Vibrations of a shallow cable with a viscous damper

The optimal tuning and effect in terms of modal damping of a viscous damper mounted near the end of a shallow cable are investigated. The damping properties of free vibrations are extracted from the complex wavenumber. The full solution for the lower modes is evaluated numerically, and an explicit and rather accurate analytical approximation is obtained, generalizing recent results for a taut cable. It is found that the effect of the damper on the nearly antisymmetric modes is independent of the sag and the stiffness parameter. In contrast, the nearly symmetric modes develop regions of reduced motion near the ends, with increasing cable stiffness, and this reduces the effect of the viscous damper. Explicit results are obtained for the modal damping ratio and for optimal tuning of the damper.