = — c<j)Zu 



proportional to z, 



f 



Jo 



X 



Jo 



sin (ut + e) cos (o)! + t)(lt 







;7) 



Thus, it is seen that the average amount of work done 

 on a fluid liy acceleration and l\v restoring forces is nil. 

 A body does the work on the fluid (huing a half cycle, 

 and the fluid does an equal amount of work on the body 

 during another half cycle. Only damping forces do a 

 net amount of work ou the fluid, and therefore take the 

 energy out of the body and dissipate it in the fluitl. In 

 an oscillation of a free body this causes grachial dim- 

 inution of the amplitude of motion, from which the 

 term "damping" has been deri\-ed. In a continued 

 forced oscillation the energy necessary to maintain it is 

 .supplied b}' the exciting forces. 



If the frecjuency of the oscillation is Icjw enough, the 

 phase lag is negligibly small and the displacement of a 

 body 2 is in phase with the exciting force. E(iuations 

 (4) show that the \elocity z is 90 deg out of phase and 

 the acceleration z is 180 deg out of phase. 



The forces caused by water pressures can be di\"ided 

 into two groups. The restoring force cz is caused by 

 hydrostatic water pressures. The hydrodynamic forces 

 63 and m,z result from the velocities and accelerations of 

 water particles. These two forces are in I'cality two 

 components of the resultant of all hydrod\ iianiic (i.e., 

 exclusive of hydrostatic) water pressures. 



Confusion has occasionally resulted from the de- 

 scriptive definitions of the damping Unre ami the inert ial 

 (acceleration of the hydrodynamic mass) force gi\en 

 earlier. In the recent literature there has been, there- 

 fore, a tendency to define these forces merely as the out- 

 of-phase (by 90 deg) and in phase (in reality ISO deg out- 

 of-phase) components of the hydrodynamic foi'ce. 



1.2 Order of Exposition. Eciuation (1) was intro- 

 duced in order to define four categories of forces acting 

 on an oscillating body, namely, inertial, damping, restor- 



ing, and exciting. The following sections of Chaptei' 2 

 will be devoteil to the e\'aluation of these forces by theo- 

 retical and experimental means. Theoretical evaluation 

 of hydrodynamic forces in harmonic oscillations has 

 been approached in three ways; 



■ a) Comparison with ellipsoids (in Section 2) 



b) Strip theory (in Sections 3, 4 and 5) 



c) Direct three-dimensional solution for mathemati- 

 cally defined ship forms (Section 6). 



In Sections 7 and 8 the forces in transient (slanuuiug) 

 conditions will be discussed. 



2 Estimates of Hydrodynamic Forces and Moments by Comparison 

 With Ellipsoids 



The problem (jf forct's and moments exerted by a fluid 

 on a body moving in it received the attention of hych'o- 

 d.ynamicists at a very early date, and chapters on this 

 subject are found in all major books on hydrodynamics 

 (see Chapter 1 : References C, pp. 353-39.3; D, pp. KiO- 

 201; F, pp. 4G4-485). The problem is usually formu- 

 lated for a liody moving within an infinite expanse of a 

 fluid initially at rest and assumed to be nonviscous. 

 Only the forces due to the fluid inertia can therefore l)e 

 present . 



The forces and moments acting on a body can be e\ alu- 

 ated bj' two methods. In the fir.st method the pressure 

 p acting on each element of a body surface is (•om])ute(l 

 by Bernoulli's theorem 



P 



d4> 



(S) 



where (t> is the velocity potential, and q is the local fluid 

 \elocity at the surface of the body, induced by its motion. 

 By taking components of pressures p in the desired di- 

 rection and integrating o\'ei' the surface of the body, the 

 total force is obtained. 



The second method consists of expressing the rate of 

 change of the kinetic energy contained in a volume of 

 fluid between the botly surface ami an imaginary control 

 surface taken at a sufficiently large distance fi-om the 

 body. The kinetic energy T is given by the expression 

 (Lamb, 1-D, p. 40) 



^I'i 



"* ,IS. 

 on 



(9) 



where n denotes the outward normal and .S the surface of 

 a body over which the integral is taken. The force is 

 then found by different iat ion of the energy with respect 

 to the body displacement; for in.stance, the force A' in 

 the direction of the .c-axis is 



X = dT/dx (10) 



In the application of either of the foregoing methods it 

 is necessary to obtain the \'elocity potential <^. Also 

 it is necessary to have the mathematical description of 

 a body in order to formulate expressions for the directions 

 of the normals, and to permit the integration o\'er a sur- 



