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- Wydawca: John Wiley & Sons
- Kategoria: Nauka i nowe technologie
- Język: angielski
- Rok wydania: 2014

Contents

Preface

Nomenclature

1: Introduction

1.1. Lubrication regimes – Stribeck curve

1.2. Topography of rough surfaces

1.3. Bibliography

2: Computing the Hydrodynamic Pressure

2.1. Patir and Cheng stochastic model

2.2. Model based on a direct computation of the flow factors

2.3. Homogenization method

2.4. Comparison between the flow factors obtained with Patir and Cheng, direct computation and homogenization models

2.5. Example of pressure profiles obtained from flow factors calculated with Patir and Cheng, direct computation and homogenization models

2.6. Comparison with deterministic computations

2.7. Bibliography

3: Computing the Contact Pressure

3.1. Concept of sum surface

3.2. Elastic contact model proposed by Greenwood and Williamson

3.3. Elasto-plastic contact model proposed by Robbe-Valloire et al.

3.4. Elasto-plastic double-layer contact model proposed by Progri et al.

3.5. Model based on discrete Fourier transformation

3.6. Deterministic model based on finite elements

3.7. Using the contact models

3.8. Influence of the roughness deformation generated by the contact pressure on the flow factors

3.9. Using the contact models in an industrial context

3.10. Bibliography

4: Wear

4.1. General concepts about wear

4.2. Running-in

4.3. Experimental determination of the Archard coefficient

4.4. Numerical modeling of the wear

4.5. Bibliography

Index

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George's RoadLondon SW19 4EUUK

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John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2014

The rights of Dominique Bonneau, Aurelian Fatu and Dominique Souchet to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2014942900

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-84821-682-2

Preface

This volume is the second part of a four part book series dedicated to hydrodynamic bearings.

The first volume describes the physical properties of lubricants that play an essential role within the hydrodynamic process; it then explains the equations of hydrodynamic lubrication as well as the models and numerical resolution. A description of “elastohydrodynamic” (EHD) models is also included in the contents of the first volume.

The numerical modeling of thin-film flows and the deformed walls which limit the film under pressure leads to the discretization of the domain occupied by the film by one of the methods described in the first volume. The more tortuous the boundary shape of the domain is, the finer the spatial discretization must be. From a sufficient distance, the surfaces that delineate the space occupied by the lubricant film appear smooth. If we retain the finite element method for the discretization of the equations, elements of only a few millimeters in size appear appropriate. However, fine profile measurements reveal flaws in the shape (plane, cylindrical, etc.) of the surfaces with amplitude of the order of micrometers and variable wavelengths ranging from a few tenths of micrometers to several millimeters. When the average local film thickness reaches the order of height of these surface flaws, the local pressure varies strongly due to the many convergent and divergent film zones generated by these flaws. The size of the elements required to describe the variations must be of the order of magnitude of the shortest wavelengths, i.e. of a few tenths of micrometers. The number of elements required by this approach would be incompatible with reasonable calculation times.

This volume describes the methods that can help us to take surface flaws of small wavelengths into account, without the need for fine discretization of the domain.

The first chapter will introduce the reader to mixed lubrication by showing the different parameters that enable characterizing rough surfaces.

When the thickness of the lubricant film becomes very small, the roughness of the surfaces has two effects which are to be taken into account.

First, as the lubricant is hampered by rough surfaces, it meets a higher resistance when flowing between the surfaces. This increase in the apparent viscosity of the lubricant becomes more significant as the surfaces become closer together. The Reynolds equation then requires a modification which, through the introduction of flow factors, helps us to take this increase in the apparent viscosity of the lubricant into account. The changes relating to the flow factors and how to calculate them are presented in Chapter 2.

The second effect produced by the roughness of the surfaces appears as follows: as the surfaces become sufficiently close to each other, contact zones appear at the top of the roughness reliefs, which produce pressure contact that is added to the hydrodynamic pressure. Calculating the flow factors and obtaining the relationship between the contact pressure and the flow factors and the distance between the average surfaces of the walls require numerical methods, which are detailed specifically in Chapter 3. The performance of these methods is compared.

The last chapter of this volume addresses the issue of the surface wear and provides a few numerical models to calculate it.

