外文翻译--板料成形中有限元仿真及相关技术的研究进展

基于MARC的板料冲压成形过程有限元模拟研究
甘辉
【期刊名称】《机械制造与自动化》
【年(卷),期】2009(038)001
【摘要】通过合理制定零件的成形工艺、模具参数来消除零件缺陷、提高成形品质是金属板料冲压成形的重要任务.通过探讨成形过程的数值模拟来分析金属板料各部分在冲压成形过程中的变形情况,预测成形缺陷,分析影响零件成形品质的因素的方法.以圆筒形拉深件的冲压成形为例,介绍应用商用有限元软件MARC对成形过程进行有限元模拟分析的实现技术,其分析结果与实验相符.
【总页数】3页(P48-50)
【作者】甘辉
【作者单位】上海交通大学,塑性成型系,上海,200030;江苏信息职业技术学院,机电工程系,江苏,无锡,214153
【正文语种】中文
【中图分类】TG386
【相关文献】
1.金属板料冲压成形过程有限元分析软件的开发 [J], 陈中奎;施法中
2.板料冲压成形过程中有限元计算模型的生成研究 [J], 黄新林
3.板料冲压成形过程有限元分析中的接触搜索法的优化 [J], 冯天飞;施法中;陈中奎
4.板料冲压成形过程有限元分析中的接触搜索算法 [J], 陈中奎;施法中
5.板料冲压成形过程有限元分析原型系统的开发 [J], 陈中奎;施法中;黄迪民
因版权原因，仅展示原文概要，查看原文内容请购买。

板料成形仿真软件− INDEED®数模软件（上海）有限公司1．引言上世纪八十年代，一个由汽车制造工程师和模具设计师组成的团队开始了INDEED软件的研发，这个软件名来源于Innovative Deep Drawing（创新的深拉延）。

顾名思义，这是一个专用于金属深拉延的有限元数值模拟分析软件。

在当时，显式有限元法非常流行，特别是涉及到高度的非线性材料响应，有限形变，以及在由于复杂的接触问题导致边界条件发生突变的情况下，显式有限元法有其独特的优势。

鉴于板料成形过程是一个准静态过程，工程师们还是决定在INDEED 中采用隐式算法；一个明显的优点是，可方便而精确地进行回弹计算。

如今，尽管显式算法在解决准静态问题时有诸多的缺点和不足，但其代表性软件（比如LS−DYNA）在板料成形数值模拟市场上似乎已经成为主导。

这种统治地位是基于一种广为流传的思想，那就是隐式算法无论是在计算的可靠性还是效率都比不上显式算法。

然而，尽管在很多情况下，显式算法可以得出令人满意的结果，但是在板料成形的模拟应用中，使用隐式算法还是有其突出的优越性。

特别是在计算冲压力和板料回弹的情况下，隐式算法得出的结果更加可信。

由于像超高强度钢、双相钢和特殊铝合金这样的新材料不断被采用，其仿真计算变得越来越重要。

在INDEED 的研发中，工程师们把重点放在了针对更加复杂工况的分析结果质量上，例如精确的回弹计算和冲压力计算。

因此，INDEED采用了隐式的求解方法，使用了一种独创的特殊壳单元，这种单元在厚度方向有一个附加的自由度，使用了领先的材料模型，将材料的变形梯度采用乘法分解为弹性和塑性两个部分，这些技术奠定了Indeed——高端成形模拟软件的基石。

过去，使用隐式算法求解问题时常常由于有限元模型非常大，必须要使用高性能的计算机。

现在，由于采用了新型的基于域分解的有限元技术，隐式算法的计算速度已经大幅提高。

对于那些需要计算大型成形模拟而又不愿购买昂贵的矢量机和大型机的小公司来讲，这是个好消息。

有限元法在材料成型过程研究中的发展、应用及作用姓名：学号：学院：班级：有限元法在材料成型过程研究中的发展、应用及作用材料加工是先进制造技术中重要的组成，它的应用涉及航空航天、汽车、石化、军事等事关国民经济的重要产业。

材料加工工艺过程中，除了运动和外力作用等因素，还涉及温度场、流场、应力应变场及内部组织的变化; 生产环境恶劣，控制因素多样。

因此，充分了解材料加工计算机模拟的重要性及其发展趋势，对于推动我国制造业的科技进步，缩短产品的开发和加工周期，快速响应市场，提高竞争能力，真正体现高速、高效、高质的制造优势，具有重要的意义。

