The Optimum Slider Bearing in Terms of Friction

1972 ◽  
Vol 94 (3) ◽  
pp. 275-279 ◽  
Author(s):  
S. M. Rohde
Keyword(s):  
Variational Method ◽  
Film Thickness ◽  
Variational Formulation ◽  
Slider Bearing ◽  
One Dimensional ◽  
First Case ◽  
Minimum Film Thickness ◽  
The Coefficient Of Friction ◽  
Constant Viscosity ◽  
Film Profile

The film profile which minimizes the coefficient of friction and the film profile which minimizes the total friction force for a given load for a one-dimensional slider bearing are determined using a variational method. The lubricant is assumed to be incompressible and of constant viscosity. The flow is assumed to be laminar, and the optimization in the first case is based upon an assumed minimum film thickness. It is shown by the use of the nonlocal variational formulation that these profiles do yield a minimum among all admissible profiles.

A Demonstrably Optimum One Dimensional Journal Bearing

1972 ◽  
Vol 94 (2) ◽  
pp. 188-192 ◽  
Author(s):  
S. M. Rohde
Keyword(s):  
Film Thickness ◽  
Carrying Capacity ◽  
Journal Bearing ◽  
Infinite Length ◽  
Load Carrying Capacity ◽  
One Dimensional ◽  
Load Carrying ◽  
Minimum Film Thickness ◽  
Constant Viscosity ◽  
Nonlocal Formulation

By the use of a new variational technique, the bearing profile which maximizes the load carrying capacity of an infinite length journal bearing is obtained. The lubricant is assumed to be incompressible and of constant viscosity. The flow is assumed to be laminar and the optimization is based upon a minimum film thickness. The solution obtained is a concentric step bearing with a film thickness ratio of 1.812 and a ridge to pad ratio of 0.328. It is mathematically shown by the use of the “nonlocal” formulation that this step profile does yield a maximum among all profiles sufficiently “close.”

A Variational Principle Applied to the Plane Slider Bearing

1965 ◽  
Vol 87 (4) ◽  
pp. 1081-1082
Author(s):  
Clarence J. Maday
Keyword(s):  
Variational Principle ◽  
Film Thickness ◽  
Power Loss ◽  
Minimum Principle ◽  
Slider Bearing ◽  
One Dimensional ◽  
Dimensional Plane ◽  
Minimum Film Thickness ◽  
The One ◽  
Made In

A minimum principle from hydrodynamics is applied to the one-dimensional plane slider bearing which is provided with a self-seeking pivot mechanism. An analysis was made in which a certain integral was minimized subject to the constraint that the load, speed, and viscosity were held fixed. This analysis showed that this corresponded exactly to that combination of minimum film thickness and inclination which would minimize the power loss subject to the above-mentioned constraint. It was also found that, in order to satisfy the minimum principle, there exists a definite numerical ratio between the slider inclination and the nondimensional minimum film thickness. This, in turn, fixed the pivot location relative to the length of the slider.

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.

The Maximum Principle Approach to the Optimum One-Dimensional Journal Bearing

1970 ◽  
Vol 92 (3) ◽  
pp. 482-487 ◽  
Author(s):  
C. J. Maday
Keyword(s):  
Maximum Principle ◽  
Film Thickness ◽  
Journal Bearing ◽  
Maximum Load ◽  
Load Direction ◽  
One Dimensional ◽  
The Maximum Principle ◽  
Minimum Film Thickness ◽  
Constant Viscosity ◽  
The One

Pontryagin’s Maximum Principle is used to determine the journal bearing which supports the maximum load for a given minimum film thickness and a specified load direction. The one-dimensional configuration which uses a constant-viscosity, incompressible lubricant is considered. Comparison shows that the optimum bearing carries a load about 13.5 percent greater than the maximum carried by the usual full-Sommerfeld bearing and about 121 percent greater than that carried by the half-Sommerfeld unit. The problem is formulated subject to the constraints of a fixed load direction and a specified minimum film thickness while the only boundary condition imposed is that the pressure must vanish at the inlet and at the outlet. The actual extent of the bearing is determined in the optimization process and it is shown that this extent is 360 deg. Further, the bearing is stepped with only two regions of different but constant film thickness.

