Parametric Optimization of Machining Parameters using Graph Theory and Matrix Approach

Transcription

1 5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India Parametric Optimization of Machining Parameters using Graph Theory and Matrix Approach L BAbhang 1, M Hameedullah 2 1*Mechanical Engineering Dept. Aligarh Muslim University, Aligarh, India , abhanglb@yahoo.co.in Abstract This study investigated the multi-performance optimization of turning process for an optimal parametric combination to yield the minimum cutting forces and surface roughness with the minimum power consumption using graph theory and matrix approach. The experiments were carried out as per L 9 orthogonal array with each experiment performed under different machining conditions of feed rate, depth of cut and lubricant temperatures. In GTMA, a performance suitable index evaluates and optimizes the multi-performance characteristics. It is registered that the performance, for which the value of PSI is highest, is the optimum choice for the given machining conditions. The index is obtained from the matrix model developed from the digraphs. Graph theory and matrix approach methodology reveals that a combination of high level of depth of cut and lubricant temperature along with feed rate in the low level is essential in order to simultaneously minimize (optimize) the main cutting force, surface roughness and power consumption during steel turning. Keywords:Graph theory and matrix approach, surface roughness, cutting force and power consumption 1Introduction The need for improving the technological performance of machining operations as assisted by the cutting forces, surface finish, and cutting power and tool life has long been recognized to increase the economic performance of the machining operations. Metal cutting process is a complicated process where the performance depends upon a number of machining and tooling conditions. In a turning operation, it is important to select cutting parameters so that high cutting performance can be achieved. Selection of desired cutting parameters by experience or using handbook does not ensure that the selected cutting parameters are optimal for a particular machine and environment. The effect of cutting parameters is reflected on surface roughness, surface texture, cutting forces and dimensional deviations of the product. Surface roughness, which is used to determine and to evaluate the quality of a product, is one of the major quality attributes of a turning product. Surface roughness is a measure of the technological quality of a product and a factor that greatly influences manufacturing cost. It describes the geometry of the machined surfaces and combined with the surface texture. The mechanism behind the formation of surface roughness is very complicated and process dependent (Nalbant et al. 2007). The surface roughness is the index of product quality and has influence on several properties, such as fatigue strength, coefficient of friction, lubrication, corrosion resistance and wear resistance of the machined components. Further, the cutting force has a significant influence on cutting process and lower cutting force is invariably preferred. The forces occurring in the metal cutting process provide information about the power requirements of the machine tool, choosing the right cutting process and the optimal technological parameters results in significant savings of energy during metal cutting. Hence, the proper selection of cutting tools and cutting parameters is an important criterion for achieving the desired surface quality and lower cutting forces in metal cutting operation. For developments in the field of adaptive control of the machining process, knowledge of the cutting force is of major importance in defining the performance limit confining the range of optimization. Traditionally the methods were used to determine the optimal cutting parameters include various graphical methods, linear programming, geometric programming, dynamic programming, goal programming, Lagrangian multipliers, fuzzy logic as well as artificial intelligence. Some researchers optimized machining parameters based on a single variable without considering any constraint. The optimization procedure was based on a general search method, where the objective function was minimized with respect to the machining conditions. The number of methods are reported in the literature for the optimization of machining parameters in turning 659-1

