- Email: [email protected]
Stochastic approach to experimental analysis of cylindrical lapping process
Int. 1. Mach. ToolsManufact.Vol. 35. No. 1. pp. 51-59. 1995 Copyright@ 1994Ekvier ScienceLtd Printedin Great Britain. All rightsreserved am-6955/95$7.ol + .oo
Pergamon
OWO-6955(94)EOOO5-4
STOCHASTIC APPROACH TO EXPERIMENTAL ANALYSIS OF CYLINDRICAL LAPPING PROCESS JEONG-DU KIMt and MIN-SEOGCHOIR (Received 25 May 1993; in final form 21 December
1993)
Abstract-Lapping is a very complicated and random process resulting from the variation of abrasive grains by its sizes and shapes and from the numerous variables which have an effect on the process quality. Thus it needs to be analyzed by experiment rather than by theory to obtain the relative effects of variables quantitatively. In this study, the cylindrical lapping experiment designed by Taguchi’s L, orthogonal array was performed and analyzed by Yates’ ANOVA table. As a result, effective variables and interaction effects were identified and discussed. Also the optimal variable combination to obtain the largest percentage improvement of surface roughness was selected and confirmatory experiments were performed.
1.
INTRODUCTION
A finish method used to obtain good surface quality. Important variables affecting lapping efficiency are abrasive grain size, lapping pressure, lapping speed, quantity of lapping compound supplied and viscosity of the compound, etc. Comparison of the effects of variables on the overall process efficiency is not yet clear owing to the complexity and randomness of the process. A generalized and simplified model of the lapping process is shown in Fig. 1, in which two motions of abrasive grains in the lapping process [l] are represented. It is assumed that grain 1 rolls and indents into the workpiece while grain 2 slides on it. The former does not produce any chips by cutting but causes the workpiece to deform plastically which may result in strainhardening and, finally, the microfracture of the workpiece. On the other hand, the latter produces microchips by the cutting operation. In addition, some small abrasive grains are driven into the workpiece by the relatively large grains. The quantitative ratio between rolling grains and sliding grains affects the lapping efficiency and it is very difficult to predict that ratio by analytical approach because it is affected by numerous environmental and process variables which are irregularly varied. It is, thus, recommended to use the experimental approach rather than only the theoretical approach to analyze the lapping process. Using the experimental design method is a typical approach to efficiently and logically identify characteristics of any complex process by experiment. In this method, experiments are generally designed by orthogonal array and the results are analyzed by ANOVA (analysis of variance).
LAPPING IS
Grain
1
Grain
2
Workpiece FIG. 1. Model of a lapping process.
tDepartment of Precision Engineering & Mechatronics, nology, Taejon, 305-701, Korea. 51
Korea Advanced Institute of Science and Tech-
52
J.-D.
KIM and M.-S.
CHOI
A number of studies related to the experimental design method have been reported in the field of quality control and statistics. Among them, Yates [2] suggested a computing method of sum-of-squares (SS) for ANOVA in 2F factorial designs. Nelson [3] developed a computer program to construct an ANOVA table and to calculate mean effects of control variables using Yates’ algorithm and backward Yates’ algorithm presented by Daniel [4]. Taguchi [5] and Konishi [5] presented a table of OAs which are directly applicable to industry and suggested a product design method using signalto-noise ratio (SIN ratio), quality-loss-function and orthogonal array. Lenth [6] suggested a evaluation method for the effectiveness of control variables using ME (margin of error) and SME (simultaneous margin of error) which are computed from Yates’ algorithm. In this study, experiments designed by Taguchi’s OA were performed. And Taguchi’s SN ratio for percentage improvement of surface roughness in a cylindrical lapping process was used as a performance (or response) variable to evaluate the process efficiency. Yates’ algorithm and the Lenth algorithm were used to analyze the effects of variables on the performance variable. It is aimed, in this study, to characterize a cylindrical lapping process efficiently by determining the important control variables which have a large effect on the percentage improvement of surface roughness, and by finding out the optimal variable combinations which provide information about the direction of later experiments to obtain an optimal lapping condition. Some confirmatory experiments are conducted. 2. EXPERIMENT
2.1. Experimental design Two-level fractional factorial design, which is used in this study is especially efficient in finding out important variables having an effect on the process performance [7]. It is a very useful tool for characterization or qualitative analysis prior to quantitative experiments of the lapping process. The efficiency of an experiment, especially in the case of lapping, can be increased by using the experimental design method as a tool for screening unimportant (or trivial) variables and, thus, reducing time, labor and other resources. Experiments at two levels of each control variable were conducted: grain size, lapping pressure, number of lapping-compound-supply and lapping speed. Control variables and their levels are shown in Table 1. Percentage improvement of surface roughness before and after lapping was taken as an evaluation or response variable. Taguchi’s fold-over type L, OA [7] is shown in Table 2. This is a resolution IV 24-’ fractional factorial design [7, 81 and all the main effects are not connected with two-factor interaction effects but are with three-factor interaction effects. Two-factor interaction effects are connected with each other: AB with CD, AC with BD and BC with AC. Therefore, if any two-factor interaction effect has been proven to be significant then other two-factor interaction effects which are connected with it should also be tested. Generally, three (or more)-factor interactions rarely occur [9]. So, in this study, only four main effects (i.e. A, B, C, D) and three two-factor interaction effects (i.e. AB, AC and BC) were considered in the experimental design. All the other effects were considered to be trivial. The first column in Table 2 represents the standard order TABLE 1. LEVELS OF
FACTORS
Levels (+) Low (-)
Symbol
Factor
High
A
Grain size Lapping pressure Number of supply Lapping speed
#5OQ 0.3 MPa 6 100 mimin
B c D
of lapping
compound
#lOO 0.15 MPa 3 50 mlmin
Stochastic Approach to Experimental TABLE
2.
