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Life analysis of coated tools using statistical methods
Surface and Coatings Technology 116–119 (1999) 654–661 www.elsevier.nl/locate/surfcoat
Life analysis of coated tools using statistical methods S.J. Dowey a, *, J. Zhang b, E.D. Doyle b, A. Matthews c a Ion Coat Ltd, PO Box 53, Hull HU6 7TW, UK b Swinburne University of Technology, John Street, Hawthorn, Victoria 3122, Australia c Research Centre in Surface Engineering, University of Hull, Hull HU6 7RX, UK
Abstract The lifetime improvements achieved using physical vapour-deposited (PVD) TiN on hexagonal washer punch tools used in the fastener industry have been investigated. Tools were coated with PVD TiN initially in an unmodified form and later with surface preparation changes prior to coating. An original analysis, relying mainly on the visual appraisal of scanning electron micrographs and a rudimentary study of the lifetime data, found that the failure modes of the tools were altered by modifying the tool manufacturing process as well as by applying a surface coating. Sample mean lifetime of the best tools was over six times that of the mean sample lifetime of the unmodified, uncoated tools. By retrospectively applying experimental design techniques to the data, the influence on the lifetime from the tool modifications and the coating application have been decoupled. This shows that the tool modifications produce a larger lifetime response than the presence or not of a PVD coating. However, the interaction of the PVD coating and the tool modification produces a stronger positive response. Simple visualisation techniques have been used to effectively present this information in the form of ANOM (analysis of means), response charts, box plots and scatter diagrams. A three-parameter Weibull analysis also revealed evidence to suggest that, as well as different failure modes between treatments, some treatments underwent a number of early life failures by different failure modes. This is revealed purely from the distribution of the data and as such greatly enhances the visual analysis and interpretation of the previous analysis. The characteristic life for the improved tooling in this particular case is 465 200 operations as opposed to 72 000 for the original tooling. This study highlights the benefits of a full statistical analysis in assessing the benefits of coatings on production tooling, especially where the use of a coating is coupled with other changes in tool design and preparation. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Experimental design; Forming tools; TiN; Weibull
1. Introduction Analysis of tooling life in industrial applications is often characterised by extensive testing, detailed metallographic or microscopic inspection of the failures, and a limited, elementary treatment of the quantifiable data (i.e., the numerical results generated by the testing programme). In order to assess the true value of coatings, industry requires techniques that give quantifiable results and an estimate of the certainty with which they are given. If statistical methods are used, it is often found that parametric limitations of the statistical theory are encountered, as typically a surface treatment not only * Corresponding author. Tel.: +44-1468-581549; fax: +44-1482-809901. E-mail address: [email protected] (S.J. Dowey)
alters the mean lifetime of a population distribution but also the variance and even the shape of the distribution. In addition to the lack of data, another pressing issue of concern to engineers is the availability of too much data. Often, multiple measurements are made on a component whilst under test, as testing can be costly and time-consuming. However, difficulties arise when the results are to be interpreted and particularly when they need to be visualised. We have investigated univariate and multivariate statistical techniques for a number of data sets from real-life industrial tests in ‘problem’ areas. These techniques include experimental design, Weibull analysis and principal component analysis. These have been shown to overcome the problems highlighted and give meaningful and quantifiable information, which then permits performance improvements to be properly measured. Therefore coatings and tool designs can be optimised.
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S.J. Dowey et al. / Surface and Coatings Technology 116–119 (1999) 654–661 Table 1 Factors and levels for the hexagonal punch experimental design Factor
Fig. 1. Schematic drawing of the hexagonal punch, showing the important working surfaces.
This paper will concentrate on the univariate methods and is illustrated by reference to results from performance tests on a coated cold-forming tool. Within the fastener industry, the hexagonal washer punch (hex punch) is a high-usage tool. The one studied in the present work comprises a hexagonal indentor with a short shoulder and two letters, ‘X’ and ‘B’, marked on its top surface. Fig. 1 is a schematic drawing showing the important working surfaces of the hex punch. In this paper, the hex punch in the as-received condition is defined as the normal hex punch, and the hex punch made by modifying the lettering procedure is defined as the modified hex punch. The aims of the initial industrial tests [1] were to see whether an improvement in punch performance could be gained by a physical vapour-deposited (PVD) TiN coating, and to identify the mechanisms of surface deterioration of both uncoated and PVD TiN-coated hex punches. Although the initial study was successful it did not fully quantify the benefits of PVD TiN coatings on the tools and although t-tests were applied to the data sets, this only demonstrated that the populations of the tools were statistically significantly different. This work reappraises the initial study in the light of recent work in the same application area [2,3] by applying experimental design techniques to the original data and analysing the failure data more thoroughly using Weibull analysis [4].
