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.

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.

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.

scholarly journals Prediction of Isolated Resonance Curves Using Nonlinear Normal Modes

2015 ◽  
Author(s):  
R. J. Kuether ◽  
L. Renson ◽  
T. Detroux ◽  
C. Grappasonni ◽  
G. Kerschen ◽  
...  
Keyword(s):  
Normal Modes ◽  
Resonance Curve ◽  
Element Model ◽  
Nonlinear Normal Modes ◽  
Initial Guess ◽  
Forced Response ◽  
Numerical Continuation ◽  
Continuation Algorithm ◽  
Resonance Curves ◽  
Nonlinear Normal

Isolated resonance curves are separate from the main nonlinear forced-response branch, so they can easily be missed by a continuation algorithm and the resonant response might be underpredicted. The present work explores the connection between these isolated resonances and the nonlinear normal modes of the system and adapts an energy balance criterion to connect the two. This approach provides new insights into the occurrence of isolated resonances as well as a method to find an initial guess to compute the isolated resonance curve using numerical continuation. The concepts are illustrated on a finite element model of a cantilever beam with a nonlinear spring at its tip. This system presents jumps in both frequency and amplitude in its response to a swept sinusoidal excitation. The jumps are found to be the result of a modal interaction that creates an isolated resonance curve that eventually merges with the main resonance branch as the excitation force increases. Excellent insight into the observed dynamics is provided with the NNM theory, which supports that NNMs can also be a useful tool for predicting isolated resonance curves and other behaviors in the damped, forced response.

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.

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 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.

Constructing the Frequency-Energy Plot of Nonlinear Vibratory Systems via the Modified Lindstedt-Poincaré Method

2013 ◽  
Vol 655-657 ◽  
pp. 547-550
Author(s):  
Xin Hua Zhang
Keyword(s):  
Energy Transfer ◽  
Nonlinear Vibration ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Numerical Results ◽  
Internal Resonances ◽  
Nonlinear Normal ◽  
Vibrating Systems

The frequency-energy plot(FEP) of nonlinear vibration systems is a powerful tool for investigating the energy transfer phenomena related wiht the internal resonances occured in multi-digreeof- freedom(Multi-DOF) nonlinear vibration systems. In this paper, the modified Lindstedt-Poincare method is employed for constructing the FEP of a two-DOF nonlinear vibrating systems. First, the original vibartion equations are modified for the application of the modified Linstedt-Poincaré method. Then, by using the modified Linstedt-Poincaé method, the nonlinear normal modes(NNMs) of the system are obtained. Finally, the frequency-energy plot of the system is constructed analytically. Numerical results show that the method adopted in this paper is effective and accurate.

Computing Nonlinear Normal Modes Using Numerical Continuation and Force Appropriation

2012 ◽  
Author(s):  
Robert J. Kuether ◽  
Matthew S. Allen
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Numerical Continuation ◽  
Gradient Based ◽  
Nonlinear Normal ◽  
Definition Of ◽  
Nonlinear Equations Of Motion ◽  
Mass Spring

Many structures can behave nonlinearly, exhibiting behavior that is not captured by linear vibration theory such as localization and frequency-energy dependence. The nonlinear normal mode (NNM) concept, developed over the last few decades, can be quite helpful in characterizing a structure’s nonlinear response. In the definition of interest, an NNM is a periodic solution to the conservative nonlinear equations of motion. Several approaches have been suggested for computing NNMs and some have been quite successful even for systems with hundreds of degrees of freedom. However, existing methods are still too expensive to employ on realistic nonlinear finite element models, especially when the Jacobian of the equations of motion is not available analytically. This work presents a new approach for numerically calculating nonlinear normal modes by combining force appropriation, numerical integration and continuation techniques. This method does not require gradients, is found to compute the NNMs accurately up to moderate response amplitudes, and could be readily extended to experimentally characterize nonlinear structures. The method is demonstrated on a nonlinear mass-spring-damper system, computing its NNMs up to a 35% shift in frequency. The results are compared with those from a gradient based algorithm and the relative merits of each method are discussed.

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