simulation of stratified surface topographies
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Available online at www.sciencedirect.com
Wear 264 (2008) 457–463
Simulation of stratified surface topographies
Pawel Pawlus ∗Rzeszow University of Technology, Department of Manufacturing Processes and Production Organisation, Poland
Accepted 28 August 2006Available online 12 February 2007
bstract
In this paper various simulation methods of isotropic and anisotropic random surfaces of Gaussian ordinate distribution with Fast Fourierransform are compared and the best of them are recommended. The method of numerical generation of arbitrarily oriented three-dimensionalough surfaces is presented. The procedure of digital simulation of two-process surface is introduced. An idea of the proposed method (and its
ossible modifications) is imposition of random surface of Gaussian ordinate distribution (second process) on the base surface (first process). Thexamples of using this method for the simulation of plateau honed cylinder surface, cylinder and piston skirt surfaces after “zero-wear” processesre given. In each case the matching criteria of real and simulated surface topographies are presented.2007 Elsevier B.V. All rights reserved.
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eywords: Random surfaces; Stratified surfaces; Simulation
. Introduction
Most engineering surfaces have height distributions whichre approximately Gaussian. But multi-process texture is moremportant from functional point of view. Plateau honed cylin-er surface is the typical example. It consists of smoothear-resistant and load-bearing plateaux with intersecting deepalleys working as oil reservoirs and debris trap. Multi-processurfaces are also formed during “zero-wear” process (when wears within the limit of original surface topography).
Today reference software and reference data are needed inrder to check accuracy of the algorithms and output parametersrom the software. Simulation techniques can be used to createeference data.
The tribological behaviour of engineering surfaces, suchs contact problem or hydrodynamic and elastohydrodynamicubrication can be predicted numerically. The numerical solu-ion of these problems involves input of surface data. It cane obtained either from digital input from profilometer or fromumerical simulation of the rough surfaces. Randomly generat-
ng surface roughness by numerical means is simpler and offersome advantages. The hardware and software requirements cane eliminated. It also removes the need to filter out the unwanted∗ Tel.: +48 17 8651588; fax: +48 17 8651184.E-mail address: [email protected].
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043-1648/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.wear.2006.08.048
avelengths from the measured surface. The numerical simula-ion is also more flexible. Furthermore, the simulation of surfaceorming during manufacturing and operating processes ensuresecrease of cost and time of experimental investigation. Theomputer-generated surfaces can be used during analysis of con-act problems, lubrication and wear of the surfaces. However, theimulated surface should be similar to real surface.
. Literature review
Researchers suggested several methods of 3D surface topog-aphy. They were mainly concentrated on random surfaces ofaussian ordinate distribution. A growing need was developed
o computer generate surfaces with desired properties, such aspectral density or autocorrelation function.
The numerical characterisation of stylus-measured data isased on the recording of 2D profiles. Surfaces of the bigarticipation of the random components do not have in theirpectra dominating components. Many investigators acceptedhe random process description of engineering surfaces, sot was possible to generate a rough surface by the randomimulator. The time series model of rough surface [1,2] was
pplied to one-dimensional profile generation by the authors ofaper [3]. The Johnson transformation system was introducednto the simulation of non-Gaussian distribution profile withhe given skewness and kurtosis [4]. Recently, fractal approach
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f profile description was introduced. The authors of papers5,6] simulated fractal rough surface profiles.
It was found that surface topography exist in three, not in twoimensions. It was a need of measuring and modelling surfaceicrogeometry in three dimensions.A numerical procedure for randomly generating any general
hree-dimensional surface topography with prescribed statisticalroperties is described in Reference [7]. This procedure is capa-le of generating Gaussian or non-Gaussian rough surfaces withny given autocorrelation function. Patir used “the linear trans-ormation on random matrixes”. However, this method requirededious effort and was impractical.
Autoregressive and Moving Average (ARMA) method isopular in generating surfaces.
Whitehouse also simulated random 3D surface [8]. AR timeeries model was used. However, only the features of the auto-orrelation function were simulated.
In 1990, Gu and Huang [9] used ARMA model to createD surfaces of desired autocorrelation function. A simulatedurface, which agrees with the actual surface, can be obtainedhrough measurement of some profiles along different direc-ions.
Hong and Ehmann [10] attempted to synthesise engineeredurfaces. The proposed method was based on two-dimensionalifference equations and two-dimensional linear autoregres-ive models. This method expressed autocorrelation or powerpectrum density functions in terms of the two-dimensionalutoregressive coefficients.
