Estimation of Chaotic and Regular (Stick–Slip and Slip–Slip) Oscillations Exhibited by Coupled Oscillators with Dry Friction

Awrejcewicz, Jan; Dzyubak, L.; Grebogi, Celso

doi:10.1007/s11071-005-7183-0

Cited by 37 publications

(16 citation statements)

References 12 publications

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“…The cannibalistic coefficients k ii are usually zero. Systems of these forms may be used to model how microscopic dry-friction leads to macroscale stick-slip or even earthquakes [3,6], or to model networks of switching in electronic, genetic, or neural circuitry (see e.g. [15,27]).…”

Section: Introductionmentioning

confidence: 99%

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

Jeffrey

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The collapse of flows onto hypersurfaces where their vector fields are discontinuous creates highly robust states called sliding modes. The way flows exit from such sliding modes can lead to complex and interesting behaviour about which little is currently known. Here we examine the basic mechanisms by which a flow exits from sliding, either along a switching surface, or along the intersection of two switching surfaces, with a view to understanding sliding and exit when many switches are involved. On a single switching surface, exit occurs via tangency of the flow to the switching surface. Along an intersection of switches, exit can occur at a tangency with a lower codimension sliding flow, or by a spiralling of the flow that exhibits geometric divergence (infinite steps in finite time). Determinacy-breaking can occur where a singularity creates a set-valued flow in an otherwise deterministic system, and we resolve such dynamics as far as possible by blowing up the switching surface into a switching layer. We show preliminary simulations exploring the role of determinacy-breaking events as organizing centres of local and global dynamics.Switching is found in dynamical models of wideranging applications, from mechanics and geophysics to biological growth and ecology. Switches occur between different dynamical laws whenever certain thresholds are encountered. In this paper we consider how systems behave when they exit from highly constrained states sliding along those thresholds or intersections thereof. For one or two switches we examine the basic mechanisms of exit. In particular we show that exit from sliding is not always deterministic, and we describe the main features of determinacy-breaking exit points. Example simulations that illustrate the theoretical results as novel dynamical phenomena are given.

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Section: Introductionmentioning

confidence: 99%

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

Jeffrey

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…. , β l , t i ) are obtained as a result of integration of the system described by the model (7). To minimise the criterion function (8) the method of gradient descent is used.…”

Section: Hysteresis Simulation By Means Of Internal Variablesmentioning

confidence: 99%

“…The presence of chaos in hysteretic systems had been demonstrated only recently [14,15]. The advantages of the approach presented in this work over standard procedures, and a comparison, can be found in [1,4,7].…”

Section: Evolution Of Chaotic Behaviour Regions In Control Parameter mentioning

confidence: 99%

“…3, one-DOF hysteretic oscillators (highly nonlinear nonautonomous Masing and Bouc-Wen hysteretic models with discontinuous right-hand sides) have been investigated using the methodology described in [1][2][3][4][5][6][7]. This methodology had already been successfully applied in particular to predict stick-slip chaos in two-DOF discontinuous systems with friction [1,4,7] and in other smooth [3,4] and non-smooth systems [2,4]. This algorithm for quantifying regular and chaotic dynamics is simpler and faster from a computational point of view than standard procedures and allows sufficiently accurate to trace the regular/irregular responses of the hysteretic systems.…”

Section: Introductionmentioning

confidence: 99%

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Hysteresis modelling and chaos prediction in one- and two-DOF hysteretic models

Awrejcewicz

Dzyubak

2006

Arch Appl Mech

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In the present work hysteresis is simulated by means of internal variables. The analytical models of different types of hysteresis loops allow the reproduction of major and minor loops and provide a high degree of correspondence with experimental data. In models of this type adding an external periodic excitation or increasing the number of dimensions can lead to the occurrence of chaotic behaviour. Using an effective algorithm based on numerical analysis of the wandering trajectories [1-7], the evolution of the chaotic behaviour regions of oscillators with hysteresis is presented in various parametric planes. The substantial influence of a hysteretic dissipation value on the form and location of these regions, as well as the restraining and generating effects of hysteretic dissipation on the occurrence of chaos, are ascertained. Conditions for pinched hysteresis are defined. Furthermore, autonomous coupled hysteretic oscillators under sliding friction are investigated. Conditions for the occurrence of chaotic behaviour in a two-degree-of-freedom (two-DOF) hysteretic system are found in the plane of maximal static friction forces of both oscillators versus belt velocity.

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“…Because vehicle quality standards are setting increasingly low thresholds for noise level and vibration, many researchers have intensively studied brake squeal, by using various analytical and experimental methods [2][3][4][5][6][7]. For example, after analyzing automotive disc brake squeal, Ouyang et al [2] concludes that chattering behavior is a self-excited vibration based on a stick-slip phenomenon and that it is induced only in a certain range of friction parameters; this feature of the stick-slip phenomenon also occurs in other physical systems [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Dynamics and Control in an Automotive Brake System

Chang¹,

Hu²

2016

Adv Automob Eng

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Brake squeal is a manifestation of friction-induced self-excited instability in disc brake systems. This study investigated non-smooth bifurcations and chaotic dynamics in disc brake systems and elucidated a chaotic control system. Decreasing squeal noise which is dependent on chaos, increases passengers comfort; consequently, suppressing chaos is crucial. First, synchronization was used to estimate the largest Lyapunov exponent to identify periodic and chaotic motions. Next, complex nonlinear behaviors were thoroughly observed for a range of parameter values in the bifurcation diagram. Rich dynamics of the disc brake system were studied using a bifurcation diagram, phase portraits, a Poincaré map, frequency spectra, and Lyapunov exponents. Finally, the proposed technique was applied to a chaotic disc brake system through the addition of an external input that is a dither signal. Simulation results demonstrated the feasibility of the proposed approach.

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Estimation of Chaotic and Regular (Stick–Slip and Slip–Slip) Oscillations Exhibited by Coupled Oscillators with Dry Friction

Cited by 37 publications

References 12 publications

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

Hysteresis modelling and chaos prediction in one- and two-DOF hysteretic models

Nonlinear Dynamics and Control in an Automotive Brake System

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