TAGUCHI
Analysis of Cylindrical Lapping
L, ORTHOGONAL
ARRAY (FOLD-OVER TYPE)
1
2
Column No and contrast 3 5
A
B
AB
tc
53
t
AC
6 BC
7 D
_
-
-
+
-
+ +
_ + +
+
a b ab
+
_
_ + -
+ _
+ + _
C
_
_
+
+
-
_
+
ac bc abc
+ +
+ +
+
+ + +
+ +
+ +
+
(1)
+
[3, 91 of experimental treatment combinations (tc). The number of combinations is eight, which is the smallest number required to satisfy resolution IV design with four variables. Experiments were performed in a random order so as not to be affected by the order of experiments and were replicated three times for each combination. The other columns represent 23-1 (=7) contrasts of which the order is the same as that of Taguchi and Konishi [5]: four main effects are assigned to the first, second, fourth and seventh columns and interaction effects to the third, fifth and sixth columns. In the table, “+” represents high-level and “-“, low-level. Though the levels in the experiment are represented as numerical quantitatives, they can also be considered to have qualitative meaning, “high” or “low”. 2.2. Experiment Experiments were performed by a manufactured lapping unit which was set on the lathe. The schematic experimental set-up is shown in Fig. 2. SM45C of +50 was used as a workpiece and was revolved by the chuck of the lathe. The lapping wheel contacts with the workpiece theoretically at a point, but actually over a small area, and oscillates in the axial direction. All of them have an initial surface roughness of R, 2.1 Pm-R, 2.7 km. The lapping compound which had been made by mixing WC-grains and grease was used. Four variables were considered as control variables: grain size (A); lapping pressure (B); number of lapping-compound-supply (C); and lapping speed (D). Experimental levels for each control variable are as follows: #lOO and #500 for A (abrasive grain size), 0.15 MPa and 0.3 MPa for B (supplied pressure to air cylinder of lapping unit), 3 and 6 for C (supply number of lapping compound), 50 m/min and 100 m/min for D (lapping speed). Total lapping distance was fixed at 20,000 revolutions for each combination. These are shown in Table 1. Surface roughness was measured at 30 points before and after lapping using Surftestof Mitutoyo and averaged.
I
I
I
L Workpiece
FIG. 2. Experimental
set-up.
Lenre
J.-D. KIM and M.-S. CHOI
54
2.3. Experimental
results
Experimental results are shown in Table 3, in which yi s (i = 1, 2, 3) are percentage improvements of surface roughness and v is the averaged value of the three. s* is variance and S/N is Taguchi’s performance or response statistic. A merit of using logtype response variables such as this in analysis of variance is to reduce the inequality of variance between treatments and, thus, to satisfy one of the basic assumptions, i.e. equal variance. In this study, we are interested in the efficiency or percentage improvement of surface roughness, so S/N ratios are calculated for the case of “Bigger Is Better” (i.e. BIB case) and the equations used are as follows:
yi =
AR, (improvement of surface roughness) x IO0 R, (surface roughness before lapping)
(AR, = R, of before lapping - R, of after lapping)
(1)
YJ3
Y = (Yl +y*+
(2)
S* = 2 (vi - y)*/(n-1) SIN= -1Olog
(
;&
(3)
,I
.
(4)
In the other two cases, SIB (smaller is better) and NIB (nominal is better), equations (5) and (6) are used, respectively [lo]. SIB case: S/N = - 10 log
(5)
NIB case: S/N = 10 log
. 3. ANALYSIS
AND DISCUSSION
3.1. Analysis of experimental results Table 4 shows the method of Yates’ computing algorithm [8] used to obtain mean effect of each variable and this is the result of Nelson’s BASIC program [9]. Column I is generated using the data (SIN ratio) column by the rule that the first four entries are created by adding adjacent pair-wise sets of data from the response variable column, and the other four entries by subtracting adjacent pair-wise sets. Columns II and III are generated by applying the same rule, column II from column I and column III from column II. The sum of squares (SS) and mean effects for each treatment are calculated from equations (7) and (8), respectively. TABLE 3. EXPERIMENTAL RESULT
tc
(1) t ab C E abc
R. improvement (%) Y3 Yl Y2
Y (%)
SIN (db)
27.15 17.10 30.09 23.81 23.98 17.53 40.86 34.79
23.96 16.60 26.21 24.78 20.23 17.91 38.51 32.51
27.4391 24.3361 28.2220 27.8715 25.8245 25.0086 31.6883 30.2091
22.44 17.69 23.29 25.41 16.57 16.71 37.38 31.66
21.69 15.01 25.24 25.12 20.14 19.49 31.29 31.08
Stochastic
Approach
to Experimental
Analysis
of Cylindrical
55
Lapping
TABLE 4. COMPUTING PROCEDUREOF YATES’ ALGORITHM
tc
(1) a b ab C
ac bc abc
Data (SIN-ratio)
I
Sum of effects II
III
27.4391 24.3361 28.2220 27.8715 25.8245 25.0086 31.6883 30.2091
51.7752 56.0935 50.8331 61.8974 -3.1030 -0.3505 -0.8159 - 1.4792
107.8687 112.7305 -3.4535 -2.2951 4.3183 11.0643 2.7525 -0.6633
220.5592 -5.7486 15.3826 2.0892 4.8618 1.1584 6.7460 -3.4158
ss
Mean effect
6083.0 4.131 29.578 0.546 2.955 0.168 5.689 1.459
27.575 - 1.438 3.846 0.522 1.216 0.290 1.686 -0.854
SS = (III)*/8
(7)
Mean effect = (111)/4, for treatments (111)/S, for overall average.
