CA6140開合螺母鉆M6鉆螺紋底孔鉆床夾具設(shè)計(jì),CA6140開合螺母鉆M6鉆螺紋底孔鉆床夾具設(shè)計(jì),ca6140,螺母,m6,螺紋,羅紋,底孔,鉆床,夾具,設(shè)計(jì) 課程設(shè)計(jì) 機(jī)械制造技術(shù)課程設(shè)計(jì)說明書 設(shè)計(jì)題目： 制定開和螺母零件的加工工藝， 設(shè)計(jì)鉆螺紋底孔鉆床夾具 序言 機(jī)械制造工藝學(xué)課程設(shè)計(jì)是我們學(xué)完了大學(xué)的全部基礎(chǔ)課、技術(shù)基礎(chǔ)課以及大部分專業(yè)課之后進(jìn)行的。這是我們?cè)谶M(jìn)行畢業(yè)設(shè)計(jì)之前對(duì)所學(xué)各課程的一次深入的綜合性的總復(fù)習(xí)。 就我個(gè)人而言，我希望能通過這次課程設(shè)計(jì)，了解并認(rèn)識(shí)一般機(jī)器零件的生產(chǎn)工藝過程，鞏固和加深已學(xué)過的技術(shù)基礎(chǔ)課和專業(yè)課的知識(shí)，理論聯(lián)系實(shí)際，對(duì)自己未來將從事的工作進(jìn)行一次適應(yīng)性訓(xùn)練，從中鍛煉自己分析問題、解決問題的能力，為今后的工作打下一個(gè)良好的基礎(chǔ)，并且為后續(xù)課程的學(xué)習(xí)大好基礎(chǔ)。 由于能力所限，設(shè)計(jì)尚有許多不足之處，懇請(qǐng)各位老師給予指教。 I 目錄 設(shè)計(jì)任務(wù)書 I 序言 II 課程設(shè)計(jì)說明書正文： 第1章 零件的分析 1 1.1 零件的作用 1 1.2 零件的工藝分析 1 第2章 工藝規(guī)程設(shè)計(jì) 2 2.1 確定毛坯的制造形式 2 2.2 幾面選擇 2 2.3 制定工藝路線 2 2.4 機(jī)械加工余量、工序尺寸及毛坯尺寸的確定 3 2.5 確定切削用量及基本工時(shí) 3 第3章 專用夾具設(shè)計(jì) 5 3.1 設(shè)計(jì)主旨 5 3.2 夾具設(shè)計(jì) 5 3.2.1 定位基準(zhǔn)的選擇 5 3.2.2 切削力和加緊力計(jì)算 5 3.2.3 定位誤差分析 5 3.2.4 夾具設(shè)計(jì)及操作的簡(jiǎn)要說明 6 參考文獻(xiàn) 7 致謝 8 第1章 零件的分析 1.1 零件的作用 題目所給定的零件是CA6140車床上的對(duì)開螺母，也稱開合螺母。其作用是為了傳遞機(jī)床梯形絲杠的動(dòng)力。當(dāng)開合螺母處于“開”的位置（即其手柄在上方）時(shí)，絲杠（或稱“絲桿”）的旋轉(zhuǎn)運(yùn)動(dòng)將無法傳遞給溜板箱；反之，則絲杠的旋轉(zhuǎn)運(yùn)動(dòng)將傳遞給溜板箱，由溜板箱將其轉(zhuǎn)換成溜板的直線運(yùn)動(dòng)，完成刀具的縱向移動(dòng)，平時(shí)機(jī)床溜板用光杠傳動(dòng)的，車螺紋的時(shí)候?qū)﹂_螺母閉合，用梯形絲杠傳動(dòng)，從而完成諸如車削螺紋等工作。 1.2 零件的工藝分析 該零件結(jié)構(gòu)比較簡(jiǎn)單，其主要加工的面只有Φ62、Φ52外圓柱面，M41.5的梯形螺紋。因此先將零件當(dāng)做軸類零件進(jìn)行加工，再擴(kuò)孔，進(jìn)行螺紋加工，鉆4個(gè)螺紋孔，最后再將零件切開一分為二。零件精度及表面粗糙度要求較高，因此要合理安排精加工，以保證達(dá)到要求。 零件圖： 第2章 工藝規(guī)程設(shè)計(jì) 2.1 確定毛坯的制造形式 零件材料為球墨鑄鐵,考慮到球墨鑄鐵的抗缺口敏感性，減震性和耐優(yōu)良等特點(diǎn),因此應(yīng)該選用鑄件,以便使開合螺母在使用中耐磨性良好,保證零件工作可靠。將融化了的金屬澆注到旋轉(zhuǎn)著的模型中，由于離心力的作用，金屬液粘貼于模型的內(nèi)壁上，凝結(jié)后所得鑄件外形與模型內(nèi)壁的形狀相同，使用這種方法可以毋需澆注口，故能顯著地減少金屬的消耗。由于免除了砂型和制模的設(shè)備，以及減少了鑄工車間的面積，生產(chǎn)成本就有了降低；這樣得來的鑄件具有緊密與微細(xì)的顆粒結(jié)構(gòu)及較好的機(jī)械性能。因此采用離心鑄造,這從提高生產(chǎn)率、保證加工精度來考慮也是應(yīng)該的。 2.2 幾面選擇 基面選擇是工藝規(guī)程設(shè)計(jì)中的重要工作之一?；孢x擇的正確與合理，可以使加工質(zhì)量得到保證、生產(chǎn)率得到提高。否則不但加工工藝過程中的問題百出，更有甚者，還會(huì)造成零件大批報(bào)廢，使生產(chǎn)無法正常進(jìn)行。 粗基準(zhǔn)的選擇：對(duì)于一般軸類零件而言，以外圓作為粗基準(zhǔn)是完全合理的。因此對(duì)本零件來說，選擇以Φ62外圓柱面作為粗基準(zhǔn)面加工Φ62外圓柱面和右端面。 精基準(zhǔn)的選擇：使用經(jīng)過加工的表面作為定位基準(zhǔn)，采用加工了的Φ62外圓柱面和右端面作為精基準(zhǔn)。 2.3 制定工藝路線 制定工藝路線的出發(fā)點(diǎn)，是應(yīng)該使零件的幾何形狀、尺寸精度及位置精度等技術(shù)要求能得到妥善的保證。在生產(chǎn)綱領(lǐng)已確定為成批生產(chǎn)的條件下，可以考慮采用萬能性機(jī)床配以專用工夾具，并盡量使工序集中來提高生產(chǎn)率。除此之外，還應(yīng)當(dāng)考慮經(jīng)濟(jì)效果，以便使生產(chǎn)成本盡量下降。 工藝路線方案 工序1：粗車Φ62外圓及端面。 工序2：粗車、半精車Φ62和Φ52外圓。 工序3：精車Φ62和Φ52外圓。 工序4：粗鏜、精鏜Φ31.5。 工序5：粗車梯形螺紋。 工序6：精鉸梯形螺紋。 工序7：切斷。 工序8：精銑R23圓弧，去除毛刺。 工序9：鉆孔M6螺紋底孔孔，去除毛刺。 工序10：攻M6螺紋底孔。 工序11：終檢。 2.4 機(jī)械加工余量、工序尺寸及毛坯尺寸的確定 由機(jī)械制造工藝簡(jiǎn)明手冊(cè)表2.2-3、2.2-5、2.2-4、2.2-1可確定毛坯尺寸及公差。 根據(jù)上述原始材料及加工工藝，分別對(duì)各加工表面的機(jī)械加工余量、工序尺寸確定如下： 一端面 粗車至113 l=2； Φ62外圓 粗車至Φ62 2z=3； 另一端面 粗車至111 l=2； Φ62外圓 粗車至Φ62 2z=3； Φ52外圓 精車至Φ52 2z=0.3； 半精車至Φ52.3 2z=1.1； 粗車至Φ53.4 2z=1.6； Φ62左軸肩面 精車至103 l=0.3； 半精車至102.7 l=0.7； 粗車至102 l=1； Φ62右軸肩面 精車至8 l=0.3； 半精車至8.3 l=0.7； 粗車至9 l=1； Φ31.5 精鏜至Φ31.5 2z=1； 粗鏜至Φ30.5 2z=2； 2.5 確定切削用量及基本工時(shí) 工序7：鉆2XΦ5孔并加工螺紋孔2XM6。本工序采用計(jì)算法確定切削用量。 （1）機(jī)床：搖臂鉆床Z35 （2）刀具選擇：選用高速鋼,錐柄標(biāo)準(zhǔn)麻花鉆Φ5，M6絲錐。 （3）確定切削用量： 2個(gè)孔共走2次刀,由切削手冊(cè)表11-2， f=0.13mm/r, V=18m/min， n=1000*18/(3.14*5)=1146.5 r/min， 根據(jù)機(jī)械制造工藝設(shè)計(jì)簡(jiǎn)明手冊(cè)表4.2-12， 取n機(jī)=1320r/min, 所以 V =3.14×1320×5/1000=20.7m/min 。 （4）計(jì)算基本工時(shí)： 鉆孔工時(shí)： tm1= 2*l/fn=2(3+16+3)/(0.13*1320) =0.26?min。 攻絲工時(shí)： t= (l+l1+l2)/fn+(l+l1+l2)/fn0 l1=(1--3)p, l2 =(2--3)p n0為絲錐或工件回程的轉(zhuǎn)速，p為加工螺紋的螺距。 根據(jù)機(jī)械加工工藝設(shè)計(jì)簡(jiǎn)明手冊(cè)表2.3-20， P=1 所以tm2=2*2(16+2+2)/(0.13*1000)=0.62 min。 