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南京理工大学泰州科技学院毕业设计(论文)外文资料翻译系部: 机械工程系 专 业: 机械工程及自动化 姓 名: 尤庆 学 号: 05010252 外文出处: AMSE 附 件: 1.外文资料翻译译文;2.外文原文。 指导教师评语:译文基本能表达原文思想,语句较流畅,条理较清晰,专业用语翻译基本准确,基本符合中文习惯,整体翻译质量一般。签名: 年 月 日附件 1:外文资料翻译译文平面度公差在自由制造时的可行性分析越来越多自由制造的功能零件正在被设计制造中。在这类应用中,SFF 也能用来创建工具(即模式铸造,低量模具等)或直接创造功能零件本身。为了矫正缺损的功能部件,在允许范围内编造部分几何公差是极为重要的。本文系统的介绍了零件生产过程中的SFF的使用方法。我们的研究预期在以下几方面帮助SFF设计者和提供商。通过评估设计公差对某一过程的帮助,这将会帮助设计者消除制造的问题和选择合适的SFF。也将有助于供应商能满足所有的设计公差要求。1 介绍固体的自由制造采用分层制造模式。先生成CAD模型的一部分切成层和几何形状图层,然后输入制作单位,建立各层次顺序直到整个部分被编写好。一些商用的SFF程序是立体的,如熔融沉积成型,坚硬的地面硬化,分层实体制造技术等等。图1表示的是如何分解成层的SFF。SFF的主要优点是,它们并不需要任何特定加工的一部分,而是完全自动化的。图1 阶梯效应由于程序的固有性质和高度自动化,SFF程序分配有益于设计和制造1。逐渐地,SFF处理在网络上正在被设计者存取。设计者公开他们的程序在网站上。设计者能下载CAD辅助设计系统,来分析制造过程,以确保制造可行性。这将会在大幅减少重复的修正。直到最近,SFF程序才主要用来创造原型零件。越来越多的SFF进程在考虑原型零件。在此类的应用中,SFF能用来创建即模式铸造,低量铸模等。或直接创造功能部分本身。为了矫正有缺陷的功能部件,在允许范围内改写部分几何公差是必要的。为了确定一个进程能否产生在该公差范围内,我们需要分析制造公差和工艺方面的限制。在本文中,我们主要侧重于制造的平面度公差分配的一部分。SFF处理中使用了图层,因此一部分产品可能会出现阶梯效应。这阶梯效应取决于层的厚度。给定进程的最低层厚度是确定的。因此给定了一个程序,最初决定的范围因数的楼梯效果是阶梯效果的夹角。如图2所示,为两个不同方向阴影面的投影,阶梯影响的部分是不同的。因此,可以实现精度的SFF进程具有高度各向异性。不同的常态不同的定向的投影面存在不确定性。如果一个部有许多不同种类的要求,这可能建立一个研究方向,满足个人需求。但可能无法找到一个研究方向满足所有人要求。图2 影响定位精度本文讨论SFF执行自动生产方案的可行性。鉴于用CAD模型的一部分来进行生产所需要的精确度,本文提出了一种新办法。我们使用二阶段的方法。首先分析每个指定的要求,并确定一套可行的方案来满足该要求。作为第二个阶段,我们占采取交叉每套可行方案的方法来建立一套能同时满足所有要求的方案。如果至少有一个方向能满足所有要求,那么,这部分就可以被制造。否则被认定为不可制造。其余的文字将介绍以下内容。第2节介绍了与以往工作相关的领域。第3节介绍了系统是如何准确的建立层的。它识别各种不同的来源的误差,并提出了一个数学模型来模拟平面度的影响。第4节描述如何有系统的评估制造过程的可行性。它描述了一能被用来运行此分析的运算法则。第5节描述我们的执行。它描述如何用几何学来处理信息,并提供了一个例子。第6节描述我们的结论和当前我们工作的局限。2 相关的工作SFF程序生产时,精度很容易被影响。精度受机械和程序的有关错误的影响。为了要生产功能零件,我们应该能够满足所有的要求。一个系统的方法是不可或缺的,以便制造特定设计方案的产品。以下小结讨论所做的工作,建模过程和影响建模质量的原因。2.1建模过程:工艺参数和零件精度Tumer 等人2审议零件表面聚碳酸酯选择性激光烧结,以确定零件的特性。用数学上的方法计算,以确定表面精度定量。对分析的差异进行研究,来揭示表面精度的一个参数。Gervasi 3已经调查统计程序控制的使用,在SLA中建立程序。他研究影响代理收缩系数和线宽补偿因子,SLA程序在一段时间内采用统计过程控制技术。Rosen 等人4提出了一个经验模型。这有助于建立各定量之间的关系和SLA程序的变量,最好的表现了这些公差。Onuh 和Hon 5已经应用 Taguchi 方法研究量化层厚度的影响、舱口间隔,舱口过度深入,深入舱口纠正SLA的错误。Tata 和Flynn 6已经广泛的研究做到试图定量,并且纠正了z误差。接下来利用z误差部分所引起的时间差,用激光扫描截面的第一部分到最后部分的多个层。由于液体树脂的特性,在受激光照射后会发生变化。使用正确的时间间隔连续的使用激光扫描z轴误差能被减少。2.2 寻找最优方案Frank 和Fadel 7开发了一个系统,只要输入用户要求的功能,并说明其相对重要性,就会产生建议的最佳方案。他们的工作并不计算阶梯效果的影响,并取决于输入的优先考虑的两个功能。它仅适用于原始方案的建立。Cheng 等人8描述的一个方法是使每一个要求都被认为是基础的一部分,建立稳定的配置。在每个这样的配置中,表面积乘以重量因素,得到了考虑阶梯效应的结构,从而得到最好的方案。如果在这些项目中有一个以上是类似的,那么就是最佳方案。Pududhai 和Dutta 9考虑了两个不同的标准,薄的切片和阶梯比率的总面积,并提出一个公式,找出最佳设计方案。最初的标准是薄的切片的数目和二级标准的阶梯比率地区总面积的一部分。而错误是因为制造过程没有考虑到他们的工作。Bablani 和Bagchi 10开发了一种算法来计算程序计划中的误差,处理误差和层的数目。输入用户所要求的旋转轴和轮换的间隔,然后错误和建议就会提出。描述错误的程序只适用于这些使用激光束和光敏聚合物的部分。Thompson 和Crawford 11提出了一个方法,建议最优方案使用四个主要目标。尽量减少方案建设高度和最大限度提高表面精度和强度的一部分。问题是目标之一,可能受其他三个目标的制约。然后的问题就是如何解决问题并优化来获得最佳方案。Suh 和Wozny 12开发了一种基于特征识别来获得最佳方案的方法。一个目标的功能与标准一起定义,分开强度,表面光洁度等,所有这些标准,不同的功能是提取模型的标准。接着是目标的功能评估,然后利用非线性优化技术找出最佳的方案。Gupta 等人14已经开发了一种计算方法,找到了最佳方案的建设方向形状沉积制造。SDM是一种固体制造方法,可以创建复杂形状的零件。他们考虑的领域中,代表所有可能的方向和形式的一套方案。他们发现了最佳方案,最后全球最好的方案的选择和比较方法出现了。3在SFF中实现数学模型的精度我们已经制作了一个模型,使一个抽象的SFF过程作为一个层来制造。该过程模型考虑到了三维错误,这两个方向的建设载体和平面层正在建造。我们确定了三个主要的错误之处: 阶梯误差:快速成型过程基于分层制造模式。是通过无数有限厚度的薄层来建立的。因此,将会有阶梯效应。阶梯误差的效果能在图2中见到。这个阶梯效果取决于层的厚度与方向上的建立。对于一个给定的程序最小层厚度是已知的。因此,对某一过程中,首要的因素是确定阶梯效应间的夹角。各种不同的叁数,如表面平整度,尺寸精度和其他几何公差,都受阶梯效应的影响。 xy误差:可能有一个变化中的层,在平面层中,有机器特性和程序特性误差。这误差是指在这总体误差层由于效果的各种参数如图3所显示的错误。图3 XY误差 z误差:由于设备和工艺的一些错误,可能有变化的层。由于各种不同参数的组合决定了z误差。图4就演示这一个误差。图4 Z误差我们进行了案例研究,建立了一个测试的进程。该个案的细节在13中解释。当时的部分测量机和偏离的层面,被称为xy和z误差的代表。对于程序的xy误差的值估计为0.05毫米是。对于程序的z误差的值估计为0.04 毫米是。