A simple numerical procedure to calculate the input

A simple numerical procedure to calculate the input data of Greenwood-Williamson model of asperity contact for actual engineering surfaces Eduardo Tomanik*, Haroldo Chacon, Giovanni Teixeira Mahle S.A. – Brazil Tech Center

Greenwood-Williamson model for asperity contact has been extensively used for prediction of tribological behaviour. However, characterization of the necessary input data, summit asperity height deviation (S), mean radius () and density () are not available from commercial roughness measuring equipment. A relatively simple numerical method is proposed to calculate the asperities data for actual surfaces. Summits were considered to be the local maximum points above the surface mean line and calculation of the standard deviation, mean radius and density of the summits are directly numerical performed. The method was applied for new and worn ICE cylinder bores. The product S.. was found to be between 0.01 to 0.05, lower than the usually accepted range 0.03 to 0.05. The lowest values were found on plateau honing finish, after 15 h test. For all cases, the calculated summit mean radius was found to be lower than previous published values. Some preliminary asperity pressures calculation with the found parameters, suggest that the usual film parameter on the Stribeck curve can lead to wrong conclusions about the tribological severity. Some discussion about the expected asperity pressures of new and worn ICE cylinder bores are also presented.

Smooth plane plano liso

1. INTRODUCTION The Greenwood-Williamson (G-W) model [1] or the later, more realistic, Greenwood-Tripp (G-T) model [2] is extensively used to calculate microcontact and pressures that arise when two rough surfaces approach each other. For simplification, the present work will focus on the G-W model, and it will be applied specially for piston ring/cylinder interactions, but the proposed method can be directly applied for other rough surfaces. It can also be easily extended for the G-T model. In the G-W model the rough surface is presumed to be covered with asperities spherical in shape, with constant radius . The summits are uniformly distributed with density  of summits per area unit. One surface is considered to have the combined roughness of the two original surfaces and the other is considered to be smooth. See figure 1.

d

E-mail address: [email protected]

zi

Surface Separation d = separação entre os corpos

linha de referência Rough surface Reference rugosa Line da superfície

zi = altura da height aspereza em Asperity relação à linha de referência

Fig. 1. Schematic contact of two surfaces according the G-W model. The mean height of summits is located above the mean height of the surface as a whole, as indicated in fig.2. The summit heights, zS, are assumed to follow a gaussian or normal distribution with standard deviation S. If two rough surfaces are brought in contact, their asperities will start to interact, and a pressure, PASP, caused by asperities deformation will appear. PASP can be calculated by:

PASP  *

regiões contato Contactdepoints

3/ 2 4    E '    (t ) c F 3/ 2 3

(1)



F

3/ 2

(t )    ( z )  ( z  t )

3/ 2

 z

(2)

d

Where: C: composite roughness of the 2 surfaces : average summit radius : summit density t: = C/d , d: distance between the two surfaces E’: Combined Modulus of Elasticity. The integral in the equation (2) basically accounts for the probability of a given summit height, zS(i), be higher than the distance d between the surfaces. (z) is the probability density function, PDF, of the summit heights. summit height distribution

z

S

surface height distribution

Fig. 2. Surface and summit mean planes and distributions. Although the G-W model is being extensively used in computer programs that deal with tribological behaviour of piston rings, except for considering S = Rq, there is little experimental data about the values of the two other surface parameters: , . The product S.. is usually taken to be in the range published by G-T: 0.03 < S.. < 0.05, although some recent works obtained different values. Table 1 reproduces some values of the G-W parameters found in the literature. Arcoumanis [3] used the composite equivalent spectral moments of the surface to obtain the summit parameters. Calculation of the spectral moments from experimental data was found difficult by the authors of the present work. See section 2.

Table 1 G-W surface parameters S   (1/m2) (m) (m) 0.12 0.24

66

3.9 E9

0.2 to 1.0 0.185

5000 275

1.0 E9 3.1 E9

S..

Ref.

