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INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol 12 No 2 pp 177 182 APRIL 2011 177 DOI 10 1007 s12541 011 0025 8 1 Introduction With the recent rapid development of industry the need has increased for precision cutting of various kinds of machine parts In particular in the cutting industry it is important to enhance cutting efficiency and precision simultaneously 1 2 In the field of metal cutting machining error in milling drilling and external lathe turning has been studied much more than internal boring Boring means enlarging a hole that was already cut by drilling or casting to the designed dimension Therefore control of dimensional tolerance and surface roughness is important 3 Boring is similar to external lathe turning in the sense of using a single point cutting tool However the shape of the boring tool has to be restricted by the workpiece hole diameter and depth This is the difference between boring and external lathe turning Generally the overhang of the boring bar has to be short to guarantee machining stability Thomas et al 4 emphasize that because of the reduced damping ratio the short overhang of the boring tool is good for tool stiffness but poor for vibration Chun and Ko5 point out that the change in dynamic stiffness of the boring tool is decided by overhang and dynamic stiffness is increased nonlinearly with overhang length The sources of machining error are tool deflection and wear thermal effects and machine tool errors Tool deflection caused by cutting forces is a dominant factor in machining errors 6 The cutting force is separated into main thrust and feed cutting forces Among these the main and thrust cutting forces induce tool deflection whereas machining leads to machining error 7 With recent enhancements in technology the shapes of cutting tools and workpieces have become more complicated Therefore it is difficult to predict the cutting force and tool deflection precisely and the experience of field operators is inevitable The purpose of this paper was to identify the effect of overhang and cutting conditions on machining error quantitatively during internal lathe boring of AISI4140 which is generally used for machine elements To this end the response surface method RSM 8 9 was applied to establish an estimation model Similar to the study of Chun and Ko5 overhang feed per revolution and cutting depth were chosen as factors for the model The cutting speed which is the main factor of built up edge BUE and tool life was kept constant at 200 m min A central composite design was used for the purpose of minimizing the number of experiments Fitness was verified by analysis of variance ANOVA residual analysis and coefficient of determination after building the first and second regression model respectively 2 The Response Surface Method RSM is a collection of mathematical and statistical techniques Study on the Response Surface Model of Machining Error in Internal Lathe Boring Se Ho Chun1 and Tae Jo Ko2 1 Graduate School of Mechanical Engineering Yeungnam University 214 1 Dae dong Gyeonsan Gyeongbuk South Korea 712 749 2 School of Mechanical Engineering Yeungnam University 214 1 Dae dong Gyeonsan Gyeongbuk South Korea 712 749 Corresponding Author E mail tjko yu ac kr TEL 82 53 810 3836 FAX 82 53 810 4627 KEYWORDS Boring Bar Machining Error Response Surface Method Central Composite Design ANOVA Residual Analysis To achieve high quality and precision of machining products the machining error must be examined The machining error defined as the difference between designed surface and the actual tool is generally caused by tool deflection and wear thermal effects and machine tool errors Among these error sources tool deflection is usually known as the most significant factor The tool deflection problem is analyzed using the instantaneous cutting forces on the cutting edge This study presents a model of the machining error caused by tool deflection in the internal boring process The machining error prediction model was described by the surface response method using overhang feed per revolution and depth of cut as the factors for the analysis The least square method revealed that overhang and depth of cut were significant factors within 90 confidence intervals Analysis of variance ANOVA and residual analysis show that the second order model is adequate Manuscript received November 23 2009 Accepted November 24 2010 KSPE and Springer 2011 178 