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Unformatted text preview: Tribol Lett
DOI 10.1007/s1124900994778 1 METHODS PAPER Measurement Uncertainties in Wear Rates 3 David L. Burris W. Gregory Sawyer 4
5 Received: 6 May 2009 / Accepted: 22 June 2009
Ó Springer Science+Business Media, LLC 2009 OO 19 1 Introduction 20
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29 Even in macroscopic tribology testing, there are numerous
experimental challenges in measuring and reporting both
friction coefﬁcients [1, 2] and wear rates [3]. And while it
is common to make comparative statements about changes
in friction coefﬁcient or wear rate of one experiment versus
another, the fact remains that in materials tribology some
portion of the sample is consumed during the experiment
and it is therefore impossible to have a standard artifact
that can be repeatedly interrogated. Thus, without the
ability to perform truly comparative analysis, it becomes A1
A2
A3 D. L. Burris
Department of Mechanical Engineering, University of Delaware,
Newark, DE, USA A4
A5
A6
A7 W. G. Sawyer (&)
Department of Mechanical and Aerospace Engineering,
University of Florida, Gainesville, FL 32611, USA
email: [email protected]ﬂ.edu PR Abstract Measurement uncertainties are vital to discussions of differences in measured values, yet they scarcely
accompany discussions of wear and wear rates in the tribology literature. In this methods article, approaches to
calculate uncertainties in singlepoint and steadystate wear
rates are presented. The analysis includes analytical treatments of uncertainties for typical macroscopic tribology
instrumentation. A fully statistical treatment using Monte
Carlo simulations is also presented and discussed for
steadystate wear rate uncertainty analysis for an arbitrary
number of interrupted measurements of volume loss. necessary to provide information regarding the uncertainty
in the measurement of friction coefﬁcient and wear rate.
Wear data are collected to provide insights into wear
phenomena and to offer guidance to design engineers.
These data are typically presented as rates, and may
involve a single measurement of wear at the end of an
experiment or multiple measurements of wear during an
experiment. These results are inﬂuenced by the intrinsic
properties of the materials, the surface preparation, the test
conditions (mechanical, chemical, electrical, etc…) and the
measurement methods. In order to make defensible statements regarding wear rate measurements, comparisons
between materials, or the dispersion of wear rate measurements, it is essential to provide details about the
uncertainties in the measurements.
There have been past efforts to examine the sources of
variability in wear results. Almond and Gee [4] and Czichos et al. [5] reported on interlaboratory wear testing of
nominally identical material pairs and nominally identical
conditions, while Guicciardi et al. [6] conducted repeat
experiments with a single tribometer under controlled
conditions. Each study demonstrated signiﬁcant variability;
not surprisingly, the results of intralaboratory repeat
experiments varied less than interlaboratory repeat tests.
Almond and Gee chose to ascribe the high interlaboratory
variability in mass loss values to differences in the types of
test machines. In general, it is very difﬁcult to determine
the relative contributions of materials properties, test conditions, instrumentation, and measurement errors to
measurand variability without a detailed analysis of each
contributor.
