Indeterminacy of a dry friction problem with viscous damping involving stiction

Abstract

We study a usual friction law for one degree of freedom solid where the dynamical friction coefficient is smaller than the static friction coefficient. We show that this law can be dangerous, because it can lose all predictive power. We first study a passage to the limit to prove that the limit is strongly depending on the data. We secondly study the obtained limit law, which is written with a non maximal monotone multivalued graph; this property implies that the differential inclusion possesses a set of solutions, whose diameter is too big to know the solution with precision. Moreover, the associated numerical scheme does not have a unique solution.

The Painlevé paradox in contact mechanics

“…This resolution is specious though because it is easy to formulate mechanisms for which µ P,min is significantly smaller. For example, if we allow the rod in the CPP to be non-uniform, then µ P,min is a function of the radius of gyration r. Taking the limit that all the mass is concentrated at the centre of mass, then r → 0 and the formula (5) shows that µ P,min becomes vanishingly small in this limit.…”

“…Instead, we shall attempt to understand the dynamical consequences of the Painlevé paradox from a mathematical modelling point of view, building on recent understanding of the theory of piecewise-smooth dynamical systems see [20,42,16,15], In fact, it is well known that piecewise-smooth systems of Filippov-type [20] can exhibit nonuniqueness or nonexistence of solutions, see e.g. [20,5,34].…”

Dynamics and bifurcations of nonsmooth systems: A survey

“…Regularisation of impact oscillators is discussed in Ivanov [182,183]. Bastien and Schatzman [26] discuss the differential inclusions that occur in the limit of the regularisation processes for dry friction oscillators and analyse the size of integral funnels of these inclusions.…”

?nalysis of energy dissipation in the impact problems of two or more bodies

“…Паралелно са тим алатом, у исто време започиње нагли развоj неглатке математичке анализе, а тиме и развоj неглатке мехнике, о чему говоре референце дате у раду Демjанова [2002,28], или монографиjама Фаjфера и Глокера [1996,73], Фремона [2002,38] и Глокера [2001,41]. Неглатка динамика jе тема коjом се баве Пидбуф и Керафел [2000,75], Матросов [2001,67], Штудер и Глокер [2007,89], Бастеjн и Ламарк [2007,12], Маркеев [2008,66], Бастеjн и Шацман [2008,13], Акари и Брољато [2008,1]. Важно jе напоменути да се тек употребом неглатких вишевредносних функциjа дошло до модела коjи коректно описуjе заустављање механичких система у коначном времену, чиме се избегаваjу проблеми сингуларног времена, видети Леин и Ниjмеиjер [2004,62], Брољата и коаутора [2002,17], или Флореса и др.…”