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    Robotics and Computer-Integrated Manufacturing 21 2005 368–378 Locating completeness evaluation and revision in fi xture plan H. Song, Y. Rong CAM Lab, Department of Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USA Received 14 September 2004; received in revised form 9 November 2004; accepted 10 November 2004 Abstract Geometry constraint is one of the most important considerations in fi xture design. Analytical formulation of deterministic location has been well developed. However, how to analyze and revise a non-deterministic locating scheme during the process of actual fi xture design practice has not been thoroughly studied. In this paper, a methodology to characterize fi xturing system’s geometry constraint status with focus on under-constraint is proposed. An under-constraint status, if it exists, can be recognized with given locating scheme. All un-constrained motions of a workpiece in an under-constraint status can be automatically identifi ed. This assists the designer to improve defi cit locating scheme and provides guidelines for revision to eventually achieve deterministic locating. r 2005 Elsevier Ltd. All rights reserved. Keywords Fixture design; Geometry constraint; Deterministic locating; Under-constrained; Over-constrained 1. Introduction A fi xture is a mechanism used in manufacturing operations to hold a workpiece fi rmly in position. Being a crucial step in process planning for machining parts, fi xture design needs to ensure the positional accuracy and dimensional accuracy of a workpiece. In general, 3-2-1 principle is the most widely used guiding principle for developing a location scheme. V-block and pin-hole locating principles are also commonly used. A location scheme for a machining fi xture must satisfy a number of requirements. The most basic requirement is that it must provide deterministic location for the workpiece [1]. This notion states that a locator scheme produces deterministic location when the workpiece cannot move without losing contact with at least one locator. This has been one of the most fundamental guidelines for fi xture design and studied by many researchers. Concerning geometry constraint status, a workpiece under any locating scheme falls into one of the following three categories 1. Well-constrained deterministic The workpiece is mated at a unique position when six locators are made to contact the workpiece surface. 2. Under-constrained The six degrees of freedom of workpiece are not fully constrained. 3. Over-constrained The six degrees of freedom of workpiece are constrained by more than six locators. In 1985, Asada and By [1] proposed full rank Jacobian matrix of constraint equations as a criterion and formed the basis of analytical investigations for deterministic locating that followed. Chou et al. [2] formulated the deterministic locating problem using screw theory in 1989. It is concluded that the locating wrenches matrix needs to be full rank to achieve deterministic location. This method has been adopted by numerous studies as well. Wang et al. [3] considered ARTICLE IN PRESS www.elsevier.com/locate/rcim 0736-5845/-see front matter r 2005 Elsevier Ltd. All rights reserved. doi10.1016/j.rcim.2004.11.012 Corresponding author. Tel. 15088316092; fax 15088316412. E-mail address hsongwpi.edu H. Song. locator–workpiece contact area effects instead of applying point contact. They introduced a contact matrix and pointed out that two contact bodies should not have equal but opposite curvature at contacting point. Carlson [4] suggested that a linear approximation may not be suffi cient for some applications such as non-prismatic surfaces or non-small relative errors. He proposed a second-order Taylor expansion which also takes locator error interaction into account. Marin and Ferreira [5] applied Chou’s formulation on 3-2-1 location and formulated several easy-to-follow planning rules. Despite the numerous analytical studies on deterministic location, less attention was paid to analyze non-deterministic location. In the Asada and By’s formulation, they assumed frictionless and point contact between fi xturing elements and workpiece. The desired location is q*, at which a workpiece is to be positioned and piecewisely differentiable surface function is gias shown in Fig. 1. The surface function is defi ned as gieqT ? 0 To be deterministic, there should be a unique solution for the following equation set for all locators. gieqT ? 0;i ? 1;2;.;n,1 where n is the number of locators and q ? ?x0;y0;z0;y0;f0;c0 represents the position and orientation of the workpiece. Only considering the vicinity of desired location q; where q ? qt Dq; Asada and By showed that gieqT ? gieqT t hiDq,2 where hiis the Jacobian matrix of geometry functions, as shown by the matrix in Eq. 3. The deterministic locating requirement can be satisfi ed if the Jacobian matrix has full rank, which makes the Eq. 2 to have only one solution q ? q rank qg1 qx0 qg1 qy0 qg1 qz0 qg1 qy0 qg1 qf0 qg1 qc0 qgi qx0 qgi qy0 qgi qz0 qgi qy0 qgi qf0 qgi qc0 qgn qx0 qgn qy0 qgn qz0 qgn qy0 qgn qf0 qgn qc0 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 8 9 ; ? 