Nonlinear oscillations of cables under harmonic loading using analytical and finite element models

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.

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“…(7) should be balanced at the same order, ε 2 . Thus, damping c and the support motion are rescaled as c → εc, z (t) → ε 2 z (t).…”

Section: Perturbation Analysis and Cable's Reduced Dynamicsmentioning

confidence: 99%

“…Due to its elasticity, initial sag, nonlinear axial stretching, and complex boundary connections, cable's dynamic behaviors are very rich and extensive investigations into cable's dynamics have been undertaken by many researchers in the past few decades [1][2][3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2015

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“…In similar cables, the effects of the dimensions of the model in the description of the cable nonlinear dynamics have been investigated through an extensive numerical campaign conducted through either path-following technique or finite element method [17].…”

Section: Control Of Spatial Motionsmentioning

confidence: 99%

“…The model has been used to define the main features of the controlled cable nonlinear dynamics under in-plane harmonic force. The system response has been described through an extensive investigation conducted by means the pseudo-arclength continuation method [16,17]. The analysis has been conducted directly on the nonlinear modal equations (9) increasing the dimension of the discretized model in order to evidence the influence of higher frequency modes in the studied dynamics due to the nonlinear modal coupling.…”

Section: Predicted Control Effectiveness By Analytical Modelmentioning

confidence: 99%

Analytical prediction and experimental validation for longitudinal control of cable oscillations

Gattulli

Alaggio

Potenza

2008

International Journal of Non-Linear Mechanics

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The paper summarizes the knowledge acquired from the analytical studies and the experimental implementation of a longitudinal noncollocated control strategy for the reduction of cable oscillations. The control is introduced by imposing a longitudinal action at one support based on the knowledge of transverse displacements and velocities of a few selected points. A spatially one-dimensional continuous model of a suspended cable has been used to describe the main features of the noncollocated longitudinal active control strategy. A discrete modal representation has permitted the introduction of suitable nonlinear state-feedback controllers. The results have been used to derive an implementable strategy, based on direct output feedback, which preserves the main previous control features. A physical model of an actively controlled cable has been used to demonstrate the control effectiveness of the proposed strategy through a large campaign of experiments, conducted in various frequency ranges and amplitude levels including meaningful external resonance conditions. The responses predicted by the analytical model and the experimental results show good qualitative agreement with one another, in both the uncontrolled and controlled experienced cable dynamics.

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“…For example, [Georgakis et al, 2001] considered multiple in-plane and out-of-plane modes for the case where auto-parametric resonance and cable nonlinearities are included. Using Mathieu-type equations to model auto-parametric resonance [Tagata, 1977;Uhrig, 1993] also capture coupling between the in-plane and out-of-plane modes of vibration [Fujino et al, 1993;Gattulli et al, 2005Gattulli et al, , 2004. The loss of stability of the semi-trivial solution, for which the response is limited to just the directly excited mode, has also been studied in detail [Berlioz &Lamarque, 2005;Gonzalez-Buelga et al, 2008;Macdonald et al, 2010;Rega, 2004a,b;Rega et al, 1999].…”

Section: Introductionmentioning

confidence: 99%

Vibration Dynamics of an Inclined Cable Excited Near Its Second Natural Frequency

Tzanov

Krauskopf

Neild

2014

Int. J. Bifurcation Chaos

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Inclined cables are essential structural elements that are used most prominently in cable stayed bridges. When the bridge deck oscillates due to an external force, such as passing traffic, cable vibrations can arise not only in the plane of excitation but also in the perpendicular plane. This undesirable phenomenon can be modelled as an auto-parametric resonance between the in-plane and out-of-plane modes of vibration of the cable. In this paper we consider a threemode model, capturing the second in-plane, and first and second out-of-plane modes, and use it to study the response of an inclined cable that is vertically excited at its lower (deck) support at a frequency close to the second natural frequency of the cable. Averaging is applied to the model and then the solutions and bifurcations of the resulting averaged differential equations are investigated and mapped out with numerical continuation. In this way, we present a detailed bifurcation study of the different possible responses of the cable. We first consider the equilibria of the averaged model, of which there are four types that are distinguished by whether each of the two out-of-plane modes is present or not in the cable response. Each type of equilibrium is computed and represented as a surface over the plane of amplitude and frequency of the forcing. The stability of the equilibria changes and different surfaces meet along curves of bifurcations, which are continued directly. Overall, we present a comprehensive geometric picture of the two-parameter bifurcation diagram of the constant-amplitude coupled-mode response of the cable. We then focused on bifurcating periodic orbits, which correspond to cable dynamics with varying amplitudes of the participating second in-plane and second out-of-plane modes. The range of excitation amplitude and frequency is determined where such whirling cable motion can occur. Further bifurcations -period-doubling cascades and a Shilnikov homoclinic bifurcation -are found that lead to a chaotic cable response. Whirling and chaotic cable dynamics are confirmed by time-step simulations of the full three-mode model. The different cable responses are characterized, and can be distinguished clearly, by the their motion at the quarter-span and by their frequency spectra.

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Nonlinear oscillations of cables under harmonic loading using analytical and finite element models

Cited by 65 publications

References 24 publications

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

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

Analytical prediction and experimental validation for longitudinal control of cable oscillations

Vibration Dynamics of an Inclined Cable Excited Near Its Second Natural Frequency

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