The contact problem in Lagrangian systems subject to bilateral and unilateral constraints, with or without sliding Coulomb’s friction: a tutorial

Blumentals, Alejandro; Brogliato, Bernard; Bertails-Descoubes, Florence

doi:10.1007/s11044-016-9527-6

Cited by 40 publications

(59 citation statements)

References 55 publications

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“…It coincides with κ 23 (l 3 ), where l 3 is the invariant line in chart κ 3 , given in (53). 2 Proof Straightforward calculation.…”

Section: Scaling Chart κmentioning

confidence: 69%

“…γ s ε is o(1)-close to γ s within U and it divides S a,ε into orbits that lift off from those that eventually stick. 2…”

Section: Resultsmentioning

confidence: 99%

“…This remarkable result also appears to be new. 2 Remark 2 Since p + < 0 within F, the new time τ reverses direction there. Therefore the manifold γ s gives a solution of (10) with respect to the original time having a smooth continuation through the singularity P (as indicated in Fig.…”

Section: Remarkmentioning

confidence: 99%

“…The estimate could probably be improved but we did not pursue this. 2 Remark 5 For the last statement of the theorem, we add the following. Fix C > 0 large and consider the following box in the (y 2 , w 2 , θ, φ)-space:…”

Section: Remarkmentioning

confidence: 99%

“…Proof By the exponential growth of La ξ (see (66)) for θ 2 → ∞ it follows that the tangent space of C e,2 is not a subspace of the tangent space of W s (C r,2 ) along l 2 for θ 2 1. Hence the intersection is transverse.…”

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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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“…It coincides with κ 23 (l 3 ), where l 3 is the invariant line in chart κ 3 , given in (53). 2 Proof Straightforward calculation.…”

Section: Scaling Chart κmentioning

confidence: 69%

“…γ s ε is o(1)-close to γ s within U and it divides S a,ε into orbits that lift off from those that eventually stick. 2…”

Section: Resultsmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

mentioning

confidence: 99%

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Le Canard de Painlevé

Kristiansen¹,

Hogan²

2018

SIAM J. Appl. Dyn. Syst.

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Selected Challenges in Realistic Multibody Modeling of Machines and Vehicles

Ambrósio

2019

IUTAM Symposium on Intelligent Multibody Systems – Dynamics, Control, Simulation

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Comparison of several formulations and integration methods for the resolution of DAEs formulations in event-driven simulation of nonsmooth frictionless multibody dynamics

et al. 2017

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International audienceThis article is devoted to the comparison of numerical integration methods for nonsmooth multibody dynamics with joints, unilateral contacts and impacts in an industrial context. With an event–driven strategy, the smooth dynamics, which is integrated between two events, may be equivalently formulated as a Differential Algebraic Equation (DAE) of index 1, 2 or 3. It is well-known that these reformulations are no longer equivalent when a numerical time–integration technique is used. The drift-off effect and the stability of the numerical scheme strongly depend on the index of the formulation. But, besides the standard properties of accuracy and stability of the DAE solvers, the event–driven context imposes some further requirements that are crucial for a robust and efficient event-driven strategy. In this article, several state–of–the–art numerical time integration methods for each formulation are compared: the generalized-α scheme for index-3 formulation and stabilized index-2 formulation, (Partitioned) Runge–Kutta Half–Explicit Method of order 5 (HEM5 and PHEM56) for index-2 DAEs with projection techniques, and Runge-Kutta explicit scheme of order 5, the Dormand-Prince scheme (DOPRI5), for index-1 DAEs with projection techniques (MDOP5). We compare these schemes in terms of efficiency, violation of the constraints and the way they handle stiff dynamics on numerous industrial benchmarks, where a CAD software is in this loop. One of the major conclusions is that the index-2 DAEs solvers prove to be better than other schemes to maintain low violations at position and acceleration levels. The best compromise allows us to design efficient event–driven solvers. When the dynamics is stiff, implicit schemes outperform explicit and half-explicit methods whichare sometimes unable to compute the dynamics when the system’s frequency range is wide. Furthermore, in industrial context, some solvers fail to reproduce the properties that they enjoy in theory. This is particularly true for half-explicit schemes when the Jacobian of the constraints has not full rank

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The contact problem in Lagrangian systems subject to bilateral and unilateral constraints, with or without sliding Coulomb’s friction: a tutorial

Cited by 40 publications

References 55 publications

Le Canard de Painlevé

Le Canard de Painlevé

Selected Challenges in Realistic Multibody Modeling of Machines and Vehicles

Comparison of several formulations and integration methods for the resolution of DAEs formulations in event-driven simulation of nonsmooth frictionless multibody dynamics

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