Uncertainty analysis makes use of the Law of Propagation of Uncertainty in order to characterize the dispersion of values that are reasonably attributed to the
measurand. In wear measurements, it is most common to Wear Á Wear rates Á Uncertainty analysis CO
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E Keywords CT ED 6
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64 Tribol Lett report a wear rate that follows Archard’s wear equation
(Eq. 1): 84 2 Uncertainties in Wear Volume Measurements 85
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93 In order to provide an uncertainty in a wear rate measurement, the uncertainties in the force, sliding distance,
and wear volume must all be accounted for. There are a
number of different techniques that are used to measure or
infer the wear volume, and three of the most common
techniques are reviewed in this section. Depending on the
instrumentation design, it is possible that the uncertainties
in the normal load and sliding distance will also require an
additional analysis. 94 2.1 The Spherical Cap Over the past 50 years, pinondisk experiments make
frequent use of a spherically capped pin as a standard
sample geometry. The pin effectively eliminates edge
contacts and provides a relatively straightforward measurement of pin wear volume under situations that produce
ﬂat circular wear scars on the end of the pin. Under such
situations, optical measurements of the resulting circular
ﬂat spots are used to calculate the removed volume of
material. The exact solution for the volume of a spherical
cap with a height (h) on a sphere of radius (R) is given by
Eq. 2, and shown schematically in Fig. 1a:
1
V
À ¼ ph2 ð3R À hÞ:
3 107
108
109 R2 ¼ ðR À hÞ2 þ a2 ¼ R2 À 2Rh þ h2 þ a2 ; ð 3Þ 2Rh ¼ h2 þ a2 : ð 4Þ 111 It is common practice to keep the circular wear scars
much smaller than the radius of the pin (a ( R). This has
lead to the orderofmagnitude assumptions made in Eqs. 5
and 6, which ultimately result in the approximate solution
given by Eq. 6. The ﬁrst step is to recognize that 2R ) h,
which simpliﬁes Eq. 4 to the approximate expression given
in Eq. 5: 113
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119 hﬃ a2
:
2R ð 5Þ Substitution of Eq. 5 into Eq. 2 gives an approximate
solution for the wear volume; this expression is further
simpliﬁed using orderofmagnitude approximations to
give the ﬁnal form of the expression:
2
1 a2
a2
pa4
V
Àﬃ p
:
ð 6Þ
3R À
ﬃ
3 2R
2R
4R 121
122
123
124 This expression always under predicts the worn volume,
and the error from this approximation monotonically
increases with an increasing ratio of a/R reaching
approximately 10% at a value of a/R = 0.5; such large
values of a/R are easily avoided in practice. For the
uncertainty analysis, the approximate solution will be used,
and Eqs. 7 and 8 derive the uncertainty in wear volume for
the spherical cap:
2
2
oÀ
V
oÀ
V
2
2
uðÀÞ ﬃ
V
uð RÞ þ
uð aÞ 2
oR
oa
p 2 a8
p2 a 6
u ð RÞ 2 þ 2 u ð a Þ 2 ;
ð 7Þ
ﬃ
16R4
R
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pa3 2
uðÀÞspherical cap ﬃ 2
V
ð 8Þ
a uðRÞ2 þ 16R2 uðaÞ2 :
4R 126
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133 CT CO
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E 95
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105 Fig. 1 Schematic and nomenclature for a the spherical cap wear
measurement presented in Sect. 2.1, and b cubiodal sample presented
in Sect. 2.2 and Sect. 2.3 OO The traditional form (and units) for the wear rate, K (mm3/
(Nm)), is the volume of material removed, À (mm3), per
V
unit normal load, FN (N), per distance of sliding, d (m). The
combined standard uncertainty, u(K), requires knowledge
of the uncertainties for each of these terms. This article will
present three of the common methods used to compute wear
volumes and will then present methods to compute the
uncertainties in wear rate for single point and multiple point
measurements. This manuscript is in noway exhaustive,
but aims to outline methods for determining the uncertainty
of the wear rate as a function of the uncertainties of the
measured input quantities. As discussed in earlier papers on
the topic by Schmitz et al. [3], information about the largest
contributors to the measurement uncertainty can be readily
used to aid in redesigning the experimental apparatus and/or
the procedure to reduce the measurement uncertainty. ED 68
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83 F ð 1Þ UN Author Proof V
À ¼ K Fn d: (b) (a) PR 65
66 ð 2Þ Since the height of the spherical cap cannot be measured
after wear tests, it is traditionally estimated using
trigonometric identities as shown in Eqs. 3 and 4: Commercially available spheres have excellent radial
tolerances (low uncertainty). It is very difﬁcult to deﬁne an
uncertainty in the measurement of the wear scar diameter/
radius, and very often an average radius is estimated from 123
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140 Tribol Lett 179
180 Assuming that all of the dimensional measurements are
similar and were measured with the same instrument, it is
reasonable
to
assume
that
uðaÞ ¼ uðbÞ ¼ uðLo Þ ¼ uðLf Þ ¼ uðLÞ; this simpliﬁes the
expressions for the uncertainty derived in Eqs. 10–12:
2
2
2
oÀ
V
oÀ
V
oÀ
V
2
2
2
uðÀÞ ﬃ
V
uðLÞ2
u ð LÞ þ
u ð LÞ þ
oa
ob
oLo
2
oÀ
V
uðLÞ2 ;
þ
oL f
uðÀÞ2 ﬃ
V CO
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E 177 ð 9Þ ð10Þ
2 2 2 2 !
