MODELING FOR RESIDUAL STRESS, SURFACE ROUGHNESS AND TOOL WEAR USING AN ADAPTIVE NEURO FUZZY INFERENCE SYSTEM
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1 CHAPTER-7 MODELING FOR RESIDUAL STRESS, SURFACE ROUGHNESS AND TOOL WEAR USING AN ADAPTIVE NEURO FUZZY INFERENCE SYSTEM 7.1 Introduction To improve the overall efficiency of turning, it is necessary to have a complete process understanding. In this chapter, on the basis of experimental results, two models are developed for axial residual stress (σ a ), surface roughness (R a ) and tool wear (V B ) estimation in hard turning operations. The first model is regression based and the second one is Neuro-Fuzzy based. The results obtained by both the models are compared with the actual experimental results. Buragohain and Mahanta (2008) have inferred that ANFIS is a useful neural network approach for the solution of function approximation problems. Gill and Singh (2009) have discussed that the parameters associated with the membership functions will keep on changing throughout the learning process. The computation of these parameters is facilitated by gradient descent algorithm during the forward pass and least square algorithm during the backward pass, which provides the measure of how well fuzzy inference system models the mapping between the input and output data for the specified parameters. Shinn Ying Ho et al. (2002) have predicted surface roughness in turning operations using ANFIS. They confirm that the proposed ANFIS based method outperforms the existing polynomial network based method in terms of modeling and predicting accuracy. Gill and Singh (2010) have inferred that the ANFIS approach is a more accurate and adaptive approach for problems associated with machining operations. Their work includes design and demonstration of the use of fuzzy logic based model to simulate the material removal rate in ultrasonic drilling of sillimanite ceramic. 7.2 Fuzzy modeling Fuzzy logic is a convenient way to map an input space to an output space. Fuzzy inference is the process of formulating the mapping from a given input to an output 147
2 using fuzzy logic. The mapping then provides a basis from which decisions can be made. The process of fuzzy inference involves membership functions, fuzzy logic operators, and if-then rules. A membership function is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. There are two types of fuzzy inference systems Mamdani-type and Sugeno-type. These two types of inference systems vary somewhat in the way outputs are determined (Roger Jang and Sun, 1995).Mamdani s fuzzy inference method is the most commonly seen fuzzy methodology. Mamdani s method was among the first control systems built using fuzzy set theory. It was proposed in 1975 by Ebrahim Mamdani (Mamdani & Assilian, 1975) as an attempt to control a steam engine and boiler combination by synthesizing a set of linguistic control rules obtained from experienced human operators. Mamdani s effort was based on Lotfi Zadeh s 1973 paper on fuzzy algorithms for complex systems and decision processes (Zadeh, 1973). Mamdanitype inference, expects the output membership functions to be fuzzy sets. After the aggregation process, there is a fuzzy set for each output variable that needs defuzzification. It s possible, and in many cases much more efficient, to use a single spike as the output membership functions rather than a distributed fuzzy set. This is sometimes known as a singleton output membership function, and it can be thought of as a pre-defuzzified fuzzy set. It enhances the efficiency of the defuzzification process because it greatly simplifies the computation required by the more general Mamdani method, which finds the centroid of a twodimensional function. Rather than integrating across the two-dimensional function to find the centroid, the weighted average of a few data points is used. Sugeno-type systems support this type of model. Sugeno, or Takagi-Sugeno-Kang method of fuzzy inference first introduced in 1985 (Sugeno, 1985). It is similar to the Mamdani method in many respects. In fact the first two parts of the fuzzy inference process, fuzzifying the inputs and applying the fuzzy operator, are exactly the same. The main difference between Mamdani-type of fuzzy inference and Sugeno-type is that the output membership functions are only linear or constant for Sugeno-type fuzzy inference. A typical fuzzy rule in a zero-order Sugeno fuzzy model has the form If x is A and y is B then z = k where A and B are fuzzy sets in the antecedent, while k is a crisply defined constant in the consequent. When the output of each rule is a 148
