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附录1：外文翻译薄壁工件夹具设计的双重优化模型摘要加工过程中必须控制变形，尤其对于薄壁零件。影响加工变形程度和布局的两个主要方面是夹具布局及夹紧力。在本文中，夹具布局和变夹紧力的双重优化模型被运用到加工薄壁零件。首先，根据变形程度和分布考虑最佳夹具布局。然后基于变夹紧力对上述夹具布局进行优化。使用有限元法分析工件变形。采用遗传算法求解优化模型。最后通过实例分析，验证了分层优化设计方法可以进一步减少工件加工变形，提高加工变形均匀度。1.引言夹具用于保证机床中工件被定位和夹紧到正确的位置和方向。设计不佳将生产出产生形变的劣品。因此，应该合理设计出定位元件的位置、夹具及支撑元件，并且能有合理的夹紧力。通常情况下，这些装置都很大程度上依赖设计中的经验，据此选择夹具等元件安装位置并计算夹紧力。因此对于给定的加工工件，不能保证得出最优和接近最优的解。因此夹具布局和夹紧力优化成为夹具设计所需考虑的两个重要因素。合理选择定位元件和夹具，并计算夹紧力以保证工件变形最小化和均匀。本文提出双重优化方案用于夹具布局设计与变夹紧力的优化，目的是使加工元件表面的最大弹性变形得以降低以及最大化的均匀性变形。有限元分析软件在给定的夹紧力和切割力下计算工件的变形。随着遗传算法的发展，采用数学软件（MATLAB）直接解决优化问题。最后根据实例研究说明拟定方案的应用。2.文献评论随着有关行业对优化方案的广泛运用，夹具布局和夹紧力的优化近年来取得一些成果。King和Hutter提出了一种使用夹具 - 工件系统的刚体模型进行最佳夹具布局设计的方法，但考虑到接触刚度1。DeMeter使用刚体模型来分析和合成最佳夹具布局和最小夹紧力2。Li和Melkote采用非线性规划方法和接触弹性模型来解决布局优化问题3。Dengand Melkote 4提出了一种基于模型的框架，用于确定最小所需的夹紧力，以确保在加工过程中夹具工件的动态稳定性。大多数上述研究使用非线性规划方法，这通常没有给出最优解。夹具设计优化的问题是非线性的，因为目标函数和设计变量之间没有直接的分析关系，即加工表面误差和夹具参数（定位元件和夹具的位置以及夹紧力）之间。以前的研究人员已经表明，遗传算法（GA）是解决这些优化问题的有用技术。Vallapuzha等人使用空间坐标在基于GA的夹具布局优化中进行编码5。Krishnakumar等人使用GA找到最小化加工表面变形的夹具布局6。Krishnakumar等人 提出了一种迭代算法，通过交替地改变夹具布局和夹紧力，使切割过程中的工件弹性变形最小化7。Kaya使用GA和FEM找到2D工件中的最佳定位器和夹紧位置8。 Zhou等 提出了一种基于GA的方法，同时优化夹具布局和夹紧力9。然而，考虑摩擦和切屑的研究很少，缺乏动态夹紧力优化。3双重优化模型薄壁工件受到反作用力，夹紧力和切割力的影响。工件的变形与作用在其上的这些力直接相关。目标函数表示为最小化加工表面的最大弹性变形，并最大化夹具 - 工件系统中变形的均匀性。对于涉及p夹具元件 - 工件接触和n个加工载荷阶段的夹具，目标函数可以数学表达如下：其中k是加工模拟的第k步的加工区域的最大弹性变形，是k的平均值。夹具 - 工件系统必须满足几个约束才能有效地执行其功能。 优化模型中使用的三个约束描述如下：其中F ni是第i个接触点的法向力，是静摩擦系数，Fi和Fi是第i个接触点的切向力，pos（i）是第i个接触点V（ i）是第i个接触点的候选区域。首先，确保在加工过程中夹具元件和工件之间没有滑动是重要的，因此在所有接触点必须满足公式（3）所写的库仑摩擦约束。其次，所有的反作用力必须是正的，以便在加工过程中保持工件与定位器接触。 这个约束可以表示为公式（4）。第三，固定件 - 工件接触点的位置必须在合理的区域，以确保正确的夹具布局设计。 这个约束可以表示为公式（5）。通常，施加到工件的夹紧力在加工过程中是固定的，并且通常大于提供夹持稳定性所必需的力。其可能导致工件弹性变形过大。 沿刀具路径的不同位置可能需要不同的力。为了减少工件的变形，在加工过程中应提供动态夹紧力并施加在工件上。对于夹紧元件 - 工件接触和n个加工载荷阶段的问题，需要搜索q个夹紧力的最佳值n次，第j个搜索模型可以数学表达如下：无滑动约束和接触约束可以表示为式 （3）和式（4）。4. 优化方案优化过程如图1所示。 1设计可行的夹具布局并优化动态夹紧力。 在切割模型中计算最大切削力，并将力发送到有限元分析（FEA）模型。 优化过程创建一些夹具布局和夹紧力，也可以发送到FEA模型。 在有限元模块中，在一定的夹具布局下计算切削力和夹紧力下的加工变形。 然后将变形发送到优化程序，以搜索最佳夹具布局和动态夹紧力。 图1 双重优化过程4.1 GA应用于夹具优化GA是模拟生物在自然环境中的遗传和进化过程而形成的一种自适应全局优化概率搜索的遗传算法。遗传算法首先随机产生若干初始群体，利用目标函数构造适应度函数，根据适应度函数计算个体适应度，较高适应度的个体被 遗传到下一代群体中的概率较大，通过杂交，变异等遗传操作作产品生进化的下一代群体，如此反复操作，不断向更优解方向 进化，直至获得问题最优解优化夹具设计的GA程序将夹具布局和夹紧力作为设计变量，生成表示不同设计的x 1 y 1 z 1 . f 1 f 2。 并将字符串与自然进化的染色体进行比较。该GA找到最佳串映射到最佳夹具设计。表1反映GA中有一些主要因素选择。表1 选择GA的参数编码真实缩放秩选择交叉突变控制参数余数中级统一自适应由于GA很可能会生成夹具设计字符串，在进行加工负载时不会完全限制夹具。这些解决方案被认为是不可行的，用于驱动GA到可行的解决方案。如果不满足方程式（2）和（3）中的约束，则设计方案被认为是不可行的。方案基本上包括将高目标函数值分配给不可行方案，从而在GA的连续迭代中将其驱动到可行区域。 对于约束（4），当生成新的个体时，需要检查它们是否满足条件。为了简化检查，使用多边形表示候选区域。 数学软件中的多边形函数可用于帮助检查。基于最佳夹具布局x 1 y 1 z 1 .，然后通过GA搜索第j个动态夹紧力（设计变量）。 夹紧力根据确定的顺序产生弦，如f j1 f j2 . f jq。 每个字符串表示第j步中的一种最佳夹紧力解。 夹紧力来自q组n步，f 11 f 21 . f n1，f 12 f 22 .，f 1q f 2q . f nq是加工过程中的q动态夹紧力。搜索动态夹紧力的过程与用于确定夹具布局的过程类似。 所需的变化是使夹具布局不变，单独添加第j步的切削力，并且只读取工件的加工变形。 像夹具布局优化一样，应该检查优化模型中的限制条件。4.2 有限元分析本研究中有限元分析软件包用于计算工件变形。在我们的研究中可以使用半弹性接触模型和无接触弹性模型。在半弹性接触模型中，每个定位器或支撑件由三个正交弹簧表示，这些弹簧在X，Y和Z方向上提供约束，每个夹具与定位器相似，但在正常方向上具有夹紧力。正常方向的弹簧称为普通弹簧，另外两个弹簧称为切向弹簧。在这项工作中，考虑到刀具路径的切屑去除。加工过程中材料的去除会改变几何形状，工件的结构刚度也会发生变化。使用元件去除废屑技术对工具运动和切屑去除分析FEA模型。为了计算适合度值，对于每个加载步骤，存储位移。然后选择最大位移作为该夹具设计方案的适合度值。当FEA模型中存在大量节点时，适应度值计算成本高昂。 因此，有必要加快GA过程的计算。 随着一代人的染色体越来越相似。 在这项工作中，计算的适应度值存储在具有染色体和适应度值的SQL Server数据库中。 GA程序首先检查当前染色体的适应度值是否已经被计算，如果没有，则将夹具设计方案发送给有限元分析软件，否则适合度值取自数据库。5. 案例分析几何和特征如图1所示。 该工件采用3-2-1夹具布局。 在两个表面上添加了四个对称的夹紧力（F3，F4，F5和F6）。 在工件中间的两个对称位置加上两个辅助夹紧力（F 1和F 2），以减少变形。 