这些值作为xy和z误差的目标在本文5.4节提到。对于平面误差,将出现两种情况,要加以区别对待。这些案例的出现取决于是否引入阶梯效果面。但是,如图5所示,当之间的夹角方向正常的面和载体接近于0或3.14,采用阶梯式可能会出现更多的错误。图5(a)显示了阶梯逼近图。图5(b)显示了水平层逼近图。图 5 夹角接近 0 和 3.14 时3.1数学模型图 6 显示的 xy 和 z 误差的平面度误差。坐标参照如图 6 所示。图 6 xy 和 z 误差的影响当建立方向的影响只有通过了z误差。当建立方向和水平面垂直,受影响的就不仅仅是是xy误差。图7中是平面度误差的允许范围,就是一个角度间隔。图 7 夹角的影响3.2阶梯效应的数学模型当没有阶段需要建立一个面的时候,平面度仅仅产生z误差。图8显示的是投影在平面上时的样子。图8 载体的投影平面4评估部分制造误差对于每一个T我们都可以建立一个可行性区域,如果我们我们选择的方向在这区域内,那么T将能实现。通过分析影响层建设进程的各种误差,是有可能用数学定义可行性区域的不同公差的。对于给定的要求列三种情况可能产生可行性区域: 没有能实行的建设方向。所给定要求很重要,而没有任何可以建设的方向来满足这些要求。 只有有限的要求能被满足的情况。 所有的建设方向都是可行的。要求比较放松,所有的要求都能产生建设方向。4.1构建可行性区域的定位误差当可行性区域的映射到达一个区域,我们应该用符号来表示它的坐标。直到可行性区域的面上,我们建立的球面坐标系沿面建立。因此,极地和方位角可以用来代表可行性的程度。在此讨论出每个被定的要求。4.1.1可行性区域的逐步介绍。如同图7提出的三种不同情况。1 如果 然后可行性区域S,会被定义为,这在单位球体中表现的是一个区域如图9。图9 球体坐标2 如果 ,那么一个区域的可行性就给出了,这是一个很大的单位球面,可以近似的看其厚度很小。3 如果 ,然后阶梯效应将是无作用的的,图10示了一个简单的平面度。图11显示了这些面的可行性。图10 平面度的简单显示图11 10中的可行性区域图12显示出这样一个区域的范围。从预定的对称面建立载体,可以看出方位角是 和 。因此,获得的是封闭区域对称的极轴。图12 可行性区域的位置查找 程序的价值。对于每个平面的要求,我们需要建造一个圆锥,如图13,如果是建立载体内锥,然后在投影面建设小于层厚度的载体。图13 计算每个 对于 的值4.2可制造性分析的方法:阶段1:构建区域方向对于每一个功能,需要控制其准确性,可行性区域的一个单位代表所有领域可能建立的方向,在4.1中我们提到过。可行性区域的功能是集所有领域的单位,如果建设载体在其中,然后指定的要求是可以被实现的。阶段2:所有区域相交的可能性 如果最后的共同可行性区域不是空的,那么有一部分就是制造的规格。 否则,该部分是不可制造的。5 执行我们已经完成了系统算法描述的一节。我们的系统由以下三个部分组成: 部分几何学 关于部分的需求资讯 工艺参数资料利用这些知识,我们的可制造性分析能够更加准确。主要有两类用户,供应商和设计师。注册中提供服务和创建的个人账户,他们可以注册或者更新。设计者在使用我们的服务前要了解分层技术。他们将上传必要的几何形状和要求信息,并选择一个进程登记名字。我们利用C+和ACIS4.0运行几何计算。我们使用Java1.2和Java3D的图形建设用户界面。正如4.1.1节中提到的,该可行性区域包含一个波段的单位球面。特定情况来决定它是否为空。我们也开发了一个算法来计算结果,然后根据结果中的一项来确定可制造的部分。5.1 例,图14中左上角的是俯视图,而右上角的是张立体图,左下角的是主视图,右下角的是左视图。图14 绘制的例子 在图15(a)显示的是不可制造的部分。图15 可制造性案例分析 在图15(b)中显示的是当要求比较放松时的本来不可制造部分。图16显示了两套可行的制造方案来实现图15(b)的情况。图 16 制造的可行性方案6 结论本文介绍了用两步的方法来分析制造可行性部分所产生的SFF与平面度要求对面的一部分。我们首先分析每个指定的要求,并确定一个可行的制造方案,来满足要求。而第二步,我们采取将相交的两套方案结合来确定出一个新的方案来满足多个要求。本文提出了准确的数学模型,如果公差是同心度,圆柱度的话还需要分析。在实践中,第二步的操作被要求符合需求。方法在本文中提出,可以满足特定的要求。此方法也可以扩大到量化公差的范围。分析算法只会产生一套可行的方案,可以用来制造的进行。没有进行优化找出最佳方案。通过选择合适的目标函数,可延长工作时间,也优化了配置,就能找出可行的建设方案。原型服务可以帮助设计者选择一个适当的生产过程。在材料的选择上没有什么要求。设计者必须每次选择一个他需要分析的程序。该系统可以分析给出设计所需的数据。但是设计者可能有一些要求在材料的选择上。在那种情况下,所有的程序,可以被建立。一个程序就能完成这些分析。在我们进行的案例分析中,我们建立了一个测试用的SFF二维坐标和测量它的测量机。获得的测量数据是用来估计xy和z错误的。上述程序的实验可以正式估算工艺参数对任何SFF程序的影响。7 致谢:这项研究已经得到了来自史丹福大学NSF的资助。任何的意见,调查结果,和结论或在这被表达的忠告只是作者的而不反映赞助者的意见。参考文献1 Rajagopalan, S., Pinilla, J. M., Losleben, P., Tian, Q, and Gupta, S. K., 1998, “综合设计制造网” 亚特兰大工程会议 9月.2 Tumer, I., Thompson, D. C., Crawford, R. H., and Wood, K. L., 1995, “表面聚碳酸酯选择性烧结”德克萨斯州奥斯汀市固体制造过程讨论专题会 八月 7-9 日.3 Gervasi, V. R., 1997, “固体制造过程的统计” 奥斯汀市固体制造过程讨论会 pp.141-148,八月 11-13 日。4 Rosen, David W., Sambu, Shiva Prasad, and West, Aaron P., 2001, “立体建造提高性能的方案规划”计算机辅助设计出版社。5 Onuh, S. O., and Hon, K. K. B., 1997,“哈奇风格精度立体参数和建造优化”奥斯汀市固体制造研讨会 pp.653-660,八月 11-13 日。6 Tata, K., and Flynn, D., 1996, “量化对z-误差的相关问题” 圣地亚哥NASUG程序会议 三月 11-13 日。7 Frank, D., and Fadel, G. F., 1994, “快速成型的推荐方法” 俄亥俄州快速成型技术第五次国际会议 pp.191-200。8 Cheng, W., Fuh, J. Y. H., Nee, A. Y. C., Wong, Y. S., Logh, H. T., and Miyazawa, T., 1995, “立体制造的多目标优化” 学者快速成型 pp.12-23。9 Pududhai, N. S., and Dutta, D., 1994,“确定最优取向”密西根大学技术报告MEAM-94-14。10 Bablani, M., and Bagchi, A., 1995, “快速成型的量化误差”中小企业研究 pp.319-324,五月。11 Thompson, D. C., and Crawford, R. H., 1995, “零件质量与优化” 奥斯汀德克萨斯大学固体制造专题研讨会 八月。12 Suh, Y. S., and Wozny, M. J., 1995, “固体制造集成” 奥斯汀德克萨斯大学固体制造专题研讨会 八月。13 Ramakrishna, A., 2000, “基于WEB的固体制造分析” 马里兰大学机械工程系。14 Gupta, S. K., Tian, Q., and Weiss, L., 1998, “沉积制造的优化”奥本山复杂曲面加工的程序 十月。