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 formulated, 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. Numerical tools and geometrical techniques from dynamical systems theory are illustrated with reference 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.

The present work is concerned with deterministic nonlinear phenomena arising in the finite-amplitude dynamics of elastic suspended cables. The underlying theoretical framework has been addressed in Part I of this Special Issue, where the mechanical system and its mathematical modeling have been presented, and different techniques for the analysis of nonlinear dynamics have been illustrated with reference to the suspended cable. Herein, we discuss the main features of system regular and complex response, and the associated bifurcational behavior. Nonlinear phenomena are considered separately for single-degree-of-freedom and multidegree-of-freedom cable models, by distinguishing between theoretical and experimental results and comparing them with each other. Regular and nonregular vibrations are considered either in the absence of internal resonance or under various internal/external, and possibly simultaneous, resonance conditions. The most robust classes of steady periodic motions, the relevant response scenarios in control parameter space, and the main features of multimodal interaction phenomena are summarized. Bifurcation and chaos phenomena are discussed for the single-dof model by analyzing the local and global features of steady nonregular dynamics. For the experimental model, the most meaningful scenarios of transition to chaos are illustrated, together with the properties of the ensuing quasiperiodic and chaotic attractors. Finally, the important issues of determining system dimensionality and identifying properly reduced-order theoretical models of cable are addressed. There are 185 references listed in this review article.

The internal resonances between the longitudinal and transversal oscillations of a forced Timoshenko beam with an axial end spring are studied in depth. In the linear regime, the loci of occurrence of 1 : ir, $$ir \in \mathbb {N}$$i r ∈ N , internal resonances in the parameters space are identified. Then, by means of the multiple time scales method, the 1 : 2 case is investigated in the nonlinear regime, and the frequency response functions and backbone curves are obtained analytically, and investigated thoroughly. They are also compared with finite element numerical simulations, to prove their reliability. Attention is paid to the system response obtained by varying the stiffness of the end spring, and it is shown that the nonlinear behaviour instantaneously jumps from hardening to softening by crossing the exact internal resonance value, in contrast to the singular (i.e. tending to infinity) behaviour of the nonlinear correction coefficient previously observed (without properly taking the internal resonance into account).