Wide frequency range 3D power spectral density analysis of plunger’s topography of a brake system A. Czifraa*, T. Godab, K. Varadib, E. Garbayoc b
a Institute of Mechanical Engineering and Safety Techniques, Budapest Tech, H-1081, Budapest, Népszínház u. 8, Hungary Department of Machine and Product Design, Budapest Uni. Technology and Economics, H-1111, Budapest, Műegyetem rkp. 3. Hungary c TRW Automotive, Pamplona, Spain
Received Date Line (to be inserted by Production)
Abstract Power spectral density (PSD) analysis is an efficient tool of “full length scale” characterisation of topographies of engineering surfaces. In case of self-affine surfaces the fractal dimension of topography can be calculated from the slope of logarithmic scaled PSD curve. Topographic measurements of plunger were performed by stylus instrument and atomic force microscope (AFM). The sampling distance was in the range 20 nm-3 m, while the largest measuring area was 3x3 mm2, thus the whole frequency range - five orders of magnitude - of surface roughness measurement was investigated. 3D PSD analysis of measured topographies was carried out with software developed by the authors. After transforming the 3D PSD topographies to 2D PSD curve the fractal dimension representing the surfaces was calculated. The results show that, in case of surfaces studied, the linear approximation of PSD curve can not be used, because the slope of the curve changes depends on frequency. Based on our observations the PSD curve – after a constant part – can be substituted with two lines having different slopes. The slopes of these two lines give the characteristic fractal dimension of the micro- and nano-topography. Keywords: Power spectral density, Topography, Fractal dimension
1.
Introduction
Surface microtopography plays a considerable role in the friction and wear processes of machine components. Efficient research in the course of the past decades has provided experts involved in surface microtopography research with a number of tools and methods. Power spectral density (PSD) analysis is an intensively investigated area of comprehensive microtopography analysis. The PSD technique is highly stable; it is slightly sensitive to measuring and sampling parameters. The fractal dimension derived from PSD topography seems to be an efficient tool for characterization. Information’s obtained from analysis of surface topography are input parameters of recent friction and wear models. Persson [1] calculates the hysteresis component of friction coefficient using the
*
Corresponding author. Tel.: +36 16665391; fax: +36 16665484. E-mail address:
[email protected] (Czifra, Árpád)
PSD curve, while [2] utilizes the PSD in simulation of adhesive and abrasive wear. The reliability of models is related to the efficiency of topographic analysis. The aim of our research was to investigate the PSD in wide frequency range using AFM and stylus instrument to measure the topography of plunger of a break system. Nomenclature APSD Df F(qx,qy) q, qx, qy M, N s x, y z(xc,yd)
2.
‘amplitude’ of power spectral density fractal dimension Fourier transformation frequency number of measured points in direction x and y slope of the Persson’s PSD curve sampling distance in direction x and y height coordinate located in xc,yd
Methodology
2.1. Measurements One plunger was measured with the parameters summarized in Table 1. Sampling distance (same in booth directions) [nm] 3000 1000 352.9 196.1 98.04 39.21 19.61
Measuring area [µm2]
Equipment
3000x3000 1000x1000 90x90 50x50 25x25 10x10 5x5
Stylus instrument
AFM
Number of measurements 1 1 3 2 2 2 2
Table 1. Measuring conditions
2.2. Characterisation technique To characterize the measured topographies an algorithm was developed and interpreted as PSD analysis software. The theoretical base of 3D PSD analysis was [1] and [3]. Discrete Fourier transformation (DFT) of 3D topography can be written as follows: N
M
F ( qx , q y ) y x z( xc , yd )e d 1 c 1
i 2 ( x c q x y d q y )
(1)
DFT gives complex results, so PSD „amplitude” is calculated: Re2 F Im2 F APSD MNxy
(2)
Showing the PSD results logarithmic scale is used. PSD topography can be reduced to Persson’s PSD curve using (3). It means 2D representation, which can be easy handled, but contains 3D information about topography.
q qx2 q 2y
(3)
The slope of fitted line to Persson’s curve has correlation with fractal dimension of surface according to (4).
Df 4
s 2
(4)
In our PSD analysis 125x125 number of frequency were used in logarithmic division. All frequencies were taken into consideration when Persson’s curve was calculated.
3.
