Warner–Sommerfeld Impedance and Mobility Maps

2015 ◽  
Vol 138 (3) ◽  
Author(s):  
S. Boedo ◽  
J. F. Booker
Keyword(s):  
Approximate Solution ◽  
Reynolds Equation ◽  
Graphical Representations ◽  
Film Pressure ◽  
Curve Fit ◽  
Leakage Factor

This note provides additional graphical representations and curve-fit expressions for characteristics of the Warner–Sommerfeld approximate solution of the Reynolds equation for cavitating journal finite bearings. In particular, curve-fit expressions for mobility components, end-leakage factor, and positive film pressure extent are provided in support of previously published graphical representations of the data.

A Calculation Method to Investigate the Effects of Geometric Parameters and Operational Conditions on the Static Characteristics of Aerostatic Spherical Bearings

2018 ◽  
Vol 141 (2) ◽  
Author(s):  
Hailong Cui ◽  
Huan Xia ◽  
Dajiang Lei ◽  
Xinjiang Zhang ◽  
Zhengyi Jiang
Keyword(s):  
Experimental Data ◽  
Calculation Method ◽  
Reynolds Equation ◽  
Geometric Parameters ◽  
Static Characteristics ◽  
Optimal Parameters ◽  
Film Pressure ◽  
Operational Conditions ◽  
Pressure Distributions ◽  
Calculation Results

In this paper, a calculation method based on matlab partial differential equations (PDE) tool is proposed to investigate the static characteristics of aerostatic spherical bearings. The Reynolds equation of aerostatic spherical bearings is transformed into a standard elliptic equation. The effects of geometric parameters and operational conditions on the film pressure, bearing film force, and stiffness are studied. The axial and radial eccentricities result in different film pressure distributions; the bearing film force and stiffness are significantly influenced by geometric parameters and operational conditions. The relative optimal parameters are confirmed based on the calculation results. A comparison between the numerical and experimental results is also presented. The highest relative error between the numerical results and the experimental data is 11.3%; the calculation results show good agreements with the experimental data, thus verifying the accuracy of the calculation method used in this paper.

Theoretical Study on Static Performance of Herringbone Grooved Aerodynamics Foil Bearing

2020 ◽  
Author(s):  
ZeDa Dong ◽  
Cheng Cheng ◽  
Fangcheng Xu
Keyword(s):  
Power Consumption ◽  
Reynolds Equation ◽  
Load Capacity ◽  
Structural Parameters ◽  
Operational Parameters ◽  
Film Pressure ◽  
Friction Power ◽  
Foil Bearings ◽  
Bearing Load ◽  
Herringbone Grooves

Abstract In this paper, the mathematical model of herringbone grooved aerodynamic foil bearings is established, and the finite difference method is used to obtain the discretized form of Reynolds equation. The static characteristics of bearings, such as film pressure, film temperature, are obtained by solving the Reynolds equation and energy equation. The bearing load capacity and friction power consumption are obtained by calculating the film thickness and film pressure distribution in the bearing gap. The influence of the bearing operational parameters, such as eccentricity and rotation speed, and the bearing structural parameters, such as groove width, groove depth ratio, groove number and helix angle, on the bearing load capacity and friction power consumption of bearings are analyzed. The methods of improving bearing load capacity and reducing friction power consumption are obtained. Simultaneously, by comparing the bearing load capacity and friction power consumption of herringbone grooved gas foil bearings and gas foil bearings (GFBs) without herringbone grooves, the influence of herringbone grooves on the bearing performance is obtained.

