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Scale-Sensitive Fractal Analysis of Turned Surfaces
Scale-Sensitive Fractal Analysis of Turned Surfaces 1
Christopher A. Brown’, William A. Johnsen?, Rachel M. Butland2 Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, USA 2 Manufacturing Engineering Program, Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, USA Submitted by Jim Bryan Received on January 5,1996
Abstract Scale-sensitive analyses using length-scale relations, which have been developed from fractal geometry (coastline, compass or Richardson methods), are applied to a series of turned surfaces with a range of feeds per revolution. This length-scale analysis method calculates the apparent lengths of profiles as a function of the scales of measurement. The profrles are acquired by conventional, stylus profiling. It is shown that this length-scale method can be used to identify the periodic components of profiles and that it is able to distinguish changes in the character of the surface roughness as a function of scale. Kewords : Machined surface texture, Topographic analysis, Surface roughness
1. Introduction The length of a profile acquired from a rough surface is not unique. It varies with the scale of measurement (Mandelbrot 1977), and this variation has been the basis for a type of fractal analysis (e.g., Richardson, compass and coastline methods Mandelbrot 1977, Dauw et al. 1990, Russ 1994). In fractal analyses, scale-sensitivity is most often ignored in the quest for compact geometric descriptions. In this paper the method used by Dauw et al. (1990) is called length-scale analysis to emphasize the scale sensitivity. The objective of the current work is to study this length-scale analysis applied to profiles acquired from turned surfaces. We would like to learn 1) if this length-scale fractal analysis is capable of differentiating the relatively large and ordered topographic features, caused by the feed of the tool in each revolution during turning, from the fine, less-ordered topographic features inside the feed marks; 2) how well it can determine the feed per revolution; and 3) if the complexity of the fine features, i.e., features below the scale of the feed, are a function of the feed. A review of fractal or of conventional analyses is not an objective, which has been done by others (Russ 1994, Whitehouse 1994, Peters et al. 1979) Length-scale type fractal analysis has been used successfully to differentiate certain manufacturing process variables and base materials in electric discharge machining (Dauw et al. 1990, Brown et al. 1990) and grinding (Brown and Savary 1991). In both cases, the distribution of the topographic features was largely random below a smooth-rough crossover scale. The differentiation was based on two parameters, one described the complexity of Annals of the ClRP Vol. 45/7/1996
the profile below the crossover and the other was the smooth-rough crossover scale. Both parameters are determined from length-scale relations. It is not clear how non-random elements would influence the results of this analysis. The power spectrum and the autocorrelation functions can find periodic structures in profiles, and can differentiate the relative contributions of large scale and fine scale details (Whitehouse 1994). Nonetheless, further study of length-scale analysis is warranted, because scale-sensitive fractal analyses can lead to parameters which have good potential for clear physical interpre!ations for the design and analysis of surface function and manufacture (Brown 1994). Russ (1994) writes that the type of length-scale fractal analysis used here is appropriate for selfsimilar shapes and not for profiles from most engineering surfaces, which are self-affine, i.e., appear smooth from a distance and rough only close to or under magnification. However, as mentioned above, the smooth-rough crossover, which results from the self-affine nature of most engineering surfaces, has been advantageous in differentiating surfaces created under different conditions and in designing rough surfaces. Whitehouse (1994) cautions against using fractal analysis for surfaces, such as turned surfaces, which are formed by essentially plastic processes and have distinctly non-fractal, geometric character. Part of the purpose of this paper is to demonstrate the utility of scale-sensitive fractal type analyses on surfaces with non-fractal elements. Here the characterization is based on more than just the fractal dimension, and therefore
515