(8)
Mean effects can be also computed by subtracting the mean value of four experimental data at low-level from those at high-level. For variable A, for example, high-level appears in rows 2, 4, 6, 8 and low-level in 1, 3, 5 and 7 rows in Table 2. So the mean effect for A can be simply computed from Table 3 as follows which is same as that in Table 4. (24.3361 + 27.8715 + 25.0086 + 30.2091)/4 - (27.4391 + 28.2220 + 25.8245 + 31.6883)/4 = - 1.438. The table of ANOVA was constructed using sum-of-square (SS) and mean effects as shown in Table 5. The error term in Table 4 was generated by the 5% pooling rule: AB, AC and D effects which are less than 5% of total sum-of-square are pooled into error. In Table 5, only the B (lapping pressure) effect has proven to be significant at a = 0.01 and B, A and BC-interaction effects are significant at (Y = 0.1. It is noted that though the C (number of lapping-compound-supply) effect is not significant by itself, it should be considered as an important variable in cylindrical lapping because the BC-interaction effect, which is shown in Fig. 3, is significant. Mean values of the SIN ratio for the mean effect of four variables are shown in Fig. 4, which indicates the optimal combination (not same as optimal lapping condition) required to obtain a larger percentage improvement of surface roughness, that is, A-B’C’D-: abrasive grain size of #lOO (coarse); lapping pressure of 0.3 MPa; number of lapping-compound-supply of six times and lapping speed of 50 m/min. It can, thus, be predicted that levels of variables A and D should be taken to minus direction (decreasing the level) and those of B and C to plus direction (increasing the level) to TABLE 5. ANOVA
tc
(1) it ab C
ac bc abc Error Total
ss
% SST
6083.0 4.131 29.578 0.546 2.955 0.168 5.689 1.459
9.28 66.43 1.23 6.64 0.38 12.78 3.28
2.172 44.524
100.00
Mean
TABLE
effect
27.575 -1.438 3.846 0.522 1.216 0.290 1.686 -0.854
YATES ALGORITHM
BY
Measures
df
MS
F
1 1 1 1 1 1 1
4.131 29.578 0.546 2.955 0.168 5.689 1.459
5.706 40.857 4.081 7.858 _
average
A B
AB c
AC BC D
7
56
KIM and M.-S.
J.-D.
CHOI
29.85
27.7 I
25.55
1 23.4
’ B-
B+
FIG. 3. Plot of BC interaction.
27.5-
25.751
24
’ A- A+
B-B+
C-C+
D-D+
Level FIG. 4. Response
graph for SIN value
obtain the maximum percentage improvement of surface roughness. Figure 5 represents each effect analyzed by Lenth’s method using Stephenson’s program [13] for the same experimental results. It shows the same trend as mentioned above.
-16
-13
-10
-7
-3
0
3
‘7
10
13
16
**
A
:
B
:
******
AR
:
* **
c AC
***
EC D
3
: SME
ME
ME
ME
margin
of error margin
SME: simultaneous FIG. 5. Plot of effects
by Lenth’s
SME
of error
method.
Stochastic Approach to Experimental Analysis of Cylindrical Lapping
57
TABLE 6, CONFIRMATORY EXPERIMENTAL RESULT FOR B C INTERACTION
B
tc
(1) b c bc
+ +
C
+ +
R. improvement ( % )
yl
y2
y3
y
S/N (dB)
20.89 22.35 18.09 41.66
18.90 27.71 19.16 38.92
25.81 24.39 21.33 43.58
21.87 24.82 19.53 41.39
26.5810 27.7943 25.7524 32.3090
3.2. Confirmatory experiment 3.2.1. BC-interaction effect. Though the BC-interaction effect in the previous section has appeared to be significant, it must to be tested by confirmatory experiments because of the possibility of connecting with the AD-interaction effect, which is a characteristic of resolution IV designs. The results of the confirmatory experiment are shown in Table 6. Mean effects of the two variables are not significantly different compared with those in Table 4 and the relative magnitude of the effects shows a similar trend as in Table 4 and Fig. 3. Therefore, it can be concluded that the BCinteraction effect, rather than the AD-interaction effect, is actually significant. 3.2.2. Optimal combination. It has been predicted that the optimal combination for increasing the percentage improvement of surface roughness is A-B+C+D -, which informs us of the direction of change of the variable level to take in the confirmatory experiment or in next stage of experiment for optimization. According to the optimal combination, the levels for A and D should be decreased and the levels for B and C should be increased to maximize the efficiency. In practice, it was attempted at new levels of the four variables: grain size of #80; lapping pressure of 0.4 MPa; number of lapping-compound-supply of eight times, lapping speed of 30 m/min instead of #100, 0.3 MPa, six times and 50 m/min, respectively. A confirmatory experiment was conducted three times and the surface roughness (average value of 30 points for each workpiece) has been improved from 2.31, 2.50 and 2.26 t~m to 1.188, 1.05 and 1.194 ~m, respectively. Percentage improvements of this are 48.59, 58.02 and 47.16. The average of these three is 51.59 (%) and the S/N ratio is 34.152. These are improved values by 34% for the average and 7.8% for the S/N ratio compared with previous experimental results, which are enough to confirm the optimal combination. 3.3. Linear regression model Three effects, A, B and BC which have proven to be significant, can be the terms of the linear regression model. Each coded variable has - 1 and +1 at its low- and high-level, respectively, in the model. In Fig. 6, the difference in response value 30
t3 ~3
28.75-
>
z
27.5-
0"3
A On
o
26.25-
>
<
25 -i
0
+1
FIG. 6. Relative effect of main contrasts on the change of response. A: Grain size; B: Lapping pressure; BC: interaction between pressure and the number of supply of lapping compound.
