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ORIGINAL ARTICLE Deformation control through fixture layout design and clamping force optimization Weifang Chen Δ 2 jj; :::; Δ j C12 C12 C12 C12 ; :::; Δ n jj C0C1 t s ; j ? 1; 2; :::; n e1T Subject to m F ni jjC21 ?????????????????? F 2 ti t F 2 hi q e2T F ni C21 0 e3T pos ieT2VieT; i ? 1; 2; :::; p e4T where Δ j refers to the maximum elastic deformation at a machining region in the j-th step of the machining operation, σ? ?????????????????????????????????????? X n j?1 Δj C0Δ C0 C16C17 2 C30 n v u u t Δ is the average of Δ j F ni is the normal force at the i-th contact point μ is the static coefficient of friction F ti ; F hi are the tangential forces at the i-th contact point pos(i) is the i-th contact point V(i) is the candidate region of the i-th contact point. The overall process is illustrated in Fig. 1 to design a feasible fixture layout and to optimize the clamping force. The maximal cutting force is calculated in cutting model and the force is sent to finite element analysis (FEA) model. Optimization procedure creates some fixture layout and clamping force which are sent to the FEA model too. In FEA block, machining deformation under the cutting force and the clamping force is calculated using finite element method under a certain fixture layout, and the deformation is then sent to optimization procedure to search for an optimal fixture scheme. 4 Fixture layout design and clamping force optimization 4.1 A genetic algorithm Genetic algorithms (GA) are robust, stochastic and heuristic optimization methods based on biological reproduction processes. The basic idea behind GA is to simulate “survival of the fittest” phenomena. Each individual candidate in the population is assigned a fitness value through a fitness function tailored to the specific problem. The GA then conducts reproduction, crossover and mutation processes to eliminate unfit individuals and the population evolves to the next generation. Sufficient number of evolutions of the population based on these operators lead to an increase in the global fitness of the population and the fittest individual represents the best solution. The GA procedure to optimize fixture design takes fixture layout and clamping force as design variables to generate strings which represent different layouts. The strings are compared to the chromosomes of natural evolution, and the string, which GA find optimal, is mapped to the optimal fixture design scheme. In this study, the genetic algorithm and direct search toolbox of MATLAB are employed. The convergence of GA is controlled by the population size (P s ), the probability of crossover (P c )andthe probability of mutations (P m ). Only when no change in the best value of fitness function in a population, N chg , reaches a pre-defined value NC max , or the number of generations, N, reaches the specified maximum number of evolutions, N max ., did the GA stop. There are five main factors in GA, encoding, fitness function, genetic operators, control parameters and con- straints. In this paper, these factors are selected as what is listed in Table 1. Since GA is likely to generate fixture design strings that do not completely restrain the fixture when subjected to machining loads. These solutions are considered infeasible and the penalty method is used to drive the GA to a feasible solution. A fixture design scheme is considered infeasible or unconstrained if the reactions at the locators are negative, in other words, it does not satisfy the constraints in equations (2)and(3). The penalty method essentially involves Machining Process Model FEA Optimization procedure cutting forces fitness Optimization result Fixture layout and clamping force Fig. 1 Fixture layout and clamp- ing force optimization process Table 1 Selection of GA’s parameters Factors Description Encoding Real Scaling Rank Selection Remainder Crossover Intermediate Mutation Uniform Control parameter Self-adapting Int J Adv Manuf Technol assigning a high objective function value to the scheme that is infeasible, thus driving it to the feasible region in successive iterations of GA. For constraint (4), when new individuals are generated by genetic operators or the initial generation is generated, it is necessary to check up whether they satisfy the conditions. The genuine candidate regions are those excluding invalid regions. In order to simplify the checking, polygons are used to represent the candidate regions and invalid regions. The vertex of the polygons are used for the checking. The “inpolygon” function in MATLAB could be used to help the checking. 4.2 Finite element analysis The software package of ANSYS is used for FEA calculations in this study. The finite element model is a semi-elastic contact model considering friction effect, where the materials are assumed linearly elastic. As shown in Fig. 2, each locator or support is represented by three orthogonal springs that provide restrains in the X, Y and Z directions and each clamp is similar to locator but clamping force in normal direction. The spring in normal direction is called normal spring and the other two springs are called tangential springs. The contact spring stiffness can be calculated according to the Herz contact theory [8] as follows k iz ? 16R C3 i E C32 i 9 C16C171 3 f iz 1 3 k iz ? k iy ? 6 E C3 i 2C0v fi G fi t 2C0v wi G wi C16C17 C01 C1 k iz 8 > : e5T where k iz , k ix , k iy are the tangential and normal contact stiffness, 1 R C3 i ? 