Shallow versus nonshallow cables: linear and nonlinear vibration performance

A mechanical model describing large motions of non-shallow cables around the initial catenary configurations is proposed. An exact kinematic formulation accounting for finite axis extensions while neglecting bending and torsional deformations is adopted whereas the material is assumed to be linearly hyperelastic. The nondimensional mechanical parameters governing the motions of non-shallow cables are obtained via a suitable nondimensionalization and the regions of their physically plausible variations are portrayed. The spectral properties of linear vibrations around the initial static configurations are first discussed. Then, the responses to primary-resonance excitations of either the lowest symmetric or skew-symmetric modes are investigated employing the method of multiple scales directly applied to the partial-differential equations of motion and boundary conditions. A detailed analysis of these responses is documented shedding light onto the importance of the quadratic non-linearities in non-shallow regimes which may entail significant non-linear spatial corrections to the leading first-order motions.