The Validity of the Reynolds Equation in Modeling Hydrostatic Effects in Gas Lubricated Textured Parallel Surfaces

2005 ◽  
Vol 128 (2) ◽  
pp. 345-350 ◽  
Author(s):  
Y. Feldman ◽  
Y. Kligerman ◽  
I. Etsion ◽  
S. Haber
Keyword(s):  
Carrying Capacity ◽  
Reynolds Equation ◽  
Stokes Equations ◽  
Surface Texturing ◽  
Laser Surface Texturing ◽  
Physical Parameters ◽  
Load Carrying Capacity ◽  
Wide Range ◽  
Load Carrying ◽  
Parallel Surfaces

Microdimples generated by laser surface texturing (LST) can be used to enhance performance in hydrostatic gas-lubricated tribological components with parallel surfaces. The pressure distribution and load carrying capacity for a single three-dimensional dimple, representing the LST, were obtained via two different methods of analysis: a numerical solution of the exact full Navier-Stokes equations, and an approximate solution of the much simpler Reynolds equation. Comparison between the two solution methods illustrates that, despite potential large differences in local pressures, the differences in load carrying capacity, for realistic geometrical and physical parameters, are small. Even at large clearances of 5% of the dimple diameter and pressure ratios of 2.5 the error in the load carrying capacity is only about 15%. Thus, for a wide range of practical clearances and pressures, the simpler, approximate Reynolds equation can safely be applied to yield reasonable predictions for the load carrying capacity of dimpled surfaces.

The Validity of the Reynolds Equation in Modeling Hydrostatic Effects in Gas Lubricated Textured Parallel Surfaces

2005 ◽  
Author(s):  
Yuri Feldman ◽  
Yuri Kligerman ◽  
Izhak Etsion ◽  
Shimon Haber
Keyword(s):  
Carrying Capacity ◽  
Reynolds Equation ◽  
Stokes Equations ◽  
Surface Texturing ◽  
Laser Surface Texturing ◽  
Navier Stokes ◽  
Load Carrying Capacity ◽  
Navier Stokes Equations ◽  
Load Carrying ◽  
Parallel Surfaces

The pressure distribution and load carrying capacity for a single 3D dimple, representing laser surface texturing (LST) of gas-lubricated tribological components with parallel surfaces, were obtained via two different methods of analysis: 1) a numerical solution of the exact full Navier-Stokes equations; 2) an approximate solution of the much simpler Reynolds equation. Comparison between the two solutions illustrated that the differences in load carrying capacity were negligible for clearances that are 3% or less of the dimple diameter. At larger realistic clearances the error in the load carrying capacity may reach a maximum of 10%.

Evolutionary Optimization of Micro-Thrust Bearings With Periodic Partial Trapezoidal Surface Texturing

2010 ◽  
Vol 133 (1) ◽  
Author(s):  
Christos I. Papadopoulos ◽  
Pantelis G. Nikolakopoulos ◽  
Lambros Kaiktsis
Keyword(s):  
Carrying Capacity ◽  
Stokes Equations ◽  
Surface Texturing ◽  
Load Carrying Capacity ◽  
Moving Wall ◽  
Stationary Wall ◽  
Optimization Study ◽  
Micro Channels ◽  
Wide Range ◽  
Load Carrying

An optimization study of trapezoidal surface texturing in slider micro-bearings, via computational fluid dynamics (CFD), is presented. The bearings are modeled as micro-channels, consisting of a moving and a stationary wall. The moving wall (rotor) is assumed smooth, while part of the stationary wall (stator) exhibits periodic dimples of trapezoidal form. The extent of the textured part of the stator and the dimple geometry are defined parametrically; thus, a wide range of texturing configurations is considered. Flow simulations are based on the numerical solution of the Navier–Stokes equations for incompressible isothermal flow. To optimize the bearing performance, an optimization problem is formulated and solved by coupling the CFD code with an optimization tool based on genetic algorithms and local search methods. Here, the design variables define the bearing geometry, while load carrying capacity is the objective function to be maximized. Optimized texturing geometries are obtained for the case of parallel bearings for several numbers of dimples, illustrating significant load carrying capacity levels. Further, these optimized texturing patterns are applied to converging bearings for different convergence ratio values; the results demonstrate that, for small and moderate convergence ratios, a substantial increase in load carrying capacity, in comparison to smooth bearings, is obtained. Finally, an optimization study performed at a high convergence ratio shows that, in comparison to the parallel slider, the optimal texturing geometry is substantially different, and that performance improvement over smooth bearings is possible even for steep sliders.