1 R wi t 1 R fi is the nominal contact radius, 1 E C3 i ? 1C0 V 2 wi E wi t 1C0 V 2 fi E fi is the nominal contact elastic modulus, R wi , R fi are radius of the i-th workpiece and fixture element, E wi , E fi are Young’s moduli for the i-th workpiece and fixture materials, ν wi , ν fi are Poisson ratios for the i-th workpiece and fixture materials, G wi , G fi are shear moduli for the i-th workpiece and fixture materials and f iz is the reaction force at the i-th contact point in the Z direction. Contact stiffness varies with the change of clamping force and fixture layout. A reasonable linear approximation of the contact stiffness can be obtained from a least-squares fit to the above equation. The continuous interpolation, which is used to apply boundary conditions to the workpiece FEA model, is Fig. 2 Semi-elastic contact model taking friction into account Spring position Fixture element position 1234567 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Fig. 3 Continuous interpolation Fig. 4 A hollow workpiece Table 2 Machining parameters and conditions Parameter Description Type of operation End milling Cutter diameter 25.4 mm Number of flutes 4 Cutter RPM 500 Feed 0.1016 mm/tooth Radial depth of cut 2.54 mm Axial depth of cut 25.4 mm Radial rake angle 10 Helix angle 30 Projection length 92.07 mm Int J Adv Manuf Technol illustrated in Fig. 3. Three fixture element locations are shown as black circles. Each element location is surrounded by its four or six nearest neighboring nodes. These sets of nodes, 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 of these nodes. For any set of nodes, the spring constant is k ij ? d ij P k2h i d ik k i e6T where k ij is the spring stiffness at the j-th node surrounding the i-th fixture element, d ij is the distance between the i-th fixture element and the j-th node surrounding it, k i is the spring stiffness at the i-th fixture element location. η i is the number of nodes surrounding the i-th fixture element location. For each machining load step, appropriate boundary conditions have to be applied to the finite element model of the workpiece. In this work, the normal springs are constrained in the three directions (X, Y, Z)andthe tangential springs are constrained in the tangential direc- tions (X, Y). Clamping forces are applied in the normal direction (Z) at the clamp nodes. The entire tool path is simulated for each fixture design scheme generated by the GA by applying the peak X, Y, Z cutting forces sequentially to the element surfaces over which the cutter passes [23]. In this work, chip removal from the tool path is taken into account. The removal of the material during machining alters the geometry, so does the structural stiffness of the workpiece. Thus, it is necessary to consider chip removal affects. The FEA model is analyzed with respect to tool movement and chip removal using the element death technique. In order to calculate the fitness value for a given fixture design scheme, displacements are stored for each load step. Then the maximum displacement is selected as fitness value for this fixture design scheme. The interaction between GA procedure and ANSYS is implemented as follows. Both the positions of locators and clamps, and the clamping force are extracted from real strings. These parameters are written to a text file. The input batch file of ANSYS could read these parameters and calculate the deformation of machined surfaces. Thus the fitness values in GA procedure can also be written to a text file for current fixture design scheme. It is costly to compute the fitness value when there are a largenumberofnodesinanFEMmodel.Thusitisnecessary to speed up the computation for GA procedure. As the generation goes by, chromosomes in the population are getting similar. In this work, calculated fitness values are stored in a SQL Server database with the chromosomes and fitness values. GA procedure first checks if current chromosome’s fitness value has been calculated before, if not, fixture design scheme are sent to ANSYS, otherwise fitness values are directly taken from the database. The meshing of workpiece FEA model keeps same in every calculating time. The difference among every calculating model is the boundary conditions. Thus, the meshed workpiece FEA model could be used repeatedly by the “resume” command in ANSYS. 5 Case study An example of milling fixture design optimization problem for a low rigidity workpiece displayed in previous research papers [16, 18, 22] is presented in the following sections. Fig. 5 Candidate regions for the locators and clamps Table 3 Bound of design variables Minimum Maximum X /mm Z /mm X /mm Z /mm L 1 0 0 76.2 38.1 L 2 76.2 0 152.4 38.1 L 3 0 38.1 76.2 76.2 L 4 76.2 38.1 152.4 76.2 C 1 0 0 76.2 76.2 C 2 76.2 0 152.4 76.2 F 1 /N 0 6673.2 F 2 /N 0 6673.2 Int J Adv Manuf Technol 5.1 Workpiece geometry and properties The geometry and features of the workpiece are shown in Fig. 4. The material of the hollow workpiece is aluminum 390 with a Poisson ration of 0.3 and Young’s modulus of 71 Gpa. The outline dimensions are 152.4 mm×127 mm× 76.2 mm. The one third top inner wall of the workpiece is undergoing an end-milling process and its cutter path is also shown in Fig. 4. The material of the employed fixture elements is alloy steel with a Poisson ration of 0.3 and Young’s modulus of 220 Gpa. 5.2 Simulating and machining operation A peripheral end milling operation is carried out on the example workpiece. The machining parameters of the operation are given in Table 2. Based on these parameters, the maximum values of cutting forces that are calculated and applied as element surface loads on the inner wall of the workpiece at the cutter position are 330.94 N (tangential), 398.11 N (radial) and 22.84 N (axial). The entire tool path is discretized into 26 load steps and cutting force directions are determined by the cutter position. 