Stochastically perturbed rotating shafts

The objective of this work is to study the long term effects of small symmetry-breaking, dissipative and noisy perturbations on the dynamics of a rotating shaft. Hamilton's principle is used to derive the equations of motion and a one mode Galerkin approximation is applied to obtain a two-degree-of-freedom (four dimensional) model. A stochastic averaging method is developed to reduce the dimension of this four dimensional system. Making use of the interaction between the gyroscopic and dissipative forces and the separation of time scales, the original system is reduced to a one dimensional Markov process. Depending on the system parameters, the reduced Markov process takes its values on a line or a graph. For the latter case, the glueing conditions required to complete the description of the problem in the reduced space are derived. This provides a qualitatively accurate and computationally feasible description of the system. Analytical results are obtained for the mean first passage time problem. The stationary probability density is obtained by solving the Fokker Planck Equation(FPE). Finally, the qualitative changes in the stationary density as a result of varying the system parameters are described.