Review Articles

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

[+] Author and Article Information
Yuri V. Mikhlin

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

Konstantin V. Avramov

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

Appl. Mech. Rev 63(6), 060802 (Jun 24, 2011) (21 pages) doi:10.1115/1.4003825 History: Published June 24, 2011; Online June 24, 2011

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.

Copyright © 2010 by American Society of Mechanical Engineers
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Figure 1

Curvilinear modal lines of NNM in a configuration space: (a) the trajectory pass through the origin; (b) the trajectory does not pass through the origin

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Figure 2

Nonlinear modes of the system 36 for different values of parameter γ

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Figure 3

The behavior of small perturbations close to modal line η1=0 in configuration space of two-DOF system

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Figure 4

Dependence of the NNMs on parameter η

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Figure 5

The forking trajectories near the rectilinear NNM

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Figure 6

Plane of stability of the Lame equation solutions: (a) finite zoning case; (b) infinite zoning case

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Figure 7

Diagram of the NNM



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