0.03 to 0.05 0.05 0.06 0.04 1 to 5 0.16 0.63 to 1.92

[1] [4] [5] [6] [7] [3] [8]

Other complication to practical utilization of the G-W model is that most of the engineering sliding surfaces are non-gaussian. E.g., ICE cylinder bores are finished to have much more valleys than peaks. Fig. 3 shows two cylinder roughness profiles from large volume, SI European engines. Notice that, for the plateau honed cylinder, the highest peaks are of the same magnitude of Rq. With the usual assumption that S = Rq, the G-W model would estimate that a smooth surface will start to contact the cylinder at a distance of 3Rq, much higher than the actual highest surface peak. To address the fact that the peaks are much smaller than the valleys, some correction is usually applied. E.g., Priest [9] used for S a value of Rq reduced by a factor from 0.32 to 0.50; Tomanik [10] assumed that S = Rpk. Those corrections were chosen to adjust specific models to experimental wear data and no general rule was given in how to estimate the input parameters.

A

B

Fig. 3. Roughness profile of gasoline ICE cylinders, as produced. A- Std Finish. Rq: 1.2 m; Rpk: 1.0 m. B- Plateau honing. Rq: 0.7 m; Rpk: 0.3 m.

2. SURFACE SPECTRAL MOMENTS Nayak [11] proposed the parameters m0, m2 and m4, spectral moments of the profile Power Spectral Density, PSD, function, to determine a large number of useful statistics for isotropic, gaussian surfaces. The PSD function is the Fourier transform of the profile autocorrelation function. According to McCool [5], these spectral moments may be calculated, for isotropic surfaces, by the expressions:

m0  E{( z ( x )) 2 }

(3)

2

 dz  m 2  E{  }  dx 

However, calculation of m2 and m4 involves practical difficulties, as sample spacing and signal processing. For this work, the authors made trials to determine the second and fourth spectral moments, calculating the variation (dz/dx) and (d²z/dx²) of the measured points by different methods, as described below. The profile file measured by the MAHR Concept Perthometer PGK equipment was used. The file contains 8064 points, 0.695 m spaced. Because, in this equipment, the data contained in the file is supplied as measured, an additional filtering software has to be developed and it is described in Annex I. This user filtering process can be omitted if the equipment supplies the profile already filtered.

(4) Three different methods for calculating the first and second derivatives were made:

2

 d 2z  m4  E{ 2  }  dx 

(5)

dz / dxi  zi1  zi / h

where z(x) is a profile in an arbitrary direction x and E{ } denotes a statistical estimator. m0 is simply the mean square height, i.e. (Rq)2. m2 is equivalent to (q)2. Both Rq and q are common roughness parameters. But m4 is approximately the mean square curvature, not yet available from commercial roughness measurement equipment. With the knowledge of m0, m2 and m4, it would be possible to calculate the asperity parameters according to random field theory, [5]:

 

 S  1 

0.8968  m   0

Simple derivation (SD):

d

2



z / dx 2 i 

zi 1  2 zi  zi 1 h2

(10) (11)

Three points derivation (3PD):

 dz  zi 1  zi 2    3h  dx  i

(12)

 d 2 z  zi 2  2 zi 1  zi 4  2   3h  dx  i

(13)

(6) Finite central differences (FCD):

3   8  m4

(7)

 dz   zi  2  8 zi 1  8 zi 1  zi 2    12h  dx  i

(14)

m4 / m2 6   3

(8)

 d 2 z   zi 2  16 zi 1  30 zi  16 zi 1  zi 2  2   12h 2  dx  i

(15)



Where: h = xi+1 - xi = 0.695 m

where:



m0  m 4 2 m2

(9)

In order to test these methods, they were applied in measurements of standard roughness profiles Mahr Perthen. These standards are used to calibrate

As expected, the calculated values of m2 agreed for the different methods as well as showed agreement (5%) with the measured q. However the calculated values of m4 showed great variation between one method and another. See table 3. Table 3 Calculated spectral moments, standard profiles. m2 m4 Profile (10-3) (10-4) SD 3PD FCD SD 3PD FCD PGN01 1.6 1.5 1.6 4.1 0.7 6.2 PGN03 3.7 3.6 3.7 5.3 1.0 7.9 PGN10 7.2 7.1 7.2 5.8 1.1 8.6 It was considered that it would be unreliable to estimate the summits radius and density with the spectral moments. A different, more direct, approach was developed to calculate the asperity parameters. It will be described in the next sections, specially in section 4.

