wearUncTL2009 - Tribol Lett DOI 10.1007/s11249-009-9477-8 1...

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Unformatted text preview: Tribol Lett DOI 10.1007/s11249-009-9477-8 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 21 22 23 24 25 26 27 28 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 e-mail: [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 single-point and steady-state 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 steady-state 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 inter-laboratory 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 intra-laboratory repeat experiments varied less than inter-laboratory repeat tests. Almond and Gee chose to ascribe the high inter-laboratory 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 RR E Keywords CT ED 6 7 8 9 10 11 12 13 14 15 16 17 18 UN Author Proof F 2 123 Journal : Large 11249 Dispatch : 1-7-2009 Pages : 7 Article No. : 9477 h LE 4 h CP h TYPESET 4 h DISK MS Code : TRIL987 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Tribol Lett report a wear rate that follows Archard’s wear equation (Eq. 1): 84 2 Uncertainties in Wear Volume Measurements 85 86 87 88 89 90 91 92 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, pin-on-disk 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 order-of-magnitude 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: 113 114 115 116 117 118 119 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 order-of-magnitude 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 126 127 128 129 130 131 132 133 CT CO RR E 95 96 97 98 99 100 101 102 103 104 105 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 no-way 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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 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 Journal : Large 11249 Dispatch : 1-7-2009 Pages : 7 Article No. : 9477 h LE 4 h CP h TYPESET 4 h DISK MS Code : TRIL987 135 137 138 139 140 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 RR 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 Dispatch : 1-7-2009 Pages : 7 Article No. : 9477 h LE 4 h CP h TYPESET 4 h DISK MS Code : TRIL987 207 208 209 210 211 212 213 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 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 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 cross-section (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: 169 170 171 172 173 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 Pin-on-disk experiments that use a sample with a uniform cross-section 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 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167  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 141 142 143 144 145 146 215 216 217 218 219 220 Tribol Lett uð K Þ ¼ ð17Þ ð23Þ 2.4 Uncertainties in the Single Point Wear Rate Calculations 232 233 234 235 236 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 steady-state wear rate. The expression for the single point wear rate K is given by Eq. 20: 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 It is desirable to report steady-state 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 y-axis and the product of the normal load and sliding distance is plotted on the x-axis. The linear region of the data is selected and the slope of the least squares regression line is the steady-state wear rate. In order to report an uncertainty in the steady-state 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 Monte-Carlo analysis on the data. Performing a Monte-Carlo simulation to compute the uncertainties in the regression analysis of the steady-state 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 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 K¼ CT ED 229 230 231 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 pin-on-disk 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 Monte-Carlo 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 steady-state. 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 RR E 238 239 240 241 242 243 244 245 246 247 248 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 Journal : Large 11249 Dispatch : 1-7-2009 Pages : 7 Article No. : 9477 h LE 4 h CP h TYPESET 4 h DISK MS Code : TRIL987 Tribol Lett (b) F CT (a) CO RR 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 Monte-Carlo simulation, and f graphical representation of the MonteCarlo simulation for wear rate uncertainty analysis (e) (d) (c) (f) 123 Journal : Large 11249 Dispatch : 1-7-2009 Pages : 7 Article No. : 9477 h LE 4 h CP h TYPESET 4 h DISK MS Code : TRIL987 317 318 319 320 321 322 323 324 325 326 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 pin-on-disk 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  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 order-of-magnitude 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 10-6 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 steady-state wear rate, and the standard deviation of these slopes is the uncertainty in the steady-state wear rate. 306 Author Proof 299 300 301 302 303 304 305 328 329 330 331 332 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 10-8mm3/ (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 Monte-Carlo 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 steady-state wear rate proceeds in the same fashion as previously discussed. Namely, the data points that represent the steady-state 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 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 ð28Þ ED uðK Þ ¼ 3  10À8 UN Author Proof 334 335 336 337 338 339 340 341 342 343 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 10-7 mm3/(Nm) with an uncertainty of u(K) = 1 9 10-7 mm3/(Nm). As previously discussed, the steady-state 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 orders-of-magnitude 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 Monte-Carlo simulations can be performed permits direct computations of uncertainties for either single-point or steady-state wear rates without mathematical derivations; this is especially important for a wide array of systems, which exhibit transient wear behavior. 381 382 383 384 385 386 387 388 389 390 391 392 393 Acknowledgments Financial support for this work was provided through an AFOSR-MURI grant FA9550-04-1-0367. We would also like to thank Prof. T. L. Schmitz and Prof. J. C. Ziegert for many helpful discussions regarding tribometry and uncertainty analysis. 394 395 396 397 123 Journal : Large 11249 Dispatch : 1-7-2009 Pages : 7 Article No. : 9477 h LE 4 h CP h TYPESET 4 h DISK MS Code : TRIL987 Tribol Lett 399 400 401 402 403 404 405 406 407 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.: Wear-rate 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 Versailles-advanced-materials-and-standards-program on wear test methods. Wear 114, 109–130 (1987) 6. Guicciardi, S., Melandri, C., Lucchini, F., de Portu, G.: On data dispersion in pin-on-disk 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 E CT ED PR Author Proof OO 398 123 Journal : Large 11249 Dispatch : 1-7-2009 Pages : 7 Article No. : 9477 h LE 4 h CP h TYPESET 4 h DISK MS Code : TRIL987 408 409 410 411 412 413 414 415 416 417 418 ...
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