（外文翻译-英）端面铣削加工夹具布局的设计与优化.pdf

ORIGINAL ARTICLE Design and optimization of machining fixture layout for end-milling operation K.A.Sundararamanx2;x3;:xn ? 1 where y is the desired response, f is the response function (or response surface), x1, x2, x3, . . . xnare the independent input variables, and is the fitting error. The expected response f, when plotted, appears as a surface. The identification of suitable approximation of f will determine the success of RSM application. The necessary data for building the re- sponse models are generally collected by the design of exper- iments. In this study, the collection of experimental data adopts a standard RSM design, central composite design (CCD) and the approximation of f will be proposed using the fitted second-order polynomial regression model known asquadraticmodel.Thequadraticmodeloffcanbewrittenas: y o X i1 k ixi X i1 k iix2 i XX iF” is less than 0.1, and the test of lack-of-fit is also insignificant as its “Prob.F” value is still greater than 0.1. The important coefficient R2, a measure of the degree of fit, approaches to unity; the response ofthe modelfits theactualdata.ThevalueofR2calculatedfor this reduced model is 0.9538 and is reasonably close to unity, which is acceptable. It denotes that about 95.38 % of the variability in the data is explained by this model. It also confirms that this model provides an excellent explanation of the relationship between the design parameters and the response (maximum deformation). The prediction error sum of squares (PRESS) provides useful residual scaling, and this model can “explain” 84.88 % of the variability in predicting a new observation. 4.4 Model development Next, regression analysis is performed with the quadratic model of response surface to derive the terms and coefficients of the approximation function. The developed quadratic mod- el reveals the relationship between design parameters and desired response. Through the backward elimination process, the final quadratic models of response equation in terms of coded factors are presented as follows: Maximum deformation Y Y 0:025597557360:00105525400 ? A0:00021043837 ? B 0:00036419956 ? C 0:00010435019 ? D 0:00037687864 ? E 0:00070642251 ? AB0:00013329988 ? BC0:00015119835 ? BE 0:00033456455 ? AA 0:00069610035 ? BB3 In terms of the actual factors, the final quadratic model of response equation is 0.17653 0.02624 0.02309 0.02594 0.025 0.02596 0.02536 0.02624 0.03033 0.02553 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 102030405060708090100 Experimental runs Minimum value of maximum deformation in mm Minimum value of maximum deformation in mm Fig. 5 Variation of minimum of maximum deformation with respect to the experimental runs Table 1 First and second minimum values of maximum deformation of four sets Set no.Number of experimental runs First minimum value of maximum deformation (mm) Second minimum value of maximum deformation (mm) 1600.026270.02637 2600.026140.02785 3600.026860.02753 4600.025960.02744 674Int J Adv Manuf Technol (2014) 73:669679 Maximum deformation Y Y 0:144265543540:00198229123 ? A0:00161575613 ? B 0:00021432897 ? C 0:00003478340 ? D 0:00015604447 ? E 0:00001381823 ? AB0:00000466083 ? BC0:00000352444 ? BE 0:00000654437 ? AA 0:00001361632 ? BB4 4.4.1 Comparison of the model results with simulation results The comparison is made between the values of maximum deformation obtained by simulation and mathematical model. The error percentage is obtained by the following equation. Error % YmodelYsimulation Ysimulation ? ? 1005 where Ymodelis the workpiece deformation obtained from the developed model and Ysimulationis the workpiece deformation obtained from the simulation. A comparison of the simulated and predicted values of maximum workpiece deformation is presented in Fig. 7. The error percentage that lies between 4.11881 and 3.71263 is smaller and acceptable. It is evident that the second-order polynomial regression model, the quadratic model, has better predictionsduetothe presenceofsquareandinteractionterms (nonlinear) in the analysis. The developed model is used to predict the maximum deformation within the limits of the factors studied. 5 Optimization of design parameters In this study, the problem concerned with minimization of maximum workpiece deformation is subjected to design pa- rameters of the position of locators L1(A), L2(B), and L3(C) and clamps C1(D) and C2(E). In other words, this work aims to find the appropriate values for these design parameters (X), which minimize the maximum workpiece deformation (Y) during the end-milling operation. Find X A;B;C;D;E?6 to minimize Y fX 7 Subject to 105:1A119:4 mm8a 0.02 0.021 0.022 0.023 0.024 0.025 0.026 0.027 0.028 0.029 0.03 1234 Set Number Minimum value of maximum deformation in mm First minimum of maximum deformation second minimum of maximum deformation 0.026268999 0.026373238 0.027854032 0.0275280350.027436314 0.026142181 0.026857117 0.025962025 Fig. 6 Consistency of minimum of maximum deformation with respect to experimental sets Table2 Inputdesignparametersfortwoconsecutiveminimumvaluesof maximum