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Research Article
 

Fourth Order Torque Prediction Model in End Milling



K. Kadirgama, M.M. Noor, M.M. Rahman, Rosli A. Bakar and Abou-El Hossein
 
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ABSTRACT

This study presents the development of mathematical models for torque in end-milling of AISI 618. Response Surface Method (RSM) was used to predict the effect of torque in the end-milling. The relationship between the manufacturing process factors including the cutting speed, feed rate, axial depth and radial depth with the torque can be developed. The effect of the factors can be investigated from the equation developed for first order to fourth order model. The acquired results show that the torque increases with decreases of the cutting speed and increases the feed rate, axial depth and radial depth. It found that the second order is more accurate based on the analysis of variance (ANOVA) and the predicted torque results is closely match with the experimental results. Third- and fourth-order model generated for the response to investigate the 3 and 4-way interaction between the factors. It’s found less significant for the variables.

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  How to cite this article:

K. Kadirgama, M.M. Noor, M.M. Rahman, Rosli A. Bakar and Abou-El Hossein, 2009. Fourth Order Torque Prediction Model in End Milling. Journal of Applied Sciences, 9: 2431-2437.

DOI: 10.3923/jas.2009.2431.2437

URL: https://scialert.net/abstract/?doi=jas.2009.2431.2437
 

INTRODUCTION

Manufacturing cost is always very important issue either manufacturing assembly or machining process. Optimizations are one of the methods to minimize the operations hours and reduce the process time itself. Response Surface Method (RSM) is a collection of statistical and mathematical methods that are useful for the modelling and optimization of the engineering problems. In this technique, the main objective is to optimize the responses that are influencing by various parameters (Montgomery, 1984). RSM also quantifies the relationship between the controllable parameters and the obtained response. In modelling of the manufacturing processes using RSM, the sufficient data is collected through designed experimentation. In general, a second order regression model is developed because of first order models often give lack off fit. The study uses the Box-Behnken design in the optimization of experiments using RSM to understand the effect of important parameters. Box-Behnken design is normally used when performing non-sequential experiments. That is, performing the experiment only once. These designs allow efficient estimation of the first and second-order coefficient. Because Box-Behnken design has fewer design points, they are less expensive to run than central composite designs with the same number of factors. The RSM is practical, economical and relatively easy for use and it was used by lot of researchers for modeling machining processes (El-Baradie, 1993; Hasegawa et al., 1976; Sundaram and Lambert, 1981). Mead and Pike (1975) and Hill and Hunter (1966) reviewed the earliest study on response surface methodology. Response surface methodology is a combination of experimental and regression analysis and statistical inferences. The concept of a response surface involves a dependent variable y called the response variable and several independent variables x1,x2,. . .,xk (Hicks and Turner, 1999). Consequentially, the RSM is utilized to describe and identify, with a great accuracy, the influence of the interactions of different independent variables on the response when they are varied simultaneously. In addition, it is one of the most widely used methods to solve the optimization problem in the manufacturing environment as studied by Grum and Slab (2004), Puri and Bhattacharyya (2003) and Kansal et al. (2005). Therefore, the quadratic model of RSM associated with the Sequential Approximation Optimization (SAO) method was used in an optimal setting of machining parameters. The main aim of the study is to develop the first and second order model by using response surface methodology. From this model, the relationship between the factors and the response can be investigated. The objective of this study is to develop the mathematical model and predicted the torque in end-milling. Yalcinkaya and Bayhana (2009) presents a modelling and solution approach based on discrete-event simulation and response surface methodology for dealing with average passenger travel time optimization problem inherent to the metro planning process. The objective is to find the headways optimizing passenger average travel time with a satisfactory rate of carriage fullness. Armarego and Wright (1984) developed a model which can be used to estimate thrust and torque for three different drill flank configurations. This model uses the findings of their analysis of cutting mechanisms and fundamental machining data such as shear stress and chip length. Design of experiments is a powerful analysis tool for modeling and analyzing the influence of process variables over some specific variable, which is an unknown function of these process variables (Paulo, 2001). The most important stage in the design of experiment lies in the selection of the control factors. As many as possible should be included, so that it would be possible to identify non-significant variables at the earliest opportunity (Nian et al., 1999). In general, the thrust and torque parameters will mainly depend on the manufacturing conditions employed, such as: feed, cutting speed, tool geometry, machine tool and cutting tool rigidity, etc.