An Approximate Solution of Muijderman's Model for Performance Calculation of Spiral Grooved Gas Seal

2017 ◽  
Vol 139 (5) ◽  
Author(s):  
Wanjun Xu ◽  
Jiangang Yang
Keyword(s):  
Approximate Solution ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Kutta Method ◽  
Film Pressure ◽  
Runge Kutta Method ◽  
Decomposition Procedure ◽  
Adomian Decomposition ◽  
Runge Kutta ◽  
Performance Calculation

This paper presents an approximate solution of Muijderman's model for compressible spiral grooved gas film. The approximate solution is derived from Muijderman's equations by Adomian decomposition method. The obtained approximate solution expresses the gas film pressure as a function of the gas film radius. The traditional Runge–Kutta method is avoided. The accuracy of the approximate solution is acceptable, and it brings convenience for performance calculation of spiral grooved gas seal. A complete Adomian decomposition procedure of Muijderman's equations is presented. The approximate solution is validated with published results.

Nonlinear Effects in a Plain Journal Bearing: Part 1—Analytical Study

1991 ◽  
Vol 113 (3) ◽  
pp. 555-561 ◽  
Author(s):  
F. K. Choy ◽  
M. J. Braun ◽  
Y. Hu
Keyword(s):  
Equilibrium Position ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Fluid Film ◽  
Liquid Oxygen ◽  
Solution Technique ◽  
Space Applications ◽  
Linear Solution ◽  
Film Pressure ◽  
Linear And Nonlinear

Hydrodynamic/hydrostatic journal bearings have been widely used in various types of high speed rotating machinery. For space applications, the issue of using cryogenic fluids as working lubricants has steadily gained in significance. The primary goal of this paper is to model the nonlinearities that occur in a hydrodynamic journal bearing with both cryogenic and oil lubricants. Results will be examined through bearing fluid film pressure distribution and bearing linear and nonlinear stiffness characteristics. The numerical model that couples a variable property Reynolds equation with the dynamics of the rotor is solved by means of a finite difference solution technique. The procedure for the fluid film pressure solution involves an iterative scheme that solves the Reynolds equation coupled with the equations of state for liquid oxygen (LO2). The pressure curve is then integrated to calculate bearing supporting forces. A two-dimensional Newton-Raphson iteration method is used to locate the journal equilibrium position from which both linear and nonlinear bearing stiffness are evaluated by means of the small perturbation technique. The effects of load on the linear/nonlinear plain journal bearing characteristics are analyzed and presented in a parametric form. The relationship between the accuracy of the linear solution and the various orders (3rd, 5th, and 7th power for ΔX) of the nonlinear approximation are also discussed. The validity of both linear and nonlinear solutions at various distances from the journal equilibrium position is also examined. A complete parametric study on the effects of load, temperature, operating speed, and shaft misalignment will be given in Part 2 of this paper.

Effect of viscosity variation on non-Newtonian lubrication of squeeze film conical bearing having porous wall operating with Rabinowitsch fluid model

2018 ◽  
Vol 233 (7) ◽  
pp. 2538-2551 ◽  
Author(s):  
Pentyala Srinivasa Rao ◽  
Amit Kumar Rahul
Keyword(s):  
Carrying Capacity ◽  
Reynolds Equation ◽  
Porous Wall ◽  
Squeeze Film ◽  
Load Carrying Capacity ◽  
Film Pressure ◽  
Pressure Load ◽  
Load Carrying ◽  
Newtonian Case ◽  
Viscosity Variation

In this study, the effect of viscosity variation of non-Newtonian lubrication on squeeze film characteristics with porous and Rabinowitsch fluid for conical bearings is analyzed. The modified Reynolds equation representing the characteristics of non-Newtonian fluid with viscosity variation on the porous wall followed by the cubic stress law condition is invoked. For lubricant flow in a bearing clearance and in a porous layer Morgan–Cameron approximation is considered. A small perturbation technique is used to compute the pressure generation using modified Reynolds equation of lubrication. Approximate analytical solutions have been obtained for the squeeze film pressure, load-carrying capacity, squeeze film time, and center of pressure. The outcomes are displayed in diagrams and tables, which show that the effect of viscosity variation and porous wall on the squeeze film lubrication of conical bearings decreases film pressure, load-carrying capacity, and response time for the Newtonian case in comparison to the non-Newtonian case.

scholarly journals Non-Newtonian Effects on the Squeeze Film Characteristics between a Sphere and a Flat Plate: Rabinowitsch Model

