Modeling And Optimization Of End Milling Machining Process - Ijret

IJRET: International Journal of Research in Engineering and Technologyof an EA to find multiple optimal solutions in one simulationrun makes EAs unique in solving multi-objective optimizationproblems.well.The method of least squares is used to estimatethe parameters in the approximating polynomials. TheRSM is then performed using the fitted surface. If the fittedsurface is the adequate approximation, of the true responsefunction, then analysis of the fitted surface will beapproximately equal to analysis of the actual system. Themodel parameters can be estimated most effectively if properexperimental design is used to collect the data. Designs forfitting response surfaces are called response surfaceresults. RSM is a sequential procedure. Often, when weare at a point on the response surface that is remote fromthe optimum, such as the current operating condition in the fig4.2, there is little curvature in the system and the first ordermodel will be appropriate. Our objective here is to leave theexperimenter rapidly and efficiently along the path ofimprovement towards the general vicinity of the optimum.Once the reason of the optimum has been found, a moreelaborate model, such as second order model, may beemployed, and an analysis will be performed to locateoptimum. From the fig 4.3 we see that the analysis of responsesurface can be thought of as „climbing a hill‟, where the topof the hill represents the point of maximum response. If thetrue optimum is a point of minimum response, then we thinkof „descending into a valley‟.The eventual objective ofRSM is to determine the optimum operating conditions forthe system or to determine a region of the factor space inwhich operating requirements are satisfied.3. RESPONSE SURFACE METHODOLOGYResponse surface methodology or RSM is a collection ofmathematical and statistical techniques that are useful for themodeling and analysis of problems in which response ofinterest is influenced by several variables and the objective isto optimize this response. For example, suppose that achemical engineer wishes to find the levels of temperature(x1) and pressure (x2) that maximizes the yield (y) of aprocess. The process yield is a function of the levels oftemperature and pressure, sayY f(x1, x2) εWhere ε represents the noise or error observed in the processy. if we denote the expected response byE(y) f f(x1, x2) η,then the surface is represented byη f(x1, x2)is called response surface.We usually represent the responsesurface graphically, such as in fig 4.1, where η is plottedversus the levels of x1, x2. To help visualize the shape of aresponse surface, we often plot the contours of the responsesurface as shown in fig 4.2. in the contour plot, lines ofkconstant responseare drawn in the x1, x2 plane. Each contourcorresponds to a particular height of the response surface.3.1 Designs For Fitting First Order ModelSuppose we wish to fit the first order model in k variablesY β0 i 1 i xi kIn most RSM problems, the form of the relationship betweenthe response and the independent variables is unknown.Thus, the first step in RSM is to find a suitableapproximation for the true functional relationship between yand the set of independent variables is employed. If theresponse is well modeled by a linear function of theindependent variables, then the approximating function is thefirst order model.Y β0 β1x1 β2x2 ---------------- βkxk εIf there is curvature in the system, then a polynomial of higherdegree must be used, such as the second order model.kY β0 j 1 j x j k i ij jxi x j x j 1 ij Almost all RSM problems use one or both of thesemodels. Of course it is unlikely that a polynomial modelwill be a reasonable approximation of the true functionrelationship over the entire space of the independent variables,but for a relatively small reason, they usually work quiteISSN: 2319-1163j2There is a unique class of designs that minimize thevariance of the regression coefficients (βi). These are theorthogonal first-order designs. A first- order design isorthogonal if the off-diagonal elements of the (X1X) matrixare all zero. This implies that the cross products of thecolumns of the X matrix sum to zero. The class oforthogonal first-order designs includes the 2k factorialand fractions of the 2k series in which main effects are notaliased with each other. In using these designs, we assume thatthe low and high levels of the k factors are coded to usual 1levels. The 2k designs do not afford an estimate of theexperimental error unless some runs are replicated. Acommon method of including replication in the 2k designsis to augment the design with several observations at thecenter (the point xi 0, i 1, 2, 3, -----, k). The addition ofcenter points to the designs does not influence the (βi) fori 1, but the estimate of β0 becomes the grand average of allobservations. Furthermore, the addition of center pointsdoes not alter the orthogonally property of the design.Volume: 01 Issue: 03 Nov-2012, Available @ http://www.ijret.org432