工件材料为铝，泊松比为0.3，杨氏模量为71 Gpa。 切割力的最大值为162.235 N（切向）和137.9 N（径向）。 加工过程由36个过程模拟，每个工况中力大小相同。F 1和F 2是夹具设计优化的重点。 F 2的位置和值作为优化参数。 优化目标是在整个加工过程中最小化加工面上的最大弹性变形f 1，并使最小化。 根据经验确定GA的控制参数。 对于该示例，P s（群体大小）= 20，P c（交叉的概率）= 0.8，P m（突变的概率）= 0.2，N max（演化的最大数）= 70。 f 1和的惩罚函数为（f v）= f v + 50，其中f v表示f 1和两者。 当最佳适应度停滞变化达到6时，交叉和突变的概率将变为0.3和0.6。 图2. 工件和FEA模型 图3. 工件最佳适应值最适合值为44.458m。 设计变量和目标函数值如表2所示多目标优化方法与本例中的经验设计相比具有优势。最大变形量减少了22.6，变形均匀度提高了22.1，最大夹紧力下降了93.7。 表2.案例优化结果根据表2所示的最佳夹具布局搜索动态夹紧力。辅助夹紧力F1和F2的值作为优化参数。优化目标是最小化当前加工区域的最大弹性变形f1。在GA中，使用以下参数值：Ps=20，Pc=0.8，Pm = 0.2，Nmax=100。搜索到的动态夹紧力如图4 。 图4. 变夹紧力为了比较夹具解决方案，图5显示了加工面的变形分布。由于GA的随机性和混合布局与夹紧力之间的耦合，在动态夹紧力下的变形在经验设计下仍然大于相应的变形。在动态夹紧力的作用下，变形分布得到改善。 图5. 变形比较6. 结论本文提出了夹具布局和变夹紧力优化方法。变夹紧力优化程序基于最佳夹具布局。 本研究的结果表明，基于最佳夹具布局的变夹紧力优化方法可以最大限度地减少变形，最有效地使变形均匀。对于NC加工中的变形控制也是有意的。参考文献1King LS, and Hutter I., Theoretical Approach for Generating Optimal Fixturing Locations for Prismatic Workparts in Automated Assembly, Journal Manufacturing System, 1993, Vol.12, No.5, pp.409-416.2De Meter EC., Min-Max Load Model for Optimizing Machine Fixture Performance, ASME- Journal of Engineering Industry, 1995, Vol.117, No.2, pp.183-186.3 Bo L, Melkote SN., Improved Workpiece Location Accuracy through Fixture Layout Optimization, International Journal of Machine Tools & Manufacture, 1999, Vol.39, No.6, pp.871-883.4Deng HY, Melkote SN. Determination of Minimum Clamping Force for Dynamically Stable Fixturing, International Journal of Machine Tools & Manufacture, 2006, Vol.46, No.7-8, pp.847-857.5Vallapuzha S, De Meter EC, Choudhuri S, et al. An Investigation into the Use of Spatial Coordinates for the Genetic Algorithm Based Solution of the Fixture Layout Optimization Problem, International Journal of Machine Tools & Manufacture, 2002, Vol.42, No.2, pp.265-275.6Kulankara K, Melkote SN., Machining Fixture Layout Optimization Using the Genetic Algorithm, International Journal of Machine Tools & Manufacture, 2000, Vol.40, No.4, pp.579-598.7Kulankara K, Satyanarayana S, Melkote SN., Iterative Fixture Layout and Clamping Force Optimization Using the Genetic Algorithm, Journal of Manufacturing Science and Engineering, 2002, Vol.124, No.1, pp.119-125.8Kaya N., Machining Fixture Locating and Clamping Position Optimization Using Genetic Algorithms, Computers in Industry, 2006, Vol.57, No.2, pp.112-120.9Zhou XL, Zhang WH, Qin GH, On Optimizing Fixture Layout and Clamping Force Simultaneously Using Genetic Algorithm, Mechanical Science and Technology, 2005, Vol.24, No.3, pp.339-342, (in Chinese).附录2：外文原文A Dual Optimization Model of Fixture Design for the Thin-walled WorkpieceWeifang Chen, Hua Chen, Lijun Ni and Jianbin XueNanjing University of Aeronautics and Astronautics,No.29, Yudao Street, Nanjing 210016 China ,AbstractThe deformation must be controlled during machining, especially for the thin-walled workpiece. Fixture layout and clamping force are the major two aspects that influence the degree and distribution of machining deformation. In this paper, a dual optimization model of fixture layout and dynamic clamping force has been established for machining the thin-walled workpiece. First, an optimal fixture layout is generated by considering the deformation degree and distribution. Thereafter, dynamic clamping force are optimized based on the optimal fixture layout. The finite element method is used to analyze the workpiece deformation. A genetic algorithm is