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Method of determination of truncation parameters from measured surface profile
Tribology International 36 (2003) 745–752 www.elsevier.com/locate/triboint
Method of determination of truncation parameters from measured surface profile Minoru Nonogaki a,∗, Takuya Morimoto a, Tsunamitsu Nakahara b a
b
Department of Mechanical Engineering, Nara National College of Technology, 22 Yata-cho, Yamatokoriyama, Nara National College of Technology, 22 Yata-cho, Yamatokoriyama, Nara 639-1080, Japan Department of Mechanical and Science Engineering, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan Received 3 September 2002; received in revised form 12 February 2003; accepted 14 February 2003
Abstract The probability density function of the roughness height of a sliding surface is not always Gaussian like that of a truncated surface caused by running-in or mild wear. Therefore, it is important for obtaining contact pressure or frictional characteristics to estimate the truncation level of the non-Gaussian distribution function. This paper presents a method of determination of the two truncation parameters in the truncation model presented by King et al. [Proceedings of the 4th Leeds–Lyon Symposium on Tribology, MEP, London, 1978, p. 333]. The two truncation parameters p and b can be determined by plotting the values of skewness Sk and kurtosis K obtained from a measured profile of surface roughness on the Sk–K diagram calculated with the truncation model for various given values of parameters p and b. The height distributions reproduced by the truncation model with the truncation parameters p and b identified by the present method is in good agreement with the original ones of the measured surfaces. 2003 Elsevier Science Ltd. All rights reserved. Keywords: Surface roughness; Running-in; Identification of truncation parameters
1. Introduction Surface topography exerts a great influence on tribological characteristics. Although the probability density function (PDF) of the roughness height of many engineering surfaces is Gaussian, it does not remain Gaussian, for an example, in the case of a rough surface sliding against a hard smoother surface [1,2]. Decrease in roughness height by wear and plastic flow is termed “runningin”. In the “running-in” period, wear usually takes place only at the tops of asperities, and as a result, the roughness profile shows a truncated shape. Many models have been presented for the process of change in topography due to sliding. Thomas [3] presented a PDF of the truncated roughness profile that is a part of the Gaussian distribution with an impulse whose integrated area is equal to that of mathematical
Corresponding author. Tel.: +81-0743-55-6079; fax: +81-074355-6089. E-mail address: [email protected] (M. Nonogaki). ∗
truncation. Golden [4] considered a wear mechanism such that the rate of decrease in height of an asperity with time is a function of the degree of its penetration. The resulting PDF remains Gaussian, but the segments of wearing heights are altered. Using a similar approach, Sugimura et al. [5] derived a “running-in equation” that described a change in the surface height distribution of a composite profile of wearing surfaces with the size distribution of wear particles. The PDF has the shape of a truncated normal distribution. King et al. [6] introduced a truncating function of Gaussian distribution whose standard deviation is less than that of the virgin surface with a Gaussian distribution. The worn height distribution is made by the truncating funcation. The model by King et al. is considered to be the most similar to a real worn surface and to be practical, because it does not need a computer simulation or curve fitting technique. The truncation model has two truncation parameters, the standard deviation p and the truncation level b. King et al. estimated the degree of running-in from the virgin surface profile known in advance. However, in most practical instances, the virgin surface pro-
0301-679X/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0301-679X(03)00055-0
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M. Nonogaki et al. / Tribology International 36 (2003) 745–752
Nomenclature c p Sk K z b s∗r s∗t sr st f(z) mr mt
distance between mean surface height of the unworn surface and that of the truncating function ratio of the standard deviation of the truncating function to that of the unworn surface skewness kurtosis general ordinate height of the worn surface truncation level unit standard deviation of a Gaussian distribution of the unworn function (s∗r = 1) standard deviation of a Gaussian distribution of the truncated surface to a unit standard deviation (s∗t = st / sr = p) standard deviation of a Gaussian distribution of the unworn surface standard deviation of a Gaussian distribution of the truncating function probability density function of the worn surface mean value of a Gaussian distribution of the unworn surface mean value of a Gaussian distribution of the truncating function
file before running-in is unknown. Thus, a method of determination of the two truncation parameters is needed. The purpose of this paper is to propose a method for the determination of the truncation parameters from a measured profile of roughness without knowing the initial roughness profile before running-in and to confirm the validity of the method.
