- Email: [email protected]
Effect of surface roughness parameters on mixed lubrication characteristics
Tribology International 39 (2006) 522–527 www.elsevier.com/locate/triboint
Effect of surface roughness parameters on mixed lubrication characteristics Wen-zhong Wang*, Hui Chen, Yuan-zhong Hu, Hui Wang The state key laboratory of Tribology, Tsinghua University, Beijing 100084, People’s Republic of China Received 14 July 2004; received in revised form 1 March 2005; accepted 20 March 2005 Available online 23 May 2005
Abstract In this paper, a computer program was developed to generate non-Gaussian surfaces with specified standard deviation, autocorrelation function, skewness and kurtosis, based on digital FIR technique. A thermal model of mixed lubrication in point contacts is proposed, and used to study the roughness effect. The area ratio, load ratio, maximum pressure, maximum surface temperature and average film thickness as a function of skewness and kurtosis are studied at different value of rms. Numerical examples show that skewness and kurtosis have a great effect on the contact parameters of mixed lubrication. q 2005 Elsevier Ltd. All rights reserved. Keywords: Non-gaussian surface; Mixed lubrication; Skewness; Kurtosis; Contact area
1. Introduction As is well known, real engineering surfaces prepared by various machining processes are not ideally smooth, but with surface roughness in different scales. Usually, the rms roughness for surfaces in most industrial application is typical in a range from 0.025 to 1.5 mm [1]. When the average film thickness is of the same order of magnitude as, or smaller than, surface roughness, a mechanical element operates in mixed lubrication, an area where hydrodynamic lubrication and asperity contacts coexist. Apparently, the surface roughness plays an important role in the contacting performance and lubricant film formation/breakdown. The statistic characteristics of real rough surface can be mathematically approximated by a stochastic process with Gaussian or non-Gaussian heights distribution. In many early numerical analyses, the assumption of Gaussian surface heights distribution was adopted. However, most of the common machining processes produce surfaces with non-Gaussian distributions [2–4]. For example, turning and * Corresponding author. Fax: C86 10 62781379. E-mail address: [email protected] (W.-z. Wang).
0301-679X/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2005.03.018
shaping produce a peaked surface with positive skewness, while grinding, honing, milling and abrasion processes produce grooved surfaces with negative skewness but high kurtosis values. In 2000, Hu and Zhu [5,6] developed an isothermal model of mixed lubrication for point contacts, in which a unified Reynolds equation approach was proposed. The model does not need to identify the types of contact, i.e. to decide whether an area is in hydrodynamic lubrication or asperity contacts. In the area where the film thickness approaches zero Reynolds equation can be modified into a reduced form and the normal pressure in the region of asperity contacts can be thus determined. As a result, a deterministic numerical solution for the mixed lubrication can be obtained through a unite system of equations and the same numerical scheme. Recently, based on the isothermal model, a thermal model has been developed [7] to consider thermal effects in mixed lubrication. Surface temperature has been determined by numerically integrating the temperature rises caused by a moving point heat source. The interactions between pressure and temperature are considered through incorporating viscosity-temperature and density-temperature relations in Reynolds equation. An FFT-based algorithm (DC-FFT) is used to speed up the calculation of surface deformation and temperature rise [8–10]. The computational practices show that
W.-z. Wang et al. / Tribology International 39 (2006) 522–527
aforementioned models enable us to simulate various lubrication conditions, from full film elastohydrodynamic lubrication (EHL) to boundary lubrication, providing a good opportunity for a better understanding of the effect of surface roughness. In the present study a method for computer generations of non-Gaussian rough surfaces is developed and mixed lubrication analyses are performed for these surfaces in relative motions. Our main interest of the study concerns the dependence of contact behavior on the non-Gaussian nature of distribution, especially the effects of the skewness and kurtosis of the height distribution.
