Modal stability of inclined cables subjected to vertical support excitation

Gonzalez-Buelga, Alicia; Neild, Simon A; Wagg, DJ; Macdonald, John H G

doi:10.1016/j.jsv.2008.04.031

Cited by 54 publications

(48 citation statements)

References 22 publications

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“…In addition to demonstrating the accuracy of the normal form technique, this example shows that although the transformed dynamic equation in u contains just the responses at the response frequencies, u rn for the nth mode, the response of the second mode is predicted accurately. Compare this to the averaging technique (see Tondl et al (2000) and Verhulst (1996) implementation on a multi-mode system demonstrated in Gonzalez-Buelga et al (2008)), for example: averaging assumes a response of each mode just at the response frequencies of the form q 1 (t) = q 1s (t) sin(u r1 t) + q 1c (t) cos(u r1 t) and q 2 (t) = q 2s (t) sin(u r2 t) + q 2c (t) cos(u r2 t), (5.23) where u r1 = U and u r2 = 2U and results in a prediction of zero response of the second mode in the steady state. The reason for this discrepancy is that the mechanism for generating a response in the second mode at twice the forcing frequency is a frequency mixing of the first-mode response at the forcing frequency with the second-mode response at the forcing frequency-which, using averaging, is assumed to be zero (the response is taken to be entirely at twice the forcing frequency).…”

Section: Two-degree-of-freedom Oscillatormentioning

confidence: 92%

Applying the method of normal forms to second-order nonlinear vibration problems

2010

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Vibration problems are naturally formulated with second-order equations of motion. When the vibration problem is nonlinear in nature, using normal form analysis currently requires that the second-order equations of motion be put into first-order form. In this paper, we demonstrate that normal form analysis can be carried out on the second-order equations of motion. In addition, for forced, damped, nonlinear vibration problems, we show that the invariance properties of the first-and second-order transforms differ. As a result, using the second-order approach leads to a simplified formulation for forced, damped, nonlinear vibration problems.

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Section: Two-degree-of-freedom Oscillatormentioning

confidence: 92%

Applying the method of normal forms to second-order nonlinear vibration problems

2010

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“…In the numerical simulations we use the parameter values given in Table 1 as chosen in Gonzalez-Buelga et al (2008) to approximately match a typical full-scale bridge cable inclined at θ = 20…”

Section: Numerical Simulationsmentioning

confidence: 99%

Robust Measurement Feedback Control of an Inclined Cable

Baudouin

Neild

Rondepierre

et al. 2013

IFAC Proceedings Volumes

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Considering the partial differential equation model of the vibrations of an inclined cable, we are interested in applying robust control technics to stabilize the system with measurement feedback when it is submitted to external disturbances. This paper focuses indeed on the construction of a standard linear infinite dimensional state space system and an H ∞ feedback control of vibrations with partial observation of the state. The control and observation are performed using an active tendon.

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“…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]. Experimental analysis of the cable response in out-of-plane modes due to vertical excitation have been reported in [Rega et al, 1997;Rega & Alaggio, 2009;Rega et al, 2008].…”

Section: Introductionmentioning

confidence: 99%

“…Our object of study is the three-mode model from [Gonzalez-Buelga et al, 2008] of an inclined cable that is vertically excited near its second natural frequency. The model takes the form of a periodically forced system of three second-order differential equations for the coupled nonlinear dynamics of the directly excited second in-plane mode and the auto-parametrically forced first and second out-of-plane modes.…”

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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Modal stability of inclined cables subjected to vertical support excitation

Abstract: This is a repository copy of Modal stability of inclined cables subjected to vertical support excitation.

Cited by 54 publications

References 22 publications

Applying the method of normal forms to second-order nonlinear vibration problems

Applying the method of normal forms to second-order nonlinear vibration problems

Robust Measurement Feedback Control of an Inclined Cable

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

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