Keywords: cables, galloping, twist in cables, aeroelasticity, instability, bifurcation.

The analysis of galloping oscillations of iced cables requires a careful formulation both of the mechanical model and of the aeroelastic forces, especially concerning nonlinear regimes [1]. The forces are usually modelled referring to the quasi-steady theory, and they depend on the mean wind speed and on the angle of attack, which in its turn descends from the velocity of the structure and from its flow exposition. The structure is generally modelled as a perfectly flexible cable, that is as a one-dimensional continuum capable of translational displacements only [2,3]. From the mechanical point of view this assumption is reliable since the torsion stiffness of the cable is usually very high and the bending stiffness is negligible with respect to the geometric one, because of the slenderness of the structure. However, simplified model of cables have highlighted the importance of the twist angle on the determination of aerodynamic forces and, therefore, on the dynamical behaviour of the system. Luongo & Piccardo [4] have tried to correct the classic model, adding to the elastic potential energy of the flexible cable an energy of pure torsion ignoring, still, every term of mechanical coupling. Therefore, the formulation of a consistent cable-beam model is a matter of great interest, able to take into account all the stiffnesses involved in the problem. To the best of authors' knowledge, similar models are usually employed in fully numerical approaches (for instance, [5]) but they are confined to the linear range and not yet employed in semi-analytical analysis, like the one here proposed. A first approach to the subject, devoted to the linear problem, has been presented by the authors in [6].

The aim of this paper concerns the formulation of a consistent cable-beam model able to considering the twist angle, which can be very important in the determination of aeroelastic behaviour of this kind of structures. Several points are worth highlighting:

1. A consistent model of a nonlinear, curved, prestressed, no-shear, elastic cable-beam has been formulated.
2. Reduced equations of motion have been deducted through a suitable magnitude order analysis; this has made possible to clarify the different role of the dynamic twist angle on symmetrical and antisymmetrical modes; as major result, the reduced equations of motion are identical to those of a flexible cable, with an additional nonlinear equation in the twist angle.
3. The dynamic twist angle is a passive variable, slave of translations.
4. The aerodynamic forces have been evaluated taking into account the angle of static rotation induced by the mean wind; the corresponding change in the configuration of the cable has been considered.
5. The effective importance of the twist velocity on the aerodynamic forces has been discussed for cables not having evanescent sag.

Numerical results are preliminarily obtained as regards the linearized reduced equations of motion, using a Galerkin procedure with translational and twist eigenfunctions, in order to study conditions of incipient instability. It has been proved that the dynamic twist angle is able to sensibly influence the critical conditions of the system, through the circulatory matrix, when symmetrical modes are taken into account, especially for small values of sag. The presence of twist angle may imply the appearance or disappearance of criticality, and may lead to remarkable differences in aeroelastic critical velocities. These alterations seem more pronounced when a cross-section in an initially non-symmetric position is considered.

1

A. Luongo, G. Piccardo, "Non-linear galloping of sagged cables in 1:2 internal resonance", Journal of Sound and Vibration, 214(5), 915-940, 1998. doi:10.1006/jsvi.1998.1583

2

A. Luongo, G. Rega, F. Vestroni, "Planar non-linear free vibrations of an elastic cable", International Journal of Non-Linear Mechanics, 19(1), 39-52, 1984. doi:10.1016/0020-7462(84)90017-9

3

C.L. Lee, N.C. Perkins, "Nonlinear oscillations of suspended cables containing a two-to-one internal resonance", Nonlinear Dynamics, 3, 465-490, 1992. doi:10.1007/BF00045648

4

A. Luongo, G. Piccardo, "On the influence of the torsional stiffness on nonlinear galloping of suspended cables", Proc., 2nd ENOC, Prague, Czech Republic, 1, 273-276, 1996.

5

G. Diana, S. Bruni, F. Cheli, F. Fossati, A. Vanenti, "Dynamic analysis of the transmission line crossing Lago de Maracaibo", Journal of Wind Engrg. and Ind. Aerodyn., 74-76, 977-986, 1998. doi:10.1016/S0167-6105(98)00089-0

6

A. Luongo, D. Zulli, G. Piccardo, "A linear model of curved beam for the galloping analysis of suspended cables", Proc., XVI AIMETA Conference, Florence (Italy), CD-Rom, 2005 (in Italian).

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