2. Generation of rough surface In order to investigate the effect of different types of rough surface on performance of mixed lubrication, in this section, a computer program is developed to generate rough surfaces with specified standard deviation, autocorrelation function, skewness and kurtosis. It is based on a digital filter technique proposed by Hu and Tonder [11]. For a Gaussian rough surface with prescribed autocorrelation function, two surface roughness parameters are used to represent its characteristics—the standard deviation of surface heights, s (or rms or Rq), and the correlation length b (bx, by for an anisotropic surface). In present study, the autocorrelation function is assumed to be exponential and is given by Rðx; yÞ Z s2 exp K 2:3½ðx=bx Þ2 C ðy=by Þ2 1=2
(1)
For a non-Gaussian rough surface, two additional parameters are needed to characterize the surface, skewness (SK) and kurtosis (K). The skewness and kurtosis are the third and fourth moments of the distribution function. A Gaussian surface has a kurtosis of 3. If KO3, the surface contains relatively fewer high peaks and low valleys while K!3 corresponds to more high peaks and low valleys over the surface. The method proposed by Hu and Tonder to generate three-dimensional rough surfaces in Gaussian or nonGaussian distribution and with specified autocorrelation functions (ACF) can be summarized as follows. Supposed h(i,j) is a sequence of random numbers having a Gaussian distribution of a known standard deviation s, the simulated rough heights z(i,j) can be taken as zði; jÞ Z
nK1 m K1 X X
hðk; lÞhði C k; j C lÞ
(2)
kZ0 lZ0
where h(i,j) is called FIR filter and is the coefficient which define the system. According to signal process theory, z(i,j) will possess the same height distribution with the input sequence h(i,j). The Fourier transformation of Eq. (2) is
523
given as
z^ði; jÞ Z
nK1 X mK1 X
^ jÞ Hði; jÞhði;
(3)
iZ0 jZ0
^ jÞ are the Fourier transformations where z^ði; jÞ, H(i,j) and hði; of z(i,j), h(i,j) and h(i,j), respectively. H(i,j) is also called the transfer function of system, and can be determined by the following relation [12] jHði; jÞj2 Z
Sz ði; jÞ Sh ði; jÞ
(4)
where Sh(i,j) denotes the spectral densities of input sequence h(i,j), which gives a constant value for a random sequence in white noise type. Sz(i,j) is the spectral densities of output sequence z(i,j), i.e. the Fourier transformation of its expected autocorrelation function R(i,j). Thus, the filter coefficients h(i,j) can be obtained by applying inverse Fourier transforms to H(i,j). When non-Gaussian rough surface is expected, the Gaussian input sequence h(i,j) should be first transformed to another input sequence h 0 (i,j) with appropriate skewness (SK 0 ) and kurtosis (K 0 ) using Johnson translator system of distribution, then let h 0 (i,j) pass through the filter to obtain the output sequence z(i,j) which possesses the specified autocorrelation function, skewness and kurtosis. The skewness (SK 0 ) and kurtosis (K 0 ) for the modified input sequence h 0 (i,j) can be obtained by the following relation SKz Z SK
0
q X
q3i
X q
iZ1
" Kz Z K
0
q X iZ1
q4i
!3=2 q2i
(5)
iZ1
C6
qK1 X q X iZ1 jZiC1
#, q2i q2j
q X
!2 q2i
(6)
iZ1
where SKz and Kz are the preset skewness and kurtosis for the rough surface to be generated. FFT technique used in this procedure may speed up the numerical process for generating rough surface, but when correlation length increases, the number of data for FFT should be increased accordingly to guarantee the accuracy. Fig. 1(a) shows a generated Gaussian rough surface. The comparison of ACF from the generated surface to its specified values is showed in Fig. 1(b). It can be found that a good agreement is reached between them. Fig. 2 shows a 3D view of a generated non-Gaussian rough surface, which looks like a typical worn surface, with desired skewness and kurtosis of K1.75 and 5.0, respectively. The real values of SK and K were calculated as K1.7827 and 5.1104. The small differences between specified and real values validate the present method.
524
W.-z. Wang et al. / Tribology International 39 (2006) 522–527
Fig. 1. Generated Gaussian rough surface. (a) 3D display of generated surface. (b) Comparison of ACF (autocorrelation function).
3. Mixed lubrication model
Q Z mpn
A thermal model of mixed lubrication in point contact developed by the present authors [7] was used to examine the contact behavior on different types of rough surfaces generated by aforementioned method. The model assumes that asperity contact occurs when the local films become sufficiently thin, so that the same equations systems are used consistently in both lubricated and contact regions. When h/0, the Poiseuille terms in the left hand of Reynolds equation vanish because of the third power of the film thickness and large viscosity. As a result, the Reynolds equation is reduced to the following form:
The friction coefficients, m, have been assigned with different values in different friction conditions of hydrodynamic lubrication, asperity contact and metal adhesion. v is the relative sliding velocity of two contacting bodies. A more detail description of the model can be found in references [5–7].
u
vrh vrh C Z0 vx vt
when h% 3
(8)
4. Results and discussions Table 1 gives the statistical parameters of generated rough surfaces, to be used in present work for investigating the effects of surface roughness on contact and lubrication
(7)
where 3 is a very small number. Below it, the lubricant flow due to the hydrodynamic pressure gradient can be neglected [5–7]. As Eq. (7) is a special case of Reynolds equation, the same iteration procedure can be used in both lubricated and contact regions to obtain the pressure distribution between contacting rough surfaces. The surface temperature is evaluated by the moving point heat source integration method based on analytical solution of Jaeger [13] for point heat source. The heat flux Q generated between two contacting bodies can be simply written as
Fig. 2. Generated non-Gaussian rough surface (SkZK1.75, KZ5.0).