J.-D. KIM and M.-S. CHOI
58
between the two levels are mean effects and their half values are the slopes of lines for each effect. A linear model for the SIN ratio of the response variable is represented by the following equation. SINratio fory (db) = 27.575 - 0.719 * A + 1.923 * B + 0.943 * BC. Here, the constant, 27.575, is the overall mean of the response. The mean S/N ratio of response (or magnitude of main effect shown in Fig. 4) for any variable, x1 or x2, in the equation can be obtained by assigning -1 and +l at low- and high-levels, respectively. 3.4. Discussion The efficiency of cylindrical lapping, and the percentage improvement of surface roughness, was increased at coarse grain size. Lapping pressure shows the largest effect of all variables considered in this study and the efficiency of lapping increased dramatically at high-level compared with that at low level. Number of lapping-compound-supply has little effect on the response even at (Y = 0.1. But it has significant interaction with lapping pressure and, so, should be treated as an important control variable. That is, the effect of lapping pressure which is most significant can be affected by the level of number of lapping-compound-supply. It can be predicted, from the experimental results, that if the lapping compound is supplied continuously the efficiency of lapping will become much higher. Lapping speed has no effect and the efficiency at low speed is a little higher than that at high-level. This is because the total lapping distances were fixed at 20000 revolutions and, thus, lapping time became longer at a low-level of lapping speed. 4. CONCLUSION
The cylindrical lapping processes of SM45C with a manufactured lapping apparatus have been characterized qualitatively by analyzing the effects of four control variables, namely: abrasive grain size; lapping pressure; number of lapping-compound-supply; and lapping speed, on the percentage improvement of surface roughness as a measure of efficiency using the experimental design method and the following results were obtained. (1) Lapping pressure has a significant effect on the efficiency of cylindrical lapping. (2) Number of lapping-compound-supply should be treated as an important variable even though it has shown no effect on the efficiency because it interacts with lapping pressure. (3) Optimal variable combination on lapping efficiency has proven to be low-levels for abrasive grain size (coarse) and lapping speed, and high-levels for lapping pressure and number of lapping-compound-supply. This optimal combination has been confirmed by a confirmatory experiment resulting in a 7.8% increase in the SIN ratio of efficiency. REFERENCES [l] E. SAIJE and R. PAULMANN, Relations between abrasive processes, Ann. CIRP 37, 641-648 (1988). [2] F. YATES,Design and analysis of factorial experiments, Technica-I Comm No. 35. Imperial Bureau of Soil Sciences, London (1937). [3] L. S. NELSON,Analysis of two-level factorial experiments, J. Qual. Technol. 14, 95-98 (1982). [4] C. DANIEL,Applications of Statistics to industrial Experimentation. John Wiley, New York (1976). [S] G. TAGUCHIand S. KONISHI, Taguchi Methods, Orthogonal Arrays and Linear Graphs. American Supplier Institute, Dearborn, Michigan (1987). [6] R. V. LENTH,Quick and easy analysis of unreplicated factorial, Technometrics 31, 469-473 (1989). [7] K. E. BULLINGTON, J. N. Hoot_ and S. MAGHSOODLOO, A simple method for obtaining resolution IV designs for use with Taguchi’s orthogonal arrays, Qual. Technof. 22, 260-264 (1990). [8] T. P. TURIEL, A computer program to determine defining contrasts and factor combinations for twolevel fractional factorial designs of resolutions III, IV, and V, J. Qual. Technol. 20, 272-287 (1988). [9] T. B. BARKER, Quality by Experimental Design. Marcel Dekker, New York (1985). [lo] K. L. D’ENTREMONT, Design for Latitude, pp. 20-28. University of Missouri-Columbia, MO (1986).
Stochastic Approach to Experimental
Analysis of Cylindrical Lapping
[ll] S. V. CROWDER, K. L. JENSEN, W. R. STEPHENSONand S. B. VARDEMAN, An interactive the analysis of data from two-level factorial experiments via probability plotting, J. Qual. 140-148 (1988). [ 121 R. L.SCHEAFFERand J. T. MCCLAVE, Probability and Sratisrics for Engineers, 3rd edition, PWS-Kent, Boston (1990). [13] W. RSTEPHENSON, A computer program for the quick and easy analysis of unreplicated Quaf. Technol. 23, 63-67 (1991).
