Bifurcations and chaotic dynamics in suspended cables under simultaneous parametric and external excitations

Abstract

The bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degree-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary resonant excitation, and principal parametric excitation of suspended cables is considered. Using the method of multiple scales, a parametrically and externally excited system is transformed to the averaged equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Five branches of dynamic solutions are found. Three of these branches that emerge from two Hopf bifurcations and the other two are isolated. The two Hopf bifurcation points, one is supercritical Hopf bifurcation point and another is primary Hopf bifurcation point. The limit cycles undergo symmetry-breaking, cyclic-fold, and perioddoubling bifurcations, whereas the chaotic attractors undergo attractor-merging, boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, homoclinic explosions and hyperchaos are also observed.

Dynamic instability of inclined cables under combined wind flow and support motion

“…In order to express the closeness of the three frequencies, two detunings σ 1 and σ 2 are taken, such that ω 1 = Ω + ε 2 σ 1 , ω 2 = Ω + ε 2 σ 2 , where ε is a dimensionless perturbation parameter (ε 1). These definitions, according to [9], are found computationally more convenient than the "classical" definitions Ω = ω 1 + ε 2 σ and ω 2 = ω 1 + ε 2 ρ where an external (σ ) and an internal (ρ) detunings appear. Of course the following relations hold between the two sets of detunings: σ = −σ 1 and ρ = σ 2 − σ 1 .…”

Free vibration of summation and difference resonance of the vertical cable and other coupled structural members

“…Since the beginning research of cable, some articles only forced on a single cable (Chen et al, 2010; Da Costa and Martins, 1996; Takahashi, 1991; Zulli and Luongo, 2014), which cannot reveal the totally nonlinear characteristics of the whole structure.…”

Revisited modelling and multimodal nonlinear oscillations of a sagged cable under support motion

“…For horizontally suspended cables, the effect of simultaneous external and parametric excitations, first addressed in [30], was considered later, e.g., in [31,32], mostly for highlighting some global dynamics aspects of system response, even though, sometimes, with weak reference to an actual physical excitation in the background. Two-d.o.f.…”