lack some of the elegance Whitehouse envisioned for fractal analysis. 2. Methods A cylindrical bar of AlSl 4140 (CrMo) steel was machined by turning with depth of cut of 2mm to a diameter of about 5Omm at a surface speed of 121mlmin. A Kennamental tool (carbide grade insert KC9025, Latrobe PA, USA) with a nose radius of 0.8mm was used to machine 9 sections, each approximately 25mm long, with 9 different feeds, with values between 162pm and 650pm. Three profiles were acquired from each feed, parallel to the axis of the bar, approximately perpendicular to the tool path. A Federal Products Surfanalyzer 5000 contact profiling device (Providence, RI, USA), with a stylus radius of 2.54pm was used. The traversing lengths were approximately 9.7mm, and consisted of approximately 7600 elevations, yielding a sampling interval of approximately 1.3pm. The evaluation lengths were approximately 6.9mm. A 2RC filter with a 0.8mm cutoff length was used for traverses from all the feeds to calculate the average maximum height of the profile, R, with 5 sampling lengths (ASMEIANSI 846, 1995). Unfiltered profiles were recorded and transferred to another computer where the length-scale analyses were performed. 2.1 Lenath-scale analvsis of Drofiles The length-scale analysis is performed using the method described by Brown and Savary (1991). This length-scale analysis is performed by a method similar to the coastline or Richardson methods (Mandelbrot 1977, Russ 1994). The length of the profile is measured as a function of scale by a virtual stepping along an unfiltered profile with virtual dividers, or a compass or ruler. The spread of the dividers or compass or the length of the ruler is the scale of measurement. Successive measurements are made at different scales or ruler lengths. For each measurement the length of the virtual rulers is kept exactly the same and linear interpolations are used to locate the virtual steps between the sampling intervals in the profile. A relative length as a function of scale is determined by dividing the measured length by projected length of the measured portion of the profile. The projected length is determined from the least squares fit and is approximately the straight line distance from the starting to the end points of the measurement at a particular scale. The relative length at a particular scale can be shown to be equal to a weighted average of the reciprocal of the cosine of the angles (€Ii) the orientation of the steps make with the least squares mean line: N
k = Z (PA)( l/CO&i) I 4
516
I
Where Rd is the relative length, L is the total projected length for all N virtual steps, pi is the projected length for virtual step i corresponding to Q. The logs of the relative lengths are plotted versus the logs of the ruler lengths. This plot is then used to derive parameters to describe geometric properties of the profile. 3. Results The average maximum heights of the profiles are shown in Fig. 1 as a function of the feed per revolution, with a theoretical curve of the maximum peak-to-valley of the profile as the square of the feed divided by eight times the tool nose radius..
I
Rz MeasuredJ
?
100
200
400 500 Feed Rate (vm)
300
600
700
Fig. 1. Average maximum heights of the profiles as a function of the feed per revolution. The solid line is Rah = t2/8r, where t is the feed and r is the tool nose radius (Whitehouse 1994).
A plot of 300 relative lengths versus step length, i.e., ruler lengths or scales of measurement, calculated from a profile acquired from a region on the bar where the feed was 467pm per revolution, is shown in Fig. 2. The relative lengths at the larger scales are close to one, indicating that the virtual steps were, on the average only slightly inclined. At the larger scales there are a series of bumps in the plot. The summit of the bump, or local maximum, occurring at the finest scale, or step length, corresponds to a step length of about 600pm and a relative length of about 1.001. The local minimum to the left of this bump is at about 466pm. For a little less than one decade of step lengths below this local minimum the log of the relative length increases linearly with a decrease in the log of the step length. Using a linear regression of the logs, the slope of this section is -0.0057, with a regression coefficient P=0.98. The absolute value of this slope plus one (1.0057) gives a fractal dimension for this region (Brown and Savary 1991). At about 70pm the increase in the relative length with respect to the decrease in step length begins to lessen.
i
:
.
I
.'.. :c
.
*
I
1.00,
.:
:
I
- . ' I
100
1,000 Step Length (mm)
Fig. 2. Length-scale plot (log-log) of a profile acquired from a region on the turned bar where the feed was 467pm per revolution. The one decade region of steepest slope is indicated and the equation for the regression line and regression coefficient are given (x = step length: y = relative length). In Fig. 3 the local minima, described above, which occur at scales just above the region of steepest slope on the length-scale plots, are plotted versus the corresponding feeds. The linear regression shows good agreement between these local minima and the feed per revolution. The regression coefficient, P, is greater than 0.98. The intercept is less than 4% of the maximum value plotted (22/650) and the slope is close to one (0.97). 700
-I
.
