Vehiole Dynamics Associated with Submarine Rescue 



where Xq, Yq, Zq are the coordinates of the vehicle center in the 

 body axis system. Neglecting t];ie velocity and acceleration of the 

 eg. with respect to the body (Xq, Yq, Zq, Xq, Yq, Zq = 0) because 

 they are small and performing the operations 



external 



^(mV,)=mt, ^,,+ 0.xmV3 



we obtain 



external 



u + qw - rv - XQ(q^ + r^) + XQ(pq - r) + Zglpr + q)"] 

 u + ru - pw - YQ(r2 + p2) + ZQ(qr - p) + XQ(qp + r) 

 _w + pv - qu - Zq(p2 + q2) + X^Crp - \) + Y (rq + p)_ 



The equations of angular motion are derived from 



M 



external 



^(i=r) 



(2) 



which states that the sum of the external moments acting on a body 

 equals the time rate of change of the angular momentum of the body 

 with both the moments and angular momentum expressed about the 

 same point. Since the hydrodynamic moments are described about 

 an off eg, axis system the development of the right-hand side of 

 the equations consists of expressing the time rate of change of the 

 angular momentum of the body about the center of the axis system 

 with the rotational vector expressed in the directions of the body 

 axis. The results of this operation [ 1] yield: 



I^P + (I^ - Iy)qr + m j Yq(w + pv - qu) - Zg(v + ru - pw)}' 



M 



external - 



\\ + (Ix - \)^ + ^ \ Zq(u + qw - rv) - Xg(w + pv - qu)[ 

 .l^r + (ly - Ix)pq + m j Xq(v+ ru - pw) - Y^{\x + qw - rv)}_ 



The moments of inertia 1^, I and I^ are the moments of 

 inertia about the center of the body axis system and not about the 

 vehicle center of gravity. Since there is near symmetry in weight 

 distribution about this body axis system, cross products of inertia 

 have been dropped from the equations of motion. 



The external forces and moments, ^^^^^^^31 and M^jj^j^^gl , 

 on the vehicle during free stream operations come from three 



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