刹车支架加工工艺及铣方槽底面夹具设计
刹车支架加工工艺及铣方槽底面夹具设计,刹车,支架,加工,工艺,铣方槽,底面,夹具,设计
装 订 线机械加工工序卡片产品型号刹车支架零件图号刹车支架产品名称刹车支架零件名称刹车支架共1页第1页 车间工序号工序名称材 料 牌 号 机加工40铣方槽底面HT20-40毛 坯 种 类毛坯外形尺寸每毛坯可制件数每 台 件 数铸造11设备名称设备型号设备编号同时加工件数铣床X52K1夹具编号切削液 Jiaju002X52K铣床 乳化液工位器具编号工位器具名称工序工时 (分)准终单件QIJU002专用器具工步号工 步 内 容工 艺 装 备主轴转速切削速度进给量切削深度进给次数工步工时r/minm/minmm/rmm机动辅助40铣方槽底面铣夹具,量具,铣刀5000.5 2 25 1 5min 1 min 设 计(日 期)校 对(日期) 审 核(日期) 标准化(日期) 会 签(日期)标记处数更改文件号签 字 日 期标记处数更改文件号签 字 日 期机械加工工艺过程卡片机械加工工艺过程卡片产品型号刹车支架零件图号刹车支架产品名称刹车支架零件名称刹车支架共 1 页第1页材料牌号HT20-40毛坯种类铸造毛坯外型尺寸每毛坯可制作件数 1每台件数1 备注工序号工序名称工序内容车 间工 段设 备工艺装备工 时准终单件10铸造铸造毛坯铸造车间一20时效时效处理铸造车间一30铣铣底面机加工二X52K铣床铣夹具,量具,铣刀101040铣铣方槽底面机加工二X52K铣床铣夹具,量具,铣刀151550铣削铣22凸台2端面机加工二X52K铣床铣夹具,量具,铣刀131360铣削铣32凸台2端面机加工二X52K铣床铣夹具,量具,铣刀131370铣削铣45度斜面机加工二X52K铣床铣夹具,量具,铣刀131380铣削铣14凸台端面机加工二X52K铣床铣夹具,量具,铣刀131390钻孔钻底面2处U型槽底孔2X12机加工二Z525钻床钻夹具,钻头1010100铣削铣2处U型槽机加工二X52K铣床铣夹具,量具,铣刀1313110钻扩铰钻扩铰14H9孔机加工二Z525钻床钻夹具,钻头,铰孔1010120钻扩铰钻扩铰20H9孔机加工二Z525钻床钻夹具,钻头,铰孔1010130钻孔攻丝钻M6底孔,然后进行攻丝M6机加工二Z525钻床钻夹具,钻头1313140钻孔攻丝钻M8底孔,然后进行攻丝M8机加工二Z525钻床钻夹具,钻头1515150去毛刺钳工去毛刺160终检按图样要求检验170入库入库设计(日期)审核(日期)标准化(日期)会签(日期)标记处数更改文件号签字日期标记处数更改文件号签字日期描图描校底图号装订号 课程设计论文刹车支架加工工艺及铣方槽底面夹具设计 所在学院专 业班 级姓 名学 号指导老师 年 月 日摘 要刹车支架零件零件加工工艺及夹具设计是包括零件加工的工艺设计、工序设计以及专用夹具的设计三部分。在工艺设计中要首先对零件进行分析,了解零件的工艺再设计出毛坯的结构,并选择好零件的加工基准,设计出零件的工艺路线;接着对零件各个工步的工序进行尺寸计算,关键是决定出各个工序的工艺装备及切削用量;然后进行专用夹具的设计,选择设计出夹具的各个组成部件,如定位元件、夹紧元件、引导元件、夹具体与机床的连接部件以及其它部件;计算出夹具定位时产生的定位误差,分析夹具结构的合理性与不足之处,并在以后设计中注意改进。关键词:工艺,工序,切削用量,夹紧,定位,误差目 录摘 要2第1章 绪论4第2章 加工a工艺规程设计52.1 零件的分析52.1.1 零件的作用52.1.2 零件的工艺分析52.2 刹车支架零件加工的主要问题和工艺过程设计所应采取的相应措施52.2.1 孔和平面的加工顺序62.2.2 孔系加工方案选择62.3 刹车支架零件加工定位基准的选择72.3.1 粗基准的选择72.3.2 精基准的选择72.4 刹车支架零件加工主要工序安排72.5 机械加工余量、工序尺寸及毛坯尺寸的确定102.6确定切削用量及基本工时(机动时间)10第3章 铣方槽底面夹具设计夹具设计263.1 研究原始质料263.2 定位、夹紧方案的选择273.3 切削力及夹紧力的计算273.4 误差分析与计算283. 5 定向键与对刀装置设计293.6 确定夹具体结构和总体结构313.7夹具设计及操作的简要说明32总 结34参考文献35致 谢3636第1章 绪论机械制造业是制造具有一定形状位置和尺寸的零件和产品,并把它们装备成机械装备的行业。机械制造业的产品既可以直接供人们使用,也可以为其它行业的生产提供装备,社会上有着各种各样的机械或机械制造业的产品。我们的生活离不开制造业,因此制造业是国民经济发展的重要行业,是一个国家或地区发展的重要基础及有力支柱。从某中意义上讲,机械制造水平的高低是衡量一个国家国民经济综合实力和科学技术水平的重要指标。刹车支架零件零件加工工艺及钻床夹具设计是在学完了机械制图、机械制造技术基础、机械设计、机械工程材料等的基础下,进行的一个全面的考核。正确地解决一个零件在加工中的定位,夹紧以及工艺路线安排,工艺尺寸确定等问题,并设计出专用夹具,保证尺寸证零件的加工质量。本次设计也要培养自己的自学与创新能力。因此本次设计综合性和实践性强、涉及知识面广。所以在设计中既要注意基本概念、基本理论,又要注意生产实践的需要,只有将各种理论与生产实践相结合,才能很好的完成本次设计。本次设计水平有限,其中难免有缺点错误,敬请老师们批评指正。第2章 加工工艺规程设计2.1 零件的分析2.1.1 零件的作用题目给出的零件是刹车支架零件。刹车支架零件的主要作用是支架轴,保证轴之间的中心距及平行度,并保证正确安装。因此刹车支架零件的加工质量,不但直接影响的装配精度和运动精度,而且还会影响工作精度、使用性能和寿命。2.1.2 零件的工艺分析由刹车支架零件零件图可知。刹车支架零件是一个刹车支架零件零件,它的外表面上有4个平面需要进行加工。支承孔系在前后端面上。此外各表面上还需加工一系列螺纹孔。因此可将其分为三组加工表面。它们相互间有一定的位置要求。现分析如下:(1)以底面为主要加工表面的加工面。这一组加工表面包括:底面的铣削加工;其中表面粗糙度要求为,(2)以支承孔14 H9、16H9为主要加工表面的加工面。(3)以宽度10mm的缺口为主要加工面。2.2 刹车支架零件加工的主要问题和工艺过程设计所应采取的相应措施由以上分析可知。该刹车支架零件零件的主要加工表面是平面及孔系。一般来说,保证平面的加工精度要比保证孔系的加工精度容易。因此,对于刹车支架零件来说,加工过程中的主要问题是保证孔的尺寸精度及位置精度,处理好孔和平面之间的相互关系。由于的生产量很大。怎样满足生产率要求也是加工过程中的主要考虑因素。2.2.1 孔和平面的加工顺序刹车支架零件类零件的加工应遵循先面后孔的原则:即先加工刹车支架零件上的基准平面,以基准平面定位加工其他平面。然后再加工孔系。刹车支架零件的加工自然应遵循这个原则。这是因为平面的面积大,用平面定位可以确保定位可靠夹紧牢固,因而容易保证孔的加工精度。其次,先加工平面可以先切去铸件表面的凹凸不平。为提高孔的加工精度创造条件,便于对刀及调整,也有利于保护刀具。刹车支架零件零件的加工工艺应遵循粗精加工分开的原则,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系加工精度。2.2.2 孔系加工方案选择刹车支架零件孔系加工方案,应选择能够满足孔系加工精度要求的加工方法及设备。除了从加工精度和加工效率两方面考虑以外,也要适当考虑经济因素。在满足精度要求及生产率的条件下,应选择价格最底的机床。根据刹车支架零件零件图所示的刹车支架零件的精度要求和生产率要求,当前应选用在组合机床上用镗模法镗孔较为适宜。(1)用镗模法镗孔在大批量生产中,刹车支架零件孔系加工一般都在组合镗床上采用镗模法进行加工。镗模夹具是按照工件孔系的加工要求设计制造的。当镗刀杆通过镗套的引导进行镗孔时,镗模的精度就直接保证了关键孔系的精度。采用镗模可以大大地提高工艺系统的刚度和抗振性。因此,可以用几把刀同时加工。所以生产效率很高。但镗模结构复杂、制造难度大、成本较高,且由于镗模的制造和装配误差、镗模在机床上的安装误差、镗杆和镗套的磨损等原因。用镗模加工孔系所能获得的加工精度也受到一定限制。(2)用坐标法镗孔在现代生产中,不仅要求产品的生产率高,而且要求能够实现大批量、多品种以及产品更新换代所需要的时间短等要求。镗模法由于镗模生产成本高,生产周期长,不大能适应这种要求,而坐标法镗孔却能适应这种要求。此外,在采用镗模法镗孔时,镗模板的加工也需要采用坐标法镗孔。用坐标法镗孔,需要将刹车支架零件孔系尺寸及公差换算成直角坐标系中的尺寸及公差,然后选用能够在直角坐标系中作精密运动的机床进行镗孔。2.3 刹车支架零件加工定位基准的选择2.3.1 粗基准的选择粗基准选择应当满足以下要求:(1)保证各重要孔的加工余量均匀;(2)保证装入刹车支架零件的零件与箱壁有一定的间隙。为了满足上述要求,应选择的主要支承孔作为主要基准。即以刹车支架零件的输入轴和输出轴的支承孔作为粗基准。也就是以前后端面上距顶平面最近的孔作为主要基准以限制工件的四个自由度,再以另一个主要支承孔定位限制第五个自由度。