Nonlinear oscillations of cables under harmonic loading using analytical and finite element models Vincenzo Gattulli a, * , Luca Martinelli b , Federico Perotti b , Fabrizio Vestroni c a Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno, Universit a di L’Aquila, P.le E. Pontieri 2, 67040 Monteluco di Roio (L’Aquila), Italy b Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, Italy c Dipartimento di Ingegneria Strutturale e Gotecnica, Universit a di Roma ‘‘La Sapienza’’, Via Eudossiana 18, 00184 Roma, Italy Received 14 April 2003; received in revised form 28 August 2003; accepted 2 September 2003 Abstract In cables, near resonance time-varying loading causes large amplitude oscillations mainly involving the resonant mode. A relevant contribution of higher modes may arise as a result of nonlinear coupling phenomena. In this work, analytical and finite element models are used to study the modal interactions in both planar and spatial responses to harmonic in-plane and out-of-plane loads. The aim of the investigation is to compare the two approaches by examining the effectiveness of the analytical model in describing the response with few degrees of freedom and the ability of the nonlinear finite element procedure adopted to capture the complex features of cable dynamics, albeit limited to stable oscillation branches. The analyses first explore a moderately taut cable and are then extended to the behavior of a slacker cable, in order to validate the simplifying kinematic assumptions introduced in the analytical models by comparing the obtained results with those furnished by the richer finite element models. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Cables; Nonlinear oscillations; Reduced order models; Finite element models; Bifurcations 1. Introduction The dynamics of suspended cables have been explored for a variety of phenomena occurring in free, harmonically forced and random oscillations related to the inherent nonlinear behavior of cables. 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 and the regions of the parameters in which coexistent solutions occur, and the interactions and energy spreading due to the occurrence of internal resonance conditions [1–6]. Although these interactions are * Corresponding author. Tel.: +39-0862-434511; fax: +39-0862-434548. E-mail addresses: gattulli@ing.univaq.it (V. Gattulli), luca.martinelli@polimi.it (L. Martinelli), federico.perotti@polimi.it (F. Perotti), vestroni@uniroma1.it (F. Vestroni). 0045-7825/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2003.09.008 Comput. Methods Appl. Mech. Engrg. 193 (2004) 69–85 www.elsevier.com/locate/cma