A large part of the content of this volume is extracted from Ramona Dragomir–Fatu's thesis [DRA 09] dedicated to the mixed lubrication issues within the engine bearings, in collaboration with Renault.

In the third volume, several thermo-hydrodynamic (THD) and thermoelastohydrodynamic (TEHD) models will be described. This volume is complemented by the description of the general algorithms for software to calculate bearings under non-stationary severe loads.

The problems specific to the calculation of engine and compressor bearings are detailed in the fourth and last volume. A chapter presents the various techniques that can be used to optimize the calculation of the bearings. An example of optimization for a connecting rod big end bearing for an internal combustion engine is described in detail.

Bibliography

[DRA 09] DRAGOMIR-FATU R., Study and modeling of mixed lubrication and types of associated damage in motor Bearings (in French), PhD Thesis, University of Poitiers, France, 2009.

M

point within the lubricant film

M

1

point on the wall 1 of the lubricant film

M

2

point on the wall 2 of the lubricant film

x

,

y

,

z

Cartesian basis for the film

Ω, Ω

F

film domain

Ω

S

domain occupied by a solid

∂Ω

boundary of the film domain

∂Ω

S

boundary of a solid

He

Hersey number

C

m

bearing radial clearance

E

1

,

E

2

N m

-2

Young modulus for solids 1 and 2 in contact

E

c

discretized contact equation

E

q

N m

-2

equivalent Young modulus (contact between two solids)

F

kg m

-2

Couette flow rate factor

G

kg (Pa.s)

-1

Poiseuille mass flow rate factor

H

1

m

level of wall 1 at point with

x

,

z

projected coordinates

H

2

m

level of wall 2 at point with

x

,

z

projected coordinates

J

,

J

2

m Pa

-1

s

-1

integrals on film thickness

K

N

,

K

T

N m

-1

penalization stiffness for a contact problem

Q

m

kg s

-1

lubricant mass flow rate

Q

v

m

3

s

-1

lubricant volume flow rate

R

m

bearing radius

U

m s

-1

shaft peripheral velocity for a bearing

U

1

m s

-1

velocity of wall 1 in

x

direction at point (

x

,

H

1

,

z

)

U

2

m s

-1

velocity of wall 1 in

x

direction at point (

x

,

H

2

,

z

)

V

m s

-1

squeezing velocity

V

1

m s

-1

velocity of wall 1 in

y

direction at point (

x

,

H

1

,

z

)

V

2

m s

-1

velocity of wall 2 in

y

direction at point (

x

,

H

2

,

z

)

W

m s

-1

shaft axial velocity for a bearing

W

1

m s

-1

velocity of wall 1 in

z

direction at point (

x

,

H

1

,

z

)

W

2

m s

-1

velocity of wall 2 in

z

direction at point (

x

,

H

2

,

z

)

f

Coulomb friction coefficient

h

m

lubricant film thickness

h

l

m

local lubricant film thickness

m

mean thickness of the lubricant film

P

Pa

pressure in the lubricant film

Pa

mean pressure in the film

q

m

kg

s

-1

m

-1

mass flow rate per arc length unit for a curve

q

v

m

2

s

-1

volume flow rate per arc length unit for a curve

u

m s

-1

circumferential velocity component at a point within the film

v

m s

-1

velocity squeeze component at a point within the film

w

m s

-1

axial velocity component at a point within the film

x

m

circumferential coordinate for a point within the film

y

m

coordinate in the thickness direction for a point within the film

z

m

axial coordinate for a point within the film

γ

roughness elongation factor

δ

1

,

δ

2

m

roughness heights for surfaces 1 and 2

ϕ

f

,

ϕ

fs

,

ϕ

fp

correction factor for the shear stress (mixed lubrication)