由于是以孔作为粗基准加工精基准面。因此,以后再用精基准定位加工主要支承孔时,孔加工余量一定是均匀的。由于孔的位置与箱壁的位置是同一型芯铸出的。因此,孔的余量均匀也就间接保证了孔与箱壁的相对位置。2.3.2 精基准的选择从保证刹车支架零件孔与孔、孔与平面、平面与平面之间的位置 。精基准的选择应能保证刹车支架零件在整个加工过程中基本上都能用统一的基准定位。从刹车支架零件零件图分析可知,它的顶平面与各主要支承孔平行而且占有的面积较大,适于作精基准使用。但用一个平面定位仅仅能限制工件的三个自由度,如果使用典型的一面两孔定位方法,则可以满足整个加工过程中基本上都采用统一的基准定位的要求。至于前后端面,虽然它是刹车支架零件的装配基准,但因为它与刹车支架零件的主要支承孔系垂直。如果用来作精基准加工孔系,在定位、夹紧以及夹具结构设计方面都有一定的困难,所以不予采用。2.4 刹车支架零件加工主要工序安排对于大批量生产的零件,一般总是首先加工出统一的基准。刹车支架零件加工的第一个工序也就是加工统一的基准。具体安排是先以孔定位粗、精加工顶平面。第二个工序是加工定位用的两个工艺孔。由于顶平面加工完成后一直到刹车支架零件加工完成为止,除了个别工序外,都要用作定位基准。因此,顶面上的螺孔也应在加工两工艺孔的工序中同时加工出来。后续工序安排应当遵循粗精分开和先面后孔的原则。先粗加工平面,再粗加工孔系。螺纹底孔在多轴组合钻床上钻出,因切削力较大,也应该在粗加工阶段完成。对于刹车支架零件,需要精加工的是支承孔前后端平面。按上述原则亦应先精加工平面再加工孔系,但在实际生产中这样安排不易于保证孔和端面相互垂直。因此,实际采用的工艺方案是先精加工孔系,然后以支承孔用可胀心轴定位来加工端面,这样容易保证零件图纸上规定的端面全跳动公差要求。各螺纹孔的攻丝,由于切削力较小,可以安排在粗、精加工阶段中分散进行。加工工序完成以后,将工件清洗干净。清洗是在的含0.4%1.1%苏打及0.25%0.5%亚硝酸钠溶液中进行的。清洗后用压缩空气吹干净。保证零件内部杂质、铁屑、毛刺、砂粒等的残留量不大于。根据以上分析过程,现将刹车支架零件加工工艺路线确定如下:工艺路线一:10铸造铸造毛坯20时效时效处理30铣削铣22凸台2端面40铣削铣32凸台2端面50铣铣底面60铣铣方槽底面70铣削铣45度斜面80铣削铣14凸台端面90钻孔钻底面2处U型槽底孔2X12100铣削铣2处U型槽110 钻扩铰钻扩铰14H9孔120钻扩铰钻扩铰20H9孔130钻孔攻丝钻M6底孔,然后进行攻丝M6140钻孔攻丝钻M8底孔,然后进行攻丝M8150去毛刺钳工去毛刺160终检按图样要求检验170入库入库工艺路线二:10铸造铸造毛坯20时效时效处理30铣铣底面40铣铣方槽底面50铣削铣22凸台2端面60铣削铣32凸台2端面70铣削铣45度斜面80铣削铣14凸台端面90钻孔钻底面2处U型槽底孔2X12100铣削铣2处U型槽110钻扩铰钻扩铰14H9孔120钻扩铰钻扩铰20H9孔130钻孔攻丝钻M6底孔,然后进行攻丝M6140钻孔攻丝钻M8底孔,然后进行攻丝M8150去毛刺钳工去毛刺160终检按图样要求检验170入库入库以上加工方案大致看来合理,但通过仔细考虑,零件的技术要求及可能采取的加工手段之后,就会发现仍有问题,以上工艺过程详见机械加工工艺过程综合卡片。综合选择方案一:工艺路线一:10铸造铸造毛坯20时效时效处理30铣铣底面40铣铣方槽底面50铣削铣22凸台2端面60铣削铣32凸台2端面70铣削铣45度斜面80铣削铣14凸台端面90钻孔钻底面2处U型槽底孔2X12100铣削铣2处U型槽110钻扩铰钻扩铰14H9孔120钻扩铰钻扩铰20H9孔130钻孔攻丝钻M6底孔,然后进行攻丝M6140钻孔攻丝钻M8底孔,然后进行攻丝M8150去毛刺钳工去毛刺160终检按图样要求检验170入库入库2.5 机械加工余量、工序尺寸及毛坯尺寸的确定“刹车支架零件”零件材料采用HT20-40制造。材料为HT20-40,硬度HB为170241,生产类型为大批量生产,采用铸造毛坯。(1)底面的加工余量。根据工序要求,顶面加工分粗、精铣加工。各工步余量如下:粗铣:参照机械加工工艺手册第1卷表3.2.23。其余量值规定为,现取。表3.2.27粗铣平面时厚度偏差取。精铣:参照机械加工工艺手册表2.3.59,其余量值规定为。(3)孔毛坯为实心,不冲孔。(4)端面加工余量。根据工艺要求,前后端面分为粗铣、半精铣、半精铣、精铣加工。各工序余量如下:粗铣:参照机械加工工艺手册第1卷表3.2.23,其加工余量规定为,现取。半精铣:参照机械加工工艺手册第1卷,其加工余量值取为。精铣:参照机械加工工艺手册,其加工余量取为。 2.6确定切削用量及基本工时(机动时间)工序30:铣底面机床:铣床X52K刀具:硬质合金端铣刀(面铣刀) 齿数10(1)粗铣刹车支架零件底面 铣削深度:每齿进给量:根据机械加工工艺手册表2.4.73,取铣削速度:参照机械加工工艺手册表2.4.81,取机床主轴转速:,取实际铣削速度:进给量:工作台每分进给量:根据机械加工工艺手册表2.4.81,被切削层长度:由毛坯尺寸可知刀具切入长度:刀具切出长度:取走刀次数为1机动时间:(2)精铣刹车支架零件底面铣削深度:每齿进给量:根据机械加工工艺手册表2.4.73,取铣削速度:参照机械加工工艺手册表2.4.81,取机床主轴转速:,取实际铣削速度:进给量:工作台每分进给量: 被切削层长度:由毛坯尺寸可知刀具切入长度:精铣时刀具切出长度:取走刀次数为1机动时间: 本工序机动时间工序30:铣方槽底面。机床:铣床X52K刀具:硬质合金可转位端铣刀(面铣刀),材料:, ,齿数,此为粗齿铣刀。因其单边余量:Z=3mm所以铣削深度:=3mm精铣该平面的单边余量:Z=1.0mm铣削深度:每齿进给量:根据参考文献3表2.473,取:根据参考文献3表2.481,取铣削速度每齿进给量:根据参考文献3表2.473,取根据参考文献3表2.481,取铣削速度机床主轴转速:按照参考文献3表3.174,取 实际铣削速度: 进给量: 工作台每分进给量: :根据参考文献3表2.481,取切削工时被切削层长度:由毛坯尺寸可知, 刀具切入长度: 刀具切出长度:取走刀次数为1机动时间: 机动时间:所以该工序总机动时间工序50:铣22凸台2端面机床:X52K铣床刀具:硬质合金端铣刀(面铣刀) 齿数10(1)粗铣铣削深度:每齿进给量:根据机械加工工艺手册表2.4.73,取铣削速度:参照机械加工工艺手册表2.4.81,取机床主轴转速:,取实际铣削速度:进给量:工作台每分进给量:根据机械加工工艺手册表2.4.81,被切削层长度:由毛坯尺寸可知刀具切入长度:刀具切出长度:取走刀次数为1机动时间:(2)精铣铣削深度:每齿进给量:根据机械加工工艺手册表2.4.73,取铣削速度:参照机械加工工艺手册表2.4.81,取机床主轴转速:,取实际铣削速度:进给量:工作台每分进给量: 被切削层长度:由毛坯尺寸可知刀具切入长度:精铣时刀具切出长度:取走刀次数为1机动时间: 本工序机动时间工序60:铣32凸台2端面机床:X52K铣床刀具:硬质合金端铣刀(面铣刀) 齿数10(1)粗铣铣削深度:每齿进给量:根据机械加工工艺手册表2.4.73,取铣削速度:参照机械加工工艺手册表2.4.81,取机床主轴转速:,取实际铣削速度:进给量:工作台每分进给量:根据机械加工工艺手册表2.4.81,被切削层长度:由毛坯尺寸可知刀具切入长度:刀具切出长度:取走刀次数为1机动时间:(2)精铣铣削深度:每齿进给量:根据机械加工工艺手册表2.4.73,取铣削速度:参照机械加工工艺手册表2.4.81,取机床主轴转速:,取实际铣削速度:进给量:工作台每分进给量: 被切削层长度:由毛坯尺寸可知刀具切入长度:精铣时刀具切出长度:取走刀次数为1机动时间: 本工序机动时间工序70:铣45度斜面机床:X52K铣床刀具:硬质合金端铣刀(面铣刀) 齿数10(1)粗铣铣削深度:每齿进给量:根据机械加工工艺手册表2.4.73,取铣削速度:参照机械加工工艺手册表2.4.81,取机床主轴转速:,取实际铣削速度:进给量:工作台每分进给量:根据机械加工工艺手册表2.4.81,被切削层长度:由毛坯尺寸可知刀具切入长度:刀具切出长度:取走刀次数为1机动时间:(2)精铣铣削深度:每齿进给量:根据机械加工工艺手册表2.4.73,取铣削速度:参照机械加工工艺手册表2.4.81,取机床主轴转速:,取实际铣削速度:进给量:工作台每分进给量: 被切削层长度:由毛坯尺寸可知刀具切入长度:精铣时刀具切出长度:取走刀次数为1机动时间: 本工序机动时间工序80:铣14凸台端面机床:X52K铣床刀具:硬质合金端铣刀(面铣刀) 齿数10(1)粗铣铣削深度:每齿进给量:根据机械加工工艺手册表2.4.73,取铣削速度:参照机械加工工艺手册表2.4.81,取机床主轴转速:,取实际铣削速度:进给量:工作台每分进给量:根据机械加工工艺手册表2.4.81,被切削层长度:由毛坯尺寸可知刀具切入长度:刀具切出长度:取走刀次数为1机动时间:(2)精铣铣削深度:每齿进给量:根据机械加工工艺手册表2.4.73,取铣削速度:参照机械加工工艺手册表2.4.81,取机床主轴转速:,取实际铣削速度:进给量:工作台每分进给量: 被切削层长度:由毛坯尺寸可知刀具切入长度:精铣时刀具切出长度:取走刀次数为1机动时间: 本工序机动时间工序90:钻底面2处U型槽底孔2X12钻孔选用机床为Z525摇臂机床,刀具选用GB1436-85直柄短麻花钻,机械加工工艺手册第2卷。