计算机模拟是制造业发展的产物。

以有限元方法为基础的计算机模拟技术是20 世纪技术发展的巨大成果，在工程物理科学的各个分支领域都起着十分重要的作用。

新材料、新工艺、新产品、高要求、高精度、低成本的现代制造模式要求深入了解和掌握材料成形机理、过程变化，在计算机上实现过程显现，开拓科学的工艺和设计方法，实现最优设计与制造。

因此，计算机数值模拟技术以及以此为基础的优化设计方法研究成为当今和今后国内研究的热点。

我们知道在工程中使用的金属材料大多数为多晶材料，材料的微观组织形态直接影响零件的机械性能和物理性能，所以选择合理的加工工艺参数十分重要。

材料加工过程微观组织的计算机模拟由于具有描述分子级尺寸水平的能力，这将对控制材料晶粒大小及分布，进一步了解位错的产生和运动、晶界结构、防止内部空洞和微裂纹的萌生和扩展等问题提供了新的方法[1-2] ，将大大推动材料微观结构研究的进展，并对确定优化材料加工的工步数和顺序、热处理方案十分有益。

此外，在金属成形过程中，适用的优化准则对材料最终的力学性能和微观组织性能具有重要的影响，通过优化坯料形状或预成形模具形状、模具速度使最终锻件具有良好的尺寸精度、少无飞边和所期望的微观组织。

为此，一方面要要研究合适的优化设计变量的选择，包括影响终锻件力学组织性能的状态变量和过程变量，即形状设计变量和速度设计变量。

板料成形有限元数值模拟中的编程语言
杨玉英;金朝海;王永志;徐伟力
【期刊名称】《塑性工程学报》
【年(卷),期】2000(7)3
【摘要】本文用自主开发的板料成形静力隐式弹塑性有限元模拟程序SHEETFORM对方盒形件拉深成形过程进行了数值模拟 ,全面比较了 FORTRAN 语言版本和 C++语言版本的有限元数值计算程序 SHEETFORM在数值计算过程中所表现的性能。

发现 :在所有参数相同的情况下 ,C++语言编写的有限元程序SHEETFORM- C和 FOR-TRAN语言编制的有限元程序 SHEETFORM- F可以得到完全相同精度的计算结果 ,而且两者在所有加载步中所需的迭代步数完全一致 ,计算效率也基本接近。

【总页数】3页(P19-21)
【关键词】弹塑性有限元;数值模拟;板料成形;程序设计
【作者】杨玉英;金朝海;王永志;徐伟力
【作者单位】哈尔滨工业大学;上海交通大学
【正文语种】中文
【中图分类】TP312;TG302
【相关文献】
1.板料成形中的有限元数值模拟技术 [J], 罗亚军;杨曦;何丹农;张永清
2.板料成形中的有限元数值模拟技术 [J], 罗亚军;杨曦;何丹农;张永清
3.板料成形数值模拟有限元求解算法 [J], 陆璐; 王辅忠; 王照旭
4.有限元模型转换及其在金属板料成形数值模拟中的应用 [J], 邱智学;黄菊花;杨雪春;杨国泰;刘志云;谢世坤
5.AutoCAD在板料拉深成形有限元数值模拟中的应用 [J], 杜拴柱;贺涞庆
因版权原因，仅展示原文概要，查看原文内容请购买。

基于DYNAFORM的板料成形研究摘要板料拉深成形是现在工业领域中一种重要的加工方法。

在拉深成形的过程中，零件容易出现开裂，起皱等问题。

随着计算机模拟和仿真技术的发展，板料拉深成形过程的分析、缺陷分布等问题都可以通过有限元模拟软件预测分析。

针对这些问题，用PRO/ENGINEER软件将零件进行三维建模，导入DYNAFORM，进行初步模拟，设置模拟控制参数，主要是修改板料厚度、板料性能、冲压速度、模具圆角半径等参数。

找出模具倒角、材料厚度、冲压速度对材料成形性能的影响，从而对于指导成形工艺的设计具有重要的意义。

关键词：DYNAFORM，拉深，模拟，参数Based on the dynaform plate formingresearchAbstract：Deep drawing of sheet metal industry is now an important processing method. In the drawing forming process, the parts prone to cracking, wrinkling and other problems.Along with the computer simulation and the simulation technology development, the process of sheet forming analysis, defects distribution problems can be simulated by FEM software prediction analysis. To solve these problems, PRO / ENGINEER software part three-dimensional modeling, import on DYNAFORM, a preliminary simulation, set the parameters of analog control, primarily to modify the sheet thickness, sheet performance, pressing speed, die fillet radius and other parameters.Identify mold chamfer, material thickness, speed of pressing forming properties of the material, which for the guidance of the design of the forming process of great significance.Key words: DYNAFORM, drawing, simulation, parameter毕业设计（论文）原创性声明和使用授权说明原创性声明本人郑重承诺：所呈交的毕业设计（论文），是我个人在指导教师的指导下进行的研究工作及取得的成果。

