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ORIGINAL ARTICLEDeformation control through fixture layout designand clamping force optimizationWeifang Chen 2jj; :; jC12C12C12C12; :; njjC0C1 s ; j 1; 2; :; n 1Subject tom FnijjC21F2ti F2hiq2FniC21 0 3pos i2Vi; i 1; 2; :; p 4where jrefers to the maximum elastic deformation at amachining region in the j-th step of the machiningoperation,Xnj1j C0C0C16C172C30nvuut is the average of jFniis the normal force at the i-th contact point is the static coefficient of frictionFti; Fhiare the tangential forces at the i-th contact pointpos(i) is the i-th contact pointV(i) is the candidate region of the i-th contact point.The overall process is illustrated in Fig. 1 to design afeasible fixture layout and to optimize the clamping force.The maximal cutting force is calculated in cutting modeland the force is sent to finite element analysis (FEA) model.Optimization procedure creates some fixture layout andclamping force which are sent to the FEA model too. InFEA block, machining deformation under the cutting forceand the clamping force is calculated using finite elementmethod under a certain fixture layout, and the deformationis then sent to optimization procedure to search for anoptimal fixture scheme.4 Fixture layout design and clamping force optimization4.1 A genetic algorithmGenetic algorithms (GA) are robust, stochastic and heuristicoptimization methods based on biological reproductionprocesses. The basic idea behind GA is to simulate “survivalof the fittest” phenomena. Each individual candidate in thepopulation is assigned a fitness value through a fitnessfunction tailored to the specific problem. The GA thenconducts reproduction, crossover and mutation processesto eliminate unfit individuals and the population evolvesto the next generation. Sufficient number of evolutions ofthe population based on these operators lead to anincrease in the global fitness of the population and thefittest individual represents the best solution.The GA procedure to optimize fixture design takesfixture layout and clamping force as design variables togenerate strings which represent different layouts. Thestrings are compared to the chromosomes of naturalevolution, and the string, which GA find optimal, ismapped to the optimal fixture design scheme. In this study,the genetic algorithm and direct search toolbox of MATLABare employed.The convergence of GA is controlled by the populationsize (Ps), the probability of crossover (Pc)andtheprobability of mutations (Pm). Only when no change inthe best value of fitness function in a population, Nchg,reaches a pre-defined value NCmax, or the number ofgenerations, N, reaches the specified maximum number ofevolutions, Nmax., did the GA stop.There are five main factors in GA, encoding, fitnessfunction, genetic operators, control parameters and con-straints. In this paper, these factors are selected as what islisted in Table 1.Since GA is likely to generate fixture design strings thatdo not completely restrain the fixture when subjected tomachining loads. These solutions are considered infeasibleand the penalty method is used to drive the GA to a feasiblesolution. A fixture design scheme is considered infeasible orunconstrained if the reactions at the locators are negative, inother words, it does not satisfy the constraints in equations(2)and(3). The penalty method essentially involvesMachiningProcess ModelFEAOptimizationprocedurecutting forcesfitnessOptimization resultFixture layout andclamping force Fig. 1 Fixture layout and clamp-ing force optimization processTable 1 Selection of GAs parametersFactors DescriptionEncoding RealScaling RankSelection RemainderCrossover IntermediateMutation UniformControl parameter Self-adaptingInt J Adv Manuf Technolassigning a high objective function value to the scheme thatis infeasible, thus driving it to the feasible region insuccessive iterations of GA. For constraint (4), when newindividuals are generated by genetic operators or the initialgeneration is generated, it is necessary to check up whetherthey satisfy the conditions. The genuine candidate regionsare those excluding invalid regions. In order to simplify thechecking, polygons are used to represent the candidateregions and invalid regions. The vertex of the polygons areused for the checking. The “inpolygon” function inMATLAB could be used to help the checking.4.2 Finite element analysisThe software package of ANSYS is used for FEAcalculations in this study. The finite element model is asemi-elastic contact model considering friction effect,where the materials are assumed linearly elastic. As shownin Fig. 2, each locator or support is represented by threeorthogonal springs that provide restrains in the X, Y and Zdirections and each clamp is similar to locator but clampingforce in normal direction. The spring in normal direction iscalled normal spring and the other two springs are calledtangential springs.The contact spring stiffness can be calculated accordingto the Herz contact theory 8 as followskiz16RC3iEC32i9C16C1713fiz13kiz kiy6EC3i2C0vfiGfi2C0vwiGwiC16C17C01C1 kiz8:5wherekiz, kix, kiyare the tangential and normal contactstiffness,1RC3i1Rwi1Rfiis the nominal contact radius,1EC3i1C0V2wiEwi1C0V2fiEfiis the nominal contact elastic modulus,Rwi, Rfiare radius of the i-th workpiece andfixture element,Ewi, Efiare Youngs moduli for the i-thworkpiece and fixture materials,wi, fiare Poisson ratios for the i-th workpieceand fixture materials,Gwi, Gfiare shear moduli for the i-th workpieceand fixture materials and fizis thereaction force at the i-th contact point inthe Z direction.Contact stiffness varies with the change of clampingforce and fixture layout. A reasonable linear approximationof the contact stiffness can be obtained from a least-squaresfit to the above equation.The continuous interpolation, which is used to applyboundary conditions to the workpiece FEA model, isFig. 