Time–frequency characterization of nonlinear normal modes and challenges in nonlinearity identification of dynamical systems

2011 ◽  
Vol 25 (7) ◽  
pp. 2358-2374 ◽  
Author(s):  
P. Frank Pai
Keyword(s):  
Dynamical Systems ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Time Frequency ◽  
Nonlinear Normal

Modal Testing of Nonlinear Vibrating Structures Based on a Nonlinear Extension of Force Appropriation

2009 ◽  
Author(s):  
M. Peeters ◽  
G. Kerschen ◽  
J. C. Golinval
Keyword(s):  
Normal Modes ◽  
Ground Vibration ◽  
Nonlinear Normal Modes ◽  
Modal Testing ◽  
Mathematical Tool ◽  
Nonlinear Dynamical ◽  
Time Frequency ◽  
Analysis Methodology ◽  
Conceptual Relation ◽  
Nonlinear Normal

Modal testing and analysis is well-established for linear systems. The objective of this paper is to progress toward a practical experimental modal analysis methodology of nonlinear mechanical structures. In this context, nonlinear normal modes (NNMs) offer a solid theoretical and mathematical tool for interpreting a wide class of nonlinear dynamical phenomena, yet they have a clear and simple conceptual relation to the classical linear normal modes (LNMs). A nonlinear extension of force appropriation techniques is investigated in this study in order to isolate one single NNM during the experiments, similarly to what is carried out for ground vibration testing. With the help of time-frequency analysis, the modal curves and the corresponding backbones are then extracted from the time series. The proposed methodology is demonstrated using a numerical benchmark, which consists of a planar cantilever beam with a cubic spring at its free end.

Global Parametrization of the Invariant Manifold Defining Nonlinear Normal Modes Using the Koopman Operator

2015 ◽  
Author(s):  
Giuseppe I. Cirillo ◽  
Alexandre Mauroy ◽  
Ludovic Renson ◽  
Gaëtan Kerschen ◽  
Rodolphe Sepulchre
Keyword(s):  
Invariant Manifold ◽  
Normal Modes ◽  
Geometric Approach ◽  
Nonlinear Normal Modes ◽  
Koopman Operator ◽  
The Past ◽  
Modes Of Vibration ◽  
Nonlinear Normal ◽  
Damped Systems

Nonlinear normal modes of vibration have been the focus of many studies during the past years and different characterizations of them have been proposed. The present work focuses on damped systems, and considers nonlinear normal mode motions as trajectories lying on an invariant manifold, following the geometric approach of Shaw and Pierre. We provide a novel characterization of the invariant manifold, that rests on the spectral theory of the Koopman operator. A main advantage of the proposed approach is a global parametrization of the manifold, which avoids folding issues arising with the use of displacement-velocity coordinates.

scholarly journals A spectral characterization of nonlinear normal modes

2016 ◽  
Vol 377 ◽  
pp. 284-301 ◽  
Author(s):  
G.I. Cirillo ◽  
A. Mauroy ◽  
L. Renson ◽  
G. Kerschen ◽  
R. Sepulchre
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Spectral Characterization ◽  
Nonlinear Normal

Multi-Point Multi-Harmonic Collocation With Continuation to Compute Branches of Nonlinear Modes of Structural Systems

2015 ◽  
Author(s):  
Sean J. Kelly ◽  
Matthew S. Allen ◽  
Hamid Ardeh
Keyword(s):  
Normal Modes ◽  
Nonlinear Behavior ◽  
Nonlinear Normal Modes ◽  
Structural Systems ◽  
Arc Length ◽  
Frequency Method ◽  
Mass System ◽  
Nonlinear Modes ◽  
Time Frequency ◽  
Nonlinear Normal

Fast numerical approaches have recently been proposed to compute the undamped nonlinear normal modes (NNMs) of discrete and structural systems, and these have enabled remarkable insights into the nonlinear behavior of more complicated systems than are tractable analytically. Ardeh et al. recently proposed the multi-point, multi-harmonic collocation (MMC) method, which finds a truncated harmonic description of a structure’s NNMs. The MMC algorithm resembles harmonic balance but doesn’t require any analytical pre-processing nor the computational expense of the alternating time-frequency method. In a previous work this method showed significantly faster performance than the shooting method and it also allows the possibility of truncating the description of the NNM to avoid having to compute every internal resonance. This work presents a pseudo-arc length continuation version of the MMC algorithm which can compute a branch of NNMs and compares its performance with the shooting/pseudo-arc length continuation algorithm presented by Peeters and Kerschen in 2009. The algorithms are compared for two systems, a two degree-of-freedom (DOF) spring-mass system and a 10-DOF model of a simply-supported beam.

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.

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