IJRET: International Journal of Research in Engineering and Technology3.2 Designs For Fitting Second Order ModelCentral composite design is the most popular class ofdesigns just for fitting second order models. Generally theCCD consists of a 2k factorial (or fractional factorial ofresolution V) with nf runs, 2k axial or star runs and nc centerruns. Figure 4.4 shows the CCD for k 2 and k 3 factors. Thepractical deployment of a CCD often arises throughsequential experimentation. That is the 2k has been used to fita first model, this model has exhibited lack of fit and the axialruns are then added to allow the quadratic terms to beincorporated in to the model. The CCD is a very efficientdesign for fitting the second order model. There are twoparameters in the design that must be specified, thedistance α of the axial runs from the design center and thenumber of center points nc. We now discuss the choice ofthese two any real-world design or decision making problems involvesimultaneous optimization of multiple objectives. In principle,multi objective optimization is very different than the singleobjective optimization. In single objective optimization, oneattempts to obtain the best design or decision, which is usuallythe global minimum or the global maximum depending on theoptimization problem is that of minimization or maximization.In the case of multiple objectives, there may not exist onesolution which is best (global minimum or maximum) withrespect to all objectives. In a typical multi objectiveoptimization problem, there exists a set of solutions which aresuperior to the rest of solutions in the search space when allobjectives are considered but are inferior to other solutions inthe space in one or more objectives. These solutions areknown as Pareto-optimal solutions or non dominated solutions(ChankongandHaimes1983;Hans1988). The rest ofthesolutions are known as dominated solutions. Since none of thesolutions in the non dominated set is absolutely better than anyother, any one of them is an acceptable solution. The choice ofone solution over the other requires problem knowledge and anumber of problem related factors. Thus, one solution chosenby a designer may not be acceptable to another designer or ina changed environment. Therefore, in multi objectiveoptimization problems, it may be useful to have a knowledgeabout alternative Pareto-optimal solutions.One way to solve multi objective problems is to scalarize thevector of objectives into one objective by averaging theobjectives with a weight vector. This process allows a simpleroptimization algorithm to be used, but the obtained solutionlargely depends on the weight vector used in the scalarizationprocess. Moreover, if available, a decision maker may beinterested in knowing alternate solutions. Since geneticalgorithms (GAs) work with a population of points, a numberof Pareto-optimal solutions may be captured using GAs. AISSN: 2319-1163nearly GA application on multi objective optimization bySchaffer (1984) opened a new avenue of research in this field.Though his algorithm, VEGA, gave encouraging results, itsuffered from biasness towards some Pareto- optimalsolutions. A new algorithm, Non dominated Sorting GeneticAlgorithm (NSGA), is presented in this paper based onGoldberg's suggestion (Goldberg1989). This algorithmeliminates the bias in VEGA and there by distributes thepopulation over the entire Pareto- optimal regions. Althoughthere exist two other implementations (Fonesca and Fleming1993; Horn, Nafpliotis, and Goldberg 1994) based on thisidea, NSGA is different from their working principles, asexplained below.In the remainder of the paper, we briefly describe difficultiesof using three common classical methods to solve multiobjective optimization problems. A brief introduction toSchaffer's VEGA and its problems are outlined. Thereafter,the non dominated sorting GA is described and applied tothree two-objective test problems. Simulation results showthat NSGA performs better than VEGA on these problems. Anumber of extensions to this work is also suggested.4.1 Multi Objective Optimization ProblemA general multi objective optimization problem consists of anumber of objectives and is associated With a number of inequality and equality constraints. Mathematically, the problemcan be written as follows (Rao1991):Minimize / Maximize f i (x) i 1,2, NSubject tog j(x) 0 j 1,2, Jh k(x) 0 k 1,2, .K4.2 GA ImplementationAs early as in 1967, Rosenberg suggested, but did notsimulate, a genetic search to the simulation of the genetics andthe chemistry of a population of single- celled organisms withmultiple properties or objectives (Rosenberg1967). The firstpractical algorithm, called Vector Evaluated GeneticAlgorithm (VEGA), was developed by Schaffer in 1984(Schaffer 1984). One of the problems with VEGA, as realizedby Schaffer himself, is its bias towards some Pareto-optimalsolutions.Later, Goldberg suggested another non dominated sortingprocedure to overcome this weakness of VEGA (Goldberg1989). Our algorithm, Non dominated Sorting GeneticAlgorithm (NSGA), is developed based on this idea. Thereexists atleast two other studies, different from our algorithm,based on Goldberg's idea. In the rest of this section, wediscuss the merits and demerits of VEGA and NSGA, and thedifferences between NSGA and the two other recentimplementations.Volume: 01 Issue: 03 Nov-2012, Available @ http://www.ijret.org433

