Chaotic solutions in differential inclusions: chaos in dry friction problems

The existence of a continuum of many chaotic solutions is shown for certain differential inclusions which are small periodic multivalued perturbations of ordinary differential equations possessing homoclinic solutions to hyperbolic fixed points. Applications are given to dry friction problems. Singularly perturbed differential inclusions are investigated as well.

     

Periodic solutions of non-smooth friction oscillators

We study the differential equation x"+g(x￠)+m(x) sgn x￠+f(x)=j(t)x''+g(x')+\mu(x)\,{\rm sgn}\, x'+f(x)=\varphi(t) with T-periodic right-hand side, which models e.g. a mechanical system with one degree of freedom subjected to dry friction and periodic external force. If, in particular, the damping term g is present and acts, up to a bounded difference, like a linear damping, we get existence of a T-periodic solution.¶In the more difficult case g = 0, we concentrate on the model equation x"+m(x) sgn x￠+x=j(t)x''+\mu(x)\,{\rm sgn}\,x'+x=\varphi(t) and obtain sufficient conditions for the existence of a T-periodic solution by application of Brouwer's fixed point theorem. For this purpose we show that a certain associated autonomous differential equation admits a periodic orbit such that the surrounded set (minus some neighborhood of the equilibria) is forward invariant for the equation above. Under additional assumptions on 7 we prove boundedness of all solutions.¶Finally, we provide a principle of linearized stability for periodic solutions without deadzones, where the "linearized" differential equation is an impulsive Hill equation.     

Periodic solutions for planar 2N-body problems

In this paper we study some necessary conditions and sufficient conditions for the nested polygonal solutions of planar 2N-body problems.

     

Second order singular periodic problems in the presence of dry friction

We prove, via an approach by ordinary differential equations, the existence of oscillations for second order inclusions with restoring terms with singularities both of repulsive and attractive type and with a dry friction term.

     

Nonlinear boundary-value problems for Boltzmann equations: Periodic solutions and their bifurcations

The statement of nonlinear boundary conditions for the Boltzmann kinetic equation is examined. On this basis, the possibility is discussed that relaxation- or preturbulent-type oscillations that are asymptotically periodic can form in gases. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 323–333, February, 1997.     

Dynamic multi-layer models of dry friction

Determining the friction law between solids as a function of material and loading parameters remains a topical and still unsolved problem of tribology. In the rail-wheel-contact the law of sliding friction determines not only the maximum tractive forces but also defines the conditions for the occurrence of instabilities. Recent simulations of irreversible processes in micro-contacts with the method of movable cellular automata (MCA) have shown that dynamic aspects do play an essential role in these processes. Inelastic processes of plastic deformation, detaching of particles and their reintegration into the surfaces as well as processes of mechanical mixing occur only in a thin surface layer called quasi-fluid layer. Studying the properties of the quasi-fluid layer is thus a key for understanding friction. To better understand the physical nature of the quasi-fluid layer and especially the physical nature of the viscosity of the quasi-fluid state, multi-layer models of an elasto-plastic solid have been studied. In the layer model it becomes particularly clear that the physical reason for occurrence of the quasi-fluid layer is the bi-stability of the friction law between layers. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with coulomb friction

In this paper, we study the problem of predicting the acceleration of a set of rigid, 3-dimensional bodies in contact with Coulomb friction. The nonlinearity of Coulomb's law leads to a nonlinear complementarity formulation of the system model. This model is used in conjunction with the theory of quasi-variational inequalities to prove for the first time that multi-rigid-body systems with all contacts rolling always has a solution under a feasibility-type condition. The analysis of the more general problem with sliding and rolling contacts presents difficulties that motivate our consideration of a relaxed friction law. The corresponding complementarity formulations of the multi-rigid-body contact problem are derived and existence of solutions of these models is established. The research of this author was based on work supported by the National, Science Foundation under grants DDM-9104078 and CCR-9213739. The research of this author was partially supported by the National Science Foundation under grant IRI-9304734, by the Texas Advanced Research Program grant 999903-078, and by the Texas Advanced Technology Program under grant 999903-095.     

Mathematical modeling of friction stir welding considering dry and viscous friction

In this paper, the investigation of a pre-immersion period of Friction Stir Welding process (FSW) from the mechanical point of view using analytical considerations is presented. This initial period is a part of the real technological process, when a FSW instrument reaches the touchdown point with the workpiece, preheats the contact zone, but does not penetrate the material. This approach enables an insight into FSW process on a level, where the system parameters are known and an experimental evaluation of theoretical results is possible. Beside these considerations an analytical estimation of the power input of dry and viscous friction during instrument rotation and translation in parallel is shown. These phenomena are investigated including friction between a FSW instrument and a workpiece. The influences of both dry and viscous friction are studied. Experiments have been done to validate the calculated results.     

Periodic solutions of difference equations

Consider the difference equation

(∗)