Modified Reynolds Equation for Aerostatic Porous Radial Bearings With Quadratic Forchheimer Pressure-Flow Assumption

2007 ◽  
Author(s):  
Rodrigo Nicoletti ◽  
Zilda C. Silveira ◽  
Benedito M. Purquerio
Keyword(s):  
Mathematical Modeling ◽  
Porous Medium ◽  
Linear Model ◽  
Dynamic Characteristics ◽  
Reynolds Equation ◽  
Dimensional Parameter ◽  
Pressure Flow ◽  
Darcy Model ◽  
Linear Effects ◽  
Modified Reynolds Equation

The mathematical modeling of aerostatic porous bearings, represented by the Reynolds equation, depends on the assumptions for the flow in the porous medium. One proposes a modified Reynolds equation based on the quadratic Forchheimer assumption, which can be used for both linear and quadratic conditions. Numerical results are compared to those obtained with the linear Darcy model. It is shown that, the non-dimensional parameter Φ, related to non-linear effects, strongly affects the bearing dynamic characteristics, but for values of Φ > 10, the results tend to those obtained with the linear model.

Modified Reynolds Equation for Aerostatic Porous Radial Bearings With Quadratic Forchheimer Pressure-Flow Assumption

2008 ◽  
Vol 130 (3) ◽  
Author(s):  
Rodrigo Nicoletti ◽  
Zilda C. Silveira ◽  
Benedito M. Purquerio
Keyword(s):  
Reynolds Equation ◽  
Load Capacity ◽  
Loading Capacity ◽  
Global Approach ◽  
Pressure Flow ◽  
Stiffness Coefficients ◽  
Porous Bearing ◽  
Bearing Load ◽  
Modified Reynolds Equation ◽  
Precision Machines

Aerostatic porous bearings are becoming important elements in precision machines due to their inherent characteristics. The mathematical modeling of such bearings depends on the pressure-flow assumptions that are adopted for the flow in the porous medium. In this work, one proposes a nondimensional modified Reynolds equations based on the quadratic Forchheimer assumption. In this quadratic approach, the nondimensional parameter Φ strongly affects the bearing load capacity, by defining the nonlinearity level of the system. For values of Φ>10, the results obtained with the modified Reynolds equation with quadratic Forchheimer assumption tend to those obtained with the linear Darcy model, thus showing that this is a more robust and global approach of the problem, and can be used for both pressure-flow assumptions (linear and quadratic). The threshold between linear and quadratic assumptions is numerically investigated for a bronze sintered porous bearing, and the effects of bearing geometry are discussed. Numerical results show that Φ strongly affects the bearing loading capacity and stiffness coefficients.

Derivation of Modified Reynolds Equation—A Porous Media Model

1999 ◽  
Vol 121 (4) ◽  
pp. 823-829 ◽  
Author(s):  
Wang-Long Li
Keyword(s):  
Porous Media ◽  
Reynolds Equation ◽  
Stokes Equations ◽  
Fluid Film ◽  
Stress Jump ◽  
Darcy Model ◽  
Viscous Shear ◽  
Porous Media Model ◽  
Modified Reynolds Equation ◽  
Film Interface

In this study, a porous media model is developed which can be applied to thin film lubrication problems. The microstructure of bearing surfaces is modeled as porous layers attached to the impermeable substrate. The Brinkman-extended Darcy equations and Stokes’ equations are utilized to model the flow in the porous region and fluid film region, respectively. The stress jump boundary condition at the porous media/fluid film interface and effects of viscous shear are included in deriving the modified Reynolds equation. The present model can correct and modify a previous study based on the Darcy model with slip-fiow effects or another based on the Brinkman-extended Darcy model with stress continuity at the porous media/fluid film interface. In the results, the effects of material properties: viscosity ratio (αi2), thickness of porous layer (Δi), permeability (Ki), stress jump parameter (βi), on the velocity distributions, and performance of one-dimensional converging wedge problems are discussed.

