Three-dimensional vibrations of a helically wound cable modeled as a Timoshenko rod

Abstract

A rod model is proposed for simultaneous tension, torsion and bending of helically wound cables. The model is formulated in the Timoshenko beam formalism by first assuming that a cable can be homogenized effectively as a 3D solid rod continuum following Spencer’s constitutive law. The cross-sectional forces and moments are obtained by integrating the stress components over the cross section and, using the constitutive relations, eigenvalue problems of the rod are set up by imposing appropriate boundary conditions, thus resulting in the eigenfrequencies and mode shapes. The model has four stiffness parameters, $E, G, C_F$ and $C_T$, which incorporate both the geometrical and material properties of the rod, and are lay-angle dependent. The applicability of the model to helically wound cables is then verified by studying the vibration of $1+6$ cables (one core and six identical helical wires). It is also positively verified that the cables can be homogenized as a helical-fiber-reinforced continuum with a slight modification to the parameter $C_T$.This paper is dedicated to the memory of Franz Ziegler