Table 1 Generated rough surfaces used in numerical analysis Surface type
rms (mm)
Skewness (original)
Kurtosis (original)
Skewness (deformed)
Kurtosis (deformed)
Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3 0.3 0.3 0.3 0.3
K1.75 K1.75 K1.75 K0.75 K0.75 K0.75 K0.75 0.30 0.30 0.30 0.30 0.75 0.75 0.75 0.75 0.75 1.75 1.75 1.75 0.75 0.75 0.75 0.75 0.75
5 8 12 2 5 8 12 2.0 5.0 8.0 12.0 2.0 4.0 5.0 8.0 12.0 5.0 8.0 12.0 2.0 4.0 5.0 8.0 12.0
0.2099 0.2158 0.2327 0.2889 0.2861 0.2799 0.2605 0.3128 0.2912 0.2685 0.2658 0.4794 0.3107 0.2649 0.1890 0.1338 0.7447 0.7216 0.4196 0.6886 0.4949 0.4247 0.2671 0.1501
1.9111 1.8751 1.9150 1.7145 1.7560 1.7812 1.8134 1.5918 1.6469 1.6772 1.6990 2.6077 2.8701 2.9711 3.2028 3.4352 3.5996 4.0397 3.4441 2.5024 2.9689 3.1468 3.5107 3.9033
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
W.-z. Wang et al. / Tribology International 39 (2006) 522–527
performance. The corresponding parameters for the same surfaces but deformed after contact are also listed in Table 1. In all the cases analyzed, the applied load is fixed at wZ800 N, the material properties are E 0 Z219.78 GPa, h0Z0.096 Pa s, and aZ1.82!10K8 m2/N. The geometric parameters are RxZRyZ19.05 mm. Therefore, the corresponding Hertzian contact radius is 0.475 mm, and the maximum Hertzian pressure Ph is 1.711 GPa. The rolling speed and the slide-to-roll ratio are uZ0.625 m/s and SZ 2.0, respectively. Fig. 3 gives the results, the area ratio (the ratio between real contact area and nominal contact area), load ratio (ratio between load shared by real contact area and the applied
load), maximum pressure and maximum surface temperature, as a function of kurtosis. The simulations were performed on rough surfaces with the same rms of 0.1 mm but different skewnesses. The variation of average film thickness is also shown in Fig. 3(e). It can be seen that with an increase of kurtosis, the area and load ratios increase accordingly, and so does the maximum pressure. The correspondence between these variables and kurtosis manifests the effects of roughness characteristic. As is known, with the increase of kurtosis, the rough surface becomes peakier so that contact will occur first at higher peaks as a smooth surface normally approaches a rough surface. As a result, the real contact area increases with ( b ) 0.20
(a) 0.35 Sk=0.75 Sk=0.3 Sk=-0.75
0.30 0.25
Load Ratio (%)
Contact area ratio (%)
525
0.20 0.15 0.10 0.05 0.00 2
4
6
8
10
0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
12
Sk=0.75 Sk=0.3 Sk=-0.75
2
4
6
Kurtosis
8
10
12
Kurtosis
(c) 1.7
(d)
1.6
414 412
1.5
Tmax, K
Pmax
410 1.4 Sk=0.75 Sk=0.3 Sk=-0.75
1.3 1.2
408
Sk=0.75 Sk=0.3 Sk=-0.75
406 404
1.1
402
1.0
400 0
2
4
6
8
10
12
14
0
2
4
6
Kurtosis
8
10
12
14
Kurtosis
Average film thickness, /a
( e) 0.00050 0.00045 0.00040 0.00035
Sk=0.75 Sk=0.3 Sk=-0.75
0.00030 0.00025 0.00020 0
2
4
6
8
10
12
14
Kurtosis
Fig. 3. The changes of contact area ratio, load ratio, maximum pressure, maximum surface temperature and average film thickness with kurtosis at sZ0.1 mm and different skewnesses. (a) Contact area ratio vs skewness. (b) Load ratio vs skewness. (c) Maximum pressure vs kurtosis. (d) Maximum temperature vs kurtosis. (e) Average film thickness vs kurtosis.
526
W.-z. Wang et al. / Tribology International 39 (2006) 522–527
(a) 0.7
(b) 0.00050
k k k k
0.5
=2 =5 =8 = 12
Average film thickness, /a
Contact area ratio, %
0.6
0.4 0.3 0.2 0.1 0.0 –2.0
–1.5
–1.0
–0.5
0.0
0.5
1.0
1.5
0.00045 0.00040
k=5.0 k=8.0 k=12.0
0.00035 0.00030 0.00025 0.00020 –2.0
2.0
–1.5
–1.0
Skewness
–0.5
0.0
0.5
1.0
1.5
2.0
Skewness
Fig. 4. The change of contact area ratio and average film thickness with skewness at sZ0.1 mm and different kurtosis. (a) Area ratio vs skewness. (b) Average film thickness vs skewness.
kurtosis, considering that the average film thickness remains largely unchanged as shown in Fig. 3(e). On the other hand, the surface with lower kurtosis has better capacity to form lubrication films than the one with higher kurtosis, which may also has an effect on the contact behavior. For this reason, the influences of kurtosis in mixed lubrication are different from those in dry contact given by the reference [14] where the maximum pressure decreases with the increase of kurtosis. In mixed lubrication, the load shared by asperities increases correspondingly with the growing contact area; while in dry contact, the load is totally applied on real contact region, which implies the increase of real contact area will decrease the maximum pressure. Fig. 3(d) shows that maximum surface temperatures are hardly affected by the increasing kurtosis. This may be because
(b) 3.0
2.0
2.5
Load ratio (%)
Contact area ratio (%)
(a)
the oil lubrication plays a predominant role at present condition. Fig. 4 shows the area ratio and average film thickness as a function of skewness. Similar trends to those shown in Fig. 3 are observed. This may also attribute to the roughness characteristics. When a smooth surface normally approaches a rough surface, a surface with positive skewness will be much more engaged in contact than the one with negative skewness. Since the average film thickness remains almost the same, as shown in Fig. 4(b), so the real contact area will increase with skewness. On the other hand, the capacity to form lubricant film is different for the surfaces with different skewnesses, which may also bring about remarkable influence on the contact behavior.