Effects of Surface Roughness and Additives in Lubrication: Generalized Reynolds Equation and its Application to Elastohydrodynamic Film

1987 ◽  
Vol 201 (1) ◽  
pp. 1-9 ◽  
Author(s):  
P Sinha ◽  
J S Kennedy ◽  
C M Rodkiewicz ◽  
P Chandra ◽  
R Sharma ◽  
...  
Keyword(s):  
Surface Roughness ◽  
Film Thickness ◽  
Reynolds Equation ◽  
Molecular Size ◽  
One Dimensional ◽  
Theoretical Investigations ◽  
Minimum Film Thickness ◽  
Porous Matrices ◽  
The Mean ◽  
Generalized Reynolds Equation

To study the effects of surface roughness and additives in lubrication, a generalized form of Reynolds equation is derived by taking into account the roughness interaction zones adjacent to the moving rough surfaces as sparsely porous matrices and purely hydrodynamic film of micropolar fluid characterizing the lubricant with additives. A particular, one-dimensional form of this equation is used to study these effects on the elastohydrodynamic (EHD) minimum film thickness at the inlet, between two rough rollers. It is shown that for the low permeability of the roughness zone, the EHD film thickness increases as the mean height of the asperities increases, whereas for the high permeability it decreases. The EHD film thickness is also found to increase with the concentration of the additives and the molecular size of the particles. These results are in conformity at least qualitatively, with various experimental and theoretical investigations, cited in the paper.

The Performance of Hydrodynamic Lubricating Films With Viscosity Variations Perpendicular to the Direction of Motion

1972 ◽  
Vol 94 (1) ◽  
pp. 44-48 ◽  
Author(s):  
E. B. Qvale ◽  
F. R. Wiltshire
Keyword(s):  
Film Thickness ◽  
Carrying Capacity ◽  
Load Carrying Capacity ◽  
End Effects ◽  
Parametric Curves ◽  
One Dimensional ◽  
Lubricating Film ◽  
Families Of Curves ◽  
Load Carrying ◽  
The Coefficient Of Friction

The effects of prescribed viscosity variations across a hydrodynamic lubricating film are studied. The film is strictly one dimensional and end effects are neglected. The viscosity variations are given by three families of curves. The considerable decreases (in the limit 100 percent) and occasional increases in the coefficient of friction that can occur for constant film thickness and load-carrying capacity are evaluated and the results are presented in terms of parametric curves. Important physical situations where these viscosity variations may be observed or produced are described.

The Optimum One-Dimensional Magnetohydrodynamic Slider Bearing

1970 ◽  
Vol 92 (3) ◽  
pp. 530-534 ◽  
Author(s):  
J. B. Shukla
Keyword(s):  
Magnetic Fields ◽  
Film Thickness ◽  
Calculus Of Variations ◽  
Bearing Surface ◽  
Slider Bearing ◽  
Bounded Control ◽  
Control Variables ◽  
Type Function ◽  
One Dimensional ◽  
Step Type

In this paper, the techniques of the calculus of variations are used to study the general problems in magnetohydrodynamic lubrication when two or more control variables are bounded. The load integral, in the case of a magnetohydrodynamic slider bearing, is maximized with film thickness and conductivity functions as bounded control variables. It is shown that if the conductivity of a bearing surface is of step-type function then uniformly applied magnetic fields are more advantageous.

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.

On the optimization of fluid film bearings

1976 ◽  
Vol 351 (1667) ◽  
pp. 481-497 ◽  
Keyword(s):  
Load Capacity ◽  
Unit Length ◽  
Turn Of The Century ◽  
Two Dimensional ◽  
Slider Bearing ◽  
One Dimensional ◽  
Fluid Film Bearings ◽  
Minimum Film Thickness ◽  
Classical Paper ◽  
Open Question

In a classical paper slightly after the turn of the century, Lord Rayleigh obtained the optimum one dimensional slider bearing profile, the Rayleigh step. His criteria for optimality was that the load capacity per unit length of the bearing be maximum among all profiles having a given minimum film thickness. Since Rayleigh, an open question has been the determina­tion of the optimum two dimensional bearing profile based on the same criteria as above. In this paper we propose a method for answering that question using algorithms which we have recently developed for this class of problems.

Combined Effects of Thermal and Non-Newtonian Character of Lubricant on Pressure, Film Profile, Temperature Rise, and Shear Stress in E.H.L.

1987 ◽  
Vol 109 (4) ◽  
pp. 666-670 ◽  
Author(s):  
S. H. Wang ◽  
H. H. Zhang
Keyword(s):  
Temperature Rise ◽  
Energy Equation ◽  
Numerical Solutions ◽  
Hydrodynamic Lubrication ◽  
Lubrication Equation ◽  
Minimum Film Thickness ◽  
The Coefficient Of Friction ◽  
Shape Equation ◽  
Rheological Character ◽  
Film Profile

A derivation of the lubrication equation suitable for nonlinear rheological model is presented. Full numerical solutions coupling the lubrication equation with film shape equation and energy equation are obtained to reveal the combined influence of thermal and non-Newtonian character of lubricant on elasto-hydrodynamic lubrication. Results show that the minimum film thickness is influenced only slightly in general. Nevertheless, the spikes of pressure and temperature rise, and the coefficient of friction are strongly affected by the rheological character of lubricant. It is concluded that both the temperature rise and non-Newtonian character of the oil film should be considered in E.H.L. analysis in order to obtain reliable results.

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