Microsoft Word - B_31_R.doc HUNGARIAN JOURNAL OF INDUSTRIAL CHEMISTRY VESZPRÉM Vol. 38(2). pp. 211-214 (2010) POWER SPECTRAL DENSITY (PSD) ANALYSIS OF MACHINED SURFACES I. BARÁNYI Donát Bánki Faculty of Mechanical Engineering and Safety Technology, Óbuda University 1081 Budapest, Népszínház u. 8., HUNGARY E-mail: baranyi.istvan@bgk.uni-obuda.hu In the first half of the 90s, computers of adequate speed of operation and processing softwares became increasingly available, made it possible to realize 3D surface characterisation. In the literature different directives can be found none of them has become widely used. Beside the extension of 2D parameter based technique to 3D many other methods have been developed. One is the power spectral density (PSD) technique when “global” surface characterisation is carried out using complex mathematical tools. Nowadays one of the most important tasks in tribology to design the surfaces optimised to the operation. In the present study topographical measurements and PSD characterizations were performed to study the engineering surfaces produced by turning and to analyse the effectiveness of 2D and 3D PSD. Keywords: power spectral density, topography, fractal dimension Introduction Efficient research in the course of the past decades has provided experts involved in surface microtopography research with a number of tools and methods to design an operationally optimized surface. At the same time, this knowledge is utilized only to a small degree in the analysis and control of tribological processes. Characteristically, designers continue to content themselves by requiring few roughness parameters [1]. In the first half of the 90s a lot of surface characterisation method has been developed. One is the power spectral density (PSD) technique when “global” surface characterisation is carried out using complex mathematical tools. PSD provides full length scale analysis, which takes into consideration not only the dominant topographic elements but also the submicro features, in contrast with traditionally surface characterisation methods. The fractal dimension derived from PSD topography seems to be an efficient tool for characterization. In the present study topographical measurements and PSD characterizations were performed to study the orientated engineering surface two and three dimensional fractal analysis. In my experience, these methods are associated with in point of view of fractal dimension. Mathematical background of characterisation To characterize the measured topographies an algorithm was developed in cooperation with Department of Machine and Industrial Product Design, Budapest University of Technology and Economics and interpreted as PSD analysis software. The theoretical base of 3D PSD analysis was [2] and [5]. Discrete Fourier transformation (DFT) of 3D topography can be written as follows: ∑∑ = = +−Δ⋅Δ= N d M c qyqxi dcyx ydxceyxzxyqqF 1 1 )(2),(),( π (1) where: qx – frequencies in x direction, qy – frequencies in y direction, z(xc, yd) – height coordinate located in xc, yd, M – number of points in profile, N – number of profiles, Δx– sampling distance in x direction, Δy – sampling distance in y direction. DFT gives complex results: )),(Im()),(Re(),( yxyxyx qqFiqqFqqF ⋅+= (2) The PSD “amplitude” is calculated (Fig. 1): yxMN FF APSD ΔΔ + = 22 ImRe (3) where: APSD – PSD amplitude, Re F – real part of the Fourier transformation, Im F – imaginary part of the Fourier transformation. 212 lgAPSD lgql lgpk Figure 1: PSD Surface PSD topography can be reduced to PSD curve using (4). It means 2D representation, which can be easy handled, but contains 3D information about topography (Fig. 2). 22 yx qqq += (4) Figure 2: PSD curve from the reduction of the PSD surface There are two possibilities of showing results. One is to represent the amplitude of PSD in the function of wavelength. The other prevalent method is logarithmic scale frequency-PSD amplitude visualization. The practical gain of the first method is that dominant wavelength components appear as a maximum point of the PSD curve. In the second method the height frequency range of the curve can be approximated by a line. The slope of the line is in correlation with the fractal dimension of surface. In the latter case, wavelengths smaller than the highest dominant wavelength play a considerable role. PSD amplitude becomes constant – the self-affinity character of the surface disappears – in a lower wavelength range. The slope of fitted line (s) to PSD curve has correlation with fractal dimension of surface according to (5). 