In paper [11] a generation procedure of surface texture usinghe non-casual 2D AR model was suggested and its efficiencyas examined. Iso- and anisotropic surface topographies ofesired shape of autocorrelation function were simulated. Max-mum error of correlation distance of isotropic surface withxponential autocorrelation function was smaller than 3%.
Fast Fourier Transform is also popular in generating surfaces.ewland, in 1984 used it. He used circular autocorrelation func-
ion [12]. In 1992, Hu and Tonder used finite impulse responselter. The procedure of generation of Gaussian surfaces hav-
ng specified autocorrelation function is described in Reference13]. Wu [14] developed a numerical procedure of three-imensional surfaces. The method was based on FFT. It canimulate surfaces with given spectral density or autocorrelationunction.
You and Ehmann [15] developed technique for the synthe-is of three-dimensional surfaces. The method was based onhe application of the discrete and continuous two-dimensionalourier transformations associated with the theory of discretend continuous time series models.
In majority of presented above papers the surfaces ofaussian normal ordinate distribution were generated. Usu-
lly, non-Gaussian random surfaces with the desired skew andurtosis were obtained by transforming the surface of nor-al ordinate distribution using the Johnson translatory system
4,8,9,13]. Interesting procedure was proposed for generat-ng non-Gaussian surfaces [16]. This method based on FFTpproach is set preserve the skewness, kurtosis together withutocorrelation function or spectral density of surfaces. The
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2008) 457–463
hase part was obtained by using Johnson translator system.owever, the proposed method did not limit to Johnson’s trans-
ator system.There are references devoted to generated cylinder surfaces
f non-symmetric ordinate distribution. Scientists from Swe-en generated cylinder surface. The method of mathematicalachining developed by the authors of paper [17] allows the user
o create a 3D synthetic surface by defining the “key” param-ters: Ra, oil volume and groove intersection angles. In ordero simulate 3D surface, naive filter was developed. The origi-ally measured profiles create a pattern of the synthetic surface.ynthetic surface profiles were created by multiplication of theormalised Ra and valley co-ordinates by the desired parameteralue. In order to create the honing angles, the filtered valleyomponent was separated into odd and even valley componentsy sorting every other valley to odd, respectively, even valleyomponents. Paper [18] describes the future concept. The sim-lation was restricted to pure abrasive cutting of the workpieceaterial. Mechanisms of ploughing fatigue and cracking were
ot yet considered in this model.Imposition of the fractal components on the random compo-
ent is another possibility [19]. It allows us to obtain topographyf ground surface with honing grooves. In this way plateau honedylinder surface was obtained.
In paper [20] a numerical method for the generation ofrbitrarily oriented three-dimensional surfaces was presented.he procedure extends the method of the linear transformationatrix, first proposed by Patir [7]. The method is capable of pro-
ucing arbitrary oriented surfaces by using an autocorrelationunction that is rotated around its origin.
The authors of a lot of papers tried to simulate fractal sur-aces. As the example in paper [21] simulation of anisotropicractal surface and Monte-Carlo simulation of fractal surfaceere shown.
. The aim and scope of the investigations
The fundamental aim of the investigation is to develop andest the method of simulation of stratified surface topography.
The scope of the paper includes:
Recommendation of existing method of random surfacetopography of Gaussian ordinate distribution simulation.Development of the method of numerical generation of arbi-tralily oriented three-dimensional rough surfaces.Description of the method of surface topography after two ormore processes simulation.Presentation of examples of using this method.
. Results and their analysis
.1. The simulation of random Gaussian surfaces
The majority of current time series models can consider fewrders system. It means that only the features near the origin areimulated well. It is known that FFT is the fast and convenientool in generating surface topography. Therefore, I analysed only
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FT methods for generating surface topographies with requiredroperties.
The methods developed by Hu and Tonder [13], Newland12] and Wu [14] were analysed.
Wu thinks that there are mathematical mistakes in Hu andonder’s method. He believes that Newland’s method is also
ncorrect. He developed his own method, which was similar toewland’s method. Wu has different ideas about spectral density
nd autocorrelation functions obtained from generated surfaces,r used to generate surfaces.
Wu generated random profiles from the given autocorrela-ion functions using methods presented in References [12] and13] and his own method. He found out that his and Newland’sethod is better than Hu and Tonder’s method.He used similar methods to generate random surface and
ompared the obtained with given autocorrelation functions. Heound out that Newland’s and his method were better than Hund Tonder’s method also in three dimensions. Ho believes thatis method is slightly better than Newland’s method. Althoughis method could not guarantee that the generated profiles havehe correct autocorrelation function, it can guarantee that theverage autocorrelation function of all profiles for the generatedurface topographies is correct.
