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Review Articles

Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

[+] Author and Article Information
Konstantin V. Avramov

A. N. Podgorny Institute for Mechanical
Engineering Problems,
National Academy of Sciences of Ukraine,
Kharkov, Ukraine 61046;
National Technical University “KhPI,”
Kharkov, Ukraine 61002
e-mail: kvavr@kharkov.ua

Yuri V. Mikhlin

National Technical University “KhPI,”
Kharkov, Ukraine 61002
e-mail: muv@kpi.kharkov.ua

Manuscript received February 9, 2012; final manuscript received January 28, 2013; published online April 11, 2013. Assoc. Editor: Chin An Tan.

Appl. Mech. Rev 65(2), 020801 (Apr 11, 2013) (20 pages) Paper No: AMR-12-1010; doi: 10.1115/1.4023533 History: Received February 09, 2012; Revised January 28, 2013

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.

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Gendelman, O. V., Manevitch, L. I., Vakakis, A. F., and Bergman, L. A., 2003, “A Degenerate Bifurcation Structure in the Dynamics of Coupled Oscillators With Essential Stiffness Nonlinearities,” Nonlinear Dyn., 33, pp. 1–10. [CrossRef]
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Gendelman, O. V., 2004, “Bifurcations of Nonlinear Normal Modes of Linear Oscillator With Strongly Nonlinear Damped Attachment,” Nonlinear Dyn., 37, pp. 115–128. [CrossRef]
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Starosvetsky, Y., and Gendelman, O. V., 2009, “Vibration Absorption in Systems Comprising Nonlinear Energy Sink: Nonlinear Damping,” J. Sound Vib., 324, pp. 916–939. [CrossRef]
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Figures

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Fig. 1

Mechanical system, consisting of linear subsystem and snap-through truss. Adapted from Ref. [17].

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Fig. 2

Absorption motion. Adapted from Ref. [17].

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Fig. 3

Mechanical system with nonlinear vibrations isolation. Adapted from Ref. [20].

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Fig. 4

The beam with nonlinear passive vibration absorber. Adapted from Ref. [22].

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Fig. 7

Outline of the one-disk rotor

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Fig. 5

NNMs of the self-sustained vibrations in plane (P0+Pt cos2Ωt)

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Fig. 6

NNM of a piecewise linear system with a shock absorber. Projections of invariant manifolds on (u, v, x2) and (u, v, x2) are shown on figures (a) and (b), respectively.

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Fig. 8

Trajectories of the resonance motions in configuration space. Points and circles denote results, which are obtained by harmonic balance method and NNM approaches, respectively. The results of numerical simulations are shown by solid lines.

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Fig. 9

Outline of rotating beams. Adapted from Ref. [52].

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Fig. 10

Pretwisted beam with variable cross section

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Fig. 11

Parametrically excited beam

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Fig. 12

Shallow shell with complex base

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Fig. 13

Base of a shallow shell with variable thickness

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Fig. 14

Coupled beams on a nonlinear elastic foundation. Adopted from Ref. [74].

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Fig. 15

System of two coupled beams. Adopted from Ref. [85].

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