A boundary modulation formulation for cable’s non-planar coupled dynamics under out-of-plane support motion

Abstract

Suspended cable's non-planar resonant coupled dynamics under out-of-plane support motion is investigated by the multiple-scale method, with a boundary modulation formulation established and nonlinear dynamic responses analyzed. Explicitly, to cope with the difficulty due to moving boundary, the small resonant support motion is properly rescaled and incorporated into cable's modulation equations as a boundary resonant modulation term, through constructing solvability conditions of the multi-scale expansions. And the boundary resonance dynamic coefficient, characterizing the boundary modulation effect, is derived analytically for cable's two-to-one resonant coupled dynamics. Numerical results for cable's non-planar coupled dynamic responses, including stability and bifurcation analysis for the equilibrium solutions of modulation equations, are obtained and presented in the end, with both saddle-node bifurcations and Hopf bifurcations detected.

Dynamic modeling and in-plane 1:1:1 internal resonance analysis of cable-stayed bridge

“…Hence, the Hopf bifurcations should be controlled in engineering. All these phenomena for cables are similar to those got by Guo et. al (Guo et al, 2016) due to the fact that the motion of cables are excited at the lower ends of cables.…”

“…5(b) for cables, one bent to the left and one bent to the right. This is associated with a double jumping phenomenon (Chen et al,2014;Guo et al, 2016) in the frequency response diagrams, which is triggered by those saddle-node bifurcations, i.e., SN1, SN2. Actually, it can be deduced that there exists two saddle-node bifurcations at the left and right side beyond the range [-1.5,1.5] of σ1, respectively.…”