6.3 Upon given a 3-2-1 locating scheme, the rank of a Jacobian matrix for constraint equations tells the constraint status as shown in Table 1. If the rank is less than six, the workpiece is under-constrained, i.e., there exists at least one free motion of the workpiece that is not constrained by locators. If the matrix has full rank but the locating scheme has more than six locators, the workpiece is over-constrained, which indicates there exists at least one locator such that it can be removed without affecting the geometry constrain status of the workpiece. For locating a model other than 3-2-1, datum frame can be established to extract equivalent locating points. Hu [6] has developed a systematic approach for this purpose. Hence, this criterion can be applied to all locating schemes. ARTICLE IN PRESS X Y Z O X’ Y’ Z’ O’ x0,y0,z0 gi UCS WCS Workpiece Fig. 1. Fixturing system model. H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 2005 368–378369 Kang et al. [7] followed these methods and implemented them to develop a geometry constraint analysis module in their automated computer-aided fi xture design verifi cation system. Their CAFDV system can calculate the Jacobian matrix and its rank to determine locating completeness. It can also analyze the workpiece displacement and sensitivity to locating error. Xiong et al. [8] presented an approach to check the rank of locating matrix WLsee Appendix. They also intro- duced left/right generalized inverse of the locating matrix to analyze the geometric errors of workpiece. It has been shown that the position and orientation errors DX of the workpiece and the position errors Dr of locators are related as follows Well-constrained DX ? WLDr,4 Over-constrained DX ? eWT LWLT 1WT LDr, 5 Under-constrained DX ? WT LeWLW T LT 1Dr t eI66 WT LeWLW T LT 1W LTl, 6 where l is an arbitrary vector. They further introduced several indexes derived from those matrixes to evaluate locator confi gurations, followed by optimization through constrained nonlinear programming. Their analytical study, however, does not concern the revision of non-deterministic locating. Currently, there is no systematic study on how to deal with a fi xture design that failed to provide deterministic location. 2. Locating completeness evaluation If deterministic location is not achieved by designed fi xturing system, it is as important for designers to know what the constraint status is and how to improve the design. If the fi xturing system is over-constrained, informa- tion about the unnecessary locators is desired. While under-constrained occurs, the knowledge about all the un- constrained motions of a workpiece may guide designers to select additional locators and/or revise the locating scheme more effi ciently. A general strategy to characterize geometry constraint status of a locating scheme is described in Fig. 2. In this paper, the rank of locating matrix is exerted to evaluate geometry constraint status see Appendix for derivation of locating matrix. The deterministic locating requires six locators that provide full rank locating matrix WL As shown in Fig. 3, for given locator number n; locating normal vector ?ai;bi;ci and locating position ?xi;yi;zi for each locator, i ? 1;2;.;n; the n 6 locating matrix can be determined as follows WL? a1b1c1c1y1 b1z1a1z1 c1x1b1x1 a1y1 aibiciciyi biziaizi cixibixi aiyi anbncncnyn bnznanzn cnxnbnxn anyn 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 .7 When rankeWLT ? 6 and n ? 6; the workpiece is well-constrained. When rankeWLT ? 6 and n46; the workpiece is over-constrained. This means there are en 6T unnecessary locators in the locating scheme. The workpiece will be well-constrained without the presence of those en 6T locators. The mathematical representation for this status is that there are en 6T row vectors in locating matrix that can be expressed as linear combinations of the other six row vectors. The locators corresponding to that six row vectors consist one ARTICLE IN PRESS Table 1 RankNumber of locatorsStatus o 6Under-constrained ? 6? 6Well-constrained ? 646Over-constrained H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 2005 368–378370 locating scheme that provides deterministic location. The developed algorithm uses the following approach to determine the unnecessary locators 1. Find all the combination of en 6T locators. 2. For each combination, remove that en 6T locators from locating scheme. 3. Recalculate the rank of locating matrix for the left six locators. 