Snap-through instability during transmission of rotation by a flexible shaft with initial curvature

Abstract

The geometric imperfection of a flexible shaft transmitting rotary motion along a curved path may result in a highly nonlinear relation between the rotation angles at the driving and driven ends. If the intrinsic curvature of the shaft surpasses a specific threshold, even snap-through instability becomes possible. Previously, the problem has been considered for the case when the shaft is confined in a rigid channel such that only the rotary motion is possible along its entire length. The present study focuses on the case of the freely deformable axis of the shaft. Just the locations and the tangential directions of its ends are fixed. We investigate the response of the shaft, in terms of rotation at the driven end, by combining analytical solutions using incremental equations from the theory of Kirchhoff rods and a finite element model that features a continuous approximation of deflections and rotations. Comparison to the constrained case shows that the impact of the deformability of the axis grows both with the intrinsic curvature of the shaft and its angular length.