I
0 100
200
400 500 Feed Rate (pm)
300
€XI0
700
Fig. 3. Local minima from the scale-length plots versus feeds. The solid line is the least squares linear regression, for which the equation and regression coefficient are given (x = feed rate; y = local minimum). 4. Discussion In fig. 1 it can be seen that the profile heights vary with the feed approximately as expected. The tendency of the simple theoretical relation to underestimate the profile heights is not unexpected, and possible causes for deviations from the predicted values have been well summarized and discussed elsewhere (Whitehouse 1994, chapter 6.3). The length-scale relation shown in Fig.2 is typical of the 27 plots generated in this study. There is a fundamental difference between the
relative length on the length-scale plots (Rel, eq. 1) and the profile height R,. They indicate that this length-scale fractal analysis can differentiate large and fine scale topographic details. Three different regions, with three distinct characters, as a function of scale, can be recognized in the scale-sensitive fractal analyses. At the scales larger than the feed the relative areas are close to 1. At these larger scales the surface is approximately smooth, at finer scales it is rough, so the feed represents a kind of smooth-rough crossover, that is easily discerned on the plots. At scales immediately finer than the feed, for about one decade, there is a linear region on the plot, the slope of which is indicative of the complexity of the profile and is found to correspond to the feed (Fig. 4). At finer scales, below about a tenth of the feed, the slope of the length-scale plot decreases, indicating less complexity in the profile data. Some of the reduction in the complexity here could be due to the interaction of the stylus geometry with the topographic features and the stylus contact stresses with the material (Brown and Savary 1991). However, the stylus is comparatively small enough, so that it Seems unlikely that all the loss in complexity can be attributed to geometric interactions. At still finer scales, below the sampling interval (not shown in Fig. 2) the slope of the plot would be zero. Scale-sensitive fractal analysis can be used to determine the feed per revolution, as shown in Fig. 3. The local minima, which might also be used to define a smooth-rough crossover, consistently occur close to the scale of the feed per revolution. The regression shows that over 98% of the variation in the local minima can be explained by the corresponding variation in feed. The feed and the local minimum will be close to the same value, since the slope in Fig 3 is near unity and the intercept is low. This result suggests that scalesensitive fractal analyses can be used to find spatial periodicity. At a scale slightly larger than the local minimum corresponding to the feed there is a local maximum (mentioned previously as about 600pm on Fig. 2). From the relative length corresponding to this local maximum (about 1.001 on Fig. 2), and using equation 1, it can be shown that the (weighted) average slope in the profile data at this scale can be determined (2.56' at 600pm in Fig. 2 - note that the slopes on the profile, like the length are not unique but depend on the scale of measurement)). An average height can be determined from Fig. 2 (step length times sin0 = 26.81pm). This value should be less than R, since it depends on measurements at more (16) locations, corresponding to the virtual steps. Furthermore, the exact coordinates for the local maxima on the summits of the bumps, like the ones used in the calculations here, are dependent on the starting point for the stepping simulation (Brown and Meacharn 1994). When the virtual stepping begins exactly on a peak on the profile separating two feed
517
marks, then the local maximum should appear at a step size slightly over 1.5 times the feed per revolution and the value above would approach R,. The fractal dimension is indicative of the geometric complexity, or intricacy of the surface (Mandelbrot 1977). The low fractal dimensions give awkward values to work with. In Fig. 4 the slope of the length-scale plot for the decade of step lengths just below the last local minimum are multiplied by -1 000 (to produce a length-scale fractalcomplexity (Isfc) parameter, a more convenient value than the fractal dimension) and are plotted versus the feed. 12
10
I
-1
:
Lengthscale FMI Ccrnpttyity i
I I -I
+
2-
0 100
-
y = 0.019x 2.2 = 0.98
4-
4
-__f__i
Mo
400 500 Feed Rate (prn)
300
600
700
Fig. 4. Length-scale fractal-complexity versus feed (x = feed rate; y = LSFC; LSFC = -1000 slope of the regression line in Figure 2). The high regression coefficient and significant slope for the plot of length-scale fractalcomplexity versus feed show that the complexity of the topographic features inside the feed marks are dependent on the feed. Fig 4 indicates that the workpiece-chip separation phenomena at the edge of the tool where the surface is formed is a function of the feed, or at least the depth-of-cut to feed ratio, since the depth of cut to feed ratio also changed for each of the feeds. 5. Conclusions For the conditions observed in this study: 0 scale-length analysis differentiates the relative large and ordered feed patterns from the fine and less ordered features inside the grooves formed by the tool nose. 0 scale-length analysis can determine the numeric value of feed per revolution on turned surfaces within a few percent. 0 the complexity of the topographic features in the grooves formed by the tool nose increases regularly with the feed per revolution.