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Udaya P. Singh ◽  
Ram S. Gupta
Keyword(s):  
Flat Plate ◽  
Carrying Capacity ◽  
Significant Variation ◽  
Reynolds Equation ◽  
Fluid Model ◽  
Squeeze Film ◽  
Load Carrying Capacity ◽  
Film Pressure ◽  
Load Carrying ◽  
Film Characteristics

The use of additives (polyisobutylene, ethylene-propylene, lithium hydroxy stearate, hydrophobic silica, etc.) changes lubricants’ rheology due to which they show pseudoplastic and dilatant nature, which can be modelled as cubic stress fluid model (Rabinowitsch fluid model). The present theoretical analysis investigates the effects of non-Newtonian pseudoplastic and dilatant lubricants on the squeezing characteristics of a sphere and a flat plate. The modified Reynolds equation has been derived and an asymptotic solution for film pressure is obtained. The results for the film pressure distribution, load carrying capacity, and squeezing time characteristics have been calculated for various values of pseudoplastic parameter and compared with the Newtonian results. These characteristics show a significant variation with the non-Newtonian pseudoplastic and dilatant behavior of the fluids.

Lubricant Inertia in Water Lubricated Bearings

2017 ◽  
Author(s):  
Xin Deng ◽  
Cori Watson ◽  
Brian Weaver ◽  
Houston Wood ◽  
Roger Fittro
Keyword(s):  
High Speed ◽  
Reynolds Equation ◽  
Current Status ◽  
Shear Stresses ◽  
Turbulent Regime ◽  
Fluid Inertia ◽  
Film Pressure ◽  
Inertia Effects ◽  
Inertia Model ◽  
Stiffness And Damping

Oil-lubricated bearings are widely used in high speed rotating machines such as those used in the aerospace and automotive industries. However, with some applications including underwater machinery and environmentally friendly applications, water lubricated bearings have become increasingly used. Due to the different fluid properties between oil and water — namely viscosity — the use of water increases the Reynolds numbers drastically and, therefore, makes water-lubricated bearings prone to turbulence and fluid inertia effects. In other words, the linear approximation of the fluid film reaction forces due to the stiffness and damping parameters — as suggested in the traditional Reynolds equation — is not adequate and should be amended to include lubricant added mass. This is because water-lubricated bearings exhibit large lubricant inertia forces on the order of viscous forces. Additionally, stiffness and damping coefficients should be calculated with the turbulence effects included. The aim of this study was to investigate the methodology of modifying the traditional Reynolds equation to include lubricant inertia effects. This paper reviews the current status of research in the lubricant inertia of bearings and explores the development of methodologies to modify the Reynolds equation to include lubricant inertia in bearings. The Reynolds equation is a partial differential equation governing the pressure distribution of thin viscous fluid films in lubrication theory. The thin film hypothesis is used to directly relate the bearing film thickness to the lubricant film pressure. Adding lubricant inertia to the Reynolds equation is vital to improving the accuracy of the bearing model and more specifically its film pressure which is essential to predicting load carrying capabilities. The film pressure relates the gradient of the velocity tensor through the Reynolds equation, and resulting shear stresses then allow the turbulent momentum equations to be written in terms of an eddy-viscosity value. An extended Reynolds equation should be developed which takes into account turbulence and both convective and temporal inertia. The most complete form of the temporal inertia effect model should be developed and applied to the turbulent regime, consisting of both primary and secondary temporal inertia terms. The convective inertia model follows Constantinescu’s approach. This analysis develops a lubricant inertia model applicable to water-lubricated bearings. The results of this study could aid in improving future designs and models of water-lubricated bearings.