金属板料成形数值模拟的研究现状一、引言金属板料成形数值模拟是现代制造业中不可或缺的一环。

通过数值模拟可以预测金属板料在成形过程中的变形、应力分布等物理量，从而优化工艺参数，提高成形质量和效率。

本文将介绍金属板料成形数值模拟的研究现状。

二、数值模拟方法1. 有限元法有限元法是目前最为常用的数值模拟方法之一。

它将连续体划分为多个小单元，在每个小单元内近似求解其物理量，最后通过组合得到整体的物理量分布。

有限元法可以考虑材料非线性、边界条件复杂等因素，适用范围广泛。

2. 边界元法边界元法是另一种常用的数值模拟方法。

它将问题转化为求解边界上的物理量分布，避免了对整个区域进行离散化计算。

边界元法适用于具有对称性或者具有复杂几何形状的问题。

3. 网格无关法网格无关法是相对于传统有限元法而言的新兴方法。

它不需要事先确定网格大小和结构，可以自动适应物理量分布的变化。

网格无关法适用于具有较大变形或者复杂几何形状的问题。

三、数值模拟在金属板料成形中的应用1. 成形过程分析数值模拟可以对金属板料成形过程进行分析，预测变形、应力分布等物理量。

通过优化工艺参数，可以避免一些不必要的缺陷和失效。

2. 模具设计数值模拟可以为模具设计提供依据。

通过对成形过程中应力和变形的预测，可以确定合适的模具结构和尺寸，从而达到更好的成形效果。

3. 材料选择数值模拟还可以为材料选择提供参考。

通过预测不同材料在成形过程中的性能表现，可以选择最为适合的材料，提高生产效率和质量。

四、数值模拟存在的问题及发展趋势1. 计算精度问题目前数值模拟存在计算精度不高、计算时间长等问题。

需要进一步发展更加高效精确的数值模拟方法。

2. 跨尺度建模问题金属板料成形涉及到多个尺度，如宏观尺度、晶体尺度等。

如何将不同尺度的模型相结合，进行跨尺度建模是一个重要的研究方向。

3. 多物理场耦合问题金属板料成形涉及到多种物理场，如力学、热学、电磁学等。

如何将这些物理场相互耦合起来进行计算，是数值模拟发展的重要方向之一。

材料成型及控制工程外文翻译文献(文档含英文原文和中文翻译)在模拟人体体液中磷酸钙涂层激光消融L. Cle`ries*, J.M. FernaHndez-Pradas, J.L. Morenza德国巴塞罗那大学,西班牙1999年七月二十八日-2000年2月文摘:三种类型的磷酸钙涂层基质,在钛合金激光烧蚀技术规定提存,沉浸在一个模拟的身体# uid为了确定条件下他们的行为类似于人的血浆。

羟基磷灰石涂层也也非晶态磷酸钙涂层和a-tricalcium磷酸盐做溶解阶段b-tricalcium磷酸盐的涂料有细微的一个阶段稍微瓦解。

一个apatitic阶段降水量偏爱在羟基磷灰石涂层的涂料磷酸b-tricalcium上有细微的一个阶段。

在钛合金基体上也有降水参考,但在大感应时代。

然而,在非晶态磷酸钙涂层不沉淀形成。

科学出版社有限公司(2000保留所有权利。

关键词:磷酸钙,脉冲激光沉积,SBF1 介绍激光消融技术用于沉积磷酸钙涂层金属基体上,将用作植体骨重建。

用这个技术,磷酸钙涂层量身定做阶段和结构也成功地研制生产了[1,2]和溶解特性鉴定海洋条件]。

然而,真正的身体条件# uid饱和对羟基磷灰石的阶段,这是钙离子的浓度高于均衡的这个阶段。

因而,这就很有趣也测试条件磷酸钙涂料接近体内的情况,以了解其完整性,在这些条件及其催化反应性质}表面沉淀过程。

因此,非晶态磷酸钙涂层(ACP),羟基磷灰石(HA)涂层,涂层中的一个阶段b-tricalcium磷酸盐较小(ba-TCP)积下激光烧蚀是沉浸在饱和溶液为迪!时间、不同的结构性演变进行了测定。

饱和溶液的使用的是身体uid(SBF模拟#),解决了其离子浓度、酸碱度几乎等于那些人类血浆[5]。

该解决方案也是一个利用在仿生(沉淀)工艺生产磷灰石层在溶胶凝胶活性钛基体。

2 实验模拟身体化学溶解试剂级严格依照以下的顺序,除氢钠,NaHCO3:)3,K2HPO4 H2O,MgCl2)6 H2O,氯化钙和Na2SO4)2 H2O,在去离子水。

