Solving the Incompressible and Isothermal Problem in Elastohydrodynamic Lubrication Through an Integral Equation

1968 ◽  
Vol 90 (1) ◽  
pp. 262-270 ◽  
Author(s):  
K. Herrebrugh
Keyword(s):  
Integral Equation ◽  
Film Thickness ◽  
Numerical Solution ◽  
Analytic Functions ◽  
Large Range ◽  
Nonlinear Function ◽  
Elastohydrodynamic Lubrication ◽  
Elasticity Equations ◽  
Single Integral ◽  
Constant Viscosity

It will be shown that the hydrodynamic and elasticity equations in elastohydrodynamic lubrication can be coupled to one single integral equation of the following form: H(x)=f(x)−T∫abK(x,ξ)F{H(ξ)}dξ in which f(x) and K(x, ξ) are both known analytic functions inside [a, b], and F(H) is in general a nonlinear function of the dimensionless film thickness. A numerical solution of this integral equation for constant viscosity is presented for a large range of loading conditions.

Solution of the Non-Newtonian Elastohydrodynamic Problem for Circular Contacts Based on a Flow Continuity Method

1996 ◽  
Vol 210 (4) ◽  
pp. 247-258 ◽  
Author(s):  
C A Holt ◽  
H P Evans ◽  
R W Snidle
Keyword(s):  
Film Thickness ◽  
Numerical Solution ◽  
Control Volume ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Theoretical Formulation ◽  
New Approach ◽  
Limiting Shear Stress ◽  
Pure Rolling ◽  
Stress Behaviour

The paper describes a numerical solution method for the point contact elastohydrodynamic lubrication (EHL) problem under non-Newtonian, isothermal conditions. The theoretical formulation of the non-Newtonian effect is general and may be applied to both shear thinning and limiting shear stress behaviour. The particular rheological model investigated in this work is the Eyring ‘sinh law’ relation. The numerical solution of the lubrication equations is based upon a control volume approach rather than the more usual methods that utilize a modified Reynolds equation. This new approach ensures that flow continuity is satisfied at the discretization level. Results are presented to show the effect of non-Newtonian behaviour on film thickness and pressure distribution in circular EHL contacts operating over a range of slide-roll ratios from 0 (pure rolling) to 1.5. Under conditions of pure rolling or low sliding there is found to be little effect of non-Newtonian behaviour, but at the highest degree of sliding the film thickness over the central, flattened area of the contact is reduced by up to 10 per cent at the highest rolling speed of 0.75 m/s.

Effects of Differential Scheme and Mesh Density on EHL Film Thickness in Point Contacts

2006 ◽  
Vol 128 (3) ◽  
pp. 641-653 ◽  
Author(s):  
Yuchuan Liu ◽  
Q. Jane Wang ◽  
Wenzhong Wang ◽  
Yuanzhong Hu ◽  
Dong Zhu
Keyword(s):  
Film Thickness ◽  
Numerical Solution ◽  
System Approach ◽  
Elastohydrodynamic Lubrication ◽  
Mesh Size ◽  
Point Contacts ◽  
Wide Range ◽  
Mesh Density ◽  
Differential Scheme

This paper investigates the effects of differential scheme and mesh density on elastohydrodynamic lubrication (EHL) film thickness based on a full numerical solution with a semi-system approach. The solution variation with different schemes and mesh sizes is revealed based on a set of numerical cases in a wide range of central film thickness from several hundred nanometers down to a few nanometers. It is observed that when the film is thick, the effects of differential schemes and mesh density are not significant. However, if the film becomes ultra-thin, e.g., below 10–20 nanometers, the influence of mesh density and differential schemes becomes more significant, and a proper dense mesh and differential scheme may be highly desirable. The present study also indicates that the solutions from the 1st-order backward scheme give the largest film thickness among all the solutions from different schemes at the same mesh size.