南京理工大学泰州科技学院毕业设计(论文)前期工作材料学 生 姓 名 : 尤庆 学 号: 05010252系 部 : 机械工程系专 业 : 机械工程及自动化设计 (论 文 )题 目 : 化妆品盒注射模设计指 导 教 师 : 丁武学 副教授材 料 目 录序号 名 称 数量 备 注1 毕业设计(论文)选题、审题表 12 毕业设计(论文)任务书 13 毕业设计(论文)开题报告含文献综述 14 毕业设计(论文)外文资料翻译含原文 15 毕业设计(论文)中期检查表 12009 年 5 月toWe use a two step approach. We first analyze each specified tol-Downloaded 24 Nov 2008 to 222.190.117.220. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmThis will help in drastically reducing the number of iterations ofmodifying the design when the manufacturing constraints areviolated.Until recently SFF processes were primarily used for creatingprototype parts. Increasingly SFF processes are being consideredfor creating functional parts. In such applications, SFF can eitherbe used for creating tooling i.e., patterns for casting, low volumemolds, etc.! or directly creating the functional part itself. In orderto create defect free functional parts, it is extremely important tofabricate the parts within allowable dimensional and geometrictolerances. In order to determine whether a process can producethe part within required tolerances, we need to analyze manufac-turability of design tolerances with respect to process constraints.In this paper we primarily focus on manufacturability of flatnesstolerances assigned on the planar faces of the part.SFF processes approximate objects using layers, therefore thepart being produced may exhibit staircase effect. The extent ofthis staircase effect depends on the layer thickness and the relativeerance on the part and identify the set of feasible build directionsthat can be used to satisfy that tolerance. As a second step, wetake the intersection of all sets of feasible build directions to iden-tify the set of build directions that can simultaneously satisfy allspecified tolerance requirements. If there is at least one build di-rection that can satisfy all tolerance requirements, then the part isconsidered manufacturable. Otherwise, the part is considered non-manufacturable.The remainder of the paper has been organized in the followingmanner. Section 2 describes the previous work in the related ar-1Corresponding author.Contributed by the General and Machine Element Design Committee for publi-cationintheJOURNAL OF MECHANICAL DESIGN. Manuscript received Sept. 1999.Associate Editor: J. M. Vance. Fig. 1 Staircase effect148 Vol. 123, MARCH 2001 Copyright 2001 by ASME Transactions of the ASMERamakrishna ArniEmail: arnirkeng.umd.eduS. K. Gupta1Email: skguptaeng.umd.eduMechanical Engineering Departmentand Institute for Systems Research,University of Maryland,College Park, MD 20742ManufacturabilityFlatnessFreeformIncreasingly, Solid Freeformating functional parts.(i.e., patterns for casting,itself. In order to createthe parts within allowablesystematic approachcesses with flatness toleranceis expected to help SFFating design tolerancesnating manufacturingIt will help process providerstolerance requirements.1 IntroductionSolid Freeform Fabrication refers to the class of processes thatbuild parts using a layered manufacturing paradigm. A three-dimensional CAD model of the part is sliced into layers and thenumerical data on the geometry of the layers is then fed into thefabrication unit, which builds each layer sequentially until theentire part is fabricated. Some of the commercially available