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离合器压盘的攻丝组合机床设计【14张CAD图纸和说明书】

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摘  要


机械制造业是一个国家经济发展的重要支柱。而制造业的生产能力主要取决于制造装备——机床的先进程度。组合机床兼有低成本和高效率的优点,在大批、大量生产中得到广泛应用,并可用以组成自动生产线。

本文对离合器压盘螺纹孔的加工工艺进行了详细的分析,就其孔的加工提出了“一次装夹,单工位加工,达到产品图样的精度要求”的思路。根据这一思路设计了四轴头单工位攻丝组合机床。

该组合机床由立柱、立柱底座、中间底座、液压滑台、动力箱、多轴箱等组成。本文对各部分的设计进行了详细的计算和论证。



关键词  组合机床,离合器压盘,主轴箱,夹具


ABSTRACT


The machinery Manufacture is an important pillar of economic development in a country. While the capability of production in trade of manufacture mostly depends on the advanced producing equipment-machine tool .modular machine tools have advantages of high efficiency and low cost. It is widely used in large batch production and can be consisted automatic production line 

This paper analyzes the manufacturing process detail of threading holes of a disc. The idea is once clamping the work. The accuracy can be satisfied the demand of drawing. According above idea, the 4-spindle modular machine tool has been designed. 

This paper presents the design and calculating of each part of this machine tool.



Key words   Modular machine tool, Clutch plate


目  录


摘  要 I

ABSTRACT II

1  绪论 1

1.1  机床在国民经济的地位及其发展简史 1

1.2  组合机床的国内、外现状 2

1.2.1  国内组合机床现状 3

1.2.2  国外组合机床现状 4

1.3  机床设计的目的、要求 5

1.3.1  设计的目的 5

1.3.2  设计要求 5

2   总体方案设计 6

2.1 工艺方案 6

2.2  配置形式 6

2.2.1 确定组合机床的配置形式和结构方案 6

2.3 “三图一卡”的编制 7

2.3.1.被加工零件图 7

2.3.2 加工示意图 8

2.3.3 绘制加工示意图的有关计算 8

2.3.5 机床联系尺寸图 12

2.3.6 生产率卡 16

3. 主轴箱设计 20

3.1  绘制主轴箱设计原始依据图 20

3.2主轴结构形式的选择及动力计算 22

3.2.1初步确定齿轮模数 22

3.2.2主轴箱的动力计算 22

3.3传动系统的设计与计算 23

3.3.1传动系统设计 23

3.3.2 验算各主轴转速 24

3.4主轴箱的坐标计算 25

3.4.1 计算各传动轴的坐标 25

3.4.2绘制坐标检查图 25

3.5 主轴箱总图设计 26

3.5.1 主视图和侧视图 26

3.5.2 展开图 27

3.5.3主轴和传动轴装配表 27

4  夹具设计 29

4.1  机床夹具的概述 29

4.1.1 机床夹具的组成 29

4.1.2 机床夹具的类型 29

4.2  工件的定位和夹具的定位设计 29

4.3  夹紧方案的设计 29

结论 31

参考文献 32

致谢 33


1  绪论

1.1  机床在国民经济的地位及其发展简史

现代社会中,人们为了高效、经济地生产各种高质量产品,日益广泛的使用各种机器、仪器和工具等技术设备与装备。为制造这些技术设备与装备,又必须具备各种加工金属零件的设备,诸如铸造、锻造、焊接、冲压和切削加工设备等。由于机械零件的形状精度、尺寸精度和表面粗糙度,目前主要靠切削加工的方法来达到,特别是形状复杂、精度要求高和表面粗糙度要求小的零件,往往需要在机床上经过几道甚至几十道切削加工工艺才能完成。因此,机床是现代机械制造业中最重要的加工设备。在一般机械制造厂中,机床所担负的加工工作量,约占机械制造总工作量的40%~60%,机床的技术性能直接影响机械产品的质量及其制造的经济性,进而决定着国民经济的发展水平。可以这样说,如果没有机床的发展,如果不具备今天这样品种繁多、结构完善和性能精良的各种机床,现代社会目前所达到的高度物质文明将是不可想象的。

一个国家要繁荣富强,必须实现工业、农业、国防和科学技术的现代化,这就需要一个强大的机械制造业为国民经济各部门提供现代化的先进技术设备与装备,即各种机器、仪器和工具等。然而,一个现代化的机械制造业必须要有一个现代化的机床制造业做后盾。机床工业是机械制造业的“装备部”、“总工艺师”,对国民经济发展起着重大作用。因此,许多国家都十分重视本国机床工业的发展和机床技术水平的提高,使本国国民经济的发展建立在坚实可靠的基础上。