59 program for Technol. 20, pp. 483-543. factorials, J.
Pergamon
OWO-6955(94)EOOO5-4
STOCHASTIC APPROACH TO EXPERIMENTAL ANALYSIS OF CYLINDRICAL LAPPING PROCESS JEONG-DU KIMt and MIN-SEOGCHOIR (Received 25 May 1993; in final form 21 December
1993)
Abstract-Lapping is a very complicated and random process resulting from the variation of abrasive grains by its sizes and shapes and from the numerous variables which have an effect on the process quality. Thus it needs to be analyzed by experiment rather than by theory to obtain the relative effects of variables quantitatively. In this study, the cylindrical lapping experiment designed by Taguchi’s L, orthogonal array was performed and analyzed by Yates’ ANOVA table. As a result, effective variables and interaction effects were identified and discussed. Also the optimal variable combination to obtain the largest percentage improvement of surface roughness was selected and confirmatory experiments were performed.
1.
INTRODUCTION
A finish method used to obtain good surface quality. Important variables affecting lapping efficiency are abrasive grain size, lapping pressure, lapping speed, quantity of lapping compound supplied and viscosity of the compound, etc. Comparison of the effects of variables on the overall process efficiency is not yet clear owing to the complexity and randomness of the process. A generalized and simplified model of the lapping process is shown in Fig. 1, in which two motions of abrasive grains in the lapping process [l] are represented. It is assumed that grain 1 rolls and indents into the workpiece while grain 2 slides on it. The former does not produce any chips by cutting but causes the workpiece to deform plastically which may result in strainhardening and, finally, the microfracture of the workpiece. On the other hand, the latter produces microchips by the cutting operation. In addition, some small abrasive grains are driven into the workpiece by the relatively large grains. The quantitative ratio between rolling grains and sliding grains affects the lapping efficiency and it is very difficult to predict that ratio by analytical approach because it is affected by numerous environmental and process variables which are irregularly varied. It is, thus, recommended to use the experimental approach rather than only the theoretical approach to analyze the lapping process. Using the experimental design method is a typical approach to efficiently and logically identify characteristics of any complex process by experiment. In this method, experiments are generally designed by orthogonal array and the results are analyzed by ANOVA (analysis of variance).
LAPPING IS
Grain
1
Grain
2
Workpiece FIG. 1. Model of a lapping process.
tDepartment of Precision Engineering & Mechatronics, nology, Taejon, 305-701, Korea. 51
Korea Advanced Institute of Science and Tech-
52
J.-D.
KIM and M.-S.
CHOI
A number of studies related to the experimental design method have been reported in the field of quality control and statistics. Among them, Yates [2] suggested a computing method of sum-of-squares (SS) for ANOVA in 2F factorial designs. Nelson [3] developed a computer program to construct an ANOVA table and to calculate mean effects of control variables using Yates’ algorithm and backward Yates’ algorithm presented by Daniel [4]. Taguchi [5] and Konishi [5] presented a table of OAs which are directly applicable to industry and suggested a product design method using signalto-noise ratio (SIN ratio), quality-loss-function and orthogonal array. Lenth [6] suggested a evaluation method for the effectiveness of control variables using ME (margin of error) and SME (simultaneous margin of error) which are computed from Yates’ algorithm. In this study, experiments designed by Taguchi’s OA were performed. And Taguchi’s SN ratio for percentage improvement of surface roughness in a cylindrical lapping process was used as a performance (or response) variable to evaluate the process efficiency. Yates’ algorithm and the Lenth algorithm were used to analyze the effects of variables on the performance variable. It is aimed, in this study, to characterize a cylindrical lapping process efficiently by determining the important control variables which have a large effect on the percentage improvement of surface roughness, and by finding out the optimal variable combinations which provide information about the direction of later experiments to obtain an optimal lapping condition. Some confirmatory experiments are conducted. 2. EXPERIMENT
2.1. Experimental design Two-level fractional factorial design, which is used in this study is especially efficient in finding out important variables having an effect on the process performance [7]. It is a very useful tool for characterization or qualitative analysis prior to quantitative experiments of the lapping process. The efficiency of an experiment, especially in the case of lapping, can be increased by using the experimental design method as a tool for screening unimportant (or trivial) variables and, thus, reducing time, labor and other resources. Experiments at two levels of each control variable were conducted: grain size, lapping pressure, number of lapping-compound-supply and lapping speed. Control variables and their levels are shown in Table 1. Percentage improvement of surface roughness before and after lapping was taken as an evaluation or response variable. Taguchi’s fold-over type L, OA [7] is shown in Table 2. This is a resolution IV 24-’ fractional factorial design [7, 81 and all the main effects are not connected with two-factor interaction effects but are with three-factor interaction effects. Two-factor interaction effects are connected with each other: AB with CD, AC with BD and BC with AC. Therefore, if any two-factor interaction effect has been proven to be significant then other two-factor interaction effects which are connected with it should also be tested. Generally, three (or more)-factor interactions rarely occur [9]. So, in this study, only four main effects (i.e. A, B, C, D) and three two-factor interaction effects (i.e. AB, AC and BC) were considered in the experimental design. All the other effects were considered to be trivial. The first column in Table 2 represents the standard order TABLE 1. LEVELS OF
FACTORS
Levels (+) Low (-)
Symbol
Factor
High
A
Grain size Lapping pressure Number of supply Lapping speed
#5OQ 0.3 MPa 6 100 mimin
B c D
of lapping
compound
#lOO 0.15 MPa 3 50 mlmin
Stochastic Approach to Experimental TABLE
2.