2 4 s Df += (5) where: Df – fractal dimension. PSD analysis of the theoretical surface In this part I solve the fractal dimension of the theoretical surface. This analysis gives a chance to investigate the surface PSD profondly. In this part I generate a topography which is constans in every direction. The PSD surface and PSD curve of this surface summarised in Fig. 3 and Fig. 4. Figure 3: Theoretical surface PSD analysis Figure 4: Approximating straight line The calculated fractal dimension from the slope of the straight line is Df = 2,51. We can see in Fig. 3, the applied program use calculating in logarithmical scale. In Fig. 4 felharmoincal points are farther position of the straight line. Two and three dimensional PSD analysis of the turned surfaces The turned surfaces have two orientation: one is perpendicular, and the other is paralell with the machining direction. The profiles of these directions are different (Fig. 5 and 6). In this part I choosed a surface, which average surface roughness war Sa = 3,02 μm. In this part of investigation was the fractal dimension in direction of paralell and perpendicular direction with machining direction. Fig. 5 shows the profile with characteristic wavelenghts. Fig. 7 representes this profile main wavelenghts. Parallel in the machining direction (Fig. 6) can’t find characteristic wavelenghts (Fig. 8). 213 Figure 5: profile in perpendicular with machining direction Figure 6: Profile in parallel in the machining direction Figure 7: Profile 2D PSD analisys in machining direction Figure 8: Profile 2D PSD analisys in perpendicular with machining direction The calculated fractal dimension of the profile is Df = 1.36 in machining direction and the turned topography fractal dimension is Df = 2.33. In this case the Persson equation associate the profile and the surface fractal dimension. The affection of small PSD amplitudes (in parallel direction of the machinining direction) isn’t considerable in the calculation of 3D fractal dimension (the 2D farctal dimension is in 1 and 2, and the 3D fractal dimension is 2 and 3). The 3D fractal dimension is one more than 2D fractal dimension. Analisation of turned surface with various roughness In this part of investigation I generated profiles with various average roughness from the original profile (Fig. 5). The profiles can be solved with constants multiplication. This generation algorythm can be warranted to waves are same in the profiles and the average roughness is more or less than the original profile’s. Table 1 summarizes the average roughness, the two dimensional and three dimensional fractal dimension and the vertical axis intersection of the generated profile. Fig. 9 shows the modification of the vertixal axis intersection. Based on these results it can be supposed that fractal dimension didn’t modify, when the profile is increased in vertical direction. This increasing modify the PSD amplitude and offset the Persson line up. Table 1: characterestic data of the profile 2D analysis 3D analysis Ra [μm] intersect. with vertical axis Df intersect. with vertical axis Df 0.6 -7.1066 1.36 -7.0378 2.33 0.75 -6.9127 1.36 -6.8631 2.33 1.51 -6.3107 1.36 -6.2931 2.33 3.02 -5.7086 1.36 -5.7114 2.33 6.03 -5.1066 1.36 -5.1487 2.33 9.05 -4.7544 1.36 -4.8218 2.33 12.06 -4.5045 1.36 -4.5895 2.33 15.08 -4.3107 1.36 -4.4379 2.33 -7,5 -7 -6,5 -6 -5,5 -5 -4,5 -4 0 5 10 15 20 Ra In te rs ec tio n 2D analysis intersection 3D analysis intersection Figure 9: the modification of the vertixal axis intersection 214 Conclusion The turned surfaces farctal dimension is different in dirrection of perpendicular and parrellel direction. The characteristic direction is parallel with the machining direction. The 3D fractal dimension is one more than the parallel profile fractal dimension. The profiles fractal dimension isn’t change, when the profile is blown up or reduced. The Persson line intersection with the vertical axis is modify in blown up or reduced profile. REFERENCES 1. B. PALÁSTI KOVÁCS: Forgácsolással megmunkált felületek mikrogeometriájának értékelése, Kandidátusi értekezés, Budapest, 1983. 2. S. HORVÁTH: A felületi hullámosság jelentősége, a hullámosság 3D-s értelmezése, GÉP 2005/9-10, 2005, 82–85, 3. VDA 2006: Oberflächenbeschaffenheit, Regeln und Verfahren zur Beurteilung der Oberflächenbeschaf- fenheit, 2002. 4. B. N. J. PERSSON, O. ALBOHR, U. TRATAGLINO, A. I. VOLOKITIN, E. TOSATTI: On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion, J. Phys, Condens. Matter, 17, R1-R62, 2005. 5. Á. CZIFRA: Sensitivity of power spectral density (PSD) analysis for measuring conditions, Rudas I. J., Fodor J., Kacprzyk J. (Eds.): Towards Intelligent Engineering and Information Technology, 2009, 505–517. 6. K. J. STOUT, L. BLUNT, P. J. SULLIVAN, W. P. DONG, E. MAINSAH, N. LUO, T. MATHIA, H. ZAHOUANI: The development of methods for characterisation of roughness in three dimensions, Printing Section, University of Birmingham Edgbuston, Birmingham, 1993. << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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