Double-armored Cable With Specified Tension and Rotation 



There are four variables of major interest: the external tension F, 

 end torque T, elongation AL, and rotation <(>. Two of these variables can 

 be fixed by prescribing boundary conditions, or external loadings. The 

 remaining two are the unknown reactions. In this case the unknown reactions 

 are end torque and elongation. These external cable reactions can be 

 determined once the geometry of the loaded helical wires is determined. 

 Each developed triangle for a loaded helix may be defined by a^ 1 and S^' 

 For double-armored cable the four unknowns are a^ ' , a^' , S^ ' and S ' . 

 Four equations are needed to define the cable configuration. 



Since both layers of wire must elongate the same axial length, based 

 on the developed triangle concept in Figure 3, there is a constraint: 



'1 



= L, 



S ' sin a ' = S ' sin a ' 



(6) 



where subscripts 1 and 2 indicate inner and outer layer, respectively. 



Another constraint may be derived from the condition that there 

 will be no relative rotation between layers, that is (j> 4 = <t>„ = <t>. From 

 Figure 3 the rotation equations are: 



'1 



S 1 ' cos a' = R 1 ' (2im 1 + <(>) 



(7) 



= R 2 ' (2to 2 + 



40 



(8) 



The pitch radius R^' is governed by Equation 5 and the pitch radius of 

 the adjacent wire layer. 



The axial force equilibrium equation requires the balance of external 

 and internal forces. 



F = m (p sin a ' + B cos a ') 



+ m„ 



(p 2 sin a 2 ' 



+ B 2 cos a ') 



(9) 



The wire tension p^ is related to S±' by Equation 1. The binormal 

 forces Bi and B2 are computed from Equation 4. 



Equations 6, 7, 8, and 9 were solved numerically for a-| ' , (*2 ' , Si ' 

 and S2 1 by iteration. First assume Si 'and a-| ' , using Equation 6, 7, and 

 8 for the iteration of cq ' . After al ' is determined, o^'and S2 ' are 

 calculated. Then Equation 9 is used to check the assumed Si ' . 