2.0 1.5 1.0 0.5 0.0
1.5 1.0 0.5 0.0
2
4
6
8
10
2
12
4
Kurtosis (c) 3.0
(d) Tmax, K
2.5
Pmax
6
8
10
12
10
12
Kurtosis
2.0 1.5
440 420 400 380 360 340 320
1.0 2
4
6
8
Kurtosis
10
12
2
4
6
8
Kurtosis
Fig. 5. The change of contact area ratio, load ratio, maximum pressure and maximum temperature with kurtosis at sZ0.3 mm, SkewnessZ0.75. (a) Contact area ratio vs kurtosis. (b) Load ratio vs kurtosis. (c) Maximum pressure vs kurtosis. (d) Maximum temperature vs kurtosis.
W.-z. Wang et al. / Tribology International 39 (2006) 522–527
SK = –0.75 K= 2 σ = 0.3µm
0.025
1.0
0.015 0.010
0.5
1.5
Pressure, /Ph
0.020
Film thickness, /a
Pressure, /Ph
1.5
0.030
(b) 2.0
0.030
SK= –0.75 K = 12 σ = 0.3µm
0.025 0.020 0.015
1.0
0.010 0.5
0.005 0.000
0.0 –1.5
–1.0
–0.5
0.0
0.5
1.0
Film thickness, /a
(a) 2.0
527
0.005 0.000
0.0 –1.5
X Axis, /a
–1.0
–0.5
0.0
0.5
1.0
X Axis, /a
Fig. 6. Profiles of film thickness and pressure at YZ0 with different kurtosis. (a) Profiles of film thickness and pressure at YZ0 with KZ2, SkZK0.75 and sZ0.3 mm. (b) Profiles of film thickness and pressure at YZ0 with KZ12, SkZK0.75 and sZ0.3 mm.
Fig. 5 shows when the rms of rough surfaces increase to 0.3 mm, the effect of kurtosis on the area ratio, load ratio, maximum pressure and maximum temperature appears slightly different from that in the case with rms of 0.1 mm. It can be seen that area ratio increases with kurtosis when kurtosis is below 4, but keeps unchanged as the kurtosis further increases. Similar trends can be observed for the load ratio. As the peak-to-mean height (the distance between the highest peak and the mean plane) increases with the kurtosis and the rms of roughness, it may be the higher peak that prevent the contact from further growth, thus real contact area remains nearly the same when kurtosis is large. The maximum surface temperature still has little variation, which is similar to one for low value of rms. Fig. 6 gives a comparison of profiles of film thickness and pressure at YZ0 in mixed lubrication for surfaces with kurtosis of 2 and 12, a slight difference can be seen.
5. Conclusions Based on FIR technique, rough surfaces with prescribed autocorrelation function, skewness and kurtosis are obtained through computer simulations. By means of the thermal model of mixed lubrication proposed by the present authors, the generated rough surfaces are then used to investigate the effect of skewness and kurtosis of rough surface on contact area ratio, load ratio, maximum pressure and temperature in mixed lubrication. It is found that skewness and kurtosis have a significant influence on the area ratio, load ratio, and maximum pressure, but little effect on the maximum temperature. The area ratio, load ratio, and maximum pressure increase with the increase of skewness and kurtosis. When the value of rms is large, however, the effect of kurtosis on the contact area ratio and load ratio become weak.
Acknowledgements The authors would like to express their appreciation for the continuous supports from General Motor Corporation USA, and postdoctoral foundation of China.
References [1] Zhu D. Effect of surface roughness on mixed EHD lubrication characteristics. Tribol Trans 2003;46(1):44–8. [2] Thomas TR. Rough surfaces. New York: Longman; 1982. [3] Whitehouse DJ. Handbook of surface metrology. Bristol: Institute of Physics Publishing; 1994. [4] Sayles RS, Thomas TR. A stochastic explanation of some structural properties of a ground surface. Int Prod Res 1976;14:641–55. [5] Hu YZ, Zhu D. A full numerical solution to the mixed lubrication in point contacts. J TRIBOL-T ASME 2000;122(1):1–9. [6] Zhu D, Hu YZ. A computer program package for the prediction of EHL and mixed lubrication characteristics, friction, subsurface stresses and flash temperatures based on measured 3-D surface roughness. TRIBOL T 2001;44(3):383–90. [7] wen-zhong Wang, yuchuan Liu, hui Wang, yuan-zhong Hu. A computer thermal model of mixed lubrication in point contacts. J TRIBOL-T ASME 2004;126(1):162–70. [8] Liu S, Wang Q, Liu G. A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses. Wear 2000;243(1–2):101–11. [9] Wang W-Z, Wang H, Liu Y-C, Hu Y-Z, Zhu D. A comparative study of the methods for calculation of surface elastic deformation. P I MECH ENG J-J ENG 2003;217(J2):145–53. [10] Yu-Chuan Liu, Hui Wang, Wen-Zhong Wang, Yuan-Zhong Hu, Dong Zhu. Methods comparison in computation of temperature rise on frictional interfaces. Tribol Int 2002;35:549–60. [11] Hu YZ, Tonder K. Simulation of 3-D random surface by 2-D digital filter and Fourier analysis. Int J Machine Tool Manuf 1992;32:82–90. [12] Oppenheim Alan V, Schafer Ronald W. Digital signal processing. Englewood Cliffs, NJ: Prentice-Hall; 1975. [13] Jaeger JC. Moving sources of heat and the temperature at sliding contacts. Proc Royal Soc NSW 1942;56:203–24. [14] Chilamakuri SK, Bhushan B. Contact analysis of non-Gaussian random surfaces. P I MECH ENG J-J ENG 1998;212(J1):19–32.
