Determining Lyapunov exponents of non-smooth systems: Perturbation vectors approach

Balcerzak, Marek; Dąbrowski, Artur; Błażejczyk-Okolewska, Barbara; Stefański, Andrzej

doi:10.1016/j.ymssp.2020.106734

Cited by 43 publications

(20 citation statements)

References 34 publications

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“…In this case, a later time instant t ¼ ði þ jÞDt was chosen such that both solutions slide, and the distance d iþj was taken into account considering 1 jDt ln jd iþj j jd i j in the sum. This approach provides adequate results, as it was pointed out in [41]. Moreover, a similar procedure was already tested on the same system in [30] and the results agreed well with the Lyapunov exponents calculated by a more specific method, published in [42].…”

Section: Two-parameter Bifurcation Analysis and Search For Chaossupporting

confidence: 76%

Asymmetric and chaotic responses of dry friction oscillators with different static and kinetic coefficients of friction

Csernák

Licskó²

2021

Meccanica

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The responses of a simple harmonically excited dry friction oscillator are analysed in the case when the coefficients of static and kinetic coefficients of friction are different. One- and two-parameter bifurcation curves are determined at suitable parameters by continuation method and the largest Lyapunov exponents of the obtained solutions are estimated. It is shown that chaotic solutions can occur in broad parameter domains—even at realistic friction parameters—that are tightly enclosed by well-defined two-parameter bifurcation curves. The performed analysis also reveals that chaotic trajectories are bifurcating from special asymmetric solutions. To check the robustness of the qualitative results, characteristic bifurcation branches of two slightly modified oscillators are also determined: one with a higher harmonic in the excitation, and another one where Coulomb friction is exchanged by a corresponding LuGre friction model. The qualitative agreement of the diagrams supports the validity of the results.

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Section: Two-parameter Bifurcation Analysis and Search For Chaossupporting

confidence: 76%

Asymmetric and chaotic responses of dry friction oscillators with different static and kinetic coefficients of friction

Csernák

Licskó²

2021

Meccanica

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“…This method is particularly suitable for smooth systems, while the results for non-smooth systems (e.g., due to frictional forces) should generally be taken with caution. The estimation of Lyapunov exponents for this type of systems is still under research, a summary of different approaches can be found for example in [20].…”

Section: Approximation Of the Energy-dependent Oscillation Frequencymentioning

confidence: 99%

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

2021

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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 in 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.

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“…It has been widely explored as regards both rigid (e.g., [14][15][16][17][18]) and soft constraints (e.g., [19][20][21][22][23][24][25][26][27]), with particular attention paid to stability of periodic solutions, bifurcations and singularities in vibro-impact dynamics. Intensive tests of impact systems have led to the development of a set of tools to identify and assess qualitatively and quantitatively the dynamic phenomena occurring in piecewise smooth systems (see, e.g., [28][29][30][31][32][33][34]).…”

Section: Introductionmentioning

confidence: 99%

Hard versus soft impact modeling of vibro-impact systems with a moving base

Okolewski

Błażejczyk-Okolewska

2021

Nonlinear Dyn

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Soft and hard impact models applied to modeling of vibro-impact systems with a moving base are discussed. The conditions under which two collision models are equivalent in terms of equal energy dissipation are derived. These conditions differ from those presented in the literature. It is shown that in the case of a stiff, harmonically moving base with a low rate of energy dissipation, both methods yield the same results, but an application of the soft impact model to either the base with low stiffness or even the stiff base with a high rate of energy dissipation leads to different results from the ones for the hard impact model.

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Determining Lyapunov exponents of non-smooth systems: Perturbation vectors approach

Cited by 43 publications

References 34 publications

Asymmetric and chaotic responses of dry friction oscillators with different static and kinetic coefficients of friction

Asymmetric and chaotic responses of dry friction oscillators with different static and kinetic coefficients of friction

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

Hard versus soft impact modeling of vibro-impact systems with a moving base

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