Davim J.P. Tribology for Engineers: A Practical Guide

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Lubrication and roughness

where 

= R/L is the bearing scale factor, L is the bearing

length, and the velocity can be deﬁ ned as u = R

The gap between journal and bearing can also be presented

in dimensionless form, using the notations e for eccentricity of

journal center, c for clearance (c = R

– R), H = h/c, ε = e/c, and

representing the cylindrical surfaces by planes, as follows:

[3.20]

This expression is justiﬁ ed usually at c/R

ratios smaller

than 0.1.

Assuming that the bearing parameters in the

-direction

(x,y-plane) are less than the bearing length; a 

< 1 (long

bearing, thus the pressure does not change along Z and

∂P/∂Z = 0); and when the gap conﬁ guration does not change

with time, ∂H/∂T = 0, eq. [3.19] simpliﬁ es to

[3.21]

The boundary conditions for this case are P(

) = P(2

) = P

and called Sommerfeld’s conditions. After the ﬁ rst integration

we have

[3.22]

and further integration of both parts of eq. [3.22] yields

[3.23]

The integration constants are determined subject to the

above boundary conditions, namely

[3.24]

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Tribology for Engineers

The integrals in eqs. [3.23] and [3.24], known as the journal-

bearing integrals, are obtainable for instance from Booker,

1965. Denoting the hydrodynamic pressure at

= 0 by

(pressure at the initial angle), we have the pressure

distribution equation as

[3.25]

This solution was ﬁ rst arrived at by Sommerfeld in 1904,

and remains topical to this day.

The redundant hydrodynamic pressures, P-P

, calculated

by eq. [3.25], are presented in Fig. 3.6 for several values of

the dimensionless eccentricity ε. The pressure in the solution

has positive and negative branches with equal maximum and

minimum. As can be seen, the pressure increases with the

p/2 p 3/2p 2p

Dimensionless pressure distribution as function

of angle at different eccentricities

Figure 3.6

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Lubrication and roughness

eccentricity; this fact should be borne in mind in the bearing

design when choosing the operating eccentricity.

3.6 Hydrodynamic lubrication of

roughened surfaces

The equations derived in the preceding sections are valid for

ideally smooth surfaces, which is not the case in practice.

Under the microscope we can see micro-deviations from the

ideal form; a schematic view of the roughness traced by

measurement stylus, together with some roughness

parameters, is presented in Fig. 3.7. Here R

is the mean

asperity height and S

and h

are the local roughness step and

height. There are also other roughness parameters speciﬁ ed

in various national and international standards (e.g. ANSI,

DIN, ISO). So, the R

-values speciﬁ ed by ISO are related

to the roughness grade number N (N1, N2, . . ., up to N11)

and range from 0.025 up to 25 μm accordingly. The mean

roughness step (interval between asperities), named

Real surface proﬁ le and some roughness

parameters

Figure 3.7

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Tribology for Engineers

sometimes roughness wavelength, varies in the wide range

3.5–75

m; both roughness parameters are connected with

the type of surface ﬁ nish, from lower values for ﬁ ne machining

to higher values for rough machining.

The size of the surface asperities often equals and even

exceeds the ﬁ lm thickness (see Section 3.2), thus they

naturally inﬂ uence the hydrodynamic behaviour of the ﬁ lm.

Knowing how the roughness parameters, asperity height R

and step S inﬂ uence the ﬁ lm pressure, the data obtained for

smooth surfaces can be adjusted with a view to more reliable

design of the machine parts.

Hydrodynamic lubrication of roughened surfaces has long

been an object of study at ﬂ uctuating levels of intensity, see for

example Dowson et al. (1978). With the advent of laser

machining, interest in it has peaked again. A ﬂ ow model was

developed by Patir and Cheng (1978, 1979) using averaged

roughness indices, and later gradually replaced by deterministic

models based on detailed roughness geometry. This approach

had already been attempted in early studies of microasperities

and microcavities (Anno and Walowit, 1968, 1969), and more

recently in that of regularly and irregularly covered porous

surfaces (Lai, 1994; Burstein and Ingman, 1999, 2000; Arghir

et al., 2003). Common to these studies is emphasis on the

integrated surface behaviour, it being assumed that the maximal

load support found for one microgroove is the same for all the

others; the cumulative effect is not considered. In general, it is

necessary to examine the entire proﬁ le or, in three-dimensional

terms, the entire real surfaces separated by the lube.

Most recently in the last few decades, we are witnessing

a new approach whereby the opposite surfaces are

deterministically described by some idealized but uniﬁ ed

expression, thus permitting in principle a theoretical solution

(Labiau et al., 2008; Letallear et al., 2002; Olver and Dini,

2007). One common model for regular structure surfaces is

the standardized version (e.g. ANSI/ASME Standard, 1985;

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Lubrication and roughness

SRM 2073a, 1995) where the roughness is presented as a

sinusoidal ripple surface with a given frequency (Tønder,

1999, 2004) obtainable experimentally, e.g. the main

frequency of the spectral density function determined from

the spectral diagram of the concrete surface.