2. Experimental The original work conveniently forms an almost full factorial experimental design [5]. The factors are described in Table 1 and the experimental treatments in Table 2. The final polished tooling was not tested in an uncoated state, leaving treatment 5 outside a two-level by two-factor experimental design. Using robust design terminology [6 ] this is in the form of a Taguchi L 4 orthogonal array; however, Table 2 does not show the interaction column between the two factors. The combination of factors to produce a particular punch will be referred to as a ‘treatment’ in this work. This terminol-
Level 1
2
3
Finish (A)
As received
2+side polished
Coating (B)
None — as received
Modified lettering procedure PVD TiN
Table 2 Experimental matrix for the hexagonal punch experiment. The treatment marked as 51 was not part of the original work and was not made or tested Treatment
Finish
Coating
Level of factor 1 2 3 4 51 5
1 1 2 2 3 3
1 2 1 2 1 2
ogy is clear when it is realised that the originator of this experimental technique applied it to agricultural studies of combinations of fertilisers, applied as treatments, to fields [7]. Surfaces of the as-received hexagonal punch (finish factor A, level 1) were microblasted with a fine aluminium oxide grit following hardening and tempering. The top surfaces of the punches were polished with diamond paste. The working side surfaces were left in the microblasted condition with measured surface finish values of 2 mm R . The marking letters ‘X, B’ on the top surface a were stamped using a 25 ton hydraulic press. The modified punches (A2) were nominally the same as A1 except the lettering process was modified, which included lowering the stamping pressure that formed the lettering on the punch. The final modification to the finish (A3) was where the working sides of the punch were polished with 1200 grit paper (prior to TiN coating). PVD TiN coating (factor B, level 2) was sourced commercially by the Swinburne University of Technology.
3. Results Data from the original tool trials are shown in Table 3 and a scatter diagram of tool lives (number of components produced per tool ) by treatment was prepared from it (Fig. 2). The same data as in Fig. 2 are presented as a box plot ( Fig. 3). The lower and upper bars represent the range of the results by treatment. The
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Table 3 Data from the hexagonal punch experiment Treatment
Tool life (number of components produced ×103)
1 2 3 4 5
34 61 68 79 205
60 70 88 164 402
65 70 91 176 406
85 85 120 200 466
96 91.5 137 211 576
139 168 224 586
195 225
274
360
Mean
95% CI
Standard error
63.75 80.5 114.5 197 412
16.99 18.45 29.87 46.05 94.21
7.19 8.00 13.20 20.92 40.85
Fig. 2. Scatter diagram of tool lives (number of components produced per tool ) by treatment. Data are from Table 3. Treatments are identified in Table 2.
Fig. 3. Box plot of tool lives (number of components produced per tool ) by treatment. The same data as in Fig. 2 are presented as a box plot. The lower and upper bars represent the range of the results by treatment. The inner box is bounded by the upper and lower quartiles (25th and 75th percentiles). The inner star marks the median value of the treatment.
inner box is bounded by the upper and lower quartiles (25th and 75th percentiles). The inner star marks the median value of the treatment. An analysis of means (ANOM ) for factors A and B, with the response of the mean for each factor individually plotted, and as a
factor combination is shown in Fig. 4. The interaction of A and B at the levels A2 and B2 shows the maximum mean response. The standard error (standard deviation of the sample mean) has been used for the error bars. Fig. 5 shows the results of the treatments based on the
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657
Fig. 4. Analysis of means (ANOM ) for the factors A and B showing the response of the mean for each factor as well as a factor combination. The interaction of A and B at the levels A2 and B2 shows the maximum mean response. The standard error (the standard deviation of the sample mean) has been used for the error bars.
estimates of the population means from the sample means of the treatments, 95% confidence intervals (CIs) are shown. This shows that the population mean (rather than the sample mean) has been significantly shifted by each successive treatment (except from 1 to 2). This is a statistically significant shift as there is no overlap in the intervals between treatments (except from 1 to 2). Therefore the shifts are real and could not have occurred by chance alone (accepting a 1-in-20 risk of error). Standard error and the 95% CI are shown in Table 3. When comparing Figs. 2, 3 and 5 it is important to realise what each is showing. Fig. 2 shows all the experimental data. Fig. 3 summarises the data but preserves the range (the measure of dispersion), as well as indicating the measure of central tendency (the median in this case) and the shape of the distribution. Figs. 2 and 3
are essentially different representations of the same data, both showing the data from the samples taken from the respective parent populations of treatments. It is important to realise that samples and the sample statistics derived from them are variable. It is the process of inference that takes the variable samples and determines, with some level of significance, the constant population parameters. It is this information, for the estimate of the population means for the respective treatments, that is shown in Fig. 5. So, although Figs 3 and 5 appear similar, they show different information. Fig. 3 clearly summarises the sample data and Fig. 5 shows the best estimate of the position of the population mean for each treatment based on the sample mean (shown as the central point), the size of the sample and the sample variance. The confidence interval of the treatment pop-
Fig. 5. Location of population mean for each treatment based on the sample estimate with 95% confidence. This shows that the population mean (rather than the sample mean) has been significantly shifted by each successive treatment (except from 1 to 2). Therefore the shifts are real and could not have occurred by chance alone (accepting a 1-in-20 risk of error).