5.3 Fixture design plan The fixture plan for holding the workpiece in the machining operation is shown in Fig. 5.Generally,the3–2–1 locator principleisusedinfixturedesign.Thebasecontrols3degrees. One side controls two degrees, and another orthogonal side controlsonedegree.Here,itusesfourlocators(L1,L2,L3and L4) on the Y=0 mm face to locate the workpiece controlling two degrees, and two clamps (C1, C2) on the opposite face where Y=127 mm, to hold it. On the orthogonal side, one locator is needed to control the remaining degree, which is neglectedintheoptimalmodel.Thecoordinateboundsforthe locating/clamping regions are given in Table 3. Since there is no simple rule-of-thumb procedure for determining the clamping force, a large value of the clamping force of 6673.2 N was initially assumed to act at each clamp, and the normal and tangential contact stiffness obtained from a least-squares fit to Eq. (5) are 4.43×10 7 N/m and 5.47×10 7 N/m separately. 5.4 Genetic control parameters and penalty function The control parameters of the GA are determined empiri- cally. For this example, the following parameter values are Fig. 6 Convergence of GA for fixture layout and clamping force optimization procedure Fig. 7 Convergence of the first function values Fig. 8 Convergence of the second function values Table 4 Result of the multi-objective optimization model Multi-objective optimization X /mm Z /mm L 1 17.102 30.641 L 2 108.169 25.855 L 3 21.315 56.948 L 4 127.846 60.202 C 1 22.989 62.659 C 2 117.615 25.360 F 1 /N 167.614 F 2 /N 382.435 f 1 /mm 0.006568 σ/mm 0.002683 Int J Adv Manuf Technol used: P s =30, P c =0.85, P m =0.01, N max =100 and N cmax = 20. The penalty function for f 1 and σ is φ f v eT?f v t 50 Here f v can be represented by f 1 or σ. When N chg reaches 6 the probability of crossover and mutation will be change into 0.6 and 0.1 separately. 5.5 Optimization result The convergence behavior for the successive optimization steps is shown in Fig. 6, and the convergence behaviors of corresponding functions (1) and (2) are shown in Fig. 7 and Fig. 8. The optimal design scheme is given in Table 4. 5.6 Comparison of the results The design variables and objective function values of fixture plans obtained from single objective optimization and from that designed by experience are shown in Table 5. The single objective optimization result in the paper [22]is quoted for comparison. The single objective optimization method has its preponderance comparing with that designed by experience in this example case. The maximum deformation has reduced by 57.5%, the uniformity of the deformation has enhanced by 60.4% and the maximum clamping force value has degraded by 49.4%. What could be drawn from the comparison between the multi-objective optimization method and the single objective optimization method is that the maximum deformation has reduced by 50.2%, the uniformity of the deformation has enhanced by 52.9% and the maximum clamping force value has degraded by 69.6%.The deformation distribution of the machined surfaces along cutter path is shown in Fig. 9. Obviously, the deformation from that of multi-objective optimization method distributes most uniformly in the deformations among three methods. With the result of comparison, we are sure to apply the optimal locators distribution and the optimal clamping force to reduce the deformation of workpiece. Figure 10 shows the configuration of a real-case fixture. 6 Conclusions This paper presented a fixture layout design and clamping force optimization procedure based on the GA and FEM. The optimization procedure is multi-objective: minimizing the maximum deformation of the machined surfaces and maximizing the uniformity of the deformation. The ANSYS software package has been used for FEM calculation of fitness values. The combinationof GAand FEM isproven to be a powerful approach for fixture design optimization problems. In this study, both friction effects and chip removal effects are considered. In order to reduce the computation time, a database is established for the chromosomes and fitness values, and the meshed workpiece FEA model is repeatedly used in the optimization process. Table 5 Comparison of the results of various fixture design schemes Experimental optimization Single objective optimization X/mm Z/mm X/mm Z/mm L 1 12.700 12.700 16.720 34.070 L 2 139.7 12.700 145.360 17.070 L 3 12.700 63.500 18.400 57.120 L 4 139.700 63.500 146.260 58.590 C 1 12.700 38.100 5.830 56.010 C 2 139.700 38.100 104.400 22.740 F 1 /N 2482 444.88 F 2 /N 2482 1256.13 f 1 /mm 0.031012 0.013178 σ/mm 0.014377 0.005696 Fig. 9 Distribution of the deformation along cutter path Fig. 10 A real case fixture configuration Int J Adv Manuf Technol Thetraditionalfixturedesignmethodsaresingleobjective optimization method or by experience. The results of this study show that the multi-objective optimization method is more effective in minimizing the deformation and uniform- ing the deformation than other two methods. It is meaningful for machining deformation control in NC machining. References 1. 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