Modelling and analysis of cutting force and surface roughness in milling operation using TSK-type fuzzy rules

Arup Kumar Nandi

Central Mechanical Engineering Research Institute (CSIR-CMERI), Advance Design and Optimization Pin-713209, Durgapur, West Bengal, India. nandiarup@yahoo.com

The present paper discusses on development of fuzzy rule based models (FRBMs) for predicting cutting force and surface roughness in milling operation. The models use TakagiSugeno-Kang-type (TSK-type) fuzzy rule to study the effect of four (input) cutting parameters (cutting speed, feed rate, radial depth of cut and axial depth of cut) on outputs (cutting force and surface roughness). The appropriate FRBM is arrived after a thorough investigation of different structures of rule-consequent function. A combined approach of genetic algorithm and multiple linear regression method is used to determine the rule-consequent parameters. Performance analysis of models by comparing with experimental data implies its potential towards practical application. Analysis of the influence of various input parameters on different outputs is carried out based on FRBMs and experimental data. It suggests that the cutting force becomes higher with increasing feed rate, axial depth of cut and radial depth of cut and lower with increase in cutting speed, whereas surface finish is improved with increase in cutting speed and gets poorer with increase in radial depth of cut.

Keywords: fuzzy rule based model, TSK-type fuzzy rule, genetic linear regression, milling, surface roughness, cutting force

Introduction

For a long time, manufacturing engineers and researchers have been realizing that in order to optimize the economic performance of metal cutting operations, efficient quantitative and predictive models are important. These models establishing the relationship between independent (input) parameters and output variable(s), are required for the wide spectrum of manufacturing processes, cutting tools and engineering materials (Armarego and Brown, 1969). Furthermore, it has been observed that the improvements in the output variables, such as tool life, cutting forces, surface roughness, etc., through the optimization of controllable/input parameters may result in a significant economic performance of machining operations (Armarego, 1994). The output variables that may have either direct or indirect indications on the performance of other variables such as tool wear rate, machining cost etc. are cutting forces and surface roughness.

Many researchers have conducted studies on predicting cutting forces produced in milling operations using theoretical and analytical approaches (Li et al., 1999; Li and Li, 2002; Yun and Cho, 2001; Yoon and Kim, 2004; Koenigsberger and Sabberwal, 1961; Sabberwal, 1960; Yun and Cho, 2000; Wang and Chiou, 2004), mechanistic model (Omar et al., 2007; Kang et al., 2007; DeVor et al., 1980; Sutherland and DeVor, 1986), etc. The problem with these approaches is that they are based on a big number of assumptions, which are not included in the analysis. This may reduce the reliability of the calculated cutting force values found by these methods. In addition, these approaches may be successfully applicable only for certain ranges of cutting condition. On the other hand, many other researchers have followed purely experimental approaches to study the relationship between cutting force and independent cutting conditions (Li et al. (2006)). It has reflected on the increased total cost of the study, as a large number of cutting experiments are required. Furthermore, with this purely experimental approach, researchers have investigated the effects of cutting parameters on output parameter(s) using machining experiments based on a one-factor-at-a-time design without having any idea about the behaviour of output parameter(s) when two or more cutting factors varied at the same time. So, some researchers had adopted the RSM (response surface methodology) technique, which is basically a group of mathematical and statistical techniques that are useful for numerical modelling the relationship between the input parameters (cutting conditions) and the output variable(s) (cutting force) (Montgomer, 2001). Although RSM saves cost and time, sometimes it becomes difficult to model the process having highly complex and non-linearity among input-output variables. For example, the 2nd order model (for cutting force in end milling operation) derived using RSM approach exhibits high mean square error value as observed during ANOVA analysis (Abou-El-Hossein et al. (2007)). There are many other approaches that have become of interest to researchers to adopt, for finding cutting force relationship in milling operation, namely, FEM analysis (Lee and Cho, 2007), Fuzzy logic (Zuperl et al., 2005), Evolutionary approach (Kovacic et al., 2004), etc.