板材成形有限元数值模拟的研究进展
霍同如;徐秉业
【期刊名称】《力学进展》
【年(卷),期】1996(026)003
【摘要】本文以不同时期引起人们广泛注意的标准考题１）为主要线索，对板材成形有限元数值模拟的发展作了简要的综述．并讨论了研究中仍存在的问题和研究的发展方向．
【总页数】10页(P379-388)
【作者】霍同如;徐秉业
【作者单位】不详;不详
【正文语种】中文
【中图分类】O34
【相关文献】
1.钛合金板材翻边成形的有限元数值模拟 [J], 郑晖;张凌云;周健
2.板材电磁成形技术研究进展 [J], 金延野;于海平
3.镁合金板材磁脉冲成形研究进展 [J], 徐俊瑞;王元丰;王宇阳;李毅;汪强昆;赵雨东;文智生;周颖奇
4.板材拉深成形中磁力压边技术的研究进展 [J], 张红升;曹丽琴;郭凯
5.板材冲压成形有限元数值模拟界面摩擦约束处理方法 [J], 杜亭;柳玉起;章志兵;李志刚
因版权原因，仅展示原文概要，查看原文内容请购买。

Prediction of workpiece deformation in afixture systemusing thefinite element methodShane P.Siebenaler,Shreyes N.Melkote*The George W.Woodruff School of Mechanical Engineering,Georgia Institute of Technology,Atlanta,GA30332-0405,USAReceived25August2004;accepted7April2005Available online23May2005AbstractKnowledge of workpiece deformations induced by loading in afixture–workpiece system is important to ensure quality part production. Suitable methods for accurately predicting such deformations are essential to the design and operation offixtures.In this regard,finite element modeling has been widely applied by researchers and practitioners.However,these studies generally neglect the role of compliance of thefixture body on workpiece deformation.Also lacking is knowledge of the effects of differentfinite element model parameters on workpiece deformation.This study usesfinite element analysis(FEA)to model afixture–workpiece system and to explore the influence of compliance of thefixture body on workpiece deformation.In addition,the effects of certainfinite element model parameters on the prediction accuracy are also examined.Experimental verification of the workpiece deformations and locator reaction forces predicted by the FEA model shows agreement within5%of the experimental data.For thefixture–workpiece system analyzed in this study,it was found that98%of all system compliance is captured by modeling just the workpiece andfixture contact tips.The remainder of deformation occurred in the other fixture components.The accuracy and computational time tradeoffs are given for variousfixture models.q2005Elsevier Ltd.All rights reserved.Keywords:Fixture-work piece system;Finite element analysis;Deformation1.IntroductionMethods for analyzingfixtures are essential to the practice and economics of machining.In particular,the ability to model and accurately predict workpiece deformation induced by fixturing loads and/or predict the unknownfixture–workpiece contact forces are crucial for designing functionalfixtures.The most common modeling and analysis approaches used for fixture–workpiece systems include the rigid body approach,the contact mechanics based approach and thefinite element modeling approach.Of these approaches,the rigid body modeling approach[1–3]is by definition incapable of predicting workpiece deformations and is therefore unsuitable for analysis of the impact offixturing on part quality.The contact mechanics approach,although attractive from a standpoint of computational effort,is limited to parts that can be approximated as elastic half-spaces.Models derived from this approach are capable of accurately predicting unknown locator reaction forces and localized contact deformations[4–6].However,they are not applicable for thin,compliant parts. Finite element models on the other hand are very powerful and are capable of accounting for all compliances and non-linearities present in the system.Although use offinite element models has been widely reported in the literature and employed in practice,a clear understanding of the role of the different fixture compliances on the prediction accuracy of workpiece deformation is lacking.Also knowledge of the effects of differentfinite element model parameters on workpiece deformation is lacking.A common assumption in application of Finite Element Analysis(FEA)to analyze afixture–workpiece system is that thefixture is completely rigid since it is much stiffer than the workpiece in many applications.In most such cases,the workpiece is modeled and nodes at the location offixture contact are completely restrained.This formulation is com-monly referred to as a single-point contact[7–12].Omitting fixture elements does not allow for the model to account for compliance in thefixture and neglects frictional contact effects between thefixture and workpiece.Other researchers[13–16] have utilized linear springs to approximate the stiffness oftheInternational Journal of Machine Tools&Manufacture46(2006)51–58/locate/ijmactool0890-6955/$-see front matter q2005Elsevier Ltd.All rights reserved.doi:10.1016/j.ijmachtools.2005.04.007*Corresponding author.Tel.:C14048948499;fax:C14048949342.E-mail address:*************************.edu(S.N.Melkote).fixture components.However,such an approach requires the stiffness to be measured or approximated,adding time and introducing potential error into the analysis.Recent work [17–19]has explored the use of surface-to-surface contact elements.Such an approach allows frictional effects to be modeled.This methodology was used for the work reported in this paper.Liao et al.