Internal resonances in whirling strings involving longitudinal dynamics and material non-linearities

Internal resonance mechanisms between near-commensurate longitudinal and tr ansverse modes of a taut spatial string are identified and studied using an asymptotic method, and the influence of material non-linearities on the re sulting solutions is considered. Geometrical non-linearities couple longitu dinal motions to in-plane and out-of-plane transverse motions, resulting in resonant and non-resonant interactions between linearly orthogonal string modes. Past studies have included only transverse modes in the description of string motions and have predicted periodic, quasi-periodic, and chaotic whirling motions arising from the geometrical non-linearities. This study c onsiders further the inclusion of longitudinal motions and a non-linear mat erial law, which are both appropriate for the study of rubber-like strings. An asymptotic analysis captures the aforementioned whirling motions, as we ll as a new class of whirling motions with significant longitudinal content . Periodic, quasi-periodic, and aperiodic (likely chaotic) responses are in cluded among these motions. Their existence, hardening-softening characteri zation, and stability are found to be highly dependent on the magnitude of the material non-linearities. (C) 2000 Academic Press.