Horng et al. (2008) have attempt to model the machinability evaluation through the Response Surface Methodology (RSM) in machining Hadfield steel. The combined effects of four machining parameters, including cutting speed, feed rate, depth of cut and tool corner radius, on the basis of two performance characteristics-flank wear (VBmax) and surface roughness (Ra), were investigated and the centered Central Composite Design (CCD) and the analysis of variance (ANOVA) were employed. The quadratic model of RSM associated with the Sequential Approximation Optimization (SAO) method was used to find optimum values of machining parameters.

MATHEMATICAL MODELING

The proposed relationship between the responses (torque and torque) and machining independent variables can be represented by the following:

Image for - Fourth Order Torque Prediction Model in End Milling
(1)

where, τ is the torque in Nm, V, F, Ax and Ar are the cutting speed (m sec-1), feed rate (mm rev-1), axial depth (mm) and radial depth (mm). C, m, n, y and z are the constants.

Equation 1 can be written in the following logarithmic form as in Eq. 2:

Image for - Fourth Order Torque Prediction Model in End Milling
(2)

Equation 2 can be written as a linear form:

Image for - Fourth Order Torque Prediction Model in End Milling
(3)

where, y = lnτ is the torque, x0 = 1 (dummy variables), x1= lnV, x2 = lnF, x3 = lnAx , x4 = lnAr and ε = ln ε’, where, ε is assumed to be normally-distributed uncorrelated random error with zero mean and constant variance, β0 = lnC and β1, β2, β3 and β4 are the model parameters.

The second model can be expressed as:

Image for - Fourth Order Torque Prediction Model in End Milling
(4)

The values of β1, β2, β3 and β4 are to be estimated using the method of least squares. The basic formula can be expressed as in Eq. 5:

Image for - Fourth Order Torque Prediction Model in End Milling
(5)

where, xT is the transpose of the matrix x and (xTx)-1 is the inverse of the matrix (xTx) and y is the value from experiment.

The details solution of this matrix approach is explained by Montgomery (1984). The parameters have been estimated by the method of least-square using a Matlab computer codes.

ENGINEERING DESIGN

To develop the first-order, a design consisting 27 experiments were conducted. Box-Behnken design method is normally used when performing the non-sequential experiments. These designs allow the efficient estimation of the first and second-order coefficients because of the Box-Behnken design has fewer design points; they are less expensive to run than central composite designs with the same number of factors. Box-Behnken design no axial points, thus all design points fall within the safe operating. Box-Behnken design also ensures that all factors are never set at their high levels simultaneously (Box and Draper, 1986; Khuri and Cornell, 1987). Preliminary tests were carried out to determine the suitable cutting speed, federate, axial and radial depth of cutting as shown in Table 1.

Table 1: Levels of independent variables
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The AISI 618 stainless steel workpieces were provided in fully annealed condition in sizes of 65x170 mm. The tools used in this study are carbide inserts PVD coated with one layer of TiN. The inserts are manufactured by Kennametal with ISO designation of KC 735 M. They are specially developed for milling applications where stainless steel is the major machined material. The end-milling tests were conducted on Okuma CNC machining centre MX-45VA. Every one passes (one pass is equal to 85 mm), the cutting test was stopped. The same experiment has been repeated for 3 times to get more accurate results.

RESULTS AND DISCUSSION

First-order torque model: The first order torque model can be expressed as in Eq. 6:

Image for - Fourth Order Torque Prediction Model in End Milling
(6)

The transforming equations for each of the independent variables are:

Image for - Fourth Order Torque Prediction Model in End Milling
(7)

The torque model can be expressed as Eq. 8:

Image for - Fourth Order Torque Prediction Model in End Milling
(8)

Table 2 shows the 95% confidence interval for the experiments and analysis of variance. For the linear model, the p-value for lack of fit is 0.196 and the F-statistics is 5.1033. Therefore, the model is adequate. The experimental and predicted torque results for the first order model are given in Table 3.

Equation 8 shows that the torque increases with decreases of the cutting speed while increases of the feed rate, axial and radial depth of cut. It also indicates that the feed rate has the most significant effect on the torque, follow by radial and axial depth of cut and cutting speed. Equation 8 is utilized to develop torque contour at selected cutting speed and feed rate. Figure 1a-c show that the torque contours in the axial-radial depth plane for different cutting speed and feed rate.