最后，將以上各工序切削用量、工時(shí)定額的計(jì)算結(jié)果，連同其他加工數(shù)據(jù)，一并填如機(jī)械加工工藝過程卡片和機(jī)械加工工序卡片中。 第3章 專用夾具設(shè)計(jì) 為了提高勞動(dòng)生產(chǎn)率，保證加工質(zhì)量，降低勞動(dòng)強(qiáng)度，通常要設(shè)計(jì)專用夾具。 經(jīng)過與指導(dǎo)師協(xié)商，決定設(shè)計(jì)第9、10道工序—鉆2M6螺紋底孔的鉆床專用夾具。 3.1 設(shè)計(jì)主旨 本夾具主要用來鉆φ5mm孔、攻M6螺紋底孔。 3.2 夾具設(shè)計(jì) 3.2.1 定位基準(zhǔn)的選擇 出于定位簡(jiǎn)單和快速的考慮，選擇底面、側(cè)面和曲面與底面相交的線為定位基準(zhǔn)，保證加工基準(zhǔn)和設(shè)計(jì)基準(zhǔn)重合，再使用手動(dòng)偏心輪夾緊機(jī)構(gòu)進(jìn)行夾緊。 3.2.2 切削力和加緊力計(jì)算 本步加工按鉆削估算夾緊力。實(shí)際效果可以保證可靠的夾緊。 鉆削軸向力 ： 扭矩 ： 夾緊力為 取系數(shù) S1=1.5 S2=S3=S4=1.1 則實(shí)際夾緊力為 F’=S1*S2*S3*S4*F=24.1N 使用偏心輪機(jī)構(gòu)進(jìn)行人工夾緊，調(diào)節(jié)偏心輪手柄，即可滿足所需的夾緊力。 3.2.3 定位誤差分析 由于加工基準(zhǔn)和設(shè)計(jì)基準(zhǔn)重合，所以能很好的保證了加工精度。 3.2.4 夾具設(shè)計(jì)及操作的簡(jiǎn)要說明 所加工的零件尺寸不大，并且所需夾緊力不是很大，為使夾具簡(jiǎn)單，操作方便靈活，故使用手動(dòng)偏心輪夾緊機(jī)構(gòu)夾緊。 裝配圖： 參考文獻(xiàn) [1]吳 拓主編．機(jī)械制造工藝與機(jī)床夾具課程設(shè)計(jì)指導(dǎo).北京：機(jī)械工業(yè)出版社，2010 [2]艾興、肖詩綱主編．切削用量簡(jiǎn)明手冊(cè)[M].北京：機(jī)械工業(yè)出版社. [3]李益民主編.機(jī)械制造工藝設(shè)計(jì)簡(jiǎn)明手冊(cè)[M] .北京：機(jī)械工業(yè)出版社，1993 [4]孟少農(nóng)主編.機(jī)械加工工藝手冊(cè).機(jī)械工業(yè)出版社，1991 [5]王 棟主編.機(jī)械制造工藝課程設(shè)計(jì)指導(dǎo)書[M].北京：機(jī)械工業(yè)出版社 [6]梅 伶主編.模具課程設(shè)計(jì)指導(dǎo)．北京：機(jī)械工業(yè)出版社，2010 [7]王衛(wèi)衛(wèi)主編.材料成形設(shè)備．北京：機(jī)械工業(yè)出版社，2010 [8]甘永立主編.幾何公差與檢測(cè)．上海：上?？茖W(xué)技術(shù)出版社，2008 [9]孫 桓、陳作模、葛文杰主編.《機(jī)械原理》．北京：高等教育出版社，2006 [10]崇 凱主編.機(jī)械制造技術(shù)基礎(chǔ).化學(xué)工業(yè)出版社，1993 [11]王紹俊主編.機(jī)械制造工藝設(shè)計(jì)手冊(cè).機(jī)械工業(yè)出版社，1987 [12]黃如林主編.切削加工簡(jiǎn)明實(shí)用手冊(cè).化學(xué)工業(yè)出版社，2004 [13]薛源順主編.機(jī)床夾具設(shè)計(jì).機(jī)械工業(yè)出版社，1995 [14]陳于萍、高曉康主編.互換性與測(cè)量技術(shù).北京高等教育出版社，2005. [15]司乃鈞，許德珠主編.熱加工工藝基礎(chǔ). 高等教育出版社，1991 [16]張龍勛主編.機(jī)械制造工藝學(xué)課程設(shè)計(jì)指導(dǎo)及習(xí)題.機(jī)械工業(yè)出版社，1999.11 致謝 課程設(shè)計(jì)是我們機(jī)械專業(yè)學(xué)生在校學(xué)習(xí)的一個(gè)總結(jié)性的理論和實(shí)踐相結(jié)合的教學(xué)環(huán)節(jié)，是綜合運(yùn)用所學(xué)知識(shí)和技能的具體實(shí)踐過程。通過這次課程設(shè)計(jì)，使我對(duì)所學(xué)的專業(yè)知識(shí)有了更深刻的理解和認(rèn)識(shí)，并為我今后從事機(jī)械相關(guān)工作樹立了堅(jiān)定的信心。 我這次課程設(shè)計(jì)的題目是CA6140開合螺母的工藝工裝設(shè)計(jì)，其中涉及到了很多曾經(jīng)學(xué)過的計(jì)算，驗(yàn)算等以及相關(guān)專業(yè)方面的知識(shí)。 在設(shè)計(jì)的過程中，通過把自己幾年來所學(xué)到的專業(yè)知識(shí)和實(shí)踐相結(jié)合，并運(yùn)用到設(shè)計(jì)中，我順利完成了課程設(shè)計(jì)的各項(xiàng)任務(wù)。但在設(shè)計(jì)過程中，我對(duì)一些理論問題掌握的不夠充足，經(jīng)過多次向指導(dǎo)老師請(qǐng)教最后才得以解決。在今后的學(xué)習(xí)和工作中，我會(huì)更多的吸取相關(guān)方面的知識(shí)，不斷提高自己的專業(yè)水平和能力。 通過本次課程設(shè)計(jì)，我了解和掌握了機(jī)械工藝等方面的諸多知識(shí)，鍛煉了自己分析問題和解決問題的能力，并積累了一定的實(shí)踐經(jīng)驗(yàn)，為今后的工作打下了堅(jiān)實(shí)的基礎(chǔ)。但由于時(shí)間緊，任務(wù)重，再加上本人的知識(shí)水平和能力有限，以及經(jīng)驗(yàn)不足，設(shè)計(jì)中難免存在一些缺點(diǎn)和錯(cuò)誤，望老師給予批評(píng)指正。 9- 機(jī) 械 加 工 工 序 卡 產(chǎn)品型號(hào) 4 零（部）件圖號(hào) 共1 頁 產(chǎn)品名稱 開和螺母 零（部）件名稱 M6 第 1 頁 車 間 工序號(hào) 工序名稱 材料牌號(hào) 金工 9 鉆M6螺紋孔 球墨鑄鐵 毛坯種類 毛坯外形尺寸 每毛坯件數(shù) 每臺(tái)件數(shù) 設(shè)備名稱 設(shè)備型號(hào) 設(shè)備編號(hào) 同時(shí)加工件數(shù) 夾 具 編 號(hào) 夾 具 名 稱 切 削 液 工序工時(shí) 準(zhǔn)終 單件 序號(hào) 工 步 內(nèi) 容 工 藝 裝 備 主軸 轉(zhuǎn)速 (r/min) 切削 速度 (m/min) 進(jìn)給 量 (mm/r) 切削 深度 (mm) 走刀次數(shù) 時(shí)間定額 機(jī)動(dòng) 輔助 1 鉆M6螺紋孔Φ5 搖臂鉆床Z35 1320 20.7 0.13 30 2 2 攻M6螺紋 搖臂鉆床Z35 1320 20.7 0.13 30 1 編制（日期） 審核（日期） 會(huì)簽（日期） 標(biāo)記 處數(shù) 更改文件號(hào) 簽字 日期 標(biāo)記 更改文件號(hào) 簽字 日期 機(jī) 械 加 工 工 藝 過 程 卡 片1 產(chǎn)品型號(hào) 零（部）件圖號(hào) 第 1 頁 產(chǎn)品名稱 開和螺母 零（部）件名稱 共 1 頁 材料牌號(hào) 球墨鑄鐵 毛坯種類 鑄件 毛坯外形尺寸 每毛坯件數(shù) 1 每臺(tái)件數(shù) 1 備注 工序號(hào) 工序名稱 工 序 內(nèi) 容 車間 工段 設(shè)備 工藝裝備 工時(shí) 準(zhǔn)終 單件 1 粗車 粗車Φ62外圓和兩端面 金工 CA6140 2 粗、半精車 粗車、半精車Φ62外圓和Φ52外圓 金工 CA6140 3 精車 精車Φ62和Φ52外圓 金工 CA6140 4 粗、精鏜 粗鏜、精鏜Φ31.5內(nèi)孔，倒角 金工 CA6140 5 粗車梯形螺紋 粗車梯形螺紋 金工 CA6140 6 精鉸梯形螺紋 精鉸梯形螺紋 金工 CA6140 7 切斷 切斷 金工 X63 8 精銑 精銑R23圓弧，去除毛刺 金工 X63 9 鉆孔 鉆孔M6螺紋孔，去除毛刺 金工 Z35 0.62min 10 攻螺紋 攻M6螺紋 金工 Z35 11 終檢 終檢 質(zhì)檢室 編制 （日期） 審核 （日期） 會(huì)簽 （日期） 、 處 數(shù) 更改文件號(hào) 簽字 日期 標(biāo)記 處數(shù) 更改文件號(hào) 簽字 日期 課程設(shè)計(jì) 機(jī)械制造技術(shù)課程設(shè)計(jì)說明書 設(shè)計(jì)題目： 制定開和螺母零件的加工工藝， 設(shè)計(jì)鉆螺紋底孔鉆床夾具 序言 機(jī)械制造工藝學(xué)課程設(shè)計(jì)是我們學(xué)完了大學(xué)的全部基礎(chǔ)課、技術(shù)基礎(chǔ)課以及大部分專業(yè)課之后進(jìn)行的。