Norrbin 



originated by Kirchhoff [ 1 1] and Lannb [ 12] . In spite of the fact that 

 these forces will be modified by the presence of viscosity in the real 

 fluid, and that new forces will also be generated by the viscous 

 effects, these ideal results should be considered when formulating 

 the mathematical model. 



If U is the velocity vector of the local fluid element the 

 total kinetic energy is given by T^_ = (p/2) j U dr, or in a potential 

 flow generated by the impuls pressure p^ 



T,= .|y*||dS (5.1. 



The integration is to be extended over the total boundary, i.e. over 

 the wetted surface of the body. Let the potential be written in 

 linearized form as 



$ = $1 u + ^gV + ^jw + ^^p + ^gq + 4>gr (5. 2) 



with respect to the six component body velocities Uj. The six 

 coefficients $j then are functions of the body geometry and of the 

 position in relation to the body. 



The condition for fluid velocity - 8$/9n at the body boundary 

 to equal the body normal velocity may be formulated by use of the 

 directional cosines for the normal in the Oxyz- system, whereby 



6 6 



2T. = ; ; AjjUjUj 

 i=i j=i 



or 



(5.3) 



2T. = - X: u^ - Y. v^ - Z. w^ - 2Y. vw - 2X.wu - 2X.uv 

 L Ci V w w w V 



- K^p^ + M. q^ - N.r^ - 2M. qr - 2K . rp - 2K.pq 



- 2(X.u + Y^v + Z. w)p - 2(X.u + Y. V + Z^w)q 



- 2(X.u + Y.v + Z.w)r 



I 



Here there are 21 different added masses (Ajj) or "accelera- 

 tion derivatives," Force derivatives with respect to a linear accelera- 

 tion are of dimension M, and moment derivatives with respect to an 

 angular acceleration are of dimension ML , as are the mass moments 

 of inertia. Cross coupling derivatives such as Xj, = - A|4 are of 

 dimension ML. 



828 