oÀ
V
oÀ
V
oÀ
V
oÀ
V
þ
þ
þ
uðLÞ2 ;
oa
ob
oLo
oLf ð11Þ
uðÀÞ2 ﬃ ðbðLo À Lf ÞÞ2 þðaðLo À Lf ÞÞ2 þðabÞ2 þðabÞ2 uðLÞ2 :
V UN 175 uðÀÞDL ﬃ
V pﬃﬃﬃ 2
2 a u ð LÞ : 2.3 Mass Loss ð12Þ Under conditions where a ﬃ b and Lo ﬃ Lf , Eq. 12 can be
further simpliﬁed to give the uncertainty in wear volume V
À¼ mo À mf Dm
:
¼
q
q 189
190 Dm
o= abLo ¼ D m a b Lo
:
mo 123
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213 ð16Þ Again, we will assume that all of the dimensional
measurements are similar, and that uðaÞ ¼ uðbÞ ¼ uðLo Þ
¼ uðLf Þ ¼ uðLÞ. We will also assume that the initial and
ﬁnal masses are similar, andpﬃﬃﬃ associated uncertainties
the
are the same (i.e., uðDmÞ ¼ 2 uðmÞ); this simpliﬁes the
expressions derived in Eqs. 17–19: Journal : Large 11249 191
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205 ð15Þ The density of the sample is computed through mass and
dimensional measurements made in the investigators’
laboratory. For this derivation, the pin sample is assumed
to have a square crosssection (as was assumed in
Sect. 2.2). Equation 15 is readily expanded to give an
expression for volume in terms of measured values as
shown in Eq. 16:
V
À¼ m 184
185
186
187 ð14Þ Mass loss measurements are widely used in polymer
tribology, where creep and plastic deformation of samples under compressive and shear stresses make dimensional measurements of the samples difﬁcult. For
materials that do not change their mass due to outgassing
or uptake of ﬂuids during testing this method can be
particularly useful. Calculations of volume loss and the
relative uncertainties associated with this method are
discussed in detail in Schmitz et al. [3] for a particular
reciprocating tribometer and laboratory instrumentation—
the importance of a high precision laboratory scale was
clearly demonstrated. In the mass loss method, the volume of material removed, À, is calculated from a meaV
surement of the change in mass of the sample,
Dm ¼ mi À mf , and the density of the sample q: CT V
À ¼ abðLo À Lf Þ:
169
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173 In low wear systems, it is quite common for the change
in length of the sample, DL, to be much smaller than the
width of the sample, a, and further simpliﬁcation of Eq. 13
can be performed: F Pinondisk experiments that use a sample with a uniform
crosssection can use a displacement measurement during
operation to calculate wear volumes assuming that all
deformations are due to wear (samples must be resistant to
creep). In low wear systems, this is tricky because of
temperature ﬂuctuations that occur during the course of an
experiment. Additionally, the approach is only ‘‘accurate’’
under conditions where wear is conﬁned to one material
(often the pin). In this derivation, the volume loss and
associated uncertainties will be calculated assuming the pin
is the material that is wearing, and it has a square crosssection with measured widths a ﬃ b and an initial length
Lo; this is shown schematically in Fig. 1b. The volume of
material removed (À) is thus the difference between the
V
initial volume measurement and current volume measurement, in which only the height of the sample is measured
again (i.e., the current volume is computed directly from a
measurement of the new specimen length Lf after assuming
that the lengths a ﬃ b are unchanged). The expression for
volume loss is given by Eq. 9: 181
182 ð13Þ OO 148
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167 pﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
uðÀÞDL ﬃ a 2 DL2 þ a2 uðLÞ:
V PR 2.2 Linear Displacement uðÀÞ in terms of the dimensions a and Lo, and the
V
difference DL ¼ Lo À Lf ; this is given in Eq. 13: ED repeat measurements of different locations across the wear
scar (at the very least the uncertainty should be as large as
the standard deviation of these measurements). The
uncertainty in the measurement of the wear scar radius is
likely the dominant uncertainty contributor in the
calculated wear volume when using this approach. 147 Author Proof 141
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146 215
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220 Tribol Lett uð K Þ ¼
ð17Þ ð23Þ 2.4 Uncertainties in the Single Point Wear Rate
Calculations 232
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236 A single point measurement, for which the sample is measured once before and once after the test, is the most common
method for quantifying wear rate, although it is inappropriate
to refer to this as a steadystate wear rate. The expression for
the single point wear rate K is given by Eq. 20: 254
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269 270
271 It is desirable to report steadystate wear rates, in part
because they are used by engineers to make life prediction,
but also because of the variability in the relatively short
lived initial transients. This is typically done by performing
a regression analysis through a set of data that has the
volume loss plotted on the yaxis and the product of the
normal load and sliding distance is plotted on the xaxis.