1 m

Table 2 Standard profiles, measured values Profile Skewness Rq q PGN01 -0.09 0.59 0.040 PGN03 0.00 1.12 0.061 PGN10 0.00 2.67 0.085

cylinder, fig 3B. As typical for a two-stage plateau honing, the surface is not gaussian, the valleys are much deeper than the peaks. Even with peaks, z>0, having an almost gaussian distribution, the usual simplification that S = Rq greatly overestimates the micro-contact occurrence and pressures. E.g., the profile of fig. 3B has Rq = 0.7 m and maximum height 1.67 m. At the height of the highest peak, the usual, gaussian, estimation would be that 0.8% of the profile is in contact*. Actually, this contact amount is only reached at a much lower distance, less than 1 m. At height of 1.0 m, the usual prediction would be that 7.6% of the profile will be in contact**, actually only 1.3% is. See fig. 5. This difference would impact in the calculated tribological parameters, as asperity pressures, minimum oil film thickness, etc.

99.9

Surface Below Specified Height - %

the equipment. They may be considered gaussian, although not isotropic. Table 2 reproduces some roughness parameters, measured by the equipment. All the measurements cited in this paper were obtained using a MAHR Perthomether equipment, with a 2 m, 90 stylus. 0.8 mm Cut-off length.

200 m

99 95

80

50

20

5

surface 1

peaks only 0.1

3. STATISTICAL DISTRIBUTION OF THE HEIGHT IN ACTUAL SURFACES. The G-W model assumes that the summit heights have a gaussian distribution. Most of the works on piston ring modelling accept this assumption, in part because there are evidences that the summit heights have gaussian distribution, even if the surface, as a whole, does not. See fig. 4. This was verified, by the present authors, for the surface finish of various ICE cylinders. E.g. fig. 5 shows the Normal plot of the cumulative height distribution of the plateau honing

0

0.5

1.0

1.5

Height above arbitrary datum - m

Fig. 4. Cumulative height distribution plotted on normal probability paper. [2]. *

0.00820 (0.8%) is the tabulated area under the standardized normal curve for z = 2.39, i.e. 1.67m divided by Rq (0.7m). ** 0.076 (7.6%) is the tabulated area under the standardized normal curve for z = 1.43, i.e. 1.0m divided by Rq (0.7m).

50.00

10.00

The equipment stylus, hence the measured profile, does not in general pass exactly at the summit of a peak. The measured profile does not exactly correspond to the surface summits. Nayak [11] proposes that, for random surfaces, the density of the summits, , is equal to 1.2 times the square of the density of peaks, DP.  = 1.2 DP2. Because the studied surfaces are not random, we chose to consider that  = DP2. It was assumed that the surface is isotropic and the density was calculated by:

99.99

Cumulative Probability (%)

Cummulative Probability (%)

99.99

99.0

90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0

 number of local maxima       profile length  

10.0

5.00 1.00

1.0

0.10

0.1 -4

-3

-2

-1

0

1

2

Height (m)

0.01 0

1.00

2.00

3.00

4.00

5.00

6.00

7.00

Height distribution. Cylinder Fig. 5. Cumulative height bore, plateau honing finish. Profile fig.3B.

To comply with the fact that actual surfaces are usually non-Gaussian, the proposed procedure is described in the next section: 4. METHOD FOR CALCULATION OF THE ASPERITIES PARAMETERS

2

(16)

There is, also, risk of finding some false local maxima. See discussion in [14], as well as not counting possible local maxima that exists between the acquired measurement points. These errors appear to be inherent of one dimensional, discrete, measurement. They were neglected in this work. 4.2. Calculation of S Following the definition given in 4.1, S is assumed to be equal to the standard deviation of the profile peak heights. The mean height of the peaks, Z S , is also calculated.