Mixed Lubrication Analysis of Piston Pin Bearing in Diesel Engine With High Power Density

2011 ◽  
Author(s):  
Xiaoli Wang ◽  
Jingfang Du ◽  
Junyan Zhang
Keyword(s):  
Diesel Engine ◽  
Film Thickness ◽  
Power Density ◽  
Elastic Deformation ◽  
High Power ◽  
Reynolds Equation ◽  
Mixed Lubrication ◽  
High Power Density ◽  
Film Pressure ◽  
Oil Film

Based on the unified Reynolds equation and Fast Fourier Transform (FFT) method, the mixed lubrication characteristics of piston pin bearing in diesel engine with high power density are numerically simulated. Firstly, the unified Reynolds equation and the elastic deformation equation are solved simultaneously, and then the effects of viscosity-pressure on the maximum film pressure, the minimum oil film thickness and the piston pin orbit are analyzed. It is shown that for the semi-floating piston pin bearing with high power density, when viscosity-pressure is taken into consideration, both the minimum oil film thickness and the maximum oil film pressure increase, while the elastic deformation of the area in which the maximum load applies decreases. The transient diagrams of the relative position between the piston pin and its bearing within a whole loading period are given. It is also indicated that the eccentricity ratio of piston pin bearing along the direction of piston stroke is greater because of the greater load exerting on the back of the semi-floating piston pin bearing and thus resulting in the obvious deformation in the back area. This result is in good agreement with the existing real failure mode of the piston pin bearing with high power density. In addition, the effects of bearing clearance and length on the minimum oil film thickness are investigated respectively. It is shown that the smaller bearing clearance and the greater width are beneficial for the increasing of the minimum oil film thickness of piston pin bearing.

Behaviors of the wedge-shaped gas-lubricated film using the finite difference lattice Boltzmann method

2016 ◽  
Vol 230 (12) ◽  
pp. 1542-1553
Author(s):  
Xueqing Zhang ◽  
Qinghua Chen ◽  
Juanfang Liu
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Reynolds Equation ◽  
Navier Stokes ◽  
Film Pressure ◽  
Gas Bearings ◽  
Modified Reynolds Equation ◽  
Boltzmann Method ◽  
Film Lubrication

In this article, the finite difference lattice Boltzmann method (FDLBM) is successfully applied to analyze the hydrodynamic properties of the wedge-shaped gas film lubrication for the high speed micro gas bearings by comparing with the macroscopic methods (solving the modified Reynolds equation coupled with the simplified energy (modified Reynolds equation) and the Navier–Stokes equations coupled with the energy equation). By comparison, it is found that the vertical flow across the gas film can weaken the gas backflow and thus improves the gas film pressure, as the Navier–Stokes equation and FDLBM are used to analyze the wedge-shaped film lubrication. The continuum assumption in the macroscopic methods leads to a larger gas film pressure, compared with the value predicted by the FDLBM. And, the high temperature and speed enlarge this difference between them. Furthermore, the FDLBM provides a good warm-up for the multiscale simulation on the complex flow in the micro gas bearings.

Theoretical and Experimental Investigations Into Spacing Characteristics Between Roller and Three Types of Webs With Different Permeabilities

2005 ◽  
Vol 128 (2) ◽  
pp. 267-274 ◽  
Author(s):  
H. Hashimoto ◽  
M. Okajima
Keyword(s):  
Film Thickness ◽  
Reynolds Equation ◽  
Numerical Solutions ◽  
Experimental Investigations ◽  
Curve Fit ◽  
Wide Range ◽  
Entrained Air ◽  
Air Diffusion ◽  
Air Film ◽  
The Web

A new theoretical model for estimating the entrained air film thickness between a web and roller is presented for both impermeable and permeable webs. A simple curve fit formula for estimating the air film thickness, which considers the effects of air leakage from the web edges and air diffusion due to the permeability of web, was obtained based on a large number of simultaneous numerical solutions of the compressible Reynolds equation and the web equilibrium equation. The variation of air film thickness with roller velocity is measured for three typical webs: polyethylene terephthalate, coated paper, and newsprint. The effects of web permeability, web width, and web tension on the air film thickness are examined theoretically and experimentally for a wide range of roller velocity. Reasonable agreement is seen both quantitatively and qualitatively between the predicted and measured results. The validity of the formula for the first-order estimation of web-roller interface problems is verified experimentally.

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