Intrinsic Localized Modes and Nonlinear Normal Modes in Micro-Resonator Arrays

2005 ◽  
Author(s):  
A. J. Dick ◽  
B. Balachandran ◽  
C. D. Mote
Keyword(s):  
Normal Modes ◽  
Nonlinear Phenomena ◽  
Strong Nonlinearity ◽  
Nonlinear Phenomenon ◽  
Solid State Physics ◽  
Localized Modes ◽  
Intrinsic Localized Modes ◽  
Micro Scale ◽  
Resonator Arrays ◽  
Nonlinear Normal

In the solid-state physics literature, nonlinear phenomena such as localizations have been studied for a number of decades in lattice structures. These localizations, which occur in periodic and strongly nonlinear discrete systems, have been found to result from a combination of the discreteness and the strong nonlinearity rather than the defects or impurities in the system. Intrinsic Localized Modes (ILMs), which are defined as spatial localization due to strong nonlinearity within arrays of oscillators, have been studied recently in the context of coupled micro-scale cantilevers. Within this paper, the hypothesis that an intrinsic localized mode may be realized as a nonlinear normal mode is explored in order to gain a better understanding of this nonlinear phenomenon. It is believed that an understanding of this phenomenon would be valuable for the design of piezoelectric micro-scale resonator arrays that are being developed for signal processing, communication, and sensor applications.

Nonlinear Normal Modes and Localization in Elastic Vibro-Impact Systems With Multiple Constraints

2007 ◽  
Author(s):  
David Wagg
Keyword(s):  
Mathematical Formulation ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Lumped Mass ◽  
Localized Modes ◽  
Impact Oscillator ◽  
Mass System ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

In this paper we consider the dynamics of compliant mechanical systems subject to combined vibration and impact forcing. Two specific systems are considered; a two degree of freedom impact oscillator and a clamped-clamped beam. Both systems are subject to multiple motion limiting constraints. A mathematical formulation for modelling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom lumped mass system are considered. We then consider sticking motions which occur when a single mass in the system becomes stuck to an impact stop, which is a form of periodic localization. Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. A numerical example of a sticking orbit for this system is shown and we discuss identifying a nonlinear normal modal basis for the system. This is achieved by extending the normal modal basis to include localized modes. Finally preliminary experimental results from a clamped-clamped vibroimpacting beam are considered and a simplified model discussed which uses an extended modal basis including localized modes.

Amplitude Nonlinear Normal Modes of Piecewise Linear Systems

2001 ◽  
Author(s):  
Dongying Jiang ◽  
Vincent Soumier ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Piecewise Linear ◽  
Autonomous Systems ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Flip Bifurcation ◽  
Strong Nonlinearity ◽  
Nonlinear Modes ◽  
Nonlinear Normal

Abstract A numerical method for constructing nonlinear normal modes for piecewise linear autonomous systems is presented. Based on the concept of invariant manifolds, a Galerkin based approach is applied here to obtain nonlinear normal modes numerically. The accuracy of the constructed nonlinear modes is checked by the comparison of the motion on the invariant manifold to the exact solution, in both time and frequency domains. It is found that the Galerkin based construction approach can represent the invariant manifold accurately over strong nonlinearity regions. Several interesting dynamic characteristics of the nonlinear modal motion are found and compared to those of linear modes. The stability of the nonlinear normal modes of a two-degree of freedom system is investigated using characteristic multipliers and Poincaré maps, and a flip bifurcation is found for both nonlinear modes.

Internal Resonance and Localization in Micro-Resonator Arrays

2007 ◽  
Author(s):  
Andrew J. Dick ◽  
Balakumar Balachandran ◽  
C. Daniel Mote
Keyword(s):  
Internal Resonance ◽  
Filter Design ◽  
Nonlinear Behavior ◽  
Vibration Modes ◽  
Intrinsic Localized Modes ◽  
Micro Scale ◽  
Localization Phenomenon ◽  
Resonator Arrays ◽  
Forced Nonlinear Vibration ◽  
The Individual

Intrinsic Localized Modes (ILMs) are localization caused by intrinsic nonlinearities within an array of perfectly periodic oscillators. This localization phenomenon is studied in the context of arrays of coupled micro-scale cantilevers and of coupled piezoelectric micro-scale resonators. Recent studies have identified the individual devices to be capable of producing nonlinear behavior and efforts are currently underway to develop resonator arrays to improve performance capabilities. It is important to determine whether an ILM would limit the performance of such an array and/or if one could take advantage of this type of localization in filter design. Previous studies have shown that it is possible to realize ILMs as forced nonlinear vibration modes. The effects of internal resonance conditions on the behavior of these localization events are explored here using analytical and numerical methods.

scholarly journals Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

2013 ◽  
Vol 65 (2) ◽  
Author(s):  
Konstantin V. Avramov ◽  
Yuri V. Mikhlin
Keyword(s):  
Asymptotic Methods ◽  
Mechanical Systems ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Nonlinear Phenomena ◽  
Finite Degree ◽  
Nonlinear Mechanical ◽  
Nonlinear Normal ◽  
Nonlinear Mechanical Systems

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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