2. Truncation model King et al. [6] have shown a realistic truncation model of abrasion by a hard surface with a Gaussian distribution, as shown in Fig. 1. The symbols sr and mr are the standard deviation and the mean value of the Gaussian distribution of the unworn surface, and st and mt are the standard deviation and the mean value of the Gaussian distribution of the truncating funcation, respectively. The height distribution of the worn surface (the height distribution of truncated surface) given by King et al. becomes:
再 冉 冊冎 冋 冕 冑 再 冉 冊冎 册 冑 再 冉 冊 冎冋 冕 冑 再 冉 冊 冎 册 f(z) ⫽
z
⫺⬁
2
1 z⫺mt exp ⫺ 2 st 2πst
1 z⫺mt ⫺ 2 st
1
z
2
1⫺
⫺⬁
1
2
dz ⫹
exp ⫺
2πsr
1 z 2 sr
1
exp 2πst
(1)
2
dz
Although the mathematical Gaussian distribution is infinitely spread, the realistic one of the roughness height is finite. Halling and Nuri [7] showed from experimental results on static contacts that the maximum heights and depth were almost three times ss, which is the standard deviation of the asperity height and is equal to s / √2. Therefore, we use 3s for the maximum height. Fig. 2 illustrates the distributions normalized by sr to define the truncation level according to wear depth. The zero truncation level (b = 0) indicates the starting point of abrasion on the unworn surface, i.e., c = 3(1 + p). If the maximum height of the truncating function reaches the bottom of the valley in the unworn surface, the original distribution does not remain, and thus, b = 1 at c = ⫺3(1⫺p). The truncation level b is defined as the ratio of the interference region to the maximum roughness height of the virgin surface: b⬅
Fig. 1. The truncation model abraded by a hard surface with a Gaussian distribution [6].
1 z exp ⫺ 2 sr 冑2πsr
1⫺
1
3(1 ⫹ p)⫺c , ⫺3(1⫺p)ⱕcⱕ3(1 ⫹ p) 6
(2)
Fig. 3 shows the height distributions of a truncated surface for various p at b = 0.7 and 0.5, and for various b at p = 0.1 and 0.3. We call “p” and “b”—standard deviation of truncating function and truncation level, respectively—“truncation parameters” in this paper. As it is not easy to determine
M. Nonogaki et al. / Tribology International 36 (2003) 745–752
Fig. 2.
Definition of truncation level b by Eq. (2) with parameters normalized by sr.
Fig. 3. Height distributions of truncated surface obtained from Eq. (1) for various given truncation parameters.
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these parameters from measured profile data of truncated surfaces, we deal with a method of determination of the truncation parameters in the next section.
Table 1 Roughness parameters of a truncated surface profile Item
Value
3. Method of determination of truncation parameters
Rmax (µm) s (µm) Sk (µm) K
2.29 0.43 ⫺1.10 3.81
We can determine the two parameters if we can identify them by comparing the PDF calculated from Eq. (1) for given values of p and b with the measured PDF. However, it is not easy to identify the two parameters simultaneously by means of direct comparison. The authors have found a simple method for determining the two truncation parameters p and b from the dia-
gram of the relationship between skewness Sk and kurtosis K, which are defined by the third and fourth moments of the resulting distribution, respectively, as indicated by Eqs. (3) and (4):
冕 冕
Sk ⫽
1 s3
K⫽
1 s4
⬁
(z⫺m)3f(z) dz
(3)
(z⫺m)4f(z) dz
(4)
⫺⬁
⬁
⫺⬁
where f(z) is the probability density function of the height distribution. Skewness represents the degree of symmetry of the height distribution. The skewness of a symmetrical distribution, e.g. Gaussian distribution, is zero. Kurtosis represents the peakedness of the height distribution and it is given as a normalized form. The kurtosis of Gaussian distribution is 3. Fig. 4(a) is a whole diagram of the relationship between skewness Sk and kurtosis K, and Fig. 4(b) shows a partly magnified diagram. Plotting the measured values of Sk and K on these diagrams, we can determine the values of p and b simultaneously. Let us demonstrate an example of the determination of the two parameters by means of the method mentioned above, and compare the reproduced distribution calculated using Eq. (1) for the parameters obtained from the Sk–K diagram with the PDF obtained from the measured surface profile. Fig. 5 shows an example of truncated surface profile and Table 1 indicates the roughness parameters. We can determine the truncation parameters by plotting the value of skewness and kurtosis in Table 1 on Fig. 6. As a result, we obtain the value of p = 0.16 and b = 0.52. Fig. 7 shows the comparison between the reproduced PDF for p = 0.16 and b = 0.52 and the original one calculated directly from the measured surface profile and indicates good agreement between them.