6. Acknowledaments The authors thank Kennametal Corporation (Latrobe, PA) for machining and Norton Company (Worcester, MA) and the CGRD at UCONN, for the use of their profiling devices. We would also like to thank Burlington Computer Systems, Waterbury Center, Vr, USA, for the use of Surfrax for lengthscale analyses. We gratefully acknowledge the financial support of NASA grant #NAG-1-1606, NSF
518
grant #EEC-9424205 Polytechnic Institute.
REU
and
Worcester
7. References 1. Brown, C.A., 1994, A Method for concurrent engineering design of chaotic surface topographies, J. Mater. Process. Technol. 44:337-344. 2. Brown,C A,. Dauw.D., Savary, G.. 1990. "Comparison of Fractal and Conventional Topographic Analysis of Electric Discharge Machined Surface Topography on Ceramics," Surface Engineering: Current Trends and Future Prospects S.A. Meguid, ed., Elsevier Applied Science, London 39-5. 3. Brown,C.A., Meacham,B., "Tiling Strategies in the Patchwork Method and the Determination of Scale-Area Relations," Fractals, Vol. 2, No. 3 (1994) 433436. 4. Brown, C.A., Savary, G., 1991, Describing ground surface texture using contact profilometry and fractal analysis, Wear 141:221226. 5. Dauw, D., Brown, C.A., vanGriethuysen, J.P., Albert, J.F.L.M, 1990 Surface Topography Investigations by Fractal Analysis of Spark Eroded Electrically Conductive Ceramics" Annals of the ClRP 39/1:161- 165. 6. Mandelbrot, B.B., 1977, Fractals, Form, Chance and Dimension, Freeman, San Francisco. 7. Peters,J., Vanherck,P., Sastrodinoto,M., 1979, Assessment of Surface Typology Analysis Techniques, Annals of the ClRP 28/2: 539-553. 8 Russ J. C. 1994 Fractal Surfaces, Plenum Press, New York. 9. Whitehouse, D.J., 1994, Handbook of Surface Metrology, Institute of Physics Publishing, Bristol.
Christopher A. Brown’, William A. Johnsen?, Rachel M. Butland2 Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, USA 2 Manufacturing Engineering Program, Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, USA Submitted by Jim Bryan Received on January 5,1996
Abstract Scale-sensitive analyses using length-scale relations, which have been developed from fractal geometry (coastline, compass or Richardson methods), are applied to a series of turned surfaces with a range of feeds per revolution. This length-scale analysis method calculates the apparent lengths of profiles as a function of the scales of measurement. The profrles are acquired by conventional, stylus profiling. It is shown that this length-scale method can be used to identify the periodic components of profiles and that it is able to distinguish changes in the character of the surface roughness as a function of scale. Kewords : Machined surface texture, Topographic analysis, Surface roughness
1. Introduction The length of a profile acquired from a rough surface is not unique. It varies with the scale of measurement (Mandelbrot 1977), and this variation has been the basis for a type of fractal analysis (e.g., Richardson, compass and coastline methods Mandelbrot 1977, Dauw et al. 1990, Russ 1994). In fractal analyses, scale-sensitivity is most often ignored in the quest for compact geometric descriptions. In this paper the method used by Dauw et al. (1990) is called length-scale analysis to emphasize the scale sensitivity. The objective of the current work is to study this length-scale analysis applied to profiles acquired from turned surfaces. We would like to learn 1) if this length-scale fractal analysis is capable of differentiating the relatively large and ordered topographic features, caused by the feed of the tool in each revolution during turning, from the fine, less-ordered topographic features inside the feed marks; 2) how well it can determine the feed per revolution; and 3) if the complexity of the fine features, i.e., features below the scale of the feed, are a function of the feed. A review of fractal or of conventional analyses is not an objective, which has been done by others (Russ 1994, Whitehouse 1994, Peters et al. 1979) Length-scale type fractal analysis has been used successfully to differentiate certain manufacturing process variables and base materials in electric discharge machining (Dauw et al. 1990, Brown et al. 1990) and grinding (Brown and Savary 1991). In both cases, the distribution of the topographic features was largely random below a smooth-rough crossover scale. The differentiation was based on two parameters, one described the complexity of Annals of the ClRP Vol. 45/7/1996
