Application to nonlinear mechanical systems with dry friction: hard bifurcation in SD oscillator

Abstract

The stochastic model approach of a nonlinear non-smooth dynamical system with the probable occurrence of stochastic P-bifurcations is devoted. The response probability density functions (PDF) for the stationary measure of a smooth and discontinuous oscillator under moving loads belt frictions is constructed. The appearance of abrupt changes and unpredictable events illustrate the complexity of the system. The stationary measure varies continuously with system’s parameters and describes various kinds of catastrophic events. In light of these facts, the behaviour of the “stochastic attractors” is examined through the stationary solution of the PDF. According to Zeeman, in the phenomenological approach in the presence of noise, “a change in character of the density function as a parameter is varied is known as p- bifurcation”. Numerous new events unique to non-smooth systems are observed under slight variation of system’s parameters. Discontinuous bifurcations are defined as the “hard bifurcations” that were the subject of Catastrophe theory. Peaks numbers increase as coefficient of friction µ (or smoothness parameter α) increases. Numerical simulations are presented that provide insights into the dynamics of these oscillators.