INTERNAL RESONANCES IN WHIRLING STRINGS INVOLVING LONGITUDINAL DYNAMICS AND MATERIAL NON-LINEARITIES

✍ Scribed by MICHAEL J. LEAMY; ODED GOTTLIEB

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 358 KB
• Volume: 236
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1006/jsvi.2000.3039

✦ Synopsis

Internal resonance mechanisms between near-commensurate longitudinal and transverse modes of a taut spatial string are identi"ed and studied using an asymptotic method, and the in#uence of material non-linearities on the resulting solutions is considered. Geometrical non-linearities couple longitudinal 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 considers further the inclusion of longitudinal motions and a non-linear material law, which are both appropriate for the study of rubber-like strings. An asymptotic analysis captures the aforementioned whirling motions, as well as a new class of whirling motions with signi"cant longitudinal content. Periodic, quasi-periodic, and aperiodic (likely chaotic) responses are included among these motions. Their existence, hardening}softening characterization, and stability are found to be highly dependent on the magnitude of the material non-linearities.

2000 Academic Press

r!1) *w *x *u *x #2 *w *x *u *x *u *x # R (r!1) *w *x *u *x # 1 2 *v *x # 3 2 *w *x # *w *x *u *x # *v *x *v *x # *w *x *u *x # *w *x *u *x # P R *w *x *u *x # *w *x *u *x . (A.3)