[17]used FEA with contact elements to model a multiple-contact fixture system.They,however,did not investigate the effects of friction and meshing parameters on the results.Satyanarayana’s [18,19]work was limited to a single fixture–workpiece contact.More importantly,these studies did not analyze the contribution of fixture body compliance to the overall deformation.This paper investigates the effects of various finite element modeling parameters,such as friction and mesh density,on workpiece deformation.In addition to modeling the workpiece and fixture tips,as is common,the effect of compliance of other fixture components such as support blocks,base plate,etc.on workpiece deformation is also examined.The FEA predictions of workpiece deformation and locator reactions are experimentally verified.2.Fixture–workpiece systemThe fixture–workpiece system used in this study consisted of a hollow block of rectangular section anduniform wall thickness restrained in a 3-2-1fixture layout as shown in Fig.1.The aluminum 6061-T6(E Z 70GPa,n Z 0.334)workpiece measured 153mm !127mm !76mm and had a fixed wall thickness (t in Fig.1)ranging from 6to 10mm.Two clamps were used to press the workpiece against six locators:three on the primary plane,two on the secondary plane,and one on the tertiary plane.Spherical and planar hardened AISI 1144steel (E Z 206GPa,n Z 0.296)fixture tips with black oxide finish were used to locate and clamp the workpiece.The global coordinates of each tip are given in Table 1.3.Model developmentFinite element models were constructed using ANSYS w Version 5.7.Solid models were assembled of the prismatic block and fixture tips.All components in the system were modeled as isotropic elastic bodies.The fixture tips,shown in Fig.2,were modeled as cylinders with either planar (clamp and locator circular contact areas 60and 127mm 2,respectively)or spherical (35mm radius of curvature)end caps.The axial lengths of the planar and spherical tips were 6.4and 10.2mm,respectively.The 10-node tetrahedral element Solid92was used to mesh all solid bodies.Contact between the workpiece and fixture was simu-lated using the quadratic surface-to-surface contact elements Targe170and Conta174.A constant static coefficient of friction was used to establish contact proper-ties at the interfaces.To simulate the locators beingrigidlyZyout of 3-2-1fixture.Table 1Coordinates of locator tipsPlanar Spherical X (mm)Y (mm)Z (mm)X (mm)Y (mm)Z (mm)L130.80.041.927.00.038.1L2132.40.041.9128.60.038.1L30.080.041.90.076.238.1L410.524.80.0 6.721.00.0L510.5112.00.0 6.7108.20.0L6154.066.60.0150.262.70.0C179.1127.041.375.3127.037.5C2153.067.341.3153.063.537.5S.P.Siebenaler,S.N.Melkote /International Journal of Machine Tools &Manufacture 46(2006)51–5852fixed in place,the surface of each locator tip opposite to the contact was restrained in all three translational degrees of freedom.A uniformly distributed pressure was applied over the surface of both clamps in opposite contact to simulate the desired clamping force.The deformation of the workpiece,analyzed in the subsequent sections,was found at two points on the workpiece.The positive directions of these two points,d C1and d C2,are shown in Fig.3.The selection of points was based upon locations on the workpiece that underwent the most deformation due to clamping.3.1.Sensitivity to friction coefficientLaboratory tests conducted by Satyanarayana [18]on the same fixture–workpiece system found an average static coefficient of friction (m )of 0.18between the workpiece and fixture tips.The average was in a range of experimental values from 0.15to 0.25.To test the effect of friction of workpiece deformation prediction,FEA models were constructed for workpiece wall thicknesses of 6–10mm restrained in the fixture previously described.A spectrum of m from 0.15to 0.30was tested in combination with the various wall thicknesses.A summary of the deformation results is given in Table 2.An average difference of 3.1%in deformation prediction was found for a change in m of 0.05.These results show that the effect of small variations in the friction coefficient on the workpiece deformation is quite small.3.2.Treatment of primary plane locatorsA series of models were constructed to determine the influence of the primary plane locators.For a properly designed 3-2-1layout in which workpiece rotation is prevented,the only normal load taken by the three primary plane locators is the weight of the workpiece.The frictional force produced by such a small load is often dwarfed by the much larger clamping loads.An analysis was performed to determine if the frictional effects at these locators must be accounted for in a modeling approach.Two sets of boundary conditions were applied to blocks with wall thicknesses of 7,8,and 9mm.The first set,Case A,included all three primary plane locators subjected to the previously described surface-to-surface contact bound-ary