SFFprocesses are Stereolithography SLA!, Fused Deposition Model-ing FDM!, Solid Ground Curing SGC!, Layered Object Manu-facturing LOM!, etc. Figure 1 shows how a given part is decom-posed into layers for SFF. The main advantages of SFF processesare that they do not require any part specific tooling and are com-pletely automated.Due to the inherent nature of the process and the high level ofautomation, SFF processes are conducive to the concept of dis-tributed design and manufacturing 1#. Increasingly, SFF pro-cesses are being accessed by designers over the network in adistributed environment. Process providers can publish their pro-cess constraints at their web sites. Designers will be able to down-load these constraints to their CAD system and perform manufac-turability analysis to make sure that the design is manufacturable.Analysis ofTolerances in SolidFabricationFabrication (SFF) processes are being considered for cre-In such applications, SFF can either be used for creating toolinglow volume molds, etc.) or directly creating the functional partdefect free functional parts, it is extremely important to fabricatedimensional and geometric tolerances. This paper describes aanalyzing manufacturability of parts produced using SFF pro-requirements on the planar faces of the part. Our researchdesigners and process providers in the following ways. By evalu-against a given process capability, it will help designers in elimi-problems and selecting the right SFF process for the given design.in selecting a build direction that can meet all designDOI: 10.1115/1.1326439#orientation of the build direction and the face normal. The mini-mum layer thickness for a given process is known. Therefore fora given process, the primary factor that determines the extent ofstaircase effect is the angle between the build orientation and theface normal. As shown in Fig. 2, for two different orientations ofthe shaded face with respect to the build orientation, the extent ofstaircase effect on the built part is different. Thus, the achievableaccuracies in SFF processes are highly anisotropic in nature. Dif-ferent faces whose direction normal is oriented differently withrespect to the build direction may exhibit different values of inac-curacies. Whether a part face or a part feature can be producedwithin the required accuracy depends on the build orientation. If apart has many different types of tolerance requirements, it may bepossible to find build orientations that can meet individual re-quirements. But it might be impossible to find a build orientationthat simultaneously satisfies all of the tolerance requirements.This paper discusses our approach to performing automaticmanufacturing feasibility analysis for SFF processes. Given theCAD model of a part to be manufactured and the requirements onthe accuracies of the planar faces on the part, this paper presents anovel approach to finding out whether the part is manufacturable.cally the staircase effect and depends on the user to input thepriority of the two features selected. It works only for primitiveDownloaded 24 Nov 2008 to 222.190.117.220. