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

Jerzy Warminski,Daniele Zulli,Jarosław Latalski,Giuseppe Rega

doi:10.1007/s11012-016-0450-y

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 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.

Highlights

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
Galerkin-based, modelling perspective commonly pursued for the investigation of nonlinear response of continuous systems, 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
The model of the considered d-sagged cable is presented in Fig. 1, with the initial static equilibrium in-plane configuration CI and the varied dynamic 3D configuration CV attained through the displacement field components u, v, w

Summary

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. 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 forcing terms possibly corresponding to involved conditions of multifrequency excitation It is worth resuming the four-d.o.f general model in Benedettini et al [1], with which no specific theoretical/numerical work has been done yet to evaluate the effect of support motion on cable nonlinear dynamics, in view of a possible confirmation and cross-validation with the independently obtained experimental outcomes. A section of conclusions and future developments ends the paper

Reduced-order model of a suspended cable

Multiple scale solution

Conclusions and further developments