deformation L1L2L3C1C2Two consecutive minimum values ofmaximumdeformation 119.40 33.60 61.00 29.00 125.00 0.02596 105.10 19.30 53.00 23.00 113.00 0.02743 Table 3 Design parameters and their ranges SymbolDesign parameterUnitLevels Low (1)High (+1) APosition of locator L1mm105.1119.4 BPosition of locator L2mm19.333.6 CPosition of locator L3mm53.061.0 DPosition of clamp C1mm23.029.0 EPosition of clamp C2mm113.0125.0 Int J Adv Manuf Technol (2014) 73:669679675 Table 4 Experimental design matrix and maximum deformation in end-milling process Exp. no. Design parametersExperimental results ABCDEMaximumdeformation (mm)Position of locator L1 Position of locator L2 Position of locator L3 Position of clamp C1 Position of clamp C2 1119.4033.6053.0023.00113.000.02555 2112.2526.4557.0018.86119.000.02543 3119.4033.6053.0029.00125.000.02596 4112.2543.4657.0026.00119.000.03028 5112.2526.4557.0026.00119.000.02564 6105.1033.6053.0029.00125.000.02665 7112.2526.4547.4926.00119.000.02474 8119.4033.6061.0029.00125.000.02596 9105.1019.3053.0029.00113.000.02779 10119.4019.3061.0023.00125.000.02607 11105.1033.6053.0023.00125.000.02636 12105.1019.3061.0029.00125.000.02992 13119.4019.3061.0029.00113.000.02531 14129.2626.4557.0026.00119.000.02481 15112.2526.4557.0033.14119.000.02602 16112.259.4457.0026.00119.000.02896 17119.4019.3061.0023.00113.000.02504 18119.4019.3053.0023.00125.000.02516 19112.2526.4557.0026.00119.000.02564 2095.2426.4557.0026.00119.000.03035 21105.1019.3053.0029.00125.000.02898 22119.4033.6061.0023.00125.000.02602 23119.4019.3053.0029.00125.000.02551 24105.1033.6053.0029.00113.000.02578 25105.1019.3053.0023.00125.000.02859 26105.1033.6061.0029.00113.000.02646 27112.2526.4557.0026.00119.000.02564 28105.1019.3053.0023.00113.000.02744 29112.2526.4557.0026.00119.000.02564 30105.1019.3061.0023.00125.000.02966 31112.2526.4557.0026.00119.000.02564 32119.4033.6053.0023.00125.000.02599 33119.4033.6053.0029.00113.000.02573 34119.4033.6061.0023.00113.000.02597 35112.2526.4557.0026.00119.000.02564 36112.2526.4557.0026.00119.000.02564 37105.1033.6061.0023.00113.000.02612 38119.4019.3053.0023.00113.000.02415 39112.2526.4557.0026.00119.000.02564 40105.1033.6061.0029.00125.000.02749 41112.2526.4557.0026.00119.000.02564 42112.2526.4557.0026.00119.000.02564 43119.4019.3053.0029.00113.000.02452 44112.2526.4557.0026.00104.730.02454 45105.1019.3061.0029.00113.000.02869 46112.2526.4566.5126.00119.000.02692 676Int J Adv Manuf Technol (2014) 73:669679 19:3B33:6 mm8b 53C61 mm8c 23D29 mm8d 113E125 mm8e 5.1 Sequential approximation optimization The optimization problem can be approximated by the Eq. (4) and then solved by means of SAO method. The SAO strategy applies the approximate procedure, and the objective function is used as desirability function. The estimated response is transformed into a scale-free value called desirability, and it ranges from zero outside of the limits to one at the goal. Desirability function has been used to determine the optimum design parameters of position of locators and clamps. The value of input design parameters that maximizes the desirabil- ity is to be considered as the optimal condition for machining fixture layout. In this research work, the goal for the response is selected as minimum and the characteristics of a goal may be changed by adjusting the weight or importance. A maxi- mum and minimum level is provided for each design param- eter. Goals and limits have been established for each design parameters individually in order to determine their impact on individual desirability. Weights have been assigned to give added emphasis to upper/lower bounds or to emphasize a target value. The value of the desirability is completely de- pendent on the closeness of lower and upper limits relative to the actual optimum value and is not always necessary to be 1. A set of conditions possessing the highest desirability value have been selected as the optimum conditions for machining fixture layout. Figure 8 shows optimized maximum workpiece deforma- tion with the corresponding values of design parameters. Minimum value of maximum deformation is represented as 0.0238 mm. The method that uses FEM directly interfaced Table 4 (continued) Exp. no. Design parametersExperimental results ABCDEMaximumdeformation (mm)Position of locator L1 Position of locator L2 Position of locator L3 Position of clamp C1 Position of clamp C2 47105.1033.6053.0023.00113.000.02611 48112.2526.4557.0026.00133.270.02611 49105.1019.3061.0023.00113.000.02856 50105.1033.6061.0023.00125.000.02722 51119.4033.6061.0029.00113.000.02606 52119.4019.3061.0029.00125.000.02631 0.0237 0.0247 0.0257 0.0267 0.0277 0.0287 0.0297 0.0307 147101316192225283134374043464952 Experiment number Maximum deformation in mm Experimental results(Simulation) Predicted results Fig. 7 Comparison of maximum deformation by simulation and prediction Int J Adv Manuf Technol (2014) 73:669679677 with GA is employed to optimize the machining fixture lay- out, the minimum value of the maximum deformation is obtained with a huge computational time, while the method that uses RSM and SAO produces very competitive results in a lesser computational time. 5.2 Optimization using LINGO Many literatures show that researchers used LINGO solver to evaluate the performance of their proposed heuristic algo- rithms in terms of the solution quality and is used to solve programmingproblems.Solutionqualityisevaluatedbycom- paring the solution obtained from SAO and LINGO. Table 5 shows the comparison of optimal results and the corresponding values of design parameters for the developed quadratic model obtained by LINGO and SAO. It is clear that the result obtained from SAO surpasses LINGO. 