The methods proposed in References [12–14] were alsohecked by the present author. Iso- and anisotropic randomurfaces of normal ordinate distribution of exponential shapef autocorrelation function were generated. The proposedrocedures were also a little modified in order to obtain surfacerofiles of proposed shape of autocorrelation function. Theesired correlation length of computer-generated profiles (theistance in which correlation between neighbouring pointsecays to 0.1 value) was compared with obtained correlationength. It was found that when correlation length was small,hree methods assured correct values. However, for bigger
orrelation length values (when neighbouring points weretrongly correlated) methods developed by authors of papers12,14] were better than by authors in Reference [13]. Theame conclusion was obtained as a result of analysis of profilestptm
Fig. 1. Anisotropic surface, when lay is parallel to referenc
2008) 457–463 459
rom simulated surfaces. However, it was difficult to say, whatethod from proposed in References [12] or [14] was the
est.The recommended procedure was used in order to obtain iso-
nd anisotropic (one-directional) Gaussian random surface. Thisurface can be also arbitrarily oriented. Bakolas [20] proposedrocedure based on change of orientation of required shape ofutocorrelation function. The author of the present paper usedifferent method.
In the first stage of the method the one-directional structures created, when lay is parallel to one of the reference axes. Thenhis surface is rotated of given angle φ. It can be done, for exam-le, by rotation of reference system. Of course the measuringrea of the initial one-directional surface should be bigger thanf the obtained surface.
Fig. 1 presents computer created one-dimensional surfacend surface rotated by angle φ.
It is possible to obtain crossed structure (like of cylinder liner)n this way. It is necessary to obtain also second surface inclinedy angle 180◦ (φ). Then the ordinates of obtained two surfacesre compared and smaller ordinate is selected. However, it isecessary to remember, that Sq parameter of obtained surfaces usually 1.25 times smaller than of inclined surfaces. So theurface height should be corrected.
.2. The simulation of profiles after two processes
The random Gaussian surfaces can be completely defined bywo parameters: the standard deviation height parameter and hor-zontal parameter derived from the autocorrelation function [22].
ost of the surfaces are manufactured by more than one process.he resulting texture is combination of the textures due to the
wo processes. Sintered materials and plateau honed cylindertructure are the examples. The understanding of two-process
extures comes from an understanding of the independent com-onents of the texture.The parameters describing surface afterwo processes can be calculated from the probability plot ofaterial ratio curve. The slope of each presented straight lines
e axis (a), this surface inclined by angle φ = 30◦ (b).
460 P. Pawlus / Wear 264 (2008) 457–463
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Fig. 2. Graphical interpretation of the probability parameters.
ives the Rq roughness of the corresponding process. Also, theransition characteristic (plateau depth Pd) from one to anotherrocess can be estimated. Negative sign represents that the trun-ation took place below the mean line of the initial profile. Intandard ISO 13565-3 Pd parameter is not calculated. Insteadf it Rmq is computed. But there is a connection between Rpq,vq, Rmq and Pd:
d = Rmq(Rpq − Rvq) (1)
The proposed method gives valuable information regardinghe functionality of the surface (see Fig. 2).
The following procedure should be done in order to simulatewo-process profile.
. Creation two Gaussian profiles, plateau (PP) and valley (PV)with correlation lengths and variances as parameters charac-terizing them.
. The choice of the distance (Pd) between the mean lines ofthe profiles (the centre of the distributions).
. For all the points “i” of two distributions (profiles): ifPP(i) > PV(i) then RP (i) (resulting profile after two pro-cesses) = PV(i), else RP(i) = PP(i).
After simulation of a lot of profiles in this way it was foundhat the Rpq, Rmq and Rqq parameters were similar to theresumptions. Dependence (1) was also confirmed. Computerreation of two-process profile is presented in Fig. 3.
It is interesting if one can predict parameters PR of profilefter two processes when parameters characterising Gaussianrofiles creating final profile Pp and Pv are known. From thisssumption the resulted profile parameter should be:
R = PpRmq + Pv(1 − Rmq)
Rmq should be in linear scale. More than 50 profilesere analysed. Rmq parameter was between 50 and 90%, butvq/Rpq ratio was between 5 and 18. It was found that errors ofbtaining average slopes (�a) were very small (average errorsbout 3.5%). The errors of parameters �q were bigger and
mounted (mean values) to about 20%. The last results wereossible to predict, rms slopes are sensitive on valleys existence.ather small average errors were found during analysing theeak curvature calculated according to 3-point equation (aboutFRg
ig. 3. Computer creation of two-process profile: (upper) plateau profile, (mid-le) valley profile, (lower) resulting two-process profile.
