A Bounded Variable Approach to the Optimum Slider Bearing

1968 ◽  
Vol 90 (1) ◽  
pp. 240-242 ◽  
Author(s):  
C. J. Maday
Keyword(s):  
Film Thickness ◽  
Load Capacity ◽  
Inequality Constraints ◽  
Maximum Load ◽  
Slider Bearing ◽  
Lagrange Equations ◽  
Variable Approach ◽  
Pressure Dependent Viscosity ◽  
One Step ◽  
Dependent Viscosity

Contemporary methods for treating inequality constraints in the calculus of variations are employed to determine the maximum load-capacity one-dimensional slider bearing using a lubricant with pressure-dependent viscosity. A lower bound on the minimum film thickness is put into equational form to facilitate the use of the Euler-Lagrange equations, the corner conditions, and the Weierstrass E-function. It is found that, for typical lubricants, the slider bearing contains only one step separting two values of the film thickness. It is shown also that there exist cases for which a solution cannot be obtained to describe a real situation.

The One-Dimensional Optimum Hydrodynamic Gas Slider Bearing

1968 ◽  
Vol 90 (1) ◽  
pp. 281-284 ◽  
Author(s):  
C. J. Maday
Keyword(s):  
Film Thickness ◽  
Load Capacity ◽  
Maximum Load ◽  
Slider Bearing ◽  
Compression Process ◽  
One Dimensional ◽  
One Step ◽  
The Difference ◽  
The One ◽  
Bearing Number

Bounded variable methods of the calculus of variations are used to determine the optimum or maximum load capacity hydrodynamic one-dimensional gas slider bearing. A lower bound is placed on the minimum film thickness in order to keep the load finite, and also to satisfy the boundary conditions. Using the Weierstrass-Erdmann corner conditions and the Weierstrass E-function it is found that the optimum gas slider bearing is stepped with a convergent leading section and a uniform thickness trailing section. The step location and the leading section film thickness depend upon the bearing number and compression process considered. It is also shown that the bearing contains one and only one step. The difference in the load capacity and maximum film pressure between the isothermal and adiabatic cases increases with increasing bearing number.

Rayleigh Step Journal Bearing, Part II—Incompressible Fluid

1969 ◽  
Vol 91 (4) ◽  
pp. 641-650 ◽  
Author(s):  
B. J. Hamrock ◽  
W. J. Anderson
Keyword(s):  
Theoretical Analysis ◽  
Film Thickness ◽  
Journal Bearing ◽  
Load Capacity ◽  
Maximum Load ◽  
Single Step ◽  
Thickness Ratio ◽  
Infinite Length ◽  
One Step ◽  
Bearing Part

A theoretical analysis of the pressure distribution, load, capacity, and attitude angle for a single-step concentric as well as a multistep infinite length eccentric Rayleigh step journal bearing is performed. The results from the single-step concentric analysis indicated that the maximum load capacity is obtained when the film thickness ratio is 1.7 and the ratio of the angle subtended by the ridge to the angle subtended by the pad is 0.35. The results from the infinite length eccentric analysis indicated that one step placed around the journal was optimal. For eccentricity ratios greater than or equal to 0.2 the maximum load occurred for a bearing without a step or a Sommerfeld bearing. For eccentricity ratios less than 0.2 the optimal film thickness ratio is 1.7 while there are three optimal ratios of angle subtended by the ridge to the angle subtended by the pad of 0.4, 0.45, and 0.5 depending on whether load capacity or stability or both load capacity and stability is more important in the application being considered.

The Optimum Rayleigh Bearing in Terms of Load Capacity or Friction for Non-Newtonian Lubricants

1985 ◽  
Vol 107 (1) ◽  
pp. 59-67 ◽  
Author(s):  
P. Bourgin ◽  
B. Gay
Keyword(s):  
Film Thickness ◽  
Load Capacity ◽  
Maximum Load ◽  
Slider Bearing ◽  
Generalized Newtonian Fluid ◽  
Newtonian Viscosity ◽  
One Dimensional ◽  
Minimum Film Thickness ◽  
The One ◽  
Numerical Program

Pontryagin’s Maximum Principle is used to show that the configuration of the one-dimensional slider bearing which carries the maximum load for a specified minimum film thickness, is a modified Rayleigh bearing. The lubricant may be any Generalized Newtonian Fluid. Having selected two optimization criteria (1: maximum load capacity for a given minimum film thickness—2: minimum friction force for a specified load), a numerical program allows one to determine the optimal step bearing associated with the lubricant non-Newtonian viscosity. Several examples are worked out, showing that significant gains are expected, in comparison with the results obtained for the classical (Newtonian) Rayleigh bearing.