TAGUCHI
Analysis of Cylindrical Lapping
L, ORTHOGONAL
ARRAY (FOLD-OVER TYPE)
1
2
Column No and contrast 3 5
A
B
AB
tc
53
t
AC
6 BC
7 D
_
-
-
+
-
+ +
_ + +
+
a b ab
+
_
_ + -
+ _
+ + _
C
_
_
+
+
-
_
+
ac bc abc
+ +
+ +
+
+ + +
+ +
+ +
+
(1)
+
[3, 91 of experimental treatment combinations (tc). The number of combinations is eight, which is the smallest number required to satisfy resolution IV design with four variables. Experiments were performed in a random order so as not to be affected by the order of experiments and were replicated three times for each combination. The other columns represent 23-1 (=7) contrasts of which the order is the same as that of Taguchi and Konishi [5]: four main effects are assigned to the first, second, fourth and seventh columns and interaction effects to the third, fifth and sixth columns. In the table, “+” represents high-level and “-“, low-level. Though the levels in the experiment are represented as numerical quantitatives, they can also be considered to have qualitative meaning, “high” or “low”. 2.2. Experiment Experiments were performed by a manufactured lapping unit which was set on the lathe. The schematic experimental set-up is shown in Fig. 2. SM45C of +50 was used as a workpiece and was revolved by the chuck of the lathe. The lapping wheel contacts with the workpiece theoretically at a point, but actually over a small area, and oscillates in the axial direction. All of them have an initial surface roughness of R, 2.1 Pm-R, 2.7 km. The lapping compound which had been made by mixing WC-grains and grease was used. Four variables were considered as control variables: grain size (A); lapping pressure (B); number of lapping-compound-supply (C); and lapping speed (D). Experimental levels for each control variable are as follows: #lOO and #500 for A (abrasive grain size), 0.15 MPa and 0.3 MPa for B (supplied pressure to air cylinder of lapping unit), 3 and 6 for C (supply number of lapping compound), 50 m/min and 100 m/min for D (lapping speed). Total lapping distance was fixed at 20,000 revolutions for each combination. These are shown in Table 1. Surface roughness was measured at 30 points before and after lapping using Surftestof Mitutoyo and averaged.
I
I
I
L Workpiece
FIG. 2. Experimental
set-up.
Lenre
J.-D. KIM and M.-S. CHOI
54
2.3. Experimental
results
Experimental results are shown in Table 3, in which yi s (i = 1, 2, 3) are percentage improvements of surface roughness and v is the averaged value of the three. s* is variance and S/N is Taguchi’s performance or response statistic. A merit of using logtype response variables such as this in analysis of variance is to reduce the inequality of variance between treatments and, thus, to satisfy one of the basic assumptions, i.e. equal variance. In this study, we are interested in the efficiency or percentage improvement of surface roughness, so S/N ratios are calculated for the case of “Bigger Is Better” (i.e. BIB case) and the equations used are as follows:
yi =
AR, (improvement of surface roughness) x IO0 R, (surface roughness before lapping)
(AR, = R, of before lapping - R, of after lapping)
(1)
YJ3
Y = (Yl +y*+
(2)
S* = 2 (vi - y)*/(n-1) SIN= -1Olog
(
;&
(3)
,I
.
(4)
In the other two cases, SIB (smaller is better) and NIB (nominal is better), equations (5) and (6) are used, respectively [lo]. SIB case: S/N = - 10 log
(5)
NIB case: S/N = 10 log
. 3. ANALYSIS
AND DISCUSSION
3.1. Analysis of experimental results Table 4 shows the method of Yates’ computing algorithm [8] used to obtain mean effect of each variable and this is the result of Nelson’s BASIC program [9]. Column I is generated using the data (SIN ratio) column by the rule that the first four entries are created by adding adjacent pair-wise sets of data from the response variable column, and the other four entries by subtracting adjacent pair-wise sets. Columns II and III are generated by applying the same rule, column II from column I and column III from column II. The sum of squares (SS) and mean effects for each treatment are calculated from equations (7) and (8), respectively. TABLE 3. EXPERIMENTAL RESULT
tc
(1) t ab C E abc
R. improvement (%) Y3 Yl Y2
Y (%)
SIN (db)
27.15 17.10 30.09 23.81 23.98 17.53 40.86 34.79
23.96 16.60 26.21 24.78 20.23 17.91 38.51 32.51
27.4391 24.3361 28.2220 27.8715 25.8245 25.0086 31.6883 30.2091
22.44 17.69 23.29 25.41 16.57 16.71 37.38 31.66
21.69 15.01 25.24 25.12 20.14 19.49 31.29 31.08
Stochastic
Approach
to Experimental
Analysis
of Cylindrical
55
Lapping
TABLE 4. COMPUTING PROCEDUREOF YATES’ ALGORITHM
tc
(1) a b ab C
ac bc abc
Data (SIN-ratio)
I
Sum of effects II
III
27.4391 24.3361 28.2220 27.8715 25.8245 25.0086 31.6883 30.2091
51.7752 56.0935 50.8331 61.8974 -3.1030 -0.3505 -0.8159 - 1.4792
107.8687 112.7305 -3.4535 -2.2951 4.3183 11.0643 2.7525 -0.6633
220.5592 -5.7486 15.3826 2.0892 4.8618 1.1584 6.7460 -3.4158
ss
Mean effect
6083.0 4.131 29.578 0.546 2.955 0.168 5.689 1.459
27.575 - 1.438 3.846 0.522 1.216 0.290 1.686 -0.854
SS = (III)*/8
(7)
Mean effect = (111)/4, for treatments (111)/S, for overall average.
