Mixed Lubrication in Hydrodynamic Bearings

By Dominique Bonneau, Aurelian Fatu and Dominique Souchet

This Series provides the necessary elements to the development and validation of numerical prediction models for hydrodynamic bearings. This book is dedicated to the mixed lubrication.

• Language: English
• Publisher: Wiley
• Release date: Aug 8, 2014
• ISBN: 9781119008057

Book preview

1

Introduction

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.

1.1. Lubrication regimes – Stribeck curve

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.

1.2. Topography of rough surfaces

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

1.2.1. 2D profile parameters

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

1.2.1.1. Definition of the 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.

1.2.1.2. Statistical treatment of the ordinate

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]

Generally, the nx points of measurement are regularly spaced from Δx along measurement line and are very numerous. In this case, the density function of the ordinates can be obtained by the construction of a histogram of the nx heights of the profile on a selection of ny heights yj, j = 1, ny, is regularly spaced from Δy (Figure 1.6):

[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:

– second-order moment: variance of the population. The standard deviation of the population, otherwise known as the quadratic average (written as Rq) or the Root Mean Square (RMS), is used;

– third-order moment: obliquity (or skewness) of profile. In the domain of surface topography, the relative skewness is preferred. This parameter expresses the asymmetry of the profile on either side of the mean line. Three different cases can be distinguished (Figure 1.7):

- Rsk = 0: the profile is symmetric (Gaussian distribution);

- Rsk > 0: the profile shows more peaks than valleys;

- Rsk < 0: the profile shows more valleys than peaks;

– fourth-order moment: peakedness of the distribution (or kurtosis). In the domain of surface topography, the normalized peakedness is preferred.

Three situations can be distinguished (Figure 1.8):

- Rku = 3: the distribution is Gaussian;

- Rku > 3: the distribution is spread;

- Rku < 3: the distribution is narrowed.

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]

1.2.1.3. Statistical treatment of the ordinate respective to the abscissa

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