Again, in case of analysis of surface roughness in end milling operation, many researchers have gone through experimental approach and mathematical relation(s) between output parameter (surface roughness) and cutting conditions allowing us to predict in general form (Dewes and Aspinwall, 1997; Alauddin et al., 1996; Chang, 1992; Kline et al., 1982; Chevrier et al., 2003; Vivancos et al., 2004). But it has been observed that such type of experimental and mathematical models result a great difference between real value(s) and theoretical value(s) due to consequence of movement error and building-ups edge as well as changes in the tool profile because of wear. Normally these causes are very difficult to maintain under precise control to obtain reproducible results. In order to overcome those difficulties, there were various approaches adopted concerning surface roughness in end milling operation, namely, Taguchi method in optimization of parameters (Ghani et al. (2004)), Computer-aided analysis for modelling (Alauddin et al. (1995)), ANN based modelling (Tsai et al. (1999)), etc.

From the above surveys, it has been observed that the prediction of surface roughness and cutting force in milling based on models which are constructed using conventional methods may not be accurate. This is so as milling process is a complex physical process, where the relationships of input-output variables are non-linear. In contrary, fuzzy logic concept is a well-established powerful tool to model physical processes, which are highly complex in nature and where the input-output relationships represent non-linearity, uncertainty and ambiguity. In the present study, cutting force and surface roughness produced during milling operation are investigated using FRBM (fuzzy rule based model) which are constructed using TSK-type fuzzy logic rule. A combined approach of multiple linear regression and genetic algorithm, so called genetic Linear Regression (GLR) approach is adopted to construct knowledge base (KB) of TSK-type FRBM. The models include four cutting (controllable) parameters: feed rate, cutting speed, axial depth of cut and radial depth of cut.

Nomenclature

FRBM Using TSK-Type Fuzzy Rule

TSK-type fuzzy logic rules are widely used in developing rulebased systems. A fuzzy rule uses the fuzzy set theory proposed by Zadeh (1965). The syntax of a TSK-type fuzzy rule looks as follows (Sugeno and Kang, 1988; Takagi, and Sugeno, 1985):

where A₁, . . . , A_n are fuzzy subsets of the input variables x₁, ..., x_n, respectively. The consequent function of each rule is described as a (linear) function, in the form

where K is the number of parameters (coefficients) associated to a function and f _j (x₁,..., x_n ) is a sub-function of the input variables x₁, . ., x_n. The overall output of the model can be obtained for the input tuple (x₁, x₂, ...., x_n) using the following empirical expression.

where n is the number of input variables that occur in the rule premise, R is the number of rules in the rule base. is the firing degree of r^th rule. ∏ is the product representing a conjunction. is the rule consequent function (y) of the r^th rule and are the function coefficients of the corresponding r^th rule consequent function. For a typical rule consequent function, say polynomial may be expressed by

The performance of this model mainly depends on the optimal values of the output function coefficients (a₁, a₂, a₃ and a₄) of the rules for a given values of the variable's exponential parameters (p₁, p₂, p₃ and p₄) and also on the choice of the type of MFDs considered for the input variables (x₁, x₂, x₃ and x₄). In addition to that the issue of having the optimized fuzzy sub-sets of each input variables is also an important concern for achieving the best performance of a model.

Model Construction

In this combined approach, the values of function coefficients are determined using linear regression method, while a GA is introduced to optimise the exponential parameters of input variables as well as optimisation of MFDs of input variables using the same GA. That means, once the values of exponential parameters of the RCFs and the parameters associated with the membership functions are obtained, the values of coefficients of the RCFs are evaluated by multiple linear regression method.

The structure of trapezoidal MFDs as considered here for the input variables is represented in Fig. 2. Two parameters, b and d are needed to describe the (semi) trapezoidal MFDs. The scaling factors (MaxV - MinV) of all input variables are kept as same during optimization of MFDs in constructing each FRBMs for surface roughness and cutting force.

