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.

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.

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 Bayesian model updating of nonlinear systems using nonlinear normal modes

2018 ◽  
Vol 25 (12) ◽  
pp. e2258 ◽  
Author(s):  
Mingming Song ◽  
Ludovic Renson ◽  
Jean-Philippe Noël ◽  
Babak Moaveni ◽  
Gaetan Kerschen
Keyword(s):  
Nonlinear Systems ◽  
Bayesian Model ◽  
Model Updating ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Nonlinear Normal

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.

scholarly journals Efficient Energy Balancing Across Multiple Harmonics of Nonlinear Normal Modes

2021 ◽  
Author(s):  
Dongxiao Hong ◽  
Thomas L. Hill ◽  
Simon A. Neild
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Analytical Framework ◽  
Phase Shifts ◽  
Net Energy ◽  
Energy Balancing ◽  
Harmonic Phase ◽  
One Step ◽  
Nonlinear Normal ◽  
Forced Responses

Abstract Predicting the forced responses of nonlinear systems is a topic that attracts extensive studies. The energy balancing method considers the net energy transfer in and out of the system over one period, and establishes connections between forced responses and nonlinear normal modes (NNMs). In this paper, we consider the energy balancing across multiple harmonics of NNMs for predicting forced resonances. This technique is constructed by combining the energy balancing mechanism with restrictions (established via excitation scenarios) on external forcing and harmonic phase-shifts; a semi-analytical framework is derived to achieve both accurate/robust results and efficient computations. With known inputs from NNM solutions, the required forcing amplitudes to reach NNMs at resonances, along with their discrepancy, i.e. the harmonic phase-shifts, are computed via a one-step scheme. Several examples are presented for different excitation scenarios to demonstrate the applicability of this method, and to show its capability in accurately predicting the existence of an isola where multiple harmonics play a significant part in the response.

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