A uniformly valid asymptotic solution of Reynolds’s equation: the finite journal bearing with small clearance

Abstract

An asymptotic expansion in terms of small δ = 1 - e , where e is the eccentricity, is constructed for the pressure distribution inside a finite journal bearing. Each term in the expansion is obtained in the form of an infinite series of eigenfunctions, and is singular along the line of minimum clearance. The method of inner and outer expansions is used to obtain a uniformly valid solution across this line and the matching determines the eigenvalues, which are obtained as series in powers of δ . The thrust and frictional couple per unit length are evaluated but are singular at the ends of the bearing. Another application of the method of inner and outer expansions is necessary but in this case the inner expansion cannot be obtained in terms of well known functions. It is however shown that the inner solution does not contribute to the first term in the asymptotic expansion of the total thrust and frictional couple. The ratio of the total thrust to that for an equivalent infinite bearing is 1+ βδ ½ / t [In δ ½ – f ( t )]+ O ( δ ½ ), where t is the slenderness ratio, f ( t ) is a known function of t and β is a known constant. Some discussion is given of the relevance of this model to the physical situation and a comparison is made with numerical results.