the profile below the crossover and the other was the smooth-rough crossover scale. Both parameters are determined from length-scale relations. It is not clear how non-random elements would influence the results of this analysis. The power spectrum and the autocorrelation functions can find periodic structures in profiles, and can differentiate the relative contributions of large scale and fine scale details (Whitehouse 1994). Nonetheless, further study of length-scale analysis is warranted, because scale-sensitive fractal analyses can lead to parameters which have good potential for clear physical interpre!ations for the design and analysis of surface function and manufacture (Brown 1994). Russ (1994) writes that the type of length-scale fractal analysis used here is appropriate for selfsimilar shapes and not for profiles from most engineering surfaces, which are self-affine, i.e., appear smooth from a distance and rough only close to or under magnification. However, as mentioned above, the smooth-rough crossover, which results from the self-affine nature of most engineering surfaces, has been advantageous in differentiating surfaces created under different conditions and in designing rough surfaces. Whitehouse (1994) cautions against using fractal analysis for surfaces, such as turned surfaces, which are formed by essentially plastic processes and have distinctly non-fractal, geometric character. Part of the purpose of this paper is to demonstrate the utility of scale-sensitive fractal type analyses on surfaces with non-fractal elements. Here the characterization is based on more than just the fractal dimension, and therefore
515
lack some of the elegance Whitehouse envisioned for fractal analysis. 2. Methods A cylindrical bar of AlSl 4140 (CrMo) steel was machined by turning with depth of cut of 2mm to a diameter of about 5Omm at a surface speed of 121mlmin. A Kennamental tool (carbide grade insert KC9025, Latrobe PA, USA) with a nose radius of 0.8mm was used to machine 9 sections, each approximately 25mm long, with 9 different feeds, with values between 162pm and 650pm. Three profiles were acquired from each feed, parallel to the axis of the bar, approximately perpendicular to the tool path. A Federal Products Surfanalyzer 5000 contact profiling device (Providence, RI, USA), with a stylus radius of 2.54pm was used. The traversing lengths were approximately 9.7mm, and consisted of approximately 7600 elevations, yielding a sampling interval of approximately 1.3pm. The evaluation lengths were approximately 6.9mm. A 2RC filter with a 0.8mm cutoff length was used for traverses from all the feeds to calculate the average maximum height of the profile, R, with 5 sampling lengths (ASMEIANSI 846, 1995). Unfiltered profiles were recorded and transferred to another computer where the length-scale analyses were performed. 2.1 Lenath-scale analvsis of Drofiles The length-scale analysis is performed using the method described by Brown and Savary (1991). This length-scale analysis is performed by a method similar to the coastline or Richardson methods (Mandelbrot 1977, Russ 1994). The length of the profile is measured as a function of scale by a virtual stepping along an unfiltered profile with virtual dividers, or a compass or ruler. The spread of the dividers or compass or the length of the ruler is the scale of measurement. Successive measurements are made at different scales or ruler lengths. For each measurement the length of the virtual rulers is kept exactly the same and linear interpolations are used to locate the virtual steps between the sampling intervals in the profile. A relative length as a function of scale is determined by dividing the measured length by projected length of the measured portion of the profile. The projected length is determined from the least squares fit and is approximately the straight line distance from the starting to the end points of the measurement at a particular scale. The relative length at a particular scale can be shown to be equal to a weighted average of the reciprocal of the cosine of the angles (€Ii) the orientation of the steps make with the least squares mean line: N
k = Z (PA)( l/CO&i) I 4
516
I
Where Rd is the relative length, L is the total projected length for all N virtual steps, pi is the projected length for virtual step i corresponding to Q. The logs of the relative lengths are plotted versus the logs of the ruler lengths. This plot is then used to derive parameters to describe geometric properties of the profile. 3. Results The average maximum heights of the profiles are shown in Fig. 1 as a function of the feed per revolution, with a theoretical curve of the maximum peak-to-valley of the profile as the square of the feed divided by eight times the tool nose radius..
I
Rz MeasuredJ
?
100
200
400 500 Feed Rate (vm)
300
600
700
Fig. 1. Average maximum heights of the profiles as a function of the feed per revolution. The solid line is Rah = t2/8r, where t is the feed and r is the tool nose radius (Whitehouse 1994).