2 Parametric Optimization of Machining Parameters using Graph Theory and Matrix Approach operation are grey relational analysis (Tosun, 2006, Abhang et al. 2011), Taguchi (Pradeepkumar, 2000), Taguchi-utility (Gaitonde et al. 2009), genetic algorithm (Suresh, 2002) and response surface methodology (Harisingh et al. 2007, Abhang et al, 2010). These tools have limitations in terms of computational time, understating, software availability, accuracy optimization experimental domain etc. However, the trend in the modeling using regression (RSM) has a low level non-linear behavior with a regular experimental domain and relatively few factors region. It is because of its limitation in developing a model to fit the data over an irregular experimental region. Graph theory [9] is one of the methodologies which can provide a solution to the number of varieties of problems. Graph theory is systematic and logical approach that has been applied in various fields of science and technology (Deo, 2000). The matrix approach is useful in analyzing the graph models expeditiously to drive the system function and index to meet the objectives. Moreover, representation of graph by a matrix offers simple, easy and convenient decision making method that involves fewer computations. In this paper, graph theory and matrix approach is used to study the multi-performance characteristics in terms of surface roughness, main cutting force and power consumption during steel turning. Graph theory and matrix approach methodology is already used by many researchers for various applications for any type of selection problem involving any number of set selection criteria (Rao et al 2006, Devim et al. 2003, Venkata, 2006, Garg et al, 2006, Paramasivam, 2008, Kulkarni, 2006, Dhar et al, 2006) etc. 2 The Digraph Representation A digraph is used to represent the factors which affect the multi-performance characteristics and their interdependencies in terms of node and edges. Multiperformance evaluation digraph models the factors, sub factors and their interrelationship. This digraph consists of a set of nodes V= {Vi}, with i=1, 2 N and a set of directed edges D= {di}. A node vi represents ith performance selection factors and edges represent the relative importance among the factors. The number of nodes N, considered is equal to the number of performance selection factors considered. If a node i is having relative importance over another node j in the performance selection (multiperformance characteristics), then a directed edge or arrow is drawn from node i to node j (i.e, dij) or (dji). In the present work the performance selection factors are considered main cutting force (Fc), surface roughness (Ra) and power consumption (Pc) as shown in Figure Figure1 Performance selection factors or PSF graph As three performance selection factors are considered, there are three nodes in the PSF graph with nodes 1, 2 and 3 representing the surface roughness, main cutting force and power consumption respectively. Surface roughness is relatively more important than cutting force (power consumption) in turning operation. A low value of surface roughness is desired for a specified workpiece material. However, power consumption is also important in performance selection during steel operation. Thus, the relative importance exists between these three factors in both the directions. PSF graph is developed based on the relation as shown in figure1. PSF graph gives graphical representation of the factors and their relative importance for quick visual appraisal. As the number of nodes and their relative importance increase, the digraph becomes complex (Rao et al. 2002, Kulkarni, 2006). In such case visual analysis of the digraph is more difficult and complex. To overcome this constraint, the digraph is represented in matrix form. 3 The Matrix representation A digraph is a visual representation so it helps in analyzing the system to a limited extent only. For the establishment of the expression for factor affecting the performance characteristics, the digraph is represented in matrix form. Matrix is a convenient in computing processing also. Matrix representation of the performance selection factor graph gives one-toone representation. This matrix (A) is called PSF matrix or variable permanent matrix. This is an N N matrix and considers all of the factors (i.e., R i ) and their relative importance (i.e., a ij ). The matrix A, for the PSF graph is represented as: Permanent matrix = A A = R a F c Ra (1) Where, ai is the value of ith factor represented by node ni and aij is the relative importance of the ith factor over the jth represented by the edge dij. The permanent of this matrix A, i.e., per (A), is defined as 3 Fc Pc R 1 a 12 a 13 a 21 R 2 a 23 a 31 a 32 R 3 (1) 659-2