An Accurate Solution of the Gas Lubricated, Flat Sector Thrust Bearing

1977 ◽  
Vol 99 (1) ◽  
pp. 82-88 ◽  
Author(s):  
I. Etsion ◽  
D. P. Fleming
Keyword(s):  
Film Thickness ◽  
Thrust Bearing ◽  
Load Capacity ◽  
Thickness Distribution ◽  
Maximum Load ◽  
Accurate Solution ◽  
Operating Conditions ◽  
Thrust Bearings ◽  
Practical Design ◽  
Minimum Film Thickness

A flat sector shaped pad geometry for gas lubricated thrust bearings is analyzed considering both pitch and roll angles of the pad and the true film thickness distribution. Maximum load capacity is achieved when the pad is tilted so as to create a uniform minimum film thickness along the pad trailing edge. Performance characteristics for various geometries and operating conditions of gas thrust bearings are presented in the form of design curves. A comparison is made with the rectangular slider approximation. It is found that this approximation is unsafe for practical design, since it always overestimates load capacity.

Squeeze Film Problems of Long Partial Journal Bearings for Non-Newtonian Couple Stress Fluids with Pressure-Dependent Viscosity

2011 ◽  
Vol 66 (8-9) ◽  
pp. 512-518 ◽  
Author(s):  
Jaw-Ren Lin ◽  
Li-Ming Chu ◽  
Chi-Ren Hung ◽  
Rong-Fang Lu
Keyword(s):  
Couple Stress ◽  
Load Capacity ◽  
Perturbation Technique ◽  
Closed Form Solution ◽  
Journal Bearings ◽  
Form Solution ◽  
Squeeze Film ◽  
Pressure Dependent Viscosity ◽  
Dependent Viscosity ◽  
Couple Stress Fluids

Abstract According to the experimental work of C. Barus in Am. J. Sci. 45, 87 (1893) [1], the dependency of liquid viscosity on pressure is exponential. Therefore, we extend the study of squeeze film problems of long partial journal bearings for Stokes non-Newtonian couple stress fluids by considering the pressure-dependent viscosity in the present paper. Through a small perturbation technique, we derive a first-order closed-form solution for the film pressure, the load capacity, and the response time of partial-bearing squeeze films. It is also found that the non-Newtonian couple-stress partial bearings with pressure-dependent viscosity provide better squeeze-film characteristics than those of the bearing with constant-viscosity situation.

On lower-dimensional models in lubrication, Part B: Derivation of a Reynolds type of equation for incompressible piezo-viscous fluids

2020 ◽  
pp. 135065012097380
Author(s):  
Andreas Almqvist ◽  
Evgeniya Burtseva ◽  
Kumbakonam Rajagopal ◽  
Peter Wall
Keyword(s):  
Film Thickness ◽  
Reynolds Equation ◽  
Constitutive Relations ◽  
Viscous Fluids ◽  
Dimensional Model ◽  
Pressure Dependent Viscosity ◽  
Constant Viscosity ◽  
Dimensional Models ◽  
Dependent Viscosity ◽  
Lower Dimensional