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmeas. Section 3 describes how SFF can be modeled as a process oflayer building and layer assembly. It identifies the various sourcesof error and presents a mathematical model of flatness accuracy ofplanar faces. Section 4 describes how a systematic assessment ofmanufacturability of the part can be performed. It describes analgorithm that can be used to perform such an analysis. Section 5describes our implementation. It describes how the geometry, tol-erance and process information is represented and provides anexample. Section 6 describes our conclusions and the current limi-tations of our work.2 Related WorkThe part accuracy produced by SFF processes is greatly af-fected by the relative orientation of the build and the face normaldirections. The accuracy is also affected by machine and processrelated errors. In order to produce functional parts, we should beable to satisfy all the tolerance specifications. A systematicmethod is needed so that the manufacturability of a particulardesign can be automatically evaluated so that the number ofdesign-manufacturing iterations can be reduced. The followingsubsections discuss the work done in the areas of process model-ing and the effect of build orientation on the part quality.2.1 Process Modeling: Relating Process Parameters toPart Accuracy. Tumer et al. 2# have considered surfaces ofpolycarbonate Selective Laser Sintering parts to determine thecharacteristics affecting the part quality. Mathematical measuresare computed for the surfaces to determine surface precisionquantitatively. An analysis-of-variance study is performed to re-veal the trends in the surface accuracy as a function of the processparameters.Gervasi 3# has investigated the use of statistical process con-trol in the SLA build process. He has studied the effect ofxy-shrinkage factor and the line width compensation factor on theSLA process over a period of time by applying statistical processcontrol techniques.Rosen et al. 4# have presented an empirical model for stere-olithography machine accuracy, as specified by geometric toler-ances. Response surface models were experimentally constructedfor evaluating accuracy, which help in establishing the quantita-tive relationships between desired tolerances and the SLA processvariables that best achieve those tolerances.Onuh and Hon 5# have applied the Taguchi Method to studyand quantify the effects of layer thickness, hatch spacing, hatchovercure depth, and hatch fill cure depth on the quality of SLAprototypes.Tata and Flynn 6# have done extensive studies in an attempt toquantify and correct down facing z-errors. A down facing z-erroris partially caused by the time difference as the laser scans thecross section of the first part and the last part in a multiple partbuild. This is due to the cure time of the liquid resin after it hasbeen exposed to the laser. Using the correct time between succes-sive laser scans, the downfacing z-error can be reduced.2.2 Finding Optimal Build Orientation. Frank and Fadel7# developed a system that takes input from the user on twofeatures of a part and also on their relative importance and sug-gests an optimal orientation. Their work does not estimate numeri-Fig. 2 Effect of orientation on accuracyJournal of Mechanical Designshapes.Cheng et al. 8# describe a method in which each face of thepart is considered as the base for fabrication if the part is stable inthat configuration. For each of such configurations, the surfacearea of the face is multiplied by a weight factor that is obtained byconsidering