3 constant like this, the similarity with Mamdani s method is striking. The only distinctions are the fact that all output membership functions are singleton spikes, and the implication and aggregation methods are fixed and cannot be edited. The implication method is simply multiplication, and the aggregation operator just includes all of the singletons. 7.3 Neuro-fuzzy model Fuzzy logic methods have been used to model various highly complex and nonlinear systems based on a set of sample data and fuzzy if-then rules. A fuzzy inference system can model the qualitative aspects of human knowledge without employing any quantitative analyses. For describing the fuzzy modelling structure for the tool wear prediction, it will be specified as follows: Linguistic variables: Form the basic concept underlying fuzzy logic, i.e. a variable whose values are words rather than numbers. The input linguistic variables to be specified herein for the specific problem of modelling are cutting speed (Vc), feed (f) and depth of cut and nose radius (r) and Axial residual stress (σ a ), Surface roughness (R a ) and Tool wear (V B ) are used as the only output variable. Fuzzy sets: In contrast to a classical set a fuzzy set does not have a crisp boundary, i.e. the transition from the case belong to a set to the case not belong to a set is gradual. Normally this smooth transition is characterized by a membership function that gives to the fuzzy sets flexibility in modelling commonly used linguistic expressions. Membership function (MF): It is a curve that defines the way that each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. The membership function type can be any appropriate parameterized membership function like triangle, Gaussian or bell-shaped. Linguistic rules: A set of linguistic if-then rules, which operate on the defined linguistic variables. A single fuzzy if-then rule assumes the form if x is A then y is B where A and B are linguistic values defined by fuzzy sets on the ranges (universe of discourse) X and Y, respectively. The if-part of the rule x is A is 149
4 called the antecedent or premise, while the then-part of the rule y is B is called the consequent or conclusion. Fuzzy if-then rules with multiple antecedents are often used e.g. as follows: The resulting output after the described fuzzy logic method has to be defuzzified or else converted to a crisp value by using any of the available defuzzification methods, like the centre of gravity method etc. The membership functions used to represent linguistic variables may have an important effect on the modeling performance as the type of the MF being used determines when a given rule is to be put in effect (in fuzzy logic the rule is fired ) or not. 7.4 Adaptive Neuro-Fuzzy Inference System (ANFIS) ANFIS architecture and its learning algorithm for the Sugeno fuzzy model (Roger Jang and Sun, 1995) is described for complete understanding let us assume that the fuzzy inference system under consideration has two inputs m and n and one output f. For a first-order Sugeno fuzzy model, a typical rule set with two fuzzy if then rules can be expressed as: Rule 1: If (m is A 1 ) and (n is B 1 ) then f 1 = p 1 m + q 1 n + r 1 Rule 2: If (m is A 2 ) and (n is B 2 ) then f 2 = p 2 m + q 2 n + r 2 Where p 1, p 2, q 1, q 2, r 1 and r 2 are linear parameters, and A 1, A 2, B 1 and B 2 are nonlinear parameters. The corresponding equivalent ANFIS architecture is as shown in Fig The entire system architecture consists of five layers, namely, a fuzzy layer, a product layer, a normalized layer, a defuzzy layer and a total output layer. The following sections discuss the relationship between the output and input of each layer in the ANFIS. Layer 1 is the fuzzy layer, in which m and n are the input of nodes A 1, B 1 and A 2, B 2, respectively. A 1, A 2, B 1, and B 2 are the linguistic labels used in the fuzzy theory for dividing the membership functions. The membership relationship between the output and input functions of this layer can be expressed as: O 1,i = µ Ai (m) (i = 1, 2) O 1,j = µ B j (n) (j = 1, 2) where O 1,i and O 1,j denote the output functions and µ Ai and µ Bj denote the membership functions. 150
5 Layer 2 is the product layer that consists of two nodes labelled П. The output w 1 and w 2 are the weight functions of the next layer. The output of this layer is the product of the input signal, which is defined as follows: O 2, i = w i = µ Ai (m) µ B j (n) (i = 1, 2) where O2, i denotes the output of Layer 2. The third layer is the normalized layer, whose nodes are labelled N. Its function is to normalize the weight function in the following process: Fig. 7.1 ANFIS architecture. where O 3,i denotes the Layer 3 output. The fourth layer is the defuzzy layer, whose nodes are adaptive. The output equation is w (pm+qn+r), where p i, q i and r i denote the linear parameters or socalled consequent parameters of the node. The defuzzy relationship between the input and output of this layer can be defined as: where O 4,i denotes the Layer 4 output. The fifth layer is the total output layer, whose node is labelled. The output of this layer is the total of the input signals, which represents the tool-state (normal or failure) detection result. The results can be written as: 151