4. Discusion
Fig. 4.
Skewness–kurtosis diagram.
Fig. 8 shows a sequence of surface profiles obtained during the running-in of a journal bearing made of phosphor bronze after finishing by abrasive paper, with about 4 µm Rmax roughness as initial roughness. The right ends in Fig. 8 indicate the standardized PDFs of the reproduction with the truncation parameters p and b determ-
M. Nonogaki et al. / Tribology International 36 (2003) 745–752
Fig. 5.
An example of surface profile after running-in.
Fig. 7. PDFs. Fig. 6.
749
Magnified graph of Fig. 4(b).
ined by the present method as well as the measured original ones. It is found that the reproduced PDFs are in good agreement with the original ones. Table 2 lists the values of roughness parameters and truncation parameters p and b in the progress of running-in, and Fig. 9 shows the variation of the truncation parameters on the diagram of K vs. Sk. It is seen that the PDF of the initial roughness (0) is approximately Gaussian, because the value of skewness is nearly zero and that of kurtosis is approximately 3. In this table, maximum height Rmax, centerline average Ra and standard deviation s decrease almost simply as the truncation by running-in progresses. Then skewness and kurtosis tend to decrease and increase, respectively, and the truncation parameters p and b tend to decrease and increase, respectively. These good agreements mean that the truncation model of King et al. with parameters p and b is practical to evaluate truncation with the help of the method
An example of comparison between original and reproduced
Table 2 Values of roughness parameters Parameter
Stage 0
Rmax (µm) Ra (µm) s (µm) s∗ (µm) Sk K p b
3.77 0.55 0.68 0.60 ⫺0.33 2.79 – –
1
2 2.70 0.35 0.45 0.29 ⫺1.30 4.87 0.21 0.60
2.17 0.22 0.31 0.17 ⫺1.99 8.08 0.18 0.67
3
4
2.04 0.15 0.23 0.11 ⫺2.62 11.93 0.13 0.69
1.60 0.10 0.17 0.08 ⫺3.00 16.29 0.13 0.73
presented in this paper for determining the truncation parameters from a measured roughness profile. Next, let us show the cases of disagreement between the reproduced and original PDFs. Not to mention, if the position of the Sk–K co-ordinates is out of the region of
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M. Nonogaki et al. / Tribology International 36 (2003) 745–752
Fig. 8.
Changes in surface profile of bearing and PDFs.
the relationship between p and b in Fig. 4, it is impossible to determine the truncation parameters using the present method. In the case of existence of deep scratches, the reproduced PDF did not agree with the original PDF of the measured surface profile, as shown in Fig. 10(a), although the truncation could be determined. In the profile curve in Fig. 10(a), there are two large scratches (the depth being about 2 and 3 µm), whereas the maximum depth of valley is 1 µm or less. If the scratch of about 3 µm depth is changed to 1 µm, as shown in the roughness profile in Fig. 10(b), then, as is
shown in Fig. 10(b), the reproduced PDF agrees well with the original one. It is noticed that the valley part of the PDF nears a Gaussian distribution with change of depth of the scratches. In the case of the profile in Fig. 11, a wide depression exists, and it deviates from Gaussian distribution in the valley part. As a result, the reproduced PDF does not coincide with the original one. Therefore, it is necessary for applying the method presented in this paper in the estimation of the truncation, that the valley part of the PDF be approximately Gaussian in distribution.
