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.

scholarly journals Identifying phase-varying periodic behaviour in conservative nonlinear systems

2020 ◽  
Vol 476 (2237) ◽  
pp. 20200028
Author(s):  
Dongxiao Hong ◽  
Evangelia Nicolaidou ◽  
Thomas L. Hill ◽  
Simon A. Neild
Keyword(s):  
Phase Difference ◽  
Normal Modes ◽  
Single Mode ◽  
Nonlinear Normal Modes ◽  
Nonlinear Mechanical ◽  
Periodic Behaviour ◽  
Single Mass ◽  
Backbone Curves ◽  
Nonlinear Normal ◽  
Nonlinear Mechanical Systems

Nonlinear normal modes (NNMs) are a widely used tool for studying nonlinear mechanical systems. The most commonly observed NNMs are synchronous (i.e. single-mode, in-phase and anti-phase NNMs). Additionally, asynchronous NNMs in the form of out-of-unison motion, where the underlying linear modes have a phase difference of 90°, have also been observed. This paper extends these concepts to consider general asynchronous NNMs , where the modes exhibit a phase difference that is not necessarily equal to 90°. A single-mass, 2 d.f. model is firstly used to demonstrate that the out-of-unison NNMs evolve to general asynchronous NNMs with the breaking of the geometrically orthogonal structure of the system. Analytical analysis further reveals that, along with the breaking of the orthogonality, the out-of-unison NNM branches evolve into branches which exhibit amplitude-dependent phase relationships. These NNM branches are introduced here and termed phase-varying backbone curves . To explore this further, a model of a cable, with a support near one end, is used to demonstrate the existence of phase-varying backbone curves (and corresponding general asynchronous NNMs) in a common engineering structure.

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.

scholarly journals Nonlinear Normal Modes in a Three-Degree-of-Freedom Vibration Isolating System with Internal Resonance.

2002 ◽  
Vol 68 (671) ◽  
pp. 1950-1958
Author(s):  
Tetsuro TOKOYODA ◽  
Noriaki YAMASHITA ◽  
Hiroyuki OISHI ◽  
Takeshi YAMAMOTO ◽  
Masatsugu YOSHIZAWA
Keyword(s):  
Internal Resonance ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Nonlinear Normal ◽  
Three Degree Of Freedom

Numerical Computation of Nonlinear Normal Modes in Multi-Mode Models of an Inertially Coupled Elastic Structure

2009 ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Small Amplitude ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Elastic Structure ◽  
Internal Resonances ◽  
Reduced Order ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom ◽  
Quadratic And Cubic Nonlinearities

There are many techniques available for the construction of nonlinear normal modes. Most studies for systems with more than one degree of freedom utilize asymptotic techniques or the method of multiple time scales, which are valid only for small amplitude motions. Previous works of the authors have investigated nonlinear normal modes in elastic structures with essential inertial nonlinearities, and considered two degree-of-freedom reduced-order models that exhibit 1:2 resonance. For small amplitude oscillations with low energy, this reduced analysis is acceptable, while for higher energy vibrations and vibrations that are away from internal resonances, this may not provide an accurate representation of NNMs. For high energy vibration and vibrations away from internal resonances, two natural issues to be addressed are the dimension of the reduced-order model used for constructing NNMs, and the order of nonlinearities retained in the truncated models. To address these issues, a comparison of NNMs computed for three different reduced degree of freedom models for the elastic structure is reported here. The reduced models considered are: (i) A two degree-of-freedom reduced model with only quadratic nonlinearities; (ii) A two degree-of-freedom reduced model with both quadratic and cubic nonlinearities; (iii) A five degrees-of-freedom model with both quadratic and cubic nonlinearities. A numerical method based on shooting technique is used for constructing the NNMs and results for system near 1:2 internal resonances between the two lowest modes and away from any internal resonance are compared.

The Construction of Nonlinear Normal Modes for Systems With Internal Resonance: Application to Rotating Beams

2002 ◽  
Author(s):  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Internal Resonance ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Reduced Order Model ◽  
Degree Of Freedom ◽  
Beam Model ◽  
Order Model ◽  
Reduced Order ◽  
Nonlinear Normal

A numerical method for constructing nonlinear normal modes for systems with internal resonances is presented based on the invariant manifold approach. In order to parameterize the nonlinear normal modes, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are constrained to these ‘seed’ variables, resulting in a system of nonlinear partial differential equations governing the constraint relationships, which must be solved numerically. The solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two nonlinear normal modes is constructed, resulting in a reduced-order model that accurately captures the system dynamics. The methodology is then applied to a more large system, namely an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the nonlinear two-mode reduced-order model is verified by time-domain simulations.

scholarly journals Application of physical function model to state estimations of nonlinear mechanical systems

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Heisei Yonezawa ◽  
Itsuro Kajiwara ◽  
Shota Sato ◽  
Chiaki Nishidome ◽  
Takashi Hatano ◽  
...  
Keyword(s):  
Physical Function ◽  
Mechanical Systems ◽  
Function Model ◽  
Nonlinear Mechanical ◽  
State Estimations ◽  
Nonlinear Mechanical Systems
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