根据机械加工工艺手册第2卷表10.4-2查得钻孔进给量为0.200.35。 则取确定切削速度,根据机械加工工艺手册第2卷表10.4-9切削速度计算公式为 (3-20)查得参数为,刀具耐用度T=35则 =1.6所以 =72选取 所以实际切削速度为=2.64确定切削时间(一个孔) =工序110:钻、扩、铰14H9孔。机床:立式钻床Z525刀具:根据参照参考文献3表4.39选高速钢锥柄麻花钻头。 钻孔14H9钻孔14H9时先采取的是钻孔,再扩到,所以。切削深度:进给量:根据参考文献3表2.438,取。切削速度:参照参考文献3表2.441,取。机床主轴转速:,按照参考文献3表3.131,取所以实际切削速度:切削工时 被切削层长度:刀具切入长度: 刀具切出长度: 取走刀次数为1机动时间: 扩孔14H9钻孔14H9时先采取的是钻孔,再扩到,所以。切削深度:进给量:根据参考文献3表2.452,取。切削速度:参照参考文献3表2.453,取。机床主轴转速:按照参考文献3表3.131,取所以实际切削速度:切削工时被切削层长度:刀具切入长度有:刀具切出长度: ,取走刀次数为1机动时间: 铰孔14H9刀具:根据参照参考文献3表4.354,选择硬质合金锥柄机用铰刀。切削深度:,且。进给量:根据参考文献3表2.458,取。切削速度:参照参考文献3表2.460,取。机床主轴转速:按照参考文献3表3.131取实际切削速度:切削工时被切削层长度:刀具切入长度,刀具切出长度: 取走刀次数为1机动时间:该工序的加工机动时间的总和是:工序3:钻、扩、铰孔。机床:立式钻床Z525刀具:根据参照参考文献3表4.39选高速钢锥柄麻花钻头。 钻孔钻孔时先采取的是钻孔,再扩到,所以。切削深度:进给量:根据参考文献3表2.438,取。切削速度:参照参考文献3表2.441,取。机床主轴转速:,按照参考文献3表3.131,取所以实际切削速度:切削工时 被切削层长度:刀具切入长度: 刀具切出长度: 取走刀次数为1机动时间: 扩孔刀具:根据参照参考文献3表4.331选择硬质合金锥柄麻花扩孔钻头。片型号:E403因钻孔时先采取的是先钻到孔再扩到,所以,切削深度:进给量:根据参考文献3表2.452,取。切削速度:参照参考文献3表2.453,取。机床主轴转速:按照参考文献3表3.131,取所以实际切削速度:切削工时被切削层长度:刀具切入长度有:刀具切出长度: ,取走刀次数为1机动时间: 铰孔刀具:根据参照参考文献3表4.354,选择硬质合金锥柄机用铰刀。切削深度:,且。进给量:根据参考文献3表2.458,取。切削速度:参照参考文献3表2.460,取。机床主轴转速:按照参考文献3表3.131取实际切削速度:切削工时被切削层长度:刀具切入长度,刀具切出长度: 取走刀次数为1机动时间:该工序的加工机动时间的总和是:工序130:钻M6底孔,然后进行攻丝M6选择钻床:Z525钻床1、刀具的选择:选择高速钢麻花钻,其直径。依据,根据表2.1及表2.2,可选择钻头的几何形状为:标准,。2、选择切削用量1)依据,根据表2.7,可得钢的强度,钻头的直径时,。 因为,所以不需要乘孔深修正系数。 2)依据,根据表2.8,根据钻头强度决定进给量:当,钻头强度允许的进给力。 3)依据,根据表2.9,按机床进给机构强度决定进给量:当,机床进给机构允许的轴向力为8330N(查,表2.35)时,进给量为。从以上三个进给量比较可以看出,受限制的进给量是工艺要求,其值为。根据Z525钻床说明书,选择。查,根据表2.19,当,时,查得轴向力。轴向力的修正系数为:故。查ZK546钻床的使用说明书,机床进给机构所允许的最大轴向力为,由于,所以可用。(2)决定钻头磨钝标准及寿命查,根据表2.12,当时,可取得钻头的后刀面最大磨损量取为0.4mm(0.40.6),刀具的寿命T=15min。(3)决定切削速度查,根据表2.30,可查得,。则。 =查,根据表2.31,可查得,故: r/min 查,根据ZK546钻头的使用说明书,可以考虑选择,但因为所选转数计算转数较高,会使刀具寿命下降,所以可将进给量降一级,即取,也可以选择较低一级的转数 ,仍用,比较这两种选择方案: 1)第一方案 , 2)第二方案 , 因为第一方案的乘积较大,基本工时较少,故第一方案好。这时, 。(4)检验机床扭矩及功率 查,根据表2.20,当,时,。扭矩的修正系数为可查得,所以。根据Z525钻床的使用说明书,当时,。 由于,故选择之切削用量可用,即 f=0.17mm/r, , 。工序140:钻M8底孔,然后进行攻丝M8选择钻床:Z525钻床1、刀具的选择:选择高速钢麻花钻,其直径。依据,根据表2.1及表2.2,可选择钻头的几何形状为:标准,。2、选择切削用量1)依据,根据表2.7,可得钢的强度,钻头的直径时,。 因为,所以不需要乘孔深修正系数。 2)依据,根据表2.8,根据钻头强度决定进给量:当,钻头强度允许的进给力。 3)依据,根据表2.9,按机床进给机构强度决定进给量:当,机床进给机构允许的轴向力为8330N(查,表2.35)时,进给量为。从以上三个进给量比较可以看出,受限制的进给量是工艺要求,其值为。根据Z525钻床说明书,选择。查,根据表2.19,当,时,查得轴向力。轴向力的修正系数为:故。查ZK546钻床的使用说明书,机床进给机构所允许的最大轴向力为,由于,所以可用。(2)决定钻头磨钝标准及寿命查,根据表2.12,当时,可取得钻头的后刀面最大磨损量取为0.4mm(0.40.6),刀具的寿命T=15min。(3)决定切削速度查,根据表2.30,可查得,。则。 =查,根据表2.31,可查得,故: r/min 查,根据ZK546钻头的使用说明书,可以考虑选择,但因为所选转数计算转数较高,会使刀具寿命下降,所以可将进给量降一级,即取,也可以选择较低一级的转数 ,仍用,比较这两种选择方案: 1)第一方案 , 2)第二方案 , 因为第一方案的乘积较大,基本工时较少,故第一方案好。这时, 。(4)检验机床扭矩及功率 查,根据表2.20,当,时,。扭矩的修正系数为可查得,所以。根据Z525钻床的使用说明书,当时,。 由于,故选择之切削用量可用,即 f=0.17mm/r, , 。第3章 铣方槽底面夹具设计夹具设计3.1 研究原始质料利用本夹具主要用来加工铣方槽底面夹具设计,加工时除了要满足粗糙度要求外,还应满足两孔轴线间公差要求。为了保证技术要求,最关键是找到定位基准。同时,应考虑如何提高劳动生产率和降低劳动强度。一、机床夹具定位元件工件定位方式不同,夹具定位元件的结构形式也不同,这里只介绍几种常用的基本定位元件。实际生产中使用的定位元件都是这些基本定位元件的组合。(一)工件以平面定位常用定位元件1支承钉 常用支承钉的结构形式如图6-1所示。平头支承钉(图a)用于支承精基准面;球头支承钉(图b)用于支承粗基准面;网纹顶面支承钉(图c)能产生较大的摩擦力,但网槽中的切屑不易清除,常用在工件以粗基准定位且要求产生较大摩擦力的侧面定位场合。一个支承钉相当于一个支承点,限制一个自由度;在一个平面内,两个支承钉限制二个自由度;不在同一直线上的三个支承钉限制三个自由度。图6-1 常用支承钉的结构形式2支承板 常用的支承板结构形式如图6-2所示。平面型支承板(图a)结构简单,但沉头螺钉处清理切屑比较困难,适于作侧面和顶面定位;带斜槽型支承板(图b),在带有螺钉孔的斜槽中允许容纳少许切屑,适于作底面定位。当工件定位平面较大时,常用几块支承板组合成一个平面。一个支承板相当于两个支承点,限制两个自由度;两个(或多个)支承板组合,相当于一个平面,可以限制三个自由度。图6-2 常用支承板的结构形式3可调支承 常用可调支承结构形式如图6-3所示。可调支承多用于支承工件的粗基准面,支承高度可以根据需要进行调整,调整到位后用螺母锁紧。一个可调支承限制一个自由度。图6-3 常用可调支承的结构形式 (二) 工件以孔定位常用定位元件1定位销 图6-6是几种常用固定式定位销的结构形式。当工件的孔径尺寸较小时,可选用图 a 所示的结构;当孔径尺寸较大时,选用图 b 所示的结构;当工件同时以圆孔和端面组合定位时,则应选用图c所示的带有支承端面的结构。用定位销定位时,短圆柱销限制二个自由度;长圆柱销可以限制四个自由度;短圆锥销(图d)限制三个自由度。图6-6 固定式定位销的结构形式3.2 定位、夹紧方案的选择由零件图可知:在对加工前,平面进行了粗、精铣加工,底面进行了钻、扩加工。因此,定位、夹紧方案有:为了使定位误差达到要求的范围之内,采用一面一销再加上一手动调节的螺丝定位的定位方式,这种定位在结构上简单易操作。一面即底平面。