机床是人类在长期生产实践中,不断改进生产工具的基础上生产的,并随着社会生产的发展和科学技术的进步而渐趋完善。最原始的机床是木制的,所有运动都是由人力或畜力驱动,主要用于加工木料、石料和陶瓷制品的泥坯,它们实际上并不是一种完整的机器。现代意义上的用于加工金属机械零件的机床,是在18世纪中叶才开始发展起来的。当时,欧美一些工业最发达的国家,开始了从工场手工业向资本主义机器大工业生产方式的过渡,需要越来越多的各种机器,这就推动了机床的迅速发展。为使蒸汽机的发明付诸实用,1770年前后创制了镗削蒸汽机汽缸内孔用的镗床。1797年发明了带有机动刀架的车床,开创了用机械代替人手控制刀具运动的先声,不仅解放了人的双手,并使机床的加工精度和工效起了一个飞跃,初步形成了现代机床的雏型。续车床之后,随着机械制造业的发展,其他各种机床也陆续被创制出来。至19世纪末,车床、钻床、镗床、刨床、拉床、铣床、磨床、齿轮加工机床等基本类型的机床已先后形成。

上世纪初以来,由于高速钢和硬质合金等新型刀具材料相继出现,刀具切削性能不断提高,促使机床沿着提高主轴转速、加大驱动功率和增强结构刚度的方向发展。与此同时,由于电动机、齿轮、轴承、电气和液压等技术有了很大的发展,使机床的转动、结构和控制等方面也得到相应的改进,加工精度和生产率显著提高。此外,为了满足机械制造业日益广阔的各种使用要求,机床品种的发展也与日俱增,例如,各种高效率自动化机床、重型机床、精密机床以及适应加工特殊形状和特殊材料需要的特种加工机床相继问世。50年代,在综合应用电子技术、检测技术、计算技术、自动控制和机床设计等各个领域最新成就的基础上发展起来的数控机床,使机床自动化进入了一个崭新的阶段,与早期发展的仅适用于大批大量生产的纯机械控制和继电器接触器控制的自动化相比,它具有很高柔性,即使在单件和小批生产中也能得到经济的使用。

综观机床的发展史,它总是随着机械工业的扩大和科学技术的进步而发展,并始终围绕着不断提高生产效率、加工精度、自动化程度和扩大产品品种而进行的,现代机床总的趋势仍然是继续沿着这一方向发展。

我国的机床工业是在1949年新中国成立后才开始建立起来的。解放前,由于长期的封锁统治和19世纪中叶以后帝国主义的侵略和掠夺,我国的工农业生产非常落后,既没有独立的机械制造业,更谈不上机床制造业。至解放前夕,全国只有少数城市的一些规模很小的机械厂,制造少量简单的皮带车间、牛头刨床和砂轮等;1949年全国机床产量仅1000多台,品种不到10个。

解放后,党和人民政府十分重视机床工业的发展。在解放初期的三年经济恢复时期,就把一些原来的机械修配厂改建为专业厂;在随后开始的几个五年计划期间,又陆续扩建、新建了一系列机床厂。经过50多年的建设,我国机床工业从无到有,从小到大,现在已经成门类比较齐全,具有一定实力的机床工业体系,能生产5000多种机床通用品种,数控机床1500多种;不仅装备了国内的工业,而且每年还有一定数量的机床出口。

我国机床行业的发展是迅速的,成就是巨大的。但由于起步晚、底子薄,与世界先进水平相比,还有较大差距。为了适应我国工业、农业、国防和科学技术现代化的需要,为了提高机床产品在国际市场上的竞争能力,必须深入开展机床基础理论研究,加强工艺试验研究,大力开发精密、重型和数控机床,使我国的机床工业尽早跻身于世界先进行列。