The Reynolds equation is a lower-dimensional model for the pressure in a fluid confined between two adjacent surfaces that move relative to each other. It was originally derived under the assumption that the fluid is incompressible and has constant viscosity. In the existing literature, the lower-dimensional Reynolds equation is often employed as a model for the thin films, which lubricates interfaces in various machine components. For example, in the modelling of elastohydrodynamic lubrication (EHL) in gears and bearings, the pressure dependence of the viscosity is often considered by just replacing the constant viscosity in the Reynolds equation with a given viscosity-pressure relation. The arguments to justify this are heuristic, and in many cases, it is taken for granted that you can do so. This motivated us to make an attempt to formulate and present a rigorous derivation of a lower-dimensional model for the pressure when the fluid has pressure-dependent viscosity. The results of our study are presented in two parts. In Part A, we showed that for incompressible and piezo-viscous fluids it is not possible to obtain a lower-dimensional model for the pressure by just assuming that the film thickness is thin, as it is for incompressible fluids with constant viscosity. Here, in Part B, we present a method for deriving lower-dimensional models of thin-film flow, where the fluid has a pressure-dependent viscosity. The main idea is to rescale the generalised Navier-Stokes equation, which we obtained in Part A based on theory for implicit constitutive relations, so that we can pass to the limit as the film thickness goes to zero. If the scaling is correct, then the limit problem can be used as the dimensionally reduced model for the flow and it is possible to derive a type of Reynolds equation for the pressure.

The oil film in a closing gap

1962 ◽  
Vol 266 (1326) ◽  
pp. 312-328 ◽  
Keyword(s):  
High Pressure ◽  
Film Thickness ◽  
Pressure Coefficient ◽  
Lubricant Film ◽  
Polished Surface ◽  
Maximum Pressure ◽  
Pressure Dependent Viscosity ◽  
Special Cases ◽  
Dry Contact ◽  
Dependent Viscosity

This paper presents a solution to the elasto-hydrodynamic problem of normal approach of two cylindrical bodies separated by a lubricating film. Analytic solutions are found for the special cases of constant viscosity and rigid material and also for pressure-dependent viscosity. The more general case accounting for elastic deformation of the bodies with constant or pressure dependent viscosity was solved by using an iterative numerical process with the help of an electronic computer. It is found that a very high pressure may be developed in the lubricant film at a finite separation of the cylinders. As the film thickness is further reduced, the value of the maximum pressure decreases and as the film thickness approaches zero, the pressure distribution converges to the Hertzian dry contact form. For a given load applied to the cylinders, the value of the maximum pressure reached depends to a large extent upon the value of the parameter oc E , i.e. the product of the pressure coefficient of viscosity and the equivalent Young’s modulus of the elastic system. Also, once the pressure has reached a sufficiently high value it becomes extremely sensitive to an increase in load; a small increase in load will produce a large increase in maximum pressure. A number of experiments were performed in order to check some of the theoretical predictions made. In these experiments a loaded steel ball was allowed to approach the polished surface of various materials whose surfaces were covered by a lubricant film, and the plastic deformations produced in the surface were then measured. These tests showed clearly the influence of the lubricant in that in every case the depth of the impressions with lubricant was significantly larger than the corresponding ones produced under Hertzian, dry contact impacts. The experimental results indicate a correlation between maximum pressure and the value of ol E and its sensitivity to increase in load at high pressure as predicted by the theory.

Design Optimization of Gas Foil Thrust Bearings for Maximum Load Capacity

2015 ◽  
Author(s):  
Tae Ho Kim ◽  
Tae Won Lee
Keyword(s):  
Experimental Data ◽  
Film Thickness ◽  
Load Capacity ◽  
Maximum Load ◽  
Drag Torque ◽  
Rotor Speed ◽  
Design Guideline ◽  
Film Pressure ◽  
Thrust Bearings ◽  
Minimum Film Thickness

Improvement of the load capacity of gas foil thrust bearings (GFTBs) is important to broadening their application in oil-free microturbomachinery (<250 kW) with high power density. Although GFTBs have the significant advantage of low friction without the use of lubrication systems compared to oil film thrust bearings, their inherently low load capacity has limited their application. The aim of the present study was to develop a design guideline for increasing the load capacity of GFTBs. The Reynolds equation for an isothermal isoviscous ideal gas was used to calculate the gas film pressure. To predict the ultimate load capacity of the GFTB, the pressure was averaged in the radial direction of the gas flow field used to deflect the foil structure. The load capacity, film pressure profile, and film thickness profile were predicted for a GFTB with an outer radius of 55 mm, inner radius of 30 mm, and eight foils each of arc length 45°. The predictions showed that the load capacity of the GFTB increased with increasing rotor speed and decreasing minimum film thickness, and was always lower than the analytically determined limit value for infinite rotor speed (obtained by simple algebraic equations). A parametric study in which the ramp extent (or inclined angle) was increased from 5° to 40°, and the ramp height from 0 to 0.320 mm, revealed that the GFTB had an optimal ramp extent of ∼22.5° and ramp height of ∼0.030 mm for maximum load capacity. Interestingly, the optimal values were also valid for a rigid-surface bearing. The predicted load capacities for a ramp extent of ∼22.5° and increasing ramp height from 0.030 to 0.320 mm were compared with experimental data obtained from a previous work. The predictions for a ramp height of 0.155 mm were in good agreement with the experimental data for all three test GFTBs with outer radii of 45, 50, and 55 mm, respectively. In addition, this paper shows that the predicted drag torque increases linearly with increasing rotor speed and decreasing minimum film thickness, and nonlinearly with decreasing ramp height. The drag torque significantly increased only for ramp heights below the optimal value. The predictions imply that the optimal ramp height improves the load capacity of the GFTB with little change in the drag torque.

