The Accuracy of Short Bearing Theory for Newtonian Lubricants

Abstract

The accuracy of the short bearing approximation is analyzed in this discussion. The results apply to Newtonian lubricants, and they can also be extended to non-Newtonian power-law lubricants. Reynolds’ lubrication equation is first solved by applying a regular perturbation expansion in pressure to the π film journal bearing; after this, a matched asymptotic expansion is applied to linear slider bearings. Approximate solutions are then compared with numerical solutions, to estimate the accuracy of the short bearing approximation. Finally, the accuracy of fluid film pressures predicted via short bearing theory is shown to depend upon three factors: the bearing aspect ratio, eccentricity ratio, and the partial-arc extent. Ocvirk’s short bearing series approximation—for finite bearing aspect ratio—is shown to become singular in the limit as the eccentricity ratio approaches unity. The one term π film Ocvirk solution is shown to be a nonconservative approximation to the journal bearing load capacity. A method to extend the accuracy of the short bearing approximation for partial-arcs and slider bearings is then presented.

Approximate analytical solution to Reynolds equation for finite length journal bearings

“…The general finite width gas slider bearing has also been studied in order to find a formal explicit uniformly valid asymptotic representation of the pressure with a low order error over the entire bearing [17][18][19]. In the case of short slider bearings, the descriptions have been done by rectifying the infinitely short bearing theory and performing an asymptotic analysis to correct the pressure field near both, the leading and the trailing edges of the slider [20,21], and by solving an Euler-Lagrange equation using a small aspect ratio singular perturbation approach [22]. The infinitely short bearing theory has been also applied to cylindrical partial-arc short bearings using a matched asymptotic perturbation method to correct the inaccuracies caused by the azimuthal edge pressure boundary conditions [20,21,23].…”

“…In the case of short slider bearings, the descriptions have been done by rectifying the infinitely short bearing theory and performing an asymptotic analysis to correct the pressure field near both, the leading and the trailing edges of the slider [20,21], and by solving an Euler-Lagrange equation using a small aspect ratio singular perturbation approach [22]. The infinitely short bearing theory has been also applied to cylindrical partial-arc short bearings using a matched asymptotic perturbation method to correct the inaccuracies caused by the azimuthal edge pressure boundary conditions [20,21,23]. The finite length JB with high-eccentricity has been analyzed in terms of inner and outer asymptotic expansions to get the pressure distribution inside the fluid film, load-carrying capacity, and frictional loss [24].…”

Nanofluids for Performance Improvement of Heavy Machinery Journal Bearings: A Simulation Study

“…where η is the coefficient of the lubricant viscosity in Pa s, U is journal surface velocity, z is the coordinate in axial direction, and = / is the eccentricity ratio. The closed form solution to Equation (1), is available as a function of slenderness ratio = L ⁄ D. For short bearings with low L/D ratio, the solution known as Ocvirk's solution, Equation (2), and for long bearings with high L/D ratio, the closed form solution known as the Sommerfeld solution, Equation (3) [31]: The analytical solution PL is valid when the slenderness ratio = L/D is bigger than 2.0, while the solution Ps is applicable for a L/D ratios smaller than 0.25 [26,31,34]. For the cases where the L/D ratio range is within 0.25 to 2.0, both approximations are inaccurate.…”

“…For these cases, there are a few models that correct Equation (4) based on the slenderness ratio. One of the best is the empirical pressure weighting correction factor proposed by Hirani et al [31] and Equation (4) The analytical solution P L is valid when the slenderness ratio = L/D is bigger than 2.0, while the solution P s is applicable for a L/D ratios smaller than 0.25 [26,31,34]. For the cases where the L/D ratio range is within 0.25 to 2.0, both approximations are inaccurate.…”

Static and dynamic characteristics of short wavy journal bearings

“…Short bearing analysis has been considered by various researchers as is reported in previous studies. [3][4][5][6] These studies have studied the accuracy and the role of axial edge effects and cavitation in lubrication of short journal bearings. To improve the static and dynamic characteristics of hydrodynamic journal bearings, many researchers have resorted to non-circular geometries.…”