The linear region of the data is selected and the slope of the
least squares regression line is the steadystate wear rate. In
order to report an uncertainty in the steadystate wear rate,
it would thus be necessary to perform the uncertainty
analysis on the least squares regression algorithm/expression. A numerical technique is a much simpler alternative,
and can be performed in a variety of ways. An accepted
technique is to perform a MonteCarlo analysis on the data.
Performing a MonteCarlo simulation to compute the
uncertainties in the regression analysis of the steadystate
wear rate requires the generation of a series of data sets
(perhaps 10,000) that have the appropriate mean and
standard deviation in volume loss and the product of normal load and sliding distance. Essentially, each data point
collected during the experiment is numerically perturbed
according to an appropriate random process. One way to do
this is to generate a ﬁeld of random numbers (remember to
use a Gaussian random number generator) with a mean
value of zero and a standard deviation that is equal to the
uncertainty in that point; the measured value is then added 272
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298 K¼ CT ED 229
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231 If the normal force is measured, then the uncertainty in
the normal load is given by the manufacturer calibration of
the force transducer. The sliding distance, d (m), is often
computed by multiplying by the sliding speed V (m/s) and
the duration of an experiment, t (s). For example, in
rotating pinondisk experiments the circumference of the
wear track 2pRdisk ðmÞ and the number of revolutions per
second (RPS) are used together to compute the sliding
speed. The uncertainty in the sliding distance can be
computed based on uncertainties in the test duration, wear
track radius, and angular speed. Our experience has shown
that the uncertainties associated with volume loss measurements dominate; to a good approximation, neglecting the
uncertainty contributions from the normal load and the
sliding distance has no discernable effect on the overall
uncertainty. F The expressions are slightly more compact if a ﬃ b, as
shown by Eq. 19, but considering the ease with which the
uncertainties can be directly computed from the above
expressions, it may not be of any tangible beneﬁt: ð19Þ 250 ﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
V
À
À
Fn2 d 2 uðÀÞ2 þ V 2 d 2 uðFnÞ2 þ V 2 Fn2 uðd Þ2 : 3 MonteCarlo Methods for Computing Wear Rate
Uncertainties in Multiple Point Measurements ð18Þ uðÀÞmass loss
V
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
aLo
Dm2 2
a2
2
2 uðLÞ2 :
2 þ 2 a uðmo Þ þ 2 þ 2 Dm
¼
mo
mo
Lo V
À
:
Fn d ð20Þ Dry sliding contacts have an initial transient period of
wear during which the rates of wear are typically higher
than those found in the steadystate. As a result, wear rate
calculations from single point measurements are often
artiﬁcially high. For this reason, and others, the method is
not generally recommended.
The uncertainty u(K) associated with single point wear
rate calculations is derived in Eqs. 21–23. The uncertainties in volume loss, uðÀÞ, would be taken from the previous
V
sections. The uncertainties in normal load, u(Fn), and
sliding distance, u(d), may also require analysis:
2
2
oK
oK 2
oK
2
2
2
uð K Þ ¼
uðÀÞ þ
V
uðFnÞ þ
uðd Þ2 ;
oÀ
V
oFn
od CO
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248 1
Fn2 d2 OO 224
225
226
227
Dm2 ða b Lo Þ2
2
uðmo Þ2
uðÀÞ ¼ 2 þ 2
V
m2
mo
o
Á2
ÀÀ 2
ÁDm2
2
þ a þ b L o þ a2 b2
u ð LÞ 2 :
m2
o UN Author Proof 222 PR
oÀ 2
V
oÀ 2
V
2
uðÀÞ ¼
V
uð m i Þ 2
uðDmÞ þ
oDm
omo
2 2 2 !
