Revisited modelling and multimodal nonlinear oscillations of a sagged cable under support motion

Meccanica (2016) 51:2541–2575 DOI 10.1007/s11012-016-0450-y NONLINEAR DYNAMICS, IDENTIFICATION AND MONITORING OF STRUCTURES Revisited modelling and multimodal nonlinear oscillations of a sagged cable under support motion Jerzy Warminski . Daniele Zulli . Giuseppe Rega . Jarosław Latalski Received: 4 March 2016 / Accepted: 4 May 2016 / Published online: 8 June 2016  The Author(s) 2016. This article is published with open access at Springerlink.com Abstract Sagged cable vibrations caused by support motion and possible external loading are investigated via the four-degree-of-freedom model proposed in Benedettini et al. (J Sound Vib 182(5):775–798, 1995). The model has a considerable potential in terms of forcing cases to be possibly addressed, with the physical motion of the supports naturally giving rise to a variety of external and parametric excitation terms. Dynamics of the system is studied close to the multiple internal resonance at cable crossover, which involves two in-plane and two out-of plane vibration modes. Solutions are found by the multiple time scale Dedicated to the memory of Francesco Benedettini, who was the first assistant and a lifelong friend of GR, as well as the unforgettable first mentor of DZ. J. Warminski (&)  J. Latalski Department of Applied Mechanics, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland e-mail: ; D. Zulli DICEAA – University of L’Aquila, Via Giovanni Gronchi 18, 67100 L’Aquila, AQ, Italy G. Rega Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Via Gramsci 53, 00197 Rome, Italy method. In the numerical investigation, attention is focused on the effects of planar support motion (symmetric and/or antisymmetric) at primary resonance, with the addition of planar symmetric external excitation entailing a nice cancellation phenomenon in the system response. Results are discussed also in the background of theoretical and experimental outcomes available in the literature. Comparison with a computer simulation of original equations of motion shows that analytical results are correct for moderately large oscillations, whereas a different scenario of multimodal responses may occur at higher excitation amplitudes. The nonlinear modal coupling is investigated through bifurcation scenarios and other dynamics tools, showing also transitions to complex response regimes. Keywords Suspended cable  Support motion  External/parametric excitations  Nonlinear oscillations  Multimodal response 1 Introduction In the last three decades, nonlinear dynamics of sagged cables has been studied in many papers under various aspects, via analytical, numerical, and geometrical approaches, as well as experimental techniques. The interest to cable nonlinear dynamics is motivated both by the richness of its theoretical 123 2542 behaviour, which has entitled the suspended cable to be considered as a meaningful archetypical model for the analysis of response of monodimensional elastic systems with initial curvature, and by the wide range of potential applications in civil, mechanical, aerospace and marine engineering. A review of nonlinear models and of the main dynamical phenomena which may appear in sagged cables under different resonance conditions is presented in [2, 3], while some more recent achievements are summarized in [4]. Among the features of specific interest there are the crossover points in the spectrum of cable natural frequencies, where conditions of multiple internal resonance involving symmetric and antisymmetric, in-plane and out-of-plane, modes occur and meaningfully affect the nonlinear dynamic response. Within the discretized, Galerkin-based, modelling perspective commonly pursued for the investigation of nonlinear response of continuous systems (for the sagged cable, see [2]), a complete four-degree-offreedom model at first crossover was formulated in Benedettini et al. [1] by considering external loading distributed along the cable and support motion. Later on, in the framework of the direct perturbation approach which also allows one to capture the spatial dependence of cable motion, a four-mode model was formulated and used in [5, 6] by considering external loading. Sole external loading was indeed considered also in the perturbation analyses of the four-d.o.f. model accomplished in [1, 7] at primary and ‘-subharmonic resonance, respectively, for dealing with the case of symmetric planar excitation and showing an already meaningful variety of classes of unimodal and multimodal responses. Excitation through support motion was instead at the basis of a considerable amount of experimental work on horizontally hanging cable dynamics performed around the turn of the millennium. Refined investigations allowed to identify a multitude of classes of cable regular and non-regular motions under different kinds of support motions and associated resonance conditions [8, 9], to characterize indepth the diverse involved mechanisms of transition to complex dynamics [10–12], and to explore features of control strategies [13]. Further experimental studies on resonant vibrations under support motion are reported in [14]. 123 Meccanica (2016) 51:2541–2575 The effect of support motion on cable regular and non-regular vibrations was indeed addressed in several theoretical/numerical (and also experimental) papers since about mid-nineties [15], however mostly with reference to the nearly taut inclined cables of cablestayed bridges subject to horizontal motion of the upper anchorage and/or to (prevailingly vertical) motion of the lower deck support, which originate simultaneous external and parametric excitations. Besides earlier minimal-order models [16–19], higher-order models accounting for three, four, or more modes have been considered as well [20–25], mostly if performing numerical simulations and using continuation techniques. Theoretical treatments under combined external and parametric excitations have been sometimes accomplished in the literature by considering also the self-excitation due, e.g., to air flow in helicopter dynamics [26, 27] or wind flow in cable-stayed bridge dynamics [28]. Experimental and numerical investigation of nonlinear vibrations of actually sagged inclined cables has been performed in [29]. For horizontally suspended cables, the effect of simultaneous external and parametric excitations, first addressed in [30], was considered later, e.g., in [31, 32], mostly for highlighting some global dynamics aspects of system response, even though, sometimes, with weak reference to an actual physical excitation in the background. Two-d.o.f. models accounting for one in-plane and one out-of-plane modes have been considered. Yet, a richer model is likely needed to observe the possible richness of interactions among vibration modes occurring around crossover due to multiple internal resonances and general nonlinear coupling, mostly in the presence of a combination of (...truncated)