The optimal values of rule-consequent coefficients and power terms are obtained using genetic linear regression approach and simultaneous optimisation of input variable's MFDs using GA, as presented in Fig. 1. The optimum values of power terms of rule consequent functions (p₁, p₂, p₃ and p₄, according to Eq. (2)) and the parameters related to MFDs (b and d, according to Fig. 2) are determined using GA, while the rule-consequent coefficient (a1, a2, a3 and a4 according to Eq. (2)) are determined using multiple linear regression method in the framework of genetic linear regression approach. As the performance of a GA depends on the GAparameters, the optimal choices of GA-parameters (namely population size, crossover probability and mutation probability) are fixed through a parametric study (Nandi, 2006) in order to achieve good results.

Linear Regression Method with TSK-Type Fuzzy Model (Nandi and Klawonn, 2004)

A general expression of linear regression system with TSK-type fuzzy model is derived here to determine the coefficients of RCFs in GLR approach. Equation (1) may be rewritten by denoting for simplicity, in the following form:

Let us assume we have a set of input-output tuple (D) of S number of sample data where the output y⁽ⁱ⁾ is assigned to the input

Now, the total quadratic error that is caused by the TSK-type FRBM with respect to the given data set is

In order to minimise E, we have to choose the following parameters appropriately:

where the parameter indicates the j^th coefficient of the output function of rth rule.

To determine the above parameters, we take the partial derivatives of E with respect to each parameter () and make them be zero, i.e.,

, where j = {1,2,....,k} and r = {1,2,.....,R}

Now, we obtain the partial derivation of E with respect to the parameter .

Thus, Eq. (3) provides the following system of linear equations from which we can compute the coefficients

In matrix form, Eq. (5) will be written as:

where

Thus Eq. (5) provides solutions of the function coefficients () of the TSK-type fuzzy rule consequents for given values of the input variable's exponential terms.

For modelling cutting force in milling, modified AISI P20 tool steel is considered as the work piece material (Abou-El-Hossein et al., 2007). It is a chromium-molybdenum alloyed which is considered as high speed steel. AISI P20 defers from normal P20 steel by containing 0.015% Sulphur, because of better machinability and more uniform hardness in all dimension. Its tensile strength is 1044 MPa and its hardness range is 280 HB to 320 HB. The cutting tool used in this study is a 00 lead-positive end milling cutter of 31.75 mm diameter and equipped with two square inserts whose all four edges can be used for cutting. Here, one insert per one experiment is mounted on the cutter. The inserts have the following specification: square shape, back rake angle of 00, clearance angle of 110, nose radius of 0.794 mm and without any chip breaker. These carbide inserts are KC735M which have a single layer of TiN. The coating is accomplished using PVD techniques to a maximum of 0.004 mm thickness. Experiments are performed in random with different cutting conditions and using a standard coolant to find the cutting force. Each experiment is stopped after 85 mm cutting length. Fc is measured with the aid of a piezoelectric cutting force dynamometer provided by Kistler. Each experiment is repeated three times using a new cutting edge every time and the average of these values is considered.

On the other hand, for surface roughness modelling, the material of workpiece used is W-Nr. 1.2344, hardened steel (50-54 HRC) (Vivancos et al., 2004). A cutting tool of KOBELCO series MIRACLE: (Al, Ti) N-coated micro grain carbide, two flute ball end mill VC2SBR0300, diameter 6 mm is used. Effective Sr is measured with a Taylor-Hobson form Taylsurf series 2 profile rugosimeter in every experiment conducted with different cutting conditions.

Now, the data collected based on experimentation are analyzed in the following sub-section to reveal the preliminary information underlying in the relationship between input-output variables. This information is used in the GLR approach to construct the KB of FRBMs.

Experimental Data Analysis

Surface roughness

In order to understand the relationship of surface roughness with cutting parameters (feed rate, radial depth of cut, axial depth of cut and cutting speed), it is essential to analyse the variation of surface roughness with respect to each of the individual cutting parameter as well as when more than one parameter are changing simultaneously. After analysing the experimental data, as shown in Figs. 3(i)-(iv) which describe the variation of surface roughness with feed rate, the following points are revealed:

Figures 4(i)-(iv) describe the variation of surface roughness with respect to radial depth of cut. After analysing the data as shown in Figs. 4(i)-(iv), it has been revealed that surface roughness get worse by increasing the value of R_d at any values of axial depth of cut, feed rate and cutting speed, and the rate deterioration (considerably high) is almost the same for all values of A_d, F_d and V_c.