Evolutionary Optimization of Micro- Thrust Bearings With Periodic Partial Trapezoidal Surface Texturing

2010 ◽  
Author(s):  
C. I. Papadopoulos ◽  
P. G. Nikolakopoulos ◽  
L. Kaiktsis
Keyword(s):  
Carrying Capacity ◽  
Stokes Equations ◽  
Surface Texturing ◽  
Load Carrying Capacity ◽  
Moving Wall ◽  
Stationary Wall ◽  
Design Variables ◽  
Optimization Study ◽  
Wide Range ◽  
Load Carrying

An optimization study of trapezoidal surface texturing in slider micro-bearings, via Computational Fluid Dynamics (CFD), is presented. The bearings are modeled as microchannels, consisting of a moving and a stationary wall. The moving wall (rotor) is assumed smooth, while part of the stationary wall (stator) exhibits periodic dimples of trapezoidal form. The extent of the textured part of the stator, and the dimple geometry are defined parametrically; thus, a wide range of texturing configurations is considered. Flow simulations are based on the numerical solution of the Navier-Stokes equations for incompressible isothermal flow. To optimize the bearing performance, an optimization problem is formulated, and solved by coupling the CFD code with an optimization tool based on genetic algorithms and local search methods. Here, the design variables define the bearing geometry, while load carrying capacity is the objective function to be maximized. Optimized texturing geometries are obtained for the case of parallel bearings, for several numbers of dimples, illustrating significant load carrying capacity levels. Further, these optimized texturing patterns are applied to converging bearings, for different convergence ratio values; the results demonstrate that, for small and moderate convergence ratios, substantial increase in the load carrying capacity, in comparison to smooth bearings, is obtained. Finally, an optimization study performed at a high convergence ratio shows that, in comparison to the parallel slider, the optimal texturing geometry is substantially different, and that performance improvement over smooth bearings is possible even for steep sliders.

Geometry Optimization of Textured Three-Dimensional Micro- Thrust Bearings

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
C. I. Papadopoulos ◽  
E. E. Efstathiou ◽  
P. G. Nikolakopoulos ◽  
L. Kaiktsis
Keyword(s):  
Carrying Capacity ◽  
Stokes Equations ◽  
Load Capacity ◽  
Three Dimensional ◽  
Load Carrying Capacity ◽  
Thrust Bearings ◽  
Micro Channels ◽  
Wide Range ◽  
Load Carrying ◽  
Bearing Load

This paper presents an optimization study of the geometry of three-dimensional micro-thrust bearings in a wide range of convergence ratios. The optimization goal is the maximization of the bearing load carrying capacity. The bearings are modeled as micro-channels, consisting of a smooth moving wall (rotor), and a stationary wall (stator) with partial periodic rectangular texturing. The flow field is calculated from the numerical solution of the Navier-Stokes equations for incompressible isothermal flow; processing of the results yields the bearing load capacity and friction coefficient. The geometry of the textured channel is defined parametrically for several width-to-length ratios. Optimal texturing geometries are obtained by utilizing an optimization tool based on genetic algorithms, which is coupled to the CFD code. Here, the design variables define the bearing geometry and convergence ratio. To minimize the computational cost, a multi-objective approach is proposed, consisting in the simultaneous maximization of the load carrying capacity and minimization of the bearing convergence ratio. The optimal solutions, identified based on the concept of Pareto dominance, are equivalent to those of single-objective optimization problems for different convergence ratio values. The present results demonstrate that the characteristics of the optimal texturing patterns depend strongly on both the convergence ratio and the width-to-length ratio. Further, the optimal load carrying capacity increases at increasing convergence ratio, up to an optimal value, identified by the optimization procedure. Finally, proper surface texturing provides substantial load carrying capacity even for parallel or slightly diverging bearings. Based on the present results, we propose simple formulas for the design of textured micro-thrust bearings.