developed to solve the optimization model. Finally, an example is used to illustrate that a satisfactory result has been obtained, which is far superior to the experiential one. This optimization method can reduce the machining deformation effectively and improve the distribution condition.1. IntroductionA fixture is used to establish and maintain the required position and orientation of a workpiece in machine tool. A poor design can lead to undesirable workpiece deformation. Consequently, the positions of the locators, clamps and supports should be strategically designed and appropriate clamping forces should be applied. Typically, it relies heavily on the designers experience to choose the positions of the fixture elements and to determine the clamping forces. Thus there is no assurance that the resultant solution is optimal or near optimal for a given workpiece. Consequently, the fixture layout and the clamping forces optimization become two important aspects in fixture design. The positions of locators and clamps should be properly selected, and the clamping forces should be calculated so that the workpiece deformation is minimized and uniformed.In this paper, a dual optimization method is presented for the fixture layout design and dynamic clamping forces optimization. The objective is to minimize the maximumelastic deformation of the machined surfaces and maximize the uniformity of the deformation. The ANSYS software package is used to calculate the deformation of the workpiece under given clamping forces and cutting force. A genetic algorithm is developed and the direct search toolbox of MATLAB is employed to solve the optimization problem. Finally a case study is given to illustrate the application of the proposed approach.2. Literature reviewWith the wide applications of optimization methods in industry, fixture layout optimization and clamping forces optimization have gained some interesting in recent years. King and Hutter presented a method for optimal fixture layout design using a rigid body model of the fixture-workpiece system but accounting for the contact stiffness 1. DeMeter used a rigid body model for the analysis and synthesis of optimal fixture layouts and minimum clamping forces 2. Li and Melkote used a nonlinear programming method and a contact elasticity model to solve the layout optimization problem 3. Deng and Melkote 4 presented a model-based framework for determining the minimum required clamping forces that ensure the dynamic stability of a fixtured workpiece during machining.Most of the above studies used nonlinear programming methods, which often did not give global or near-global optimum solutions. The problem of fixture design optimization is nonlinear because there is no direct analytical relationship between the objective function and design variables, i.e., between the machined surface error and the fixture parameters (positions of locator and clamp, and clamping forces).Previous researchers had shown that genetic algorithm (GA) was a useful technique in solving such optimization problems. Vallapuzha et al. used spatial coordinates to encode in a GA based fixture layout optimization 5. Krishnakumar et al. used GA to find the fixture layout that minimized the deformation of the machined surface 6. Krishnakumar et al. presented an iterative algorithm that minimized the workpiece elastic deformation for