IJRET: International Journal of Research in Engineering and TechnologyISSN: 2319-11634.2.1. Schaffer's VEGA4.3. Non-Dominating setSchaffer modified the simple tripartite genetic algorithm byperforming independent selection cycles according to eachobjective. He modified Grefenstette's GENESIS program(Schaffer 1984) by creating a loop around the traditionalselection procedure so that the selection method is repeated foreach individual objective to fill up a portion of the matingpool. Then the entire population is thoroughly shuffled toapply cross over and mutation operators. This is performed toachieve the mating of individuals of different sub populationgroups. The algorithm worked efficiently for some generationsbut in some cases suffered from its bias towards someindividuals or regions. The independent selection of specialistsresulted in speciation in the population. The out come of thiseffect is the convergence of the entire population towards theindividual optimum regions after a large number ofgenerations. Being a decision maker, we may not like to haveany bias towards such middling individuals, rather we maywant to find as many non dominated points as possible.If S is the non dominating set then following two conditionmust hold Any two solutions of S must be non dominated with respectto each other. Any solution not belonging to S is dominated by at least onemember of S.Schaffer tried to minimize this speciation by developing twoheuristics the non dominated selection heuristic (a wealth redistribution scheme), and the mate selection heuristic (acrossbreeding scheme) (Schaffer 1984). In the non dominatedselection heuristic, dominated individuals are penalized bysubtracting a small fixed penalty from their expected numberof copies during selection. Then the total penalty fordominated individuals was divided among the non dominatedindividuals and was added to their expected number of copiesduring selection. But this algorithm failed when the populationhas very few non dominated individuals, resulting in a largefitness value for those few non dominated points, eventuallyleading to a high selection pressure. The mate selectionheuristic was intended to promote the cross breeding ofspecialists from different sub groups. This was implementedby selecting an individual, as a mate to a randomly selectedindividual, which has the maximum Euclidean distance in theperformance space from its mate. But it failed too to preventthe participation of poorer individuals in the mate selection.This is because of random selection of the rest mate and thepossibility of a large Euclidean distance between a championand a mediocre. Schaffer concluded that the random mateselection is far superior than this heuristic.Figure 4.1.Concept of DominanceOne method to minimize speciation is through anondominated sorting procedure in conjunction with a sharingtechnique, as suggested by Goldberg (1989). Recently Fonescaand Fleming (1993) and Horn, Nafpliotis, and Goldberg(1994) implemented that suggestion, and successfully appliedto some problems. These methods are briefly discussed later.But before that, we discuss our algorithm NSGA which is alsodeveloped based on Goldberg's suggestions.4.4 Identifying Non Dominating setThere are several approaches proposed in the literature[?]likeNaive and Slow approach,Continuously update,Kung et al.sEfficient Method etc.We use continuously update approach tocompute non-dominating set of solution.Identifying Non Dominating setStep1: Initialize PI {1}.Set solution counter i 2.Step2: Set j 1.Step3: Compare solution i with j from P0 for domination.Step4: If i dominates j ,delete the j th member from P I or updatePI PI \{ PI (j)}.If j PI , increment j by one and then go tostep3.Otherwisego to step5.Alternatively ,if the jth member if P0 dominatesi,increment iby one and then go to step2.Step5: Insert i in PI or Update PI PI [ {i}.If i N, incrementi by oneand go to step2.Otherwise ,Stop and declare PI as the nondominated set.4.5 Non-Dominated sortingIn MOO there are sets of optimal solutions which are NonDominated with respected to each other. Such solutions arearranged in the ascending level of non domination. Procedureto find various level of domination is as follows:Volume: 01 Issue: 03 Nov-2012, Available @ http://www.ijret.org434

IJRET: International Journal of Research in Engineering and TechnologyNon-Dominated Sorting AlgorithmStep1: Set all non dominated sets Pj ,(j 1, 2.J) as emptysets. Set nondomination level counter j 1.Step2: Use any approach to find the non-dominated set P0 ofpopulationP.Step3: Update Pj P0 and P P\P0.Step4: If P Φ ;,increment j by one and go tostep2.Otherwise,stop anddeclare all non-dominated sets Pi ,for i 1, 2.j.ISSN: 2319-1163euclidian distance dij from another solution j in the same frontis calculated as follows:where η is the number of objectives5.6 Pareto OptimalityAll non dominated solutions are important in the context ofMulti Objective Optimization. All these solution are calledpareto optimal solution and curve joining such points is calledpareto optimal front.5.7 Non-Dominated Sorting Genetic AlgorithmThe multi-objective genetic algorithm used in our work is ahybrid genetic algorithm, where the initial population isformed as a combination of first fit (80% of initial populationsize) and random (20% of initial population size) instead ofentirely random manner. We shall try to find a set of solutionas close as possible to the pareto optimal front and as diverseas possible. We have implemented multi-objective geneticalgorithm for traffic grooming problem using Non-dominatingSorting Genetic Algorithm (NSGA). The first step of anNSGA is to sort the population P according to nondomination. This classifies the population into a number ofmutually exclusive equivalent classes (or non dominated sets)Pj , i.e.,The above function takes value between [0,1],depending onthe values of d(euclidean distance) andshare.If d iszero(means two solutions are identical or their distance iszero),Sh(d) 1.On the other hand, if dshare (meaning that twosolutions are at least a distance of share away from eachother),Sh(d) 0.This means that two solutions which are adistance ofshare away from each other do not have any sharing effecton each other.Any other distance will have partial effect oneach.Hence,we compute niche count (assuming 2) as:Niche count of i(nci) is an estimate measure of crowdingaround a solution i.Whereis the number of non-domination levels.The fitness assignment procedure begins from the first nondominated set and successively proceeds to dominated sets.Any solution i of the first (or best) non-dominated set isassigned a fitness equal to Fi P (population size). Thisspecific value of P is used for a particular purpose. Since allsolutions in the first non-dominated set are equally importantin terms of their closeness to the pareto optimal front relativeto the current population, we assign the same fitness to all ofthem. Assigning more fitness to a solution belonging to abetter non-dominated set ensures a selection pressure towardthe pareto optimal front. However, in order to achieve thesecond goal, diversity among the solutions in a front must alsobe maintained.The sharing function method is used front-wise.That is, for each solution i in the front F, the normalizedVolume: 01 Issue: 03 Nov-2012, Available @ http://www.ijret.org435

