Nonsmooth Dynamics by Path Integration: An Example of Stochastic and Chaotic Response of a Meshing Gear Pair

The probability density function (PDF) of the solution process of a nonlinear stochastic differential equation (SDE) is found in this paper using the path integration technique. The SDE is a piecewise linear system representing a model of an imperfectly mounted spur gear pair with a small stochastic noise added to the driving force. It is known that the system model for a particular choice of parameters shows chaotic behavior (Kahraman and Singh, 1990, “Non-Linear Dynamics of a Spur Gear Pair,” J. Sound Vibrat., 142(1), pp. 49–75). The PDF is compared with the Poincaré map of the deterministic system and it is shown that the stochastic and deterministic attractors are very similar. Then it is shown that although the stochastic attractor appears clearly after just a few iterations, the probability density over the attractor depends on the initial condition. The system does converge to one unique periodic PDF eventually but the convergence is fairly slow. However, the transient is almost periodic with a period that is twice that of the forcing, which can be utilized to obtain a much higher convergence rate. The advantage of using a SDE to study this rattling problem is that it can provide a very detailed picture of the dynamics and the most likely states of the system can immediately be identified.