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沈阳理工大学学士学位论文附录二 ：中文翻译 通过夹具布局设计和夹紧力的优化控制变形摘 要工件变形必须控制在数值控制机械加工过程之中。夹具布局和夹紧力是影响加工变形程度和分布的两个主要方面。在本文提出了一种多目标模型的建立，以减低变形的程度和增加均匀变形分布。有限元方法应用于分析变形。遗传算法发展是为了解决优化模型。最后举了一个例子说明，一个令人满意的结果被求得， 这是远优于经验之一的。多目标模型可以减少加工变形有效地改善分布状况。关键词：夹具布局；夹紧力； 遗传算法；有限元方法1 引言夹具设计在制造工程中是一项重要的程序。这对于加工精度是至关重要。一个工件应约束在一个带有夹具元件，如定位元件，夹紧装置，以及支撑元件的夹具中加工。定位的位置和夹具的支力，应该从战略的设计，并且适当的夹紧力应适用。该夹具元件可以放在工件表面的任何可选位置。夹紧力必须大到足以进行工件加工。通常情况下，它在很大程度上取决于设计师的经验，选择该夹具元件的方案，并确定夹紧力。因此，不能保证由此产生的解决方案是某一特定的工件的最优或接近最优的方案。因此，夹具布局和夹紧力优化成为夹具设计方案的两个主要方面。 定位和夹紧装置和夹紧力的值都应适当的选择和计算，使由于夹紧力和切削力产生的工件变形尽量减少和非正式化。 夹具设计的目的是要找到夹具元件关于工件和最优的夹紧力的一个最优布局或方案。在这篇论文里， 多目标优化方法是代表了夹具布局设计和夹紧力的优化的方法。 这个观点是具有两面性的。一，是尽量减少加工表面最大的弹性变形； 另一个是尽量均匀变形。 ANSYS软件包是用来计算工件由于夹紧力和切削力下产生的变形。遗传算法是MATLAB的发达且直接的搜索工具箱，并且被应用于解决优化问题。最后还给出了一个案例的研究，以阐述对所提算法的应用。2 文献回顾随着优化方法在工业中的广泛运用，近几年夹具设计优化已获得了更多的利益。夹具设计优化包括夹具布局优化和夹紧力优化。King 和 Hutter提出了一种使用刚体模型的夹具-工件系统来优化夹具布局设计的方法。DeMeter也用了一个刚性体模型，为最优夹具布局和最低的夹紧力进行分析和综合。他提出了基于支持布局优化的程序与计算质量的有限元计算法。李和melkote用了一个非线性编程方法和一个联络弹性模型解决布局优化问题。两年后， 他们提交了一份确定关于多钳夹具受到准静态加工力的夹紧力优化的方法。他们还提出了一关于夹具布置和夹紧力的最优的合成方法，认为工件在加工过程中处于动态。相结合的夹具布局和夹紧力优化程序被提出，其他研究人员用有限元法进行夹具设计与分析。蔡等对menassa和devries包括合成的夹具布局的金属板材大会的理论进行了拓展。秦等人建立了一个与夹具和工件之间弹性接触的模型作为参考物来优化夹紧力与，以尽量减少工件的位置误差。Deng和melkote 提交了一份基于模型的框架以确定所需的最低限度夹紧力，保证了被夹紧工件在加工的动态稳定。大部分的上述研究使用的是非线性规划方法，很少有全面的或近全面的最优解决办法。所有的夹具布局优化程序必须从一个可行布局开始。此外，还得到了对这些模型都非常敏感的初步可行夹具布局的解决方案。夹具优化设计的问题是非线性的，因为目标的功能和设计变量之间没有直接分析的关系。例如加工表面误差和夹具的参数之间（定位、夹具和夹紧力）。以前的研究表明，遗传算法（ GA ）在解决这类优化问题中是一种有用的技术。吴和陈用遗传算法确定最稳定的静态夹具布局。石川和青山应用遗传算法确定最佳夹紧条件弹性工件。vallapuzha在基于优化夹具布局的遗传算法中使用空间坐标编码。他们还提出了针对主要竞争夹具优化方法相对有效性的广泛调查的方法和结果。这表明连续遗传算法取得最优质的解决方案。krishnakumar和melkote 发展了一个夹具布局优化技术，用遗传算法找到夹具布局，尽量减少由于在整个刀具路径的夹紧和切削力造成的加工表面的变形。定位器和夹具位置被节点号码所指定。krishnakumar等人还提出了一种迭代算法，尽量减少工件在整个切削过程之中由不同的夹具布局和夹紧力造成的弹性变形。Lai等人建成了一个分析模型，认为定位和夹紧装置为同一夹具布局的要素灵活的一部分。Hamedi 讨论了混合学习系统用来非线性有限元分析与支持相结合的人工神经网络（ ANN ）和GA。人工神经网络被用来计算工件的最大弹性变形，遗传算法被用来确定最佳锁模力。Kumar建议将迭代算法和人工神经网络结合起来发展夹具设计系统。Kaya用迭代算法和有限元分析，在二维工件中找到最佳定位和夹紧位置，并且把碎片的效果考虑进去。周等人。提出了基于遗传算法的方法，认为优化夹具布局和夹紧力的同时，一些研究没有考虑为整个刀具路径优化布局。一些研究使用节点数目作为设计参数。一些研究解决夹具布局或夹紧力优化方法，但不能两者都同时进行。 有几项研究摩擦和碎片考虑进去了。碎片的移动和摩擦接触的影响对于实现更为现实和准确的工件夹具布局校核分析来说是不可忽视的。因此将碎片的去除效果和摩擦考虑在内以实现更好的加工精度是必须的。在这篇论文中，将摩擦和碎片移除考虑在内，以达到加工表面在夹紧和切削力下最低程度的变形。一多目标优化模型被建立了。一个优化的过程中基于GA和有限元法提交找到最佳的布局和夹具夹紧力。最后，结果多目标优化模型对低刚度工件而言是比较单一的目标优化方法、经验和方法。3 多目标优化模型夹具设计一个可行的夹具布局必须满足三限制。首先，定位和夹紧装置不能将拉伸势力应用到工件；第二，库仑摩擦约束必须施加在所有夹具-工件的接触点。夹具元件-工件接触点的位置必须在候选位置。为一个问题涉及夹具元件-工件接触和加工负荷步骤，优化问题可以在数学上仿照如下： 这里的表示加工区域在加工当中j次步骤的最高弹性变形。其中是的平均值；是正常力在i次的接触点；是静态摩擦系数；fhi是切向力在i次的接触点；pos(i)是i次的接触点；是可选区域的i次接触点；整体过程如图1所示，一要设计一套可行的夹具布局和优化的夹紧力。最大切削力在切削模型和切削力发送到有限元分析模型中被计算出来。优化程序造成一些夹具布局和夹紧力，同时也是被发送到有限元模型中。在有限元分析座内，加工变形下，切削力和夹紧力的计算方法采用有限元方法。根据某夹具布局和变形，然后发送给优化程序，以搜索为一优化夹具方案。图1 夹具布局和夹紧力优化过程4 夹具布局设计和夹紧力的优化4.1 遗传算法遗传算法（ GA ）是基于生物再生产过程的强劲，随机和启发式的优化方法。基本思路背后的遗传算法是模拟“生存的优胜劣汰“的现象。每一个人口中的候选个体指派一个健身的价值，通过一个功能的调整，以适应特定的问题。