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.

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.

Forced Oscillations of the Discrete Membrane Under Conditions of “Sonic Vacuum”

2020 ◽  
Vol 87 (11) ◽  
Author(s):  
S. S. Kevorkov ◽  
I. P. Koroleva ◽  
V. V. Smirnov ◽  
L. I. Manevitch
Keyword(s):  
Frequency Response ◽  
External Force ◽  
Normal Modes ◽  
Single Mode ◽  
Nonlinear Normal Modes ◽  
Shaking Table ◽  
Forced Oscillations ◽  
Amplitude Frequency Response ◽  
Nonlinear Normal ◽  
External Harmonic Force

Abstract This study presents a new analytical model for nonlinear dynamics of a discrete rectangular membrane that is subjected to external harmonic force. It has recently been shown that the corresponding autonomous system admits a series of nonlinear normal modes. In this paper, we describe stationary and non-stationary dynamics on a single mode manifold. We suggest a simple formula for the amplitude-frequency response in both conservative and non-conservative cases and present an analytical expression (in parametric space) for thresholds for all possible bifurcations. Theoretical results obtained through asymptotic approach are confirmed by the experimental data. Experiments on the shaking table show that amplitude-frequency response to external force in a real system matches our theory. Substantial hysteresis is observed in the regimes with increasing and decreasing frequency of external force. The obtained results may be used in designing nonlinear energy sinks.

scholarly journals Nonlinear Normal Modes of a Piecewise Linear Continuous Structure with a Regular State

2018 ◽  
Author(s):  
Németh Róbert Károly ◽  
Geleji B. Borbála
Keyword(s):  
Free Vibration ◽  
Modal Analysis ◽  
Piecewise Linear ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Vibration Modes ◽  
Continuous Structure ◽  
Periodic Behaviour ◽  
Active Vibration ◽  
Nonlinear Normal

In this paper, the dynamic behaviour of a family of piecewise linear structures, namely the vibration of beams on block-and-tackle suspension system is analysed. The regularity of the vibration modes in one of the linear states induces non-harmonic, yet periodic free vibration modes. The periodicity constraint of the continuous structure is formulated using modal analysis in the regular state. The required number of modes in the finite modal analysis is specified so that the numerical damping caused by the omitted modes does not change the periodic or non-periodic nature of the free vibration of the continuous structure. It is shown, that the application of five excess passive modes allows to draw conclusions about the behaviour of the continuous structure. The periodic behaviour depends on the number and position of the suspension points and the number of the active vibration modes. Analysis of the limits of the periodic behaviour reveals that suspension points close to the middle of the beam, or first few active vibration modes result in periodic vibration of the nonlinear system.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

Localized and Non-Localized Normal Modes in a Strongly Nonlinear Discrete System

1991 ◽  
Author(s):  
Alexander F. Vakakis
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Linear Vibration ◽  
Mode Localization ◽  
Linear Mode ◽  
Strongly Nonlinear ◽  
System A ◽  
Coupling Stiffness ◽  
Nonlinear Normal ◽  
Nonlinear Discrete System

Abstract The free oscillations of a strongly nonlinear, discrete oscillator are examined by computing its “nonsimilar nonlinear normal modes.” These are motions represented by curves in the configuration space of the system, and they are not encountered in classical, linear vibration theory or in existing nonlinear perturbation techniques. For an oscillator with weak coupling stiffness and “mistiming,” both localized and nonlocalized modes are detected, occurring in small neighborhoods of “degenerate” and “global” similar modes of the “tuned” system. When strong coupling is considered, only nonlocalized modes are found to exist. An interesting result of this work is the detection of mode localization in the “tuned” periodic system, a result with no counterpart in existing theories on linear mode localization.

Nonlinear Modal Analysis and Energy Localization in a Bladed Disk Assembly

2008 ◽  
Author(s):  
F. Georgiades ◽  
M. Peeters ◽  
G. Kerschen ◽  
J. C. Golinval ◽  
M. Ruzzene
Keyword(s):  
Modal Analysis ◽  
Nonlinear Oscillations ◽  
Periodic Structures ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Energy Localization ◽  
Bladed Disk ◽  
Bladed Disk Assembly ◽  
Nonlinear Modal Analysis ◽  
Nonlinear Normal

The objective of this study is to carry out modal analysis of nonlinear periodic structures using nonlinear normal modes (NNMs). The NNMs are computed numerically with a method developed in [18] that is using a combination of two techniques: a shooting procedure and a method for the continuation of periodic motion. The proposed methodology is applied to a simplified model of a perfectly cyclic bladed disk assembly with 30 sectors. The analysis shows that the considered model structure features NNMs characterized by strong energy localization in a few sectors. This feature has no linear counterpart, and its occurrence is associated with the frequency-energy dependence of nonlinear oscillations.

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