Effect of surface roughness parameters on mixed lubrication characteristics Wen-zhong Wang*, Hui Chen, Yuan-zhong Hu, Hui Wang The state key laboratory of Tribology, Tsinghua University, Beijing 100084, People’s Republic of China Received 14 July 2004; received in revised form 1 March 2005; accepted 20 March 2005 Available online 23 May 2005
Abstract In this paper, a computer program was developed to generate non-Gaussian surfaces with specified standard deviation, autocorrelation function, skewness and kurtosis, based on digital FIR technique. A thermal model of mixed lubrication in point contacts is proposed, and used to study the roughness effect. The area ratio, load ratio, maximum pressure, maximum surface temperature and average film thickness as a function of skewness and kurtosis are studied at different value of rms. Numerical examples show that skewness and kurtosis have a great effect on the contact parameters of mixed lubrication. q 2005 Elsevier Ltd. All rights reserved. Keywords: Non-gaussian surface; Mixed lubrication; Skewness; Kurtosis; Contact area
1. Introduction As is well known, real engineering surfaces prepared by various machining processes are not ideally smooth, but with surface roughness in different scales. Usually, the rms roughness for surfaces in most industrial application is typical in a range from 0.025 to 1.5 mm [1]. When the average film thickness is of the same order of magnitude as, or smaller than, surface roughness, a mechanical element operates in mixed lubrication, an area where hydrodynamic lubrication and asperity contacts coexist. Apparently, the surface roughness plays an important role in the contacting performance and lubricant film formation/breakdown. The statistic characteristics of real rough surface can be mathematically approximated by a stochastic process with Gaussian or non-Gaussian heights distribution. In many early numerical analyses, the assumption of Gaussian surface heights distribution was adopted. However, most of the common machining processes produce surfaces with non-Gaussian distributions [2–4]. For example, turning and * Corresponding author. Fax: C86 10 62781379. E-mail address: [email protected] (W.-z. Wang).
0301-679X/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2005.03.018
shaping produce a peaked surface with positive skewness, while grinding, honing, milling and abrasion processes produce grooved surfaces with negative skewness but high kurtosis values. In 2000, Hu and Zhu [5,6] developed an isothermal model of mixed lubrication for point contacts, in which a unified Reynolds equation approach was proposed. The model does not need to identify the types of contact, i.e. to decide whether an area is in hydrodynamic lubrication or asperity contacts. In the area where the film thickness approaches zero Reynolds equation can be modified into a reduced form and the normal pressure in the region of asperity contacts can be thus determined. As a result, a deterministic numerical solution for the mixed lubrication can be obtained through a unite system of equations and the same numerical scheme. Recently, based on the isothermal model, a thermal model has been developed [7] to consider thermal effects in mixed lubrication. Surface temperature has been determined by numerically integrating the temperature rises caused by a moving point heat source. The interactions between pressure and temperature are considered through incorporating viscosity-temperature and density-temperature relations in Reynolds equation. An FFT-based algorithm (DC-FFT) is used to speed up the calculation of surface deformation and temperature rise [8–10]. The computational practices show that
W.-z. Wang et al. / Tribology International 39 (2006) 522–527
aforementioned models enable us to simulate various lubrication conditions, from full film elastohydrodynamic lubrication (EHL) to boundary lubrication, providing a good opportunity for a better understanding of the effect of surface roughness. In the present study a method for computer generations of non-Gaussian rough surfaces is developed and mixed lubrication analyses are performed for these surfaces in relative motions. Our main interest of the study concerns the dependence of contact behavior on the non-Gaussian nature of distribution, especially the effects of the skewness and kurtosis of the height distribution.
2. Generation of rough surface In order to investigate the effect of different types of rough surface on performance of mixed lubrication, in this section, a computer program is developed to generate rough surfaces with specified standard deviation, autocorrelation function, skewness and kurtosis. It is based on a digital filter technique proposed by Hu and Tonder [11]. For a Gaussian rough surface with prescribed autocorrelation function, two surface roughness parameters are used to represent its characteristics—the standard deviation of surface heights, s (or rms or Rq), and the correlation length b (bx, by for an anisotropic surface). In present study, the autocorrelation function is assumed to be exponential and is given by Rðx; yÞ Z s2 exp K 2:3½ðx=bx Þ2 C ðy=by Þ2 1=2
(1)