遗传算法，然后进行复制，交叉和变异过程消除不适宜的个人和人口的演进给下一代。人口足够数目的演变基于这些经营者引起全球健身人口的增加和优胜个体代表全最好的方法。遗传算法程序在优化夹具设计时需夹具布局和夹紧力作为设计变量，以生成字符串代表不同的布置。字符串相比染色体的自然演变，以及字符串，它和遗传算法寻找最优，是映射到最优的夹具设计计划。在这项研究里，遗传算法和MATLAB的直接搜索工具箱是被运用的。 收敛性遗传算法是被人口大小、交叉的概率和概率突变所控制的 。只有当在一个人口中功能最薄弱功能的最优值没有变化时，nchg达到一个预先定义的价值ncmax ，或有多少几代氮，到达演化的指定数量上限nmax， 没有遗传算法停止。有五个主要因素，遗传算法，编码，健身功能，遗传算子，控制参数和制约因素。 在这篇论文中，这些因素都被选出如表1所列。表1 遗传算法参数的选择由于遗传算法可能产生夹具设计字符串，当受到加工负荷时不完全限制夹具。这些解决方案被认为是不可行的，且被罚的方法是用来驱动遗传算法，以实现一个可行的解决办法。1夹具设计的计划被认为是不可行的或无约束，如果反应在定位是否定的。在换句话说，它不符合方程（2）和（3）的限制。罚的方法基本上包含指定计划的高目标函数值时不可行的。因此，驱动它在连续迭代算法中的可行区域。对于约束（4），当遗传算子产生新个体或此个体已经产生，检查它们是否符合条件是必要的。真正的候选区域是那些不包括无效的区域。在为了简化检查，多边形是用来代表候选区域和无效区域的。多边形的顶点是用于检查。“inpolygon ”在MATLAB的功能可被用来帮助检查。4.2 有限元分析ANSYS软件包是用于在这方面的研究有限元分析计算。有限元模型是一个考虑摩擦效应的半弹性接触模型，如果材料是假定线弹性。如图2所示，每个位置或支持，是代表三个正交弹簧提供的制约。图2 考虑到摩擦的半弹性接触模型在x ， y和z 方向和每个夹具类似，但定位夹紧力在正常的方向。弹力在自然的方向即所谓自然弹力，其余两个弹力即为所谓的切向弹力。接触弹簧刚度可以根据向赫兹接触理论计算如下：随着夹紧力和夹具布局的变化，接触刚度也不同，一个合理的线性逼近的接触刚度可以从适合上述方程的最小二乘法得到。连续插值，这是用来申请工件的有限元分析模型的边界条件。在图3中说明了夹具元件的位置，显示为黑色界线。每个元素的位置被其它四或六最接近的邻近节点所包围。图3 连续插值这系列节点，如黑色正方形所示，是（37，38，31和30 ），（9，10 ，11 ， 18，17号和16号）和（ 26，27 ，34 ， 41，40和33 ）。这一系列弹簧单元，与这些每一个节点相关联。对任何一套节点，弹簧常数是：这里，kij 是弹簧刚度在的j -次节点周围i次夹具元件，Dij 是i次夹具元件和的J -次节点周围之间的距离，ki是弹簧刚度在一次夹具元件位置,i 是周围的i次夹具元素周围的节点数量为每个加工负荷的一步，适当的边界条件将适用于工件的有限元模型。在这个工作里，正常的弹簧约束在这三个方向（X ， Y ， Z ）的和在切方向切向弹簧约束，（X ， Y ）。夹紧力是适用于正常方向（Z）的夹紧点。整个刀具路径是模拟为每个夹具设计计划所产生的遗传算法应用的高峰期的X ，Y ，z切削力顺序到元曲面，其中刀具通行证。在这工作中，从刀具路径中欧盟和去除碎片已经被考虑进去。在机床改变几何数值过程中，材料被去除，工件的结构刚度也改变。 因此，这是需要考虑碎片移除的影响。有限元分析模型，分析与重点的工具运动和碎片移除使用的元素死亡技术。在为了计算健身价值，对于给定夹具设计方案，位移存储为每个负载的一步。那么，最大位移是选定为夹具设计计划的健身价值。遗传算法的程序和ANSYS之间的互动实施如下。定位和夹具的位置以及夹紧力这些参数写入到一个文本文件。那个输入批处理文件ANSYS软件可以读取这些参数和计算加工表面的变形。 因此， 健身价值观，在遗传算法程序，也可以写到当前夹具设计计划的一个文本文件。当有大量的节点在一个有限元模型时，计算健身价值是很昂贵的。因此，有必要加快计算遗传算法程序。作为这一代的推移，染色体在人口中取得类似情况。在这项工作中，计算健身价值和染色体存放在一个SQL Server数据库。遗传算法的程序，如果目前的染色体的健身价值已计算之前，先检查；如果不，夹具设计计划发送到ANSYS，否则健身价值观是直接从数据库中取出。啮合的工件有限元模型，在每一个计算时间保持不变。每计算模型间的差异是边界条件，因此，网状工件的有限元模型可以用来反复“恢复”ANSYS 命令。5 案例研究一个关于低刚度工件的铣削夹具设计优化问题是被显示在前面的论文中，并在以下各节加以表述。5.1 工件的几何形状和性能工件的几何形状和特点显示在图4中，空心工件的材料是铝390与泊松比0.3和71Gpa的杨氏模量。外廓尺寸152.4mm127mm*76.2mm.该工件顶端内壁的三分之一是经铣削及其刀具轨迹，如图4 所示。夹具元件中应用到的材料泊松比0.3和杨氏模量的220的合金钢。图4 空心工件5.2 模拟和加工的运作举例将工件进行周边铣削，加工参数在表2中给出。基于这些参数，切削力的最高值被作为工件内壁受到的表面载荷而被计算和应用，当工件处于330.94 n（切）、398.11 N （下径向）和22.84 N （下轴） 的切削位置时。整个刀具路径被26个工步所分开，切削力的方向被刀具位置所确定表2加工参数和条件。5.3 夹具设计方案夹具在加工过程中夹紧工件的规划如图5所示。图5 定位和夹紧装置的可选区域一般来说， 3-2-1定位原则是夹具设计中常用的。夹具底板限制三个自由度，在侧边控制两个自由度。这里，在Y=0mm截面上使用了4个定点（L1，L2 ， L3和14 ），以定位工件并限制2自由度；并且在Y=127mm的相反面上，两个压板（C1,C2）夹紧工件。在正交面上，需要一个定位元件限制其余的一个自由度，这在优化模型中是被忽略的。在表3中给出了定位加紧点的坐标范围。表3 设计变量的约束由于没有一个简单的一体化程序确定夹紧力，夹紧力很大部分（6673.2N）在初始阶段被假设为每一个夹板上作用的力。且从符合例5的最小二乘法，分别由4.43107 N/m 和5.47107 N/m得到了正常切向刚度。5.4 遗传控制参数和惩罚函数在这个例子中，用到了下列参数值：Ps=30, Pc=0.85, Pm=0.01, Nmax=100和Ncmax=20.关于f1和的惩罚函数是这里fv可以被F1或代表。当nchg达到6时，交叉和变异的概率将分别改变成0.6和 优化结果连续优化的收敛过程如图6所示。且收敛过程的相应功能（1）和（2）如图7、图8所示。优化设计方案在表4中给出。