M. Nonogaki et al. / Tribology International 36 (2003) 745–752
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5. Conclusion This paper has presented the method of determination of the two parameters in the truncation model proposed by King et al. with the diagram of the relationship between skewness and kurtosis calculated from the measured surface profile. The PDF reproduced using Eq. (1) and the truncation parameters determined by the present method are in good agreement with the original ones if the valley part of the original distribution is approximately Gaussian. The authors consider that truncation parameter b indicates the degree of progress of truncation and parameter p shows the quality of truncation. To demonstrate the use of the two parameters, we should indicate the quantitative effect of the two parameters on the lubrication characteristics of various tribo-elements. Fig. 9. Changes in truncation parameters p and b on the Sk–K diagram.
Fig. 10.
Influence of deep scratch on difference between original and reproduced PDFs.
Fig. 11. Influence of depression on difference between original and reproduced PDFs.
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M. Nonogaki et al. / Tribology International 36 (2003) 745–752
References [1] Thomas TR. The characterizations of changes in surface topography in running-in. In: Proceedings of the 4th Leeds–Lyon Symposium on Tribology. London: MEP; 1978. p. 99–108. [2] Rowe GW, Kaliszer H, Trmal G, Cotter A. Running-in of plain bearings. Wear 1975;34:1–14. [3] Thomas TR. Computer simulation of wear. Wear 1972;22:83–90. [4] Golden JM. The evolution of asperity height distribution of a surface subjected to wear. Wear 1976;39:25–44. [5] Sugimura J, Kimura Y, Amino K. Analysis of the topographic
changes due to wear—Geometry of the running-in process. J Jpn Soc Lubr Eng 1986;31(11):813–20. [6] King TG, Watson W, Stout KJ. Modelling the micro-geometry of lubricated wear. In: Proceedings of the 4th Leeds–Lyon Symposium on Tribology. London: MEP; 1978. p. 333–43. [7] Halling J, Nuri KA. The elastic contact of rough surfaces and its importance in the reduction of wear. Proc Inst Mech Eng 1985;199(C2):139–44.
Method of determination of truncation parameters from measured surface profile Minoru Nonogaki a,∗, Takuya Morimoto a, Tsunamitsu Nakahara b a
b
Department of Mechanical Engineering, Nara National College of Technology, 22 Yata-cho, Yamatokoriyama, Nara National College of Technology, 22 Yata-cho, Yamatokoriyama, Nara 639-1080, Japan Department of Mechanical and Science Engineering, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan Received 3 September 2002; received in revised form 12 February 2003; accepted 14 February 2003
Abstract The probability density function of the roughness height of a sliding surface is not always Gaussian like that of a truncated surface caused by running-in or mild wear. Therefore, it is important for obtaining contact pressure or frictional characteristics to estimate the truncation level of the non-Gaussian distribution function. This paper presents a method of determination of the two truncation parameters in the truncation model presented by King et al. [Proceedings of the 4th Leeds–Lyon Symposium on Tribology, MEP, London, 1978, p. 333]. The two truncation parameters p and b can be determined by plotting the values of skewness Sk and kurtosis K obtained from a measured profile of surface roughness on the Sk–K diagram calculated with the truncation model for various given values of parameters p and b. The height distributions reproduced by the truncation model with the truncation parameters p and b identified by the present method is in good agreement with the original ones of the measured surfaces. 2003 Elsevier Science Ltd. All rights reserved. Keywords: Surface roughness; Running-in; Identification of truncation parameters
1. Introduction Surface topography exerts a great influence on tribological characteristics. Although the probability density function (PDF) of the roughness height of many engineering surfaces is Gaussian, it does not remain Gaussian, for an example, in the case of a rough surface sliding against a hard smoother surface [1,2]. Decrease in roughness height by wear and plastic flow is termed “runningin”. In the “running-in” period, wear usually takes place only at the tops of asperities, and as a result, the roughness profile shows a truncated shape. Many models have been presented for the process of change in topography due to sliding. Thomas [3] presented a PDF of the truncated roughness profile that is a part of the Gaussian distribution with an impulse whose integrated area is equal to that of mathematical
