Internal Resonance of Coupled Non-planar Vibrations of an Axially Moving Cable with Small Sag

摘要

The coupled nonlinear response in in-plane and out-of-plane vibrations is investigated for a small sagged axially moving cable in the present paper. A two-degree-of-freedom model is obtained by neglecting the longitudinal displacement of the cable. A group of ordinary differential equations are derived by using the Galerkin's discretization. The frequency analysis shows that internal resonance occurs for a range of small sag-to-span ratio and cable tension. Under an in-plane harmonic excitation, the three-to-one internal resonance between in-plane and out-of-plane modes is analyzed using the modified Lindstedt-Poincare method. In the region of internal resonance several branches of large frequency-amplitude response curves of the out-of-plane motion are found. The influence of excitation amplitude and transport speed on the energy exchange between the in-plane and the out-of-plane modes is discussed.