First step is to verify if the peaks follow a gaussian distribution. This was verified to be true for most of the analyzed, new and worn, cylinder bores. Some form and waviness filtering was necessary in the profile file supplied by the equipment used for this work. See Annex I.

4.3. Calculation of 

4.1. Calculation of 

It was also calculated the radius that passes through the maximum and its second nearest neighbours: (zi-2, zi+2). In average the mean radius increased three times, while the density decreased 46% for the std bores and 38% for the plateau ones. These results are shown as 2, 2 in table 4.

The summit density is calculated by computing the number of occurrences of positive local maxima, only the portion of the filtered profile that is above or in the mean line is considered. I.e. the number of occurrences in the profile that:

zi  ( zi 1, zi 1 ) and zi  0

For each positive local maximum, it is calculated the radius of the circle that passes through the maximum, zi, and its nearest neighbours: (zi-1, zi+1).The mean radius of the summits, , is the average of these peak radii.

- some cases, including all PU, had the S.. lower than the usual limit 0.03. This may be due to the fact that ICE cylinders are produced to have smaller peaks, and may be expected for worn surfaces, like the ones after 15 h engine test. Or it can be caused by the calculated low mean radius, see discussion ahead. - the calculated summit radius, , is much lower than the values previously published.  increased if the second, not the first, nearest neighbours were used. But still to values lower than literature. Preliminary, it was decided to use the value calculated with the nearest neighbours, but deeper studies are necessary. A visual examination of the profile local maxima, where the radius were calculated, supports the choice of 1. Fig. 6 shows a measured profile detail, with the same magnification in both axis. The calculated radius for this peak is 5.0 m.

4.4. Application of the method on actual cylinder bores The proposed methodology was applied for 2 engine blocks of large production, European SI 4 cylinder engine. One block had standard bore finish. The other, plateau honing. The blocks were measured, as produced (new) and after 15 hour dynamometer test. Topography parameters were calculated, according to the proposed method, for each measurement. See table 4, where SN: Std new, SU: Std after 15h test, PN: Plateau New, PU: Plateau after 15 h test. As expected, S decreased after test, and it was lower on the plateau honed cylinders. This may be understood as a preliminary indication of the lower tribological severity which is achieved with the plateau and with worn surfaces. In average, the product S.. also decreased after test and is lower on the plateau ones. In comparison with the literature, see table 1, the following comments can be done about the calculated values: Table 4 ICE cylinders – Measured and Calculated Values

