126 



THEORY OF SEAKEEPING 



Diograrrr A-H M.S. Sul + on 

 + + Angles given by Evperitncn+s a+ Berehovisn \Q13 



Calculated Curve No.l, {"rO-OgS^lj 



Calculoted CurvsNo.2, b- 0.0550 



Calculated Curve No.3, a.0.03911 



Diagram B -H.M.S." Volage" 

 + + Angles given b^ Experiments at Portsmouth 1871 



calculated Curve .10,1,(0^: 0-0^8_^ 



Calculoted Curve b=O.OI39 



Calculoted Curve 0=0.05193 



Count of Osc'llations 



Fig. 20 Curves of declining angles in rolling oscillations of two ships in still water (from W. 



Froude, 1874) 



data is derived from analysis of ship iiiotion wliich is as- 

 sumed to be generally expressed by such differential 

 equations as (25). The literature on the subject is 

 divided into: (a) Rolling in still water (i.e., RH term of 

 equation 25 equal to zero), and (6) rolling in waves. 



When a ship rolls freely in smooth water, a small heav- 

 ing motion and side swaying occur. It has been shown, 

 however, (Ueno, 1942; Grim, 1950) that heaving in this 

 case is small, and side sway, although more pronounced, 

 is also small. It is believed permissible, therefore, to 

 treat the motion as a pure rotation about the fore-and- 

 aft axis passing through the center of gravity. This is 

 believed true providing the centers of gravity of all ship 

 sections lie on this axis. If the principal axis of inertia is 

 inclined appreciably to the fore-and-aft axis of a ship, a 

 significant yawing component of motion also will de- 

 velop. 



5.1 Restoring Moment. The restoring mom ent i s de- 

 fined l)y the concepts of "metacentric height" GM and 

 of "righting arm" GZ. These are familiar to naval archi- 

 tects and will not be elaborated on here. The righting 

 moment at small roll angles is 



L = AGZ = AGAI sin </> (26) 



where the right-hand part of the equality is \'alid only 

 for constant GM. Here A is the weight of a ship. 



For small angles 4>, sin 4> = 4>- Substitution of equa- 

 tion (26) with sin <i> = 4>, for C4> in equation (25) yields 

 a linear differential equation that is simple to solve. 

 The initial height of the metacenter over the center of 

 buoyancy of a ship BM can be computed by the expres- 

 sion 



BM = IJV (27) 



where /„ is the second moment of the waterplane about 

 the axis of symmetry, and V is the displaced volume. 

 If the height of the center of gravity over the center of 

 buoyancy GB is known, the GM is determined. Equa- 

 tion (26) is the basis for experimental verification of GM 

 by inclining a ship through small angles. If a known 

 heeling moment L is applied and the resulting heel angle 

 is measured, GM is determined. 



For larger heel angles, GM does not remain constant 

 and the righting arm GZ follows some other law than sin 

 4>. In order to use the simple form of equation (25), 

 Froude (1861) assumed GZ = GM 4>, and showed that 

 this corresponds with reality for a ship with slight tumble- 

 home sides. This was the usual hull form of the battle- 

 ship of his day. For a wall-sided ship 



GZ = sin (/> ( G:M -I- ^ BM tan= 



(28) 



and in general 



GZ = GMo 4> + F{<j>) 



(29) 



where GMo is the metacentric height for very small 

 angles </>. 



5.2 Inertial Moments. Once a metacentric height is 

 known, the effective moment of inertia of a ship in rolling 

 is defined by 



T = ^' (30) 



(gGUyi- 



where T is the natural rolling period and k the radius of 

 gyration. Once GM is known from static inclining ex- 

 periments and the natural period of rolling T is observed, 

 k and the virtual moment of inertia (/ = k-A/g) can be 