In parallel, 1D and 2D transient lubrication was

theoretically studied for surfaces with equal asperity sizes,

and ﬁ nally generalized in a model representing unequal

roughnesses and unequal wavelengths along the axes – of

which the equal-roughness version is a particular case (see

Burstein, 2006, 2008, 2009). In these studies the opposite

surfaces also had equal wave numbers along the identical

axes. Under the generalized approach, sinusoidal roughness

and the same number of waves were assumed on both

surfaces. Lately non-sinusoidal 1D and 2D roughness models

were developed (Burstein, 2010), in which it was possible to

study the unequal case for both the asperity heights and

wavelengths. Hereafter two roughness types are brieﬂ y

presented: sinusoidal and triangular wave conﬁ gurations.

3.6.1 Roughness parameters and surface

roughness models

As was noted earlier, even perfectly ﬁ nished machine parts

have haphazardly distributed numerous micro or nano peaks

and hollows reﬂ ected in a similar pattern in the gap width

and correspondingly in the ﬁ lm thickness geometry, with the

attendant difﬁ culties in theoretical and computerized

treatment. The main problem is that the randomly generated

opposite surfaces (see Fig. 3.8, surface generated by Matlab)

are in relative motion with the gap changing from moment

to moment, so that Reynolds’ equation cannot be solved

analytically. Numerical 1D or 2D solutions are also highly

problematic because of the large numbers of asperities on the

real surfaces.

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Tribology for Engineers

The problem becomes manageable if the proﬁ le is

approximated by some regular conﬁ guration, mostly by a

sinusoid or by a triangular wave function. A computer-

generated view of such surfaces, modelled in Matlab, is

presented in Fig. 3.9(a) and (b). In spite of the outward

similarity of these surfaces, the triangular pattern is maybe

more realistic than the sinusoidal – see Fig. 3.7. In any event,

applying these models permits better evaluation of the

surface indices, selecting those suitable for each speciﬁ c case.

Computer image of surface with randomly

generated asperity heights and roughness step

Figure 3.8

Note: Asperity heights 2

m, roughness step 2.5R

, standard deviation 0.2 of the

asperity height; ﬁ ve asperities in each dimension; azimuth –31, elevation 86

Surface with sinusoidal (a) and triangular

(b) roughness

Figure 3.9

Note: Asperity heights 2

m and roughness step 2.5R

, ﬁ ve waves in each

dimension; azimuth –17, elevation 66

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Lubrication and roughness

3.6.2 Hydrodynamic lubrication for

sinusoidal roughness

Reynolds’ equation can be rewritten in dimensionless form

as follows:

[3.26]

where

, and subscripts 1

and 2 denote the stationary and moving surfaces respectively.

The x-plane of the proﬁ le geometry and the 3D view of the

surfaces are presented in Fig. 3.10.

With the surfaces in sinusoidal form, the dimensionless

gap is given by

[3.27]

where A

and A

are the dimensionless asperity heights,

= R

/(Δ + R

+ R

), A

= R

/(Δ + R

+ R

), and 

= k

and 

= k

are the wave ratios; k

, k

are the

wave numbers along the X and Z axes for surfaces 1 and 2;

Φ =

/(2

) – the phase angle of the moving surface. Here

the roughnesses – R

, R

and the wave numbers k

, k

– differ between the surfaces; and the origin of coordinates

is set such that the phase angle Φ

= 0.

Assuming that the surfaces have the same width and length

= L

= L or 

= 1, and the same wave ratios 

= 

= ,

we can concentrate on two cases – (a) the wave numbers in

the X- and Z-directions being equal (scale factor  = 1) and

(b) the wave numbers unequal (

≠ 

, scale factor  ≠ 1).

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Tribology for Engineers

Solution for equal wave numbers

In the case of identical wavelength characteristics in both

the X- and Z-directions for the two surfaces, 

= 

= ,

and bearing in mind the trigonometric expressions for a

general sinusoid (see, for example, Bronshtein et al., 2007)

with amplitudes A

and A

, period 2

k, and phases Φ

= 0

Schematic of unequally roughened surfaces

(a) and gap geometry in X,H plane (b) at wave

number k = 5, roughness height ratio R

0.5, and phase displacement Φ = 1/(4k)

Figure 3.10

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Lubrication and roughness

and (Φ

– T) for the moving surface, the gap as per eq. [3.27]

can be rewritten as

[3.28]

where

[3.29]

[3.30]

and

[3.31]

Here A

and Φ

represent the correlated roughness

and correlated phase angle respectively. For the case of

equal roughnesses A

= A

= A, the trigonometric relations

were used.

The derivatives on the right-hand side of eq. [3.26] can be

rewritten in terms of

(eq. [3.30]) as:

where

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Tribology for Engineers

Bearing in mind the chain rule ,

eq. [3.26] can evidently be reduced to quasi-one-dimensional

form, namely to the ordinary differential equation

[3.32]

This operation is known in mathematical physics as the

‘travelling-wave’ solution (see, e.g., Polyanin et al., 2006), and

represents transition from the two- and three-dimensional

relations of types

(x,y),

(x,t),

(x,y,t) to the one-dimensional

relation W(θ), in the case of the linear

(x,y,t) form.

Expression [3.32] contains the derivatives of the correlated

roughness A

and phase angle Φ

; differentiating eqs. [3.29]

and [3.31], we have

[3.33]

Now, double integration of eq. [3.32] with respect to θ yields

[3.34]

where C

is an integration constant invariant in

, and C

represents the pressure at the initial

, also invariant in

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