6 Conclusions Inthisstudy,aneffectiveprocedureofRSMisusedtodevelop the mathematical model to determine the value of maximum workpiece deformation caused during end-milling operation. The developed model correlates the maximum workpiece deformation and position of the locators, clamps in the end- Fig. 8 Optimal position for the design variables in SAO Table 5 Comparison of the re- sults obtained by SAO and by LINGO for the case study ParametersUnitSAOLINGO Initial valueOptimal valueOptimal value Position of locator L1, Amm105.1119.4000119.4000 Position of locator L2, Bmm19.322.443621.85473 Position of locator L3, Cmm5353.000053.0000 Position of clamp C1, Dmm2323.000023.0000 Position of clamp C2, Emm113113.0000113.0000 Maximum deformation, Ymm0.03180.02380.02632 678Int J Adv Manuf Technol (2014) 73:669679 milling operation. It is used to predict the maximum deforma- tionYandtofindappropriatepositionsforthelocators,clamps in order to minimize the maximum deformation Y of the workpiece caused during end-milling operation. Based on the above study, the following conclusions are made. 1.The developed model correlates the maximum workpiece deformation with the position of the locators and clamps, with a good degree of approximation. 2.For minimizing the value of maximum workpiece defor- mation, the main effect of the position of locator L1 and L2 plays a significant role as an influencing factor. Interaction effect of the position of locator L1 with L2 and that of locator L2 with the position of clamp C2 and the second-order effect of position of locator L1 and L2 also affect the workpiece deformation in a significant manner. 3.The predictive model can be used for designing fixture layout without undergoing costly trial and error procedures. 4.Using SAO method, the five different design parameters with the optimum adjustment are acquired and the mini- mal values of Y have been predicted and verified. From the comparison of the predicted and simulated results, the predictive model developed using RSM is reasonably accurate and can be used for describing minimum value of maximum deformation of the workpiece within the limits of the factors studied. 5.Optimal results of the developed model attained by SAO method is compared against the results obtained by LINGO solver, and the comparison shows that the result of SAO is better than LINGO. 6.The limitation of this model is that it is valid only within the studied range of the design parameters. Also, the results obtained from the model are only applicable to the given workpiece (shape and size) and magnitude of the verticaland horizontalcutting forcesconsidered inthe present work. Though the results are specific, the ap- proach is generic and it can further be extended to solve fixture layout problems having any complex geometrical shapes of the workpiece. 7.Without tedious computational complexity, this model can be easily employed in the real production environ- ment to find the suitable position of the locators and clamps in order to minimize maximum workpiece defor- mation during clamp actuation in end-milling operation. Thus, large volume of experimentation work is reduced which results in very short computational time for this prediction method. AcknowledgmentsThe authors would like to thank the anonymous reviewers and the editor for their insightful comments and suggestions. References 1. Kang X, Peng Q (2009) Recent research on computer-aided fixture planning. Recent Patents Mech Eng 2:818 2. Vallapuzha S, De Meter EC, Choudhuri S, Khetan RP (2002) An investigation of the effectiveness of fixture layout optimization methods. Int J Mach Tools Manuf 42(2):251263 3. Menassa RJ, DeVries WR (1991) Optimization methods applied selecting support positions in fixture design. J Eng Ind ASME Trans 113:412418 4. Jayaram S, El-Khasawneh BS, Beutel DE, Merchant ME (2000) A fast analytical method to compute optimum stiffness of fixturing locators. CIRPAnn Manuf Technol 49(1):317320 5. Wang XC, LiuQ,Gindy N (2006) Optimisation ofmachiningfixture layout under multi-constraints. Int J Mach Tools Manuf 46:1291 1300 6. Pelinescu DM, Wang MY (2002) Multi-objective optimal fixture layout design. Robot Comput Integr Manuf 18:365372 7. Marin RA, Ferreira PM (2003) Analysis of the influence of fixture locator errors on the compliance of work part features to geometric tolerance specifications. J Manuf Sci Eng 125(3):609, 8 pages 8. ANSYS 8.0, ANSYS Inc., 2003 9. Kaya N (2005) Machining fixture locating and clamping position optimizat