scholarly journals Singular perturbation approach to an elastic dry friction problem with non-monotone coefficient

2000 ◽  
Vol 58 (2) ◽  
pp. 303-324 ◽  
Author(s):  
Yves Renard
Keyword(s):  
Singular Perturbation ◽  
Dry Friction ◽  
Perturbation Approach ◽  
Monotone Coefficient

Singular perturbation approach to stability of a SIRS epidemic system

2009 ◽  
Vol 10 (5) ◽  
pp. 2688-2699 ◽  
Author(s):  
Zhonghua Zhang ◽  
Yaohong Suo ◽  
Jigen Peng ◽  
Weihua Lin
Keyword(s):  
Singular Perturbation ◽  
Perturbation Approach

Analysis of the non-periodic oscillations of a self-excited friction-damped system with closely-spaced modes

2021 ◽  
Author(s):  
Lukas Woiwode ◽  
Alexander F. Vakakis ◽  
Malte Krack
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Dry Friction ◽  
The Self ◽  
Perturbation Approach ◽  
Friction Damping ◽  
Periodic Oscillations ◽  
Analytical Results ◽  
Fold Bifurcation ◽  
Phase Motion

Abstract It is widely known that dry friction damping can bound the self-excited vibrations induced by negative damping. The vibrations typically take the form of (periodic) limit cycle oscillations. However, when the intensity of the self-excitation reaches a condition of maximum friction damping, the limit cycle loses stability via a fold bifurcation. The behavior may become even more complicated in the presence of any internal resonance conditions. In this work, we consider a two-degree-of-freedom system with an elastic dry friction element (Jenkins element) having closely spaced natural frequencies. The symmetric in-phase motion is subjected to self-excitation by negative (viscous) damping, while the symmetric out-of-phase motion is positively damped. In a previous work, we showed that the limit cycle loses stability via a secondary Hopf bifurcation, giving rise to quasi-periodic oscillations. A further increase of the self-excitation intensity may lead to chaos and finally divergence, long before reaching the fold bifurcation point of the limit cycle. In this work, we use the method of Complexification-Averaging to obtain the slow flow in the neighborhood of the limit cycle. This way, we show that chaos is reached via a cascade of period doubling bifurcations on invariant tori. Using perturbation calculus, we establish analytical conditions for the emergence of the secondary Hopf bifurcation and approximate analytically its location. In particular, we show that non-periodic oscillations are the typical case for prominent nonlinearity, mild coupling (controlling the proximity of the modes) and sufficiently light damping. The range of validity of the analytical results presented herein is thoroughly assessed numerically. To the authors' knowledge, this is the first work that shows how the challenging Jenkins element can be treated formally within a consistent perturbation approach in order to derive closed-form analytical results for limit cycles and their bifurcations.

Reaction rates when barriers fluctuate: A singular perturbation approach

1998 ◽  
Vol 235 (1-3) ◽  
pp. 11-26 ◽  
Author(s):  
Peter Reimann ◽  
Roland Bartussek ◽  
Peter Hänggi
Keyword(s):  
Singular Perturbation ◽  
Reaction Rates ◽  
Perturbation Approach
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