Corresponding author. Tel.: +81-0743-55-6079; fax: +81-074355-6089. E-mail address: [email protected] (M. Nonogaki). ∗
truncation. Golden [4] considered a wear mechanism such that the rate of decrease in height of an asperity with time is a function of the degree of its penetration. The resulting PDF remains Gaussian, but the segments of wearing heights are altered. Using a similar approach, Sugimura et al. [5] derived a “running-in equation” that described a change in the surface height distribution of a composite profile of wearing surfaces with the size distribution of wear particles. The PDF has the shape of a truncated normal distribution. King et al. [6] introduced a truncating function of Gaussian distribution whose standard deviation is less than that of the virgin surface with a Gaussian distribution. The worn height distribution is made by the truncating funcation. The model by King et al. is considered to be the most similar to a real worn surface and to be practical, because it does not need a computer simulation or curve fitting technique. The truncation model has two truncation parameters, the standard deviation p and the truncation level b. King et al. estimated the degree of running-in from the virgin surface profile known in advance. However, in most practical instances, the virgin surface pro-
0301-679X/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0301-679X(03)00055-0
746
M. Nonogaki et al. / Tribology International 36 (2003) 745–752
Nomenclature c p Sk K z b s∗r s∗t sr st f(z) mr mt
distance between mean surface height of the unworn surface and that of the truncating function ratio of the standard deviation of the truncating function to that of the unworn surface skewness kurtosis general ordinate height of the worn surface truncation level unit standard deviation of a Gaussian distribution of the unworn function (s∗r = 1) standard deviation of a Gaussian distribution of the truncated surface to a unit standard deviation (s∗t = st / sr = p) standard deviation of a Gaussian distribution of the unworn surface standard deviation of a Gaussian distribution of the truncating function probability density function of the worn surface mean value of a Gaussian distribution of the unworn surface mean value of a Gaussian distribution of the truncating function
file before running-in is unknown. Thus, a method of determination of the two truncation parameters is needed. The purpose of this paper is to propose a method for the determination of the truncation parameters from a measured profile of roughness without knowing the initial roughness profile before running-in and to confirm the validity of the method.
2. Truncation model King et al. [6] have shown a realistic truncation model of abrasion by a hard surface with a Gaussian distribution, as shown in Fig. 1. The symbols sr and mr are the standard deviation and the mean value of the Gaussian distribution of the unworn surface, and st and mt are the standard deviation and the mean value of the Gaussian distribution of the truncating funcation, respectively. The height distribution of the worn surface (the height distribution of truncated surface) given by King et al. becomes:
再 冉 冊冎 冋 冕 冑 再 冉 冊冎 册 冑 再 冉 冊 冎冋 冕 冑 再 冉 冊 冎 册 f(z) ⫽
z
⫺⬁
2
1 z⫺mt exp ⫺ 2 st 2πst
1 z⫺mt ⫺ 2 st
1
z
2
1⫺
⫺⬁
1
2
dz ⫹
exp ⫺
2πsr
1 z 2 sr
1
exp 2πst
(1)
2
dz
Although the mathematical Gaussian distribution is infinitely spread, the realistic one of the roughness height is finite. Halling and Nuri [7] showed from experimental results on static contacts that the maximum heights and depth were almost three times ss, which is the standard deviation of the asperity height and is equal to s / √2. Therefore, we use 3s for the maximum height. Fig. 2 illustrates the distributions normalized by sr to define the truncation level according to wear depth. The zero truncation level (b = 0) indicates the starting point of abrasion on the unworn surface, i.e., c = 3(1 + p). If the maximum height of the truncating function reaches the bottom of the valley in the unworn surface, the original distribution does not remain, and thus, b = 1 at c = ⫺3(1⫺p). The truncation level b is defined as the ratio of the interference region to the maximum roughness height of the virgin surface: b⬅
Fig. 1. The truncation model abraded by a hard surface with a Gaussian distribution [6].