3.3 切削力及夹紧力的计算刀具:铣刀(硬质合金) 刀具有关几何参数: 由参考文献55表129 可得铣削切削力的计算公式: 有:根据工件受力切削力、夹紧力的作用情况,找出在加工过程中对夹紧最不利的瞬间状态,按静力平衡原理计算出理论夹紧力。最后为保证夹紧可靠,再乘以安全系数作为实际所需夹紧力的数值,即: 安全系数K可按下式计算: 式中:为各种因素的安全系数,查参考文献5121可知其公式参数: 由此可得: 所以 根据工件受力切削力、夹紧力的作用情况,找出在加工过程中对夹紧最不利的瞬间状态,按静力平衡原理计算出理论夹紧力。最后为保证夹紧可靠,再乘以安全系数作为实际所需夹紧力的数值。即:安全系数K可按下式计算有:式中:为各种因素的安全系数,查参考文献5表可得: 所以有: 该孔的设计基准为中心轴,故以回转面做定位基准,实现“基准重合”原则;参考文献,因夹具的夹紧力与切削力方向相反,实际所需夹紧力F夹与切削力之间的关系F夹KF轴向力:F夹KF (N)扭距:Nm3.4 误差分析与计算该夹具以一底面一侧面,两支撑钉和一个调节螺丝定位,为了满足工序的加工要求,必须使工序中误差总和等于或小于该工序所规定的尺寸公差。与机床夹具有关的加工误差,一般可用下式表示: 由参考文献5可得:销的定位误差 : 其中:, 夹紧误差 : 其中接触变形位移值: 查5表1215有。 磨损造成的加工误差:通常不超过 夹具相对刀具位置误差:取误差总和:3. 5 定向键与对刀装置设计定向键安装在夹具底面的纵向槽中,一般使用两个。其距离尽可能布置的远些。通过定向键与铣床工作台T形槽的配合,使夹具上定位元件的工作表面对于工作台的送进方向具有正确的位置。定向键可承受铣削时产生的扭转力矩,可减轻夹紧夹具的螺栓的负荷,加强夹具在加工中的稳固性。根据GB220780定向键结构如图所示: 图5.1 夹具体槽形与螺钉根据T形槽的宽度 a=18mm 定向键的结构尺寸如表5.4:表5.4 定向键 BLHhD夹具体槽形尺寸公称尺寸允差d允差公称尺寸允差D180.0120.03525124124.518+0.0195对刀装置由对刀块和塞尺组成,用来确定刀具与夹具的相对位置。塞尺选用平塞尺,其结构如图5.3所示: 图5.3 平塞尺塞尺尺寸参数如表5.5:表5.5 塞尺公称尺寸H允差dC30.0060.25上的分析可见,所设计的夹具能满足零件的加工精度要求。3.6 确定夹具体结构和总体结构对夹具体的设计的基本要求(1)应该保持精度和稳定性在夹具体表面重要的面,如安装接触位置,安装表面的刀块夹紧安装特定的,足够的精度,之间的位置精度稳定夹具体,夹具体应该采用铸造,时效处理,退火等处理方式。(2)应具有足够的强度和刚度保证在加工过程中不因夹紧力,切削力等外力变形和振动是不允许的,夹具应有足够的厚度,刚度可以适当加固。(3)结构的方法和使用应该不错夹较大的工件的外观,更复杂的结构,之间的相互位置精度与每个表面的要求高,所以应特别注意结构的过程中,应处理的工件,夹具,维修方便。再满足功能性要求(刚度和强度)前提下,应能减小体积减轻重量,结构应该简单。(4)应便于铁屑去除在加工过程中,该铁屑将继续在夹在积累,如果不及时清除,切削热的积累会破坏夹具定位精度,铁屑投掷可能绕组定位元件,也会破坏的定位精度,甚至发生事故。因此,在这个过程中的铁屑不多,可适当增加定位装置和夹紧表面之间的距离增加的铁屑空间:对切削过程中产生更多的,一般应在夹具体上面。(5)安装应牢固、可靠夹具安装在所有通过夹安装表面和相应的表面接触或实现的。当夹安装在重力的中心,夹具应尽可能低,支撑面积应足够大,以安装精度要高,以确保稳定和可靠的安装。夹具底部通常是中空的,识别特定的文件夹结构,然后绘制夹具布局。图中所示的夹具装配。加工过程中,夹具必承受大的夹紧力切削力,产生冲击和振动,夹具的形状,取决于夹具布局和夹具和连接,在因此夹具必须有足够的强度和刚度。在加工过程中的切屑形成的有一部分会落在夹具,积累太多会影响工件的定位与夹紧可靠,所以夹具设计,必须考虑结构应便于铁屑。此外,夹点技术,经济的具体结构和操作、安装方便等特点,在设计中还应考虑。在加工过程中的切屑形成的有一部分会落在夹具,切割积累太多会影响工件的定位与夹紧可靠,所以夹具设计,必须考虑结构应便排出铁屑。3.7夹具设计及操作的简要说明为提高生产率,经过方案的认真分析和比较,选用了手动夹紧方式(螺旋机构)。这类夹紧机构结构简单、夹紧可靠、通用性大,在机床夹具中很广泛的应用。此外,当夹具有制造误差,工作过程出现磨损,以及零件尺寸变化时,影响定位、夹紧的可靠。为防止此现象,选用可换定位销。以便随时根据情况进行调整换取。总 结加工工艺的编制和专用夹具的设计,使对零件的加工过程和夹具的设计有进一步的提高。在这次的设计中也遇到了不少的问题,如在编写加工工艺时,对所需加工面的先后顺序编排,对零件的加工精度和劳动生产率都有相当大的影响。在对某几个工序进行专用夹具设计时,对零件的定位面的选择,采用什么方式定位,夹紧方式及夹紧力方向的确定等等都存在问题。这些问题都直接影响到零件的加工精度和劳动生产率,为达到零件能在保证精度的前提下进行加工,而且方便快速,以提高劳动生产率,降低成本的目的。通过不懈努力和指导老师的精心指导下,针对这些问题查阅了大量的相关资料。最后,将这些问题一一解决,并夹紧都采用了手动夹紧,由于工件的尺寸不大,所需的夹紧力不大。完成了本次设计,通过做这次的设计,使对专业知识和技能有了进一步的提高,为以后从事本专业技术的工作打下了坚实的基础。参考文献参考文献1.机床夹具设计 第2版 肖继德 陈宁平主编 机械工业出版社2.机械制造工艺及专用夹具设计指导 孙丽媛主编 冶金工业出版社3.机械制造工艺学 周昌治、杨忠鉴等 重庆大学出版社4. 机械制造工艺设计简明手册李益民 主编 机械工业出版社5. 工艺师手册 杨叔子主编 机械工业出版社3. 机床夹具设计手册王光斗、王春福主编 上海科技出版社7. 机床专用夹具设计图册南京市机械研究所 主编 机械工业出版社8. 机械原理课程设计手册 邹慧君主编 机械工业出版社9.金属切削手册第三版 上海市金属切削技术协会 上海科学技术出版社10.几何量公差与检测第五版 甘永立 主编 上海科学技术出版社11机械设计基础 第三版 陈立德主编 高等教育出版社12工程材料 丁仁亮主编 机械工业出版社13机械制造工艺学课程设计指导书, 机械工业出版社14机床夹具设计 王启平主编 哈工大出版社15.现代机械制图 吕素霞 何文平主编 机械工业出版社致 谢经过了的很长时间,终于比较圆满完成了设计任务。回顾这日日夜夜,感觉经过了一场磨练,通过图书、网络、老师、同学等各种可以利用的方法,巩固了自己的专业知识。对所学知识的了解和使用都有了更加深刻的理解。此时此刻,我要特别感谢我的导师的精心指导,不仅指导我们解决了关键性技术难题,更重要的是为我们指引了设计的思路并给我们讲解了设计中用到的实际工程设计经验,从而使我们设计中始终保持着清晰的思维也少走了很多弯路,也使我学会综合应用所学知识,提高分析和解决实际问题的能力。不仅如此,老师的敬业精神更是深深的感染了我,鞭策着我在以后的工作中爱岗敬业,导师是真真正正作到了传道、授业、解惑。同时也要感谢其他同学、老师和同事的热心帮助,感谢院系领导对我们课程设计的重视和关心,为我们提供了作图工具和场所,使我们能够全身心的投入到设计中去,为更好、更快的完成课程设计提供了重要保障。43Robotics and Computer-Integrated Manufacturing 21 (2005) 368378Locating completeness evaluation and revision in fixture planH. Song?, Y. RongCAM Lab, Department of Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USAReceived 14 September 2004; received in revised form 9 November 2004; accepted 10 November 2004AbstractGeometry constraint is one of the most important considerations in fixture design. Analytical formulation of deterministiclocation has been well developed. However, how to analyze and revise a non-deterministic locating scheme during the process ofactual fixture design practice has not been thoroughly studied. In this paper, a methodology to characterize fixturing systemsgeometry constraint status with focus on under-constraint is proposed. An under-constraint