這是我們?cè)谶M(jìn)行畢業(yè)設(shè)計(jì)之前對(duì)所學(xué)各課程的一次深入的綜合性的總復(fù)習(xí)。 就我個(gè)人而言，我希望能通過這次課程設(shè)計(jì)，了解并認(rèn)識(shí)一般機(jī)器零件的生產(chǎn)工藝過程，鞏固和加深已學(xué)過的技術(shù)基礎(chǔ)課和專業(yè)課的知識(shí)，理論聯(lián)系實(shí)際，對(duì)自己未來將從事的工作進(jìn)行一次適應(yīng)性訓(xùn)練，從中鍛煉自己分析問題、解決問題的能力，為今后的工作打下一個(gè)良好的基礎(chǔ)，并且為后續(xù)課程的學(xué)習(xí)大好基礎(chǔ)。 由于能力所限，設(shè)計(jì)尚有許多不足之處，懇請(qǐng)各位老師給予指教。 I 目錄 設(shè)計(jì)任務(wù)書 I 序言 II 課程設(shè)計(jì)說明書正文： 第1章 零件的分析 1 1.1 零件的作用 1 1.2 零件的工藝分析 1 第2章 工藝規(guī)程設(shè)計(jì) 2 2.1 確定毛坯的制造形式 2 2.2 幾面選擇 2 2.3 制定工藝路線 2 2.4 機(jī)械加工余量、工序尺寸及毛坯尺寸的確定 3 2.5 確定切削用量及基本工時(shí) 3 第3章 專用夾具設(shè)計(jì) 5 3.1 設(shè)計(jì)主旨 5 3.2 夾具設(shè)計(jì) 5 3.2.1 定位基準(zhǔn)的選擇 5 3.2.2 切削力和加緊力計(jì)算 5 3.2.3 定位誤差分析 5 3.2.4 夾具設(shè)計(jì)及操作的簡(jiǎn)要說明 6 參考文獻(xiàn) 7 致謝 8 第1章 零件的分析 1.1 零件的作用 題目所給定的零件是CA6140車床上的對(duì)開螺母，也稱開合螺母。其作用是為了傳遞機(jī)床梯形絲杠的動(dòng)力。當(dāng)開合螺母處于“開”的位置（即其手柄在上方）時(shí)，絲杠（或稱“絲桿”）的旋轉(zhuǎn)運(yùn)動(dòng)將無法傳遞給溜板箱；反之，則絲杠的旋轉(zhuǎn)運(yùn)動(dòng)將傳遞給溜板箱，由溜板箱將其轉(zhuǎn)換成溜板的直線運(yùn)動(dòng)，完成刀具的縱向移動(dòng)，平時(shí)機(jī)床溜板用光杠傳動(dòng)的，車螺紋的時(shí)候?qū)﹂_螺母閉合，用梯形絲杠傳動(dòng)，從而完成諸如車削螺紋等工作。 1.2 零件的工藝分析 該零件結(jié)構(gòu)比較簡(jiǎn)單，其主要加工的面只有Φ62、Φ52外圓柱面，M41.5的梯形螺紋。因此先將零件當(dāng)做軸類零件進(jìn)行加工，再擴(kuò)孔，進(jìn)行螺紋加工，鉆4個(gè)螺紋孔，最后再將零件切開一分為二。零件精度及表面粗糙度要求較高，因此要合理安排精加工，以保證達(dá)到要求。 零件圖： 第2章 工藝規(guī)程設(shè)計(jì) 2.1 確定毛坯的制造形式 零件材料為球墨鑄鐵,考慮到球墨鑄鐵的抗缺口敏感性，減震性和耐優(yōu)良等特點(diǎn),因此應(yīng)該選用鑄件,以便使開合螺母在使用中耐磨性良好,保證零件工作可靠。將融化了的金屬澆注到旋轉(zhuǎn)著的模型中，由于離心力的作用，金屬液粘貼于模型的內(nèi)壁上，凝結(jié)后所得鑄件外形與模型內(nèi)壁的形狀相同，使用這種方法可以毋需澆注口，故能顯著地減少金屬的消耗。由于免除了砂型和制模的設(shè)備，以及減少了鑄工車間的面積，生產(chǎn)成本就有了降低；這樣得來的鑄件具有緊密與微細(xì)的顆粒結(jié)構(gòu)及較好的機(jī)械性能。因此采用離心鑄造,這從提高生產(chǎn)率、保證加工精度來考慮也是應(yīng)該的。 2.2 幾面選擇 基面選擇是工藝規(guī)程設(shè)計(jì)中的重要工作之一。基面選擇的正確與合理，可以使加工質(zhì)量得到保證、生產(chǎn)率得到提高。否則不但加工工藝過程中的問題百出，更有甚者，還會(huì)造成零件大批報(bào)廢，使生產(chǎn)無法正常進(jìn)行。 粗基準(zhǔn)的選擇：對(duì)于一般軸類零件而言，以外圓作為粗基準(zhǔn)是完全合理的。因此對(duì)本零件來說，選擇以Φ62外圓柱面作為粗基準(zhǔn)面加工Φ62外圓柱面和右端面。 精基準(zhǔn)的選擇：使用經(jīng)過加工的表面作為定位基準(zhǔn)，采用加工了的Φ62外圓柱面和右端面作為精基準(zhǔn)。 2.3 制定工藝路線 制定工藝路線的出發(fā)點(diǎn)，是應(yīng)該使零件的幾何形狀、尺寸精度及位置精度等技術(shù)要求能得到妥善的保證。在生產(chǎn)綱領(lǐng)已確定為成批生產(chǎn)的條件下，可以考慮采用萬能性機(jī)床配以專用工夾具，并盡量使工序集中來提高生產(chǎn)率。除此之外，還應(yīng)當(dāng)考慮經(jīng)濟(jì)效果，以便使生產(chǎn)成本盡量下降。 工藝路線方案 工序1：粗車Φ62外圓及端面。 工序2：粗車、半精車Φ62和Φ52外圓。 工序3：精車Φ62和Φ52外圓。 工序4：粗鏜、精鏜Φ31.5。 工序5：粗車梯形螺紋。 工序6：精鉸梯形螺紋。 工序7：切斷。 工序8：精銑R23圓弧，去除毛刺。 工序9：鉆孔M6螺紋底孔孔，去除毛刺。 工序10：攻M6螺紋底孔。 工序11：終檢。 2.4 機(jī)械加工余量、工序尺寸及毛坯尺寸的確定 由機(jī)械制造工藝簡(jiǎn)明手冊(cè)表2.2-3、2.2-5、2.2-4、2.2-1可確定毛坯尺寸及公差。 根據(jù)上述原始材料及加工工藝，分別對(duì)各加工表面的機(jī)械加工余量、工序尺寸確定如下： 一端面 粗車至113 l=2； Φ62外圓 粗車至Φ62 2z=3； 另一端面 粗車至111 l=2； Φ62外圓 粗車至Φ62 2z=3； Φ52外圓 精車至Φ52 2z=0.3； 半精車至Φ52.3 2z=1.1； 粗車至Φ53.4 2z=1.6； Φ62左軸肩面 精車至103 l=0.3； 半精車至102.7 l=0.7； 粗車至102 l=1； Φ62右軸肩面 精車至8 l=0.3； 半精車至8.3 l=0.7； 粗車至9 l=1； Φ31.5 精鏜至Φ31.5 2z=1； 粗鏜至Φ30.5 2z=2； 2.5 確定切削用量及基本工時(shí) 工序7：鉆2XΦ5孔并加工螺紋孔2XM6。本工序采用計(jì)算法確定切削用量。 （1）機(jī)床：搖臂鉆床Z35 （2）刀具選擇：選用高速鋼,錐柄標(biāo)準(zhǔn)麻花鉆Φ5，M6絲錐。 （3）確定切削用量： 2個(gè)孔共走2次刀,由切削手冊(cè)表11-2， f=0.13mm/r, V=18m/min， n=1000*18/(3.14*5)=1146.5 r/min， 根據(jù)機(jī)械制造工藝設(shè)計(jì)簡(jiǎn)明手冊(cè)表4.2-12， 取n機(jī)=1320r/min, 所以 V =3.14×1320×5/1000=20.7m/min 。 （4）計(jì)算基本工時(shí)： 鉆孔工時(shí)： tm1= 2*l/fn=2(3+16+3)/(0.13*1320) =0.26?min。 攻絲工時(shí)： t= (l+l1+l2)/fn+(l+l1+l2)/fn0 l1=(1--3)p, l2 =(2--3)p n0為絲錐或工件回程的轉(zhuǎn)速，p為加工螺紋的螺距。 根據(jù)機(jī)械加工工藝設(shè)計(jì)簡(jiǎn)明手冊(cè)表2.3-20， P=1 所以tm2=2*2(16+2+2)/(0.13*1000)=0.62 min。 