内容简介:
Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 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 systems 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 /locate/rcim 0736-5845/$-see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2004.11.012 ?Corresponding author. Tel.: +15088316092; fax: +15088316412. E-mail address: hsong (H. Song). locatorworkpiece 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 Chous 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 Bys 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 gi(as shown in Fig. 1). The surface function is defi ned as giq? 0: To be deterministic, there should be a unique solution for the following equation set for all locators. giq 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 q? Dq; Asada and By showed that giq giq? 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) 368378369 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 WL(see 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 WT LWL ?1WT LDr, (5) Under-constrained :DX WT LWLW T L ?1Dr I6?6 ? WT LWLW T L ?1W Ll, (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 rankWL 6 and n 6; the workpiece is well-constrained. When rankWL 6 and n46; the workpiece is over-constrained. This means there are n ? 6 unnecessary locators in the locating scheme. The workpiece will be well-constrained without the presence of those n ? 6 locators. The mathematical representation for this status is that there are n ? 6 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) 368378370 locating scheme that provides deterministic location. The developed algorithm uses the following approach to determine the unnecessary locators: 1. Find all the combination of n ? 6 locators. 2. For each combination, remove that n ? 6 locators from locating scheme. 3. Recalculate the rank of locating matrix for the left six locators. 4. If the rank remains unchanged, the removed n ? 6 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 rankWLo6; 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 workpieces 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) 368378371 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 w11:w1i:w1n w21:w2i:w2n w31:w3i:w3n w41:w4i:w4n w51:w5i:w5n w61:w6i:w6n 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 LWLW T L ?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 rankDXo6; 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 ? rankWL basic solutions. Following analysis is dedicated to fi nd out the basic solutions. From Eqs. (8) and (9) VX dxDx dyDy dzDz daxDax dayDay dazDaz dx X n i1 w1iDri dy X n i1 w2iDri dz X n i1 w3iDri dax X n i1 w4iDri day X n i1 w5iDri daz X n i1 w6iDri X n i1 Vw1i;w2i;w3i;w4i;w5i;w6i?TDri 0.10 Eq. (10) holds for 8Driif and only if Eq. (11) is true for 8i1pipn: Vw1i;w2i;w3i;w4i;w5i;w6i?T 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 row(s) from locating matrix. Let r rank WL; n number of locator. If ron; create a vector in n ? r dimension space U u1:uj:un?r hi 1pjpn ? r; 1pujpn: Select ujin the way that rankWL r still holds after setting all the terms of all the uj th row(s) 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 r?6 , where i 1;2;:;niauj: ARTICLE IN PRESS H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378372 2. Compute the 6 ? n right generalized inverse of the modifi ed locating matrix Wr WT LMWLMW T LM ?1 w11:w1i:w1r w21:w2i:w2r w31:w3i:w3r w41:w4i:w4r w51:w5i:w5r w61:w6i:w6r 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 6?r 3. Trim Wrdown to a r ? rfull rank matrix Wrm: r rankWLo6: Construct a 6 ? r dimension vector Q q1:qj:q6?r hi 1pjp6 ? r; 1pqjpn: Select qjin the way that rankWr r still holds after setting all the terms of all the qj th row(s) equal to zero. Set r ? r modifi ed inverse matrix Wrm w11:w1i:w1r : wl1:wli:wlr : w61:w6i:w6r 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 6?6 , where l 1;2;:;6 laqj: 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 Q1pkp6 ? r: Set Vqk ?1; Vqj 0 j 1;2;:;6 ? r;jak; ( 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 ? w11:w1i:w1r : wl1:wli:wlr : w61:w6i:w6r 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ?1 , where s 1;2;:;6saqj;saqk;l 1;2;:;6 laqj: 6. Repeat step 4 (select another term from Q) and step 5 until all 6 ? r 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, 10, 1, 3, 00, L2:0, 0, 10, 3, 3, 00, L3:0, 0, 10, 2, 1, 00, L4:0, 1, 00, 3, 0, 20, L5:0, 1, 00, 1, 0, 20. Consequently, the locating matrix is determined. WL 0013?10 0013?30 0011?20 010?203 010?201 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) 368378373 This locating system provides under-constrained positioning since rankWL 5o6: The program then calculates the right generalized inverse of the locating matrix. Wr 00000 0:50:5?1?0:51:5 0:75?1:251:500 0:250:25?0:500 0:5?0:5000 0000:5?0:5 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, 00, 896, ?877, ?5150, L2:0, 1, 00, 1060, ?875, ?3780, L3:0, 1, 00, 1010, ?959, ?6120, L4:0.9955, ?0.0349, 0.0880, 977, ?902, ?6240, L5:0.9955, ?0.0349, 0.0880, 977, ?866, ?6240, L6:0.088, 0.017, ?0.9960, 1034, ?864, ?3590. The locating matrix of this confi guration is WL 010515:000:8960 010378:001:0600 010612:001:0100 0:9955?0:03490:0880?101:2445?707:26640:8638 0:9955?0:03490:0880?98:0728?707:26640:8280 0:08800:0170?0:9960866:6257998:24660:0936 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 , rankWL 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) 368378374 modifi ed locating matrix turns into WLM 010378:001:0600 010612:001:0100 0:9955?0:03490:0880?101:2445?707:26640:8638 0:9955?0:03490:0880?98:0728?707:26640:8280 0:08800:0170?0:996866:6257998:24660:0936 2 6 6 6 6 6 6
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