APRIL 2011 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol 12 No 2 useful for modeling and analyzing problems in which a response of interest is influenced by several variables and the objective is to optimize this response A response surface is a functional relation between response variable and factors RSM assumes a statistical model with respect to the response surface Then the response surface model is estimated with regression analysis of test data generated by several conditions composed from the design factors Generally it is difficult to know the response surface formula and therefore the approximated model is assumed first After that this model is verified by lack of fit In RSM the first and second order regression models are normally used The third order regression model can be but is seldom used 8 Central composite design is a representative experimental design of RSM To estimate the experimental surface with the minimum number of experiments central and axial points are added in the 2k experiments where k means the number of factors Therefore sequential experiments are possible here If the 2k factor experiments are lack of fit with the first order regression model the second order regression model do not need new experiments but need to adding new data points on the center and axes of the 2k experiments To analyze the first and second order regression models simultaneously in this study the experimental design including experimental points 2k axial points 2k and central points nc was selected Therefore the total number of experiments was 2k 2k nc 8 11 3 Machining Error Mechanism The cutting force induces deflection in the cutting tool and workpiece The cutting force is a dominant factor in analyzing machining error from the deflection of cutting tool and workpiece The cutting tool deflection is analyzed as a response to the instantaneous cutting force 12 In the case of the boring tool the cutting force model for analyzing a cutting tool deflection is simplified as the cantilever beam see Fig 1 The expression for the cutting tool deflection x at position x from the free end point is as follows 3 3 FLx x E I 1 where F is the cutting force L is tool overhang E and I are the elasticity modulus and moment of inertia of the tool The deflection of the boring tool is determined by the tool material diameter and overhang Obviously overhang changes according to the clamping position as shown in Fig 2 In this case the tool deflection is composed of deflections m by the main cutting force and t by the thrust cutting force Deflection by the main cutting force moves the cutting edge under the tool center line Therefore the radial rake angle becomes negative and consequently the relief angle decreases which induces large flank wear 7 In this paper the difference between tool diameter before machining designed surface D and workpiece diameter machined surface M is defined as machining error Fig 3 shows the simulation analysis of cutting force variation in Fig 1 Tool deflection model DM Error Fig 2 Deflection of the boring tool Fig 3 Cutting force analysis by AdvantEdge INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol 12 No 2 APRIL 2011 179 the internal boring using the commercial cutting analysis software of AdvantEdge ThirdWave Co The simulation was conducted with the cutting condition of 0 25 mm rev depth of cut of 1 mm and cutting velocity of 200 m min The main y direction cutting force is the largest while the thrust x direction cutting force is ranked second The feed z direction force is the smallest and it acts on the axial direction of the boring bar The axial stiffness of the boring bar is sufficient i e the influence of the feed force is negligible 4 Experiment 4 1 Experimental Design The boring bar and insert were S16R SCLCR 09 and CCMT 09T308 MT TT3500 from TaeguTec respectively The boring bar was clamped into the tool holder with a sleeve on the tool post Overhang was defined as the distance from the end of the insert to the front of the sleeve The workpiece outer and inner diameter were 80 and 40 mm respectively and the inner hole depth was 50 mm The mechanical properties of the workpiece material AISI4140 are specified in Table 1 Consistency of the experiment was kept by brand new tools and changing new specimen at each time The horizontal lathe was a DC 2 model DMC Co machine and the power on the main spindle was 5 5 kW The deflection of boring bar is affected by the cutting force The