Two Dimensional Modified Reynolds Equation Including Pressure Dependent Viscosity Effect

2012 ◽  
Author(s):  
Jung Gu Lee ◽  
Alan Palazzolo
Keyword(s):  
Reynolds Equation ◽  
Elastohydrodynamic Lubrication ◽  
Fluid Film ◽  
Maximum Pressure ◽  
Elastic Solids ◽  
Pressure Distributions ◽  
Pressure Dependent Viscosity ◽  
Modified Reynolds Equation ◽  
Dependent Viscosity ◽  
Modified Equation

The Reynolds equation plays an important role for predicting pressure distributions for fluid film bearing analysis, One of the assumptions on the Reynolds equation is that the viscosity is independent of pressure. This assumption is still valid for most fluid film bearing applications, in which the maximum pressure is less than 1 GPa. However, in elastohydrodynamic lubrication (EHL) where the lubricant is subjected to extremely high pressure, this assumption should be reconsidered. The 2D modified Reynolds equation is derived in this study including pressure-dependent viscosity, The solutions of 2D modified Reynolds equation is compared with that of the classical Reynolds equation for the ball bearing case (elastic solids). The pressure distribution obtained from modified equation is slightly higher pressures than the classical Reynolds equations.

A Review on Active Bearings and Perspectives of Using Them in Rotating Machinery

2014 ◽  
Vol 630 ◽  
pp. 181-187
Author(s):  
Denis Shutin ◽  
Leonid Savin ◽  
Alexander Babin
Keyword(s):  
Mathematical Modeling ◽  
Artificial Intelligence ◽  
Control System ◽  
Motion Control ◽  
Dynamic Characteristics ◽  
Rotating Machinery ◽  
Fluid Film ◽  
Motion Control System ◽  
Fluid Film Bearings ◽  
System Components

The paper examines the issues of improving the rotor units by means of using support units with actively changeable characteristics. An overview of the known solutions related to the use of active bearings in various types of turbomachinery is provided. A closer look is given at the design and features of active radial bearings, the main elements of which are fluid film bearings. The results of mathematical modeling of active hybrid bearings are presented. The prospects of the use of this type of supports to improve the dynamic characteristics of rotating machinery, including reducing vibrations caused by various factors, are analyzed. Promising directions of development of active bearings are considered, which primarily involves the modification of system components and rotor motion control system algorithms, including intelligent technologies and artificial intelligence methods.

Roughness Influence on Turbulent Flow Through Annular Seals

1994 ◽  
Vol 116 (2) ◽  
pp. 321-328 ◽  
Author(s):  
Victor Lucas ◽  
Sterian Danaila ◽  
Olivier Bonneau ◽  
Jean Freˆne
Keyword(s):  
Surface Roughness ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Turbulent Flow ◽  
Dynamic Characteristics ◽  
Reynolds Equation ◽  
Mixing Length ◽  
Local Shear ◽  
Wall Shear ◽  
Local Shear Stress

This paper deals with an analysis of turbulent flow in annular seals with rough surfaces. In this approach, our objectives are to develop a model of turbulence including surface roughness and to quantify the influence of surface roughness on turbulent flow. In this paper, in order to simplify the analysis, the inertial effects are neglected. These effects will be taken into account in a subsequent work. Consequently, this study is based on the solution of Reynolds equation. Turbulent flow is solved using Prandtl’s turbulent model with Van Driest’s mixing length expression. In Van Driest’s model, the mixing length depends on wall shear stress. However there are many numerical problems in evaluating this wall shear stress. Therefore, the goal of this work has been to use the local shear stress in the Van Driest’s model. This derived from the work of Elrod and Ng concerning Reichardt’s mixing length. The mixing length expression is then modified to introduce roughness effects. Then, the momentum equations are solved to evaluate the circumferential and axial velocity distributions as well as the turbulent viscosity μ1 (Boussinesq’s hypothesis) within the film. The coefficients of turbulence kx and kz, occurring in the generalized Reynolds’ equation, are then calculated as functions of the flow parameters. Reynolds’ equation is solved by using a finite centered difference method. Dynamic characteristics are calculated by exciting the system numerically, with displacement and velocity perturbations. The model of Van Driest using local shear stress and function of roughness has been compared (for smooth seals) to the Elrod and Ng theory. Some numerical results of the static and dynamic characteristics of a rough seal (with the same roughness on the rotor as on the stator) are presented. These results show the influence of roughness on the dynamic behavior of the shaft.

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