Dynamics of coupled mechanical oscillators with friction

Complex phenomena such as stick-slip vibrations, chaos and self-organized dynamics are frequently encountered in several mechanical systems with friction. Some applications include control of robot manipulators, distribution of earthquakes, suspension dynamics in vehicles, among others. These systems are strongly nonlinear. Spring-mass oscillators with friction have emerged as a simple jet effective model capturing the dynamics of much more complex system. In this dissertation, we study stability and dynamics of single and coupled mechanical oscillators with friction, mathematically described by differential equations with discontinuous right-hand sides. One particular problem in discontinuous systems is the computation of the basins of attraction of their stable equilibria or other attractors; for example, they provide important information about complex behavior caused by friction or damping, useful in the design of mechanical devices. To cope with this problem, we implemented an algorithm for the computation of basins of attraction in discontinuous systems based on the Simple Cell Mapping method, which has been evaluated via a set of representative applications. In the second part of the thesis, a piecewise smooth analysis of two coupled oscillators was carried out, finding out some conditions that guarantee the stability of the sliding dynamics in the presence of one or more intersecting surfaces. Finally, the dynamics of a network of N mechanical oscillators was studied from the point of view of synchronization, where the goal was to steer the positions and velocities of each oscillator in the network towards a common behavior. In particular, an extensive numerical analysis for studying synchronization in chaotic friction oscillators was performed, characterizing the influence of dynamic coupling and providing an estimation of the synchronization region in terms of the coupling parameters. Initially, we considered the simple case of two coupled oscillators, then we extended the analysis to the case of larger networks of coupled systems with different network topologies. Moreover, preliminary analytical results of the convergence on a network of N friction oscillators based on contraction analysis are investigated. The results were also validated through a representative example.