4. If the rank remains unchanged, the removed en 6T locators are responsible for over-constrained status. This method may yield multi-solutions and require designer to determine which set of unnecessary locators should be removed for the best locating performance. When rankeWLTo6; the workpiece is under-constrained. 3. Algorithm development and implementation The algorithm to be developed here will dedicate to provide information on un-constrained motions of the workpiece in under-constrained status. Suppose there are n locators, the relationship between a workpiece’s position/ ARTICLE IN PRESS Fig. 2. Geometry constraint status characterization. X Z Y a1,b1,c1 2,b2,c2 x1,y1,z1 x2,y2,z2 ai,bi,ci xi,yi,zi a Fig. 3. A simplifi ed locating scheme. H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 2005 368–378371 orientation errors and locator errors can be expressed as follows DX ? Dx Dy Dz ax ay az 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ? w11w1iw1n w21w2iw2n w31w3iw3n w41w4iw4n w51w5iw5n w61w6iw6n 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 Dr1 Dri Drn 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ,8 where Dx;Dy;Dz;ax;ay;azare displacement along x, y, z axis and rotation about x, y, z axis, respectively. Driis geometric error of the ith locator. wij is defi ned by right generalized inverse of the locating matrix Wr? WT LeWLW T LT 1 [5]. To identify all the un-constrained motions of the workpiece, V ? ?dxi;dyi;dzi;daxi;dayi;dazi is introduced such that V DX ? 0.9 Since rankeDXTo6; there must exist non-zero V that satisfi es Eq. 9. Each non-zero solution of V represents an un- constrained motion. Each term of V represents a component of that motion. For example, ?0;0;0;3;0;0 says that the rotation about x-axis is not constrained. ?0;1;1;0;0;0 means that the workpiece can move along the direction given by vector ?0;1;1 There could be infi nite solutions. The solution space, however, can be constructed by 6 rankeWLT basic solutions. Following analysis is dedicated to fi nd out the basic solutions. From Eqs. 8 and 9 VX ? dxDx t dyDy t dzDz t daxDaxt dayDayt dazDaz ? dx X n i?1 w1iDrit dy X n i?1 w2iDrit dz X n i?1 w3iDri t dax X n i?1 w4iDrit day X n i?1 w5iDrit daz X n i?1 w6iDri ? X n i?1 V?w1i;w2i;w3i;w4i;w5i;w6iTDri ? 0.e10T Eq. 10 holds for 8Driif and only if Eq. 11 is true for 8ie1pipnT V?w1i;w2i;w3i;w4i;w5i;w6iT? 0.11 Eq. 11 illustrates the dependency relationships among row vectors of Wr In special cases, say, all w1jequal to zero, V has an obvious solution [1, 0, 0, 0, 0, 0], indicating displacement along the x-axis is not constrained. This is easy to understand because Dx ? 0 in this case, implying that the corresponding position error of the workpiece is not dependent of any locator errors. Hence, the associated motion is not constrained by locators. Moreover, a combined motion is not constrained if one of the elements in DX can be expressed as linear combination of other elements. For instance, 9w1ja0;w2ja0; w1j? w2jfor 8j In this scenario, the workpiece cannot move along x- or y-axis. However, it can move along the diagonal line between x- and y-axis defi ned by vector [1, 1, 0]. To fi nd solutions for general cases, the following strategy was developed 1. Eliminate dependent rows from locating matrix. Let r ? rank WLeT; n ? number of locator. If ron; create a vector in en rT dimension space U ? u1ujunr hi e1pjpn r; 1pujpnT Select ujin the way that rankeWLT ? r still holds after setting all the terms of all the uj th rows equal to zero. Set r 6 modifi ed locating matrix WLM? a1b1c1c1y1 b1z1a1z1 c1x1b1x1 a1y1 aibiciciyi biziaizi cixibixi aiyi anbncncnyn bnznanzn cnxnbnxn anyn 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 r6 , where i ? 1;2;;neiaujT ARTICLE IN PRESS H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 2005 368–378372 2. Compute the 6 n right generalized inverse of the modifi ed locating matrix Wr? WT LMeWLMW T LMT 1 ? w11w1iw1r w21w2iw2r w31w3iw3r w41w4iw4r w51w5iw5r w61w6iw6r 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 6r 3. Trim Wrdown to a r rfull rank matrix Wrm r ? rankeWLTo6 Construct a e6 rT dimension vector Q ? q1qjq6r hi e1pjp6 r; 1pqjpnT Select qjin the way that rankeWrT ? r still holds after setting all the terms of all the qj th rows equal to zero. Set r r modifi ed inverse matrix Wrm? w11w1iw1r wl1wliwlr w61w6iw6r 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 66 , where l ? 1;2;;6 elaqjT 4. Normalize the free motion space. Suppose V ? ?V1;V2;V3;V4;V5;V6 is one of the basic solutions of Eq. 10 with all six terms undetermined. Select a term qkfrom vector Qe1pkp6 rT Set Vqk? 1; Vqj? 0 j ? 1;2;;6 r;jakeT; 5. Calculated undetermined terms of V V is also a solution of Eq. 11. The r undetermined terms can be found as follows. v1 vs v6 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ? wqk1 wqki wqkr 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 w11w1iw1r wl1wliwlr w61w6iw6r 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 1 , where s ? 