Geometry Optimization of Textured 3-D Micro-Thrust Bearings

2011 ◽  
Author(s):  
C. I. Papadopoulos ◽  
E. E. Efstathiou ◽  
P. G. Nikolakopoulos ◽  
L. Kaiktsis
Keyword(s):  
Carrying Capacity ◽  
Stokes Equations ◽  
Optimization Problems ◽  
Load Capacity ◽  
Computational Cost ◽  
Load Carrying Capacity ◽  
Thrust Bearings ◽  
Wide Range ◽  
Load Carrying ◽  
Bearing Load

The paper presents an optimization study of the geometry of three-dimensional micro-thrust bearings, in a wide range of convergence ratios. The optimization goal is the maximization of the bearing load carrying capacity. The bearings are modeled as microchannels, consisting of a smooth moving wall (rotor), and a stationary wall (stator) with partial periodic rectangular texturing. The flow field is calculated from the numerical solution of the Navier-Stokes equations for incompressible isothermal flow; processing of the results yields the bearing load capacity and friction coefficient. The geometry of the textured channel is defined parametrically for several width-to-length ratios. Optimal texturing geometries are obtained by utilizing an optimization tool based on genetic algorithms, which is coupled to the CFD code. Here, the design variables define the bearing geometry and convergence ratio. To minimize the computational cost, a multi-objective approach is proposed, consisting in the simultaneous maximization of the load carrying capacity and minimization of the bearing convergence ratio. The optimal solutions, identified based on the concept of Pareto dominance, are equivalent to those of single-objective optimization problems at different convergence ratio values. The present results demonstrate that the characteristics of the optimal texturing patterns depend strongly on both the convergence ratio and the width-to-length ratio. Further, the optimal load carrying capacity increases at increasing convergence ratio, up to an optimal value, identified by the optimization procedure. Finally, proper surface texturing provides substantial load carrying capacity even for parallel or slightly diverging bearings. Based on the present results, we propose simple formulas for the design of textured micro-thrust bearings.

Hydrodynamic Lubrication of Finite Slider Bearings: Effect of One Dimensional Film Shape, and Their Computer Aided Optimum Designs

1983 ◽  
Vol 105 (1) ◽  
pp. 48-63 ◽  
Author(s):  
C. Bagci ◽  
A. P. Singh
Keyword(s):  
Numerical Solution ◽  
Carrying Capacity ◽  
Reynolds Equation ◽  
Hydrodynamic Lubrication ◽  
Load Carrying Capacity ◽  
Slider Bearing ◽  
Design Data ◽  
Slider Bearings ◽  
Load Carrying ◽  
Computer Aided

The effect of the film shape on the load carrying capacity of a hydrodynamically lubricated bearing has not been considered an important factor in the past. Flat-faced tapered bearing and the Raileigh’s step bearing of constant film thickness have been the primary forms of film shapes for slider bearing studies and design data developments. This article, by the computer aided numerical solution of the Reynolds equation for two dimensional incompressible lubricant flow, investigates hydrodynamically lubricated slider bearings having different film shapes and studies the effect of the film shape on the performance characteristics of finite bearings; and it shows that optimized bearing with film shapes having descending slope toward the trailing edge of the bearing has considerably higher load carrying capacity than the optimized flat-faced tapered bearing of the same properties. For example the truncated cycloidal film shape yields 26.3 percent higher load carrying capacity for Lz/Lx = 1 size ratio, and 44 percent higher for Lz/Lx = 1/2. The article then presents charts for the optimum designs of finite slider bearings having tapered, exponential, catenoidal, polynomial, and truncated-cycloidal film shapes, and illustrates their use in numerical bearing design examples. These charts also furnish information on flow rate, side leakage, temperature rise, coefficient of friction, and friction power loss in optimum bearings. Appended to the article are analytical solutions for infinitely wide bearings with optimum bearing characteristics. The computer aided numerical solution of the Reynolds equation in most general form is presented by which finite or infinitely wide hydrodynamically or hydrostatically lubricated bearings, externally pressurized or not, can be studied. A digital computer program is made available.