最后，將以上各工序切削用量、工時(shí)定額的計(jì)算結(jié)果，連同其他加工數(shù)據(jù)，一并填如機(jī)械加工工藝過程卡片和機(jī)械加工工序卡片中。 第3章 專用夾具設(shè)計(jì) 為了提高勞動(dòng)生產(chǎn)率，保證加工質(zhì)量，降低勞動(dòng)強(qiáng)度，通常要設(shè)計(jì)專用夾具。 經(jīng)過與指導(dǎo)師協(xié)商，決定設(shè)計(jì)第9、10道工序—鉆2M6螺紋底孔的鉆床專用夾具。 3.1 設(shè)計(jì)主旨 本夾具主要用來鉆φ5mm孔、攻M6螺紋底孔。 3.2 夾具設(shè)計(jì) 3.2.1 定位基準(zhǔn)的選擇 出于定位簡(jiǎn)單和快速的考慮，選擇底面、側(cè)面和曲面與底面相交的線為定位基準(zhǔn)，保證加工基準(zhǔn)和設(shè)計(jì)基準(zhǔn)重合，再使用手動(dòng)偏心輪夾緊機(jī)構(gòu)進(jìn)行夾緊。 3.2.2 切削力和加緊力計(jì)算 本步加工按鉆削估算夾緊力。實(shí)際效果可以保證可靠的夾緊。 鉆削軸向力 ： 扭矩 ： 夾緊力為 取系數(shù) S1=1.5 S2=S3=S4=1.1 則實(shí)際夾緊力為 F’=S1*S2*S3*S4*F=24.1N 使用偏心輪機(jī)構(gòu)進(jìn)行人工夾緊，調(diào)節(jié)偏心輪手柄，即可滿足所需的夾緊力。 3.2.3 定位誤差分析 由于加工基準(zhǔn)和設(shè)計(jì)基準(zhǔn)重合，所以能很好的保證了加工精度。 3.2.4 夾具設(shè)計(jì)及操作的簡(jiǎn)要說明 所加工的零件尺寸不大，并且所需夾緊力不是很大，為使夾具簡(jiǎn)單，操作方便靈活，故使用手動(dòng)偏心輪夾緊機(jī)構(gòu)夾緊。 裝配圖： 參考文獻(xiàn) [1]吳 拓主編．機(jī)械制造工藝與機(jī)床夾具課程設(shè)計(jì)指導(dǎo).北京：機(jī)械工業(yè)出版社，2010 [2]艾興、肖詩綱主編．切削用量簡(jiǎn)明手冊(cè)[M].北京：機(jī)械工業(yè)出版社. [3]李益民主編.機(jī)械制造工藝設(shè)計(jì)簡(jiǎn)明手冊(cè)[M] .北京：機(jī)械工業(yè)出版社，1993 [4]孟少農(nóng)主編.機(jī)械加工工藝手冊(cè).機(jī)械工業(yè)出版社，1991 [5]王 棟主編.機(jī)械制造工藝課程設(shè)計(jì)指導(dǎo)書[M].北京：機(jī)械工業(yè)出版社 [6]梅 伶主編.模具課程設(shè)計(jì)指導(dǎo)．北京：機(jī)械工業(yè)出版社，2010 [7]王衛(wèi)衛(wèi)主編.材料成形設(shè)備．北京：機(jī)械工業(yè)出版社，2010 [8]甘永立主編.幾何公差與檢測(cè)．上海：上海科學(xué)技術(shù)出版社，2008 [9]孫 桓、陳作模、葛文杰主編.《機(jī)械原理》．北京：高等教育出版社，2006 [10]崇 凱主編.機(jī)械制造技術(shù)基礎(chǔ).化學(xué)工業(yè)出版社，1993 [11]王紹俊主編.機(jī)械制造工藝設(shè)計(jì)手冊(cè).機(jī)械工業(yè)出版社，1987 [12]黃如林主編.切削加工簡(jiǎn)明實(shí)用手冊(cè).化學(xué)工業(yè)出版社，2004 [13]薛源順主編.機(jī)床夾具設(shè)計(jì).機(jī)械工業(yè)出版社，1995 [14]陳于萍、高曉康主編.互換性與測(cè)量技術(shù).北京高等教育出版社，2005. [15]司乃鈞，許德珠主編.熱加工工藝基礎(chǔ). 高等教育出版社，1991 [16]張龍勛主編.機(jī)械制造工藝學(xué)課程設(shè)計(jì)指導(dǎo)及習(xí)題.機(jī)械工業(yè)出版社，1999.11 致謝 課程設(shè)計(jì)是我們機(jī)械專業(yè)學(xué)生在校學(xué)習(xí)的一個(gè)總結(jié)性的理論和實(shí)踐相結(jié)合的教學(xué)環(huán)節(jié)，是綜合運(yùn)用所學(xué)知識(shí)和技能的具體實(shí)踐過程。通過這次課程設(shè)計(jì)，使我對(duì)所學(xué)的專業(yè)知識(shí)有了更深刻的理解和認(rèn)識(shí)，并為我今后從事機(jī)械相關(guān)工作樹立了堅(jiān)定的信心。 我這次課程設(shè)計(jì)的題目是CA6140開合螺母的工藝工裝設(shè)計(jì)，其中涉及到了很多曾經(jīng)學(xué)過的計(jì)算，驗(yàn)算等以及相關(guān)專業(yè)方面的知識(shí)。 在設(shè)計(jì)的過程中，通過把自己幾年來所學(xué)到的專業(yè)知識(shí)和實(shí)踐相結(jié)合，并運(yùn)用到設(shè)計(jì)中，我順利完成了課程設(shè)計(jì)的各項(xiàng)任務(wù)。但在設(shè)計(jì)過程中，我對(duì)一些理論問題掌握的不夠充足，經(jīng)過多次向指導(dǎo)老師請(qǐng)教最后才得以解決。在今后的學(xué)習(xí)和工作中，我會(huì)更多的吸取相關(guān)方面的知識(shí)，不斷提高自己的專業(yè)水平和能力。 通過本次課程設(shè)計(jì)，我了解和掌握了機(jī)械工藝等方面的諸多知識(shí)，鍛煉了自己分析問題和解決問題的能力，并積累了一定的實(shí)踐經(jīng)驗(yàn)，為今后的工作打下了堅(jiān)實(shí)的基礎(chǔ)。但由于時(shí)間緊，任務(wù)重，再加上本人的知識(shí)水平和能力有限，以及經(jīng)驗(yàn)不足，設(shè)計(jì)中難免存在一些缺點(diǎn)和錯(cuò)誤，望老師給予批評(píng)指正。 9- Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 Keywords: Fixture design; Geometry constraint; Deterministic locating; Under-constrained; Over-constrained constraint status, a workpiece under any locating scheme falls into one of the following three categories: 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 0736-5845/$-see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2004.11.012 C3 Corresponding author. Tel.: +15088316092; fax: +15088316412. E-mail address: hsongwpi.edu (H. Song). 