This paper presents a theoretical analysis of the vibration response of a rotating rigid shaft supported by two radial deep‐groove lubricated ball bearings. The bearings and their oil films are approximated to a set of non‐linear elastic springs and dampers rotating relative to the shaft when it is subjected to a rotating unbalance or inner race surface waviness. Under gravity load, the shaft's initial vibration is damped down to a limit cycle operating at a quasi‐simple harmonic frequency of small amplitude. Rotating unbalance and surface features introduce further significant frequencies which influence the output response.

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The Vibrations of Radial Ball Bearings

The regime known as ‘mixed’ lubrication lies between those of hydrodynamic and elastohydrodynamic. This paper reviews work on this subject, in order to recognise the key areas of progress, and of ignorance. Film thickness and surface roughness are discussed, to conclude that below certain values of λ, there are limitations to earlier models. Computational solution of the micro-EHD problem has largely succeeded, as well as more analytical approaches. The mixed lubrication regime has been studied by electrical contact resistance, friction measurement, and, more recently, analysis has been aided by optical interferometry — the implications of this work, and future prospects, are addressed.

Mixed lubrication — an overview

This paper presents experimental measurements of the approach and rebound of a particle colliding obliquely with a wall in a viscous fluid. Steel and glass particles 12.7 mm in diameter were used. The experiments were performed using a thick Zerodur wall (a glass-like material) with various mixtures of glycerol and water. Normal and tangential coefficients of restitution were defined from the ratios of the respective velocity components at the point of contact just prior to and after impact. These coefficients account for losses due to lubrication effects and inelasticity. A third parameter, a coefficient of sliding friction, provides a measure of the tangential force acting on the particle as it slides during a collision. 

Oblique collisions in a fluid are qualitatively similar to oblique collisions in a dry system, with a lowered friction coefficient dependent on surface roughness. For smooth surfaces the friction coefficient is drastically reduced due to lubrication effects. A theoretical model that takes into account the dependence of viscosity on pressure is proposed to explain the observed tangential force acting on a smooth sphere during an oblique collision. The model relies on an inferred uniform temperature increase within the lubrication layer, a consequence of viscous heating during impact. The tangential force felt by the particle is expressed as a friction coefficient dependent on the viscosity within the lubrication layer. The viscosity increases owing to pressure effects and decreases owing to thermal effects. For rough surfaces the friction coefficient is comparable to that measured in dry systems, since the surface asperities may interact with each other through the lubrication layer.

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Oblique particle–wall collisions in a liquid

Elastohydrodynamic (EHD) film thickness measurements using optical interferometry have been made of artificially produced dents and grooves under rolling and sliding conditions. These measurements are compared to stylus traces of the dent and groove profiles to determine the local deformation associated with micro-EHD pressure generation. The surface geometry associated with the dents and grooves is seen to become intimately involved in the lubrication process itself, creating local pressure variations that substantially deform the local surface geometry, particularly under sliding conditions. The rolling results have implications concerning surface initiated fatigue and the sliding results show clearly the EHD surface interactions that must occur prior to scuffing failure. Presented as an American Society of Lubrication Engineers paper at the ASLE/ASME Lubrication Conference in Minneapolis, Minnesota, October 24–26, 1978

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Elastohydrodynamic Film Thickness Measurements of Artificially Produced Surface Dents and Grooves

Elastohydrodynamic film thickness measurements of artificially produced surface dents and grooves. [using optical interferometry

The Hydrodynamic Lubrication of Rough Bearing Surfaces of Finite Width

The Hydrodynamic Lubrication of Rough Journal Bearings

: The elastohydrodynamic problem of normal approach of two spherical bodies is studied and the lubrication and elasticity equations governing this type of motion are established. Numerical solutions to the general case accounting for elastic deformation of the bodies and pressure dependent viscosity are presented. (Author)

H. Christensen

Papers

The Hydrodynamic Lubrication of Rough Bearing Surfaces of Finite Width

The Hydrodynamic Lubrication of Rough Journal Bearings

Elastohydrodynamic Theory of Spherical Bodies in Normal Approach