#

Rq

Rz

m

m

SN

1 2 3 4

1.2 1.4 1.3 1.1

6.4 8.9 8.1 6.8

SU

1 2 3 4

0.7 0.6 0.6 0.6

PN

1 2 3 4

PU

1 2 3 4

RSk

Measured Rpk Rk

Rvk

MR1

MR2

ZS

S

Calculated 1 1

2

2

m

7.5 6.7 5.5 4.1

10 1/m² 9.7 10 7.2 7.7

21.2 15.4 19.4 15.2

109 1/m² 4.6 5.2 3.9 4.1

0.22 0.17 0.22 0.19

16.1 20.4 20.4 9.5

8.5 9.5 8.5 7.2

28.6 22.8 40.8 33.5

3.7 5.0 4.9 4.3

0.50 0.54 0.51 0.61

0.26 0.31 0.26 0.32

4.6 9.3 8.5 4.1

16.4 17.2 14.4 16.0

20.0 27.9 22.3 18.0

9.8 10.0 9.1 9.8

0.30 0.28 0.31 0.32

0.14 0.17 0.13 0.13

15.0 19.0 8.8 6.6

10.3 7.1 8.2 10.2

35.2 55.2 42.2 28.7

5.8 4.2 5.5 6.4

m

m

m

%

%

m

m

m

-0.7 -1.6 0.1 -0.2

1.0 1.0 1.4 1.0

2.4 2.8 2.8 2.8

1.8 2.9 1.7 1.4

9.8 6.0 11.4 8.9

85.0 85.0 88.0 89.8

0.90 1.05 1.11 1.00

0.65 0.66 0.90 0.66

4.0 3.3 3.9 3.2

-3.6 -2.5 -2.6 -2.4

0.1 0.1 0.1 0.1

1.0 0.9 1.0 1.1

1.3 1.2 1.3 1.1

1.9 3.5 6.3 4.9

75.3 81.3 82.4 84.8

0.43 0.34 0.39 0.39

0.7 0.9 0.6 0.9

3.9 6.4 3.9 4.8

-1.7 -4.1 -1.8 -2.0

0.3 0.4 0.3 0.3

1.1 1.0 1.2 1.3

1.4 2.3 1.3 1.8

7.4 8.3 7.2 7.6

82.4 84.8 84.0 79.3

0.5 0.7 0.6 0.6

3.0 4.1 4.0 4.1

-2.2 -7.0 -4.0 -3.3

0.1 0.2 0.1 0.1

0.6 0.5 0.4 0.4

1.2 1.0 1.1 1.2

4.7 6.2 6.0 3.9

75.4 75.7 64.8 69.5

9

0.8

(m)

1.0

1.0

0.6

0.8

0.4

0.7

0.2

0.6

(m)

0.0 1842.5

S (m)

0.9

0.5 1843.0

1843.5

1844.0

1844.5

0.4

Fig. 6. Profile detail showing a typical peak. Same magnification on both axes.

0.3 0.2 0.1

The calculated S was compared with some common measured roughness parameters. One of them was obviously Rq. See fig.7. The correlation is good, R2 = 0.88, in a relatively large range, covering all the 16 studied bores. But not for smaller values of Rq. In the spotted region (Rq < 1.0) that considers all cases except the SN, R2 decreases to 0.64. Much better correlation of S, was found with Rpk + Rk/2. See fig. 8. For all 16 cases, R2 = 0.98. For the spotted region, R2 = 0.91. This very good correlation can be explained by the fact that both parameters, S and (Rpk + Rk/2), are basically derived from the portion above the surface mean line.

Rq (m)

0.0 0.0

0.4

0.8

1.2

1.6

Fig. 7. Correlation between the calculated S and the measured Rq.

1.0

s (m)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Rpk + Rk/2 (m)

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Fig. 8. Correlation between the calculated S and the measured Rpk + Rk/2.

5. CALCULATION OF MICRO-CONTACT PRESSURES WITH THE CALCULATED PARAMETERS. Some numerical data treatment was necessary to address the usual simplifying hypothesis that the summit mean line coincides with the surface mean line. This hypothesis is present in several current commercial, as well proprietary, computer codes for piston ring simulation. In these codes, it is suggested to use not the calculated S, but a value double of the originally calculated. This is done to accommodate all the summit distribution, even neglecting its mean height, Z s . See fig. 9. It should be emphasized that this is only a numerical artifice to be used in computer codes that do not consider the summit mean height. True summit height distribution

Zs

“equivalent” distribution for Zs= 0

+3 3’ -3

2100

2200

2300

Fig. 9. Schematic equivalent summit height distribution (dotted line) for computer codes that do not consider the Summit mean Height ( Z s ). The micro-contact pressures, that arise when a lubricated piston ring slides against the cylinder bore, were calculated with the computer program described in [15]. The asperity pressures were calculated according the G-W model. The hydrodynamic pressures are calculated by a simplified Reynolds equation where the calculation of the squeeze effect was turned off . Increase of oil viscosity by pressure was not considered, neither shear thinning effects. The following condition was investigated:

-

-

1.2 mm wide ring, with 10 m barrel drop, symmetrical. The ring face barrel profile was approximated by a parabola. E’ = 100 GPa oil SAE 40 at 100 C (viscosity= 0.013 Pa.s) loading pressure = 3.5 MPa (35 bar) piston speed = 1 m/s