the978 -1-4244-1579-3/07/$25.00 2007 IEEEcutting process by alternatively varying the fixture layout and clamping forces 7. Kaya used the GA and FEM to find the optimal locator and clamping positions in 2D workpiece 8. Zhou et al. presented a GA based method that optimizes fixture layout and clamping forces simultaneously 9. However, there were few studies taking friction and chip removal into account and had lack on dynamic clamping forces optimization.3. A dual optimization modelA thin-walled workpiece is subject to reaction forces, clamping forces and cutting force. The deformation of the workpiece is directly related to these forces acting on it. The objective function is expressed as minimize the maximum elastic deformation of the machined surfaces and maximize the uniformity of the deformation in the fixtureworkpiece system.For a fixture involving p fixture element- workpiece contacts and n machining load steps, the objective function can be mathematically stated as followsmin(max(1,2,k,n) ,k = 1,n (1)n2= min( )(2) ( k )min/ nk =1where k refers to maximumelasticdeformation at amachining region in the kth step of the machiningsimulation,is the average ofk.A fixture-workpiece system has to satisfy several constraints to effectively perform its functions. Three constraints used in the optimization model are describedas follows(3)22Fni Fi+ Fi(4)Fni 0pos(i)V (i) , i = 1, 2, , p(5)where Fni is the normal force at the ith contact point, is the static coefficient of friction, Fi and Fi are the tangential force at the ith contact point, pos(i) is the ith contact point, V(i) is the candidate region of the ith contact point.First, it is important to ensure that there is no slip between the fixture elements and workpiece during machining, so the Coulomb friction constraint written as Eq. (3) must be satisfied at all contact points. Secondly, all of the reaction forces must be positive in order to keep the workpiece in contact with the locators during the machining process. This constraint can be expressed as Eq. (4) . Thirdly, positions of fixture element-workpiece contact points must be in the reasonable regions to ensure proper fixture layout design. This constraint can be expressed as Eq. (5).Typically, the clamping forces applied to the workpieceare fixed during the machining process and usually larger than necessary to provide fixturing stability. It may cause excessive workpiece elastic deformation. Different forces may be required at different positions along the tool path. In order to decrease the deformation of the workpiece, dynamic clamping forces should be provided and applied on the workpiece during the machining process.For a problem involving q clamping element-workpiece contacts and n machining load steps, the optimal values of q clamping forces need searched n times, and the jth searching model can be mathematically stated as followsmin(j) j=1,2,n(6)The no slip constraint and the contact constraint can be expressed as Eq. (3) and Eq. (4).4. Optimization methodThe optimization process is illustrated in Fig. 1 to design a feasible fixture layout and optimize the dynamic clamping forces. The maximum cutting force is calculated in cutting model and the forces are sent to finite element analysis (FEA) model. Optimization procedure creates some fixture layout and clamping forces which are sent to the FEA model too. In FEA