the effects of staircase effect, support structures, etc.The sum of these quantities over all the faces of the part is used tosuggest the best configuration. The configuration with the maxi-mum sum is the best orientation. If more than one of these sumsare similar, then the orientation with the smallest build time isselected.Pududhai and Dutta 9# have considered two different criteria;number of slices and ratio of the staircase area to the total surfacearea, and suggested an algorithm to find out the optimum buildorientation. The primary criterion was the number of slices andthe secondary criterion was the ratio of the staircase area to thetotal surface area of the part. The errors that result from the manu-facturing process itself are not considered in their work.Bablani and Bagchi 10# developed an algorithm to calculatethe process planning error, process error and the number of layers.Input is sought from the user on the axis of rotation and theinterval of rotation and for each of the orientations the error isquantified and the preferred orientation is suggested. The charac-terization of process errors was applied only to those processesthat use laser beam and photosensitive polymers to build parts.Thompson and Crawford 11# proposed a method that suggeststhe optimal orientation using four main objectives, viz. minimiz-ing the build height and the number of supports and maximizingthe surface accuracy and the strength of the part. The problem isformulated by choosing one of the four of these objectives as thefinal objective function and the remaining three as constraints.The problem is then solved optimally to get the best orientation.Suh and Wozny 12# developed a method based on feature rec-ognition principles to obtain the optimal part orientation. An ob-jective function is defined with criteria like the height of thetrapped liquid, area of the part requiring supports, part strength,surface finish, etc. For each of these criteria, different features areextracted from the model to evaluate the criterion. The objectivefunction is then evaluated using non-linear optimization tech-niques to find out the best build orientation.Gupta et al. 14# have developed an algorithm for finding near-optimal build orientation for Shape Deposition ManufacturingSDM! process. SDM is a solid freeform fabrication process thatallows the creation of complex shaped parts by the iterative ap-plication of numerically controlled material deposition and mill-ing operations. The idea behind their algorithm to find a near-optimal build orientation is as follows. They consider a unitsphere that represents all possible orientations and form the set ofspherical polygons by taking the intersection of all great circlescorresponding to various direction normals in the part. They findthe best build orientation within each spherical polygon. Finally,the global best build orientation is selected by comparing bestorientations from individual spherical polygons.3 Mathematical Model of Achievable Accuracy in SFFProcessesWe have developed a process model by making an abstractionof the SFF process as a layer manufacturing and layer assemblyprocess. The process model takes into account the dimensionalerrors, both in the direction of the build vector and in the plane ofthe layer being built. We identified the following three primarysources of part inaccuracies: Staircase errors: Solid Freeform Fabrication processes arebased on the layered manufacturing paradigm. Layers of finitethickness are used to build the part. As a result, there will be astaircase effect on the part. The effect of staircase error can beMARCH 2001, Vol. 123 149Downloaded 24 Nov 2008 to 222.190.117.220. