Two principal concepts of nonlinear normal vibrations modes (NNMs), namely the Kauderer–Rosenberg and Shaw–Pierre concepts, are analyzed. Properties of the NNMs and methods of their analysis are presented. NNMs stability and bifurcations are discussed. Combined application of the NNMs and the Rauscher method to analyze forced and parametric vibrations is discussed. Generalization of the NNMs to continuous systems dynamics is also described.

This review article addresses the dynamics and control of low-frequency unsteadiness, as observed in some aerodynamic applications. It presents a coherent and rigorous linearized approach, which enables both to describe the dynamics of commonly encountered open-flows and to design open-loop and closed-loop control strategies, in view of suppressing or delaying instabilities. The approach is global in the sense that both cross-stream and streamwise directions are discretized in the evolution operator. New light will therefore be shed on the streamwise properties of open-flows. In the case of oscillator flows, the unsteadiness is due to the existence of unstable global modes, i.e., unstable eigenfunctions of the linearized Navier–Stokes operator. The influence of nonlinearities on the dynamics is studied by deriving nonlinear amplitude equations, which accurately describe the dynamics of the flow in the vicinity of the bifurcation threshold. These equations also enable us to analyze the mean flow induced by the nonlinearities as well as the stability properties of this flow. The open-loop control of unsteadiness is then studied by a sensitivity analysis of the eigenvalues with respect to base-flow modifications. With this approach, we manage to a priori identify regions of the flow where a small control cylinder suppresses unsteadiness. Then, a closed-loop control approach was implemented for the case of an unstable open-cavity flow. We have combined model reduction techniques and optimal control theory to stabilize the unstable eigenvalues. Various reduced-order-models based on global modes, proper orthogonal decomposition modes, and balanced modes were tested and evaluated according to their ability to reproduce the input-output behavior between the actuator and the sensor. Finally, we consider the case of noise-amplifiers, such as boundary-layer flows and jets, which are stable when viewed in a global framework. The importance of the singular value decomposition of the global resolvent will be highlighted in order to understand the frequency selection process in such flows.

The general theory of elastic stability invented by Koiter (1945, “On the Stability of Elastic Equilibrium,” Ph.D. thesis, Delft, Holland) motivated the development of a series of asymptotic approaches to deal with the initial postbuckling behavior of structures. These approaches, which played a pivotal role in the precomputer age, are somewhat overshadowed by the progress of computational environment. Recently, the importance of the asymptotic approaches has been revived through the extension of their theoretical background and the combination with the framework of finite element method and with group-theoretic bifurcation theory in nonlinear mathematics. The approaches serve as an efficient and insightful strategy to tackle probabilistic scatter of critical loads. We review, through the perspective of theoretical engineers, the historical development and recent revival of the asymptotic approaches for buckling of imperfection-sensitive structures and materials.

The significant advances in nonlinear stochastic dynamics and control in Hamiltonian formulation during the past decade are reviewed. The exact stationary solutions and equivalent nonlinear system method of Gaussian-white -noises excited and dissipated Hamiltonian systems, the stochastic averaging method for quasi Hamiltonian systems, the stochastic stability, stochastic bifurcation, first-passage time and nonlinear stochastic optimal control of quasi Hamiltonian systems are summarized. Possible extension and applications of the theory are pointed out. This review article cites 158 references.

In this paper, research on transverse vibrations of axially moving strings and their control is thoroughly reviewed. In the last few decades, there have been extensive studies on analysis and control of transverse vibrations of axially moving strings because of the wide applications of many engineering devices that axially moving strings represent. In the investigations adopting linear models of moving strings, the paper summarizes recent studies on modal analysis, complicatedly constrained strings, coupled vibrations, and parametric vibration, as well as some early results. In the investigations adopting nonlinear models of moving strings, the paper presents the governing equations with large amplitude, and reviews progress on discretized or direct approximate analytical analyses and numerical approaches based on the Galerkin method or the finite difference method. Furthermore, investigations are reported on modeling of damping mechanisms as viscoelastic materials, coupled vibration of power transmission systems, and bifurcation and chaos. The state of the art of active control of moving strings is surveyed on controllability and observability, the Laplace transform domain analysis and the energy analysis, nonlinear vibration control and adaptive vibration control. Finally, future research directions are suggested such as nonlinear vibration of moving strings under complex constraints and couplings, energetics of nonlinear and time-varying strings, bifurcation and chaos in transverse motion of moving strings, control of hybrid systems containing moving strings, robust and adaptive controls of nonlinear moving strings, and experimental investigations. In this review article there are 242 references cited.

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.

This review article deals with the random excitation of nonlinear strings and suspended cables in air and fluid flow. For strings and 1D cables, the system dynamics is governed by different forms of Duffing oscillator. A brief review is devoted to the stochastic excitation of a Duffing oscillator. Under random excitation, this oscillator may or may not possess multiple solutions depending on the excitation bandwidth and level. One may be interested in estimating response statistics, first passage problem, and power spectral density. Particular attention is given to the complex response phenomena associated with increasing the spectral density level of excitation. The numerical results of the problem of nonlinear modal interaction in suspended cables will be discussed in the neighborhood of multiple internal resonance conditions. For a unimodal response, the linear theory fails to predict nonzero mean response and underestimates the mean square response under white noise excitation. Complex response phenomena such as “on-off” intermittency, energy transfer, and stochastic bifurcation are reviewed. The dynamic behavior of suspended cables in still air is different from that in flowing fluid or severe wind current due to the action of vortices, fluid normal forces, added fluid inertia force, and fluid drag force. Aeolian and galloping vibration of suspended cables in air and their dynamics in fluid flow are discussed, together with the influence of dynamic tension. In the absence of external excitation, the action of fluid forces induces vibration to the cable. The dynamics of cables subjected to steady and random fluid flow is reviewed for mooring systems. Depending on the flow speed, the cable may experience divergence or flutter similar to the case of aeroelastic structures. While the deterministic theory of strings and cables has reached an advanced stage, the reader will realize that these systems need further investigations under random excitations. There are 297 references cited in this review article.

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.