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ulation mean, CI( m), based on the treatment sample mean, x: , is calculated from:
C
CI( m): x: ±t(df, a)
s 앀n
D
,
(1)
where t, the inverse of the two-sided Student’s t distribution, is a function of the degrees of freedom of the sample mean (equal to n−1) and a, the level of significance required in the result (for 95% confidence interval a=5%), s is the sample variance and n is the sample size. As the sample size approaches infinity the value of t approaches 1.96 for a two-sided 95% confidence interval, the number of standard deviations required to encompass 95% of a normally distributed population. The last term, s/앀n, is often referred to as the standard error of the mean. It is the standard deviation of the sample mean, scaled from the original distribution’s standard deviation by 1/앀n. A three-parameter Weibull analysis [8] was carried out by treatment and the results are plotted in Fig. 6. Weibull probability paper can also be used but Fig. 6 has been plotted using a computer spreadsheet. Weibull probability paper or a more configurable graphing package would have displayed cumulative per cent failure explicitly as the abscissa and the age of failure explicitly as the ordinate. Although a three-parameter analysis (t , b, g) was carried out, t (the locating constant) is 0 0 zero in all the plots. The Weibull distribution, expressed as failure, F, as a function of time, t, is given by:
C A BD
F(t)=1−exp −
t−t b 0 . g
(2)
It can represent many shapes of population, depending on the shape factor, b; for example, b=3.44 is close to a normal distribution. The scaling constant, g, determines the spread of the distribution. Weibull analysis has been used as a graphical method of determining the population distribution (provided the data fit) as well as determining the mean life. The original t lifetime data are used to calculate t , where practical, which represents 0 a life before any failures are likely to occur. It is referred to as a life of intrinsic reliability. Depending on the shape of the distribution it may also be possible to estimate what type of failure a component underwent [8]. For example, with t >0 and b<1, failure is consis0 tent with fatigue — where there are no failures initially and the failure rate suddenly increases and then decreases with age — rather than wear-out — where the failure rate increases with time (consistent with normally distributed populations). With the axes arranged as in Fig. 6, the form of the Weibull equation when the natural logarithm of Eq. (2) is taken twice and the equation rearranged, a straight-line plot fits the twoparameter Weibull distribution. Therefore the linear regression equation has also been given, with R2, the
coefficient of determination, used to indicate the goodness of fit. The equation is in the form y=mx+c where the slope, m, is equal to b and the intercept, c, a function of g. It is clear from Fig. 6(b) that either the data do not have a Weibull distribution or a non-zero value of t is required; initially assuming the latter, t has been 0 0 calculated to be close to 59.6×103 by an iterative curvefitting technique. This is equivalent to an intrinsic life of reliability of about 60 000 operations, which means no failures of a tool in this failure mode in this population are expected before 60 000 components have been produced. Compare this with no clear intrinsic reliability for most of the other treatments.
4. Discussion The findings and sequential improvements from the initial study were based on the observations of failures in the tools of the previous treatment. The uncoated hex punch failed by chipping associated with the stamped letter markings on the punches and a subsequent PVD TiN coating did very little to improve the tool life. The lettering process was then modified. Modification of the tool prior to PVD TiN coating brought about a significant improvement in tool life both with and without a PVD coating, and the failure appeared then to be away from the lettering on the working sides of the punches. Examination by scanning electron microscopy (SEM ) showed that the edges of the punches were being undermined by cracks running at 30° to the sides of the punch [1]. This then led to the final modifications to the uncoated punch (referred to as level 3 of the ‘finish’ factor). The failure mode of these tools was found to change to chipping on the shoulders of the punch. Fig. 7 shows typical failures of TiN-coated and uncoated punch tools [2,3]. Fig. 7(a) shows a coated tool with a machined surface where the coating has worn as the tool has been used. Figs. 7(b) and (c) show polished tools that show little wear where the coating remains but wear consistent with wear on uncoated tools [Fig. 7(d)] where the coating has failed. For the polished tools, coating failure does not appear to be a gradual, ‘wear-out’ process. Interestingly, although Student’s t-test had been applied to the treatments and used in hypothesis testing to show that there was a statistical difference between the (constant) population means of most of the treatments (except between 1 and 2), the confidence intervals for each treatment were not calculated. However, referring to Fig. 5 and Table 3, a CI does give a quantitative means of describing the improvements and the location of the treatments. Confidence intervals are based on the product of the standard error of the sample mean and the t-value, where t is a function of the degrees of freedom of the sample and the level of significance
S.J. Dowey et al. / Surface and Coatings Technology 116–119 (1999) 654–661
(a)
(b)
(c)
(d)
659
(e)
Fig. 6. Weibull analysis by treatment. (a) Treatment 1 — normal uncoated hexagonal punch; (b) treatment 2 — PVD TiN-coated normal hexagonal punch; (c) treatment 3 — uncoated modified hexagonal punch; (d) treatment 4 — PVD TiN-coated modified hexagonal punch; (e) treatment 5 — PVD TiN-coated polished modified haxagonal punch. The equation for a ‘best-fit’ straight line is also given.
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(a)
(b)
(c)
(d)
Fig. 7. Typical failures of TiN-coated and uncoated punch tools. (a) A coated tool with a machined surface, in this case the coating has worn as the tool has been used (equivalent to treatment 2). (b) and (c) Polished tools that show little or no wear where the coating remains (equivalent to treatment 5) but normal wear where the coating has failed. This is comparable to the wear on uncoated tools [(d ), equivalent to treatment 3].
