Instantaneous Center Manifolds and Nonlinear Modes of Vibration

2012 ◽  
Author(s):  
Hamid A. Ardeh ◽  
Matthew S. Allen
Keyword(s):  
Nonlinear System ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Oscillatory System ◽  
Nonlinear Normal Modes ◽  
Center Manifolds ◽  
Invariance Properties ◽  
Modes Of Vibration ◽  
Instantaneous Center ◽  
Nonlinear Normal

Nonlinear Normal Modes (NNM) have been defined in various ways; first by Rosenberg as a subset of periodic solutions of a nonlinear system and then by Shaw and Pierre as invariant manifolds tangent to the vector field of a nonlinear system at its equilibrium point. This work presents an alternative approach, namely Instantaneous Center Manifold (ICM), that extends the concept of modes of vibration to nonlinear systems, using both periodicity and invariance properties. Instantaneous Center Manifolds are invariant manifolds that contain all of the periodic invariant solutions of the nonlinear oscillatory system. The ICM approach is explained through three simple analytical examples, and is shown to be capable of finding solutions that have been remaining latent using the aforementioned approaches. New branches of nonlinear normal modes, separate from the main branches that are a continuation of linear modes, are illustrated. It is shown that these new branches connect the main branches of Rosenberg’s NNMs, and make it possible to travel from one main branch to another. Some natural extensions and applications of the ICM approach are briefly discussed in the conclusion.

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.

419 Nonlinear Normal Modes in 2-DOF Nonlinear System under Forced Excitation

2003 ◽  
Vol 2003 (0) ◽  
pp. _419-1_-_419-6_
Author(s):  
Ryo KANDA ◽  
Hiroshi YABUNO ◽  
Md. Zahid Hossain ◽  
Tsuyoshi INOUE ◽  
Yukio ISHIDA
Keyword(s):  
Nonlinear System ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Nonlinear Normal ◽  
Forced Excitation

scholarly journals Isolated Response Curves in a Base-Excited, Two-Degree-of-Freedom, Nonlinear System

2015 ◽  
Author(s):  
J. P. Noël ◽  
T. Detroux ◽  
L. Masset ◽  
G. Kerschen ◽  
L. N. Virgin
Keyword(s):  
Nonlinear System ◽  
Harmonic Balance ◽  
Global Analysis ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Basins Of Attraction ◽  
Response Curves ◽  
Restoring Force ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

In the present paper, isolated response curves in a nonlinear system consisting of two masses sliding on a horizontal guide are examined. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imposed by prescribing the displacement of their supports. Numerical simulations are carried out to study the conditions of existence of isolated solutions, their bifurcations, their merging with the main response branch and their basins of attraction. This is achieved using tools including nonlinear normal modes, energy balance, harmonic balance-based continuation and bifurcation tracking, and global analysis.

scholarly journals Identifying the significance of nonlinear normal modes

2017 ◽  
Vol 473 (2199) ◽  
pp. 20160789 ◽  
Author(s):  
T. L. Hill ◽  
A. Cammarano ◽  
S. A. Neild ◽  
D. A. W. Barton
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Analytical Investigation ◽  
Nonlinear Normal ◽  
Definition Of ◽  
Forced Responses ◽  
Insight Into

Nonlinear normal modes (NNMs) are widely used as a tool for understanding the forced responses of nonlinear systems. However, the contemporary definition of an NNM also encompasses a large number of dynamic behaviours which are not observed when a system is forced and damped. As such, only a few NNMs are required to understand the forced dynamics. This paper firstly demonstrates the complexity that may arise from the NNMs of a simple nonlinear system—highlighting the need for a method for identifying the significance of NNMs. An analytical investigation is used, alongside energy arguments, to develop an understanding of the mechanisms that relate the NNMs to the forced responses. This provides insight into which NNMs are pertinent to understanding the forced dynamics, and which may be disregarded. The NNMs are compared with simulated forced responses to verify these findings.

scholarly journals Numerical Computation of Nonlinear Normal Modes of Nonconservative Systems

2013 ◽  
Author(s):  
L. Renson ◽  
G. Kerschen
Keyword(s):  
Periodic Solutions ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Numerical Techniques ◽  
Nonlinear Beam ◽  
Conservative Systems ◽  
Linear Damping ◽  
Nonlinear Normal ◽  
Element Technique

Since linear modal analysis fails in the presence of non-linear dynamical phenomena, the concept of nonlinear normal modes (NNMs) was introduced with the aim of providing a rigorous generalization of linear normal modes to nonlinear systems. Initially defined as periodic solutions, numerical techniques such as the continuation of periodic solutions were used to compute NNMs. Because these methods are limited to conservative systems, the present study targets the computation of NNMs for non-conservative systems. Their definition as invariant manifolds in phase space is considered. Specifically, the partial differential equations governing the manifold geometry are considered as transport equations and an adequate finite element technique is proposed to solve them. The method is first demonstrated on a conservative nonlinear beam and the results are compared to standard continuation techniques. Then, linear damping is introduced in the system and the applicability of the method is demonstrated.

Nonlinear Normal Modes for Vibratory Systems Under Periodic Excitation

2003 ◽  
Author(s):  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Normal Mode ◽  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Beam Model ◽  
Periodic Excitation ◽  
Nonlinear Normal Mode ◽  
Nonlinear Normal

This paper considers the use of numerically constructed invariant manifolds to determine the response of nonlinear vibratory systems that are subjected to periodic excitation. The approach is an extension of the nonlinear normal mode formulation previously developed by the authors for free oscillations, wherein an auxiliary system that models the excitation is used to augment the equations of motion. In this manner, the excitation is simply treated as an additional system state, yielding a system with an extra degree of freedom, whose response is known. A reduced order model for the forced system is then determined by the usual nonlinear normal mode procedure, and an efficient Galerkin-based solution method is used to numerically construct the attendant invariant manifolds. The technique is illustrated by determining the frequency response for a simple two-degree-off-reedom mass-spring system with cubic nonlinearities, and for a discretized beam model with 12 degrees of freedom. The results show that this method provides very accurate responses over a range of frequencies near resonances.

Energy Transfer and Dissipation in a Duffing Oscillator Coupled to a Nonlinear Attachment

2009 ◽  
Vol 4 (4) ◽  
Author(s):  
R. Viguié ◽  
M. Peeters ◽  
G. Kerschen ◽  
J.-C. Golinval
Keyword(s):  
Energy Transfer ◽  
Hamiltonian System ◽  
Nonlinear System ◽  
Duffing Oscillator ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Numerical Continuation ◽  
Damped System ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

The dynamics of a two-degree-of-freedom nonlinear system consisting of a grounded Duffing oscillator coupled to an essentially nonlinear attachment is examined in the present study. The underlying Hamiltonian system is first considered, and its nonlinear normal modes are computed using numerical continuation and gathered in a frequency-energy plot. Based on these results, the damped system is then considered, and the basic mechanisms for energy transfer and dissipation are analyzed.

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