Debris Effects on EHL Contact

2000 ◽  
Vol 122 (4) ◽  
pp. 711-720 ◽  
Author(s):  
Young S. Kang ◽  
Farshid Sadeghi ◽  
Xiaolan Ai
Keyword(s):  
Film Thickness ◽  
Reynolds Equation ◽  
Elastohydrodynamic Lubrication ◽  
Force Balance ◽  
Time Dependent ◽  
Elasticity Equations ◽  
Minimum Film Thickness ◽  
Bounding Surfaces ◽  
Force Balance Equation ◽  
Ehl Contact

A model was developed to study the effects of a rigid debris on elastohydrodynamic lubrication of rolling/sliding contacts. In order to achieve the objectives the time dependent Reynolds equation was modified to include the effects of an ellipsoidal shaped debris. The modified time dependent Reynolds and elasticity equations were simultaneously solved to determine the pressure and film thickness in EHL contacts. The debris force balance equation was solved to determine the debris velocity. The model was then used to obtain results for a variety of loads, speeds, and debris sizes. The results indicate that the debris has a significant effect on the pressure distribution and causes a dent on the rolling/sliding bounding surfaces. Depending on the size and location of the debris the pressure generated within the contact can be high enough to plastically deform the bounding surfaces. Debris smaller than the minimum film thickness do not enter the contact and only large and more spherical debris move toward the contact. [S0742-4787(11)00501-7]

Thermal Elastohydrodynamic Lubrication of Rider Rings

1984 ◽  
Vol 106 (4) ◽  
pp. 492-498 ◽  
Author(s):  
Vilmos Simon
Keyword(s):  
Film Thickness ◽  
Numerical Solution ◽  
Temperature Variation ◽  
Elastohydrodynamic Lubrication ◽  
Operating Conditions ◽  
Performance Characteristics ◽  
Oil Film ◽  
The Real ◽  
Ring Geometry

The full thermal elastohydrodynamic analysis of the lubrication of rider rings is presented. A numerical solution of the coupled Reynolds, elasticity, energy, and Laplace’s equations for the oil film thickness, pressure, and temperature and rider rings temperatures is obtained. The temperature variation across the oil film is included. The real rider ring geometry is treated. The effect of the operating conditions on the performance characteristics is discussed.

scholarly journals Non-Newtonian Fluid Model Incorporated Into Elastohydrodynamic Lubrication of Rectangular Contacts

1984 ◽  
Vol 106 (2) ◽  
pp. 275-282 ◽  
Author(s):  
B. O. Jacobson ◽  
B. J. Hamrock
Keyword(s):  
Shear Stress ◽  
Shear Strength ◽  
Film Thickness ◽  
Numerical Solution ◽  
Newtonian Fluid ◽  
Fluid Model ◽  
Elastohydrodynamic Lubrication ◽  
Sliding Velocity ◽  
Minimum Film Thickness ◽  
Limiting Value

A procedure is outlined for the numerical solution of the complete elastohydrodynamic lubrication of rectangular contacts incorporating a non-Newtonian fluid model. The approach uses a Newtonian model as long as the shear stress is less than a limiting shear stress. If the shear stress exceeds the limiting value, the shear stress is set equal to the limiting value. The numerical solution requires the coupled solution of the pressure, film shape, and fluid rheology equations from the inlet to the outlet. Isothermal and no-side-leakage assumptions were imposed in the analysis. The influence of dimensionless speed U, load W, materials G, and sliding velocity U* and limiting-shear-strength proportionality constant γ on dimensionless minimum film thickness Hmin was investigated. Fourteen cases were investigated for an elastohydrodynamically lubricated rectangular contact incorporating a non-Newtonian fluid model. The influence of sliding velocity (U*) and limiting shear strength (γ) on minimum film thickness was observed to be small. Hence the film thickness equation obtained for a Newtonian fluid is sufficient for calculations considering non-Newtonian effects. Computer plots are also presented that indicate in detail pressure distribution, film shape, shear stress at the surfaces, and flow throughout the conjunction.

Elastohydrodynamic Squeeze Films Between Two Cylinders in Normal Approach

1970 ◽  
Vol 92 (2) ◽  
pp. 292-301 ◽  
Author(s):  
K. Herrebrugh
Keyword(s):  
Integral Equation ◽  
Bifurcation Point ◽  
Present Problem ◽  
Proper Treatment ◽  
Elasticity Equations ◽  
Single Integral ◽  
The One

In this approach the fundamental hydrodynamic and elasticity equations that govern the problem are reduced to one single integral equation. Although the resulting integral equation in the present problem is at first sight similar in form to the one governing the steady rolling problem, on closer consideration it appears that the former constitutes a different type of equation, which is mainly characterized by the existence of a bifurcation point not occurring in the equation for steady rolling. By proper treatment of this complication, solutions also in the range of conditions where deformation may be expected to be large are obtained.

