MEASUREMENT SCIENCE REVIEW, Volume 1, Number 1, 2001
CUMULATIVE CURVES CHARACTERISING LOAD CARRYING
CAPABILITIES AND WEAR PROPERTIES OF SURFACES
Paweł Pawlus, Jerzy Jabłoński
Rzeszów University of Technology
35-959 Rzeszów, ul. W. Pola 2, Poland
fax: +48 17 8543112
email: ppawlus@prz.rzeszow.pl
Abstract
The possibilities of generating various cumulative curves were studied. The mutual positions of these
curves were analysed basing on computer-generated profiles of various shapes and real profiles of
cylinders.
1. Introduction
Surface topography plays an important role in various functions of many kinds of components. The
attempts of contact and wear predictions are based on two kinds of parameters/functions: material/air
ratio as the function of height and parameters based on peaks. The material ratio curve was established
by Abbott and Firestone [1]. It describes the transition of air to material though the surface profile. It is
also a useful tool in characterising a large group of surfaces of some practical importance. These
surfaces are not produced in a single operation but in sequence of machining operations [2]. Such
processes are termed as multi-process [3] or stratified [4].
Greenwood and Williamson looked for a criterion, which would determine the mode of deformation of
an array of asperities of varying heights. Their versions of so-called plasticity index describe the
roughness in terms of the mean asperity radius of curvature and the standard deviations of the asperity
height distribution [5]. Other workers used different formulae based on a slope [6, 7, and 8]. The
conventional way to calculate average peak height is to split the profile vertically into equal height
intervals and the count the number of peak within this interval. The peak counts (peak frequency) is
sometimes plotted as a function of height, which is called peak distribution histogram. Whitehouse in
new paper [9] proposed another method of estimation of peak probability. It does not derive the peak
probability curve via the peak count but the “extent” of profile surrounding each peak is measured and
summed for each height. In practice each peak is “weighted” by its radius of curvature. This method is
called “peak probability”; not “peak frequency”. The proposed curve could be a better indication of the
load carrying capability and wear properties of a surface than the material ratio curve and the
cumulative peak count, which can give information about spatial correlation of peaks.
2. The scope of the investigation
In work [9] profile of Gaussian ordinate distribution was analysed. The present authors should confirm
the theoretical investigations based on the computer-generated profiles of Gaussian height distribution.
Profiles of exponential shape of auto correlation function were analysed. The method of computation
originated from paper [10] and adapted from 3D surface topography to 2D profile. Other two curves
were analysed: the cumulative HSC curve (high spot count) and average slope cumulative curve. The
positions of these 5 curves were compared for irregularity profiles of the same height, defined by Rq
parameter, but various correlation lengths CL (the distance at which auto correlation function decays
to 0.1). New aspect is the study of computer generated profiles after two processes. The method of this
profile generation is described in [11]. Finally, the results of practical experiments were analysed.
3. Methods
The curvature of a peak [12] is simply expressed by the three-ordinate Lagrangiam differential
formula:
c  (2 yo  y1  y1) / h2
y-1,0,+1 are the ordinates of neighbouring points, h is the sampling interval. A point is considered as the
peak if his ordinate yo is bigger than ordinates of points y-1, y+1. Each peak was counted (when yo was
between y and y +y) and weighted by the measured value of radius of curvature 1/c. Cumulative
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Measurement of Physical Quantities ● P. Pawlus, J. Jabłoński
probability peak curve was then obtained in a way similar to material ratio curve. Cumulative
frequency peak curve was received, when the weighting factors were equal to 1. In addition, the
cumulative high spot count curve was analysed. Because the slope is very important factor in contact
analyses, the cumulative slope curve was obtained. The local slope [13] was calculated based on the
following formula:
dy/dx = (yi+3 –9yi+2 +45yi+1-45yi-1+9yi-2-yi-3)/(60 x)
The slope was counted when the point yo was between y and y+y. The average slope is the mean
value of absolute local slopes. The last curve analysed was the material ratio curve MR.