A plot of 300 relative lengths versus step length, i.e., ruler lengths or scales of measurement, calculated from a profile acquired from a region on the bar where the feed was 467pm per revolution, is shown in Fig. 2. The relative lengths at the larger scales are close to one, indicating that the virtual steps were, on the average only slightly inclined. At the larger scales there are a series of bumps in the plot. The summit of the bump, or local maximum, occurring at the finest scale, or step length, corresponds to a step length of about 600pm and a relative length of about 1.001. The local minimum to the left of this bump is at about 466pm. For a little less than one decade of step lengths below this local minimum the log of the relative length increases linearly with a decrease in the log of the step length. Using a linear regression of the logs, the slope of this section is -0.0057, with a regression coefficient P=0.98. The absolute value of this slope plus one (1.0057) gives a fractal dimension for this region (Brown and Savary 1991). At about 70pm the increase in the relative length with respect to the decrease in step length begins to lessen.
i
:
.
I
.'.. :c
.
*
I
1.00,
.:
:
I
- . ' I
100
1,000 Step Length (mm)
Fig. 2. Length-scale plot (log-log) of a profile acquired from a region on the turned bar where the feed was 467pm per revolution. The one decade region of steepest slope is indicated and the equation for the regression line and regression coefficient are given (x = step length: y = relative length). In Fig. 3 the local minima, described above, which occur at scales just above the region of steepest slope on the length-scale plots, are plotted versus the corresponding feeds. The linear regression shows good agreement between these local minima and the feed per revolution. The regression coefficient, P, is greater than 0.98. The intercept is less than 4% of the maximum value plotted (22/650) and the slope is close to one (0.97). 700
-I
.
I
0 100
200
400 500 Feed Rate (pm)
300
€XI0
700
Fig. 3. Local minima from the scale-length plots versus feeds. The solid line is the least squares linear regression, for which the equation and regression coefficient are given (x = feed rate; y = local minimum). 4. Discussion In fig. 1 it can be seen that the profile heights vary with the feed approximately as expected. The tendency of the simple theoretical relation to underestimate the profile heights is not unexpected, and possible causes for deviations from the predicted values have been well summarized and discussed elsewhere (Whitehouse 1994, chapter 6.3). The length-scale relation shown in Fig.2 is typical of the 27 plots generated in this study. There is a fundamental difference between the
relative length on the length-scale plots (Rel, eq. 1) and the profile height R,. They indicate that this length-scale fractal analysis can differentiate large and fine scale topographic details. Three different regions, with three distinct characters, as a function of scale, can be recognized in the scale-sensitive fractal analyses. At the scales larger than the feed the relative areas are close to 1. At these larger scales the surface is approximately smooth, at finer scales it is rough, so the feed represents a kind of smooth-rough crossover, that is easily discerned on the plots. At scales immediately finer than the feed, for about one decade, there is a linear region on the plot, the slope of which is indicative of the complexity of the profile and is found to correspond to the feed (Fig. 4). At finer scales, below about a tenth of the feed, the slope of the length-scale plot decreases, indicating less complexity in the profile data. Some of the reduction in the complexity here could be due to the interaction of the stylus geometry with the topographic features and the stylus contact stresses with the material (Brown and Savary 1991). However, the stylus is comparatively small enough, so that it Seems unlikely that all the loss in complexity can be attributed to geometric interactions. At still finer scales, below the sampling interval (not shown in Fig. 2) the slope of the plot would be zero. Scale-sensitive fractal analysis can be used to determine the feed per revolution, as shown in Fig. 3. The local minima, which might also be used to define a smooth-rough crossover, consistently occur close to the scale of the feed per revolution. The regression shows that over 98% of the variation in the local minima can be explained by the corresponding variation in feed. The feed and the local minimum will be close to the same value, since the slope in Fig 3 is near unity and the intercept is low. This result suggests that