(8)
Mean effects can be also computed by subtracting the mean value of four experimental data at low-level from those at high-level. For variable A, for example, high-level appears in rows 2, 4, 6, 8 and low-level in 1, 3, 5 and 7 rows in Table 2. So the mean effect for A can be simply computed from Table 3 as follows which is same as that in Table 4. (24.3361 + 27.8715 + 25.0086 + 30.2091)/4 - (27.4391 + 28.2220 + 25.8245 + 31.6883)/4 = - 1.438. The table of ANOVA was constructed using sum-of-square (SS) and mean effects as shown in Table 5. The error term in Table 4 was generated by the 5% pooling rule: AB, AC and D effects which are less than 5% of total sum-of-square are pooled into error. In Table 5, only the B (lapping pressure) effect has proven to be significant at a = 0.01 and B, A and BC-interaction effects are significant at (Y = 0.1. It is noted that though the C (number of lapping-compound-supply) effect is not significant by itself, it should be considered as an important variable in cylindrical lapping because the BC-interaction effect, which is shown in Fig. 3, is significant. Mean values of the SIN ratio for the mean effect of four variables are shown in Fig. 4, which indicates the optimal combination (not same as optimal lapping condition) required to obtain a larger percentage improvement of surface roughness, that is, A-B’C’D-: abrasive grain size of #lOO (coarse); lapping pressure of 0.3 MPa; number of lapping-compound-supply of six times and lapping speed of 50 m/min. It can, thus, be predicted that levels of variables A and D should be taken to minus direction (decreasing the level) and those of B and C to plus direction (increasing the level) to TABLE 5. ANOVA
tc
(1) it ab C
ac bc abc Error Total
ss
% SST
6083.0 4.131 29.578 0.546 2.955 0.168 5.689 1.459
9.28 66.43 1.23 6.64 0.38 12.78 3.28
2.172 44.524
100.00
Mean
TABLE
effect
27.575 -1.438 3.846 0.522 1.216 0.290 1.686 -0.854
YATES ALGORITHM
BY
Measures
df
MS
F
1 1 1 1 1 1 1
4.131 29.578 0.546 2.955 0.168 5.689 1.459
5.706 40.857 4.081 7.858 _
average
A B
AB c
AC BC D
7
56
KIM and M.-S.
J.-D.
CHOI
29.85
27.7 I
25.55
1 23.4
’ B-
B+
FIG. 3. Plot of BC interaction.
27.5-
25.751
24
’ A- A+
B-B+
C-C+
D-D+
Level FIG. 4. Response
graph for SIN value
obtain the maximum percentage improvement of surface roughness. Figure 5 represents each effect analyzed by Lenth’s method using Stephenson’s program [13] for the same experimental results. It shows the same trend as mentioned above.
-16
-13
-10
-7
-3
0
3
‘7
10
13
16
**
A
:
B
:
******
AR
:
* **
c AC
***
EC D
3
: SME
ME
ME
ME
margin
of error margin
SME: simultaneous FIG. 5. Plot of effects
by Lenth’s
SME
of error
method.
Stochastic Approach to Experimental Analysis of Cylindrical Lapping
57
TABLE 6, CONFIRMATORY EXPERIMENTAL RESULT FOR B C INTERACTION
B
tc
(1) b c bc
+ +
C
+ +
R. improvement ( % )
yl
y2
y3
y
S/N (dB)
20.89 22.35 18.09 41.66
18.90 27.71 19.16 38.92
25.81 24.39 21.33 43.58
21.87 24.82 19.53 41.39
26.5810 27.7943 25.7524 32.3090
3.2. Confirmatory experiment 3.2.1. BC-interaction effect. Though the BC-interaction effect in the previous section has appeared to be significant, it must to be tested by confirmatory experiments because of the possibility of connecting with the AD-interaction effect, which is a characteristic of resolution IV designs. The results of the confirmatory experiment are shown in Table 6. Mean effects of the two variables are not significantly different compared with those in Table 4 and the relative magnitude of the effects shows a similar trend as in Table 4 and Fig. 3. Therefore, it can be concluded that the BCinteraction effect, rather than the AD-interaction effect, is actually significant. 3.2.2. Optimal combination. It has been predicted that the optimal combination for increasing the percentage improvement of surface roughness is A-B+C+D -, which informs us of the direction of change of the variable level to take in the confirmatory experiment or in next stage of experiment for optimization. According to the optimal combination, the levels for A and D should be decreased and the levels for B and C should be increased to maximize the efficiency. In practice, it was attempted at new levels of the four variables: grain size of #80; lapping pressure of 0.4 MPa; number of lapping-compound-supply of eight times, lapping speed of 30 m/min instead of #100, 0.3 MPa, six times and 50 m/min, respectively. A confirmatory experiment was conducted three times and the surface roughness (average value of 30 points for each workpiece) has been improved from 2.31, 2.50 and 2.26 t~m to 1.188, 1.05 and 1.194 ~m, respectively. Percentage improvements of this are 48.59, 58.02 and 47.16. The average of these three is 51.59 (%) and the S/N ratio is 34.152. These are improved values by 34% for the average and 7.8% for the S/N ratio compared with previous experimental results, which are enough to confirm the optimal combination. 3.3. Linear regression model Three effects, A, B and BC which have proven to be significant, can be the terms of the linear regression model. Each coded variable has - 1 and +1 at its low- and high-level, respectively, in the model. In Fig. 6, the difference in response value 30
t3 ~3
28.75-
>
z
27.5-
0"3
A On
o
26.25-
>
<
25 -i
0
+1
FIG. 6. Relative effect of main contrasts on the change of response. A: Grain size; B: Lapping pressure; BC: interaction between pressure and the number of supply of lapping compound.