1. Well-constrained (deterministic): The workpiece ismatedat auniqueposition when six locatorsare madeto 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 1. Introduction A xture is a mechanism used in manufacturing operations to hold a workpiece rmly in position. Being a crucial step in process planning for machining parts, 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. Alocationschemeforamachiningxturemustsatisfyanumberofrequirements.Themostbasicrequirementisthat 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 xture design and studied by many researchers. Concerning geometry Abstract Geometry constraint is one of the most important considerations in 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 xture design practice has not been thoroughly studied. In this paper, a methodology to characterize xturing systems geometry constraint status with focus on under-constraint is proposed. An under-constraint status, if it exists, can be recognized withgiven locatingscheme.All un-constrainedmotionsofaworkpiece inanunder-constraintstatuscanbeautomaticallyidentied. This assists the designer to improve decit locating scheme and provides guidelines for revision to eventually achieve deterministic locating. r 2005 Elsevier Ltd. All rights reserved. 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 Locating completeness evaluation and revision in xture plan H. Song C3 , Y. Rong 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 sufcient for some applications such as non-prismatic surfaces or non-small relative errors.Heproposed asecond-order Taylor expansionwhichalsotakes locatorerror 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 xturing elements and workpiece. The desired location is q*, at which a workpiece is to be positioned and piecewisely differentiable surface function is g i (as shown in Fig. 1). The surface function isdened as g i q C3 0: To be deterministic, there should be a unique solution for the following equation set for all locators. g i q0; i 1;2; .; n, (1) where n is the number of locators and q x 0 ; y 0 ; z 0 ;y 0 ;f 0 ;c 0 C138 represents the position and orientation of the workpiece. Only considering the vicinity of desired location q C3 ; where q q C3 Dq; Asada and By showed that ARTICLE IN PRESS H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 369 g i qg i q C3 h i Dq, (2) where h i is the Jacobian matrix of geometry functions, as shown by the matrix in Eq. (3). The deterministic locating requirement can be satised if the Jacobian matrix has full rank, which makes the Eq. (2) to have only one solution q q C3 : rank qg 1 qx 0 qg 1 qy 0 qg 1 qz 0 qg 1 qy 0 qg 1 qf 0 qg 1 qc 0 : qg i qx 0 qg i qy 0 qg i qz 0 qg i qy 0 qg i qf 0 qg i qc 0 : qg n qx 0 qg n qy 0 qg n qz 0 qg n qy 0 qg n qf 0 qg n qc 0 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) Upongivena3-2-1locatingscheme, therankofaJacobianmatrixforconstraintequationstellstheconstraintstatus 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. X Y Z O X Y Z O (x 0 ,y 0 ,z 0 ) g i UCS WCS Workpiece Fig. 1. Fixturing system model. They further introduced several indexes derived from those matrixes to evaluate locator congurations, followed by optimization through constrained nonlinear programming. Their analytical study, however, does not concern the ARTICLE IN PRESS revision of non-deterministic locating. Currently, there is no systematic study on how to deal with a xture design that failed to provide deterministic location. 2. Locatingcompletenessevaluation If deterministic location is not achieved by designed xturing system, it is as important for designers to know what the constraint status is and how to improve the design. If the 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 efciently. 