Explain initial winding before unwinding in freely rotating helical springs under extension

Explain the mechanism underlying the non-monotonic rotational response of a helical spring with freely rotating ends when it is extended from its relaxed state (pitch parameter χ = χ0 and number of turns N = N0), in which the spring first slightly increases its number of turns (winds) and then decreases its number of turns (unwinds).

Background

In the paper, the authors derive an exact elastic potential energy expression for helical springs accounting for wire torsion and bending, and allow the moving end to both translate and rotate. Minimizing this energy with respect to the number of turns yields a closed-form expression N(χ) predicting how a spring with freely rotating ends winds or unwinds as its height changes.

When analyzing the prediction and plotting N(χ), the authors observe that, starting from the relaxed state (χ = χ0, N = N0), an extension causes the spring to first wind slightly before subsequently unwinding, a counterintuitive, non-monotonic behavior. Although they note the effect is known in the spring mechanics literature, they explicitly state they lack a simple explanation for it, motivating the open question.