IJRET: International Journal of Research in Engineering and TechnologyThe steps of NSGA as given in is as followsNSGA Fitness AssignmentStep1: Choose the sharing parameter share and a smallpositive numberand initialize Fmin P . Set front counter j 1.Step2: Classify population P according to non-domination:{P1, P2, P } Sort(PStep3: For each q)PjStep3.1: Assign Fitness Fj(q) Fmin .Step3.2: Calculate niche count ncq using the above equationamong solutions of Pj only.Step3.3: Calculate shared fitness Fj/(q) Fj(q)Step4: Fmin min(Fj/(q) : qStep5: If jcompletePj) and set j j 1., to go Step3. Otherwise, the process isISSN: 2319-1163spindle speed of 12,000 rpm, feed rate of 10 m/min and a 15kW drive motor. CNC part programs for tool paths werecreated. The workpiece material used was AISI 1040 steel inthe form of a 60mmΧ60mmΧ40mm block. Tables 1 and 2provide detailed information on chemical composition andmechanical properties of this AISI 1040 steel. A flat end mill(10mm diameter, 451 helix angle, TiAlN coated solid carbide,4-flutes) produced by Sandvik(R216.34-10045-AC22N 1620)was used in the tests. The up milling cutting method andcompressed cooling oil as the cutting environment were used.The same tool was used until maximum flank wear reachedVBmax 0.1 mm.The setup of the workpiece and flat end millis shown in Fig. 6.1.5.1.2. Surface roughness measurementSurface roughness Ra was measured using a portableMitutoyo Surf Test 301. A minimum of 10 measurement in thetraverse direction were taken, the highest and lowest valueswere discarded and the average value was recorded. In thisstudy, Ra values were measured between 0.55 and 2.74 mm.The repeatability of the measurements was found to be in therange of 2–5%, which was considered satisfactory forgenerating empirical models.5.1.3. Experimental designIn this study, the experimental plan has four controllablevariables namely, spindle speed, feed rate, depth of cut andstep over. Thus, a minimum of 16 runs is required to develop afull second-order model. Meanwhile, plans with some highlydesirable properties such as rotatability, orthogonal or uniformprecision require more runs. Among various designs, therotatable central composite design has the most popularpromising outstanding benefits. In this study, a rotatablecentral composite (uniform precision) design with six centralreplicates was selected, with five different levels for eachvariable, as shown in Table 3. Variable ranges weredetermined on the basis of a cutting tool catalog. As presentedin Table 4, the experimental plan was composed of a full 24factorial with four central replicates (runs 1–20), augmentedby eight axial runs with two central replicates (runs 21–30) toestimate second-order effects.Fig.4.2. NSGA Flow chart5.IMPLEMENTATIONOFPROPOSEDFor the selection of the best model, the adjusted coefficient ofmultiple correlations.METHODOLOGY5.1. EXPERIMENTAL DETAILS5.1.1. Work piece material, cutting tools andequipmentThe experimental study was carried out in wet cuttingconditions on a DECKEL MAHO DMU 60 P five-axis,highspeed CNC milling machine equipped with a maximumVolume: 01 Issue: 03 Nov-2012, Available @ http://www.ijret.org436

IJRET: International Journal of Research in Engineering and TechnologyISSN: 2319-11635.3. FINDING THE UPPER AND LOWER LIMITSOF THE CONTROL VARIABLESThe upper and

The optimization process involves the optimal selection of machining parameters such as cutting speed, feed and depth of cut, subjected to practical constraints of surface finish, tool wear, dimensional accuracy and machine tool capabilities. Several researches have used different techniques in literature to optimize machining process