1;2;;6esaqj;saqkT;l ? 1;2;;6 elaqjT 6. Repeat step 4 select another term from Q and step 5 until all e6 rT basic solutions have been determined. Based on this algorithm, a C program was developed to identify the under-constrained status and un- constrained motions. Example 1. In a surface grinding operation, a workpiece is located on a fi xture system as shown in Fig. 4. The normal vector and position of each locator are as follows L1[0, 0, 1]0, [1, 3, 0]0, L2[0, 0, 1]0, [ 3, 3, 0]0, L3[0, 0, 1]0, [2, 1, 0]0, L4[0, 1, 0]0, [3, 0, 2]0, L5[0, 1, 0]0, [1, 0, 2]0. Consequently, the locating matrix is determined. WL? 001310 001330 001120 010203 010201 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 . ARTICLE IN PRESS H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 2005 368–378373 This locating system provides under-constrained positioning since rankeWLT ? 5o6 The program then calculates the right generalized inverse of the locating matrix. Wr? 00000 050510515 0751251500 0250250500 0505000 0000505 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 . The fi rst row is recognized as a dependent row because removal of this row does not affect rank of the matrix. The other fi ve rows are independent rows. A linear combination of the independent rows is found according the requirement in step 5 of the procedure for under-constrained status. The solution for this special case is obvious that all the coeffi cients are zero. Hence, the un-constrained motion of workpiece can be determined as V ? ?1; 0; 0; 0; 0; 0 This indicates that the workpiece can move along x direction. Based on this result, an additional locator should be employed to constraint displacement of workpiece along x-axis. Example 2. Fig. 5 shows a knuckle with 3-2-1 locating system. The normal vector and position of each locator in this initial design are as follows L1[0, 1, 0]0, [896, 877, 515]0, L2[0, 1, 0]0, [1060, 875, 378]0, L3[0, 1, 0]0, [1010, 959, 612]0, L4[0.9955, 0.0349, 0.088]0, [977, 902, 624]0, L5[0.9955, 0.0349, 0.088]0, [977, 866, 624]0, L6[0.088, 0.017, 0.996]0, [1034, 864, 359]0. The locating matrix of this confi guration is WL? 0105150008960 0103780010600 0106120010100 0995500349008801012445707266408638 099550034900880980728707266408280 0088000170099608666257998246600936 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 , rankeWL T ? 5o6 reveals that the workpiece is under-constrained. It is found that one of the fi rst fi ve rows can be removed without varying the rank of locating matrix. Suppose the fi rst row, i.e., locator L1is removed from WL; the ARTICLE IN PRESS X Z Y L3 L4 L5 L2 L1 Fig. 4. Under-constrained locating scheme. H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 2005 368–378374 modifi ed locating matrix turns into WLM? 0103780010600 0106120010100 0995500349008801012445707266408638 099550034900880980728707266408280 008800017009968666257998246600936 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 . The right generalized inverse of the modifi ed locating matrix is Wr? 187681860720666521371604995 30551205513244483244480 109561086212064812476402916 000440004400061000610 0002500025000650006900007 000040000400284002840 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 . The program checked the dependent row and found every row is dependent on other fi ve rows. Without losing generality, the fi rst row is regarded as dependent row. The 5 5 modifi ed inverse matrix is Wrm? 30551205513244483244480 109561086212064812476402916 000440004400061000610 0002500025000650006900007 000040000400284002840 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 . The undetermined solution is V ? ?1; v2; v3; v4; v5; v6 To calculate the fi ve undetermined terms of V according to step 5, 18768 18607 206665 213716 04995 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 T 30551205513244483244480 109561086212064812476402916 000440004400061000610 0002500025000650006900007 000040000400284002840 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 1 ? ?0; 1713; 00432; 00706; 004. Substituting this result into the undetermined solution yields V ? ?1;0; 1713; 00432; 00706; 004 ARTICLE IN PRESS Fig. 5. Knuckle 610 modifi ed from real design. H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 2005 368–378375 This vector represents a free motion defi ned by the combination of a displacement along [1, 0, 1.713] direction combined and a rotation about [0.0432, 0.0706, 0.04]. To revise this locating confi guration, another locator should be added to constrain this free motion of the workpiece, assuming locator L1was removed in step 1. The program can also calculate the free motions of the workpiece if a locator other than L1was removed in step 1. This provides more revision options for designer. 4. Summary Deterministic location is an important re
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    摘 要

