DYNAMICS AND KINEIVLITICS OF SUBMARINE CABLE 1203 



[n the (7, u,v) system the vector n becomes for example 



- _ w sin 6 cos rp — ?; sin ^ 

 [sin^ 6 cos- xp + sin^ \p]^ ' 



Imagine the origin of the {i,u,v) triad to traverse the cable at unit 

 velocity. The triad during this traverse rotates hke a rigid body with 

 respect to the fixed (^, rj, f ) frame. The rotation, which we denote by fi, 

 is seen from Fig. 27 to be 



Q ='j ^ -^p = u cos d \p -\- V 6 + 1 sin 6 4^. 



Here the dot denotes differentiation with respect to time, or since 

 ds/dt = 1 it may be interpreted as differentiation with respect to distance 

 along the cable. The vector 7 is a fixed vector of constant magnitude in 

 the rotating (t, u,v) triad, hence 



^ = fi X T =ue-'vcosd^. (105) 



ds 



From (101), (102), (104), and (105) we obtain for (103) 

 {T - pcVc') Qd - I cos 6 4^) +l(^ + Drj 



+ -^ — (sin^ 6 cos^ \l/ + sin" \p)'iu sin 6 cos ^ — y sin ^) (106) 



— w{u COS 6 -{- t sind) = 0, 



which gives the three scalar equations, (47). 

 Further, let r(s) be the cable configuration, i.e., 



r(s) = Uis) +J vis) + A^r(s), 



where ^(s), r}{s), and f (s) are the ^, 77, f coordinates of a point s of the 

 cable. Then 



1 = i ^ii^ + ^ ^^ + A- ^^ 

 ds ds ds 



Forming the scalar product of t with i, j ,k respectively, we get (48) of 

 Section 7.1. 



In Si 6, xj/, T space the solution trajectories of (47) are given by the 

 solutions of 



, jt2\ dd _ A(cos^ rp sin^ 6 + sin^ ^) ' cos xp sin. 6 — cos 6 



V -' Per c ) -Jjf, — p; 1 : 2 ' 



dT Dt/w — sm 6 



, -.r2\d\l/_ A (cos \(/ sm 6 + sin" \l/)' sm ^ 



- P'^^'^)-^ cos d{Dr/w - sin d) ' 