conditions.A value of 0.18was specified for m .The second configuration,Case B,removed all three of the primary plane locators and simply restrained the bottom surface of the workpiece from translating in the z -direction.A summary of the results is given in Table 3.The FEA showed that the predicted deformations differed between the two boundary condition sets by an average of only 1.31%.This small impact of the bottom locators allows the FEA models to be constructed without the primary plane locators,thus saving considerable computational time.This (Case B)boundary condition set used only 77and 69%of the complete model computational time for the planar and spherical tips,respectively.The omission of the three primary plane locators coupled with restraining the bottom surface of the workpiece was the configuration used for subsequent models.It should be pointed out that this approximation may not be valid when machining loads are also taken intoaccount.Fig.2.Fixture tip geometry.Table 2Effect of variation in friction coefficient w (mm)m d C1(m m)d C2(m m)100.1515.018.00.2015.016.50.2515.015.60.3015.114.790.1520.022.60.2020.021.10.2519.918.90.3020.118.980.1528.030.10.2027.828.60.2527.827.20.3028.225.770.1541.042.20.2040.640.40.2540.738.50.3041.336.360.1564.965.90.2064.463.30.2564.960.2Fig.3.Workpiece deformation points.S.P.Siebenaler,S.N.Melkote /International Journal of Machine Tools &Manufacture 46(2006)51–58534.Effect of mesh densityWhile some published literature uses FEA to analyze workpiece deformation,a rigorous study of the influence of mesh density on the modeling of fixture–workpiece compliance is lacking.A key factor in determining an appropriate FEA model for this investigation is to select the ideal mesh density.A coarse mesh might yield inaccurate results.However,too fine a mesh might be unnecessary as well as computationally expensive.For this study,the SMRT smart-meshing function of ANSYS w was utilized to construct the solid mesh.Discrete values ranging from 1(most dense mesh)to 10(least dense mesh)were assigned to the various solid parts.A wide spectrum of combinations of fixture and workpiece mesh sizes was used to determine the optimal mesh size.To test the accuracy of the results,the deformation of the workpiece,d C1and d C2,at the location of the two clamps was computed.Experimental results were used as a benchmark for assessing the validity of the simulated results.A test fixture with dimensions identical to the models was constructed.The workpiece for this study had a uniform wall thickness of 7mm.The fixture elements were fastened to a 15mm thick steel base plate.The threaded fixture tips on the primary plane were screwed directly into the base plate.The other locators were screwed into steel support blocks that were in turn each fastened to the base plate via four bolts and two press-fit dowel pins.A picture of the fixture is given in Fig.4.The two clamps,which were similarly fastened to the base plate via steel support blocks,were actuated by a hydraulic hand pump.The order of clamp actuation can influence workpiece deflection as shown by otherresearchers [20,21]and in previous work by the authors [22].However,in the current study,the two clamps were actuated simultaneously by a single hydraulic pump.The difference in actuation times for the two clamps was deemed to be negligible.The deformation at each point was measured using an eddy current proximity probe.The change in magnetic flux of a target patch detected by the sensor was converted to a displacement value by the data acquisition system.Average deformation results over five trials for each tip and load pair are given in Table 4.The standard deviation for the measurements was 0.43m m.Figs.5and 6show the percentage error between the experimentally measured values and the FEA results for the planar and spherical tip cases,respectively.Plots of the 250N load case at Clamp 1for both spherical and planar tips are given for each wall thickness tested.A similar trend was found for the deformation at Clamp 2as well as for a load of 350N.As depicted in the plots,the workpiece mesh density has the dominant effect on model accuracy.A coarse mesh density can produce results with as much as 20%error.However,as can be observed in Fig.6,the influence of fixture tip mesh density on the accuracy of the results is much less.The workpiece and fixture SMRT density levels for the planar case,which provided the most accurate results,were five and two,respectively.No finite element meshes denser than this combination altered the predicted deformations.For the spherical tip case,a SMRT mesh density level of six for the workpiece and two for the fixture components wasTable 3Effect of primary plane locators w (mm)md C1(m m)d C2(m m)AB %Diff.A B %Diff.100.1515.014.6 2.7618.017.8 1.020.2015.014.6 2.9216.516.40.740.2515.014.5 3.0615.615.50.540.3015.114.6 3.2114.714.60.1690.1520.019.4 2.8922.622.50.810.2020.019.5 2.6321.120.9 1.160.2519.919.4 