162 



THEORY OF SEAKEEPING 



2.3 Six-Component Ship Motions in Waves: 2.31 

 Motions with rudder fixed. In Sections 2.1 and 2.2 two 

 simple and yet realistic cases were considered in which a 

 ship's motions are confined either to the longitudinal 

 plane of symmetry or to the transverse yz-plane. For 

 both of these cases the motion was described comiiletely 

 by three coupled differential etiuations, and was well ap- 

 proximated by two. A ship traveling ol)li(iue!y to the 

 direction of wave crests, on the other hand, moves in all 

 six modes. Assuming that the rudder is hxetl in neutral 

 position, such a motion can be described by six coupled 

 differential ecjuations. Krilotf (1898) formulated the.se 

 ecjuations and presented a thorough discussion of their 

 solution together with numerical examples. This analy- 

 sis appears to have been aliead of its time and no further 

 use of it was made in naval architecture. Later, the sub- 

 ject was developed by Bairstow, et al (1913-1914, 1916- 

 1917) in connection with aeronautical engineering and the 

 information on it can be found conveniently in .Jones 

 (1933). These methods of analysis have sul)se<iuently 

 been applied to the investigation of stability of sub- 

 marines and torpedoes. A reference to the SNAME T. 

 and R. Bulletin No. 1-5" will be u.seful in this connection. 

 It appears, however, that in all references certain features 

 not essential to the particular problem on hand were left 

 out and the reader is warned to be cautious in this connec- 

 tion. 



The large number of parameters in\'olved in the six- 

 mode motion makes it necessary to de\'iate from the pre- 

 viously used simple notation for the coefficients and to 

 follow a notation generally similar to the one used in the 

 above-mentioned references. The symbols will corre- 

 spond to the XAC'A convention included in the list of 

 symbols in Chapter 2. The complexity of the motion and 

 the many angles involved make it advisaljle to establish 

 a co-ordinate axes system fixed in the ship with the 

 origin at the center of gravity. The ai'celerations, de- 

 fining the inertial forces in the following eiiuations, are, 

 therefore, those measured by accelerometers installed in a 

 ship so as to read translational accelerations along the .r, 

 ij, and £-axes of the ship's co-ordinate system or rotational 

 accelerations about these axes. Eciuations (1) repre- 

 senting Newton's second law for heaving and pitching, 

 are now extended to : 



(12) 



m. (it — VI -\- wq) = X 

 m [i) — ivp -\- ur) = Y 

 m (w — iiq + rp) = Z 

 hp + (/, - I„)qr = L 

 14 + (h- h)rp = .1/ 

 hr + (J, - lApq = -V 



The left-hand sides of the e(|uations now include dy- 

 namic (often referred to as gyroscopic) coupling terms. In 

 the case of a surface .ship, the hydrodynamic forces A', Y, 

 Z and moments L, M, N depend on positions, velocities 

 and accelerations so that, for instance. 



A' = X(x,!j,z,6,(i>,4',u,r,w,p,q,r,uJ\w.p,q.r) (13)'° 



and similarly for Y, Z, L, M , N. 



If a hydrodynamic force, for in.stance A', is known and 

 designated A'u at a certain instant t, its value at I + dl is 

 expressed by a linearized Taylor expansion as 



.' , dA' , , dA , , bX , 



A = Ao 4- — - dx + ~~ dij + -—- dz ...etc. 

 ox oij oz 



(14) 



covering all terms in etiuation (13). 



There are, therefore, 18 coefficients (derivatives) de- 

 fining hydrodynamic forces on the right-hand side of 

 each of equations (12) or 108 coefficients in the set of six 

 simultaneous equations. These coefficients refer to a 

 ship oscillating in smooth water, .\dditional terms must 

 be added to represent the wave-excited forces. 



Equati(jns (12) are written for the body axes, so that, 

 for instance, moments of inertia remain constant despite 

 changing attitude of a ship. The quantities on the right- 

 hand sides of these eciuations depend on the relative 

 instantaneous orientation of body axes with respect to the 

 moving axes in which the .(-//-plane remains parallel to the 

 mean water surface, or, in other words, on the angle be- 

 tween the e-axes of these two co-ordinate systems. A 

 suitable transformation of the hydrodynamic and hydro- 

 static forces into the forces with respect to body co- 

 ordinates must, therefore, be carried out. For instance, 

 an added vertical buoyancy force acting on a rolled and 

 heaved ship must be resolved into its components along 

 the ship's z and y-axes. ' ' 



In application to airplanes the problem is simplified in 

 that the forces do not depend on position, and aerody- 

 namic forces acting on wings (with certain exceptions) 

 do not depend on accelerations. Coefficients of the 

 form dA'/d.f, and bX/bi'i \-anish, therefore, and only six 

 coefficients depending on linear and angular velocities u, 

 r, w, p, q, r remain in each equation. 



For a submarine the forces do not depentl on position 

 but do depend on accelerations. Derivatives with re- 

 spect to li, i', w, p, q, f are therefore retained. 



For a ship floating on the water surface, the forces also 

 depend on position, and the complete set of deri\'atives in- 

 dicated by equations (13) and (14) must be retained in 

 principle. In all applications, however, a few deriva- 

 tives can be omitted, by inspection, as ha\'ing zero or near 

 zero value. 



Even in the simplest case of an airplane which in- 

 volves only 37 derivatives (.Jones, 1933), it is not possible 

 to solve formally the set of six equations (12) as this has 

 ijeen done for the simple two-mode system shown by 

 equation (2). Only a numerical step-l)y-step integration 

 is possible, the laboriousness of which did not permit 



'Technical and Research Bulletin No. 1-5, "Nomenclature tor 

 Treatinf? the Motion of a Sulimerged Bodv Through a Fluid," 

 SNAiME, April 1950. 



i» The reader's attention is called to the two s\-.«tems of fre- 

 (inently used notations: » ^ x, u ^ .r; q "^ 6. q ^ B: and so on. 



" Tiie data on <-n-nrcliiiate transformations will lie found in 

 .SXAME T. and R. Bulletin 1-5 and in St. Denis and Craven, 1958. 

 A thorough discussion of co-ordinates will be found in Kriloff 

 (1898). 



