Periodic solutions of dry friction problems

Dry friction problems lead to discontinuous differential equations, e.g. to

Periodic solutions for neutral coupled oscillators network with feedback and time-varying delay

This paper is considered to be about the existence of periodic solutions for neutral coupled oscillators network with feedback control and time-varying delay (NCONFT). Based on the systematic method which is firstly applied for NCONFT and consisting of coincidence degree theory, graph theory, and Lyapunov method, some sufficient criteria are obtained to verify the existence of periodic solutions for NCONFT. What’s more, how coupling topology, feedback control, and time-varying delay affect the existence of periodic solutions for NCONFT can be shown by these sufficient criteria. Finally, a numerical simulation is offered to illustrate the effectiveness of our results.     

Bifurcation analysis of a stochastic non-smooth friction model

Non-smooth systems with stochastic parameters are important models e.g. for brake and cam follower systems. They show special bifurcation phenomena, such as grazing bifurcations. This contribution studies the influence of stochastic processes on bifurcations in non-smooth systems. As an example, the classical mass on a belt system is considered, where stick-slip vibrations occur. Measurements indicate that the friction coefficient which plays a large role in the system behavior is not deterministic but can be described as a friction characteristic with added white noise. Therefore, a stochastic process is introduced into the non-smooth model and its influence on the bifurcation behavior is studied. It is shown that the stochastic process may alter the bifurcation behavior of the deterministic system. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation analysis of a stochastic non-smooth friction model

Non-smooth friction systems such as systems with dry friction show several bifurcation phenomena. The discontinuity of these so called slip-stick vibrations makes these systems interesting and there has been a lot of research in this field, see for example Hinrichs [1]. Due to the non-smooth friction force even the deterministic system shows a rich bifurcation behavior. Measurements indicate that the friction coefficient which plays a large role in the system behavior is not deterministic but can be described as a friction characteristic with added white noise. Therefore, the stochastic characteristic is introduced into the non-smooth system and the change of the bifurcation behavior is studied. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Method of homogeneous solutions under non-smooth loads

We consider a two-dimensional problem concerned with static deformation of an elastic halfstrip under the action of a self-equilibrated load at its end. A method is proposed for regularization of series over homogeneous solutions in the case of non-smooth prescribed loads. An example is given of a load for which divergence obtains for a series over homogeneous solutions for the stress field at the end of the halfstrip.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 111–116, 1988.     

On a class of non-smooth dynamical systems: a sufficient condition for smooth versus non-smooth solutions

In this paper we present a possible classification of the elements of a class of dynamical systems, whose underlying mathematical models contain non-smooth components. For this purpose a sufficient condition is introduced. To illustrate and motivate this classification, three nontrivial and realistic examples are considered.     

The periodic solutions of certain non-linear oscillators

A method is presented for the analysis of single degree of freedom non-linear oscillators characterized by second order non-linear ordinary differential equations containing a parameter ε which need not be small. The method is based on expanding the solution as a sine power series. We explain our results as an application to the Duffing oscillators.     

Remark on periodic solutions of nonlinear oscillators

We contribute to the method of trigonometric series for solving differential equations of certain nonlinear oscillators.     

Periodic solutions of difference equations

We give an overview of results on the existence of periodic solutions of difference equations that have been obtained in the last two decades. The survey covers both ordinary and Volterra difference systems. Some extensions and generalizations of known result are also presented.     

Periodic solutions of evolutionary equations

An analog of L. Cesari's method for a very general class of evolutionary equations is investigated. A result is proved concerning the coincidence of rotation of a Cesari field with the rotation of a translation operator (on boundaries of naturally corresponding regions), and Galerkin's method of obtaining periodic solutions is studied. The results obtained are new for ordinary differential equations.Translated from Matematicheskie Zametki, Vol. 9, No. 6, pp. 651–662, June, 1971.     

Periodic solutions of difference equations

Consider the difference equation

(∗)     

Attractivity theory for second order non-smooth dynamical systems with application to dry friction

In this paper, we study the attractivity properties of the set of stationary solutions for a general class of second order non-smooth dynamical system involving friction term. Sufficient conditions for the local attractivity of the set of stationary solutions are given in the case of dry friction and negative viscous damping. An estimation of the attraction domain is also given in this case. Applications can be found in unilateral mechanics.