Table 2: Analysis of variance (ANOVA) for first-order torque model with 95% confidence interval
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Table 3: The predicted result for first order torque model
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Image for - Fourth Order Torque Prediction Model in End Milling
Fig. 1:
Torque contours in the axial-radial depth plane for different cutting speed and feed rate; (a) Vc = 100 m min-1 and f = 0.1 mm rev-1; (b) Vc = 140 m min-1 and f = 0.15 mm rev-1 and (c) Vc = 180 m min-1 and f = 0.2 mm rev-1

It is helpful to predict the torque at any experimental zone. It is clearly shown that the cutting speed, feed rate, axial depth of cut, radial depth of cut and feed rate are strongly related with the torque in end-milling. It can be seen that the increases of torque with increases of cutting speed and feed rate. The torque obtained the highest value about 25 N at cutting speed 180 m min-1.

Second-order torque model: The second-order model was postulated in obtaining the relationship between the cutting force and the machine independent variables. The second order model equation can be expressed as in Eq. 9:

Image for - Fourth Order Torque Prediction Model in End Milling
(9)

Table 4 is given the 95% confidence interval for the experiments and analysis of variance. For the second-order model, the p-value for lack of fit is 0.221 and the F-statistics is 4.5249. Therefore, the model is adequate. The second-order model is more adequate because of the predicted result is much more accurate than the first model. The p-value higher than the first order predicted value. The predicted torque results for second order model are given Table 5.

Third-order torque model: The third-order model as shown below was use is obtained to investigate the 3-way interaction between the variables:

Image for - Fourth Order Torque Prediction Model in End Milling
(10)

Table 4: Analysis of variance (ANOVA) for second-order model with 95% confidence interval
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Table 5: The predicted result for second order torque model
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From this model the most important points are the main effects, 2-way interaction and 3-way interaction. The third order torque model can be presented as in Eq. 11:

Image for - Fourth Order Torque Prediction Model in End Milling
(11)

The third-order model parameters can be solved using least-squares method. β’s are the model parameters, x1 = cutting speed, x2 = feedrate, x3 = axial depth and x4 = radial depth. The third order model for torque can be rewrite in Eq. 12:

Image for - Fourth Order Torque Prediction Model in End Milling
(12)

The model adequate and significant of 3-way interaction can be seen form Table 6. From the ANOVA both model not significant to the 3-way interaction since the p>0.05. The third-order model adequate for torque since the p-value for lack of fit for torque is 0.818 and the F-statistics is 0.52. It indicates that this model is not suitable as much as second-order torque model.

Fourth-order torque model: The fourth-order model as shown below is obtained to investigate the 4-way interaction between the variables.

Image for - Fourth Order Torque Prediction Model in End Milling
(13)

From this model the most important points are the main effects, 2-way, 3-way and 4-way interactions. So, the fourth order model can be reduced as in Eq. 14:

Image for - Fourth Order Torque Prediction Model in End Milling
(14)

Table 6: Analysis of Variance (ANOVA) for third-order torque model
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Table 7: Analysis of Variance (ANOVA) for fourth-order torque model
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This model parameters can be solved using least squares method. β’s are the model parameters, x1 = cutting speed, x2 = feed rate, x3 = axial depth and x4 = radial depth. The fourth order torque model can be presented as in Eq. 15:

Image for - Fourth Order Torque Prediction Model in End Milling
(15)

The model adequate and significant of 4-way interaction for both model are also presented in Table 7. It can be seen that from the ANOVA analysis both model not significant to 4-way interaction since the p>0.05. The third-order model adequate for torque since the p-value for lack of fit for torque is 0.818 and the F-statistics is 0.52. It indicates that this model is not suitable as much as second-order torque model.

CONCLUSION

Reliable torque model have been developed and utilized to enhance the efficiency of the milling 618 stainless steel. The torque equation shows that feed rate, cutting speed, axial depth and radial depth plays the major role to produce the torque. The higher the feed rate, axial depth and radial depth, the torque generates very high compare with low value of feed rate, axial depth and radial depth. Contours of the torque outputs were constructed in planes containing two of the independent variables. These contours were further developed to select the proper combination of cutting speed, feed, axial depth and radial depth to produce the optimum torque. The third order model and fourth order model are very important to investigate the 3-way interaction and 4-way interaction. The third order model and fourth order model, shows that the 3-way interaction and 4-way interaction not significant.

ACKNOWLEDGMENTS

The authors would like to thanks the Universiti Tenaga National for provided laboratory facilities and Universiti Malaysia Pahang for providing financial support for provided research grant (03-99-03-0011-EA 0041).

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