3 5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India the PSF function. The permanent is a standard matrix function and is used in combinatorial mathematics(kulkarni, 2006). Application of permanent concept leads to a better appreciation of performance selection factors. The permanent function is obtained in a similar manner as its determinant but with all signs positive (Venkata, 2006). The performance selection factor function for matrix expression (1) is written as 3 i= 1 Per (A) = Ri + ( aijaji ) Rk + ( aijajkaki ) + aikakjaji ) i, j, k i, j, k (2) Expression (2) is the complete expression for the considered performance selection problem, as it considers the presence of all factors and all of the possible relative importance between the factors. Thus, the PSF function characteristics the considered performance selection problem and their relative importance. It is to be mentioned that this expression is nothing but the determinant of a 3 3 matrix but considering all the terms as positive terms. A computer program is developed in MATLAB software for calculating the permanent function values of a matrix and the program can be used for calculating the value of permanent function of a square matrix of N N size. 4 Methodology The main steps of the methodology are as follows: Graph theory and matrix approach evaluates the impact of different factors which affects the multiperformance characteristics, in terms of a single numerical index. Step1. Identify the performance selection factors for then given application and short-list the performance on the basis of identified factors satisfying the requirements. Step2. Identify sub factors affecting each other. For each factor, logically develop a digraph among the sub factors based on interaction between them. This is the digraph at each subsystem level. Step3. a) After short-listing the performances, it is to find out the aij relations between the factors and normalize the values Ri for different alternatives. b) Logically develop a digraph between the main factors depending on their interdependencies. The number of nodesshould be equal to the number of factors (Ri) and direction of edges should correspond to their interdependencies (aij). Step4. Develop a variable permanent matrix (eq n 1) at the system level based on the digraph developed in step (b). This is an N N square matrix with diagonal elements of Ri and off-diagonal elements of rij. In permanent matrix diagonal elements represent the contribution of factors (events) and the off-diagonal elements represent the relative importance among the factors. Step5. Obtain the permanent function of the variable permanent matrix at system level using eq n (2). Substitute the values of rij and normalize values of Ri, obtained in step1, in performance selection factors function to evaluate the performance suitability index for the considered performance. Compare the effect of different factors on multi-performance characteristics in terms of the index values. The performance having the highest value of index is the best choice for the given machining operation. Step6 Select the suitable machining condition for which numerical index is higher. 5 Experimentalprocedure In the present work, three parameters, namely, feed rate, depth of cut and lubricant temperatures are considered and the ranges of the parameters were selected based on the earlier investigation carried out by author (Abhang et al, 2010). The suitable values for the cutting conditions for this work were selected based on this work and the specifications of the lathe used. The parameters identified in the present study are multilevel parameters and their outcome effects are not linearly related, and hence, it has been decided to use three level tests for the cutting parameters. The experiments were performed as follows-nine experiments were carried out with parameters at different levels. The experimental conditions so far discussed have been summarized below in the following Table1. In this study the experimental work was carried out by turning EN-31 steel alloy by using tungsten carbide tool with minimum quantity lubrication (solid-liquid). The reason for selecting EN-31 steel alloy as work material is that this alloy is widely used in the automotive industry for the parts made by turning operations such as roller bearing, ball bearing, spline shaft and shearing blades. The surface roughness was measured on an optical microscope (Carl-zesis, Japan made lens factor is 0.89). The surface roughness was taken perpendicular to the turning direction. In this work the average surface roughness (Ra, µm)) values were measured by taking average of the three readings. The surface roughness was measured at three equally spaced locations around the circumference of the work-piece. The average surface roughness (Ra), which is commonly used in the manufacturing industry, is considered for the present investigation. Ra is the arithmetic value of the departure of profile from the centerline along sampling length. Cutting forces are measured with the help of calibrated lathe tool dynamometer.the cutting forces were measured and 659-3

4 Parametric Optimization of Machining Parameters using Graph Theory and Matrix Approach power consumption calculated. Power consumption is calculated by (Pc = Fc * velocity (v)), where, P is the power in watt, V is the cutting speed in m/min and Fc is the main cutting force in N. The identified process parameters and their levels are presented in Table2. The experiments were conducted as per the Taguchi design (L 9 orthogonal array). The responses (surface roughness, main cutting force and power consumption) were recorded. The recorded values of cutting force, surface roughness and calculated power consumption are summarized in Table3. Machine Tool Work material Tool holder Insert configura tions Cutting speed Nose radius Environm ents Work material:c ompositio n Table 1: Experimental conditions 10 HP. Lathe LTM-20 EN-31 steel, 500 mm in length and 60 mm Ф WIDAX, SCLCR1212FO9T3, INDIA Lit. CNMA , (diamond shape),(α = 6º, γ 0 = -6º, λ = -6º, Kr = 95º, Єr = 80 º, r = 1.2 mm) rpm 1.2 mm MQL(10% boric acid + SAE-40 base oil) with different degree of temperatures EN-31 steel, (C=0.95 to 1.2%, Si = 0.10 to 0.35%, MN = 0.30 to 0.75%, Cr = 1.0 to 1.6%, Co = 0.025%, S = %, P = 0.04%) Table 2: Cutting parameters and their levels Symbol Cutting parameter Unit Level 1 Level 2 Level 3 A Feed rate mm/rev B Depth of cut mm C Lubricant temperature C Table 3 Normalization data Surface roughness (µm) Main cutting force (N) Power consumption (W) The various steps of the methodology were carried out as described below: Step1. In the present investigation the three factors considered i.e surface roughness, main cutting force and power consumption. The quantitative values by the multi-performance characteristics, which are not presented here due to space problem and these, were normalized in the interval scale 0to1. Surface roughness, cutting force and power consumption are non-beneficial factors. Values of these three factors are normalized and are given in Table 3 in the respective columns. Relative importance of factors (aij) is also assigned the values. The relative importance of factors (i.e., aij) given in Table4. Let the decision maker select the following assignments: Table 4Relative importance of attributes (aij) Ra Fc Pc Ra Fc Pc Permanent matrix (A) = Ra Fc Pc Ra Fc Pc (3) However, it is observed that, in actual practice, the designer depending on the requirements can judiciously decide the values of relative importance. The assigned values in this work are for demonstration purpose only. Step2. The attribute graph, showing the presence as well as relative importance of the factors (surface roughness, cutting force and power consumption). Step3. The matrix 3 3 of this graph is written based on expression (1) Step4. The value of permanent function of each factor is evaluated by using eq n (2). However, it may be added that as computer program is developed in MATLAB software for calculating the permanent function values of a matrix. Step5. The performance selection index is calculated using the values of Ri and aij for each alternative of performance. The PSI values of different alternatives are given in Table6. From the table6 values of the PSI index, it is observed that the machining parameter 659-4