Stokes Roughness Effects on Hydrodynamic Lubrication. Part II—Effects Under Slip Flow Boundary Conditions

1986 ◽  
Vol 108 (2) ◽  
pp. 159-166 ◽  
Author(s):  
Y. Mitsuya
Keyword(s):  
Flow Regime ◽  
Carrying Capacity ◽  
Stokes Equations ◽  
Hydrodynamic Lubrication ◽  
Slip Flow ◽  
Load Carrying Capacity ◽  
Roughness Effects ◽  
Random Roughness ◽  
Load Carrying ◽  
Flow Effects

Stokes roughness effects on hydrodynamic lubrication are studied in the slip flow regime. Slip flow boundary conditions for Navier-Stokes equations are derived, assuming that the fluid on a surface slips due to the molecular mean free path along the surface, even if the surface is rough. The perturbation method for Navier-Stokes equations, which was derived in Part I of this report, is then applied. Slip flow effects on load carrying capacity and frictional force are numerically clarified for both Stokes and Reynolds roughnesses. In the slip flow regime, second-order quantities induced by Stokes effects, such as flow rate, load carrying capacity, and frictional force are in proportion to the wavenumber squared. This phenomenon relative to the quantities being proportional is also the same as that in the continuum flow regime. As a result of velocity slippage, the load carrying capacity in Stokes roughness is found to decrease more than in Reynolds roughness for incompressible films, while the relationship is reversed for compressible films having a high compressibility number. The simulation of random roughness, which is generated by numerical means, clarifies one important result: the average slip flow effects associated with random Stokes roughness become similar to the slip flow effects in deterministic sinusoidal Stokes roughness, whose wavelength and height are statistically equivalent to those of random roughness. Although attention should be given to the fact that Stokes effects on random roughness demonstrate considerable scattering with the continuum flow, such scattering diminishes with the slip flow.

Theoretical Analysis of Externally Pressurized Air Journal Bearing

1970 ◽  
Vol 12 (2) ◽  
pp. 123-129 ◽  
Author(s):  
B. C. Majumdar
Keyword(s):  
Experimental Data ◽  
Carrying Capacity ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Design Parameters ◽  
Load Carrying Capacity ◽  
Load Carrying ◽  
Dimensional Parameters ◽  
Bearing Design

A theoretical investigation is made to predict the performance of an externally pressurized air journal bearing having several pressure sources. The pressure distribution, which leads to the determination of load-carrying capacity and flow requirement, is obtained by solving Reynolds equation numerically. The load and flow, expressed in non-dimensional parameters, are presented for different bearing design parameters (dimensionless). The results predicted by this method are compared with others' experimental data.

Constraint-Based Fracture Mechanics Analysis of Cylinders With Circumferential Cracks

2009 ◽  
Author(s):  
Michael Bach ◽  
Xin Wang ◽  
Robert Bell
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Carrying Capacity ◽  
Load Carrying Capacity ◽  
Element Analysis ◽  
Fracture Parameters ◽  
Wide Range ◽  
Load Carrying ◽  
T Stress ◽  
Low Constraint

In this paper, the fracture behaviour of hollow cylinders with internal circumferential crack under tensile loading is examined extensively. Finite element analysis of the cracked cylinders is conducted to determine the fracture parameters including stress intensity factor, T-stress, and J-integral. Linear elastic finite element analysis is conducted to obtain K and T-stress, and elastic plastic analysis is conducted to obtain fully plastic J-integrals. A wide range of cylinder geometries are studied, with cylinder thickness ratios of ri/ro = 0.2 to 0.8 and crack depth ratio a/t = 0.2 to 0.8. These fracture parameters are then used to construct conventional and constraint-based failure assessment diagrams (FADs) to determine the maximum load carrying capacity of cracked cylinders. It is demonstrated that these tensile loaded cylinders with circumferential cracks are under low constraint conditions, and the load carrying capacity are higher when the low constraint effects are properly accounted for, using constraint-based FADs, comparing to the predictions from the conventional FADs.

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