oÀ
V
oÀ
V
oÀ
V
þ
þ
þ
u ð LÞ 2 ;
oa
ob
oLi
2 uð K Þ 2 ¼ 2 1
V
À
uðÀÞ2 þ
V
Fn d
Fn2 d
2
V
À
þ
uð d Þ 2 ;
Fn d2 2 ð21Þ uðFnÞ2
ð22Þ 252 123
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E Fig. 2 Experimental data taken
from Burris and Sawyer [7]: a
experimental sample geometry,
nomenclature, nominal
measurements, and associated
uncertainties, b plot of ﬁve
interrupted measurements, c
graphical representation of the
single point uncertainty
calculation, d simulated data of
the ﬁnal measurement, e steadystate analysis requires a set of
simulated data for each
experimental data point
included in the MonteCarlo
simulation, and f graphical
representation of the MonteCarlo simulation for wear rate
uncertainty analysis (e) (d) (c) (f) 123
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320
321
322
323
324
325
326 ð24Þ Following the analysis presented in Sect. 2.3, the
uncertainty in the volume of material removed, uðÀÞ, is
V
0.056 mm3; this is approximately 3% of the calculated
mass loss (À ¼ 1:84 mm3 ). This calculation is given by
V
Eqs. 25 and 26: ED Example calculations of uncertainty in wear rate are shown
graphically in Fig. 2. Data from Burris and Sawyer [7] are
used. Brieﬂy, the experimental setup was a reciprocating
pinondisk tribometer that had an average normal load of
250 N and a sliding speed of 50 mm/s. The samples were
machined from a bulk nanocomposite of PTFE and alumina into cubiodal samples (shown in Fig. 2a). The
dimensions of the sample and the associated uncertainties
are given in Fig. 2a. The 50 lg (5 9 the scale resolution)
mass measurement uncertainty reported by Burris and UN 307
308
309
310
311
312
313
314
315
316 V
À
Dm a b Lo
¼
Fn d
mo Fn d
3:83 mg 6:3 mm 6:3 mm 10:8 mm
¼
890:79 mg
250 N 3; 048 m
mm3
¼ 2:42 Â 10À6
:
Nm K¼ OO 4 Example Calculations Sawyer [7] is used in this analysis. The uncertainties in the
sliding distance and normal load are exaggerated here by
nearly an orderofmagnitude to make them visible in the
graphical representation of the data shown in Fig. 2b.
Examining Fig. 2b, it is clear that this sample experienced a substantial transient wear process. The data are
comprised ﬁve sequential mass measurements. If the last
measurement were used to compute a single point wear
rate, the wear rate would be K = 2.42 9 106 mm3/(Nm)
as given by Eq. 24: PR to each of these numbers, which results in a ﬁeld of
numbers with the appropriate statistics. Once this ﬁeld of
numbers has been generated, a series of regression analyses
are completed. The mean slope from this regression analysis is the steadystate wear rate, and the standard deviation
of these slopes is the uncertainty in the steadystate wear
rate. 306 Author Proof 299
300
301
302
303
304
305 328
329
330
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332 Tribol Lett ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
s
3:832
6: 32
2þ
6:32 0:0502 þ 2 þ
3:832 0:12 mg2 mm2 ;
890:732
10:82 Following the analysis for the uncertainty in single point
wear rates presented and discussed in Sect. 3.0, the
uncertainty in the wear rate is u(K) = 3 9 108mm3/
(Nm). In this example, the uncertainty in normal load is
uðFnÞ ¼ 2:5 N and the uncertainty in the sliding distance is
uðdÞ ¼ 2N uðsÞ ¼ 30 m, where the number of reciprocating
cycles, N, is known exactly but there is uncertainty in the
reciprocating stroke length u(s) = 250 lm. These
calculations are shown in Eqs. 27 and 28, where Eq. 27
is taken from Eq. 23:
mm3
:
Nm F pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð2502 3; 0482 0:0562 þ 1:842 3; 0482 2:52 þ 1:842 2502 302 ÞN2 m2 mm6
;
uð K Þ ¼
2502 N2 3; 0482 m2 ð27Þ CT This analysis is shown graphically in Fig. 2c, where a
scatter plot of simulated data with the appropriate statistics
is shown in place of the measured value and experimental
uncertainties (Fig. 2d). This, in effect, is the technique that
would be used for a MonteCarlo simulation of a single
point wear rate assuming no uncertainty in the starting
value. The average wear rate would then be the average of
the slopes of the lines from the origin to the simulated data
point and the uncertainty would be the standard deviation
in these slopes; these converge upon the analytical values
from Eqs. 20 and 23 as the number of simulated points
approaches inﬁnity.
