Planar and non-planar non-linear dynamics of suspended cables under random in-plane loading—I. Single internal resonance

• Published in: International journal of non-linear mechanics 1996, Vol.31 (6), p.837-859
• Main Authors: Chang, W.K., Ibrahim, R.A., Afaneh, A.A.
• Format: Article
• Language: English
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/S0020-7462(96)00110-2

The non-linear interaction of the in-plane and out-of-plane motions of a suspended cable in the neighbourhood of 2:1 internal resonance under random loading is studied. The random loading acts externally on the in-plane mode, while the out-of-plane mode is non-linearly coupled with the in-plane mode. Any non-trivial motion of the out-of-plane mode is mainly due to this non-linear coupling, which becomes significant in the neighbourhood of internal resonance. The response statistics are estimated by employing the Fokker-Planck equation together with Gaussian and non-Gaussian closures. Monte-Carlo simulation is also used for numerical verification. Away from the internal resonance condition, the response is governed by the inplane motion, and the non-Gaussian closure solution is found to be in good agreement with numerical simulation results. The stochastic bifurcation of the out-of-plane mode is predicted by Gaussian and non-Gaussian closures, and by Monte-Carlo simulation. The non-Gaussian closure can only predict bounded solutions within a limited region. The on-off intermittency of the second mode is observed in the Monte-Carlo simulation over a small range of excitation level. The influence on response statistics of excitation level and cable parameters, such as internal detuning, damping ratios, and sag-to-span ratio, is reported.