%). Average relative errors of prediction of Rq were also small5.5%). It was difficult to forecast spacing parameters of thenal profile. The predicted values were similar to the param-ters of the final profile only when spacing parameters of thease (valley) part were bigger than of fine (plateau) part. In thisase the average error of Sm was 20%, and of correlation length0%. In the other case the accuracy of parameters prediction wasmaller.
One should remember that the shape of two-process profileepends not only on height but also on spacing parameter (seeig. 4). SPv and SPp are mean spacing parameters characterising
he plateau and valley part (correlation lengths).It is also possible to create stratified profile from Gaussian real
rofiles and presumed distance between their mean lines. Onean also impose Gaussian profile on deterministic (real or sim-
ig. 4. Profiles characterised by the following parameters: Rpq = 1 �m,vq = 5 �m, Pd = −3 �m; in upper graph, SPv = 100 �m, SPp = 10 �m; in lowerraph, SPv = 10 �m, SPp = 100 �m.
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Fig. 5. View of simulated surface after two processes.
.3. Creation of two-process surface
The method of simulation of 3D surface after two processess similar to described above. It is also based on descriptionf surface stereometry using probability graph. The followingossibilities are possible:
Imposition of simulated random plateau surface on simulatedrandom valley surface.Imposition of simulated random surface on real random sur-face.Imposition of simulated random surface on simulated deter-ministic (or mixed) surface.Imposition of simulated random surface on real deterministic(or mixed) surface.Imposition of real random surface on simulated random sur-face.Imposition of real random surface on real random surface.Imposition of real random surface on simulated deterministic(or mixed) surface.Imposition of real random surface on real deterministic (ormixed) surface.
The base (valley, virgin) surface can be after one or tworocesses. The plateau (smooth) surface is usually one-processandom surface. It can be isotropic or anisotropic (Fig. 5).
SSf
Fig. 6. View of real (a) and simulated p
2008) 457–463 461
.4. Examples of using the proposed method
During investigation of honed cylinder surface topographyt was found that it cane be described by the following set ofarameters: Smq, Spq, Svq (as the parameters describing theaterial ratio curve), Sq as the statistical height parameter,
wo spacing parameters in two perpendicular directions SPaxaxial) and SPcir (circumferential) and in addition mean profileverage slopes in these directions P�aax, P�acir. SP parameterhould be explained. It is equal to maximum sampling intervalecommended in Reference [23]. This sampling interval washosen equal to 1/(2fk), where fk is the minimum frequency forhich the vertical distance between the accumulation spectrum
urve and the straight line describing (fitted to) its upper (almostinear) part does not exceed some fraction (0.02) of the totalower. The author found that the proposed frequency is verylose to frequency of maximum radius of curvature of functionpproximating cumulative spectrum curve. From the principlef simulation Smq, Spq and Svq parameters of modelled andeal surface should be the same. So the emptiness coefficientp/St was chosen as the parameter describing the shape of
he material ratio curve. The same parameters describe wornylinder surface. In addition the other height parameters werebtained: Sa, Ssk, Sku, St, Sk, summit density Sds, summiturvature SSc, summit curvature in axial and circumferentialirections SScax and SSccir as well as average rms surfacelope S�q. Based on the analysis of cylinder surfaces theollowing limiting errors of the fundamental parameters werebtained: Sp/St = 0.1, Sq = 0.1 �m, SP = 2.6 �m, P�a = 0.01.he simulation of two-process cylinder surface was done.oth the base (valley) and plateau parts were digitally simu-
ated. The result was compared with the results of measuredylinder. The simulated honing angle was equal to realngle.
The views of real and simulated surface are presented inig. 6.
The real surface was characterised by the following parame-ers: Sa = 0.96 �m, Sq = 1.28 �m, St = 14.98 �m, Sp/St = 0.26,k = 2.35 �m, SPax = 18.5 �m, SPcir = 31.2 �m, SSk = −1.4,
Sc = 0.013 �m , SScax = 0.016 �m , SSccir = 0.009 �m ,ds = 0.0071 �m−2. The simulated surface topography has theollowing parameters: Sa = 0.97 �m, Sq = 1.29 �m, St = 9.7 �m,
lateau honed cylinder surface (b).
