A numerical approach to predict cavitation and contact in misaligned journal bearings

Résumé

This dissertation presents a numerical approach to reach the equilibrium position of a misaligned journal bearing with radial loading. We consider the hydrodynamic Reynolds equation with cavitation phenomenon, through both Reynolds and Elrod–Adams models. The device consists of an external cylinder surrounding a rotating shaft, both separated by a lubricant to prevent contact. The unknowns are the pressure of the lubricant satisfying a proper cavitation model, and the shaft position which could be misaligned. We couple the hydrodynamic model to Newton’s second law, which describes the position of the shaft. This is consid- ered an inverse problem where the coefficient of the equation is given by the unknown shaft position, which depends on the pressure. Considering the Reynolds cavitation model, we solve the direct problem minimiz- ing a convex functional. We use a preconditioned conjugate gradient method modified with projection and restarting strategies to account for cavitation. To solve the associated inverse problem we use an interior, trust-region algorithm subject to bounds, through which we transform the constrained optimization problem into an unconstrained one. The simulations show the existence of contact points for finite loading when misalignment occurs. We provide a mathematical proof for the point contact case. Similarly, we present the admissible range of misalignment angle projections for prescribed values of the shaft eccentricity and angular coordinate. With the Elrod–Adams model, we approximate the Heaviside function by a third order Hermite polynomial and solve the direct problem by minimizing a convex and lower semi continuous func- tional. To solve the associated inverse problem we introduce Ant Colony Optimization, an approach inspired by the ants’ foraging behaviour and communication through pheromone trails. This approach, originally de- veloped to solve discrete optimization problems has been now successfully extended to continuous domains. Numerical results are presented to verify the different numerical approaches. To validate the proposal the predicted pressure values, in the bearing mid-plane, are compared to published experimental data.