J.-D. KIM and M.-S. CHOI
58
between the two levels are mean effects and their half values are the slopes of lines for each effect. A linear model for the SIN ratio of the response variable is represented by the following equation. SINratio fory (db) = 27.575 - 0.719 * A + 1.923 * B + 0.943 * BC. Here, the constant, 27.575, is the overall mean of the response. The mean S/N ratio of response (or magnitude of main effect shown in Fig. 4) for any variable, x1 or x2, in the equation can be obtained by assigning -1 and +l at low- and high-levels, respectively. 3.4. Discussion The efficiency of cylindrical lapping, and the percentage improvement of surface roughness, was increased at coarse grain size. Lapping pressure shows the largest effect of all variables considered in this study and the efficiency of lapping increased dramatically at high-level compared with that at low level. Number of lapping-compound-supply has little effect on the response even at (Y = 0.1. But it has significant interaction with lapping pressure and, so, should be treated as an important control variable. That is, the effect of lapping pressure which is most significant can be affected by the level of number of lapping-compound-supply. It can be predicted, from the experimental results, that if the lapping compound is supplied continuously the efficiency of lapping will become much higher. Lapping speed has no effect and the efficiency at low speed is a little higher than that at high-level. This is because the total lapping distances were fixed at 20000 revolutions and, thus, lapping time became longer at a low-level of lapping speed. 4. CONCLUSION
The cylindrical lapping processes of SM45C with a manufactured lapping apparatus have been characterized qualitatively by analyzing the effects of four control variables, namely: abrasive grain size; lapping pressure; number of lapping-compound-supply; and lapping speed, on the percentage improvement of surface roughness as a measure of efficiency using the experimental design method and the following results were obtained. (1) Lapping pressure has a significant effect on the efficiency of cylindrical lapping. (2) Number of lapping-compound-supply should be treated as an important variable even though it has shown no effect on the efficiency because it interacts with lapping pressure. (3) Optimal variable combination on lapping efficiency has proven to be low-levels for abrasive grain size (coarse) and lapping speed, and high-levels for lapping pressure and number of lapping-compound-supply. This optimal combination has been confirmed by a confirmatory experiment resulting in a 7.8% increase in the SIN ratio of efficiency. REFERENCES [l] E. SAIJE and R. PAULMANN, Relations between abrasive processes, Ann. CIRP 37, 641-648 (1988). [2] F. YATES,Design and analysis of factorial experiments, Technica-I Comm No. 35. Imperial Bureau of Soil Sciences, London (1937). [3] L. S. NELSON,Analysis of two-level factorial experiments, J. Qual. Technol. 14, 95-98 (1982). [4] C. DANIEL,Applications of Statistics to industrial Experimentation. John Wiley, New York (1976). [S] G. TAGUCHIand S. KONISHI, Taguchi Methods, Orthogonal Arrays and Linear Graphs. American Supplier Institute, Dearborn, Michigan (1987). [6] R. V. LENTH,Quick and easy analysis of unreplicated factorial, Technometrics 31, 469-473 (1989). [7] K. E. BULLINGTON, J. N. Hoot_ and S. MAGHSOODLOO, A simple method for obtaining resolution IV designs for use with Taguchi’s orthogonal arrays, Qual. Technof. 22, 260-264 (1990). [8] T. P. TURIEL, A computer program to determine defining contrasts and factor combinations for twolevel fractional factorial designs of resolutions III, IV, and V, J. Qual. Technol. 20, 272-287 (1988). [9] T. B. BARKER, Quality by Experimental Design. Marcel Dekker, New York (1985). [lo] K. L. D’ENTREMONT, Design for Latitude, pp. 20-28. University of Missouri-Columbia, MO (1986).
Stochastic Approach to Experimental
Analysis of Cylindrical Lapping
[ll] S. V. CROWDER, K. L. JENSEN, W. R. STEPHENSONand S. B. VARDEMAN, An interactive the analysis of data from two-level factorial experiments via probability plotting, J. Qual. 140-148 (1988). [ 121 R. L.SCHEAFFERand J. T. MCCLAVE, Probability and Sratisrics for Engineers, 3rd edition, PWS-Kent, Boston (1990). [13] W. RSTEPHENSON, A computer program for the quick and easy analysis of unreplicated Quaf. Technol. 23, 63-67 (1991).
59 program for Technol. 20, pp. 483-543. factorials, J.