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 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.

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 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.

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.

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.

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.

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.

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.

Dynamic Analysis of Tilting-Pad Journal Bearing—Influence of Pad Deformations

1994 ◽  
Vol 116 (3) ◽  
pp. 621-627 ◽  
Author(s):  
H. Desbordes ◽  
M. Fillon ◽  
C. Chan Hew Wai ◽  
J. Frene
Keyword(s):  
Film Thickness ◽  
Pressure Field ◽  
Journal Bearing ◽  
Maximum Pressure ◽  
Tilting Pad ◽  
Minimum Film Thickness ◽  
Dimensional Finite Element ◽  
The One ◽  
Tilting Pad Journal Bearing ◽  
Tilting Pad Journal Bearings

A theoretical nonlinear analysis of tilting-pad journal bearings is presented for small and large unbalance loads under isothermal conditions. The radial displacements of internal pad surface due to pressure field are determined by a two-dimensional finite element method in order to define the actual film thickness. The influence of pad deformations on the journal orbit, on the minimum film thickness and on the maximum pressure is studied. The effects of pad displacements are to decrease the minimum film thickness and to increase the maximum pressure. The orbit amplitude is also increased by 20 percent for the large unbalance load compared to the one obtained for rigid pad.

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.

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