This paper is an extension of the previous review, done by the same authors (Mikhlin, Y., and Avramov, K. V., 2010, “Nonlinear Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments,” ASME Appl. Mech. Rev., 63(6), p. 060802), and it is devoted to applications of nonlinear normal modes (NNMs) theory. NNMs are typical regimes of motions in wide classes of nonlinear mechanical systems. The significance of NNMs for mechanical engineering is determined by several important properties of these motions. Forced resonances motions of nonlinear systems occur close to NNMs. Nonlinear phenomena, such as nonlinear localization and transfer of energy, can be analyzed using NNMs. The NNMs analysis is an important step to study more complicated behavior of nonlinear mechanical systems.This review focuses on applications of Kauderer–Rosenberg and Shaw–Pierre concepts of nonlinear normal modes. The Kauderer–Rosenberg NNMs are applied for analysis of large amplitude dynamics of finite-degree-of-freedom nonlinear mechanical systems. Systems with cyclic symmetry, impact systems, mechanical systems with essentially nonlinear absorbers, and systems with nonlinear vibration isolation are studied using this concept. Applications of the Kauderer–Rosenberg NNMs for discretized structures are also discussed. The Shaw–Pierre NNMs are applied to analyze dynamics of finite-degree-of-freedom mechanical systems, such as floating offshore platforms, rotors, piece-wise linear systems. Studies of the Shaw–Pierre NNMs of beams, plates, and shallow shells are reviewed, too. Applications of Shaw–Pierre and King–Vakakis continuous nonlinear modes for beam structures are considered. Target energy transfer and localization of structures motions in light of NNMs theory are treated. Application of different asymptotic methods for NNMs analysis and NNMs based model reduction are reviewed.

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.

The possibility of absorption of linear elastic system forced oscillations by means of an essentially nonlinear absorber (snap-through truss), which is attached to the linear subsystem, is analyzed. The simplest mass–spring linear subsystem is chosen to study the problem of forced oscillations absorption. The combination of the nonlinear normal vibrations modes method, the Rauscher approach and the asymptotic analysis is used to study forced oscillations of the two-dof system. The absorption vibrations mode in the form of nonlinear normal vibrations mode with small oscillations amplitudes of the linear subsystem and large amplitudes of the snap-through truss is studied. It is shown that this mode is stable over the wide range of the excitation frequency.

The equations of nonlinear flexural-flexural-torsional vibrations of rotating beams with asymmetric cross-sections are derived. As the cross-section is asymmetric the centre of gravity and shear centre of a beam cross-section are not assumed to coincide. The nonlinear vibrations are expanded into flexural-torsional vibrational mode series. Using the Galerkin technique, the large-scale system of ordinary differential equations is derived. The non-linear modes are used to study teh free vibrations of a rotating beam, and the backbone curves of the free vibrations and the surfaces of the non-linear modes are also presented.