2 Semi-elastic contact model taking friction into accountSpring positionFixture element position12345678 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31 32 33 34 3536 37 38 39 40 41 4243 44 45 46 47 48 49Fig. 3 Continuous interpolationFig. 4 A hollow workpieceTable 2 Machining parameters and conditionsParameter DescriptionType of operation End millingCutter diameter 25.4 mmNumber of flutes 4Cutter RPM 500Feed 0.1016 mm/toothRadial depth of cut 2.54 mmAxial depth of cut 25.4 mmRadial rake angle 10Helix angle 30Projection length 92.07 mmInt J Adv Manuf Technolillustrated in Fig. 3. Three fixture element locations areshown as black circles. Each element location is surroundedby its four or six nearest neighboring nodes. These sets ofnodes, which are illustrated by black squares, are 37, 38,31 and 30, 9, 10, 11, 18, 17 and 16 and 26, 27, 34, 41,40 and 33. A set of spring elements are attached to each ofthese nodes. For any set of nodes, the spring constant iskijdijPk2hidikki6wherekijis the spring stiffness at the j-th node surrounding thei-th fixture element,dijis the distance between the i-th fixture element and thej-th node surrounding it,kiis the spring stiffness at the i-th fixture elementlocation.iis the number of nodes surrounding the i-th fixtureelement location.For each machining load step, appropriate boundaryconditions have to be applied to the finite element model ofthe workpiece. In this work, the normal springs areconstrained in the three directions (X, Y, Z)andthetangential springs are constrained in the tangential direc-tions (X, Y). Clamping forces are applied in the normaldirection (Z) at the clamp nodes. The entire tool path issimulated for each fixture design scheme generated by theGA by applying the peak X, Y, Z cutting forces sequentiallyto the element surfaces over which the cutter passes 23.In this work, chip removal from the tool path is takeninto account. The removal of the material during machiningalters the geometry, so does the structural stiffness of theworkpiece. Thus, it is necessary to consider chip removalaffects. The FEA model is analyzed with respect to toolmovement and chip removal using the element deathtechnique. In order to calculate the fitness value for a givenfixture design scheme, displacements are stored for eachload step. Then the maximum displacement is selected asfitness value for this fixture design scheme.The interaction between GA procedure and ANSYS isimplemented as follows. Both the positions of locators andclamps, and the clamping force are extracted from realstrings. These parameters are written to a text file. Theinput batch file of ANSYS could read these parameters andcalculate the deformation of machined surfaces. Thus thefitness values in GA procedure can also be written to a textfile for current fixture design scheme.It is costly to compute the fitness value when there are alargenumberofnodesinanFEMmodel.Thusitisnecessaryto speed up the computation for GA procedure. As thegeneration goes by, chromosomes in the population aregetting similar. In this work, calculated fitness values arestored in a SQL Server database with the chromosomes andfitness values. GA procedure first checks if currentchromosomes fitness value has been calculated before, ifnot, fixture design scheme are sent to ANSYS, otherwisefitness values are directly taken from the database.The meshing of workpiece FEA model keeps same inevery calculating time. The difference among everycalculating model is the boundary conditions. Thus, themeshed workpiece FEA model could be used repeatedly bythe “resume” command in ANSYS.5 Case studyAn example of milling fixture design optimization problemfor a low rigidity workpiece displayed in previous researchpapers 16, 18, 22 is presented in the following sections.Fig. 5 Candidate regions for thelocators and clampsTable 3 Bound of design variablesMinimum MaximumX /mm Z /mm X /mm Z /mmL10 0 76.2 38.1L276.2 0 152.4 38.1L30 38.1 76.2 76.2L476.2 38.1 152.4 76.2C10 0 76.2 76.2C276.2 0 152.4 76.2F1/N 0 6673.2F2/N 0 6673.2Int J Adv Manuf Technol5.1 Workpiece geometry and propertiesThe geometry and features of the workpiece are shown inFig. 