Bending of helically twisted cables under variable ... - Pfisterer

To determine the curvature at which the wire starts slipping, we must set ϕ = 0 in (2.52) (see
Section 2.3).
Using the series expansion:
we find, for small values of ϕ and ignoring higher order terms that are small:
Under the same assumptions, we also find:
Substituting this, together with (2.53) 53) and (2.54) in (2.52), yields:
and
Analogously, the transition curvature for the end of the wire slippage at ϕ = /2, i.e. for the entire length
of the considered wire, can be found from (2.52):
or, since sin ( /2) = 1:
31
(equation (2.24)) q.e.d.
(equation (2.23)) q.e.d.