On the regularization of impact without collision: the Painlevé paradox and compliance

Hogan, S. J.; Kristiansen, Kristian

doi:10.1098/rspa.2016.0773

Cited by 16 publications

(36 citation statements)

References 37 publications

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“…For the system in Fig. 1, Painlevé paradoxes occur when v > 0 and θ ∈ (0, π 2 ), provided p + (θ) < 0 [21]. From (7), it is straightforward to show that p + (θ) < 0 requires…”

Section: Constraint-based Methodsmentioning

confidence: 96%

“…In order to maintain the constraint y = 0, at most one of F N and y can be positive [21] and so F N and y must satisfy…”

Section: Constraint-based Methodsmentioning

confidence: 99%

“…, or leave the box U through the {y 2 = −C}-face with w 2 < 0 such that sticking and IWC occurs (like S in Proposition 1), as described in [21].…”

Section: Remarkmentioning

confidence: 99%

“…Their results showed a strong dependence on the ratio of these times. More recently, the current authors presented [21] the first rigorous analysis of compliant IWC in both the inconsistent and indeterminate cases and gave explicit asymptotic expressions in the limiting cases of small and large damping. For the indeterminate case, we presented a formula for conditions that separate compliant IWC and lift-off.…”

Section: Introductionmentioning

confidence: 99%

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Le Canard de Painlevé

Kristiansen¹,

Hogan²

2018

SIAM J. Appl. Dyn. Syst.

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We consider the problem of a slender rod slipping along a rough surface. Painlevé [44,45,46] showed that the governing rigid body equations for this problem can exhibit multiple solutions (the indeterminate case) or no solutions at all (the inconsistent case), provided the coefficient of friction µ exceeds a certain critical value µ P . Subsequently Génot and Brogliato [19] proved that, from a consistent state, the rod cannot reach an inconsistent state through slipping. Instead there is a special solution for µ > µ C > µ P , with µ C a new critical value of the coefficient of friction, where the rod continues to slip until it reaches a singular "0/0" point P . Even though the rigid body equations can not describe what happens to the rod beyond the singular point P , it is possible to extend the special solution into the region of indeterminacy. This extended solution is very reminiscent of a canard [1]. To overcome the inadequacy of the rigid body equations beyond P , the rigid body assumption is relaxed in the neighbourhood of the point of contact of the rod with the rough surface. Physically this corresponds to assuming a small compliance there. It is natural to ask what happens to both the point P and the special solution under this regularization, in the limit of vanishing compliance.In this paper, we prove the existence of a canard orbit in a reduced 4D slow-fast phase space, connecting a 2D focus-type slow manifold with the stable manifold of a 2D saddle-type slow manifold. The proof combines several methods from local dynamical system theory, including blowup. The analysis is not standard, since we only gain ellipticity rather than hyperbolicity with our initial blowup. * K.Remark 3 For µ P < µ < µ C so that ξ > 1, then the direction θ = θ 1 is strong while (18) is weak. However, by evaluating the slope of the curve b(θ, φ) = 0 at the point P and comparing the result with s/(1 − ξ), it is straightforward to show that the weak eigendirection (18) is not contained within F ∪ P ∪ T . The reduced problem is only defined within F ∪ T and hence the classical Painlevé problem does therefore not support weak singular canards. 2 7 The result µ C (3) = 8 3 √ 3 does appear in [19].

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“…For the system in Fig. 1, Painlevé paradoxes occur when v > 0 and θ ∈ (0, π 2 ), provided p + (θ) < 0 [21]. From (7), it is straightforward to show that p + (θ) < 0 requires…”

Section: Constraint-based Methodsmentioning

confidence: 96%

“…In order to maintain the constraint y = 0, at most one of F N and y can be positive [21] and so F N and y must satisfy…”

Section: Constraint-based Methodsmentioning

confidence: 99%

“…, or leave the box U through the {y 2 = −C}-face with w 2 < 0 such that sticking and IWC occurs (like S in Proposition 1), as described in [21].…”

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Le Canard de Painlevé

Kristiansen¹,

Hogan²

2018

SIAM J. Appl. Dyn. Syst.

Self Cite

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“…The expression (C.14) follows directly from (C.13) and the fact that the right hand side of (C.10) is independent of r 2 . To obtain (C.15) we consider dε dx 2 = −ε 2 (2x 2 + m(x 2 )), (C. 16) obtained by inserting (C.14) into (C.10) We can then integrate (C.16) from…”

Section: Appendix B Proof Of Theorem 35 By Direction Integrationmentioning

confidence: 99%

Blowup for flat slow manifolds

Kristiansen

2017

Nonlinearity

120

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In this paper we present a method for extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic order. Although these manifolds have infinite co-dimension, they do appear naturally in certain settings. For example in (a) the regularization of piecewise smooth systems by tanh, (b) a model of aircraft landing dynamics, and finally (c) in a model of earthquake faulting. We demonstrate the approach on a simple model system and the examples (a) and (b).

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Nonsmooth analysis of three-dimensional slipping and rolling in the presence of dry friction

Antali

Stépàn

2019

Nonlinear Dyn

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In this paper, the nonsmooth dynamics of two contacting rigid bodies is analysed in the presence of dry friction. In three dimensions, slipping can occur in continuously many directions. Then, the Coulomb friction model leads to a system of differential equations, which has a codimension-2 discontinuity set in the phase space. The new theory of extended Filippov systems is applied to analyse the dynamics of a rigid body moving on a fixed rigid plane to explore the possible transitions between the slipping and rolling behaviour. The paper focuses on finding the so-called limit directions of the slipping equations at the discontinuity. This leads to a complete qualitative description of the possible scenarios of the dynamics in the vicinity of the discontinuity. It is shown that the new approach consistently extends the information provided from the static friction force of the rolling behaviour. The methods are demonstrated on an application example.

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On the regularization of impact without collision: the Painlevé paradox and compliance

Cited by 16 publications

References 37 publications

Le Canard de Painlevé

Le Canard de Painlevé

Blowup for flat slow manifolds

Nonsmooth analysis of three-dimensional slipping and rolling in the presence of dry friction

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