Some Hydrodynamic Aspects of Ship Maneuverability 



Appendix 

 NOMENCLATURE AND EQUATIONS OF MOTION 



A common notation has become almost universal in the field of maneuvering 

 and control. The principal elements are given below, and further details are 

 given in S.N.A.M.E. Bulletin 1-5 (anon., 1950) and in the Proceedings of the 10th 

 I.T.T.C. Conference (Goodrich, ed., 1963). 



Cartesian coordinates (xq.Yq, z^) are taken to be fixed in space or, nomi- 

 nally, with respect to the earth, such that x^ coincides with the general direc- 

 tion of the "initial" motion of the ship and Zg is vertically downward. The as- 

 sociated coordinates fixed with respect to the ship are denoted (x,y,z), the 

 right-hand convention is applied so that y is positive to starboard, and y = o is 

 taken to be the plane of symmetry of the ship. The three components of force 

 (x,Y,z), moment (k,m,n), linear velocity (u,v,w), and angular velocity (p,q,r) 

 are all defined in relation to the ship coordinates (x,y, z). Angular orientation 

 of the ship is defined by the symbols ^ (roll), e (trim), and si- (yaw), in accord- 

 ance with the following coordinate transformation: 



Xq = X COS 6 cos + y (sin 6 sin cp cos - cos sin \p) 



+ z (cos 4> sin 6 cos + sin (p sin i/^) 

 yg = X cos 6 sin + y (cos (p cos + sin 6 sin sin 0) 



+ z (cos 4> sin 6 sin - sin <p cos 0) 

 Zq = - X sin (9 + y cos 9 sin <p + z cos 6 cos . 



Alternatively, these (finite) angles can be defined by prescribing their order: if 

 the (x,y,z) coordinates coincide initially with the fixed (x^.y^.z^) system, then 

 the final orientation is obtained by first a yaw angular displacement, secondly a 

 trim, and lastly a roll, all with respect to the (x,y, z) axis and in the right- 

 handed sense. 



It will be noted that the above systems are not sufficiently general to de- 

 scribe translations as well as rotations between the fixed and moving coordinate 

 systems. In fact, it is customary to consider that the spatially "fixed" coordi- 

 nates (xg.yg.zo) are in fact translating with the origin of the ship's (x.y,z) 

 system but that the first system is fixed in space at each point in time to allow 

 the application of Newton's laws in this system. Such a shortcut is expedient if 

 free surface and viscous effects are ignored, but for a complete physical de- 

 scription it is necessary to consider the effects of translations as well as rota- 

 tions between the two coordinate systems. 



Since the translational velocity components (u.v.w) are defined in reference 

 to the ship-fixed (x,y,z) coordinates it follows that a steady "yawed" motion 

 must be associated not with the yaw angle 4> but with a constant value of the 

 "drift" or "sideslip" angle /3 = tan" ' (v u). The drift angle is used frequently 



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