The application of integral equation methods to the numerical solution of some exterior boundary-value problems

1971 ◽  
Vol 323 (1553) ◽  
pp. 201-210 ◽  
Keyword(s):  
Integral Equation ◽  
Wave Equation ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Integral Operators ◽  
Exterior Boundary ◽  
Integral Equation Methods ◽  
Singular Kernels ◽  
Single Integral

The application of integral equation methods to exterior boundary-value problems for Laplace’s equation and for the Helmholtz (or reduced wave) equation is discussed. In the latter case the straightforward formulation in terms of a single integral equation may give rise to difficulties of non-uniqueness; it is shown that uniqueness can be restored by deriving a second integral equation and suitably combining it with the first. Finally, an outline is given of methods for transforming the integral operators with strongly singular kernels which occur in the second equation.

Elastohydrodynamic Lubrication of Hypoid Gears

1981 ◽  
Vol 103 (1) ◽  
pp. 195-203 ◽  
Author(s):  
V. Simon
Keyword(s):  
Film Thickness ◽  
Numerical Solution ◽  
Elastohydrodynamic Lubrication ◽  
Operating Conditions ◽  
Performance Characteristics ◽  
Temperature Variations ◽  
Hypoid Gears ◽  
Oil Film ◽  
Gear Teeth ◽  
Energy Equations

The full thermal elastohydrodynamic analysis of the lubrication of hypoid gears is presented. A numerical solution of the coupled Reynolds, elasticity and energy equations for the pressure, temperature and film thickness is obtained. The temperature variations across the oil film and in the pinion and gear teeth are included. The real tooth geometry of the modified hypoid gears is treated. The effect of the operating conditions on the performance characteristics is discussed.

Influence of piezo-viscous behavior and load on the effectiveness of inlet zone bump in unidirectional pure sliding EHL line contacts

2018 ◽  
Vol 70 (9) ◽  
pp. 1766-1773
Author(s):  
Punit Kumar
Keyword(s):  
Film Thickness ◽  
Elastohydrodynamic Lubrication ◽  
Governing Equations ◽  
Local Pressure ◽  
Content Type ◽  
Elasticity Equations ◽  
Line Contacts ◽  
Lower Load ◽  
Inlet Zone ◽  
Lubricant Viscosity

Purpose The purpose of this paper is to introduce the concept of stationary inlet zone bump (IZB) for film thickness enhancement in unidirectional pure sliding elastohydrodynamic lubrication (EHL) line contacts and to investigate the effects of maximum Hertzian pressure (load) and piezo-viscous response on the effectiveness of IZB. Design/methodology/approach The numerical analysis involves the solution of Reynolds and elasticity equations. The well-established Doolittle–Tait equations are used herein to determine the lubricant viscosity and density as functions of local pressure, while the Carreau model is used to describe the lubricant rheology. The IZB is assumed to have a sinusoidal profile and it is present on the stationary surface. The governing equations are discretized using finite difference scheme and solved using the Newton–Raphson technique. Findings Two test oils, L7808 and SR600, with linear and exponential piezo-viscous responses in the inlet zone are considered here for comparison. The effectiveness of IZB in terms of film thickness enhancement is found to be more for SR600. Besides, IZB is found to be more effective at lower values of maximum Hertzian pressure. The bump needs to shift downstream at higher load to be as effective as at lower load. Originality/value This is the first paper to simulate EHL characteristics in the presence of a stationary IZB and to study the effect of various parameters on EHL effectiveness. The film thickness enhancement obtained here is remarkable and hence it is a novel and valuable contribution.

Non-Newtonian Elastohydrodynamic Lubrication of Point Contact

1991 ◽  
Vol 113 (4) ◽  
pp. 703-711 ◽  
Author(s):  
Kyung Hoon Kim ◽  
Farshid Sadeghi
Keyword(s):  
Film Thickness ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Two Dimensional ◽  
Point Contacts ◽  
Dimensionless Velocity ◽  
Simultaneous System ◽  
Elasticity Equations ◽  
Minimum Film Thickness ◽  
Pressure Spike

A numerical solution to the problem of isothermal non-Newtonian elastohydrodynamic lubrication of rolling/sliding point contacts has been obtained. The multigrid technique is used to solve the simultaneous system of two-dimensional modified Reynolds and elasticity equations. The effects of various loads, speeds, and slide to roll ratios on the pressure distribution, film thickness, and friction force have been investigated. Results for the dimensionless load W = 4.6 × 10−6 and 1.1 × 10−6, and the dimensionless velocity U = 3 × 10−10 and 3 × 10−11 are presented. The results indicate that slide to roll ratio has negligible effect on the minimum film thickness, however, it significantly reduces the pressure spike.

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