The variations of surface roughness with respect to axial depth of cut are illustrated in Figs. 5(i)-(iv). Analysis of data as presented in Figs. 5(i)-(iv) implies the following points:

After analysing the data as shown in Figs. 6(i)-(iv), which describe the variation of surface roughness with cutting speed, the following points are revealed:

From the above analyses, it is stated that change in radial depth of cut influences much on surface roughness than other cutting parameters, namely axial depth of cut, cutting velocity and feed rate.

Cutting Force

Like surface roughness, the influences of different cutting parameters (F_d, A_d, R_d and V_c) on cutting force generated during milling operation are illustrated in graphical manner based on experimental data. This underlying information in the cutting force relation with cutting parameters extracted from experimental data is later utilized during learning of FRBM for constructing cutting force model.

Figures 7(i)-(iii) show the graphs representing the variation of cutting force with axial depth of cut. It is observed that cutting force increases with increasing axial depth of cut at almost equal rate at any values of V_c, F_d and R_d. Again it is observed in Fig. 7(iii) that, when cutting velocity is decreased, the amount of cutting force value is comparatively higher for the constant values of F_d and R_d.

Figures 8(i)-(iii) represent the variation cutting force with feed rate. It is found that cutting force increases with increase in feed rate for any values of V_c, A_d and R_d, but the increasing rate varies in different cases. Again in Fig. 8(i), when Ad changes the value from 1 mm to 2 mm, with increase in feed rate, the cutting force increases but it starts from a high value as well as with higher rate.

In Figs. 9(i)-(ii), the graphs are drawn showing the variation of cutting force with radial depth of cut. It is observed that cutting force increases with increase in radial depth of cut. It is observed that, if the value of A_d changes from 1 mm to 2 mm (Fig. 9(i)) and V_c changes value from 180 m/min to 100 m/min (Fig. 9(ii)), with increase in R_d, the cutting force value becomes high and it increases with almost equal rate.

In Figs. 10(i)-(iii), the curves are drawn representing the variation of cutting force with cutting speed. Here it is observed that with increase in cutting speed, the cutting force decreases for any values of F_d, A_d and R_d, i.e. proportionally inverse. For a given cutting speed, the cutting force value becomes high if R_d changes from 2 mm to 5 mm and A_d changes from 1 mm to 2 mm, as shown in Fig. 10(i) and Fig. 10(ii), respectively.

From the above analysis of experimental data, it is clearly observed that the outputs (surface roughness and cutting force) in milling are not linearly related with the cutting parameters and ambiguity is involved when more than one cutting parameters vary simultaneously.

Mathematical Model

The mathematical model between cutting parameters (cutting velocity, feed rate, axial depth of cut and radial depth of cut) and the cutting force in milling operation (with workpiece material of AISI P20) was derived by using Box-Behnken design (one type of RSM) and it is defined by:

The regression model of surface roughness with cutting parameters for (climb) milling (with workpiece material of W-Nr) is derived by Vivancos et al. (2004), as follows:

Training Data and Fitness Evaluation of GA

Training data

In order to determine the rule consequent function coefficients and power terms of a TSK-type FRBM, a huge number of example data are required. In the present study, 81 numbers of data (Fig. 11 and Fig. 12 related to cutting force and surface roughness, respectively) are considered for constructing KB of FRBMs. These data are obtained through real experimentation as well as based on empirical correlation models (as stated in the section "Mathematical Model"). However, those empirical models are not accurate. Hence, the results obtained using the empirical models do not follow the real characteristics of the relationships among input-output variables in milling process. For this reason, it is required to modify the data obtained using mathematical models to suit the process input-output relationship as discussed in experimental data analysis (in subsection "Experimental Data Analysis").

