Nonlinear vibrations of suspended cables—Part I: Modeling and analysis Giuseppe Rega Dipartimento di Ingegneria Strutturale e Geotecnica, Universita ` di Roma ‘‘La Sapienza,’’ Rome, Italy; giuseppe.rega@uniroma1.it This review article is the first of three parts of a Special Issue dealing with finite-amplitude oscillations of elastic suspended cables. This part is concerned with system modeling and methods of analysis. After shortly reporting on cable historical literature and identifying the topic and scope of the review, the article begins with a presentation of the mechanical system and of the ensuing mathematical models. Continuum equations of cable finite motion are for- mulated, their linearized version is reported, and nonlinear discretized models for the analysis of 2D or 3D vibration problems are discussed. Approximate methods for asymptotic analysis of either single or multi-degree-of-freedom models of small-sag cables are addressed, as well as asymptotic models operating directly on the original partial differential equations. Numeri- cal tools and geometrical techniques from dynamical systems theory are illustrated with refer- ence to the single-degree-of-freedom model of cable, reporting on measures for diagnosis of nonlinear and chaotic response, as well as on techniques for local and global bifurcation analysis. The paper ends with a discussion on the main features and problems encountered in nonlinear experimental analysis of vibrating suspended cables. This review article cites 226 references. DOI: 10.1115/1.1777224 1 INTRODUCTION Suspended cables are lighweight, flexible structural elements used in numerous applications in mechanical, civil, electri- cal, ocean, and space engineering due to their capability of transmitting forces, carrying payloads, and conducting sig- nals across large distances. At the same time, the suspended cable is a basic element of theoretical interest in applied mechanics and an archetypal model of various phenomena in dynamics. Cable dynamics has a long and rich history documented in the classic monograph by Irvine 1, and summarized in the review articles by Triantafyllou 2–4 and Starossek 5. The early work on cable dynamics goes back as early as the eighteenth century, when d’Alembert, Euler, Bernoulli, and Lagrange see Routh 6 made considerable effort in under- standing the vibrations of taut strings and of a sagged cable with concentrated masses, as well as in the development of the theory of partial differential equations. In the nineteenth century, equations of motion of a cable element were derived by Poisson 1820, see Routh 6, and were then used by Rohrs 7 and Routh 6 to obtain, respectively, approximate and closed form solutions for the natural frequencies of small oscillations of an inextensible sagged cable with distributed mass bare cable. After a rather long period in which the topic was set aside, there was a renewal of interest towards cable dynamics around the middle of the twentieth century, just after the failure of the Tacoma Narrows Bridge. The topic was dealt with, among other papers, in the articles of Pugsley 8 and of Saxon and Cahn 9, whose solutions, however, still failed to reproduce the spectrum for the clas- sical taut string in the limit of vanishing cable sag. After- wards, the sagged cable models developed by Simpson 10 and Soler 11 were capable of a correct transition to the taut string, whose nonlinear vibrations were first considered by Carrier 12. Nevertheless, the whole matter was clarified just in the fundamental contribution of Irvine and Caughey 13—together with the basic crossover phenomenon of the in-plane frequencies—by introducing the effect of cable elas- ticity see also 14. Actually, the great diversity of cable applications has led to different elastic cable theories. They refer to either small- sag parabolic or large-sag catenary cables, used as over- head transmission lines, mooring lines, hanging systems, and cableways, as well as in connection with other structural el- ements in various civil engineering applications eg, suspen- sion bridges, cable-supported structures, etc. Analytical so- lutions are available for the natural frequencies and mode shapes of small-sag cables, wherein continuum formulations are adopted and the analysis is greatly simplified by the para- bolic approximation. In contrast, discrete formulations and mostly numerical methods finite differences, finite elements, lumped parameter techniques, multibody dynamics are used to deal with large-sag cables. Former numerically oriented Transmitted by Associate Editor P Pfeiffer Appl Mech Rev vol 57, no 6, November 2004 © 2004 American Society of Mechanical Engineers 443