status, if it exists, can be recognizedwith given locating scheme. All un-constrained motions of a workpiece in an under-constraint status can be automatically identified.This assists the designer to improve deficit locating scheme and provides guidelines for revision to eventually achieve deterministiclocating.r 2005 Elsevier Ltd. All rights reserved.Keywords: Fixture design; Geometry constraint; Deterministic locating; Under-constrained; Over-constrained1. IntroductionA fixture is a mechanism used in manufacturing operations to hold a workpiece firmly in position. Being a crucialstep in process planning for machining parts, fixture design needs to ensure the positional accuracy and dimensionalaccuracy of a workpiece. In general, 3-2-1 principle is the most widely used guiding principle for developing a locationscheme. V-block and pin-hole locating principles are also commonly used.A location scheme for a machining fixture must satisfy a number of requirements. The most basic requirement is thatit must provide deterministic location for the workpiece 1. This notion states that a locator scheme producesdeterministic location when the workpiece cannot move without losing contact with at least one locator. This has beenone of the most fundamental guidelines for fixture design and studied by many researchers. Concerning geometryconstraint 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 contactthe 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 thebasis of analytical investigations for deterministic locating that followed. Chou et al. 2 formulated the deterministiclocating problem using screw theory in 1989. It is concluded that the locating wrenches matrix needs to be full rank toachieve deterministic location. This method has been adopted by numerous studies as well. Wang et al. 3 consideredARTICLE IN PRESS 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: hsongwpi.edu (H. Song).locatorworkpiece contact area effects instead of applying point contact. They introduced a contact matrix andpointed out that two contact bodies should not have equal but opposite curvature at contacting point. Carlson 4suggested that a linear approximation may not be sufficient for some applications such as non-prismatic surfaces ornon-small relative errors. He proposed a second-order Taylor expansion which also takes locator error interaction intoaccount. Marin and Ferreira 5 applied Chous formulation on 3-2-1 location and formulated several easy-to-followplanning rules. Despite the numerous analytical studies on deterministic location, less attention was paid to analyzenon-deterministic location.In the Asada and Bys formulation, they assumed frictionless and point contact between fixturing elements andworkpiece. The desired location is q*, at which a workpiece is to be positioned and piecewisely differentiable surfacefunction is gi(as shown in Fig. 1).The surface function is defined as giq? 0: To be deterministic, there should be a unique solution for the followingequation 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 theworkpiece.Only considering the vicinity of desired location q?; where q q? Dq; Asada and By showed thatgiq giq? hiDq,(2)where hiis the Jacobian matrix of geometry functions, as shown by the matrix in Eq. (3). The deterministic locatingrequirement can be satisfied if the Jacobian matrix has full rank, which makes the Eq. (2) to have only one solutionq q?:rankqg1qx0qg1qy0qg1qz0qg1qy0qg1qf0qg1qc0:qgiqx0qgiqy0qgiqz0qgiqy0qgiqf0qgiqc0:qgnqx0qgnqy0qgnqz0qgnqy0qgnqf0qgnqc026666666664377777777758:9=; 6.