Hydrodynamic Lubrication of Slider Bearings With Two-Dimensionally Varying Contoured Surfaces and Optimum Design Data

1989 ◽  
Author(s):  
C. Bagci ◽  
C. J. McClure ◽  
S. K. Rajavenkateswaran
Keyword(s):  
Optimum Design ◽  
Hydrodynamic Lubrication ◽  
Load Capacity ◽  
Maximum Load ◽  
Load Flow ◽  
Effect Of Temperature ◽  
Temperature Dependent Viscosity ◽  
Design Data ◽  
Slider Bearings ◽  
Dependent Viscosity

Abstract Contoured bearing surfaces forming continuous surface pockets in slider bearings increase the load carrying capacity considerably. Article investigates the effect of two dimensionally contoured surfaces of a few sample shapes on the performance characteristics of hydrodynamic slider bearings. Exponential and trigonometric film shapes are considered. Computer-aided numerical finite difference solution of the two-dimensional Reynolds equation is used via a self mesh generating computer program, which also generates optimum design data including dimensionless load-, flow-, temperature rise-, power loss-, stiffness-, damping, and friction coefficients. Optimum bearings are defined as the maximum load capacity bearings. Optimum design charts are given, where dimensional ratios and powers of exponents are optimized. Lubricant is incompressible with temperature dependent viscosity. The effect of temperature is considered by maintaining energy balance via iterative thermal loop. In comparison to optimum one-dimensional tapered film bearings gains of over 200% in the load capacities of contoured narrow bearings are observed.

Design Optimization of Gas Foil Thrust Bearings for Maximum Load Capacity1

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Tae Ho Kim ◽  
Moonsung Park ◽  
Tae Won Lee
Keyword(s):  
Film Thickness ◽  
Load Capacity ◽  
Maximum Load ◽  
Ideal Gas ◽  
Outer Radius ◽  
Test Results ◽  
Rotor Speed ◽  
Design Guideline ◽  
Film Pressure ◽  
Thrust Bearings

The aim of the present study is to develop a design guideline to improve the load capacity of gas foil thrust bearings (GFTBs). The Reynolds equation for an isothermal isoviscous ideal gas calculates the gas film pressure. The film pressure averaged in the radial direction determines the ultimate load capacity. The load capacity, film pressure profile, and film thickness profile are predicted for a GFTB with an outer radius of 55 mm, inner radius of 30 mm, and eight foils each of arc length 45 deg. The predictions show that the load capacity of the GFTB increases with increasing rotor speed and decreasing minimum film thickness. A parametric study, in which the ramp extent (or inclined angle) is increased from 5 deg to 40 deg, and the ramp height from 0 to 320 μm, reveals that GFTBs have an optimal ramp extent of ∼22.5 deg and ramp height of 30 μm for maximum load capacity. A series of maximum load capacity measurements are conducted on four test GFTBs with ramp heights of 50, 150, 250, and 350 μm at the speeds of 12, 15, and 18 krpm. To estimate the maximum load capacity, the applied load is increased until the drag torque rises suddenly with a sharp peak. The test results show that the maximum load capacity generally increases for decreasing ramp height and for increasing rotor speed. The GFTB with a ramp height of 50 μm shows the largest maximum load capacity of 510 N, for example. Test results are in good agreement with model predictions.

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