applied with the test (1). The CI can be reduced in two ways, either by reducing the variance of the data or by increasing the number of data points (reducing the level of significance defeats the object of applying the test to a certain degree). The t-test is applicable when the data populations are assumed to be normal. However, when applying a two-sample t-test, as a hypothesis test for example, the populations are assumed to have the same variances. This can be tested using the F-test; applying this shows, for treatments 4 and 5, that they have significantly different variances from the lower number tests and that any test for homogeneity of variance will fail between non-adjacent treatments (except 1 to 3). However, it is clear from the confidence intervals that hypothesis testing is unnecessary and that differences in the population means (rather than the sample means) relate to significant shifts by each successive treatment (except from 1 to 2). Therefore the changes in population mean are real and could not have occurred by chance alone (accepting a 1-in-20 risk of error). Although the use of CIs gives the required quantitative measurement for improvements, it does not let the data tell their full story. When the variance visible in the test sample results is seen to be so large, it could leave doubts as to the quality benefits of this combina-
tion of surface preparation and coating [3,6 ]. This is, in some cases, just a perception as demonstrated in [3], but it is also dependent on the quality measure that the data are compared against. The results initially do not appear to support some of the conclusions in [2]; namely, that the failure mode of polished and TiNcoated tools is a fatigue-like process (although clearly they are not just wearing out, Fig. 7). With this type of failure and with rough loading it is expected that most of the tools will fail relatively early, with the remainder, possibly only slightly stronger, lasting a very long time, giving a characteristic, highly skewed distribution [8]. This shape of distribution is only clear in treatment 2, the TiN-coated normal hexagonal punches. Treatments 4 and 5, though both coated, appear normally distributed and the initial values of the shaping parameters from the Weibull analysis support this. This analysis is therefore not as cut-and-dried as that in [2]. However, Weibull analysis ( Fig. 6) again gives some good indications to the answers to these problems; i.e., the unexpected shapes of the distributions (regarding the coated tools) and the large variance in the sample data. As stated in the Results, a non-zero value of t 0 has been calculated for treatment 2 as being close to 59.6×103. This is equivalent to an intrinsic life of
S.J. Dowey et al. / Surface and Coatings Technology 116–119 (1999) 654–661
reliability of about 60 000; i.e., no failures of a tool in this failure mode in this population before 60 000 components have been produced. When this is applied the shape factor changes from nearly normal ( b=3.3) to highly skewed (as seen in Fig. 2; b=0.83), a typical high-cycle ‘fatigue-like’ failure [8]. Interestingly this makes treatment 2, although appearing to last no longer than the uncoated tools, more reliable than the original, uncoated tools of treatment 1. This could be argued to be purely due to the change in the failure mode from the application of the surface coating. To use a value for the locating constant (t ) it is necessary to recognise 0 curvature in the initial data plot; curvature of the same sense as in Fig. 6(b) will be referred to as positive, giving a positive value of t . Negative curvature, on the 0 other hand, does not necessarily mean that the tools were wearing out before they came into service. The presence of this, and a discontinuity or ‘knee’ as in Fig. 6(c), does however strongly indicate that an existing defect was present and that these first four tools failed by another failure mode [9]. By inspecting the tools with this possibility in mind, a more thorough metallurgical analysis could be carried out. Once these anomalies are seen in Fig. 6(c), then — although more subtle — there are similar features observable in the Weibull plots for the remaining treatments. These are enough to prompt further investigation into the failures modes of the actual tools by further, more detailed inspection. This discussion could raise the question of the benefits of industrial testing, as the difficulties of demonstrating small but cost-effective savings may, in some minds, be too great and lead to false conclusions. It is our contention that industrial or ‘real-life’ testing is precisely where experimentation should take place and that mature statistical tools are available to aid the experimenter to find real improvements through the inherent ‘noise’ in the results drawn from an industrial environment and help prevent false conclusions.
5. Conclusions The use of confidence intervals to estimate the location of the population mean for various treatments (tabulated in Table 2) has been carried out. This shows that the population mean of treatment 5 lies within the range 317 800 to 506 200 components as compared with the uncoated normal tools in treatment 1, which have a population mean within the range 46 750 to 80 750. In the worst case, this makes on average nearly 240 000 more components per tool. Alternatively, the Weibull
661
characteristic life can be used to quantify the benefits and this has been found to be 465 200 operations for the improved tooling as opposed to 72 000 for the original tooling using the original data (with no data ‘censoring’). The key practical outcome for the analysis has been to quantify the benefits of the process changes; this is very important for any company as it ensures that a cost for that performance increase can be calculated. In general in industry, if a change cannot be costed then it will not be introduced. Simple statistical analysis has again been shown to have positive benefits in an area of engineering normally associated with complex material analysis techniques, as it enables use of the entire numerical data sets to make valid inferences. Weibull analysis gives the experimenter the ability to ‘look inside’ these sets and see if they are giving the whole picture. Recognising that different failure modes could exist within a treatment allows the real benefits of that treatment to be seen without unexpected failures clouding the picture. This has immediately increased the value of all the repeated testing. Intuitively we know to test more than one product to make a valid statement on any performance benefits, a statistical approach provides the tools to plan and carry out experiments and lets the experimenter work with the ‘noise’ of uncertainty. Recognising where some of this uncertainty comes from, for example by using Weibull analysis, is of great benefit for the practical experimenter in the application of surface engineered products, and we believe that such statistical methods should be used much more widely than is presently the case.
References [1] J. Zhang, Ph.D. thesis, Swinbourne University of Technology, 1996. [2] S.J. Dowey, B. Ra¨hle, A. Matthews, Surf. Coat. Technol. 99 (1998) 213–221. [3] S.J. Dowey, A. Matthews, Taguchi and TQM: quality issues for surface engineered applications, Surf. Coat. Technol. 110 (1998) 86–93. [4] W. Weibull, J. Appl. Mech. 18 (1951) 293. [5] N. Logothetis, Managing for Total Quality, From Deming to Taguchi and SPC, 1st ed., Prentice-Hall International, UK, 1992. [6 ] M.S. Phadke, Quality Engineering using Robust Design, 1st ed., Prentice-Hall International Editions, London, 1989. [7] R.A. Fisher, The Design of Experiments, 8th ed., Hafner Publishing Company, Inc, 1971. reprinted. [8] A.D.S. Carter, Mechanical Reliability, 2nd ed., MacMillan, London, 1986. [9] A.D.S. Carter, Mechanical Reliability and Design, 1st ed., MacMillan, London, 1997.