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 W L : As shown in Fig. 3, for given locator number n; locating normal vector a i ; b i ; c i C138 and locating position x i ; y i ; z i C138 for each locator, i 1;2; .; n; the n C26 locating matrix can be determined as follows: a 1 b 1 c 1 c 1 y 1 C0 b 1 z 1 a 1 z 1 C0 c 1 x 1 b 1 x 1 C0 a 1 y 1 : : : : 2 6 3 7 Kang et al. 7 followed these methods and implemented them to develop a geometry constraint analysis module in their automated computer-aided xture design verication 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 W L (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 W L Dr, (4) Over-constrained : DX W T L W L C01 W T L Dr, (5) Under-constrained : DX W T L W L W T L C01 Dr I 6C26 C0 W T L W L W T L C01 W L l, (6) where l is an arbitrary vector. Table 1 Rank Number of locators Status o 6 Under-constrained 6 6 Well-constrained 6 46 Over-constrained H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378370 W L a i b i c i c i y i C0 b i z i a i z i C0 c i x i b i x i C0 a i y i : : : : a n b n c n c n y n C0 b n z n a n z n C0 c n x n b n x n C0 a n y n 6 6 6 6 6 4 7 7 7 7 7 5 .(7) When rankW L 6 and n 6; the workpiece is well-constrained. When rankW L 6 and n46; the workpiece is over-constrained. This means there are n C06 unnecessary locators in the locating scheme. The workpiece will be well-constrained without the presence of those n C06 locators. The mathematical representationforthisstatusisthat thereare n C06 rowvectorsinlocating matrix thatcanbeexpressed as linear combinations of the other six row vectors. The locators corresponding to that six row vectors consist one ARTICLE IN PRESS locat determ 1. 2. 3. 4. be 3. workpi H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 371 ing scheme that provides deterministic location. The developed algorithm uses the following approach to ine the unnecessary locators: Find all the combination of n C06 locators. For each combination, remove that n C06 locators from locating scheme. Recalculate the rank of locating matrix for the left six locators. If the rank remains unchanged, the removed n C06 locators are responsible for over-constrained status. This method may yield multi-solutions and require designer to determine which set of unnecessary locators should removed for the best locating performance. When rankW L o6; the workpiece is under-constrained. Algorithmdevelopmentandimplementation The algorithm to be developed here will dedicate to provide information on un-constrained motions of the ece in under-constrained status. Suppose there are n locators, the relationship between a workpieces position/ Fig. 2. Geometry constraint status characterization. X Z Y (a 1 ,b 1 ,c 1 ) 2 ,b 2 ,c 2 ) (x 1 ,y 1 ,z 1 ) (x 2 ,y 2 ,z 2 ) (a i ,b i ,c i ) (x i ,y i ,z i ) (a Fig. 3. A simplied locating scheme. orient ij L L L ARTICLE IN PRESS 372 5. To identify allthe un-constrained motions oftheworkpiece, V dx i ;dy i ;dz i ;da xi ;da yi ;da zi C138 isintroducedsuchthat V DX 0. (9) Since rankDXo6; there must exist non-zero V that satises 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;0C138 says that the rotation about x-axisisnotconstrained. 0;1;1;0;0;0C138 meansthat theworkpiececanmovealongthedirection given by vector 0;1;1C138: There could be innite solutions. The solution space, however, can be constructed by 6C0 rankW L basic solutions. Following analysis is dedicated to nd out the basic solutions. From Eqs. (8) and (9) VX dxDx dyDy dzDz da x Da x da y Da y da z Da z dx X n i1 w 1i Dr i dy X n i1 w 2i Dr i dz X n i1 w 3i Dr i da x X n i1 w 4i Dr i da y X n i1 w 5i Dr i da z X n i1 w 6i Dr i X n i1 Vw 1i ; w 2i ; w 3i ; w 4i ; w 5i ; w 6i C138 T Dr i 0. 