5 5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India designated as alternative 3, is the right choice (optimum) for the given problem of selection of a suitable machining parameter for turning EN-31 steel. The second choice is alternative two and last choice is alternative nine. Thus, for optimal process parameter setting for the present investigation is A1, B3 and C3. Hence, the best combination values for simultaneously achieving higher surface finish, lower cutting force and power consumption. The optimum parameters are lower feed rate (0.05mm/rev), higher depth of cut (0.6mm) and high lubricant temperature (50 0 c). Table 5 PSI index Exp. No. PSI Index Discussions From the results of graph theory and matrix approach, it is found that the combinations required for simultaneously optimizing (minimizing) the cutting force, power consumption and surface roughness in turning of En-31 steel are, feed rate at lower level and the depth of cut and lubricant temperature at higher levels. As seen from the PSI index (Table 6), the requirement of feed rate is low for minimizing both surface roughness and cutting force. The reason might be, at higher feed rate, more heat is generated between tool and work piece interface because of high material removal rate, consequently increase in chip-tool interface temperature and thus increase in tool wear, which in turn increases the surface roughness (poor surface finish). Increase in feed rate also increases chatter, providing an incomplete machining at a faster traverse and hence higher surface roughness. However, the surface roughness increases with an increase in feed rate, as the surface roughness being proportional to the square of the feed rate. This result was in consistent with the literature (Abhang et al, 2010, Davim, 2003, Dhar et al, 2006). As the lubricant temperature increases the cutting force decreases. Comparatively higher forces at low lubricant temperature (below atmospheric temperature) are due to increase of local hardness due to which more force is required to remove material. Whereas heated (higher lubricant temperature, above atmospheric temperature) lubricant raises the cutting zone temperature so easy deformation of workpiece takes place and thus force and power consumption requirement decreases in studied ranges of the process parameters selected. On the other hand, high lubricant temperature (solid-liquid) and depth of cut with low feed rate there is a reduction in adhesion of work-material and consequently friction forces, which in turn reduces the cutting force and power consumption. In this study, globally, the depth of cut affects very less influence on surface roughness and cutting force as compared to other two parameters, when applying heated lubricant during steel turning operation. So the operator must choose these parameters depending on the surface roughness and productivity desired. 7 Conclusions 1. A methodology based on digraph and matrix approach was suggested for the multi-performance optimization of machining parameters in steel turning operation. It helps in selecting the most suitable choice from among a large number of candidate alternatives for a given problem. This is a general method and is applicable to any type of metal machining operation. 2. Graph theory and matrix approach method is relatively new, and offers a generic, simple, easy and convenient decision-making method that involve less computation. The method enables a more critical analysis and any number of objective and subjective attributes can be considered. The measures of the factors and their relative importance are used together to rank the alternatives. 3. This study should help the operator to choice the cutting parameters depending on the surface quality (surface roughness), desired productivity and consumed power (cutting force). The results of this study are valid for En-31 steel alloy and selected parameters and their specified ranges. Acknowledgement The authors would like to express their deep gratitude to the Department of Mechanical engineering of Aligarh Muslim University (AMU) for providing the laboratory facilities and financial support

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