Life analysis of coated tools using statistical methods S.J. Dowey a, *, J. Zhang b, E.D. Doyle b, A. Matthews c a Ion Coat Ltd, PO Box 53, Hull HU6 7TW, UK b Swinburne University of Technology, John Street, Hawthorn, Victoria 3122, Australia c Research Centre in Surface Engineering, University of Hull, Hull HU6 7RX, UK
Abstract The lifetime improvements achieved using physical vapour-deposited (PVD) TiN on hexagonal washer punch tools used in the fastener industry have been investigated. Tools were coated with PVD TiN initially in an unmodified form and later with surface preparation changes prior to coating. An original analysis, relying mainly on the visual appraisal of scanning electron micrographs and a rudimentary study of the lifetime data, found that the failure modes of the tools were altered by modifying the tool manufacturing process as well as by applying a surface coating. Sample mean lifetime of the best tools was over six times that of the mean sample lifetime of the unmodified, uncoated tools. By retrospectively applying experimental design techniques to the data, the influence on the lifetime from the tool modifications and the coating application have been decoupled. This shows that the tool modifications produce a larger lifetime response than the presence or not of a PVD coating. However, the interaction of the PVD coating and the tool modification produces a stronger positive response. Simple visualisation techniques have been used to effectively present this information in the form of ANOM (analysis of means), response charts, box plots and scatter diagrams. A three-parameter Weibull analysis also revealed evidence to suggest that, as well as different failure modes between treatments, some treatments underwent a number of early life failures by different failure modes. This is revealed purely from the distribution of the data and as such greatly enhances the visual analysis and interpretation of the previous analysis. The characteristic life for the improved tooling in this particular case is 465 200 operations as opposed to 72 000 for the original tooling. This study highlights the benefits of a full statistical analysis in assessing the benefits of coatings on production tooling, especially where the use of a coating is coupled with other changes in tool design and preparation. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Experimental design; Forming tools; TiN; Weibull
1. Introduction Analysis of tooling life in industrial applications is often characterised by extensive testing, detailed metallographic or microscopic inspection of the failures, and a limited, elementary treatment of the quantifiable data (i.e., the numerical results generated by the testing programme). In order to assess the true value of coatings, industry requires techniques that give quantifiable results and an estimate of the certainty with which they are given. If statistical methods are used, it is often found that parametric limitations of the statistical theory are encountered, as typically a surface treatment not only * Corresponding author. Tel.: +44-1468-581549; fax: +44-1482-809901. E-mail address: [email protected] (S.J. Dowey)
alters the mean lifetime of a population distribution but also the variance and even the shape of the distribution. In addition to the lack of data, another pressing issue of concern to engineers is the availability of too much data. Often, multiple measurements are made on a component whilst under test, as testing can be costly and time-consuming. However, difficulties arise when the results are to be interpreted and particularly when they need to be visualised. We have investigated univariate and multivariate statistical techniques for a number of data sets from real-life industrial tests in ‘problem’ areas. These techniques include experimental design, Weibull analysis and principal component analysis. These have been shown to overcome the problems highlighted and give meaningful and quantifiable information, which then permits performance improvements to be properly measured. Therefore coatings and tool designs can be optimised.
0257-8972/99/$ – see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S0 2 5 7- 8 9 7 2 ( 9 9 ) 0 0 07 4 - 2
655
S.J. Dowey et al. / Surface and Coatings Technology 116–119 (1999) 654–661 Table 1 Factors and levels for the hexagonal punch experimental design Factor
Fig. 1. Schematic drawing of the hexagonal punch, showing the important working surfaces.
This paper will concentrate on the univariate methods and is illustrated by reference to results from performance tests on a coated cold-forming tool. Within the fastener industry, the hexagonal washer punch (hex punch) is a high-usage tool. The one studied in the present work comprises a hexagonal indentor with a short shoulder and two letters, ‘X’ and ‘B’, marked on its top surface. Fig. 1 is a schematic drawing showing the important working surfaces of the hex punch. In this paper, the hex punch in the as-received condition is defined as the normal hex punch, and the hex punch made by modifying the lettering procedure is defined as the modified hex punch. The aims of the initial industrial tests [1] were to see whether an improvement in punch performance could be gained by a physical vapour-deposited (PVD) TiN coating, and to identify the mechanisms of surface deterioration of both uncoated and PVD TiN-coated hex punches. Although the initial study was successful it did not fully quantify the benefits of PVD TiN coatings on the tools and although t-tests were applied to the data sets, this only demonstrated that the populations of the tools were statistically significantly different. This work reappraises the initial study in the light of recent work in the same application area [2,3] by applying experimental design techniques to the original data and analysing the failure data more thoroughly using Weibull analysis [4].