10 Eq. (10) holds for 8Dr i if and only if Eq. (11) is true for 8i1pipn: Vw 1i ; w 2i ; w 3i ; w 4i ; w 5i ; w 6i C138 T 0. (11) Eq. (11) illustrates the dependency relationships among row vectors of W r : In special cases, say, all w 1j equal 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, 9w 1j a0;w 2j a0; w 1j C0w 2j for 8j: Inthisscenario,theworkpiece cannotmovealong x-ory-axis.However,it can move along the diagonal line between x-andy-axis dened by vector 1, 1, 0. To nd solutions for general cases, the following strategy was developed: 1. Eliminate dependent row(s) from locating matrix. Let r rank W L ; n number of locator. If ron; create a vector in n C0 r dimension space U u 1 : u j : u nC0r hi 1pjpn C0 r; 1pu j pn: Select u j in the way that rankW L r still holds after setting all the terms of all the u j th row(s) equal to zero. Set r C26 modied locating matrix W LM a 1 b 1 c 1 c 1 y 1 C0 b 1 z 1 a 1 z 1 C0 c 1 x 1 b 1 x 1 C0 a 1 y 1 : : : : a i b i c i c i y i C0 b i z i a i z i C0 c i x i b i x i C0 a i y i : : : : a n b n c n c n y n C0 b n z n a n z n C0 c n x n b n x n C0 a n y n 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 rC26 , wher geomet ation errors and locator errors can be expressed as follows: DX Dx Dy Dz a x a y a z 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 w 11 : w 1i : w 1n w 21 : w 2i : w 2n w 31 : w 3i : w 3n w 41 : w 4i : w 4n w 51 : w 5i : w 5n w 61 : w 6i : w 6n 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 C1 Dr 1 : Dr i : Dr n 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 , (8) e Dx;Dy;Dz;a x ;a y ;a z are displacement along x, y, z axis and rotation about x, y, z axis, respectively. Dr i is ric error of the ith locator. w is dened by right generalized inverse of the locating matrix W r W T W W T C01 H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 where i 1;2; :; niau j : 4. 6. constr Exampl vector ARTICLE IN PRESS L 3 : 0, 0, 1 0 , 2, 1, 0 0 , L 4 : 0, 1, 0 0 , 3, 0, 2 0 , L 5 : 0, 1, 0 0 , 1, 0, 2 0 . Consequently, the locating matrix is determined. W L 001 3 C010 001 3 C030 001 1 C020 010C0203 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 . L L v s : v 6 6 6 6 4 7 7 7 5 w q k i : w q k r 6 6 6 4 7 7 7 5 C1 w l1 : w li : w lr : w 61 : w 6i : w 6r 6 6 6 4 7 7 7 5 , where s 1;2; :;6saq j ; saq k ; l 1;2; :;6 laq j : Repeat step 4 (select another term from Q) and step 5 until all 6C0 r basic solutions have been determined. Based on this algorithm, a C+ program was developed to identify the under-constrained status and un- ained motions. e1. In a surface grinding operation, a workpiece is located on a xture system as shown in Fig. 4. The normal and position of each locator are as follows: 1 : 0, 0, 1 0 , 1, 3, 0 0 , 2 : 0, 0, 1 0 ,3,3,0 0 , Calculated undetermined terms of V: V is also a solution of Eq. (11). The r undetermined terms can be found as follows. v 1 : 2 6 6 6 3 7 7 7 w q k 1 : 2 6 6 6 3 7 7 7 w 11 : w 1i : w 1r : 2 6 6 6 3 7 7 7 C01 5. W rm w l1 : w li : w lr : w 61 : w 6i : w 6r 6 6 6 4 7 7 7 5 6C26 , where l 1;2; :;6 laq j : Normalize the free motion space. Suppose V V 1 ; V 2 ; V 3 ; V 4 ; V 5 ; V 6 C138 is one of the basic solutions of Eq. (10) with all six terms undetermined. Select a term q k from vector Q1pkp6C0 r: Set V q k C01; V q j 0 j 1;2; :;6C0 r; jak; ( 2. Compute the 6C2 n right generalized inverse of the modied locating matrix W r W T LM W LM W T LM C01 w 11 : w 1i : w 1r w 21 : w 2i : w 2r w 31 : w 3i : w 3r w 41 : w 4i : w 4r w 51 : w 5i : w 5r w 61 : w 6i : w 6r 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 6C2r 3. Trim W r down to a r C2 rfull rank matrix W rm : r rankW L o6: Construct a 6C0 r dimension vector Q q 1 : q j : q 6C0r hi 1pjp6C0 r; 1pq j pn: Select q j in the way that rankW r r still holds after setting all the terms of all the q j th row(s) equal to zero. Set r C2 r modied inverse matrix w 11 : w 1i : w 1r : 2 6 6 6 3 7 7 7 H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 373 010C0201 ARTICLE IN