1 z exp ⫺ 2 sr 冑2πsr
1⫺
1
3(1 ⫹ p)⫺c , ⫺3(1⫺p)ⱕcⱕ3(1 ⫹ p) 6
(2)
Fig. 3 shows the height distributions of a truncated surface for various p at b = 0.7 and 0.5, and for various b at p = 0.1 and 0.3. We call “p” and “b”—standard deviation of truncating function and truncation level, respectively—“truncation parameters” in this paper. As it is not easy to determine
M. Nonogaki et al. / Tribology International 36 (2003) 745–752
Fig. 2.
Definition of truncation level b by Eq. (2) with parameters normalized by sr.
Fig. 3. Height distributions of truncated surface obtained from Eq. (1) for various given truncation parameters.
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M. Nonogaki et al. / Tribology International 36 (2003) 745–752
these parameters from measured profile data of truncated surfaces, we deal with a method of determination of the truncation parameters in the next section.
Table 1 Roughness parameters of a truncated surface profile Item
Value
3. Method of determination of truncation parameters
Rmax (µm) s (µm) Sk (µm) K
2.29 0.43 ⫺1.10 3.81
We can determine the two parameters if we can identify them by comparing the PDF calculated from Eq. (1) for given values of p and b with the measured PDF. However, it is not easy to identify the two parameters simultaneously by means of direct comparison. The authors have found a simple method for determining the two truncation parameters p and b from the dia-
gram of the relationship between skewness Sk and kurtosis K, which are defined by the third and fourth moments of the resulting distribution, respectively, as indicated by Eqs. (3) and (4):
冕 冕
Sk ⫽
1 s3
K⫽
1 s4
⬁
(z⫺m)3f(z) dz
(3)
(z⫺m)4f(z) dz
(4)
⫺⬁
⬁
⫺⬁
where f(z) is the probability density function of the height distribution. Skewness represents the degree of symmetry of the height distribution. The skewness of a symmetrical distribution, e.g. Gaussian distribution, is zero. Kurtosis represents the peakedness of the height distribution and it is given as a normalized form. The kurtosis of Gaussian distribution is 3. Fig. 4(a) is a whole diagram of the relationship between skewness Sk and kurtosis K, and Fig. 4(b) shows a partly magnified diagram. Plotting the measured values of Sk and K on these diagrams, we can determine the values of p and b simultaneously. Let us demonstrate an example of the determination of the two parameters by means of the method mentioned above, and compare the reproduced distribution calculated using Eq. (1) for the parameters obtained from the Sk–K diagram with the PDF obtained from the measured surface profile. Fig. 5 shows an example of truncated surface profile and Table 1 indicates the roughness parameters. We can determine the truncation parameters by plotting the value of skewness and kurtosis in Table 1 on Fig. 6. As a result, we obtain the value of p = 0.16 and b = 0.52. Fig. 7 shows the comparison between the reproduced PDF for p = 0.16 and b = 0.52 and the original one calculated directly from the measured surface profile and indicates good agreement between them.
4. Discusion
Fig. 4.
Skewness–kurtosis diagram.
Fig. 8 shows a sequence of surface profiles obtained during the running-in of a journal bearing made of phosphor bronze after finishing by abrasive paper, with about 4 µm Rmax roughness as initial roughness. The right ends in Fig. 8 indicate the standardized PDFs of the reproduction with the truncation parameters p and b determ-
M. Nonogaki et al. / Tribology International 36 (2003) 745–752
Fig. 5.
An example of surface profile after running-in.
Fig. 7. PDFs. Fig. 6.
749
Magnified graph of Fig. 4(b).
