Stick-slip transition and dynamic cyclic response of friction damped systems

This thesis is subdivided into two parts: in the first we analyze the transition from static to dynamic friction with some emphasis on the implication of using more refined friction laws (with respect to the simple Coulomb model) while in the second part we study the cyclic response of dynamical systems that experience friction. Particularly, in the first part we will take inspiration from some recent experiments from the group of Prof. Fineberg to tackle some partial slip contact problems, with the idea in mind of providing analytical models that can, in some extent, interpreter some of the numerous experimental evidences that came from the direct observation of the sliding phenomena. In the chapters 1-2 a brief introduction of the equations that govern the contact of elastic bodies and the experimental test rig used in the experiments is presented. In chapter 3 the partial slip problem of a flat square-ended punch pressed against an half-plane and tangentially loaded above the contact interface is studied, then a FEM of the Prof. Fineberg experimental test rig will be proposed to avoid the hypothesis of half-plane elasticity, with good agreement between numerical and experimental results. In chapter 4 the implications of using a slip weakening friction law instead of the classical Coulomb law are discussed and an energetic criterion for slip inception is derived, which we will call "Griffith friction model". In chapter 5, using this "Griffith friction", the partial slip problem for different plane geometries (power law punches and sinusoidal profile) is solved. In the second part of the thesis we will focus our attention on the dynamic response of mechanical systems subjected to friction. In chapter 7 a very simple model of structure subjected to dry friction is studied, constituted by a single degree of freedom system subjected to a periodical tangential excitation and a (possibly) varying normal load. First we compare the quasi-static solution with the dynamic solution in the limit of very low excitation frequency, then we study (in the bounded regime) how the peak displacement and dissipation is related to the phase shift between the normal and the tangential load. In chapter 8 the dynamical behavior of a mass-spring-viscous damper structure linked to a massless Coulomb damper is studied with attention to the regime that minimize the vibration amplitude of the mass. Finally in chapter 9, we study a friction-excited nonlinear oscillator chain, where a polynomial nonlinearity is introduced in the system. We focus our attention on the multiplicity of solutions that are proven to exist in certain parameter rangeS which leads to a bifurcation pattern similar to the snaking bifurcations. In the end conclusions and possible developments of the present work are proposed.