6 where O 5,i denotes the Layer 5 output. 7.5 The application of Adaptive Neuro-Fuzzy Inference System (ANFIS) for Axial residual stress (σ a ), Surface roughness (R a ) and Tool wear (V B ) MATLAB version (R2009b) - ANFIS editor is used for the ANFIS modeling purpose. The 23 Runs are used to construct and train the models out of total 29 runs and 6 Runs are used to test the model. The Run No s which are taken for training and testing the models with the actual measured values of the responses (axial residual stress, surface roughness and tool wear) are shown in the Table 7.1 and 7.2. After this, the same set of experiments is used for regression models and ANOVA tables are formulated for axial residual stress, surface roughness and tool wear. Further the results obtained by both the models are compared with the actual experimental values. 7.6 Clustering of Data Clustering of numerical data is the basis of many classifications and system modelling algorithms. The purpose of clustering is to identify natural groupings of data from a large data set to produce a concise representation of the system behavior. Clustering technique can be used to generate a Sugeno type fuzzy inference system that best models the data behavior using a minimum number of fuzzy rules and thus prevents the explosion of rules. The rules partition themselves according to the fuzzy qualities associated with each one of the data clusters. Various methods of clustering have been described in the literature. The subtractive clustering method, which is an extension of the mountain clustering method, has been used in this paper to estimate the number of clusters and cluster centres in the fatigue life data. This method assumes each data point as a potential cluster centre and calculates a measure of the likelihood that each data point would define the cluster centre, based on the density of surrounding data points. The steps of the fuzzy model algorithm can be summarized as: (1) select the data point with the highest potential to be the first cluster centre, (2) remove all data points in the vicinity of the first cluster centre as determined by the range of influence (radius), and (3) iterate this process until all the data are within the radii of a cluster centre. 152
7 Data clustering was performed herein in order to assist ANFIS modeling performance. Table 7.1 Run No.s used for training the ANFIS Model Run no. V c m/min f mm/ rev. mm r mm Table 7.2 Run No.s used for testing the ANFIS Model Run No. V c m/min f mm/ rev. mm r mm
8 7.7 Adaptive Neuro-Fuzzy Inference System (ANFIS) for Axial Residual Stress (σ a ) The ANFIS architecture that was used in the present study was based on the first order Takagi and Sugeno model and is schematically presented in Fig.7.1. It was assumed that the axial residual stress (σ a ) is a function of cutting speed (V c ), feed (f) depth of cut ( ) and nose radius (r). Thus V c, f, and r are considered as the input parameters, while the axial residual stress (σ a ) which corresponds to each combination of the four input parameters was considered as the unique output of the ANFIS model. The model shown in Figure 7.2 is ANFIS model for axial residual stress (σ a ) estimation. It considers the cutting speed, feed and depth of cut and nose radius as input parameters and axial residual stress as output. Figures 7.3, 7.4, 7.5, 7.6 show various membership functions of cutting speed, feed and depth of cut and nose radius for the proposed axial residual stress estimation model. These membership functions are computed based on the input and output data, which is used to train the system. Figure 7.2 ANFIS model for axial residual stress (σ a ) Input variable cutting speed V c (m/min) Figure 7.3 MF membership function (V c ) for axial residual stress estimation model 154
9 Input variable feed f (mm/rev.) Figure 7.4 MF membership function (f) for axial residual stress estimation model Input variable depth of cut (mm) Figure 7.5 MF membership function ( ) for axial residual stress estimation model Input variable nose radius r (mm) Figure 7.6 MF membership function (r) for axial residual stress estimation model 155