ined by the present method as well as the measured original ones. It is found that the reproduced PDFs are in good agreement with the original ones. Table 2 lists the values of roughness parameters and truncation parameters p and b in the progress of running-in, and Fig. 9 shows the variation of the truncation parameters on the diagram of K vs. Sk. It is seen that the PDF of the initial roughness (0) is approximately Gaussian, because the value of skewness is nearly zero and that of kurtosis is approximately 3. In this table, maximum height Rmax, centerline average Ra and standard deviation s decrease almost simply as the truncation by running-in progresses. Then skewness and kurtosis tend to decrease and increase, respectively, and the truncation parameters p and b tend to decrease and increase, respectively. These good agreements mean that the truncation model of King et al. with parameters p and b is practical to evaluate truncation with the help of the method
An example of comparison between original and reproduced
Table 2 Values of roughness parameters Parameter
Stage 0
Rmax (µm) Ra (µm) s (µm) s∗ (µm) Sk K p b
3.77 0.55 0.68 0.60 ⫺0.33 2.79 – –
1
2 2.70 0.35 0.45 0.29 ⫺1.30 4.87 0.21 0.60
2.17 0.22 0.31 0.17 ⫺1.99 8.08 0.18 0.67
3
4
2.04 0.15 0.23 0.11 ⫺2.62 11.93 0.13 0.69
1.60 0.10 0.17 0.08 ⫺3.00 16.29 0.13 0.73
presented in this paper for determining the truncation parameters from a measured roughness profile. Next, let us show the cases of disagreement between the reproduced and original PDFs. Not to mention, if the position of the Sk–K co-ordinates is out of the region of
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M. Nonogaki et al. / Tribology International 36 (2003) 745–752
Fig. 8.
Changes in surface profile of bearing and PDFs.
the relationship between p and b in Fig. 4, it is impossible to determine the truncation parameters using the present method. In the case of existence of deep scratches, the reproduced PDF did not agree with the original PDF of the measured surface profile, as shown in Fig. 10(a), although the truncation could be determined. In the profile curve in Fig. 10(a), there are two large scratches (the depth being about 2 and 3 µm), whereas the maximum depth of valley is 1 µm or less. If the scratch of about 3 µm depth is changed to 1 µm, as shown in the roughness profile in Fig. 10(b), then, as is
shown in Fig. 10(b), the reproduced PDF agrees well with the original one. It is noticed that the valley part of the PDF nears a Gaussian distribution with change of depth of the scratches. In the case of the profile in Fig. 11, a wide depression exists, and it deviates from Gaussian distribution in the valley part. As a result, the reproduced PDF does not coincide with the original one. Therefore, it is necessary for applying the method presented in this paper in the estimation of the truncation, that the valley part of the PDF be approximately Gaussian in distribution.
M. Nonogaki et al. / Tribology International 36 (2003) 745–752
751
5. Conclusion This paper has presented the method of determination of the two parameters in the truncation model proposed by King et al. with the diagram of the relationship between skewness and kurtosis calculated from the measured surface profile. The PDF reproduced using Eq. (1) and the truncation parameters determined by the present method are in good agreement with the original ones if the valley part of the original distribution is approximately Gaussian. The authors consider that truncation parameter b indicates the degree of progress of truncation and parameter p shows the quality of truncation. To demonstrate the use of the two parameters, we should indicate the quantitative effect of the two parameters on the lubrication characteristics of various tribo-elements. Fig. 9. Changes in truncation parameters p and b on the Sk–K diagram.
Fig. 10.
Influence of deep scratch on difference between original and reproduced PDFs.
Fig. 11. Influence of depression on difference between original and reproduced PDFs.
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M. Nonogaki et al. / Tribology International 36 (2003) 745–752
References [1] Thomas TR. The characterizations of changes in surface topography in running-in. In: Proceedings of the 4th Leeds–Lyon Symposium on Tribology. London: MEP; 1978. p. 99–108. [2] Rowe GW, Kaliszer H, Trmal G, Cotter A. Running-in of plain bearings. Wear 1975;34:1–14. [3] Thomas TR. Computer simulation of wear. Wear 1972;22:83–90. [4] Golden JM. The evolution of asperity height distribution of a surface subjected to wear. Wear 1976;39:25–44. [5] Sugimura J, Kimura Y, Amino K. Analysis of the topographic
changes due to wear—Geometry of the running-in process. J Jpn Soc Lubr Eng 1986;31(11):813–20. [6] King TG, Watson W, Stout KJ. Modelling the micro-geometry of lubricated wear. In: Proceedings of the 4th Leeds–Lyon Symposium on Tribology. London: MEP; 1978. p. 333–43. [7] Halling J, Nuri KA. The elastic contact of rough surfaces and its importance in the reduction of wear. Proc Inst Mech Eng 1985;199(C2):139–44.