Fitness Evaluation of GA

During the iteration process of genetic algorithm, the GA population (individuals/chromosomes) having lower fitness value (for error minimization) is chosen in order to reproduce the child chromosomes in the next iteration using the three GA-operators, namely selection, cross-over and mutation. On the other hand, to have a better reliability of FRBM, the performance of FRBM is to be uniform throughout the entire input space. To achieve such consistent result of an FRBM, in every region of the input space the errors of all training data samples that are considered to be uniformly distributed over the whole range of the input variable's space should be equally important for minimization in finding a lower fitness value. Thus, the fitness value of a GA solution is estimated based on the percentage error (instead of simple error) of each training data sample. The error of each set of training data is the deviation of the result (surface roughness) of the FRBM from that of the desired one. Since the error may be positive or negative, absolute value of the error is considered in determining average percentage error as a fitness value of GA-solution.

For cutting force, the fitness value of GA-solution during model construction is calculated in the same way as discussed above for surface roughness.

TSK-Type FRBM for Milling Process

Model of Cutting Force

In order to develop a FRBM for cutting force in milling process, the structure of rule consequent function (as shown in Eq. (9)) considered here has four coefficients and four power terms. Thus the RB, with a maximum of 16 rules in the rule premise, would have a total of 64 (16x4) coefficients and 64 power terms.

A GA-string of 720-bits long is considered for finding the RCFs parameters using GLR approach as well as optimization of MFDs of input variables. First 80 bits (10 bits for each variable) of the GAstring carry information of the eight continuous variables (two variables related to MFDs, b and d for each of the four inputs). The remaining 640 bits (10 bits for each variable) are used to obtain the values of 64 power terms. It is noted that during optimization of MFDs of input variables, the scaling factors (length of input range) of all input variables are not changed.

During GA-based optimization, the parameters related to MFDs - b₁ and d₁ (for cutting speed); b₂ and d₂ (for feed rate); b₃ and d₃ (axial depth of cut) and b₄ and d₄ (radial depth of cut), as shown in Fig. 2, are varied in the range of {(20, 60) and (0, 20)}; {(0.02, 0.05) and (0, 0.02)}; {(0.2, 0.8) and (0, 0.2)} and {(1, 2) and (0, 1)}, respectively. The values of power terms lie in the range of 0.0 to 3.0. The fitness values of GA solution are calculated using the procedure as discussed in sub-section "Fitness Evaluation of GA". The optimal choices of GA-parameters (namely population size, crossover probability and mutation probability) are fixed through a parametric study in order to achieve good results.

After a parametric study of GA, the following GA parameters are selected for the best optimization during training of FRBM for cutting force prediction:

The optimized data base and rule base of FRBM for cutting force in milling obtained using Eq. (9) are shown in Fig. 13 and Table 1, respectively.

Model of Surface Roughness

Like cutting force, the structure of rule consequent function for surface roughness (as shown in Eq. (10)) has four coefficients and four power terms. Since the rule base consists of a maximum 16 rule in the rule premise, there would be a total of 64 (16x4) coefficients and 64 power terms in the RB. A GA-string of 720-bits long is considered here for the GLR technique as well as the optimization of MFDs of input variables. The first 80 bits (10 bits for each variable) are used to carry information of the eight continuous variables related MFDs of input variables. The remaining 640 bits (10 bits for each variable) are used to obtain the values of 64 power terms. It is noted that during optimization of MFDs of input variables, the scaling factors are not changed.

During GA-based optimization, the parameters related to MFDs - b₁ and d₁ (for cutting speed), b₂ and d₂ (for feed rate), b₃ and d₃ (for axial depth of cut), and b₄ and d₄ (for radial depth of cut), as shown in Fig. 2, are varied in the range of {(55.359, 105.359) and (0, 55.359)}, {(0.012, 0.052) and (0, 0.012)}, {(0.111, 0.211) and (0, 0.111)}, and {(0.111, 0.211) and (0, 0.111)}, respectively. In this case, the values of power terms are kept in the range of 0.0 to 2.0. The fitness values of GA solution are calculated using the same procedure as used in case of cutting force. After a parametric study of GA, the following GA parameters are selected for best optimization during tuning of FRBM used for power prediction in milling:

The optimized data base and rule base of the TSK-type FRBM for surface roughness obtained using Eq. (10) are shown in Fig. 14 and Table 2, respectively.