图6 夹具布局和夹紧力优化程序的收敛性遗传算法 图7 第一个函数值的收敛图8第二个函数值的收敛性表4 多目标优化模型的结果 表5 各种夹具设计方案结果进行比较，5.6 结果的比较 从单一目标优化和经验设计中得到的夹具设计的设计变量和目标函数值，如表5所示。单一目标优化的结果，在论文中引做比较。在例子中，与经验设计相比较，单一目标优化方法有其优势。最高变形减少了57.5 ，均匀变形增强了60.4 。最高夹紧力的值也减少了49.4 。从多目标优化方法和单目标优化方法的比较中可以得出什么呢？最大变形减少了50.2 ，均匀变形量增加了52.9 ，最高夹紧力的值减少了69.6 。加工表面沿刀具轨迹的变形分布如图9所示。很明显，在三种方法中，多目标优化方法产生的变形分布最均匀。与结果比较，我们确信运用最佳定位点分布和最优夹紧力来减少工件的变形。图10示出了一实例夹具的装配。图9沿刀具轨迹的变形分布图10 夹具配置实例6 结论本文介绍了基于GA和有限元的夹具布局设计和夹紧力的优化程序设计。优化程序是多目标的：最大限度地减少加工表面的最高变形和最大限度地均匀变形。ANSYS软件包已经被用于健身价值的有限元计算。对于夹具设计优化的问题，GA和有限元分析的结合被证明是一种很有用的方法。 在这项研究中，摩擦的影响和碎片移动都被考虑到了。为了减少计算的时间，建立了一个染色体的健身数值的数据库，且网状工件的有限元模型是优化过程中多次使用的。 传统的夹具设计方法是单一目标优化方法或经验。此研究结果表明，多目标优化方法比起其他两种方法更有效地减少变形和均匀变形。这对于在数控加工中控制加工变形是很有意义的。参考文献1、 King LS，Hutter（ 1993年） 自动化装配线上棱柱工件最佳装夹定位生成的理论方法。De Meter EC (1995) 优化机床夹具表现的Min - Max负荷模型。2、 De Meter EC (1998) 快速支持布局优化。Li B, Melkote SN (1999) 通过夹具布局优化改善工件的定位精度。3、 Li B, Melkote SN (2001) 夹具夹紧力的优化和其对工件的定位精度的影响。4、 Li B, Melkote SN (1999) 通过夹具布局优化改善工件的定位精度。5、 Li B, Melkote SN (2001) 夹具夹紧力的优化和其对工件定位精度的影响。6、 Li B, Melkote SN (2001) 最优夹具设计计算工件动态的影响。7、 Lee JD, Haynes LS (1987) 灵活装夹系统的有限元分析。8、 Menassa RJ, DeVries WR (1991) 运用优化方法在夹具设计中选择支位。9、 Cai W, Hu SJ, Yuan JX (1996) 变形金属板材的装夹的原则、算法和模拟。10、 Qin GH, Zhang WH, Zhou XL (2005) 夹具装夹方案的建模和优化设计。11、Deng HY, Melkote SN (2006) 动态稳定装夹中夹紧力最小值的确定。12、Wu NH, Chan KC (1996) 基于遗传算法的夹具优化配置方法。13、Ishikawa Y, Aoyama T(1996) 借助遗传算法对装夹条件的优化。14、Vallapuzha S, De Meter EC, Choudhuri S, et al (2002) 一项关于空间坐标对基于遗传算法的夹具优化问题的作用的调查。15、Vallapuzha S, De Meter EC, Choudhuri S, et al (2002) 夹具布局优化方法成效的调查。16、Kulankara K, Melkote SN (2000) 利用遗传算法优化加工夹具的布局。17、Kulankara K, Satyanarayana S, Melkote SN (2002) 利用遗传算法优化夹紧布局和夹紧力。18、Lai XM, Luo LJ, Lin ZQ (2004) 基于遗传算法的柔性装配夹具布局的建模与优化。19、Hamedi M (2005) 通过一种人工神经网络和遗传算法混合的系统设计智能夹具。20、Kumar AS, Subramaniam V, Seow KC (2001) 采用遗传算法固定装置的概念设计。21、Kaya N (2006) 利用遗传算法优化加工夹具的定位和夹紧点。22、Zhou XL, Zhang WH, Qin GH (2005) 遗传算法用于优化夹具布局和夹紧力。23、Kaya N, ztrk F (2003) 碎片位移和摩擦接触的运用对工件夹具布局的校核。62ORIGINAL ARTICLEDeformation control through fixture layout designand clamping force optimizationWeifang Chen&Lijun Ni&Jianbin XueReceived: 2 February 2007 /Accepted: 4 July 2007#Springer-Verlag London Limited 2007Abstract Workpiece deformation must be controlled in thenumerical control machining process. Fixture layout andclamping force are two main aspects that influence thedegree and distribution of machining deformation. In thispaper, a multi-objective model was established to reducethe degree of deformation and to increase the distributinguniformity of deformation. The finite element method wasemployed to analyze the deformation. A genetic algorithmwas developed to solve the optimization model. Finally, anexample illustrated that a satisfactory result was obtained,which is far superior to the experiential one. The multi-objective model can reduce the machining deformationeffectively and