cutting force is dependent on tool workpiece contact area chip area which is composed of feed per revolution and depth of cut In this experiment overhang feed per revolution and depth of cut were selected as the design factors in the sense of the tool deflection Cutting velocity was set as constant A central composite design was selected to construct the response model with respect to characteristic values Uncoded variables were notated as 1 Overhang 2 Feed and 3 Cutting depth The experimental range was assumed to be 1L 1H 2L 2H and 3L 3H and the relations between real and coded variables were determined by Eqs 2 3 and 4 as follows The inverted data is shown in Table 2 9 11 1 11 2 4 HL X 2 22 2 22 2 4 HL X 3 33 3 33 2 4 HL X 4 where 11 1 2 LH 22 2 2 LH and 33 3 2 LH For all the cutting conditions the machining power was limited below 70 of the main spindle power To avoid the ploughing force13 16 that is mainly influenced by the size effect the minimum feed per revolution was set higher than 0 03 mm rev considering the size of the honing dimension at the insert edge Because the distance from the central to the axial point in experiment design was 1 2 all the cutting conditions were in the allowable range As shown in Fig 4 the total number of experiments was 18 based on experiment points 1 8 axial points 6 14 and central points 15 18 4 2 Experimental Results Table 3 shows the experimental results according to the designed cutting conditions As defined in Fig 2 machining error Table 1 Mechanical properties of AISI4140 Specification Value Yield strength kgf mm2 85 Tensile strength kgf mm2 100 Elongation 12 Reduction of area 45 Charpy impact value kgf m cm2 6 Hardness HB 285 352 Table 2 Levels of the variables in the experiment Coding 1 2 1 0 1 1 2 Overhang mm 30 4 32 40 48 49 6 Feed mm rev 0 03 0 050 15 0 25 0 27 Depth of cut mm 0 12 0 2 0 6 1 1 08 Fig 4 Central composite design for experiment14 Table 3 Design of experiment and results X1 2 X3 Error 1 1 1 1 0 105 2 1 1 1 0 281 3 1 1 1 0 144 4 1 1 1 0 289 5 1 1 1 0 162 6 1 1 1 0 275 7 1 1 1 0 153 8 1 1 1 0 304 9 1 2 0 0 0 192 10 1 2 0 0 0 336 11 0 1 2 0 0 190 12 0 1 2 0 0 189 13 0 0 1 2 0 133 14 0 0 1 2 0 287 15 0 0 0 0 184 16 0 0 0 0 180 17 0 0 0 0 191 18 0 0 0 0 187 180 APRIL 2011 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol 12 No 2 was defined as the difference between the virtual diameter generated by the air cutting state and the measured internal workpiece diameter after machining The measurement position for the internal diameter was 10 mm engaging distance from the start of machining which was selected as the representative data for analysis If the aspect ratio defined as diameter to length L D in the lathe process is less than 4 the machining accuracy is almost the same in all the machined areas because the workpiece is strong enough 15 In this experiment the aspect ratio was smaller than 2 so the variation in measurement data is negligible along with the other measuring places 5 Analysis of Results and Prediction Model 5 1 First Order Regression Model Equation 5 shows the first order regression model that is composed of newly defined independent variables by Eqs 2 3 and 4 The coefficients estimated by the least squares method are shown in Table 4 0112233 121213132323 YXXX X XX XX X 5 Considering the coefficients within 90 significance level 2 12 13 23 are factors that decrease the model s accuracy The effects of feed per revolution and its interaction terms are insignificant To improve the model estimation has to be done without insignificant factors Expression 6 shows the re estimated model after removing the insignificant factors such as X2 and its interaction effects 13 0 2101110 0696510 023879YXX 6 Table 5 shows the ANOVA for the estimated model and the coefficient of determination is 0 77 for the first order regression To verify the regression model we performed residual analysis The normal probability plot and residual histogram from this analysis are depicted in Figs 5 and 6 respectively As shown in Fig 5 the departures are scattered It indicates the abnormalities in the residual distribution Alternatively the residual histogram shows that frequency of the residual is not satisfied with the normal distribution and the frequency of residual is highest between 0 01 and 0 03 This means that the first order regression model is weak in explaining the machining errors 5 2 Second Order Regression Model The second order regression model is expressed