For a non-Gaussian rough surface, two additional parameters are needed to characterize the surface, skewness (SK) and kurtosis (K). The skewness and kurtosis are the third and fourth moments of the distribution function. A Gaussian surface has a kurtosis of 3. If KO3, the surface contains relatively fewer high peaks and low valleys while K!3 corresponds to more high peaks and low valleys over the surface. The method proposed by Hu and Tonder to generate three-dimensional rough surfaces in Gaussian or nonGaussian distribution and with specified autocorrelation functions (ACF) can be summarized as follows. Supposed h(i,j) is a sequence of random numbers having a Gaussian distribution of a known standard deviation s, the simulated rough heights z(i,j) can be taken as zði; jÞ Z
nK1 m K1 X X
hðk; lÞhði C k; j C lÞ
(2)
kZ0 lZ0
where h(i,j) is called FIR filter and is the coefficient which define the system. According to signal process theory, z(i,j) will possess the same height distribution with the input sequence h(i,j). The Fourier transformation of Eq. (2) is
523
given as
z^ði; jÞ Z
nK1 X mK1 X
^ jÞ Hði; jÞhði;
(3)
iZ0 jZ0
^ jÞ are the Fourier transformations where z^ði; jÞ, H(i,j) and hði; of z(i,j), h(i,j) and h(i,j), respectively. H(i,j) is also called the transfer function of system, and can be determined by the following relation [12] jHði; jÞj2 Z
Sz ði; jÞ Sh ði; jÞ
(4)
where Sh(i,j) denotes the spectral densities of input sequence h(i,j), which gives a constant value for a random sequence in white noise type. Sz(i,j) is the spectral densities of output sequence z(i,j), i.e. the Fourier transformation of its expected autocorrelation function R(i,j). Thus, the filter coefficients h(i,j) can be obtained by applying inverse Fourier transforms to H(i,j). When non-Gaussian rough surface is expected, the Gaussian input sequence h(i,j) should be first transformed to another input sequence h 0 (i,j) with appropriate skewness (SK 0 ) and kurtosis (K 0 ) using Johnson translator system of distribution, then let h 0 (i,j) pass through the filter to obtain the output sequence z(i,j) which possesses the specified autocorrelation function, skewness and kurtosis. The skewness (SK 0 ) and kurtosis (K 0 ) for the modified input sequence h 0 (i,j) can be obtained by the following relation SKz Z SK
0
q X
q3i
X q
iZ1
" Kz Z K
0
q X iZ1
q4i
!3=2 q2i
(5)
iZ1
C6
qK1 X q X iZ1 jZiC1
#, q2i q2j
q X
!2 q2i
(6)
iZ1
where SKz and Kz are the preset skewness and kurtosis for the rough surface to be generated. FFT technique used in this procedure may speed up the numerical process for generating rough surface, but when correlation length increases, the number of data for FFT should be increased accordingly to guarantee the accuracy. Fig. 1(a) shows a generated Gaussian rough surface. The comparison of ACF from the generated surface to its specified values is showed in Fig. 1(b). It can be found that a good agreement is reached between them. Fig. 2 shows a 3D view of a generated non-Gaussian rough surface, which looks like a typical worn surface, with desired skewness and kurtosis of K1.75 and 5.0, respectively. The real values of SK and K were calculated as K1.7827 and 5.1104. The small differences between specified and real values validate the present method.
524
W.-z. Wang et al. / Tribology International 39 (2006) 522–527
Fig. 1. Generated Gaussian rough surface. (a) 3D display of generated surface. (b) Comparison of ACF (autocorrelation function).
3. Mixed lubrication model
Q Z mpn
A thermal model of mixed lubrication in point contact developed by the present authors [7] was used to examine the contact behavior on different types of rough surfaces generated by aforementioned method. The model assumes that asperity contact occurs when the local films become sufficiently thin, so that the same equations systems are used consistently in both lubricated and contact regions. When h/0, the Poiseuille terms in the left hand of Reynolds equation vanish because of the third power of the film thickness and large viscosity. As a result, the Reynolds equation is reduced to the following form:
The friction coefficients, m, have been assigned with different values in different friction conditions of hydrodynamic lubrication, asperity contact and metal adhesion. v is the relative sliding velocity of two contacting bodies. A more detail description of the model can be found in references [5–7].
u
vrh vrh C Z0 vx vt
when h% 3
(8)
4. Results and discussions Table 1 gives the statistical parameters of generated rough surfaces, to be used in present work for investigating the effects of surface roughness on contact and lubrication
(7)
where 3 is a very small number. Below it, the lubricant flow due to the hydrodynamic pressure gradient can be neglected [5–7]. As Eq. (7) is a special case of Reynolds equation, the same iteration procedure can be used in both lubricated and contact regions to obtain the pressure distribution between contacting rough surfaces. The surface temperature is evaluated by the moving point heat source integration method based on analytical solution of Jaeger [13] for point heat source. The heat flux Q generated between two contacting bodies can be simply written as
Fig. 2. Generated non-Gaussian rough surface (SkZK1.75, KZ5.0).