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmseen in the Fig. 2. Every layered manufacturing technique has abuild direction associated with it. Build direction is the direction,normal to which, the part to be manufactured is sliced into layers.The extent of this staircase effect depends on the layer thicknessand the relative orientation of the build direction and the facenormal. The minimum layer thickness for a given process isknown. Therefore, for a given process, the primary factor thatdetermines the extent of staircase effect is the angle between thebuild orientation and the face normal. Various parameters like thesurface flatness, dimensional accuracy, and other geometric toler-ances, are affected by staircase errors. xy-errors: There could be a variation in the layer, in the planeof the layer, due to machine specific and process specific errors.For instance, as an electro-mechanical system drives the mirrorsor the head seating the source of fused material, every time a newlayer is built, it may not be exactly located as desired. Dependingupon the particular process, there would be other parameters likethe width of the laser beam and the post-cure shrinkage in the caseof SLA, which will affect dimensional accuracy in the plane of thelayer. The xy-error denotes this overall inaccuracy in the layerdue to the effect of the various parameters mentioned. Figure 3shows this error. z-errors: There could be a variation in the layer, normal tothe plane of the layer, due to machine specific and process specificerrors. For instance, as and when a new layer needs to be formed,the platform which supports the already formed part of the objectbeing manufactured is lowered so as to accommodate the newlayer that is to be built. The system that controls the lowering ofthe platform would have its own limitations and thus would intro-duce error in the z-direction movement and this would in turnaffect the thickness of the layer being built. As in the case of thexy-error, depending upon the particular manufacturing process,other parameters like the post-cure shrinkage in the case of SLAcause dimensional inaccuracies in the layer normal to its plane.This value of the inaccuracy of the layer thickness in the builddirection, due to the combined effect of the various parametersstated, is the z-error. Figure 4 shows this error.We conducted a case study in which we built a test part usingSanders ModelMaker II process. The details of the case study areexplained in 13#. The part was then measured on a coordinatemeasuring machine and using the deviations from the nominaldimensions, representative values of the xy and z errors wereestimated. The value of xy error for the process was estimated tobe 0.05 mm. The value of z-error for the process was estimated tobe 0.04 mm. These values were used for the xy and z errors forthe purpose of the example in Sections 5.4 of this paper.In deriving the mathematical model for the flatness error on aplanar face, two cases would arise which need to be treated sepa-Fig. 3 xy-error in a layerFig. 