On the effect of twist angle on nonlinear galloping of suspended cables
Introduction
It is well known that the aerodynamic forces acting on a non-circular rigid cylinder, subjected to a wind flow, depend, besides on the mean wind velocity, on the exposure to the flow of the body, that is on the attitude of the cylinder cross-section. When an elastic beam is analyzed in the framework of the quasi-steady theory of the aerodynamic forces, the loads are usually evaluated referring to the initial attitude of the section (see [1]), and rotations taken into account only to determine, in an approximate manner (see [2]), the fluid to structure relative velocity. A more refined analysis, however, is possible, in which the time-dependent actual attitude of the cross-section is considered, as described by the so-called twist angle (i.e. by the rotation of the section around its normal axis). In contrast, when a pretensioned string is studied, such an angle is usually not included among the kinematic descriptors of the body, since all the rotations are believed to be unimportant in capturing the main structural behavior. Therefore, a model of perfectly flexible string is adopted (one-dimensional not polar continuum), and the aerodynamic forces evaluated with reference to the initial attitude of the section, which is assumed to remain immutable in time. The problem is made even more complicated when a sagged cable is considered. Indeed, due to the steady part of the aerodynamic forces and to the high flexibility of the structure, the cable significantly changes its equilibrium configuration and, therefore, its exposure to the flow. Hence, in addition to a dynamic rotation, a static velocity-dependent rotation of the section must be considered in evaluating the aerodynamic forces.
The aeroelastic instability of sagged cables has been widely studied in the literature. Luongo and Piccardo [3] have studied the nonlinear galloping of cables in 2:1 internal resonance condition, by using a perfectly flexible cable model [4], [5] and accounting for the static rotation only. In a successive work [6], they have tentatively corrected the classical cable model to account for the twist, by using a quite simplified model. Yu et al. [7], McConnel and Chang [8], White et al. [9] have employed a model of cable-beam, accounting for twisting but not for bending, and neglecting the cable initial curvature in defining the torsion strain. In contrast, they have considered a realistically coupled extension–torsion constitutive law, based on experimental results. Recently, Luongo et al. [10], have formulated a consistent linear model of cable-beam accounting for the (small) curvature of the cable, as well as for bending and torsional stiffness. By retaining only the leading terms in each equations, they obtained linear reduced equations, amenable to an analytical solution, identical to that of the perfectly flexible model, plus an additional equation accounting for both bending and torcent moments. The model permitted to detect the influence of the dynamic twist on the critical wind velocity.
In this paper, the model presented in [10] is reformulated in the nonlinear range and nonlinear, reduced equations are derived along the same lines. These equations are obviously a particular case of more complete models, as, for example, that of Lu and Perkins [11]; these models, however, are composed by very complex equations and suffer of some numerical problems related to the existence of boundary layers, caused by the smallness of the flexural terms (nearly-singular equations). A simple two-degree-of-freedom nonlinear system is then derived from the continuous model via a Galerkin procedure, and the critical and post-critical aeroelastic behavior of the cable investigated in resonant and non-resonant cases via a Multiple Scale perturbation approach (see [12]). The role played by the dynamic twist is finally highlighted.
The paper is organized as follows. The reduced equations of motion are formulated in Section 2. The discretization is performed in Section 3. The perturbation analysis is carried out in Section 4, where the amplitude modulation equations are derived. These latter are numerically studied in Section 5 for some sample systems. Finally, some conclusions are drawn in Section 6.
Section snippets
Model
The cable is modeled as a body made of a flexible centerline and rigid cross-sections restrained to remain orthogonal to the axis (shear-undeformable beam). It is assumed to be uniformly iced and loaded by a wind flow of mean velocity , blowing horizontally. Three different configurations are considered, described in the following (Fig. 1). (a) The initial configuration , taken by the body at the time , under the action of its self-weight (including the ice accretion). This
Discrete model
A discrete model is drawn by Eq. (12) via a Galerkin procedure. The displacement field is discretized as follows:where , , are the unknown amplitudes of the in-plane trial functions , and , , are the unknown amplitudes of the out-of-plane trial functions . The translational modes are deduced from the associated linearized Hamiltonian problem (, [15]) whereas the torsional mode is derived from the later Eq.
Amplitude modulation equations
The Multiple Scale perturbation method (MSM) is employed to attack Eq. (17). As a first hypothesis, let us assume that in-plane and out-of-plane natural frequencies are different, , i.e. the cable has a small but finite sag. Then, a dimensionless perturbation parameter is introduced (), and the unknowns are expanded in series of :Two independent slow time scales, and , are introduced, in addition to the fast scale (, ), so that the
Numerical results
In order to numerically illustrate the proposed theory, the mechanical and aerodynamic cable properties are selected, together with the eigenfunctions to be used in the discrete model. Then, the critical conditions are analyzed with particular attention to the influence of the twist angle and to the accuracy of the perturbation solutions. Finally, the post-critical equilibrium patterns are investigated, pointing out the alterations due to the torsional effects.
Conclusions
In this paper, a nonlinear model of cable-beam has been formulated, accounting for both torsion and bending. After a consistent analysis of the order-of-magnitude of the terms (mainly based on the hypotheses of large slenderness ratio, small initial curvature and quasi-steady stretching and twisting), a set of reduced integro-differential equations of motion has been obtained, which captures the essential dynamics of the cable. As a main result, it has been proved that, while the bending
Acknowledgements
This work has been partially supported by a PRIN grant (http://www.disg.uniroma1.it/fendis) and by the INTAS Project no. 06-1000013-9019 (www.intas.be).
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