2.5820.119.9 1.120.3020.119.6 2.6318.918.80.8180.1528.027.4 2.2330.130.00.060.2027.827.3 1.8328.628.50.490.2527.827.2 2.0027.227.20.110.3028.227.6 1.9825.725.70.0770.1541.040.3 1.5842.242.20.110.2040.639.9 1.8040.440.60.610.2540.740.0 1.5238.538.80.690.3041.340.6 1.5736.336.70.9960.1564.964.30.9265.965.50.640.2064.463.7 1.1563.363.20.120.2564.964.30.9160.260.10.23Table 4Experimental datad C1(m m)d C2(m m)250N Planar 26.520.5Spherical 29.428.3350NPlanar 36.228.9Spherical41.338.5Fig.4.Experimental 3-2-1fixture.S.P.Siebenaler,S.N.Melkote /International Journal of Machine Tools &Manufacture 46(2006)51–5854chosen.Table 5summarizes these results.As the table shows,the finite element model provides solutions with less than 5%error for the designated mesh density levels.The selected mesh densities were used in all subsequent analyses.A summary of the element properties corresponding to the selected mesh density levels is presented in Table 6.5.Validation of selected mesh parametersThe general validity of the finite element mesh parameters obtained above for the fixture–workpiece system was established by applying the same mesh guidelines to a different load–wall thickness combination.An identical experimental set-up was used as in Section 4.The wallthickness of the block was 8mm and a 375N clamping load was utilized.Table 7presents the corresponding exper-imental and FEA results.As the table shows,there is less than 5%error between the FEA and experimental values,–20.00%–18.00%–16.00%–14.00%–12.00%–10.00%–8.00%–6.00%–4.00%–2.00%0.00%1234567SMRT Size (Fixture)% E r r o r i n d C 1Fig.5.Percentage error for various workpiece density levels (planar tips).12345678SMRT Size (Fixture)–30.00%–25.00%–20.00%–15.00%–10.00%–5.00%0.00%% E r r o r i n d C 1Fig.6.Percentage error for various workpiece density levels (spherical tips).Table 5Predicted vs.measured results for selected mesh density levels250N 350N d C1(m m)d C2(m m)d C1(m m)d C2(m m)Planar Measured 26.520.536.228.9FEA 25.320.335.428.5%Error K 4.55K 0.91K 2.19K 1.55Spherical Measured 29.428.341.338.5FEA 28.727.739.438.8%ErrorK 2.34K 2.17K 4.530.71S.P.Siebenaler,S.N.Melkote /International Journal of Machine Tools &Manufacture 46(2006)51–5855thus establishing the adequacy of the selected mesh parameters for the given fixture–workpiece system.6.Reaction force predictionAnother point of interest was the reaction forces between the locators and workpiece.Knowledge of reaction forces is important to ensure stability and equilibrium in the system.A series of experiments were run using the set-up shown in Fig.4to determine the reaction forces at Locators 1–3.Pressure-sensitive (Fuji Pre-scale)film was used at each of the three locator–workpiece contacts to produce a color intensity map.The individual images were mapped into a density array via a regression analysis to determine the contact pressure [23].The area of the imprint was found using a photo-editing program,leading to the determination of the reaction forces.The mesh parameters that were established in Section 5and a coefficient of friction of 0.18were used to predict reaction forces at the same locations.Table 8gives the results of the analysis.As can be seen from the table,FEA predictions are within 5%of the experimen-tal values thereby further validating the model.7.Effect of fixture body complianceThe models described in the previous sections only took into account the workpiece and the fixture tips.Most published analyses of fixtures assume that the fixture is rigid due to its considerably higher stiffness compared to that of the workpiece.Thus,these studies do not model the fixture itself and cannot account for any deformation effects of the fixture components.This paper,however,sought to investigate the contribution of fixture compliance to the overall predicted workpiece deformation by modeling various fixture elements.The components that were modeled in addition to the contact tips were the fixture support blocks,base plate,and clamp supports as shown in Fig.7.A series of models were constructed incorporating various combinations of these components.The support blocks and base plate were constructed to be solid prismatic blocks with dimensions matching those in the physical set-up.The support blocks had dimensions 63.5mm !50.8mm !73.8mm.The steel base plate had dimensions of 305mm !305mm !15mm.The planar and spherical tip fixture element dimensions were the same as those mentioned earlier in the paper.The fixture tips were modeled as fastened to the support blocks via the vglue command of ANSYS w .The support blocks were similarly affixed to the base plate and the bolt and dowel holes wereTable 6Mesh density parametersPlanarSpherical No.of WP elements 43807343WP element size (mm 3)65.2738.93No.of fixture elements 539423,408Fixture element size (mm 3)0.540.32Contact element size (mm 3)1.500.12Table 7Model validation resultsd C1(m m)d C2(m m)PlanarFEA26.321.4Measured 27.322.3%Error K 3.56K 