as Eq 7 and the estimated coefficients by the least squares method are shown in Table 6 222 0112233111222333 121213132323 YXXXXXX X XX XX X 7 Here the influence of overhang is larger than the other factors to the machining errors Since its squared term is also significant the response surface will be curved with respect to the change in the factor level Similarly with the first order regression model the terms of feed per revolution and its interaction are insignificant within the 90 significance level Therefore those factors are pulled down to error terms and finally a new estimation model shown in Eq 8 was taken The coefficient of determination estimated from Eq 8 was 0 852 which was an improvement over the first order regression model 2 131 0 189790 069650 023880 03362YXXX 8 Table 7 shows the ANOVA result of the second order regression model and on the other hand Figs 7 and 8 show the results of Table 4 First order regression coefficient Coef Coef SE T P 0 0 210111 0 009199 22 840 0 000 1 0 069651 0 011832 5 887 0 000 2 0 006048 0 011832 0 511 0 619 3 0 023879 0 011832 2 018 0 069 12 0 000875 0 013799 0 063 0 951 13 0 007215 0 013799 0 516 0 616 23 0 003375 0 013799 0 245 0 811 Notes Coef SE Standard error of a coefficient T T test P Probability of type I error Table 5 ANOVA of first regression model DF Seq SS Adj MS F P Regression 2 0 0589 0 02949 25 05 0 000 Linear 2 0 0589 0 02949 25 05 0 000 Residual Error 15 0 0176 0 00117 Pure Error 9 0 0013 0 00014 Sum 17 0 0766 Notes DF Degree of freedom Seq SS Sequential sum of squares Adj MS Adjusted mean squares F Fisher statistic F test Fig 5 Normal probability plot of the residuals Fig 6 Histogram of the residuals INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol 12 No 2 APRIL 2011 181 normal probability and histogram of the residuals respectively As shown in Fig 7 the result is enhanced more than the first order regression model Also the frequency of the residual was satisfied with the normal distribution as shown in Fig 8 and the second order regression model was suitable for explaining the machining errors 5 3 Contour Plot and Surface Plot A contour plot expresses the response surface in the second dimensional plane On the other hand a surface plot expresses the response surface in the third dimensional space to explain the response values Figures 9 and 10 respectively show the contour and surface plots of the second order regression model All points on the contour plot were experimental points and the factor X2 was fixed to zero as a median The contours of X1 direction are denser than X3 direction Alternatively in Fig 10 the machining error rises sharply with increase of X1 6 Conclusions The purpose of this study was to build an estimation model for machining errors during internal boring of SCM440 materials The experiment was performed according to a central composite design with three factors that were believed to be parameters in machining errors RSM was adopted to estimate machining errors Alternatively through ANOVA and residual analysis the significance of factors and the fitness of the designed model were verified From the experimental results and model analysis the following conclusions were drawn 1 The second order regression model is more suitable than the first order regression model for describing the internal boring process from the view point of ANOVA and residual analysis In this case the second regression model s coefficient of determination was 0 852 2 Overhang and depth of cut were relatively more significant than feed per revolution in terms of machining errors Table 6 Second order regression coefficient Coef Coef SE T P 0 0 196336 0 01475 13 308 0 000 1 0 069651 0 01038 6 712 0 000 2 0 006048 0 01038 0 583 0 576 3 0 023879 0 01038 2 301 0 050 11 0 037342 0 01532 2 437 0 041 22 0 014394 0 01532 0 939 0 375 33 0 000158 0 01532 0 010 0 992 12 0 000875 0 01210 0 072 0 944 13 0 007125 0 01210 0 589 0 572 23 0 003375 0 01210 0 279 0 787 Table 7 ANOVA of second regression model DF Seq SS Adj MS F P Regression 3 0 0652 0 02176 26 82 0 000 Linear 2 0 0589 0 02949 36 35 0 000 Square 1 0 0062 0 00629 7 76 0 015 Residual Error 14 0 0113 0 00081 Pure Error 9 0 0013 0 00014 Sum 17 0 0766 Fig 7 Normal probability plot of the residuals Fig 8 Histogram of the residuals Fig 9 Contour plot of machining error Fig 10 Surface plot of machining error 182 APRIL 2011 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol 12 No 2 3 It is recommended tha