Computing the wear rate and the associated uncertainties in the steadystate wear rate proceeds in the same
fashion as previously discussed. Namely, the data points
that represent the steadystate region are selected; for this
example, points 2, 3, 4, and 5 are chosen. Simulated data
with the appropriate statistics are created in an uncorrelated fashion (a random number with the appropriate CO
RR
E 345
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379 ð28Þ ED uðK Þ ¼ 3 Â 10À8 UN Author Proof 334
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343 ð26Þ Gaussian distribution is used) and plotted as shown in
Fig. 2e (for this example, a thousand points were generated but it is not unusual to generate 10,000 or more
points). Regression analyses are performed on the 1,000
data sets that are selected randomly from the simulated
data points. The average wear rate is found to be
K = 3.2 9 107 mm3/(Nm) with an uncertainty of
u(K) = 1 9 107 mm3/(Nm). As previously discussed,
the steadystate wear rate is less than the single point
wear rate, and interestingly has a larger uncertainty primarily due to the dispersions of data in the points 2, 3,
and 4, which so strongly inﬂuence the regression analysis. An approach that would improve this experimental
design would be to collect more points between points 4
and 5, or space the collected data more evenly throughout the experiment. OO 68:04 mm2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðð2Þ0:1 þ ð2:34Þ0:146Þmm2 mg2
890:73 mg
¼ 0:056 mm3 : uðÀÞ ¼
V ð25Þ PR 6:3 mm 10:8 mm
uðÀÞ ¼
V
890:73 mg 5 Concluding Remarks 380 A defensible statement of measurement uncertainty is critical to the analysis of data dispersion; this is especially
important for wear rates, for which ordersofmagnitude
variations are common. Frequently, the uncertainties in low
wear rate materials are on the same order of magnitude as
the measurement; these uncertainties should be reduced
before comparisons and discussions of differences are discussed. The relative ease with which MonteCarlo simulations can be performed permits direct computations of
uncertainties for either singlepoint or steadystate wear
rates without mathematical derivations; this is especially
important for a wide array of systems, which exhibit transient wear behavior. 381
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393 Acknowledgments Financial support for this work was provided
through an AFOSRMURI grant FA95500410367. We would also
like to thank Prof. T. L. Schmitz and Prof. J. C. Ziegert for many
helpful discussions regarding tribometry and uncertainty analysis. 394
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Journal : Large 11249 Dispatch : 172009 Pages : 7 Article No. : 9477 h LE
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h CP h TYPESET
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h DISK MS Code : TRIL987 Tribol Lett 399
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407 1. Schmitz, T.L., Action, J.E., Ziegert, J.C., Sawyer, W.G.: The
difﬁculty of measuring low friction: uncertainty analysis for
friction coefﬁcient measurements. J. Tribol. 127, 673–678 (2005)
2. Burris, D.L., Sawyer, W.G.: Addressing practical challenges of
low friction coefﬁcient measurements. Tribol. Lett. 35, 17–23
(2009)
3. Schmitz, T.L., Action, J.E., Burris, D.L., Ziegert, J.C., Sawyer,
W.G.: Wearrate uncertainty analysis. J. Tribol. 126, 802–808
(2004) 4. Almond, E.A., Gee, M.G.: Results from a UK interlaboratory
project on dry sliding wear. Wear 120, 101–116 (1987)
5. Czichos, H., Becker, S., Lexow, J.: Multilaboratory tribotesting:
results from the Versaillesadvancedmaterialsandstandardsprogram on wear test methods. Wear 114, 109–130 (1987)
6. Guicciardi, S., Melandri, C., Lucchini, F., de Portu, G.: On data
dispersion in pinondisk wear tests. Wear 252, 1001–1006 (2002)
7. Burris, D.L., Sawyer, W.G.: Tribological sensitivity of PTFE/
alumina nanocomposites to a range of traditional surface ﬁnishes.
Tribol. Trans. 48, 147–153 (2005) F References UN CO
RR
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This note was uploaded on 08/22/2011 for the course EGM 4313 taught by Professor Mei during the Spring '08 term at University of Florida.
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