ϕ

x

,

ϕ

z

,

ϕ

xx

,

ϕ

xz

,

ϕ

zx

,

ϕ

zx

Poiseuille flow factors

ϕ

s

,

ϕ

sx

,

ϕ

sz

Couette flow factors

μ

Pa.s

lubricant dynamic viscosity

1

,

2

Poisson coefficients for solids 1 and 2 in contact

σ

m

combined roughness of film walls

σ

1

,

σ

2

m

roughness of walls 1 and 2

τ

xy

,

τ

zy

Pa

shear stress within the lubricant film

h

/

σ

d

e

m

elastic deformation normal to the film wall

n

unit vector orthogonal to a domain boundary

x

unit vector in the direction of the shaft surface displacement (developed bearing)

x

c

,

y

c

,

z

c

unit vectors for a bearing;

z

c

parallel to the bearing axis

y

unit vector in the direction of the film thickness (developed bearing)

z

unit vector equal to

x

∧

y

p

, {

p

}

Pa

vector of pressure nodal values

p

c

Pa

contact pressure

[C]

m Pa

-1

compliance matrix

[K]

N m

-1

stiffness matrix

1, 2

surfaces delimiting the film

F

film or lubricant

S

shaft, solid

CPV

contact pressure × velocity product

DFT

discrete Fourier transform

EHD

elastohydrodynamic

FE, FEM

finite element method

FFT

fast Fourier transform

IDFT

inverse discrete Fourier transform

MFT, MOFT

minimum (oil) film thickness

MOFP

maximum oil film pressure

The numerical modeling of thin film flows and the deformation under pressure of the walls that bound the film requires a discretization of the domain occupied by the film using the methods described in Chapter 3 of [BON 14] for research into the field of pressure (Reynolds equation) and in Chapter 4 of [BON 14] for the deformations (elasticity equations). These numerical methods require even finer spatial discretizations when the shape of the walls that delimit the film domain is rough. Seen from a certain distance, the surfaces of a shaft and a sleeve appear smooth. When the finite element method is used to discretize equations, elements that are a few millimeters in length seem suitable. However, fine profilometric measurements reveal defects in the forms (flatness, cylindrical shape, etc.) whose amplitude is of the order of micrometers and wavelengths varying from a few tens of micrometers to several millimeters. When the average local thickness of the film becomes equivalent to the height of the surface defects, the local pressure varies considerably under the influence of the numerous convergents and divergents that cause these defects. The size of the subdomains required to describe these variations should be of the order of magnitude of the shortest wavelengths – in other words some tens of micrometers. The numerous elements involved in such an approach mean that the computation time becomes prohibitive.

This chapter describes the main parameters used in the modeling of rough surfaces.

An average reference surface is defined for each facing surface (see section 1.2.1.1). The roughness of each surface is characterized by its standard deviation (see section 1.2.1.2), which allows us to define an equivalent roughness σ for the pair of the two surfaces. Therefore, the dimensionless average distance between the two surfaces is defined by:

[1.1]

where h is the distance between the average surfaces of each surface. Three lubrication regimes are distinguished, depending on the value of (Figure 1.11):

– > 3: hydrodynamic regime;

– 3 ≥ > 0.5: mixed regime;

– ≤ 0.5: boundary regime.

Passage from one regime to another can be characterized by a graph representing the friction as a function of Hersey’s number written as He, a dimensionless characteristic involving the viscosity μ, of the lubricant in Pa.s, the relative velocity of the surfaces, the average pressure p in Pa. For a bearing, this is expressed as:

where ω is the frequency of rotation of the bearing in revolutions per second (rps).

Figure 1.1.Lubrication regimes as a function of the film thickness: a) hydrodynamic; b) mixed; c) boundary

The resulting graph, of which an example is shown in Figure 1.2, is known as the Stribeck curve.

Figure 1.2.Stribeck curve and lubrication regimes

After intense friction at low values of He (low speed or major stress) due to frequent contact between the surface asperities typical in a boundary regime, the friction diminishes as the hydrodynamic aspect increases (the mixed regime). When the thickness has increased sufficiently, the effect of roughness is no longer detectable, and the friction coefficient increases linearly with the speed, as the shear stress in the case of a hydrodynamic regime. In the case of boundary lubrication regimes, the friction coefficient remains markedly less than that obtained for dry surfaces because of the molecular layers of additives that remain adsorbed on them.

The bearings of internal combustion engines function principally in hydrodynamic and mixed modes.

Surface properties play an essential role in all processes where the surface forms an interface. The characterization of surface properties constitutes a vast discipline, which includes physical, chemical and geometric characteristics, among others. Only geometric characteristics will be considered below.

In light of the importance of relative parameters to the notion of surface finish, it is important to clarify the definitions of key parameters used in formulating equations for mixed lubrication. Only commonly used systems are shown.