PRESS This locating system provides under-constrained positioning since rankW L 5o6: The program then calculates the right generalized inverse of the locating matrix. W r 00 000 0:50:5 C01 C00:51:5 0:75 C01:25 1:50 0 0:25 0:25 C00:50 0 0:5 C00:5000 0000:5 C00:5 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 . The rst row is recognized as a dependent row because removal of this row does not affect rank of the matrix. The other ve rows are independent rows. A linear combination of the independent rows is found according the requirementinstep5oftheprocedureforunder-constrainedstatus.Thesolutionforthisspecialcaseisobviousthatall the coefcients are zero. Hence, the un-constrained motion of workpiece can be determined as V C01; 0; 0; 0; 0; 0C138: 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. X Z Y L 3 L 4 L 5 L 2 L 1 Fig. 4. Under-constrained locating scheme. H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378374 Example2. 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: L 1 : 0, 1, 0 0 , 896, C0877, C0515 0 , L 2 : 0, 1, 0 0 , 1060, C0875, C0378 0 , L 3 : 0, 1, 0 0 , 1010, C0959, C0612 0 , L 4 : 0.9955, C00.0349, 0.088 0 , 977, C0902, C0624 0 , L 5 : 0.9955, C00.0349, 0.088 0 , 977, C0866, C0624 0 , L 6 : 0.088, 0.017, C00.996 0 , 1034, C0864, C0359 0 . The locating matrix of this conguration is W L 0 1 0 515:000:8960 01 0378: 1:0600 0 1 0 612:00:0100 0:9955 C00:0349 0:0880 C0101:2445 C0707:2664 0:8638 0:9955 C00:0349 0:0880 C098:0728 C0707:2664 0:8280 0:0880 0:0170 C00:9960 866:6257998 :2466 0:0936 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 , rankW L 5o6 reveals that the workpiece is under-constrained. It is found that one of the rst ve rows can be removed without varying the rank of locating matrix. Suppose the rst row, i.e., locator L 1 is removed from W L ; the ARTICLE IN PRESS modied locating matrix turns into W LM 010378:001:0600 0 1 0 612: :0100 0:9955 C00:0349 0:0880 C0101:2445 C0707:2664 0:8638 0:9955 C00:0349 0:0880 C098:0728 C0707:2664 0:8280 0:0880 0:0170 C00:996 866:6257998 :2466 0:0936 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 . The right generalized inverse of the modied locating matrix is W r 1:8768 C01:8607 C020:6665 21:3716 0:4995 3:0551 C02:0551 C032:4448 32:4448 0 C01:0956 1:0862 12:0648 C012:4764 C00:2916 C00:0044 0:0044 0:0061 C00:0061 0 0:0025 C00:0025 0:0065 C00:0069 0:0007 C00:0004 0:0004 0:0284 C00:0284 0 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 ve rows. Without losing generality, the rst row is regarded as dependent row. The 5C25 modied inverse matrix is 2 3 Fig. 5. Knuckle 610 (modied from real design). H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 375 W rm 3:0551 C02:0551 C032:4448 32:4448 0 C01:0956 1:0862 12:0648 C012:4764 C00:2916 C00:0044 0:0044 0:0061 C00:0061 0 0:0025 C00:0025 0:0065 C00:0069 0:0007 C00:0004 0:0004 0:0284 C00:0284 0 6 6 6 6 6 6 4 7 7 7 7 7 7 5 . The undetermined solution is V C01; v 2 ; v 3 ; v 4 ; v 5 ; v 6 C138: To calculate the ve undetermined terms of V according to step 5, 1:8768 C01:8607 C020:6665 21:3716 0:4995 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 T C1 3:0551 C02:0551 C032:4448 32:4448 0 C01:0956 1:0862 12:0648 C012:4764 C00:2916 C00:0044 0:0044 0:0061 C00:0061 0 0:0025 C00:0025 0:0065 C00:0069 0:0007 C00:0004 0:0004 0:0284 C00:0284 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 C01 0; C01:713; C00:0432; C00:0706; 0:04C138. Substituting this result into the undetermined solution yields V C01;0; C01:713; C00:0432; C00:0706; 0:04C138 This vector represents a free motion dened by the combination of a displacement along C01, 0, C01.713 direction combine