258 



THEORY OF SEAKEEPING 



leads to the development of a sagging moment which in- 

 creases with a ship's speed. None of these time-inde- 

 pendent moments will he considered here. Their exist- 

 ence, however, should be remembered and ai)prf)priate 

 measurements in still water should he included in model 

 tests and ship observations in waves. Only the time- 

 dependent loads added by waves and by a ship's hea\'ing 

 and pitching will be considered. 



The loading, p, per foot of a ship's length is equal to 

 the algebraic sum of acceleration forces and water 

 pressures : 



P 



lllZo 



dF/dt 



(1) 



where p is the time-dependent vertical force per foot of 

 ship's length, w; the mass of the ship's structure, equip- 

 ment, and load apportioned to this length and dF'd^ 

 the vertical force due to water pressures per unit length. 

 The symbol Zo is used for the heaving displacement of an 

 element d^ of a ship's length. When the heaving mo- 

 tion of an entire ship is considered, the sum of all water 

 pressures is balanced by the inertia force and the sum 

 of all forces is equal to zero. At a particular ship's 

 section of unit length, however, the inertial and water 

 pressures are usually not balanced, p y^ 0, and the nec- 

 essary balance of forces results from the application of 

 shear forces at the ends of the section. The double in- 

 tegration of these shear forces gives the bending moment. 



The inertial force, mSo, depends on the value of the 

 acceleration, so. The force, dF/d^, of water pressures 

 depends on the acceleration, velocity and displacement 

 of the ship and water. The solution of the equations of 

 motion is, therefore, a necessary prerequisite to the evalua- 

 tion of the loading cuire and a ship's bending moment. 

 For the structural analysis the load per foot, p, is to be 

 evaluated at a certain instant; i.e., with certain specific 

 values of z, z, and S, as well as wave elevation rj. 



2.1 Linear Theory. The linear theory of a ship's 

 heaving and pitching motions in waves antl oi the 

 resulting bending moments was originally developed by 

 Kriloff (3-1896, 1898fl and b) and was further elalxirated 

 by Horn (1910). In both cases the heaving and pitching 

 motions were considered independently; i.e., their 

 coupling was neglected. Horn's work is particularly 

 important in demonstrating the significance of phase 

 relationships (between ship and wave motions) in the 

 summation of water pressure and ship inertia loads. 



The phase relationships are strongly affected by the 

 coupling of heaving and pitching motions, and the con- 

 sideration of this coupling is therefore important. The 

 differential equations of coupled pitch-heave motion 

 were formulated and a method of evaluating the co- 

 efficients of these equations w'as developed by Korvin- 

 Kroukovsky and Jacobs (3-1957). The application of 

 this material to the evaluation of bending monients will 

 be outlined briefly. The equations of motion are 



The nomenclature u.sed in these equations and the 

 evaluation of coefficients are given in Appendi.x C. 



In the process of derivation of equations (1) and (2), 

 it was shown that the total hydrodynamic force per unit 

 length, dF/d^, can be considered as the sum of two forces, 

 (dF'dt), -f (dF/d^),,. The first of the.se forces is de- 

 veloped when a ship oscillates in smooth water and the 

 second when waves encounter a restrained (non-oscillat- 

 ing) ship. Corresponding to this subdivision, all terms 

 of the left-hand sides of equations (2) represent forces 

 and moments obtained by different forms of integration 

 of (dF/d^)t„ with respect to a ship's length. The terms 

 on the right-hand sides of the equations result from inte- 

 grations of the force exerted by waves on a restrained 

 ship, {dF/di)w These latter terms are usually referred 

 to as "exciting functions." 



The forces cau.sed by ship's oscillations in smooth water 

 will be considered first. An isolated section of a ship's 

 length, d^, has on'y a simple heaving motion, z^, which 

 results from the contributions of a ship's heaving, z, 

 and pitching, Q, .so that 



2o = ~- + id 



(3) 



az + bz + cz + de + ee + g9 = Fe'"' 

 A9 + Be + Cd + Dz + Ez + Gz = Me" 



(2) 



The total force and moment acting on a ship, oscillating 

 in smooth water, are obtained by the integration of the 

 sectional forces {dF/d^)t,, which result from the fore- 

 going composite motion. This integration leads to 12 

 terms. The coefficients of these terms are divided 

 into four groups, {a,b,c), {A,B,C), {d,e,g), and {D,E,G). 

 The first two groups define pure heaving and pure pitch- 

 ing motions and the last two define the effect of the 

 pitching on heaving and vice ver.sa. 



Application of the foregoing material to the evaluation 

 of the loading curve requires a regrouping of terms. 

 Instead of referring forces, acting on the entire ship, to 

 isolated heaving and pitching motions, they are now to 

 be expres.sed as acting on an isolated section, di^, in a 

 composite vertical motion of the section, Za. Equation 

 (3), however, indicates only the displacement Zq. It 

 is necessary to derive the corresponding velocity iu and 

 the acceleration So- lu doing so, one must remember 

 the dual nature of the co-ordinate f. It is a constant 

 defining the distance from the ship's center of gravity 

 of a mass w, apportioned to a certain ship section. On 

 the other hand, it is a time-dependent co-ordinate when 

 it refers to a stationar.y slice of water d^, and its time 

 derivative in this ca.se is d^/dt = —V. 



Since only the simple vertical motion of an isolated 

 section d^ is now to be considered, the integrands of the 

 coefficients a, b, and c will suffice. These are obtained 

 by adding integrands of coefficients listed in groups of 

 eciuations (34) and (41) of Appendix C. Certain co- 

 efficients vanish in the process of integration with re- 

 spect to ship length and are omitted in the final summary 

 of equations (42), which, therefore, is not to be u.sed in 

 this connection. Use of this material now permits 

 evaluation of three components of the force caused by 

 the ship's motion, {dF/d^)t, as follows: 