The term “surface roughness”, means the geometric deviation of the actual surface of a part from a geometrically ideal or flawless surface, whether that is on a macroscopic or microscopic level. In engineering, this is what is usually meant by “surface finish”.

Defects in the surface do not all have the same influence on the performance of a workpiece. Three types of defects can be distinguished using experimental techniques for measuring the microgeometrics of the surface and standard signal processing techniques (numerical filters, statistical concepts and shape recognition). These are defects of form, waviness and roughness.

Surface metrology techniques may or may not require contact with the surface:

– measurement involving contact: a stylus is applied to the part using a standard, constant pressure. This stylus ends in a pyramidal point made of diamond, tipped with a spherical cap which is 2 to 10 μm in radius. The speed at which the stylus moves is usually less than a few millimeters per second;

– non-contact measurement: optical profilometers are used. To provide localized measurements, the optical technique usually uses converging beams that are reflected by the surface under examination. The absence of contact means that an optical profilometer can operate much more quickly than a contact profilometer.

Figure 1.3.Types of defects

The zones under examination are of relatively limited dimensions, only rarely exceeding a hundred millimeters. There are two types of examinations (Figure 1.4)

– examination following a line or generatrix (profilometry);

– examination of a zone or surface (surfometry).

Figure 1.4.Examples of 2D and surface profiles

A profilometric measurement following a line or a generatrix is characterized by two types of parameters:

– parameters issuing from statistical treatment of the heights measured without reference to their distribution along the measurement line;

– parameters issuing from statistical treatment of heights measured in correlation with their distribution along the measurement line.

Figure 1.5.Reference height

There are many ways of defining a reference height [HAM 04]. The simplest way is to take a mean line (or mean plane for a surface measurement) such that the area of the zone situated below this line (or the volume below this surface) is equal to the area of the zone above it (Figure 1.5). If the nx points of measurement are regularly spaced along the line (or plane) of measurement, it becomes very simple to calculate the arithmetic average of the heights:

[1.2]

The level of reference thus calculated does not compensate for errors in measurement, particularly for defects in gradient. A study of the line (or plane for surface measurements) that minimizes the quadratic error for the set of points of measurement (least squares method) allows us to correct defects in gradient. In the case of a linear measurement, the resulting equation of the “mean” line is written as:

The quadratic error between the points measured and this line is expressed as:

The minimization of error Eq results in the cancellation of the derivatives of Eq in relation to the coefficients a and b. The solution to the system of the two linear equations obtained gives the values of a and b. A “correction” of the set of points measured:

results in a set of aligned data whose average is zero.

This technique can also be used to eliminate defects in the measurement of more complex forms. For example, surfometric measurements of bearings for internal combustion engines involve the cylindrical form of the bearing. Since the surface measured is limited (a few square millimeters in area), the form of the cylinder can be represented by a quadratic equation in terms of x and y whose coefficients can easily be determined by the least squares method.

This treatment uses centered moments of the distribution of the ordinate values in the profile. It consists of statistical treatments used for discretized variables.

The centered variable is the height y(x) of the profile in relation to the mean line. This variable is centered, as its mean value is null by definition.

The p(y) density function of the ordinates on the profile creates a bell curve that can be quantified by means of different parameters, which are known as centered moments. The n-order centered moment Mn of the distribution is defined as follows:

[1.3]

[1.4]

Figure 1.6.Discretization of the roughness profile and roughness distribution

Whichever method of measurement is used, the heights noted are situated in a finite interval of [hmin, hmax]. Therefore, the number of ny levels of selection is finite. In practice, 99.9% of heights are situated in the interval [ – 3σ, + 3σ] [HAM 04] where and σ, respectively represent the arithmetic mean and standard deviation of the measured values:

[1.5]

and the values of hmin and hmax can be, respectively, replaced by h – 3σ and h + 3σ.