block, machining deformation under the cutting force and clamping forces are calculated under a certain fixture layout. And the deformation is then sent to optimization procedure to search for an optimal fixture layout and dynamic clamping forces.Figure 1. A dual optimization process4.1. GA applied to fixture optimizationGA is robust, stochastic and heuristic optimization method based on biological reproduction processes. Each individual candidate is assigned a fitness value through a fitness function tailored to the specific problem. The GA then uses 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 leads to an increase in the global fitness of the population and the fittest individuals represent the best solutions.The GA procedure to optimize fixture design takes fixture layout and clamping forces as design variables to generate strings x1y1z1 f1f2 which represent different design. And the strings are compared to chromosomes of natural evolution. The optimal string which GA finds ismapped to the optimal fixture design.There are some main factors in GA which are selected as what is listed in Table 1.Table 1. Selection of GAs parametersEncodingRealScalingRankSelectionRemainderCrossoverIntermediateMutationUniformControl parameterSelf-adaptingSince 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 if it does not satisfy the constraints in Eqs. (2) and (3). The penalty method essentially involves assigning a high objective function value to the infeasible scheme, thus driving it to the feasible region in successive iterations of GA. For constraint (4), when new individuals are generated, it is necessary to check up whether they satisfy the condition. The genuine candidate regions are those candidate regions excluding invalid regions. In order to simplify the checking, polygons are used to represent the candidate regions. The polygon function in MATLAB could be used to help the checking.Based on the optimal fixture layout x1y1z1, the jth dynamic clamping forces(design variables) is then searched by GA. The clamping forces generate strings according to determinate order, such as fj1 fj2fjq . Each string represents a kind of optimal clamping forces solution in the jth step. Clamping forces got from qgroups of n steps, f11 f21fn 1，f12 f22，f1q f2qfnq, are the q dynamic clamping forces in the machining process.The process for searching dynamic clamping forces is similar to which used to determine fixture layout. Changes needed are making fixture layout a constant, adding cutting force of the jth step alone, and only reading out machining deformation of workpiece. Like fixture layout optimization, restriction conditions in optimization model should be checked.4.2. Finite element analysisThe software package of ANSYS is used for calculating the workpiece deformation in this study. The semi-elastic contact model and no contact elastic model can be used in our study. In a semi-elastic contact model, each locator or support is represented by three orthogonal springs that provide restrain 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.In this work, chip removal from the tool path is takeninto account. The removal of the material during machining alters the geometry, so does the structural stiffness of the workpiece. 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, displacements are stored for each load step. Then the maximum displac