This paper focuses on a comparison of cable non-linear dynamic responses obtained with the kinematically non-condensed and condensed modeling. Planar non-linear interactions involving simultaneous primary external and internal resonance in horizontal suspended cables are analytically investigated. 1:1 or 2:1 internal resonance is considered. The governing partial-differential non-linear equations of motion of the non-condensed cable model account for the effects of dynamic extensibility, i.e., dynamic tension spatio-temporal variation, and capture the non-linear coupling and contributions of longitudinal/transversal modal displacements. On the contrary, in the condensed cable model, a single integro-partial-differential equation of motion is obtained by neglecting the longitudinal inertia according to a quasi-static stretching assumption of cable in motion. This entails linking the longitudinal displacement to the transversal one and considering a space-independent dynamic tension. This simplified model is typically considered in the literature involving cable nonlinear dynamics. Based on a multi-dimensional Galerkin-based discretization and a second-order multiple scales approach accounting for higher-order non-linear effects and resonant/non-resonant modal contributions, the ensuing dynamic responses and their stability are evaluated by means of force- and frequency-response diagrams with stability analyses. Moreover, the corresponding spacetime non-linear coupled configurations and dynamic tension distributions are analyzed. The numerical explorations highlight that, depending on cable elasto-geometric properties, internal resonance condition and system control parameters, the condensed model may lead to quantitative and/or qualitative discrepancies in the non-linear dynamic responses, with respect to the non-condensed model. The results allow us to point out such meaningful effects of disregarding the system longitudinal dynamics via the kinematic condensation procedure, and to identify cases where the parametric investigation has to be pursued with the more accurate non-condensed model.
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- Resonant Non-Linear Dynamic Responses of Horizontal Cables via Kinematically Non-Condensed/Condensed Modeling
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