improve the distribution condition.Keywords Fixturelayout.Clampingforce.Geneticalgorithm.Finiteelementmethod1 IntroductionFixture design is an important procedure in manufacturingengineering. It is critical to machining accuracy. Aworkpiece should be constrained in a fixture duringmachining with fixture elements such as locators, clamps,and supports. The positions of locators, clamps andsupports should be strategically designed and appropriateclamping forces should be applied. The fixture elementscan be placed anywhere within the candidate regions on theworkpiece surfaces. Clamping force must be large enoughto hold the workpiece during machining. Typically, it reliesheavily on the designers experience to choose the positionsof the fixture elements and to determine the clampingforces. Thus there is no assurance that the resultant solutionis optimal or near optimal for a given workpiece.Consequently, the fixture layout and the clamping forceoptimization become two main aspects in fixture design.The positions of locators and clamps, and the values ofclamping force should be properly selected and calculatedso that the workpiece deformation due to clamping andcutting force is minimized and uniformed.The objective of fixture design is to find an optimallayout or positions of the fixture elements around theworkpiece and optimal clamping force. In this paper, amulti-objective optimization method is presented for thefixture layout design and clamping force optimization.The objective is two folded. One is to minimize themaximum elastic deformation of the machined surfaces,and another is to maximize the uniformity of deforma-tion. The ANSYS software package is used to calculatethe deformation of the workpiece under given clampingforce and cutting force. A genetic algorithm is devel-oped, and the direct search toolbox of MATLAB isemployed to solve the optimization problem. Finally, acase study is given to illustrate the application of theproposed approach.2 Literature reviewWith the wide applications of optimization methods inindustry, fixture design optimization has gained moreinterests in recent years. Fixture design optimizationincludes fixture layout optimization and clamping forceoptimization. King and Hutter presented a method forInt J Adv Manuf TechnolDOI 10.1007/s00170-007-1153-2W. Chen:L. Ni:J. Xue (*)College of Mechanical and Electronical Engineering,Nanjing University of Aeronautics and Astronautics,No. 29, Yudao Street,Nanjing 210016, Chinae-mail: optimal fixture layout design using a rigid body model of thefixture-workpiece system 1. DeMeter also used a rigidbody model for the analysis and synthesis of optimalfixture layouts and minimum clamping force 2. Hepresented a finite element method (FEM) based supportlayout optimization procedure with computationally attrac-tive qualities 3. Li and