Two-process profile was characterised by the following parameters: Rqp, Rqv, CLp, CLv and plateau
depth DIS [14].
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4. Results
Figures 1 and 2 present the cumulative curves for computer-generated profiles of Gaussian ordinate
distribution but various correlation lengths. From Fig. 2 one can see that the mutual positions of MR,
CPP and CFP curves are the same that Whitehouse suggested [9]. Once correlation length was bigger,
CPP and CFP were more similar to each other (the same positions). In Figures 1 and 2 the other curves
(HSC and CS) were similar and in the same position to MR curve, but in Fig. 2 distance between
these curves and CPP, CFP was smaller.
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Fig. 1. Material ratio curve (MR) of Gaussian profile (solid), cumulative probability peak (CPP) curve
(points) and cumulative frequency peak (CFP) curve (dashed) - left, cumulative HSC curve (solid),
cumulative slope (CS) curve (dashed) – right, CL = 10 m
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Fig. 3 presents computer-generated profiles after two processes. In two profiles Rqp =1 m, Rqv = 8
m, distance between fundamental profiles of Gaussian distribution (sometimes called plateau depth
[17]) DIS = -5 m (it means that the mean line of the upper profile is 5 m below mean line of the
lower), but in the upper figure CLv = 10 m, CLp = 200 m, in lower figure CLv = 200 m, CL:p = 10
m. Figures 4 and 5 present cumulative curves from profiles shown in Fig. 3.
From Figs. 4 and 5 one can see that the functional properties of surface obtained from CPP and CFP
curves are better than that obtained from MR curve. CS curve is similar to MR curve to some abscissa
(see Fig. 4). In Fig. 5 the cumulative HSC and CS curves are in the same positions.
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Fig.2. Similar curves to Fig. 1, but CL = 200 m
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MEASUREMENT SCIENCE REVIEW, Volume 1, Number 1, 2001
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Fig. 3(a) Profile: Rqp = 1 m, Rqv = 8 m, CLp = 200 m, CLv = 10 m, DIS = -5 m, (b) profile: Rqp =
1 m, Rqv = 8 m, CLp = 10 m, CLv = 200 m, DIS = -5 m
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Fig. 4. Various cumulative curves of profile from Fig. 3a
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Generally, for two-process texture, HSC and CS curves lie below MR curve, differently to Gaussian
profiles. Some real profiles after honing, plateau honing and running in were analysed. For profiles of
Gaussian ordinate distribution the positions of MR, CPP, and CFP curves were similar to
Withehouse’s prediction. The mutual positions of MR, cumulative HSC and CS curves were rather
similar. For stratified profiles HSC and CS curves lie below MR curve, so theoretical prediction was
confirmed. For some profiles after running in the analysed curves are similar to Fig.4, which suggests
that fine profile can have bigger spacing parameter than base profile. For all the profiles generally, the
MR curves were below CPP and CFP curves. Figure 6 presents various cumulative curves for axial
profile of plateau honed cylinder.
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Fig. 5. Various cumulative curves of profile from Fig. 3b
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Measurement of Physical Quantities ● P. Pawlus, J. Jabłoński
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Fig. 6. Various cumulative curves of axial profile of plateau honed cylinder
5. Conclusions
The possibilities of generating various cumulative curves were studied. In addition to proposal of prof.
D.J. Whitehouse other curves: HSC and slope cumulative curves were analysed. For real and
computer generated profiles of normal ordinate distribution the positions of these curves was in
accordance to Whitehouse’s prediction. For multi-process profiles the mutual positions of these curves
is a little different from Gaussian profiles. From the shape of the analysed curve it is possible to obtain
more information of functional properties of surface. The next step of investigation will be the analysis
of 3D surface topography.
References
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Surfaces having stratified functional properties: Part 1: Filtering and general measurement
conditions, Part 2: Height characterisation using the material ratio curve.
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