Keywords: cables, nonlinear oscillations, reduced-order models, FEM, bifurcations.

The dynamics of suspended cables have been explored for a variety of phenomena related to the inherent nonlinear behaviour of cables [1,2]. Different theoretical and experimental methods have been used to describe, for example, the frequency-amplitude dependence of the system, the jump phenomenon, the coupling between in-plane and out-of-plane motions. Although these interactions are more evident for cables at the so-called crossover point, due to multiple internal resonances, they have nevertheless also been found to be relevant in the more common non-resonant cables. This behaviour can be ascribed to the richness of cable nonlinearities; it is therefore easy to encounter resonance conditions even when the primary 1:2 resonance between the first in-plane and out-of-plane modes (well explored in the literature) is avoided.. Most of papers use a low dimension model, obtained by expanding the displacement functions in terms of the eigenfunctions of the modes involved by the resonant conditions and, more in general, by nonlinear coupling. In the analytical investigations the dimension of ODEs describing the motion has been usually limited so as to give a synthetic description of the problem. Although recent continuation methods facilitate the task of describing the solution of ODEs of larger dimensions, efforts have been made to produce more refined reduced order models based on nonlinear normal modes, suitable orthogonal bases and direct approaches.

The other fundamental numerical procedure is based on the finite element method combined with direct integration of the equations of motion, which is able to handle small and large displacements and can be used for complex structures as well. A satisfactory accuracy is usually achieved by increasing either the number of degrees-of-freedom or the richness of the kinematics description at the element level. Thus, even in the case of a simple cable, a high number of equations is obtained and to obtain asymptotic solutions is not straightforward.

The present study aims to investigate the effectiveness and limitations of analytical Galerkin models and finite element models; a comparison is made between the results obtained by each model in describing the most important nonlinear phenomena of suspended cables under in- and out-of-plane harmonic loading. The analytical model is based on the discretization of the two integro- differential equations of motion in the two transverse displacement components while the longitudinal component has been eliminated by static condensation. The finite element model is instead based on the discretization via a three-node element and contains a full dynamic description of the problem [3].

Planar oscillations are first analysed in order to investigate the contribution of higher modes. The 3D motions are then considered; they are either the result of the bifurcation of planar oscillations under in-plane loading or directly excited by harmonic out-of-plane loading.

For the analytical models, frequency response curves are evaluated through direct use of the pseudoarclength continuation algorithm on the modal equations of motion. These results are compared with those obtained through the finite element model, where the direct time integration of the motion equations in the global nodal coordinates has been performed. To facilitate the comparison with the modal amplitudes of the analytical model, the displacement solutions of the finite element model are expanded in the space of the linearized eigenvectors.

The main part of the analysis is conducted on a taut cable and then extended to a slacker cable in order to verify the kinematics assumptions in the case of larger span-to-length ratios. A comparison of the results obtained makes it possible to draw conclusions regarding the two approaches, in particular, the effectiveness of the reduced analytical model and the ability of the adopted finite element numerical procedure to capture quantitatively even complex bifurcation phenomena, even though they can only describe stable branches. The two approaches offer a more refined picture of the behaviour of a cable and furnishes novel findings.

1

Luongo, A., Rega, G., Vestroni, F., "Planar Non-Linear Free Vibrations of an Elastic Cable", Int. J. of Non-Linear Mechanics, 19, 39-52, 1984. doi:10.1016/0020-7462(84)90017-9

2

Perkins, N.C., "Modal interactions in the non-linear response of elastic cables under parametric/external excitation", International Journal of Non-Linear Mechanics, 27, 233-250, 1992. doi:10.1016/0020-7462(92)90083-J

3

Gattulli, V., Martinelli, L., Perotti, F., Vestroni, F.,"Nonlinear oscillations of cables under harmonic loading by analytical and finite element models", Computer Methods in Applied Mechanics and Engineering, 193(1), 69-85, 2004. doi:10.1016/j.cma.2003.09.008

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