Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

Konstantin V. Avramov,Yuri V. Mikhlin

doi:10.1115/1.4023533

Abstract

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.

Highlights

Nonlinear normal modes (NNMs) are a generalization of normal vibrations of linear systems
This paper provides a classification of coherent intrinsic dynamics of one-dimensional homogeneous granular chains with no precompression
Different applications of nonlinear normal modes (NNMs) are considered in this review

Summary

Introduction

Nonlinear normal modes (NNMs) are a generalization of normal vibrations of linear systems. The Kauderer–Rosenberg concept of NNMs, which is based on a construction of trajectories in the system configuration space, was developed to analyze large amplitude vibrations of finite-degreeof-freedom systems without dissipation. Later Shaw and Pierre proposed another concept of NNMs, based on the invariant manifolds theory. The Shaw–Pierre NNMs are used to investigate nonlinear dynamics of dissipative mechanical systems. The Kauderer–Rosenberg NNMs are used to analyze nonlinear dynamics of systems with cyclic symmetry, impact systems, systems with essentially nonlinear absorbers, systems with nonlinear vibrations isolation, arches, beams, and cylindrical shells. The Shaw–Pierre NNMs are used to analyze nonlinear dynamics of floating offshore platform, double-tracked road vehicle, rotors, piece-wise linear systems, beams structures, plates and shells, and thin-walled structures interacting with fluids. A detailed review of theoretical investigations on NNMs is published in Ref. [1]

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