Nonlinear Phenomena in the Single-Mode Dynamics of a Cable-Supported Beam

Abstract

In this paper we discuss the practical usefulness of nonlinear dynamical analysis for the design of a planar cable-supported beam: we refer to a feasible case, assuming the value of the parameters corresponding to a realistic pedestrian footbridge. We consider a one degree of freedom model, obtained by the classical Galerkin reduction technique: the ensuing ordinary differential equation has both quadratic and cubic terms, due to geometric nonlinearities. Extensive numerical simulations are performed: they point out that this model, in spite of its apparent simplicity, is able to highlight the complex dynamics of the cable-supported beam, describing several common and uncommon nonlinear phenomena. Each of them is interpreted in terms of oscillations of the considered mechanical system; we explain the relevance of all the obtained results in the design of the examined structure under steady loads as wind and pedestrians, but also under transient phenomena as earthquake and gust; the ensuing issues, the most dangerous ranges and also the sensibility to perturbations are discussed in detail. In particular we deal with the importance, for an engineering design, of a careful interpretation of: isola bifurcation, transition to chaos both by period doubling cascade and reverse boundary crisis, multistability with coexistence of chaotic and periodic attractors, fractal basins boundaries, erosion of immediate basins, interrupted sequence of period doubling bifurcations. Also the effects of secondary attractors are analyzed, and it is shown that in general they cannot be neglected even if their range of existence is very small. We underline that all these investigations are performed choosing the excitation frequency far from resonances in order to alert the designer that the system dynamics may be complex independently of the activation mechanism due to resonance

Experiment Study on Dynamic Effects of Tower‐Line Systems Induced by Ice Shedding

“…As the transmission line is a large-span flexible structure, it has properties similar to the cable. So, in the study of the dynamics of transmission lines, we use the relevant dynamics theory of the cable for reference (see [3][4][5][6][7][8]). In theory, Rega and Lacarbonara performed many studies on the nonlinear free and forced vibrations of the transmission lines (see [9,10]).…”

Non-linear interactions in the flexible multi-body dynamics of cable-supported bridge cross-sections

“…In this respect, a number of low-dimensional analytical models have been proposed to parametrically investigate a single or a few specific phenomena. In particular, different monodimensional continuum models have been formulated to describe paradigmatic simple systems, typically composed of a non-linear Irvine cable joined with a linear simplysupported beam or cantilever [13][14][15][16][17]. The latest enrichments of these models include up to four cables sustaining a multi-span beam deck [18].…”

“…On the one hand, as primary research outcomes in the linear field, significant frequency veering and modal hybridization phenomena have been recognized [12,14]. On the other hand, as remarkable findings in the rich non-linear scenario, some aspects of the well-known mechanism of cable parametric excitation have been enlightened [13,19,20], new autoparametric sources of high-amplitude local vibrations have been identified [20,21] and transitions to chaotic dynamics have been described [16,17,22]. According to this well-established literature, high-amplitude cable oscillations can be determined by either linear or non-linear excitation mechanisms, generally regulated by commensurable frequencies of global and local modes.…”

Application of control-based continuation to a nonlinear structure with harmonically coupled modes

“…Isola can arise through a number of mechanisms such as sub-and super-harmonic resonances on simple single-degree-of-freedom (SDOF) nonlinear systems [6,8,9] as well as through modal interactions. As such, isola have been observed across a wide range of nonlinear mechanical systems, for instance, suspended cables [10], pedestrian bridges [11], vibration absorbers [12,13] and a satellite structure [14]. The reader is referred to [15] for a more detailed review on this topic.The aforementioned challenges associated with nonlinear MDOF structures mean that a robust systematic method for experimental testing is highly desirable.…”