Melkote used a nonlinearprogramming method and a contact elasticity model tosolve the layout optimization problem 4. Two years later,they presented a method for determining the optimalclamping force for a multiple clamp fixture subjected toquasi-static machining force 5. They also presented anoptimal synthesis approach of fixture layout and clampingforce that considers workpiece dynamics during machining6. A combined fixture layout and clamping forceoptimization procedure was presented. Other researchers7, 8 used the FEM for fixture design and analysis. Cai etal. 9 extended the work of Menassa and DeVries 8 toinclude synthesis of fixture layout for sheet metal assembly.Qin et al. 10 established an elastic contact model betweenclamp and workpiece to optimize the clamping force withan objective to minimize the position error of theworkpiece. Deng and Melkote 11 presented a model-based framework for determining the minimum requiredclamping force, which ensures the dynamic stability of afixtured workpiece during machining.Most of the above studies used nonlinear programmingmethods, which seldom gave global or near-global opti-mum solutions. All of the fixture layout optimizationprocedures must start with an initial feasible layout. Inaddition, solutions obtained from these models are verysensitive to the initial feasible fixture layout. The problemof fixture design optimization is nonlinear because there isno direct analytical relationship between the objectivefunction and design variables, i.e. between the machinedsurface error and the fixture parameters (positions of locatorand clamp, and clamping forces).Previous researchers had shown that genetic algorithm(GA) was a useful technique in solving such optimiza-tion problems. Wu and Chan 12 used the GA todetermine the most statically stable fixture layout. Ishikawaand Aoyama 13 applied GA to determine the optimalclamping condition for an elastic workpiece. Vallapuzha etal. 14 used spatial coordinates to encode in the GA basedoptimization of fixture layout. They also presented themethodology and results of an extensive investigation intothe relative effectiveness of the main competing fixtureoptimization methods, which showed that continuous GAyielded the best quality solutions 15. Krishnakumar andMelkote 16 developed a fixture layout optimizationtechnique that used GA to find the fixture layout thatminimized the deformation of the machined surface due toclamping and cutting force over the entire tool path.Locator and clamp positions were specified with nodenumbers. Krishnakumar et al. 17 presented an iterativealgorithm that minimized the workpiece elastic deformationfor the entire cutting process by alternatively varying thefixture layout and clamping force. Lai et al. 18 set up ananalysis model that treated locator and clamps as the samefixture