2. Experimental The original work conveniently forms an almost full factorial experimental design [5]. The factors are described in Table 1 and the experimental treatments in Table 2. The final polished tooling was not tested in an uncoated state, leaving treatment 5 outside a two-level by two-factor experimental design. Using robust design terminology [6 ] this is in the form of a Taguchi L 4 orthogonal array; however, Table 2 does not show the interaction column between the two factors. The combination of factors to produce a particular punch will be referred to as a ‘treatment’ in this work. This terminol-
Level 1
2
3
Finish (A)
As received
2+side polished
Coating (B)
None — as received
Modified lettering procedure PVD TiN
Table 2 Experimental matrix for the hexagonal punch experiment. The treatment marked as 51 was not part of the original work and was not made or tested Treatment
Finish
Coating
Level of factor 1 2 3 4 51 5
1 1 2 2 3 3
1 2 1 2 1 2
ogy is clear when it is realised that the originator of this experimental technique applied it to agricultural studies of combinations of fertilisers, applied as treatments, to fields [7]. Surfaces of the as-received hexagonal punch (finish factor A, level 1) were microblasted with a fine aluminium oxide grit following hardening and tempering. The top surfaces of the punches were polished with diamond paste. The working side surfaces were left in the microblasted condition with measured surface finish values of 2 mm R . The marking letters ‘X, B’ on the top surface a were stamped using a 25 ton hydraulic press. The modified punches (A2) were nominally the same as A1 except the lettering process was modified, which included lowering the stamping pressure that formed the lettering on the punch. The final modification to the finish (A3) was where the working sides of the punch were polished with 1200 grit paper (prior to TiN coating). PVD TiN coating (factor B, level 2) was sourced commercially by the Swinburne University of Technology.
3. Results Data from the original tool trials are shown in Table 3 and a scatter diagram of tool lives (number of components produced per tool ) by treatment was prepared from it (Fig. 2). The same data as in Fig. 2 are presented as a box plot ( Fig. 3). The lower and upper bars represent the range of the results by treatment. The
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Table 3 Data from the hexagonal punch experiment Treatment
Tool life (number of components produced ×103)
1 2 3 4 5
34 61 68 79 205
60 70 88 164 402
65 70 91 176 406
85 85 120 200 466
96 91.5 137 211 576
139 168 224 586
195 225
274
360
Mean
95% CI
Standard error
63.75 80.5 114.5 197 412
16.99 18.45 29.87 46.05 94.21
7.19 8.00 13.20 20.92 40.85
Fig. 2. Scatter diagram of tool lives (number of components produced per tool ) by treatment. Data are from Table 3. Treatments are identified in Table 2.
Fig. 3. Box plot of tool lives (number of components produced per tool ) by treatment. The same data as in Fig. 2 are presented as a box plot. The lower and upper bars represent the range of the results by treatment. The inner box is bounded by the upper and lower quartiles (25th and 75th percentiles). The inner star marks the median value of the treatment.
inner box is bounded by the upper and lower quartiles (25th and 75th percentiles). The inner star marks the median value of the treatment. An analysis of means (ANOM ) for factors A and B, with the response of the mean for each factor individually plotted, and as a
factor combination is shown in Fig. 4. The interaction of A and B at the levels A2 and B2 shows the maximum mean response. The standard error (standard deviation of the sample mean) has been used for the error bars. Fig. 5 shows the results of the treatments based on the
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Fig. 4. Analysis of means (ANOM ) for the factors A and B showing the response of the mean for each factor as well as a factor combination. The interaction of A and B at the levels A2 and B2 shows the maximum mean response. The standard error (the standard deviation of the sample mean) has been used for the error bars.
estimates of the population means from the sample means of the treatments, 95% confidence intervals (CIs) are shown. This shows that the population mean (rather than the sample mean) has been significantly shifted by each successive treatment (except from 1 to 2). This is a statistically significant shift as there is no overlap in the intervals between treatments (except from 1 to 2). Therefore the shifts are real and could not have occurred by chance alone (accepting a 1-in-20 risk of error). Standard error and the 95% CI are shown in Table 3. When comparing Figs. 2, 3 and 5 it is important to realise what each is showing. Fig. 2 shows all the experimental data. Fig. 3 summarises the data but preserves the range (the measure of dispersion), as well as indicating the measure of central tendency (the median in this case) and the shape of the distribution. Figs. 2 and 3
are essentially different representations of the same data, both showing the data from the samples taken from the respective parent populations of treatments. It is important to realise that samples and the sample statistics derived from them are variable. It is the process of inference that takes the variable samples and determines, with some level of significance, the constant population parameters. It is this information, for the estimate of the population means for the respective treatments, that is shown in Fig. 5. So, although Figs 3 and 5 appear similar, they show different information. Fig. 3 clearly summarises the sample data and Fig. 5 shows the best estimate of the position of the population mean for each treatment based on the sample mean (shown as the central point), the size of the sample and the sample variance. The confidence interval of the treatment pop-
Fig. 5. Location of population mean for each treatment based on the sample estimate with 95% confidence. This shows that the population mean (rather than the sample mean) has been significantly shifted by each successive treatment (except from 1 to 2). Therefore the shifts are real and could not have occurred by chance alone (accepting a 1-in-20 risk of error).
