Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator

Kowalczyk, Piotr; Piiroinen, Petri T.

doi:10.1016/j.physd.2007.12.007

Cited by 74 publications

(62 citation statements)

References 40 publications

(73 reference statements)

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“…Typical behaviour of piecewise-smooth systems suggests that these will take the form of stickslip periodic orbits for parameters close to N µs = |q|, undergoing brief intervals of sticking between intervals of right and/or left slipping motion, though a complete study is beyond our scope here. Stick-slip orbits were found in the case of the undamped c = 0 and nonhysteretic β = 0 oscillator, and for a sinusoidal forcingq, in [25,34]. Crucially these and other previous works that use Filippov-type solutions assume linear sticking, i.e.…”

Section: Closing Remarks and Friction-inspired Modelsmentioning

confidence: 94%

“…for z > −1/ρ, the quantity z being constant in the system (34). Thus by Fenichel's theorem there exist locally invariant manifolds B ε (z) that lie within O (ε) of and diffeomorphic to B(z).…”

Section: Piecewise-smooth Dynamicsmentioning

confidence: 95%

“…Proof The equilibria of the limiting (ε = 0) fast subsystem (34) are hyperbolic in a region satisfying Γ 1 (u) > 0, since the system is one-dimensional and has eigenvalue…”

Section: Piecewise-smooth Dynamicsmentioning

confidence: 99%

“…Experimental, numerical, and analytical studies of similar oscillator models can be found in [20,34,25,27,43,12], but in analytical considerations these predominantly consider ρ = 0 only.…”

Section: Proposition 4 Sticking Dynamicsmentioning

confidence: 99%

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On the mathematical basis of solid friction

Jeffrey

2015

Nonlinear Dyn

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms Noname manuscript No.(will be inserted by the editor)The mathematical description of friction between solid bodies Mike R. Jeffrey May 8, 2014 Abstract A piecewise-smooth ordinary differential equation model of a dry-friction oscillator is studied, as a paradigm for the role of nonlinear and hysteretic terms in discontinuities of dynamical systems. The friction discontinuity is a switch in direction of the contact force in the transition between left and rightward slipping motion. Nonlinear terms introduce dynamics that is novel in the context of piecewise-smooth dynamical systems theory (in particular they are outside the standard Filippov convention), but are shown to account naturally for static friction, and moreover provide a simple route to including hysteresis. The nonlinear terms are understood in terms of dummy dynamics at the discontinuity, given a formal derivation here. The result is a three-parameter model built on the minimal mathematical features necessary to account for the key characteristics of dry friction. The effect of compliance can be distinguished from the contact model, and numerical simulations reveal that all behaviours persist under smoothing and under small random perturbations, but nonlinear effects can be made to disappear abruptly amid sufficient noise.

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Section: Closing Remarks and Friction-inspired Modelsmentioning

confidence: 94%

Section: Piecewise-smooth Dynamicsmentioning

confidence: 95%

“…Proof The equilibria of the limiting (ε = 0) fast subsystem (34) are hyperbolic in a region satisfying Γ 1 (u) > 0, since the system is one-dimensional and has eigenvalue…”

Section: Piecewise-smooth Dynamicsmentioning

confidence: 99%

Section: Proposition 4 Sticking Dynamicsmentioning

confidence: 99%

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On the mathematical basis of solid friction

Jeffrey

2015

Nonlinear Dyn

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“…in [16][17][18][19][20][21][22][23][24][25]. Bifurcations of limit cycles of discontinuous systems are studied using a return map; see [16,17,22,24].…”

Section: Introductionmentioning

confidence: 99%