layout elements for the flexible part deformation.Hamedi 19 discussed a hybrid learning system that usednonlinear FEA with a supportive combination of artificialneural network (ANN) and GA. The ANN was used tocalculate workpiece maximum elastic deformation, the GAwas used to determine the optimum clamping forces.Kumar 20 proposed to combine the GA and ANN todevelop a fixture design system. Kaya 21 used the GAand FEM to find the optimal locators and clampingpositions in 2D workpiece and took chip removal effectsinto account. Zhou et al. 22 presented a GA based methodthat optimized fixture layout and clamping force simulta-neously. Some of the studies did not consider theoptimization of the layout for entire tool path. Some ofthe studies used node numbers as design parameters.Some of the studies addressed fixture layout or clampingforce optimization methods but not both simultaneously.And there were few studies taking friction and chipremoval into account. The effects of chip removal andfrictional contact cannot be neglected for achieving amore realistic and accurate workpiece-fixture layoutverification analysis 23, so it is essential to take chipremoval effects and friction effect into account to achieve abetter machining accuracy.In this paper, the friction and chip removal are takeninto account to achieve the minimum degree of themaximum deformation of the machined surfaces underclamping and cutting force and to uniform the deforma-tion. A multi-objective optimization model is established.An optimization process based on GA and FEM ispresented to find the optimal fixture layout and clampingforce. Finally, the result of the multi-objective optimiza-tion model is compared with the single objectiveoptimization method and the experience method for a lowrigidity workpiece.3 A multi-objective optimization model for fixturedesignA feasible fixture layout has to satisfy three constraints.First, the locators and clamps cannot apply tensile forces onthe workpiece. Second, the Coulomb friction constraintmust be satisfied at all fixture-workpiece contact points.The positions of fixture element-workpiece contact pointsmust be in the candidate regions. For a problem involving pfixture element-workpiece contacts and n machining loadInt J Adv Manuf Technolsteps, the optimization problem can be mathematicallymodeled as followsmin max1jj; 2jj;:; j?;:; njj? s?;j 1;2;:;n1Subject tom Fnijj ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF2ti F2hiq2Fni? 03pos i 2 V i ;i 1;2;:;p4where jrefers to the maximum elastic deformation at amachining region in the j-th step of the machiningoperation,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXnj1j ? ?2?nvuutis the average of jFniis the normal force at the i-th contact pointis 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) and theprobability 