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ulation mean, CI( m), based on the treatment sample mean, x: , is calculated from:
C
CI( m): x: ±t(df, a)
s 앀n
D
,
(1)
where t, the inverse of the two-sided Student’s t distribution, is a function of the degrees of freedom of the sample mean (equal to n−1) and a, the level of significance required in the result (for 95% confidence interval a=5%), s is the sample variance and n is the sample size. As the sample size approaches infinity the value of t approaches 1.96 for a two-sided 95% confidence interval, the number of standard deviations required to encompass 95% of a normally distributed population. The last term, s/앀n, is often referred to as the standard error of the mean. It is the standard deviation of the sample mean, scaled from the original distribution’s standard deviation by 1/앀n. A three-parameter Weibull analysis [8] was carried out by treatment and the results are plotted in Fig. 6. Weibull probability paper can also be used but Fig. 6 has been plotted using a computer spreadsheet. Weibull probability paper or a more configurable graphing package would have displayed cumulative per cent failure explicitly as the abscissa and the age of failure explicitly as the ordinate. Although a three-parameter analysis (t , b, g) was carried out, t (the locating constant) is 0 0 zero in all the plots. The Weibull distribution, expressed as failure, F, as a function of time, t, is given by:
C A BD
F(t)=1−exp −
t−t b 0 . g
(2)
It can represent many shapes of population, depending on the shape factor, b; for example, b=3.44 is close to a normal distribution. The scaling constant, g, determines the spread of the distribution. Weibull analysis has been used as a graphical method of determining the population distribution (provided the data fit) as well as determining the mean life. The original t lifetime data are used to calculate t , where practical, which represents 0 a life before any failures are likely to occur. It is referred to as a life of intrinsic reliability. Depending on the shape of the distribution it may also be possible to estimate what type of failure a component underwent [8]. For example, with t >0 and b<1, failure is consis0 tent with fatigue — where there are no failures initially and the failure rate suddenly increases and then decreases with age — rather than wear-out — where the failure rate increases with time (consistent with normally distributed populations). With the axes arranged as in Fig. 6, the form of the Weibull equation when the natural logarithm of Eq. (2) is taken twice and the equation rearranged, a straight-line plot fits the twoparameter Weibull distribution. Therefore the linear regression equation has also been given, with R2, the
coefficient of determination, used to indicate the goodness of fit. The equation is in the form y=mx+c where the slope, m, is equal to b and the intercept, c, a function of g. It is clear from Fig. 6(b) that either the data do not have a Weibull distribution or a non-zero value of t is required; initially assuming the latter, t has been 0 0 calculated to be close to 59.6×103 by an iterative curvefitting technique. This is equivalent to an intrinsic life of reliability of about 60 000 operations, which means no failures of a tool in this failure mode in this population are expected before 60 000 components have been produced. Compare this with no clear intrinsic reliability for most of the other treatments.
4. Discussion The findings and sequential improvements from the initial study were based on the observations of failures in the tools of the previous treatment. The uncoated hex punch failed by chipping associated with the stamped letter markings on the punches and a subsequent PVD TiN coating did very little to improve the tool life. The lettering process was then modified. Modification of the tool prior to PVD TiN coating brought about a significant improvement in tool life both with and without a PVD coating, and the failure appeared then to be away from the lettering on the working sides of the punches. Examination by scanning electron microscopy (SEM ) showed that the edges of the punches were being undermined by cracks running at 30° to the sides of the punch [1]. This then led to the final modifications to the uncoated punch (referred to as level 3 of the ‘finish’ factor). The failure mode of these tools was found to change to chipping on the shoulders of the punch. Fig. 7 shows typical failures of TiN-coated and uncoated punch tools [2,3]. Fig. 7(a) shows a coated tool with a machined surface where the coating has worn as the tool has been used. Figs. 7(b) and (c) show polished tools that show little wear where the coating remains but wear consistent with wear on uncoated tools [Fig. 7(d)] where the coating has failed. For the polished tools, coating failure does not appear to be a gradual, ‘wear-out’ process. Interestingly, although Student’s t-test had been applied to the treatments and used in hypothesis testing to show that there was a statistical difference between the (constant) population means of most of the treatments (except between 1 and 2), the confidence intervals for each treatment were not calculated. However, referring to Fig. 5 and Table 3, a CI does give a quantitative means of describing the improvements and the location of the treatments. Confidence intervals are based on the product of the standard error of the sample mean and the t-value, where t is a function of the degrees of freedom of the sample and the level of significance
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(a)
(b)
(c)
(d)
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(e)
Fig. 6. Weibull analysis by treatment. (a) Treatment 1 — normal uncoated hexagonal punch; (b) treatment 2 — PVD TiN-coated normal hexagonal punch; (c) treatment 3 — uncoated modified hexagonal punch; (d) treatment 4 — PVD TiN-coated modified hexagonal punch; (e) treatment 5 — PVD TiN-coated polished modified haxagonal punch. The equation for a ‘best-fit’ straight line is also given.
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(a)
(b)
(c)
(d)
Fig. 7. Typical failures of TiN-coated and uncoated punch tools. (a) A coated tool with a machined surface, in this case the coating has worn as the tool has been used (equivalent to treatment 2). (b) and (c) Polished tools that show little or no wear where the coating remains (equivalent to treatment 5) but normal wear where the coating has failed. This is comparable to the wear on uncoated tools [(d ), equivalent to treatment 3].
