CALCULATION OF HYDRODYNAMIC FORCES BY STRIP THEORY 



341 



It is clear that the factor 



'2pgrh sin 27r(.r — ct)/\ = 2pgrr] 



represents the change in displacement force with wave 

 rise and fall; r in this case then is to be taken as the half- 

 beam B/2, at the load waterline. Next it is noted that 

 when the body-wave interaction is taken into account the 

 Smith effect on a circular body is doubled. This factor 

 of 2 may be interpreted as (1 -|- A'2) by analogy with G. I. 

 Taylor's exjircssion (1928) for the force acting on a body 

 placed in a fluid flow with a velocity gradient. Here A'2 is 

 the coefficient of accession to inertia in vertical flow and is 

 equal to 1 for a circular .section.' 



With a free water surface and the formation of a 

 standing-wave system, the value of A-o = 1 for the circular 

 cylinder is modified by a factor which is designated as A4. 

 Ursell (1954, and answer to discussion, Korvin-Kroukov- 

 sky, 1955fe) has computed the following values of A4 versus 

 w-r/g (or a}''B/2g) for the circular cylinder: 



k, 



CO 



0.818 

 0,632 

 0.592 

 0.673 

 0.738 

 0.762 

 0.818 

 0.859 

 0.883 



In the absence of more complete information it will be 

 assumed that this table of corrections applies to non- 

 circular sections as well. 



From experiments with an oscillator, Golovato (1956) 

 derived the coefficients of added (virtual) mass in heav- 

 ing oscillation for a ship form symmetrical fore and aft 

 with U-sections almost wall-sided at the load waterline. 

 Fig. 8 of that reference shows a curve quite similar in 

 trend to the coefficients of the foregoing table but with 

 values about 20 per cent higher and with the minimum 

 shifted to a somewhat higher frequency. The effect of 

 these differences on the ship response are expected to be 

 small. 



The factor (1 + A-2A4)/2 will then be applied to all 

 terms of equation (22) after the first displacement-force 

 term. In the earlier paper, A:4 was estimated for the ship 

 as a whole and it was applied only to the integrated vir- 

 tual mass and inertia effects due to body motion in 

 smooth water; it was omitted in the expression for the 

 exciting forces. Subsequently, it was found necessary 

 to apply the ^4 correction to each section for the calcula- 

 tion of bending moments and highly advisable to adopt 

 this more accurate procedure for the motion calculations. 

 Therefore, this omission has been corrected in the 

 present paper. 



Since the modified Smith-effect terms are connected 

 with virtual masses and since the effect of section shape 



' O. Grim (3-1957) proved this interpretation on basis of F. M. 

 Lewis' (19291 transformation. 



is defined by the coefficient As, the factor r in this case is 

 interpreted as a measure of sectional area; i.e.. 



r = (25/7r)"2 



(23) 



where ^S' is sectional area below the load waterline, and 

 therefore 



tan (3 



cir 



(IS 



(24) 



(27r,S)'/2(/$ 



With the substitutions just indicated, the distribution 

 of vertical heaving forces due to the action of waves on 

 a ship at a particular instantaneous position of the ship 

 on the wave (t = 0) is expre.s.sed as 



(IF 



= pghB 



,, . 27r,r , ,, 27r.r 



Ai sui \- A2 cos 



X X 



(l.r 

 where A'l and A'o are nondimensional coefficients 



1 + hh -^ (2. 

 2 X 



(25) 



A', 



1 



^y/2 



+ 4 (1 + A2A4) f„ ,S' 



A" 



A, = 



7r(l + A2A4) 



1 - 



l(i 



;5x 



2.S' 



1/2" 



Y 



1 



dS 



Tt J J c {:2Trsyi^di, 



These coefficients depend on the sectional shape and area, 

 on the wave length and also, because of the presence of 

 the coefficient A;4, on the frequency of wave encounter. 

 The distribution of forces along the length of the ship 

 given by equation (25) can be used directly in the com- 

 putation of the bending moments exerted on a ship by 

 waves. 



For an analysis of ship motions the force distribution 

 must be integrated to provide the total heaving force F 

 and the total pitching moment M 



F = 



/ 



^ dF J 

 — - ax 

 dx 



and 



M 



Js dx 



I (IS 



TT (/$ 



dx 



(26) 



(27) 



where the limits of integration s and b are the values of 

 X at the stern and the bow, respectively. The second term 

 of eciuati(jn (27) results from the consideration that the 

 water pressure acts normally to the body surface ; in the 

 case of a body of varying circular section the moment 

 arm is (^ -|- r tan 13) and, by the use of the relatioiisliips 

 (23) and (24), for a normal ship form the moment arm 

 may be assumed to be (J -|- dS/wd^). Equations (25), 

 (26), and (27) replace Equations (33) and (34) of the 

 earlier paper. 



The integrals of equations (26) and (27) can be evalu- 

 ated readily by Simp.son's rule. By changing the ship's 

 position relative to the wave the maximum values or 

 amplitudes of the exciting force and moment can be 

 found as well as the phase lags a and t. Calculations of 

 these amplitudes Fo and i1/o, were made for a 5-ft-long 

 model of the Series 60, 0.60-block-coefficient hull (ETT 

 Model No. 1445) in waves of ship length (X/L = 1) and 

 wave height of 1.5 in., for comparison with the ex- 