Table 1 Generated rough surfaces used in numerical analysis Surface type
rms (mm)
Skewness (original)
Kurtosis (original)
Skewness (deformed)
Kurtosis (deformed)
Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3 0.3 0.3 0.3 0.3
K1.75 K1.75 K1.75 K0.75 K0.75 K0.75 K0.75 0.30 0.30 0.30 0.30 0.75 0.75 0.75 0.75 0.75 1.75 1.75 1.75 0.75 0.75 0.75 0.75 0.75
5 8 12 2 5 8 12 2.0 5.0 8.0 12.0 2.0 4.0 5.0 8.0 12.0 5.0 8.0 12.0 2.0 4.0 5.0 8.0 12.0
0.2099 0.2158 0.2327 0.2889 0.2861 0.2799 0.2605 0.3128 0.2912 0.2685 0.2658 0.4794 0.3107 0.2649 0.1890 0.1338 0.7447 0.7216 0.4196 0.6886 0.4949 0.4247 0.2671 0.1501
1.9111 1.8751 1.9150 1.7145 1.7560 1.7812 1.8134 1.5918 1.6469 1.6772 1.6990 2.6077 2.8701 2.9711 3.2028 3.4352 3.5996 4.0397 3.4441 2.5024 2.9689 3.1468 3.5107 3.9033
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
W.-z. Wang et al. / Tribology International 39 (2006) 522–527
performance. The corresponding parameters for the same surfaces but deformed after contact are also listed in Table 1. In all the cases analyzed, the applied load is fixed at wZ800 N, the material properties are E 0 Z219.78 GPa, h0Z0.096 Pa s, and aZ1.82!10K8 m2/N. The geometric parameters are RxZRyZ19.05 mm. Therefore, the corresponding Hertzian contact radius is 0.475 mm, and the maximum Hertzian pressure Ph is 1.711 GPa. The rolling speed and the slide-to-roll ratio are uZ0.625 m/s and SZ 2.0, respectively. Fig. 3 gives the results, the area ratio (the ratio between real contact area and nominal contact area), load ratio (ratio between load shared by real contact area and the applied
load), maximum pressure and maximum surface temperature, as a function of kurtosis. The simulations were performed on rough surfaces with the same rms of 0.1 mm but different skewnesses. The variation of average film thickness is also shown in Fig. 3(e). It can be seen that with an increase of kurtosis, the area and load ratios increase accordingly, and so does the maximum pressure. The correspondence between these variables and kurtosis manifests the effects of roughness characteristic. As is known, with the increase of kurtosis, the rough surface becomes peakier so that contact will occur first at higher peaks as a smooth surface normally approaches a rough surface. As a result, the real contact area increases with ( b ) 0.20
(a) 0.35 Sk=0.75 Sk=0.3 Sk=-0.75
0.30 0.25
Load Ratio (%)
Contact area ratio (%)
525
0.20 0.15 0.10 0.05 0.00 2
4
6
8
10
0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
12
Sk=0.75 Sk=0.3 Sk=-0.75
2
4
6
Kurtosis
8
10
12
Kurtosis
(c) 1.7
(d)
1.6
414 412
1.5
Tmax, K
Pmax
410 1.4 Sk=0.75 Sk=0.3 Sk=-0.75
1.3 1.2
408
Sk=0.75 Sk=0.3 Sk=-0.75
406 404
1.1
402
1.0
400 0
2
4
6
8
10
12
14
0
2
4
6
Kurtosis
8
10
12
14
Kurtosis
Average film thickness, /a
( e) 0.00050 0.00045 0.00040 0.00035
Sk=0.75 Sk=0.3 Sk=-0.75
0.00030 0.00025 0.00020 0
2
4
6
8
10
12
14
Kurtosis
Fig. 3. The changes of contact area ratio, load ratio, maximum pressure, maximum surface temperature and average film thickness with kurtosis at sZ0.1 mm and different skewnesses. (a) Contact area ratio vs skewness. (b) Load ratio vs skewness. (c) Maximum pressure vs kurtosis. (d) Maximum temperature vs kurtosis. (e) Average film thickness vs kurtosis.
526
W.-z. Wang et al. / Tribology International 39 (2006) 522–527
(a) 0.7
(b) 0.00050
k k k k
0.5
=2 =5 =8 = 12
Average film thickness, /a
Contact area ratio, %
0.6
0.4 0.3 0.2 0.1 0.0 –2.0
–1.5
–1.0
–0.5
0.0
0.5
1.0
1.5
0.00045 0.00040
k=5.0 k=8.0 k=12.0
0.00035 0.00030 0.00025 0.00020 –2.0
2.0
–1.5
–1.0
Skewness
–0.5
0.0
0.5
1.0
1.5
2.0
Skewness
Fig. 4. The change of contact area ratio and average film thickness with skewness at sZ0.1 mm and different kurtosis. (a) Area ratio vs skewness. (b) Average film thickness vs skewness.
kurtosis, considering that the average film thickness remains largely unchanged as shown in Fig. 3(e). On the other hand, the surface with lower kurtosis has better capacity to form lubrication films than the one with higher kurtosis, which may also has an effect on the contact behavior. For this reason, the influences of kurtosis in mixed lubrication are different from those in dry contact given by the reference [14] where the maximum pressure decreases with the increase of kurtosis. In mixed lubrication, the load shared by asperities increases correspondingly with the growing contact area; while in dry contact, the load is totally applied on real contact region, which implies the increase of real contact area will decrease the maximum pressure. Fig. 3(d) shows that maximum surface temperatures are hardly affected by the increasing kurtosis. This may be because
(b) 3.0
2.0
2.5
Load ratio (%)
Contact area ratio (%)
(a)
the oil lubrication plays a predominant role at present condition. Fig. 4 shows the area ratio and average film thickness as a function of skewness. Similar trends to those shown in Fig. 3 are observed. This may also attribute to the roughness characteristics. When a smooth surface normally approaches a rough surface, a surface with positive skewness will be much more engaged in contact than the one with negative skewness. Since the average film thickness remains almost the same, as shown in Fig. 4(b), so the real contact area will increase with skewness. On the other hand, the capacity to form lubricant film is different for the surfaces with different skewnesses, which may also bring about remarkable influence on the contact behavior.
