Distributed-Parameters Base Modeling and Vibration Analysis of Microcantilevers Used in Atomic Force Microscopy

Friction-induced self-sustained oscillation result in a very robust limit cycle that characterizes stick-slip motion. This motion should be avoided because it creates unwanted noise, diminishes accuracy, and increases wear. The stick-slip motion produced by a mass-spring-damper on a moving belt is analyzed using Lyapunov second method, which is based on constructing a positive definite function and checking the condition for which its time derivative is negative semi-definite. From this condition an estimate of the amplitude of the velocity of the limit cycle of the stick-slip motion is obtained. This estimate is found to be the zero of a certain function derived from the Coulomb friction model. An estimate of the amplitude of the displacement is also found. It is shown that the simulation results of the amplitude and the estimated amplitude are in a good match.