Free vibration of a sagged cable supporting a discrete mass

Abstract

A continuum model is presented that describes the nonlinear and three-dimensional response of an elastic cable that supports a single attached mass. Two asymptotic forms of this model are derived for the free, linear response of sagged suspensions having small equilibrium curvature (sag) and level supports. The first model, which is valid for relatively small attached masses, assumes that the cable stretches quasi-statically and results in uniform dynamic cable tension. The quasi-static stretching assumption is partially relaxed in the second model, which accounts for spatially varying dynamic tension in an approximate manner. In particular, the second model captures the discontinuous change in dynamic tension across the attached mass and the resulting tangential mass acceleration. The eigensolutions governing free response are compared for the two models. The comparison reveals that the first (simpler) model provides excellent approximations to the natural frequency spectrum for all cable modes having natural frequencies less than that of an (approximate) elastic mode.