       

       本设计是杠杆零件的加工工艺规程及铰削孔的专用夹具设计。杠杆零件的主要加工表面是平面及孔。一般来说,保证平面的加工精度要比保证孔的加工精度容易。因此,本设计遵循先面后孔的原则。并将孔与平面的加工明确划分成粗加工和精加工阶段以保证孔的加工精度?;嫉难≡褚愿芨送庠裁孀魑只?,以孔及其下表面作为精基准。先将底面加工出来,然后作为定位基准,在以底面作为精基准加工孔。整个加工过程选用专用机床。在夹具方面选用专用夹具??悸堑搅慵慕峁钩叽缂虻?,夹紧方式多采用手动夹紧,夹紧简单,机构设计简单,且能满足设计要求。

       

       

       关键词: 杠杆零件  加工工艺  夹具  定位  夹紧

       

    目 录

    第1章 绪论1

    1.1设计背景及发展趋势1

    1.2夹具的基本结构及夹具设计的内容1

    第2章 零件的工艺分析2

    2.1零件的作用2

    2.2技术要求分析2

    2.3审查杠杆的工艺性3

    2.4加工表面的分析3

    2.4.1主要加工面3

    2.4.2主要基准面3

    第3章 杠杆机械加工工艺规程设计4

    3.1计算生产纲领及确定生产类型4

    3.2毛坯选择与毛坯图说明4

    3.2.1选择毛坯4

    3.2.2绘制毛坯图5

    3.3工艺路线的确定5

    3.3.1选择定位基准5

    3.3.2零件表面加工方法的确定6

    3.3.3加工阶段的划分6

    3.3.4工序集中与分散7

    3.3.5确定工艺路线7

    3.4工序尺寸的确定及计算9

    3.5确定切削用量(切削速度、进给量、切削深度)及时间定额9

    3.5.1确定切削用量9

    3.5.2确定基本工时19

    第4章 铰削Φ10H7孔用钻床夹具设计24

    4.1 夹具类型的确定24

    4.1.1通用夹具24

    4.1.2专用夹具24

    4.1.3可调夹具24

    4.1.4组合夹具24

    4.1.5拼装夹具25

    4.2确定工件的定位方案25

    4.3确定工件的夹紧形式26

    4.4确定刀具的导向方式或对刀装置27

    4.5.1定位基准分析29

    4.5.2钻削力计算29

    4.5.3夹紧力计算30

    4.6.1铸造夹具体30

    4.6.2锻造夹具体31

    4.6.3焊接夹具体31

    4.6.4装配夹具体31

    4.7夹具精度校核31

    4.7.1验算中心距4831

    4.7.2验算两孔平行度精度32

    第5章 绘制夹具装配图34

    第6章 总结35

    参考文献36


    第1章 绪论

       机床夹具已成为机械加工中的重要装备?;布芯叩纳杓坪褪褂檬谴俳⒄沟闹匾ひ沾胧┲?。随着我国机械工业生产的不断发展,机床夹具的改进和创造已成为广大机械工人和技术人员在技术革新中的一项重要任务。

    1.1设计背景及发展趋势

       材料、结构、工艺是产品设计的物质技术基础,一方面,技术制约着设计,另一方面,技术也推动着设计。从设计美学的观点看,技术部仅仅是物质基础还具有其本身的“功能”作用,只要善于应用材料的特性,予以相应的结构和适当的加工工艺,就能够创造出适用,美观,经济的产品,即在产品中发挥技术的潜在的“功能”。

       技术是产品形态发展的先导,新材料,新工艺的出现,必然给产品带来新的结构,新的形态和新的造型风格。材料,加工工艺,结构,产品形象有机地联系在一起的,某个环节的变革,便会引起整个机体的变化。

       工业的迅速发展,对产品的品种和生产效率提出了越来越高的要求,使多品种,对中小批生产为机械生产的主流,为了适应机械生产的这种发展趋势,必然对机床夹具提出更高的要求。


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    本文标题:等臂杠杆工艺及铰削Φ10H7孔用钻床夹具设计 版本2[含CAD图纸,工艺工序卡,说明书等资料全套]
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