10 The training patterns have been selected from a population of patterns such that they represent all possible output values in the population. These are tuned using a hybrid system that contains the combination of back propagation and least squares type method. The error tolerance of 0 is used and number of epochs are 3. The ifthen rule statements are used to formulate the conditional statements that comprise fuzzy logic. By the model, 5 rules have been obtained which are sufficient to match the requirements of the data. The ANFIS rule structure is shown in Figure 7.7 and the manner in which defuzzification occurs is shown in Figure 7.8. Corresponding to each rule there is one output membership function. Subtractive clustering has been used in this paper for estimating the number of clusters and the cluster centres in a set of data. This algorithm is single pass and fast. Here 5 cluster centres were located and for each cluster separate membership function and rule is created as described below: 1. If (V c is 1) and (f is 1) and ( is 1) and (r is 1) then (axial residual stress is out1 cluster 1) (1) 2. If (V c is 2) and (f is 2) and ( is 2) and (r is 2) then (axial residual stress is out1 cluster 2) (1) 3. If (V c is 3) and (f is 3) and ( is 3) and (r is 3) then (axial residual stress is out1 cluster 3) (1) 4. If (V c is 4) and (f is 4) and ( is 4) and (r is 4) then (axial residual stress is out1 cluster 4) (1) 5. If (V c is 5) and (f is 5) and ( is 5) and (r is 5) then (axial residual stress is out1 cluster 5) (1) Figure 7.7 Rules Network for axial residual stress estimation model 156
11 Figure 7.8 ANFIS Defuzzifier for axial residual stress estimation model 7.8 Regression Model for Axial Residual Stress (σ a ) The objective is to determine which factors and factor interactions are statistically significant in affecting the axial residual stress. The ANOVA table also indicates the significance of the model obtained. ANOVA table is formulated as shown in the Table 7.3. The Model F-value of implies the model is significant. Values of Prob > F less than indicate model terms are significant. The regression equation obtained for the axial residual stress in terms of cutting parameter is as follows: Axial Residual Stress (σ a ) = * V c * f * * r Source Table 7.3 ANOVA Table for Axial Residual Stress (σ a ) Sum of DF Mean F Value Prob > F Squares Square Model significant A B C D Residual Lack of Fit not significant Pure Error Cor Total
12 7.9 Adaptive Neuro-Fuzzy Inference System (ANFIS) for Surface Roughness (R a ) The same procedure was repeated for constructing and training the surface roughness model shown in Figure 7.9. The Figures 7.10, 7.11, 7.12 and 7.13 show various membership functions of cutting speed, feed and depth of cut and nose radius for the proposed surface roughness estimation model. Figure 7.9 ANFIS model for surface roughness (R a ) By the model, 4 rules have been obtained which are sufficient to match the requirements of the data. The manner in which defuzzification occurs is shown in Figure Corresponding to each rule there is one output membership function. The 4 rules are given below: 1. If (V c is 1) and (f is 1) and ( is 1) and (r is 1) then (R a is out1 cluster 1) (1) 2. If (V c is 2) and (f is 2) and ( is 2) and (r is 2) then (R a is out1 cluster 2) (1) 3. If (V c is 3) and (f is 3) and ( is 3) and (r is 3) then (R a is out1 cluster 3) (1) 4. If (V c is 4) and (f is 4) and ( is 4) and (r is 4) then (R a is out1 cluster 4) (1) 158
13 Input variable cutting speed V c (m/min) Figure 7.10 MF membership function (V c ) for surface roughness estimation model Input variable feed f (mm/rev.) Figure 7.11 MF membership function (f) for surface roughness estimation model Input variable depth of cut Figure 7.12 MF membership function ( estimation model (mm) ) for surface roughness 159
14 Input variable nose radius r (mm) Figure 7.13 MF membership function (r) for surface roughness estimation model Figure 7.14 ANFIS Defuzzifier for surface roughness estimation model 7.10 Regression Model for Surface Roughness (R a ) The Model F-value of implies that the model is significant as shown in ANOVA Table 7.4. Values of Prob > F less than indicate model terms are significant. The regression equation obtained for the surface roughnesss in terms of cutting parameter is as follows: R a = * V c * f * * r E-004 * V 2 c * f * * r * (V c * f) E-003 * (V c * ) *(V c * r) * (f * ) * (f * r) * ( * r) 160