scalesensitive fractal analyses can be used to find spatial periodicity. At a scale slightly larger than the local minimum corresponding to the feed there is a local maximum (mentioned previously as about 600pm on Fig. 2). From the relative length corresponding to this local maximum (about 1.001 on Fig. 2), and using equation 1, it can be shown that the (weighted) average slope in the profile data at this scale can be determined (2.56' at 600pm in Fig. 2 - note that the slopes on the profile, like the length are not unique but depend on the scale of measurement)). An average height can be determined from Fig. 2 (step length times sin0 = 26.81pm). This value should be less than R, since it depends on measurements at more (16) locations, corresponding to the virtual steps. Furthermore, the exact coordinates for the local maxima on the summits of the bumps, like the ones used in the calculations here, are dependent on the starting point for the stepping simulation (Brown and Meacharn 1994). When the virtual stepping begins exactly on a peak on the profile separating two feed
517
marks, then the local maximum should appear at a step size slightly over 1.5 times the feed per revolution and the value above would approach R,. The fractal dimension is indicative of the geometric complexity, or intricacy of the surface (Mandelbrot 1977). The low fractal dimensions give awkward values to work with. In Fig. 4 the slope of the length-scale plot for the decade of step lengths just below the last local minimum are multiplied by -1 000 (to produce a length-scale fractalcomplexity (Isfc) parameter, a more convenient value than the fractal dimension) and are plotted versus the feed. 12
10
I
-1
:
Lengthscale FMI Ccrnpttyity i
I I -I
+
2-
0 100
-
y = 0.019x 2.2 = 0.98
4-
4
-__f__i
Mo
400 500 Feed Rate (prn)
300
600
700
Fig. 4. Length-scale fractal-complexity versus feed (x = feed rate; y = LSFC; LSFC = -1000 slope of the regression line in Figure 2). The high regression coefficient and significant slope for the plot of length-scale fractalcomplexity versus feed show that the complexity of the topographic features inside the feed marks are dependent on the feed. Fig 4 indicates that the workpiece-chip separation phenomena at the edge of the tool where the surface is formed is a function of the feed, or at least the depth-of-cut to feed ratio, since the depth of cut to feed ratio also changed for each of the feeds. 5. Conclusions For the conditions observed in this study: 0 scale-length analysis differentiates the relative large and ordered feed patterns from the fine and less ordered features inside the grooves formed by the tool nose. 0 scale-length analysis can determine the numeric value of feed per revolution on turned surfaces within a few percent. 0 the complexity of the topographic features in the grooves formed by the tool nose increases regularly with the feed per revolution.
6. Acknowledaments The authors thank Kennametal Corporation (Latrobe, PA) for machining and Norton Company (Worcester, MA) and the CGRD at UCONN, for the use of their profiling devices. We would also like to thank Burlington Computer Systems, Waterbury Center, Vr, USA, for the use of Surfrax for lengthscale analyses. We gratefully acknowledge the financial support of NASA grant #NAG-1-1606, NSF
518
grant #EEC-9424205 Polytechnic Institute.
REU
and
Worcester
7. References 1. Brown, C.A., 1994, A Method for concurrent engineering design of chaotic surface topographies, J. Mater. Process. Technol. 44:337-344. 2. Brown,C A,. Dauw.D., Savary, G.. 1990. "Comparison of Fractal and Conventional Topographic Analysis of Electric Discharge Machined Surface Topography on Ceramics," Surface Engineering: Current Trends and Future Prospects S.A. Meguid, ed., Elsevier Applied Science, London 39-5. 3. Brown,C.A., Meacham,B., "Tiling Strategies in the Patchwork Method and the Determination of Scale-Area Relations," Fractals, Vol. 2, No. 3 (1994) 433436. 4. Brown, C.A., Savary, G., 1991, Describing ground surface texture using contact profilometry and fractal analysis, Wear 141:221226. 5. Dauw, D., Brown, C.A., vanGriethuysen, J.P., Albert, J.F.L.M, 1990 Surface Topography Investigations by Fractal Analysis of Spark Eroded Electrically Conductive Ceramics" Annals of the ClRP 39/1:161- 165. 6. Mandelbrot, B.B., 1977, Fractals, Form, Chance and Dimension, Freeman, San Francisco. 7. Peters,J., Vanherck,P., Sastrodinoto,M., 1979, Assessment of Surface Typology Analysis Techniques, Annals of the ClRP 28/2: 539-553. 8 Russ J. C. 1994 Fractal Surfaces, Plenum Press, New York. 9. Whitehouse, D.J., 1994, Handbook of Surface Metrology, Institute of Physics Publishing, Bristol.