2.0 1.5 1.0 0.5 0.0
1.5 1.0 0.5 0.0
2
4
6
8
10
2
12
4
Kurtosis (c) 3.0
(d) Tmax, K
2.5
Pmax
6
8
10
12
10
12
Kurtosis
2.0 1.5
440 420 400 380 360 340 320
1.0 2
4
6
8
Kurtosis
10
12
2
4
6
8
Kurtosis
Fig. 5. The change of contact area ratio, load ratio, maximum pressure and maximum temperature with kurtosis at sZ0.3 mm, SkewnessZ0.75. (a) Contact area ratio vs kurtosis. (b) Load ratio vs kurtosis. (c) Maximum pressure vs kurtosis. (d) Maximum temperature vs kurtosis.
W.-z. Wang et al. / Tribology International 39 (2006) 522–527
SK = –0.75 K= 2 σ = 0.3µm
0.025
1.0
0.015 0.010
0.5
1.5
Pressure, /Ph
0.020
Film thickness, /a
Pressure, /Ph
1.5
0.030
(b) 2.0
0.030
SK= –0.75 K = 12 σ = 0.3µm
0.025 0.020 0.015
1.0
0.010 0.5
0.005 0.000
0.0 –1.5
–1.0
–0.5
0.0
0.5
1.0
Film thickness, /a
(a) 2.0
527
0.005 0.000
0.0 –1.5
X Axis, /a
–1.0
–0.5
0.0
0.5
1.0
X Axis, /a
Fig. 6. Profiles of film thickness and pressure at YZ0 with different kurtosis. (a) Profiles of film thickness and pressure at YZ0 with KZ2, SkZK0.75 and sZ0.3 mm. (b) Profiles of film thickness and pressure at YZ0 with KZ12, SkZK0.75 and sZ0.3 mm.
Fig. 5 shows when the rms of rough surfaces increase to 0.3 mm, the effect of kurtosis on the area ratio, load ratio, maximum pressure and maximum temperature appears slightly different from that in the case with rms of 0.1 mm. It can be seen that area ratio increases with kurtosis when kurtosis is below 4, but keeps unchanged as the kurtosis further increases. Similar trends can be observed for the load ratio. As the peak-to-mean height (the distance between the highest peak and the mean plane) increases with the kurtosis and the rms of roughness, it may be the higher peak that prevent the contact from further growth, thus real contact area remains nearly the same when kurtosis is large. The maximum surface temperature still has little variation, which is similar to one for low value of rms. Fig. 6 gives a comparison of profiles of film thickness and pressure at YZ0 in mixed lubrication for surfaces with kurtosis of 2 and 12, a slight difference can be seen.
5. Conclusions Based on FIR technique, rough surfaces with prescribed autocorrelation function, skewness and kurtosis are obtained through computer simulations. By means of the thermal model of mixed lubrication proposed by the present authors, the generated rough surfaces are then used to investigate the effect of skewness and kurtosis of rough surface on contact area ratio, load ratio, maximum pressure and temperature in mixed lubrication. It is found that skewness and kurtosis have a significant influence on the area ratio, load ratio, and maximum pressure, but little effect on the maximum temperature. The area ratio, load ratio, and maximum pressure increase with the increase of skewness and kurtosis. When the value of rms is large, however, the effect of kurtosis on the contact area ratio and load ratio become weak.
Acknowledgements The authors would like to express their appreciation for the continuous supports from General Motor Corporation USA, and postdoctoral foundation of China.
References [1] Zhu D. Effect of surface roughness on mixed EHD lubrication characteristics. Tribol Trans 2003;46(1):44–8. [2] Thomas TR. Rough surfaces. New York: Longman; 1982. [3] Whitehouse DJ. Handbook of surface metrology. Bristol: Institute of Physics Publishing; 1994. [4] Sayles RS, Thomas TR. A stochastic explanation of some structural properties of a ground surface. Int Prod Res 1976;14:641–55. [5] Hu YZ, Zhu D. A full numerical solution to the mixed lubrication in point contacts. J TRIBOL-T ASME 2000;122(1):1–9. [6] Zhu D, Hu YZ. A computer program package for the prediction of EHL and mixed lubrication characteristics, friction, subsurface stresses and flash temperatures based on measured 3-D surface roughness. TRIBOL T 2001;44(3):383–90. [7] wen-zhong Wang, yuchuan Liu, hui Wang, yuan-zhong Hu. A computer thermal model of mixed lubrication in point contacts. J TRIBOL-T ASME 2004;126(1):162–70. [8] Liu S, Wang Q, Liu G. A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses. Wear 2000;243(1–2):101–11. [9] Wang W-Z, Wang H, Liu Y-C, Hu Y-Z, Zhu D. A comparative study of the methods for calculation of surface elastic deformation. P I MECH ENG J-J ENG 2003;217(J2):145–53. [10] Yu-Chuan Liu, Hui Wang, Wen-Zhong Wang, Yuan-Zhong Hu, Dong Zhu. Methods comparison in computation of temperature rise on frictional interfaces. Tribol Int 2002;35:549–60. [11] Hu YZ, Tonder K. Simulation of 3-D random surface by 2-D digital filter and Fourier analysis. Int J Machine Tool Manuf 1992;32:82–90. [12] Oppenheim Alan V, Schafer Ronald W. Digital signal processing. Englewood Cliffs, NJ: Prentice-Hall; 1975. [13] Jaeger JC. Moving sources of heat and the temperature at sliding contacts. Proc Royal Soc NSW 1942;56:203–24. [14] Chilamakuri SK, Bhushan B. Contact analysis of non-Gaussian random surfaces. P I MECH ENG J-J ENG 1998;212(J1):19–32.