462 P. Pawlus / Wear 264 (2008) 457–463
Fig. 7. View of real (a) and simulated worn cylinder surface (b).
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p/St = 0.27, Sk = 2.24 �m, SPax = 18.3 �m, SPcir = 30.1 �m,Sk = −1.3, Sku = 5, S�q = 0.11, P�aax = 0.047, P�acir =.067, SSc = 0.016 �m−1, SScax = = 0.02 �m−1, SSccir = =.012 �m−1, Sds = 0.0073 �m−2. As seen the matching errorsf fundamental parameters were much smaller than the limitingrrors. Some differences of maximum surface height whichas influenced by individual peaks of real surface (however,
he corrected surface heights corresponding to material ratioange 0.13–99.87% were 7.7 and 7.5 �m for real and simulatedurfaces) was observed as well as of kurtosis (the individualeaks had great effect on it) and peak curvature.
The similar procedure was done for worn cylinder surface.owever, in this case the simulated digital surface was imposedn the real surface after honing. The views of real and simulatedurface are presented in Fig. 7. The real surface was characterisedy the following parameters: Sa = 0.24 �m, Sq = 0.3 �m, St =.4 �m, Sp/St = 0.38, Sk = 0.57 �m, SPax = 23.2 �m, SPcir =4.3 �m, SSk = −1.9, Sku = 12, S�q = 0.045, P�aax = 0.014,�acir = 0.028, SSc = 0.01 �m−1, SScax = 0.005 �m−1, SSccir =.015 �m−1, Sds = 0.0026 �m−2 and the simulated surfacey the following parameters: Sa = 0.26 �m, Sq = 0.31 �m,t = 4.6 �m, Sp/St = 0.32, Sk = 0.62 �m, SPax = 23.8 �m,Pcir = 16.0 �m, SSk = −2.1, Sku = 13, S�q = 0.045, P�aax =.013, P�acir = 0.028, SSc = 0.011 �m−1, SScax = 0.006 �m−1,Sccir = 0.016 �m−1, Sds = 0.0023 �m−2. In this case also theatching errors were smaller than the limiting errors.
The proposed method was used for simulation of piston skirturfaces after “zero-wear” process. The analysis of theses sur-ace type is more complicated because the base (valley) structureas deterministic, but the plateau structure—random character.
sdis
orn piston skirt surface (b).
o the special programme was used in order to determine Sqparameter. It was found that these surfaces can be characterisedy the following set of parameters: Sq, S�q, Sp/St (the simplestescription) and S�q/Sq (or Sds) and P�aax/P�acir (widerescription). The limiting errors of these parameters are:q 0.25 �m, Sp/St = 0.16, S�q = 0.01, S�q/Sq = 0.01 �m−1,ds = 0.0016 �m−2, P�aax/P�acir = 0.24. During simulationrocedure the simulated surface was imposed into the realurface. The real surface was characterised by the followingarameters: Sa = 2.1 �m, Sq = 2.5 �m, St = 11.6 �m, SSk =0.63, Sku = 2.2, S�q = 0.087, P�aax = 0.053, P�acir =
.028, SSc = 0.015 �m−1, SScax = 0.014 �m−1, SSccir =
.016 �m−1, Sds = 0.0019 �m−2. The simulated surface washaracterised by the following parameters: Sa = 2.2 �m,q = 2.6 �m, St = 10.1 �m, SSk = −0.6, Sku = 1.9, S�q = 0.09,�aax = 0.056, P�acir = 0.034, SSc = 0.018 �m−1, SScax = =.018 �m−1, SSccir = = 0.019 �m−1, Sds = 0.0015 �m−2.ig. 8 presents the real and simulated piston skirt surfaces.
. Conclusions
It was found that methods based on FFT are good for mod-lling the random surface of Gaussian ordinate distribution. It isossible to simulate arbitrarily oriented random surface basingn the rotation of the co-ordinate system. Two-process surfacean be modelled by imposition of random plateau Gaussian
tructure on base (valley) structure. The valley part can haveeterministic or random character. This procedure can be donen two and three dimensions. The simulation results proved howome parameters like average slope and standard deviation of
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eight of two-process surface depend on these parameters ofwo basic surface parts. This information is helpful in simula-ion process. It was found that simulation method can be usedor various surface types: plateau honed surface and surfacesfter “zero-wear” process; worn cylinder liners and worn pistonkirts are typical examples.
cknowledgement
I thank Mr. Zenon Krzyzak for collaboration.
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