On the dynamics of a high-speed coned fluid-lubricated bearing

An incompressible air-flow model for a fluid film bearing is derived using a modified Reynolds equation for the thin-film dynamics of a rapidly rotating rotor and stator. Mathematical and numerical modelling is applied to the coupled processes of the fluid flow through the bearing and the axial motion of the rotor and stator. This work focuses on extending previous studies to incorporate the dynamics of a coned rotor operating at high speeds and an incompressible lubrication approximation. The dynamics of fully coupled, unsteady bearing motion and associated forcing of the rotor with axial periodic oscillations are studied. The axial motion of the stator is modelled as a spring–mass–damper system that responds to the rotor displacement through the film dynamics. In order to solve the modified Reynolds equation and stator equation simultaneously, a new variable is introduced, namely the time-dependent face clearance. This leads to explicit analytical expressions for the pressure and force in terms of the face clearance and the stator equation is transformed to a non-linear, second-order, non-autonomous, ordinary differential equation for the face clearance. Applying a transient solver gives solutions settling down to a stable periodic behaviour which motivates seeking a solver for periodic solutions. A Fourier spectral collocation scheme is derived to compute the periodic time-dependent face clearance. Both solvers have matching periodic solutions of O(1) with an absolute error of order of magnitude 10−5. The dynamics of the unsteady bearing are examined for a range of pressure gradients and configurations including an asymptotic investigation of small face clearance associated with a start-up transient. Results are provided relating to changes in the width of the bearing, strength of the spring holding stator to its housing, damping of the stator and strength of the force coupling and rotor mass. The dynamics of the bearing are also investigated relative to values of key system parameters including the coning of the rotor, rotation speed and value of the bearing squeeze number. A parameter investigation is undertaken to highlight ideal bearing configurations to maximize the load-carrying capacity, fluid stiffness and damping.