To study only the influence of the bore roughness, it was assumed that the combined roughness is equal to 2 S (due to the cited correction because Z s = 0) and the ring was assumed to be smooth. The pressures were calculated with the average values found for each bore condition of table 4. For comparison, the same ring profile and operation condition, but with the S, ,  values given by [3], was also calculated. Table 5 summarises the results showing the calculated values of minimum lube film (Hm) between ring and bore; the film parameter (t = Hm/C) and the percentage of the load supported by hydrodynamic pressures, PHYD. A fully hydrodynamic regime, t > 3, would have PHYD = 100%. Table 5 Calculated Hydrodynamic indicators. Hm t C   9 10 m m m 1.44 5.9 8.3 3.13 2.2 SN 0.40 16.6 8.1 0.64 1.6 SU 0.58 6.6 15.8 1.08 1.9 PN 0.28 12.4 8.7 0.41 1.5 PU 0.185 275 3.0 0.30 1.6 [3]

PHYD % 4 35 18 58 81

It is interesting to notice that, at least for the calculated cases, two parameters that can be seen as indicators of the tribological severity didn’t follow the same trend: the percentage of the load supported by the hydrodynamic pressures, P HYD, was lowest for the Standard New (SN) case that at the same time presented the highest parameter film, t. The latter would be an indicative of a more hydrodynamic, less severe regime. This didn’t happen because the relatively large lubricant film (3.13 m) was unable to produce the same load capacity as in the other cases, with lower lubricant film thickness.

So, according to the percentage of the load supported by the Hydrodynamic pressures P HYD, the tribological severity increases from:

in the profile, can be expanded to a 3D surface. More investigations of the correction applied on the calculated  used in simulation codes, ’ = 2S, is also recommended.

PU < SU < PN < SN while just the opposite was found for the film parameter, t: SN < PN < SU < PU The lowest lube film thickness was obtained with the data published in [3], mainly due to the very low C (0.185 m). However this low oil film thickness produced the highest PHYD (81%), but with the second lowest film parameter (t= 1.6).

6. CONCLUSIONS AND DISCUSSION A numerical method, relatively easy to implement, was developed to calculate the input parameters, S, , , of the G-W and G-T microcontact models. The calculation was made from the usually available measurement profile from commercial equipment. The method was applied for new and worn ICE cylinder bores. The product S.. was found to be between 0.01 to 0.05, lower than the usually accepted range 0.03 to 0.05. The lowest values were obtained on plateau honing finish, after 15 h test. For all cases, the calculated summit mean radius was found to be lower than previous published values. The method allows a more realistic modelling of actual, non-gaussian surfaces with engineering use, as e.g. ICE cylinders with different bore finish. Some preliminary asperity pressure calculations with the found parameters, suggest that the usual film parameter, t= Hm/C, on the Stribeck curve can lead to wrong conclusions about the tribological severity. More investigations are necessary to expand the method to consider the non-isotropy of the surfaces. The basic geometrical calculations, that were made

NOMENCLATURE   S ’ Hm m0 m2 m4 MR1 MR2 PHYD q Rpk Rk Rvk Rq Rsk t z Zs

Summit mean radius Summit density Summit standard deviation Corrected  for use in simulation codes that assume that Zs= 0, ’ = 2 Minimum lube oil film thickness 0th Spectral moment 2nd Spectral moment 4th Spectral moment Material ratio of peaks Material ratio of valleys Percentage of the Load that is supported by Hidrodynamic pressures Root mean slope of the profile Reduced peak height Core roughness depth Reduced valley depth Root mean square of profile heights Skewness Oil film parameter, t = Hm / C Asperities height Average Asperity height

REFERENCES 1. Greenwood, J.A.; Williamson, J. “Contact of nominally flat surfaces”. Proc. R. Soc. London, Series A, v.295, p.300-319, 1966 2. Greenwood, J.A.; Tripp, J.H. “The Contact of Two Nominally Flat Rough Surfaces”. Proc Inst Mech Engrs 1970-71 v.185, n.48, p.625-633 , 1970 3. Arcoumanis, C. et al. “Mixed lubrication Modelling of Newtonian and Shear Thinning