4.14SphericalFEA32.826.4Measured 33.927.7%ErrorK 3.17K 4.71Table 8Comparison of measured and predicted locator reaction forcesMeasured (N)FEA (N)%Diff.F L1(N)Spherical 2508986K 2.9350126121K 3.9Planar2508481K 4.0350117113K 3.8F L2(N)Spherical 2509895K 2.9350138133K 3.6Planar25010299K 3.1350145139K 4.5F L3(N)Spherical 250170177 4.1350250248K 0.9Planar250177173K 2.1350244243K0.6Fig.7.Fixture components.S.P.Siebenaler,S.N.Melkote /International Journal of Machine Tools &Manufacture 46(2006)51–5856not modeled.Models that included the base plate used the vglue command to affix primary plane locating elements to the plate.For the models including the support blocks but not the base plate,the bottom surface of the blocks was constrained in all three translational degrees of freedom to simulate the blocks being fastened firmly to the base plate.The cylindrical clamp supports were modeled as solid cylinders that could only translate along the axis of the clamping force.The clamping pressure produced by the hydraulic pump was modeled as a uniformly distributed pressure across the area on the back surface of the clamp.FEA models of workpiece wall thicknesses of 7,8,and 9mm were used for the analysis.The loading conditions used are the same as those in Table 5.Workpiece deformation was also evaluated at the same locations as in the mesh density study.Results for the planar tip case are given in Table 9.The percent prediction error,averaged over both loading conditions and deformation locations,is given for FEA models including various fixture com-ponents.Also given in the table are computational times relative to the model containing only the fixture tips and workpiece.It is evident that modeling the additional individual fixture elements improved the accuracy of the FEA only slightly.The complete fixture model showed significant improvement in the simulated results,yielding results within 1%of the experimental values.However,the complete model required 653%of the computational time needed by the model consisting of just the workpiece and fixture tips.For the fixture with spherical tips,modeling of fixture components did not improve the accuracy of the results.The modeling of additional fixture components gave a more accurate representation of the fixture–workpiece interaction for the planar contact case.The deformation of the fixture itself,while small in comparison with that of the workpiece,influenced the overall system compliance.The FEA showed that 2.4%of the measured deformation is actually deformation in the fixture.The simulations showed that 22.7%of the fixture compliance occurs in the tips and the remaining 77.3%in the support system.Thus,the original model only containing the workpiece and fixture tips captured 98.1%of the overall system compliance.7.1.Significance of support block complianceAn analysis was performed to see how changes in the fixture compliance altered the overall system compliance.Complete fixture models of the initial configuration as well as cases for which the support blocks had different depths,D ,were run.The initial depth,D 0,is shown in Fig.8.Fig.9plots the percentage of system deformation occurring in the fixture itself for various normalized values of D .It would be expected that thinner blocks would deflect more than larger ones,shifting more deformation to the fixture.This assumption is confirmed in the plot.8.ConclusionsThis paper focused on the factors influencing the prediction of workpiece deformation when using the finite element method.Specifically,it analyzed the effects of different finite element modeling parameters such as contact friction,mesh density and fixture body compliance on the prediction of workpiece deformation.Predicted workpieceTable 9Effect of fixture compliance:planar tip caseAvg.error (%)Time (%)Fixture tips 2.30100Support blocks 1.76153Clamp supports 1.52305Blocks and supports 1.69346Complete fixture0.93653Fig.8.Support blockdepth.0.000.020.040.060.080.100.120.140.160.180.250.500.751D/D o d F /d T O TFig.9.Effect of support block compliance.S.P.Siebenaler,S.N.Melkote /International Journal of Machine Tools &Manufacture 46(2006)51–5857deformations and locator reaction forces were then experimentally verified.The simulation studies showed that models of the workpiece andfixture contacts based on surface-to-surface contact elements could predict workpiece deformations and reaction forces to within5%of the experimental values.The mesh density of the workpiece was found to be more vital to model accuracy than thefixture tip density.For thefixture–workpiece system analyzed in this paper,over98%of all system deformation could be captured by models containing only the workpiece andfixture tips.The remainder of system deformation is present in thefixture itself.The capturing of such compliance is possible by modeling the entirefixture body.However,a completefixture model requires over six-fold computational time. 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