(3)Upon given a 3-2-1 locating scheme, the rank of a Jacobian matrix for constraint equations tells the constraint statusas shown in Table 1. If the rank is less than six, the workpiece is under-constrained, i.e., there exists at least one freemotion of the workpiece that is not constrained by locators. If the matrix has full rank but the locating scheme hasmore than six locators, the workpiece is over-constrained, which indicates there exists at least one locator such that itcan 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 6has developed a systematic approach for this purpose. Hence, this criterion can be applied to all locating schemes.ARTICLE IN PRESSX 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) 368378369Kang et al. 7 followed these methods and implemented them to develop a geometry constraint analysis module intheir automated computer-aided fixture design verification system. Their CAFDV system can calculate the Jacobianmatrix and its rank to determine locating completeness. It can also analyze the workpiece displacement and sensitivityto 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 hasbeen shown that the position and orientation errors DX of the workpiece and the position errors Dr of locators arerelated as follows:Well-constrained :DX WLDr,(4)Over-constrained :DX WTLWL?1WTLDr,(5)Under-constrained :DX WTLWLWTL?1Dr I6?6? WTLWLWTL?1WLl,(6)where l is an arbitrary vector.They further introduced several indexes derived from those matrixes to evaluate locator configurations, followed byoptimization through constrained nonlinear programming. Their analytical study, however, does not concern therevision of non-deterministic locating. Currently, there is no systematic study on how to deal with a fixture design thatfailed to provide deterministic location.2. Locating completeness evaluationIf deterministic location is not achieved by designed fixturing system, it is as important for designers to knowwhat the constraint status is and how to improve the design. If the fixturing 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 locatingscheme more efficiently. A general strategy to characterize geometry constraint status of a locating scheme is describedin Fig. 2.In this paper, the rank of locating matrix is exerted to evaluate geometry constraint status (see Appendixfor derivation of locating matrix). The deterministic locating requires six locators that provide full rank locatingmatrix WL:As shown in Fig. 3, for given locator number n; locating normal vector ai;bi;ci? and locating position xi;yi;zi? foreach locator, i 1;2;.;n; the n ? 6 locating matrix can be determined as follows:WLa1b1c1c1y1? b1z1a1z1? c1x1b1x1? a1y1:aibiciciyi? biziaizi? cixibixi? aiyi:anbncncnyn? bnznanzn? cnxnbnxn? anyn2666666437777775.(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 locatorsin the locating scheme. The workpiece will be well-constrained without the presence of those n ? 6 locators. Themathematical representation for this status is that there are n ? 6 row vectors in locating matrix that can be expressedas linear combinations of the other six row vectors. The locators corresponding to that six row vectors consist oneARTICLE IN PRESSTable 1RankNumber of locatorsStatuso 6Under-constrained 6 6Well-constrained 646Over-constrainedH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378370locating scheme that provides deterministic location. The developed algorithm uses the following approach todetermine 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 shouldbe removed for the best locating performance.When rankWLo6; the workpiece is under-constrained.3. Algorithm development and implementationThe algorithm to be developed here will dedicate to provide information on un-constrained motions of theworkpiece in under-constrained status. Suppose there are n locators, the relationship between a workpieces position/ARTICLE IN PRESSFig. 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) (aFig. 3. A simplified locating scheme.