4 z-error in a layer150 Vol. 123, MARCH 2001rately. These cases arise depending on whether or not a stair-stepis introduced to approximate the face. Whether or not a face isapproximated by a stair-step depends upon the value of the layerthickness, the angle between the normal to the face and the buildvector and the dimensions of the face. In general, faces are builtby stair-step approximation. However, as shown in Fig. 5, whenthe angle between direction normal of the face and the build vec-tor is close to zero or p, introducing a stair step may introducemore error than approximating the face by a horizontal face. Fig-ure 5a! shows the stair-step approximation and Fig. 5b! showsthe horizontal layer approximation. The combination of the mod-els in these two cases for a planar face would give the overallmodel for the flatness error for that face. The following two sec-tions discuss how each of these models are derived to build thecomposite mathematical model.3.1 Mathematical Model When a Stair-Step is Introducedon the Face. Figure 6 shows the effect of xy and z errors on theflatness error of a planar face. The coordinate frame of referenceis as shown in Fig. 6a!. Based on the errors shown in Fig. 6, thefollowing mathematical model is used to describe the flatness er-ror of a planar face.5h1dz!cos u1dxy!sin uucr,udz, then as explained in Section 3.2 and the Fig. 8,for a given value of the azimuth angle, the polar angle could bevaried, starting from a value of zero, to find out the value of thecritical polar angle ucrsuch that the projection of the face on thebuild vector is equal to the layer thickness. Similarly, differentvalues of critical polar angles can be found out for different valuesof the azimuth angle. The values of the critical polar angle fordifferent values of the azimuth angle can be plotted on a sphere. Aclosed curve can be obtained by joining all such points. Thiswould denote a region such that if the build vector is inside it,then no step will be introduced on that particular face.Figure 13 shows such a region on a sphere. From the consider-ations of symmetry of the projection of the face on the buildvector, it can be seen that the value of ucris the same when theazimuth angle is w and p1w. Hence, the closed region obtained issymmetrical about the polar axis. Moreover, for the same azimuthangle w, if the angle between the build vector and the face normalis in the range p2ucr,p#, then it can be seen that projection ofthe face on the build vector is again less than the layer thickness.Hence, there is another exactly similar region diametrically oppo-site to the first region. These two regions constitute the feasibilityregion for the case when a stair-step is not introduced on the face.4.1.3 Procedure for Finding the Value of ucr. For every pla-nar face on which a flatness tolerance has been specified, we needto construct a cone, as shown in Figure 14, around the face normalsuch that if the build vector is inside the cone, then the projectionof the face on the build vector is less than the layer thickness. InFig. 13 Feasibility region for case 2 in Section 4.1.2Fig. 14 Computing the value of ucrfor a given value of gJournal of Mechanical DesignFig. 14, f denotes the planar face, n is
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