Newman 



as a substitute for the dependent variable v , and plays a particularly important 

 role in describing motions in the horizontal plane. 



If the origin coincides with the ship's center of gravity the six equations of 

 motion of the ship are as follows: 



X = m(u - rv + q\v) + mg sin 6 



Y = m(v-pw+ru) - mg cos 9 sin 4> 



Z = m(w - qu + pv) - mg cos 6 cos <p 



K = IxxP + (Izz-Iyy) ^' + Ixy^ + ^^z^ 



M = lyyq + (I^^-I,,) pr + ly^r + I^yP 



N=I r+(I -I )pq+I p+I q. 



zz ^ yy ■x.-x.' ^^ xzt^ y z^ 



Here a dot denotes time differentiation, m is the mass of the ship, g is the grav- 

 itational acceleration, (ixx- ^yy ^zz) ^i"6 the moments of inertia of the ship's 

 mass, and (ixy Ixz- ^yz) ^^^ ^^e corresponding products of inertia. If the mass 

 distribution is symmetrical with respect to the plane y = (i.e., port and star- 

 board) the products of inertia I^^y and ly^ will vanish, and in practice for con- 

 ventional ships the fore-and-aft symmetry is sufficiently dominant that the re- 

 maining product I,^^ can probably be ignored. However, this assumption does 

 not appear to have been verified, and it should be emphasized that, in general, 

 the coordinates cannot be taken to coincide with the principal axes of inertia un- 

 less the usual assumptions that x is horizontal and z is vertical are sacrificed. 



Various manipulations can be performed with the above system of equations. 

 Coordinate transformations can be made to a more convenient origin than the 

 center of gravity, several additional terms thus being introduced to the above 

 equations; these transformed equations can be found, for example, in Bulletin 

 1-5 (anon., 1950). The degrees of freedom can be limited to the horizontal plane, 

 so that the number of terms in the equations of motion is substantially reduced. 

 Also, these equations may be linearized, on the assumption that the unsteady 

 motions are small perturbations of an initial steady longitudinal velocity; this 

 step is justifiable in many instances, particularly in performing a dynamic sta- 

 bility analysis, but not necessarily in predicting maneuvering characteristics 

 which may be inherently nonlinear, and certainly not in studying the motions of 

 unstable ships. 



The left sides of the equations of motion, the external force and moment 

 applied to the ship, include all of the hydrodynamic effects which act on the hull. 

 For a conventional self-propelled ship there are no other components of the ex- 

 ternal force and moment if we assume that aerodynamic effects on the above- 

 water portion of the ship can be neglected and that there exist no internal 

 changes of the mass or its distribution. On the other hand, towed or towing 

 bodies such as barges or tugs and captive ship models will have additional 



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