Effect of random forcing on fluid-lubricated bearing

Abstract

A model for a fluid lubricated bearing is derived for operation under conditions where external forces are subject to random fluctuations that may act to destabilise the bearing. The fluid flow through the bearing is described by a Reynolds equation for incompressible flow and is coupled to the axial displacement of the bearing faces as modelled by spring-mass-damper systems. Representative dynamics of a highly rotating bearing subject to external potentially destabilising random forcing is developed. An external force characterised by a noise term is imposed on the rotor, where both white noise and coloured noise are considered. For industrial applications it is important to evaluate potential bearing failure that can arise when the face clearance becomes sufficiently small. Therefore, a quantity of interest is the average time for the face clearance to reach a prescribed tolerance. A computational technique to evaluate the bearing characteristics is implemented based on a simple random walk for a Dirichlet problem for a linear parabolic partial differential equation combined with a Monte Carlo technique. Results of numerical experiments are presented, to give indicative predictions of possible face contact, which has the potential to result in bearing failure.

Effect of Uncertainty in External Forcing on a Fluid Lubricated Bearing

“…A similar trend is seen for coloured noise, with α = 100. Investigations show increasing the coloured noise parameter α, results in the average time at which the face clearance becomes as small as the prescribed tolerance E[τ δ ] tending to the values obtained for white noise [19]; results for white noise and coloured noise in Figure 5 are similar. These results can help aid bearing design by giving an indication of average bearing lifetime for types of external disturbances and force coupling parameters.…”

“…The associated solution notation for ( 14) is g(t) = g s,g0,z0 (t), z(t) = z s,g0,z0 (t), t ≥ s; g(s) = g 0 , z(s) = z 0 are the initial condition at t = s. If s = 0 the we write g g0,z0 (t), z g0,z0 (t) for compactness. Existence and uniqueness of the solution to (14), as well as boundedness of its moments was shown in [19]; this analysis formally demonstrated that g g0,z0 (t) can never become zero, thus that a positive face clearance is always maintained, although it may become arbitrary small. The time at which the face clearance g(t) first reaches a small given tolerance, δ > 0, is of interest.…”

“…These random variables are independent of the Wiener process w(t), t ≥ 0. The expectation of a quantity of interest can be expressed, for some functions φ and f , as [19]:…”

“…Using the Ornstein-Uhlenbeck process, the following system of SDEs is obtained (see, e.g. [15,17,19]):…”

“…Analogous to the white noise case, we obtain an algorthim for approximating (15) based on simple random walks and the Monte Carlo technique (see further details in [19]).…”