Hyperchaos co-existing with periodic orbits in a frictional oscillator

Stender, Merten; Jahn, Martin; Hoffmann, Norbert; Wallaschek, Jörg

doi:10.1016/j.jsv.2020.115203

Cited by 15 publications

(10 citation statements)

References 10 publications

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“…Here, the damping parameters are reduced by a factor of 10 to d x = d y = d z = d lin = 0.002. This system configuration has already been studied in [13,14] where the authors found an isolated solution branch resulting from the damping variation. Figure 7 (a) displays the bifurcation diagram for the horizontal stiffness k x .…”

Section: Bi-stable Oscillator With Isolated Periodic Solutionmentioning

confidence: 93%

“…For the nonlinear joint element, a cubic stiffness nonlinearity k nl is chosen [11]. The equations of motion and parameter values are given in Appendix B and the model is displayed in branches [11,13,14]. In this study, a variation of the horizontal stiffness k x is performed.…”

Section: Bi-stable Oscillator With Mode-couplingmentioning

confidence: 99%

“…through self-excitation [8][9][10]. Sub-critical Hopf bifurcations [11,12] and isolated solution branches [13][14][15] are a common observation in those systems, such that bi-and multi-stable systems have been reported numerously [14,16,17]. The computation of those nonlinear responses (periodic, quasi-period orbits, chaotic trajectories) is a well-established field of research [18][19][20][21], mostly resulting in the identification of complicated bifurcation diagrams [11,13,[22][23][24].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Basin Stability of Bi-Stable Friction-Excited Oscillators

Stender¹,

Hoffmann²,

Papangelo³

2020

Preprint

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Stability considerations play a central role in structural dynamics to determine states that are robust against perturbations during the operation. Linear stability concepts, such as the complex eigenvalue analysis, constitute the core of analysis approaches in engineering reality. However, most stability concepts are limited to local perturbations, i.e. they can only measure a state’s stability against small perturbations. Recently, the concept of basin stability has been proposed as a global stability concept for multi-stable systems. As multi-stability is a well-known property of a range of nonlinear dynamical systems, this work studies the basin stability of bi-stable mechanical oscillators that are affected and self-excited by dry friction. The results indicate how the basin stability complements the classical binary stability concepts for quantifying how stable a state is given a set of permissible perturbations.

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Section: Bi-stable Oscillator With Isolated Periodic Solutionmentioning

confidence: 93%

Section: Bi-stable Oscillator With Mode-couplingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Basin Stability of Bi-Stable Friction-Excited Oscillators

Stender¹,

Hoffmann²,

Papangelo³

2020

Preprint

Self Cite

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“…The E-PMC is thus an excellent choice for limit cycle detection of self-excited systems with non-conservative nonlinear forces. Details considering implementation and further results can be found in [6] and [7]. Limit cycles of non-conservative nonlinear systems can be represented by 1D-submanifolds for a certain parameter variation.…”

Section: Concept Of Nonlinear Modesmentioning

confidence: 99%

Limit cycle computation of self‐excited dynamic systems using nonlinear modes

Tatzko

Stender

Jahn

et al. 2021

Proc Appl Math and Mech

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A self‐excited dynamic system is able to oscillate periodically by itself. Corresponding solutions of the autonomous differential equation are called limit cycles or periodic attractors. To find these solutions, a simple approach would be brute‐force search for the corresponding basins of attraction. However, grid searching might become unfeasible with increasing number of degrees of freedom. Instead, solution path continuation techniques are often used to keep computational costs low. As the continuation of solution branches and their bifurcations provides only solutions which are connected to each other, isolas and detached branches are missed out. We present a method for fast limit cycle detection of self‐excited systems with isolas based on nonlinear modes. A nonlinear mode, often referred to as nonlinear normal mode, is defined as a periodic motion of the undamped and unforced mechanical system. For nonconservative systems however, e.g. with friction nonlinearity, damping cannot be neglected as it is characteristic for the oscillators nonlinear dynamics. Therefore, the Extended Periodic Motion Concept (E‐PMC) was proposed recently to find periodic solutions of nonconservative nonlinear systems. In this work, the E‐PMC is applied to self‐excited dynamic systems in order to find periodic attractors along its nonlinear modes. Zero crossings of the nonlinear damping curve indicate autonomous solutions which can be used as starting points for single parameter continuation. Thus, solutions corresponding to the main branch and detached curves in the solution space are connected by nonlinear modes. The proposed method is applied to a frictional oscillator with cubic stiffness and proves to be robust in the search for isolated periodic solutions that are already known from literature.

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“…Depending on the initial condition, multi-stable systems exhibit different time-asymptotic behavior. Recently, multistability has obtained attention in mechanical systems which are not necessarily cyclic, nor of network form: fluttering airfoils [19], few-degree-offreedom friction oscillators [20][21][22], friction oscillator chains [23], and (hyper-chaotic) self-excited oscillators [24] amongst others. Experimental observations of multi-stable mechanical systems range from bi-stable automotive friction brake vibrations [25] , bi-stable responses of helicopter blades [26], windtunnel airfoil tests [27,28], to small cyclic mechanical structures with gap-induced nonlinear vibration localization [29].…”

Section: Introductionmentioning

confidence: 99%

bSTAB: an open-source software for computing the basin stability of multi-stable dynamical systems

Stender

Hoffmann

2021

Nonlinear Dyn

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The pervasiveness of multi-stability in nonlinear dynamical systems calls for novel concepts of stability and a consistent quantification of long-term behavior. The basin stability is a global stability metric that builds on estimating the basin of attraction volumes by Monte Carlo sampling. The computation involves extensive numerical time integrations, attractor characterization, and clustering of trajectories. We introduce , an open-source software project that aims at enabling researchers to efficiently compute the basin stability of their dynamical systems with minimal efforts and in a highly automated manner. The source code, available at https://github.com/TUHH-DYN/bSTAB/, is available for the programming language featuring parallelization for distributed computing, automated sensitivity and bifurcation analysis as well as plotting functionalities. We illustrate the versatility and robustness of for four canonical dynamical systems from several fields of nonlinear dynamics featuring periodic and chaotic dynamics, complicated multi-stability, non-smooth dynamics, and fractal basins of attraction. The projects aims at fostering interdisciplinary scientific collaborations in the field of nonlinear dynamics and is driven by the interaction and contribution of the community to the software package.

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Hyperchaos co-existing with periodic orbits in a frictional oscillator

Cited by 15 publications

References 10 publications

The Basin Stability of Bi-Stable Friction-Excited Oscillators

The Basin Stability of Bi-Stable Friction-Excited Oscillators

Limit cycle computation of self‐excited dynamic systems using nonlinear modes

bSTAB: an open-source software for computing the basin stability of multi-stable dynamical systems

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