On the nonlinear effects of the mean wind force on the galloping onset in shallow cables

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ORIGINAL PAPER

On the nonlinear effects of the mean wind force on the galloping onset in shallow cables Daniele Zulli · Giuseppe Piccardo Angelo Luongo

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Received: 26 April 2020 / Accepted: 5 August 2020 © The Author(s) 2020

Abstract The critical aeroelastic behavior of horizontal, suspended, and shallow cables is analyzed via a continuous model accounting for both external and internal damping. Quasi-steady aerodynamic forces are considered, including their stationary contribution (mean wind force). This latter induces a rotation of the cable (steady swing) around the line connecting the suspension points, together with a deformation of the initial equilibrium profile under self-weight. First, by using perturbation methods, the nontrivial equilibrium configuration is analytically determined as a nonlinear function of the wind velocity. Then, the wind critical values at which bifurcations take place and the corresponding modal shapes are determined by solving a boundary value problem in the complex field. Numerical investigations are carried out to validate the perturbation solution. A preliminary nonlinear galloping analysis is also performed to verify the galloping onset in terms of non-trivial equilibrium path and critical modes. The nonlinear terms related to the fundaD. Zulli · G. Piccardo (B) · A. Luongo M&MoCS - University of L’Aquila, Loc. Monteluco di Roio, 67100 L’Aquila, AQ, Italy e-mail: [email protected] G. Piccardo e-mail: [email protected] A. Luongo e-mail: [email protected] G. Piccardo DICCA - University of Genoa, Via Montallegro 1, 16145 Genova, GE, Italy

mental path, from which bifurcations take place, play a key role revealing new insights. They are able to heavily influence the system bifurcation, making unstable configurations which instead would be aerodynamically stable without considering the effect of the mean wind force. Keywords Shallow cable · Galloping onset · Swing · Bifurcation from a nontrivial path · Perturbation solutions

1 Introduction Cables are widely used structures in civil and industrial applications. They assume a fundamental role in the carrying capacity of suspended and stayed bridges, as well as in the realization of cable-cars, electrical transmission lines and many other structures. A general characterization of statics and dynamics of cables can be found in the classic Irvine’s book [1], where several applications are analyzed as well. Recent advances in the statics of cables under forces exclusively vertical but not necessary uniform are proposed in [2], then extended to the case of general 3D forces in [3], where a perturbation asymptotic expression for the elastic problem solution is proposed. At the same time, the dynamic characterization of suspension cable systems is very important (e.g., [4]). Concerning nonlinear vibrations of suspended cables, the quadratic terms play a fundamental role in presence of internal resonances (e.g.,

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[5,6]), differently from taut strings for which only cubic terms are present (e.g.,