Results and Discussions

Cutting force

The developed FRBM will be used for prediction cutting force and parameter optimization to achieve a desired objective in milling operation. In order to demonstrate the prediction capability of FRBM, both the results of FRBM and mathematical correlation model (available in the literature) are compared with the experimental data. For this comparative study, 22 numbers of cases are considered at random and the results of FRBM, mathematical model and experimentation for the 22 cases are enlisted in Table 3. In Table 3, Error I is the deviation (in percentage) of the result obtained using FRBM from that of the experimental value. Whereas, Error II is the percentage deviation of the result obtained using mathematical correlation model (Eq. (7), as shown in the section "Mathematical model") from that of the experimental value.

In Table 3, it is observed that for almost all the cases, FRBM outperforms over the mathematical correlation model. For 11 cases (case no 1, 2, 7, 9, 10, 12, 14, 16, 17, 20 and 22), it is found that the results obtained by the FRBM are much better than the corresponding mathematical correlation results. Moreover, it is observed that RMS (root mean square) value (4.097) of Error I (evaluated in TSK-type FRBM model) is less than the RMS value (4.248) of Error-II (evaluated in mathematical model).

Thus, the developed FRBM may be adopted for prediction of cutting force to achieve a desired objective in drilling. The performance of FRBM may be improved by considering the interaction effect(s) of the four cutting parameters in the rule consequent functions. But, in such cases, the computational complexity during model construction will be higher. For this reason, it is important to investigate the level of contribution(s) of the independent parameter's interactions toward cutting force, which may be achieved using statistical approach such as analysis of variance (ANOVA).

Surface roughness

The developed FRBM for surface roughness will be used for prediction and parameter optimization to achieve a desired surface roughness in milling operation. The prediction capability of FRBM is verified by comparing the results of FRBM and mathematical correlation model with the experimental results. For this comparative study, 25 cases are considered at random and the results of FRBM, mathematical model and that of experimentation for the 25 cases are enlisted in Table 4. In Table 4, Error I is the deviation (in percentage) of the result obtained using FRBM from that of the experimental value. Whereas, Error II is the percentage deviation of the result obtained using mathematical model (Eq. (8), as shown in the section "Mathematical model") from that of the experimental value.

In Table 4, it can be seen that in most of the cases, FRBM gives better results than mathematical correlation model, except in cases no. 5 and 9. Moreover, it is observed that RMS value (3.410) exhibited by the TSK-type FRBM model is less than that found by mathematical correlation model (RMS value = 11.65456173).

Conclusion

In this work an attempt has been made to develop suitable TSKtype FRBMs for modelling of surface roughness and cutting force in milling operation.

In order to carry out these objectives, the present research work is carried out in three successive stages:

From experimental study, it is found that change in radial depth of cut influences much on surface roughness than other cutting parameters such as axial depth of cut, cutting velocity and feed rate. On the other hand, surface roughness and cutting force in milling are not linearly related to the cutting parameters and ambiguity happens by varying multiple cutting parameters simultaneously. For constructing the TSK-type FRBM, a combined approach of multiple linear regression method and genetic algorithm is utilized. The function coefficients are determined by linear regression whereas the optimized values of the exponential parameters are obtained by using GA. In addition to that, the MFDs of input variables (cutting speed, feed rate, axial depth of cut and radial depth of cut) are simultaneously optimized in order to improve the performances of the FRBMs. After validation of each of the models corresponding to different outputs (surface roughness and cutting force) with the experimental data, it is suggested that both the FRBMs give satisfactory results showing excellent trade-off and practical implementation.

Acknowledgements

The author is thankful to DIT (Department of Information Technology), New Delhi, India for financial support of the grant-inaid project (Ref No. 31(1)/2007-IEAD).

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Paper received 20 May 2011.
Paper accepted 19 August 2011.

Technical Editor: Alexandre Abrão