1144 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



Fig. 9 shows the forces acting on an element of the cable, with tension 

 at the point s being denoted by 7\ The normal drag force per iniit length 

 Dff may, by virtue of (4) and (5), be written in the foi-m 



^, Copy d . a ] • a \ 



Dn = ^ sm 6 \su\d \. 



It is necessary to introduce here | sin 9 \ in order for D^ to have the proper 

 sign foi' all 6. We note, however, that if Vc ^ V, we have from (13) 



Vt = Vc - V cos ^ ^ 0. 



Hence in normal laying and recovery the unit tangential drag force 

 Dr is always in the positive s direction. 



,2de 



jAs 

 wAs 

 Fig. 9 — Diagram of forces acting on a cable element. 



The forces acting on an element produce a centrifugal acceleration 

 Vcdd/ds. Hence, summing forces along the directions t (tangential) and 

 n (normal), dividing by As and sending As to zero, we obtain 



(r - pA\ ')'^ + CDpV__d ^.^^ ^ ^.^^ e\ - wcose = 0, (a) 



ds 2 ' ^ ^ 



(18) 



^ -\r Dr - w sin 6 = 0, (b) 



as 



where pc is the mass density per unit length of cable. 



It is seen at the out.set that = a is a solution of (18a). It is in fact 

 the important straight-line solution which has been discussed in Sec- 

 tion 3.1. 



li 6 ^ a and Dt varies only with Vt we may divide (18b) into (18a) 

 and integrate to obtain the solution for T in the following form: 



^^^ jT - P.7/) ^ r {w sin ^ - Dr) ^^^ ^^^ 



(To — pcVc') ho w(cos f — A sin ^ | sin ^ I) ' 



C'dP dV cos a ., X 



A = — 5 = ^-r-, (•>) 



2iv sm- a 



where To is the tension corresponding to the angle 9^ 