applied with the test (1). The CI can be reduced in two ways, either by reducing the variance of the data or by increasing the number of data points (reducing the level of significance defeats the object of applying the test to a certain degree). The t-test is applicable when the data populations are assumed to be normal. However, when applying a two-sample t-test, as a hypothesis test for example, the populations are assumed to have the same variances. This can be tested using the F-test; applying this shows, for treatments 4 and 5, that they have significantly different variances from the lower number tests and that any test for homogeneity of variance will fail between non-adjacent treatments (except 1 to 3). However, it is clear from the confidence intervals that hypothesis testing is unnecessary and that differences in the population means (rather than the sample means) relate to significant shifts by each successive treatment (except from 1 to 2). Therefore the changes in population mean are real and could not have occurred by chance alone (accepting a 1-in-20 risk of error). Although the use of CIs gives the required quantitative measurement for improvements, it does not let the data tell their full story. When the variance visible in the test sample results is seen to be so large, it could leave doubts as to the quality benefits of this combina-
tion of surface preparation and coating [3,6 ]. This is, in some cases, just a perception as demonstrated in [3], but it is also dependent on the quality measure that the data are compared against. The results initially do not appear to support some of the conclusions in [2]; namely, that the failure mode of polished and TiNcoated tools is a fatigue-like process (although clearly they are not just wearing out, Fig. 7). With this type of failure and with rough loading it is expected that most of the tools will fail relatively early, with the remainder, possibly only slightly stronger, lasting a very long time, giving a characteristic, highly skewed distribution [8]. This shape of distribution is only clear in treatment 2, the TiN-coated normal hexagonal punches. Treatments 4 and 5, though both coated, appear normally distributed and the initial values of the shaping parameters from the Weibull analysis support this. This analysis is therefore not as cut-and-dried as that in [2]. However, Weibull analysis ( Fig. 6) again gives some good indications to the answers to these problems; i.e., the unexpected shapes of the distributions (regarding the coated tools) and the large variance in the sample data. As stated in the Results, a non-zero value of t 0 has been calculated for treatment 2 as being close to 59.6×103. This is equivalent to an intrinsic life of
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reliability of about 60 000; i.e., no failures of a tool in this failure mode in this population before 60 000 components have been produced. When this is applied the shape factor changes from nearly normal ( b=3.3) to highly skewed (as seen in Fig. 2; b=0.83), a typical high-cycle ‘fatigue-like’ failure [8]. Interestingly this makes treatment 2, although appearing to last no longer than the uncoated tools, more reliable than the original, uncoated tools of treatment 1. This could be argued to be purely due to the change in the failure mode from the application of the surface coating. To use a value for the locating constant (t ) it is necessary to recognise 0 curvature in the initial data plot; curvature of the same sense as in Fig. 6(b) will be referred to as positive, giving a positive value of t . Negative curvature, on the 0 other hand, does not necessarily mean that the tools were wearing out before they came into service. The presence of this, and a discontinuity or ‘knee’ as in Fig. 6(c), does however strongly indicate that an existing defect was present and that these first four tools failed by another failure mode [9]. By inspecting the tools with this possibility in mind, a more thorough metallurgical analysis could be carried out. Once these anomalies are seen in Fig. 6(c), then — although more subtle — there are similar features observable in the Weibull plots for the remaining treatments. These are enough to prompt further investigation into the failures modes of the actual tools by further, more detailed inspection. This discussion could raise the question of the benefits of industrial testing, as the difficulties of demonstrating small but cost-effective savings may, in some minds, be too great and lead to false conclusions. It is our contention that industrial or ‘real-life’ testing is precisely where experimentation should take place and that mature statistical tools are available to aid the experimenter to find real improvements through the inherent ‘noise’ in the results drawn from an industrial environment and help prevent false conclusions.
5. Conclusions The use of confidence intervals to estimate the location of the population mean for various treatments (tabulated in Table 2) has been carried out. This shows that the population mean of treatment 5 lies within the range 317 800 to 506 200 components as compared with the uncoated normal tools in treatment 1, which have a population mean within the range 46 750 to 80 750. In the worst case, this makes on average nearly 240 000 more components per tool. Alternatively, the Weibull
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characteristic life can be used to quantify the benefits and this has been found to be 465 200 operations for the improved tooling as opposed to 72 000 for the original tooling using the original data (with no data ‘censoring’). The key practical outcome for the analysis has been to quantify the benefits of the process changes; this is very important for any company as it ensures that a cost for that performance increase can be calculated. In general in industry, if a change cannot be costed then it will not be introduced. Simple statistical analysis has again been shown to have positive benefits in an area of engineering normally associated with complex material analysis techniques, as it enables use of the entire numerical data sets to make valid inferences. Weibull analysis gives the experimenter the ability to ‘look inside’ these sets and see if they are giving the whole picture. Recognising that different failure modes could exist within a treatment allows the real benefits of that treatment to be seen without unexpected failures clouding the picture. This has immediately increased the value of all the repeated testing. Intuitively we know to test more than one product to make a valid statement on any performance benefits, a statistical approach provides the tools to plan and carry out experiments and lets the experimenter work with the ‘noise’ of uncertainty. Recognising where some of this uncertainty comes from, for example by using Weibull analysis, is of great benefit for the practical experimenter in the application of surface engineered products, and we believe that such statistical methods should be used much more widely than is presently the case.
References [1] J. Zhang, Ph.D. thesis, Swinbourne University of Technology, 1996. [2] S.J. Dowey, B. Ra¨hle, A. Matthews, Surf. Coat. Technol. 99 (1998) 213–221. [3] S.J. Dowey, A. Matthews, Taguchi and TQM: quality issues for surface engineered applications, Surf. Coat. Technol. 110 (1998) 86–93. [4] W. Weibull, J. Appl. Mech. 18 (1951) 293. [5] N. Logothetis, Managing for Total Quality, From Deming to Taguchi and SPC, 1st ed., Prentice-Hall International, UK, 1992. [6 ] M.S. Phadke, Quality Engineering using Robust Design, 1st ed., Prentice-Hall International Editions, London, 1989. [7] R.A. Fisher, The Design of Experiments, 8th ed., Hafner Publishing Company, Inc, 1971. reprinted. [8] A.D.S. Carter, Mechanical Reliability, 2nd ed., MacMillan, London, 1986. [9] A.D.S. Carter, Mechanical Reliability and Design, 1st ed., MacMillan, London, 1997.