Therefore, the centered moments are given by:

[1.6]

It can easily be seen that the centered moments are also given by:

[1.7]

When the distribution of heights is Gaussian, the function p(z) is expressed as:

[1.8]

and gives:

Thus, it can be deduced that for a Gaussian distribution the odd-ordered centered moments are null and the even-ordered centered moments are given by:

[1.9]

Figure 1.7.Illustration of different skewness values

Figure 1.8.Illustration of different kurtosis values

The centered moments used for the description of the topology of profiles or surfaces are:

Three situations can be distinguished (Figure 1.8):

For a description of the topology of profiles or surfaces, we should add the following parameter, which describes the average deviation between the profile and the mean line:

[1.10]

For a discretized profile in nx points with a constant step Δx, the arithmetic average can be obtained using the following expression:

[1.11]

The types of analysis described previously have privileged the ordinate values over the abscissas. The following treatments, however, concern the spacing of the asperities. This type of analysis generally uses one of the three following functions: autocorrelation, structure function or the spectral density. These three functions differ in form, but they contain the same quantity of data.

– Autocorrelation: the autocovariance R(t) quantifies the degree of correlation existing between the points of the profile situated a distance t apart:

[1.12]

The autocorrelation r(t) is deduced from the autocovariance by the relationship:

[1.13]

If we use the discretized form, the density function of the ordinates, or autocorrelation is given by:

[1.14]

Figure 1.9.Roughness profile and correspondent autocorrelation function

The observation from a profile and graph of its autocorrelation function (Figure 1.9) allows us to discern better the usefulness of this mathematical tool. The autocorrelation function is characterized by a single maximum value (the peak) equal to 1 when the correlation is maximal, and a rapid decrease of its value to zero, if the profile is not periodic. For a periodic profile the autocorrelation function is also periodic. The autocorrelation length λ (sometimes simply called correlation length) is the distance calculated between an abscissa whose autocorrelation value has diminished by a certain degree and the abscissa of the maximum of the autocorrelation function. In the examination of rough surfaces, autocorrelation lengths are generally chosen in such a way so as to delimit the entire form of the roughness motif as precisely as possible. The correlation length can be assimilated to the scale of roughness in the direction in which it is evaluated, thus defining the distance beyond which no point retains any relation with the other points in the profile. Correlation lengths are most often measured at a decrease of 50%, 80% or 90% of the maximal value of the autocorrelation function. This value fluctuates, depending on the others, but according to Stout [STO 00] it is preferable to truncate the autocorrelation function at an 80% decrease because beyond this point the disturbances can skew the representativeness of the autocorrelation length.

– The structure function S(t) is defined by:

[1.15]

The structure function S(t) is linked to the autocorrelation r(t) by the relationship:

[1.16]

– The power spectral density: this reveals the periodicities of the profile as a result of a decomposition of the profile y(x) into a series of sinusoids (Fourier transform). Y(k) is the Fourier transform of the profile y(x):

[1.17]

Therefore, the power spectral density D(k) is given by:

[1.18]

The principle of fractal analysis relies on the fact that beyond a certain level of detail, we can identify the surface profile of a part perfectly. Relatively recent measurements of real surfaces have proven that technical surfaces have a fractal dimension and that this dimension could be representative of certain properties of the surfaces.

The following standards only concern measurements taken on generatrices (profilometry). They cover all aspects of measurement.

The procedure for characterization consists of three main operations:

– measurement of the profile of the part on a generatrix;

– treatment of the actual profile in order to extract either the roughness profile or the waviness profile;

– calculation of the parameters of the surface topography.

The treatment of the profile and the calculation of the parameters of the surface can be conducted according to three international standards: EN ISO 4287, EN ISO 4288 [ISO 96a, STA 98a, STA 98b] and EN ISO 12085 [ISO 96b, STA 98c].

Standard EN ISO 12085 was established as a result of cooperation between French carmakers.

The wear on rough surfaces is often described according to the parameters of the Abbot curve. This is defined by standard EN ISO 13565 [ISO 96c, STA 95].

The specification and measurement of three-dimensional (3D) surface texture parameters was defined in 2012 by standard ISO 25178 [ISO 12].

The roughness profile is obtained by filtering the surface profile. The aim of this filtering is to isolate the contribution of the various defects from the profile of the part. The frontier currently used to separate the defects (waviness and roughness) is situated at 0.8 mm. The following values can also be used: 0.08, 0.25 and 2.5 mm. This parameter is called the base length.

After this filtering, we can calculate the parameters on the different profiles:

– form profile;