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378371orientation errors and locator errors can be expressed as follows:DX DxDyDzaxayaz2666666666437777777775w11:w1i:w1nw21:w2i:w2nw31:w3i:w3nw41:w4i:w4nw51:w5i:w5nw61:w6i:w6n2666666666437777777775?Dr1:Dri:Drn2666666437777775,(8)where Dx;Dy;Dz;ax;ay;azare displacement along x, y, z axis and rotation about x, y, z axis, respectively. Driisgeometric error of the ith locator. wijis defined by right generalized inverse of the locating matrix Wr WTLWLWTL?15.To identify all the un-constrained motions of the workpiece, V dxi;dyi;dzi;daxi;dayi;dazi? is introduced such thatV DX 0.(9)Since rankDXo6; there must exist non-zero V that satisfies 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 therotation about x-axis is not constrained. 0;1;1;0;0;0? means that the workpiece can move along the direction given byvector 0;1;1?: There could be infinite solutions. The solution space, however, can be constructed by 6 ? rankWLbasic solutions. Following analysis is dedicated to find out the basic solutions.From Eqs. (8) and (9)VX dxDx dyDy dzDz daxDax dayDay dazDaz dxXni1w1iDri dyXni1w2iDri dzXni1w3iDri daxXni1w4iDri dayXni1w5iDri dazXni1w6iDriXni1Vw1i;w2i;w3i;w4i;w5i;w6i?TDri 0.10Eq. (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 tounderstand because Dx 0 in this case, implying that the corresponding position error of the workpiece is notdependent of any locator errors. Hence, the associated motion is not constrained by locators. Moreover, a combinedmotion is not constrained if one of the elements in DX can be expressed as linear combination of other elements. Forinstance, 9w1ja0;w2ja0; w1j ?w2jfor 8j: In this scenario, the workpiece cannot move along x- or y-axis. However, itcan move along the diagonal line between x- and y-axis defined by vector 1, 1, 0.To find 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 vectorin n ? r dimension space U u1:uj:un?rhi1pjpn ? r; 1pujpn: Select ujin the way that rankWL r still holds after setting all the terms of all the ujth row(s) equal to zero. Set r ? 6 modified locating matrixWLMa1b1c1c1y1? b1z1a1z1? c1x1b1x1? a1y1:aibiciciyi? biziaizi? cixibixi? aiyi:anbncncnyn? bnznanzn? cnxnbnxn? anyn2666666437777775r?6,where i 1;2;:;niauj:ARTICLE IN PRESSH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 3683783722. Compute the 6 ? n right generalized inverse of the modified locating matrixWr WTLMWLMWTLM?1w11:w1i:w1rw21:w2i:w2rw31:w3i:w3rw41:w4i:w4rw51:w5i:w5rw61:w6i:w6r26666666664377777777756?r3. Trim Wrdown to a r ? rfull rank matrix Wrm: r rankWLo6: Construct a 6 ? r dimension vector Q q1:qj:q6?rhi1pjp6 ? r; 1pqjpn: Select qjin the way that rankWr r still holds after setting all theterms of all the qjth row(s) equal to zero. Set r ? r modified inverse matrixWrmw11:w1i:w1r:wl1:wli:wlr:w61:w6i:w6r26666664377777756?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) withall six terms undetermined. Select a term qkfrom vector Q1pkp6 ? r: SetVqk ?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 asfollows.v1:vs:v62666666437777775wqk1:wqki:wqkr2666666437777775?w11:w1i:w1r:wl1:wli:wlr:w61:w6i:w6r2666666437777775?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 fixture system as shown in Fig. 4. The normalvector 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.WL0013?100013?300011?20010?203010?2012666666437777775.ARTICLE IN PRESSH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378373This locating system provides under-constrained positioning since rankWL 5o6: The program then calculatesthe right generalized inverse of the locating matrix.Wr000000:50:5?1?0:51:50:75?1:251:5000:250:25?0:5000:5?0:50000000:5?0:526666666643777777775.The first row is recognized as a dependent row because removal of this row does not affect rank of the matrix. Theother five rows are independent rows. A linear combination of the independent rows is found according therequirement in step 5 of the procedure for under-constrained status. The solution for this