singular Perturbation Problems in Ship Hydro dynamics 



The following terms are purely hydrodynamic; they will be expressed 

 in terms of added-mass and damping coefficients, as follows: Let: 



m(x) +-i-n(x)H p \ di n,(|>,, (3-31) 



^"^ -^C(x) ^ ^ 



where C(x) is the contour of the immersed part of the cross section 

 of X, Cf. (2-82). We call m(x) the "added mass per unit length" 

 and n(x) the "damping coefficient per unit length." Using the 

 slender-body approximations that <^5*^ - x4>, and n^*^ - xn,, we 

 find for Fj"(t): 



F3(t) = - Pg J dS n^{i^-^i^) - (ico)^ ^ dx (e3-xe5)[ m(x) + n(x)/ico] ; 



(3-32) 

 F5(t) = Pg J dS xn3(e3-x^) + (ico)^ J dx x(e3-xe5)[ m(x) + n(x)/ico] . 



Sn L 



Finally, we abbreviate these formulas: 



Fj'"(t) = - y [(ia))^a.j +(ia))bji +C3i]ej(t) , (3-33) 



i=3,5 

 where 



^55 



= \ dx m(x); ^33 ~ \ ^^ i^(x)j 



> = ^53 = - J ^^ ™^^^' ^35 = ^53 = - J^ ^ ""^^^ 



= \ dx xm(x); bgg = \ dx X n(x) ; 



= pg \ dSrL=2pg\ dxb(x,0); 



- pg \ dS xn = - 2pg \ dx xb(x,0); 



\ dS xV = 2pg \ dx x^b(x,0); 



b(x,z) is the hull offset at a point (x,z) on the centerplane. 



So 



C = C 



35 53 



S5 = pg .. 



So 



755 



