/pdf/dynamics-and-bifurcations-of-nonsmooth-systems-a-survey-1qecz49b7p.pdf

Dynamics and bifurcations of nonsmooth systems: A survey

This review article focuses on the problems related to numerical simulation of finite dimensional nonsmooth multibody mechanical systems. The rigid body dynamical case is examined here. This class of systems involves complementarity conditions and impact phenomena, which make its study and numerical analysis a difficult problem that cannot be solved by relying on known Ordinary Differential Equation (ODE) or Differential Algebraic Equation (DAE) integrators only. The main techniques, mathematical tools, and existing algorithms are reviewed.

https://hal.archives-ouvertes.fr/hal-01349852/document

Numerical simulation of finite dimensional multibody nonsmooth mechanical systems

Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for colli- sions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of vari- ational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated.

/pdf/nonsmooth-lagrangian-mechanics-and-variational-collision-uhz1dwuolg.pdf

Nonsmooth Lagrangian Mechanics and Variational Collision Integrators

In this paper, a novel discrete-time implementation of sliding-mode control systems is proposed, which fully exploits the multivaluedness of the dynamics on the sliding surface. It is shown to guarantee a smooth stabilization on the discrete sliding surface in the disturbance-free case, hence avoiding the chattering effects due to the time-discretization. In addition, when a disturbance acts on the system, the controller attenuates the disturbance effects on the sliding surface by a factor h (where h is the sampling period). Most importantly, this holds even for large h . The controller is based on an implicit Euler method and is very easy to implement with projections on the interval [-1, 1] (or as the solution of a quadratic program). The zero-order-hold (ZOH) method is also investigated. First- and second-order perturbed systems (with a disturbance satisfying the matching condition) without and with dynamical disturbance compensation are analyzed, with classical and twisting sliding-mode controllers.

/pdf/chattering-free-digital-sliding-mode-control-with-state-af3lzcfy08.pdf

Chattering-Free Digital Sliding-Mode Control With State Observer and Disturbance Rejection

https://hal.inria.fr/inria-00423576/document

Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems

An analysis of the modified Dahl and Masing models: Application to a belt tensioner

In this article, we show the convergence of a class of numerical schemes for certain maximal monotone evolution systems; a by-product of this results is the existence of solutions in cases which had not been previously treated. The order of these schemes is 1/2 in general and 1 when the only non Lipschitz continuous term is the subdifferential of the indicatrix of a closed convex set. In the case of Prandtl's rheological model, our estimates in maximum norm do not depend on spatial dimension.

/pdf/numerical-precision-for-differential-inclusions-with-3fktf7547z.pdf

Numerical precision for differential inclusions with uniqueness

A large number of rheological models can be covered by the existence and uniqueness theory for maximal monotone operators. Numerical simulations display hysteresis cycles when the forcing is periodic. A given shape of hysteresis cycle in an appropriate class of polygonal cycles can always be realized by adjusting the physical parameters of the rheological model.

Study of some rheological models with a finite number of degrees of freedom

Persoz’s gephyroidal model, which consists of elementary rheological models (dry friction element and linear spring), can be covered by the existence and uniqueness theory for maximal monotone operators. Moreover, classical results of numerical analysis allow one to use a numerical implicit Euler scheme, with convergence order of the scheme equal to one. Some numerical simulations are presented.

Persoz’s gephyroidal model described by a maximal monotone differential inclusion

Resume Soit V↪H↪V′ un triplet d'espaces de Hilbert separables. Dans cette Note, on etudie l'ordre de convergence d'une approximation numerique du probleme u +Bu+Au∋f(·,u), u(0)=u 0 , ou B est un operateur lipschitzien et V-elliptique de V dans V′ et A est un graphe maximal monotone dans V×V′. Si f est lipschitzienne de [0,T]×H dans V′ par rapport a son deuxieme argument et si la section A0 de A est bornee, alors le schema a un pas U p+1 −U p h +B(U p+1 )+A(U p+1 )∋f(ph,U p ), est d'ordre 1/2. Sous des hypotheses particulieres, il est d'ordre 1.

Jérôme Bastien

Papers

An analysis of the modified Dahl and Masing models: Application to a belt tensioner

Numerical precision for differential inclusions with uniqueness

Study of some rheological models with a finite number of degrees of freedom

Persoz’s gephyroidal model described by a maximal monotone differential inclusion

Schéma numérique pour des inclusions différentielles avec terme maximal monotone