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Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

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.

https://hal.archives-ouvertes.fr/hal-01338616/document

Nonlinears Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

The present investigation deals with the dynamics of a two-degrees-of-freedom system which consists of a main linear oscillator and a strongly nonlinear absorber with small mass. The nonlinear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model. The periodic solutions of this system are studied and their calculation is performed through an averaging procedure. The study of nonlinear modes and their stability shows, under specific conditions, the existence of localization which is responsible for a passive irreversible energy transfer from the linear oscillator to the nonlinear one. The dissipative effect of the nonlinearity appears to play an important role in the energy transfer phenomenon and some design criteria can be drawn regarding this parameter among others to optimize this energy transfer. The free transient response is investigated and it is shown that the energy transfer appears when the energy input is sufficient in accordance with the predictions from the nonlinear modes. Finally, the steady-state forced response of the system is investigated. When the input of energy is sufficient, the resonant response (close to nonlinear modes) experiences localization of the vibrations in the nonlinear absorber and jump phenomena.

https://hal.archives-ouvertes.fr/docs/00/21/82/44/PDF/NLM_2006_41_969-978.pdf

Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber

Nonlinear modes of snap-through motions of a shallow arch

Nonlinear modes of parametric vibrations and their applications to beams dynamics

http://users.kpi.kharkov.ua/ndynamics/index_files/Papers/Mikhlin&ReshetnikovaJournalSoundVibration2005.pdf

Dynamical interaction of an elastic system and essentially nonlinear absorber

The nonlinear two-degree-of-freedom system under consideration consists of the linear oscillator with a relatively big mass, which is an approximation of some continuous elastic system, and of the vibro-impact oscillator with a relatively small mass, which is an absorber of the linear system vibrations. Analysis of nonlinear normal vibration modes shows that a stable localized vibration mode, which provides the vibration regime appropriate for the elastic vibration absorption, exists in a large region of the system parameters. In this regime, amplitudes of vibrations of the linear system are small, simultaneously vibrations of the absorber are significant.

http://emis.maths.adelaide.edu.au/journals/HOA/MPE/Volume2006/37980.pdf

S. N. Reshetnikova

Papers

Dynamical interaction of an elastic system and essentially nonlinear absorber

Dynamical interaction of an elastic system and a vibro-impact absorber

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