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 and clamping force Fig. 1 Fixture layout and clamp-ing force optimization processTable 1 Selection of GAs parametersFactorsDescriptionEncodingRealScalingRankSelectionRemainderCrossoverIntermediateMutationUniformControl parameterSelf-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 followskiz16R?iE?2i9?13fiz13kiz kiy6E?i2?vfiGfi2?vwiGwi?1? kiz8:5wherekiz, kix, kiyare the tangential and normal contactstiffness,1R?i1Rwi1Rfiis the nominal contact radius,1E?i1?V2wiEwi1?V2fiEfiis 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 position12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849Fig. 3 Continuous interpolationFig. 4 A hollow workpieceTable 2 Machining parameters and conditionsParameterDescriptionType of operationEnd millingCutter diameter25.4 mmNumber of flutes4Cutter RPM500Feed0.1016 mm/toothRadial depth of cut2.54 mmAxial depth of cut25.4 mmRadial rake angle10Helix angle30Projection length92.07 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) and thetangential 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 alargenumber of nodes in an FEM model.Thus itis necessaryto 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 variablesMinimumMaximumX /mmZ /mmX /mmZ /mmL10076.238.1L276.20152.438.1L3038.176.276.2L476.238.1152.476.2C10076.276.2C276.20152.476.2F1/N06673.2F2/N06673.2Int 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, the 321 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 isneglectedinthe optimalmodel.The coordinateboundsfor thelocating/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 /mmZ /mmL117.10230.641L2108.16925.855L321.31556.948L4127.84660.202C122.98962.659C2117.61525.360F1/N167.614F2/N382.435f1/mm0.006568/mm0.002683Int J Adv Manuf Technolused: Ps=30, Pc=0.85, Pm=0.01, Nmax=100 and Ncmax=20. The penalty function for f1and is fv fv 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 22 isquoted 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 combination of GA and FEM is proven 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 optimizationSingle objective optimizationX/mmZ/mmX/mmZ/mmL112.70012.70016.72034.070L2139.712.700145.36017.070L312.70063.50018.40057.120L4139.70063.500146.26058.590C112.70038.1005.83056.010C2139.70038.100104.40022.740F1/N2482444.88F2/N24821256.13f1/mm0.0310120.013178/mm0.0143770.005696Fig. 9 Distribution of the deformation along cutter pathFig. 10 A real case fixture configurationInt J Adv Manuf TechnolThe traditionalfixturedesignmethodsaresingleobjectiveoptimization 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. It is meaningfulfor machining deformation control in NC machining.References1. King LS, Hutter I (1993) Theoretical approach for generatingoptimal fixturing locations for prismatic workparts in automatedassembly. J Manuf Syst 12(5):4094162. D