Liquids in a Piston-Ring Configuration”. paper SAE 972924, 1997 4. Johnson, K. L. et al “A simple theory of asperity contact in Elastohydrodynamic lubrication” Wear, 19 (1972) p.91-108 5. McCool, J.I. “Comparison of models for the contact of rough surfaces”. Wear, v.107, 1986 6. Hu, Y. et al. “Numerical Simulation of Piston Ring in Mixed Lubrication- A Nonaxisymetrical Analysis”. Trans. of the ASME v.116, July 1994, p.470-478, 1994 7. Gulwadi, S.D. “Analysis of Lubrication, Friction, Blow-by and Oil Consumption in a Piston Ring Pack”. ASME-ICE Division, Spring Conference. Fort Collins, Colorado, USA, 1997. 8. Thomas, T.; Rosén, B.-G “Determination of the optimum sampling interval for rough contact mechanics” Tribology International 33 (2000) p.601-610, 2000 9. Priest, M. et al. "Predictive wear modelling of lubricated piston rings in a diesel engine" Wear v.231, p.89-101, 1999

15. Tomanik, E; Ferrarese, A. “Use of a microcontact model to optimize SI engine’s 3-piece oil ring profiles regarding wear and lubrication” – ASME ICE-Vol. 39, p.427-435, 2002. ANNEX I- PROFILE FILTERING In order to calculate the surface parameters, it was necessary first to filter the profile file available by the equipment. This was done by an user made program, using MatLab routines. The purpose was to replicate the filter usually applied by the equipment for roughness measurement. This program would not be necessary in an equipment that supplies a file, with the data already filtered. The filter process has two steps: Form Removal Elimination of the nominal shape of the component from the assessment of texture. The nominal shape may be defined by using various mathematical functions such as circle, conical and aspheric functions. The form removal of the signal was preliminary implemented by a least-square curve fitting of 2nd order polynomial. Waviness Filtering

10. Tomanik, E.; Nigro, F. “Piston Ring Pack and Cylinder Wear Modelling” SAE paper 2001-010572 11. Nayak, P.R.; “Random Process Model of Rough Surfaces” Journal of Lubrication Technology, ASME transactions p.398-407, July 1971 12. Standard DIN 4777 – Phase correct filters for use in electrical stylus instruments.

Corresponds to a high-pass filter which attenuates long-wave signals, depending on its characteristic, and transmits higher-frequency or short wave signals. This was implemented according to DIN4777 [12]. The filter characteristic of the waviness profile (i.e. of the mean line), is determined from the weighting function by means of the Fourier transformation. It is expressed by the following equation: a1  0.4697   c   e    a0   

13. Form Talysurf series – Appendix: Instrument theory, parameters and definitions.

2

Where,

14. Greenwood, J.A. “A unified theory of surface roughness” Proc. R. Soc. London A 393, p.133-157, 1984.

a0 amplitude of the profile before filtering. a1 amplitude of the profile in the mean line. c cut-off wavelength of the profile filter.

S u rface R ou gh n ess P rofile

8 7 6 5 4 3 2 1 0

S u rface W avin ess P rofile

Profile [  m ]

1.0

0.5

0.0 800

1000

2000

3000

4000

P rofile L en g th [m m ]

5000

2800

3800

4800

-1.0

Profile Length [m m ]

Fig. II. User calculated Surface waviness. F iltered, Roughness R ou gh n es P rofile Filtered, Profile

2 1 0 -1 -2 -3 -4 -5 800

0

1800

-0.5

Profile [  m]

Profile [ m]

Fig. I to III illustrate the filtering process. Fig. I reproduces the profile, as supplied by the equipment. 0.8 mm Cut-off, 5.6 mm measurement length, 8064 points. Fig. II shows the user calculated waviness profile, first and last cut-off were considered in the calculation of the waviness, but not plotted. Fig. III shows the profile after form removal and waviness filtering. First and last cut-off length are discarded, equivalent of using a 2CR PC filter [13]. The result is a filtered text file, that for the studied cases, a file containing 5760 points, 0.695 m spaced, that was then used for calculating the G-W surface parameters.

1800

2800 Profile Length [m m ]

3800

4800

6000

Fig. I. Surface Roughness Profile, as supplied by the equipment, before form removal and filtering.

Fig. III. User roughness profile, after form removal and waviness filtering.