15 Table 7.4 ANOVA Table for Surface Roughness (R a ) Source Sum of Squares DF Mean Square F Value Prob > F Model significant A B < C D A B C D AB AC AD BC BD CD Residual Lack of Fit significant Pure Error Cor Total Adaptive Neuro-Fuzzy Inference System (ANFIS) for Tool Wear (V B ) The Figures 7.15, 7.16, 7.17 and 7.18 show various membership functions of cutting speed, feed and depth of cut and nose radius for the proposed tool wear estimation model. By the model, 5 rules have been obtained which are sufficient to match the requirements of the data. The manner in which defuzzification occurs is shown in Figure Corresponding to each rule there is one output membership function. The 5 rules are given below: 1. If(V c is 1) and (f is 1) and ( is 1) and (r is 1) then (V B is out1 cluster 1) (1) 2. If(V c is 2) and (f is 2) and ( is 2) and (r is 2) then (V B is out1 cluster 2) (1) 3. If(V c is 3) and (f is 3) and ( is 3) and (r is 3) then (V B is out1 cluster 3) (1) 4. If(V c is 4) and (f is 4) and ( is 4) and (r is 4) then (V B is out1 cluster 4) (1) 161
16 5. If(V c is 5) and (f is 5) and ( is 5) and (r is 5) then (V B is out1 cluster 5) (1) Input variable cutting speed V c (m/min.) Figure 7.15 MF membership function V c for tool wear estimation model Input variable feed f (mm/rev.) Figure 7.16 MF membership function f for tool wear estimation model Input variable depth of cut (m/min.) Figure 7.17 MF membership function for tool wear estimation model 162
17 Input variable nose radius r (mm) Figure 7.18 MF membership function r for tool wear estimation model Figure 7.19 ANFIS Defuzzifier for tool wear estimation model 7.12 Regression Model for Tool Wear (V B ) The Model F-value of implies that the model is significant as shown in ANOVA Table 7.4. Values of Prob > F less than indicate model terms are significant. The regression equation obtained for the tool wear in terms of cutting parameter is as follows: V B = E-003 * V c * f * * r Table 7.5 ANOVA Table for Tool Wear (V B ) Source Sum of squares DF Mean Square F Value Prob > F Model Significant A B C D Residual Lack of Fit not significant Pure Error Cor Total
18 7.13 Comparison of Regression and ANFIS Models for Axial residual stress (σ a ), Surface roughness (R a ) and Tool wear (V B ) The experimental and predicted results obtained for axial residual stress (σ a ), surface roughness (R a ) and tool wear (V B ) from regression and ANFIS models are indicated in Table 7.5 for all the 29 experiments. These models gave good estimation capabilities as compared to the actual values. Thus it can be concluded that there is close relation between the simulated results and the practical results obtained at similar cutting conditions for predicting tool wear. The accuracy of the models depends upon the data point selection used for creating the model. Figures 7.20, 7.21 and 7.22 show the comparison of models for axial residual stress, surface roughness and the tool wear estimation by using regression model and ANFIS model with actual experimental values. To check the effectiveness of both the modelling techniques a Chi-Square (χ2) test for goodness of fit was conducted on the data as per Table 7.6 and the vaueles for individual outputs are tabulated in Tables 7.7, 7.8, 7.9. The larger the value of Chi-Square (χ2), the greater is the discrepancy between observed and expected frequencies. So from Chi-Square (χ2) tests for goodness of fit it is clear that the value of Chi-Square (χ2) in the case of ANFIS model for axial residual test is low as compared with regression model Thus it can be deduced that the axial residual stress values of ANFIS model are close to the experimental values. Hence the model prepared through ANFIS outperforms marginally the regression model. In case of surface roughness and tool wear the values of Chi-Square (χ2) for regression model is less as compared to ANFIS model so regression model is more close the experimental values. 164
19 Table 7.6 Experimental and predicted values of regression and ANFIS models Run No. Axial residual stress (σ a ) Surface roughness (R a ) Tool wear (V B ) EXP. ANFIS REG. EXP. ANFIS REG. EXP. ANFIS REG
20 Table 7.7 Chi-Square (χ2) test for axial residual stress (σ a ) (σ a ) E Experimental (σ a ) A ANFIS (σ a ) R Regression [ (σ a ) E - (σ a ) A ] 2 / (σ a ) A χ2 value of ANFIS [(σ a ) E - (σ a ) R] 2 / (σ a ) R χ2 value of Regression = =
21 Table 7.8 Chi-Square (χ2) test for surface roughness (R a ) (R a ) E Experimental (R a ) A ANFIS (R a ) R Regression [(R a ) E - (R a ) A] 2 / (R a ) A χ2 value of ANFIS [(R a ) E - (R a ) R ] 2 / (R a ) R χ2 value of Regression E = =
22 Table 7.9 Chi-Square (χ2) test for tool wear (V B ) (V B ) E Experimental (V B ) A ANFIS (V B ) R Regression [(V B ) E - (V B)A ] 2 / (V B ) A χ2 value of ANFIS [(V B ) E - (V B ) R ] 2 / (V B ) R χ2 value of Regression E = =
23 Figure 7.20 Comparison of models for axial residual stress (σ a ) Figure 7.21 Comparison of models for surface roughness (R a ) Figure 7.22 Comparison of models for tool wear (V B ) 169
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