special case is obvious that allthe coefficients 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 beemployed 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 thisinitial 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 configuration isWL010515:000:8960010378:001:0600010612:001:01000:9955?0:03490:0880?101:2445?707:26640:86380:9955?0:03490:0880?98:0728?707:26640:82800:08800:0170?0:9960866:6257998:24660:093626666666643777777775,rankWL 5o6 reveals that the workpiece is under-constrained. It is found that one of the first five rows can beremoved without varying the rank of locating matrix. Suppose the first row, i.e., locator L1is removed from WL; theARTICLE IN PRESSXZYL3L4L5L2L1Fig. 4. Under-constrained locating scheme.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378374modified locating matrix turns intoWLM010378:001:0600010612:001:01000:9955?0:03490:0880?101:2445?707:26640:86380:9955?0:03490:0880?98:0728?707:26640:82800:08800:0170?0:996866:6257998:24660:09362666666437777775.The right generalized inverse of the modified locating matrix isWr1:8768?1:8607?20:666521:37160:49953:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:0284026666666643777777775.The program checked the dependent row and found every row is dependent on other five rows. Without losinggenerality, the first row is regarded as dependent row. The 5 ? 5 modified inverse matrix isWrm3:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:028402666666437777775.The undetermined solution is V ?1; v2; v3; v4; v5; v6?:To calculate the five undetermined terms of V according to step 5,1:8768?1:8607?20:666521:37160:499526666666643777777775T?3:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:0284026666666643777777775?1 0; ?1:713; ?0:0432; ?0:0706; 0:04?.Substituting this result into the undetermined solution yields V ?1;0; ?1:713; ?0:0432; ?0:0706; 0:04?ARTICLE IN PRESSFig. 5. Knuckle 610 (modified from real design).H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378375This vector represents a free motion defined by the combination of a displacement along ?1, 0, ?1.713 directioncombined and a rotation about ?0.0432, ?0.0706, 0.04. To revise this locating configuration, another locator shouldbe added to constrain this free motion of the workpiece, assuming locator L1was removed in step 1. The program canalso calculate the free motions of the workpiece if a locator other than L1was removed in step 1. This provides morerevision options for designer.4. SummaryDeterministic location is an important requirement for fixture locating scheme design. Analytical criterion fordeterministic status has been well established. To further study non-deterministic status, an algorithm for checking thegeometry constraint status has been developed. This algorithm can identify an under-constrained status and indicatethe un-constrained motions of workpiece. It can also recognize an over-constrained status and unnecessary locators.The output information can assist designer to analyze and improve an existing locating scheme.Appendix. Locating matrixConsider a general workpiece as shown in Fig. 6. Choose reference frame fWg fixed to the workpiece. Let fGg andfLig be the global frame and the ith locator frame fixed relative to it. We haveFiXw;Hw;rwi fiXli;Hli;rli,(12)where Xw2 3?1and Hw2 3?1(Xli2 3?1and Hli2 3?1) are the position and orientation of the workpiece(the ith locator) in the global frame fGg; rwi2 3?1(rli2 3?1) is the position of the ith contact point between theworkpiece and the ith locator in the workpiece frame fWg (the ith locator frame fLig).Assume that DXw2 3?1(DHw2 3?1) and Drwi2 3?1are the deviations of the position Xw2 3?1(orientationHw2 3?1) of the workpiece and the position of the ith contact point rwi2 3?1; respectively. Then we have the actualcontact on the wor
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