4. The material of the hollow workpiece is aluminum390 with a Poisson ration of 0.3 and Youngs modulus of71 Gpa. The outline dimensions are 152.4 mm127 mm76.2 mm. The one third top inner wall of the workpiece isundergoing an end-milling process and its cutter path is alsoshown in Fig. 4. The material of the employed fixtureelements is alloy steel with a Poisson ration of 0.3 andYoungs modulus of 220 Gpa.5.2 Simulating and machining operationA peripheral end milling operation is carried out on theexample workpiece. The machining parameters of theoperation are given in Table 2. Based on these parameters,the maximum values of cutting forces that are calculatedand applied as element surface loads on the inner wall ofthe workpiece at the cutter position are 330.94 N(tangential), 398.11 N (radial) and 22.84 N (axial). Theentire tool path is discretized into 26 load steps and cuttingforce directions are determined by the cutter position.5.3 Fixture design planThe fixture plan for holding the workpiece in the machiningoperation is shown in Fig. 5.Generally,the321 locatorprincipleisusedinfixturedesign.Thebasecontrols3degrees.One side controls two degrees, and another orthogonal sidecontrolsonedegree.Here,itusesfourlocators(L1,L2,L3andL4) on the Y=0 mm face to locate the workpiece controllingtwo degrees, and two clamps (C1, C2) on the opposite facewhere Y=127 mm, to hold it. On the orthogonal side, onelocator is needed to control the remaining degree, which isneglectedintheoptimalmodel.Thecoordinateboundsforthelocating/clamping regions are given in Table 3.Since there is no simple rule-of-thumb procedure fordetermining the clamping force, a large value of theclamping force of 6673.2 N was initially assumed to actat each clamp, and the normal and tangential contactstiffness obtained from a least-squares fit to Eq. (5) are4.43107N/m and 5.47107N/m separately.5.4 Genetic control parameters and penalty functionThe control parameters of the GA are determined empiri-cally. For this example, the following parameter values areFig. 6 Convergence of GA for fixture layout and clamping forceoptimization procedureFig. 7 Convergence of the first function valuesFig. 8 Convergence of the second function valuesTable 4 Result of the multi-objective optimization modelMulti-objective optimizationX /mm Z /mmL117.102 30.641L2108.169 25.855L321.315 56.948L4127.846 60.202C122.989 62.659C2117.615 25.360F1/N 167.614F2/N 382.435f1/mm 0.006568/mm 0.002683Int J Adv Manuf Technolused: Ps=30, Pc=0.85, Pm=0.01, Nmax=100 and Ncmax=20. The penalty function for f1and is fvfv 50Here fvcan be represented by f1or . When Nchgreaches 6the probability of crossover and mutation will be changeinto 0.6 and 0.1 separately.5.5 Optimization resultThe convergence behavior for the successive optimizationsteps is shown in Fig. 6, and the convergence behaviors ofcorresponding functions (1) and (2) are shown in Fig. 7 andFig. 8. The optimal design scheme is given in Table 4.5.6 Comparison of the resultsThe design variables and objective function values offixture plans obtained from single objective optimizationand from that designed by experience are shown in Table 5.The single objective optimization result in the paper 22isquoted for comparison. The single objective optimizationmethod has its preponderance comparing with that designedby experience in this example case. The maximumdeformation has reduced by 57.5%, the uniformity of thedeformation has enhanced by 60.4% and the maximumclamping force value has degraded by 49.4%. What couldbe drawn from the comparison between the multi-objectiveoptimization method and the single objective optimizationmethod is that the maximum deformation has reduced by50.2%, the uniformity of the deformation has enhanced by52.9% and the maximum clamping force value hasdegraded by 69.6%.The deformation distribution of themachined surfaces along cutter path is shown in Fig. 9.Obviously, the deformation from that of multi-objectiveoptimization method distributes most uniformly in thedeformations among three methods.With the result of comparison, we are sure to apply theoptimal locators distribution and the optimal clamping forceto reduce the deformation of workpiece. Figure 10 showsthe configuration of a real-case fixture.6 ConclusionsThis paper presented a fixture layout design and clampingforce optimization procedure based on the GA and FEM.The optimization procedure is multi-objective: minimizingthe maximum deformation of the machined surfaces andmaximizing the uniformity of the deformation. The ANSYSsoftware package has been used for FEM calculation offitness values. The combinationof GAand FEM isproven tobe a powerful approach for fixture design optimizationproblems.In this study, both friction effects and chip removaleffects are considered. In order to reduce the computationtime, a database is established for the chromosomes andfitness values, and the meshed workpiece FEA model isrepeatedly used in the optimization process.Table 5 Comparison of the results of various fixture design schemesExperimental optimization Single objective optimizationX/mm Z/mm X/mm Z/mmL112.700 12.700 16.720 34.070L2139.7 12.700 145.360 17.070L312.700 63.500 18.400 57.120L4139.700 63.500 146.260 58.590C112.700 38.100 5.830 56.010C2139.700 38.100 104.400 22.740F1/N 2482 444.88F2/N 2482 1256.13f1/mm 0.031012 0.013178/mm 0.014377 0.005696Fig. 9 Distribution of the deformation along cutter pathFig. 10 A real case fixture configurationInt J Adv Manuf TechnolThetraditionalfixturedesignmethodsaresingleobjectiveoptimization method or by experience. The results of thisstudy show that the multi-objective optimization method ismore effective in minimizing the deformation and uniform-ing the deformation than other two methods. 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