Results
Fig. 1. shows two topographies of measured surface. One is measured by stylus while the other by AFM. Based on topographic images it can be proved that dominant topographic elements are relatively height, they are in micrometric range. It is important in “traditional” topography characterisation for instance in case of parametric method. The surface parameters are connected to dominant topographic elements, so the measuring area can greatly influence them. In our case (see Fig. 1.) the stylus topography “z” scale extent is about 4500 nm, while AFM measurement is only a part of it, about 380 nm. Power spectral density analysis reveal the fractal character of the surface, so – if the studied surface is a fractal – it does not depend on the measuring area, and take all frequency information into consideration. [nm] 1490
[nm] 129
1 mm -3190
1 mm
10 µm -159
10 µm
Fig. 1. Topographies measured by stylus (left) and measured by AFM (right)
Fig. 2. shows the results of PSD analysis of 1 by 1 mm stylus measured topography. The “conical” shape of PSD surface facilitates the appliance of Persson’s simplification. The frequency range covered by the measurement is wide enough to find the break point of curve. It is –2.15, that means 0.0071/µm frequency, so the highest wavelength is about 140 µm. The dominant topographic elements can be characterised with this
“dimension”. All AFM measurements areas are under this size, so the Persson’s curves of AFM measurements can not reach the break point. The linear character of Persson’s curve of stylus measurement is unambiguous, out of accordance in AFM measurements (see Fig. 3.). In that case the PSD surface differs from “cone” and the Persson’s curve has major deviation.
Fig. 2. PSD surface (left) and Persson’s curve (right) of 1 by 1 mm stylus measured topography
Fig. 3. PSD surface (left) and Persson’s curve (right) of 10 by 10 µm AFM measured topography
Table 2. summarise the results. The frequency ranges are overlapping each other, but the fractal dimensions are different. The difference of fractal dimension values is so high that we can suppose that not only one fractal dimension belong to the surface. It seems that higher frequency range has higher fractal dimension. The break point may be in range 0.05635 – 0.8239 1/μm (90x90 µm AFM measurments). Under this frequency range (stylus measurements) “micro” fractal dimension, while over it “nano” fractal dimension can be assigned. It means that surface is not an ideal self-affin fractal, and to do a correct analysis wide frequency range must be performed. It is also important that smaller topographies have higher standard deviation in fractal dimension. The reason is that in small AFM measurements are included only a very local part of topography, actually only a part of a dominant micro-topographic elements.
Sampling distance [μm] 3 1
Frequency range1 [1/μm]
90x90 μm
0.3529
0.0111 – 1.416
50x50 μm
0.1961
0.02 – 2.549
25x25 μm
0.0980
0.04 – 5.102
10x10 μm
0.0392
0.1 – 12.75
5x5 μm
0.0196
0.2 – 25.5
Measurement Stylus
AFM
3x3mm 1x1mm
0.00033 – 0.16 0.001 – 0.5
Analyzed frequency Fractal range2 dimension [1/μm] 0.0118 – 0.1012 2.218 0.0164 – 0.2479 2.215 0.05635 – 0.9907 2.398 0.0670 – 0.8239 2.356 0.05635 – 0.8239 2.420 0.1014 – 1.483 2.445 0.1260 – 1.483 2.478 0.202 – 3.589 2.572 0.252 – 2.967 2.458 0.507 – 7.415 2.552 0.630 – 7.415 2.478 1.014 – 12.5 2.595 1.260 – 12.5 2.498
Average fractal dimension 2.22 2.39 2.46 2.52 2.51 2.55
Table 2. Fractal dimensions of topographies frequency range depends on the measured area and sampling distance 2 analyzed frequency range is the range where the line is fitted to PSD curve 1
4.
Conclusions
Based on the results following conclusions can be drawn: PSD is an effective tool for surface characterization; Reliable characterization of surface needs wide frequency range analysis; Two different fractal dimension (Df) can be calculated depending on frequency range. Measuring area effects the standard deviation of Df: smaller topographies have higher deviation.
Acknowledgements The authors wish to acknowledge the support of the Partners of the KRISTAL project and the European Commission for their support in the integrated project “Knowledge-based Radical Innovation Surfacing for Tribology and Advanced Lubrication” (EU project Reference NMP3-CT-2005-515837). Authors return thanks to Péter Nagy (Chemical Research Center, Hungarian Academy of Sciences) for atomic force microscope measurements.
References [1] 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 (2005) R1-R62 [2] M. Schargott, V. Popov, Diffusion as a model of formation and development of surface topography, Trib. Inter., 39 (2006) 431-436 [3] Stout, Sullivan, Dong, Mainsah, Luo, Mathia, Zahouni, The development of methods for characterisation of roughness in three dimensions, Printing Section, University of Birmingham Edgbuston, Birmingham, 1993
