Vortex Theory for Bedies Moving in Water 
A, B, A, , B, are constants relatively close to unity, 
and C, C, are functions of — ;. As Ej is practically invariable, 
C and C, may also be considered as constants. 
In fact all the coefficients depend on the hull shape and must 
be experimentally determined. Let - © , = @ _ bethe y-compo- 
nent and the moment of the hydrodynamic force exerted on the rudder, 
M, the added mass for the double model calculated by neglecting the 
free vortices effect for a non-uniform motion in the y-direction, 
and I, the similar inertial moment in a non-uniform rotation about 
the z-axis. We put: 
1 L 
2 
—— T € k ’ ke = ke tk 
fe) 2 fo) fe) 
Furthermore, because the circulation can never take immediately 
its asymptotic value corresponding to the steady motion defined by 
the present values of a and Baer the present value of Y is not 
given by the above expression, but by Y - Y, . Similarly the hydro- 
dynamic momentisnot N, but N-N,_. Y, and ny are the defi- 
ciencies due to the history of the motion. 
Finally the equations of the unsteady motion are as follows : 
b+Y 
L L da 16; L i | 2 
= - — —-=Aa+B —+C — |—|- 
we on ae Se ae es Last. 
Guba es 
eee 
18 d 6 L i i; LZ 
= (ees) Ss A Ye geet ogi (Rae ae Cas 
av > at ‘aR? iy. else toe | I 2 
aoe ae 
Let 
= A = 
S ete Oke Ee Be 
: : 4 L : : 
One sees that in a steady motion, the ratio oR 28 given by 
@ E; ie t 
A +A,)——_>~———_ = SO (A GC + A —— _ j___}. 
( )4 oe Slater! the tee Ci) aR aC 
2 
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