傳動(dòng)軸的機(jī)械加工工藝及夾具設(shè)計(jì)

傳動(dòng)軸的機(jī)械加工工藝及夾具設(shè)計(jì),傳動(dòng)軸,機(jī)械,加工,工藝,夾具,設(shè)計(jì) 重慶大學(xué)網(wǎng)絡(luò)教育學(xué)院 學(xué)生畢業(yè)設(shè)計(jì)（論文）開(kāi)題報(bào)告 一、課題的目的及意義（含國(guó)內(nèi)外的研究現(xiàn)狀分析）： 傳動(dòng)軸零件是傳動(dòng)系統(tǒng)的重要部件，是組成機(jī)器的重要的、基本的構(gòu)件。其加工工藝過(guò)程的技術(shù)水平直接決定了零件質(zhì)量的好壞。與西方發(fā)達(dá)國(guó)家相比，我國(guó)機(jī)械行業(yè)起步較晚，加工設(shè)備及方法相對(duì)落后，這使得我國(guó)的機(jī)械制造技術(shù)整體水平不高，同時(shí)我國(guó)現(xiàn)階段零件制造企業(yè)眾多，制造水平良莠不齊。在目前我國(guó)倡導(dǎo)節(jié)能減排，綠色環(huán)保的今天，研究如何用科學(xué)合理的、省時(shí)省力的方法來(lái)制造傳動(dòng)軸零件是擺在面前的任務(wù)之一。 本課題的目的是研究傳動(dòng)軸的機(jī)械加工工藝過(guò)程及鍵槽加工用夾具設(shè)計(jì)，其重點(diǎn)在于培養(yǎng)學(xué)生面對(duì)本專業(yè)相關(guān)的一般工程技術(shù)問(wèn)題的獨(dú)立思考能力、分析問(wèn)題和解決問(wèn)題的能力，加深學(xué)生的知識(shí)面和所學(xué)課程之間的聯(lián)系。綜合運(yùn)用機(jī)械設(shè)計(jì)課程及其相關(guān)課程的理論和生產(chǎn)實(shí)際進(jìn)行零件工藝設(shè)計(jì)及夾具設(shè)計(jì)，是在老師的指導(dǎo)下進(jìn)行的一次實(shí)踐。 二、課題任務(wù)、重點(diǎn)研究?jī)?nèi)容、實(shí)現(xiàn)途徑、條件： 1、課題任務(wù)：傳動(dòng)軸的機(jī)械加工工藝及夾具設(shè)計(jì) 2、重點(diǎn)研究?jī)?nèi)容：研究如何實(shí)現(xiàn)傳動(dòng)軸零件機(jī)械加工和加工夾具設(shè)計(jì)。 3、實(shí)現(xiàn)途徑、條件 （1）傳動(dòng)軸生產(chǎn)類(lèi)型分析 （2）傳動(dòng)軸零件分析 （3）傳動(dòng)軸加工方案分析 （4）傳動(dòng)軸毛坯分析 （5）工藝規(guī)程設(shè)計(jì) （6）機(jī)床夾具設(shè)計(jì) ? ??????????????????????????????????????????????????????????報(bào)告人簽名??張小波??? ??????????????????????????????????????????????????????????日??????期??2011-9-19 ? 1.引言 軸類(lèi)零件的加工工藝及鍵槽的夾具設(shè)計(jì)是制造這類(lèi)零件的兩個(gè)基本要素，充分理解和掌握軸類(lèi)零件加工工藝是提高制造能力和工藝水平的關(guān)鍵，同時(shí)配合優(yōu)良的夾具設(shè)計(jì)來(lái)達(dá)到現(xiàn)代機(jī)械制造業(yè)要求的高效率、高品質(zhì)的目的。 2.傳動(dòng)軸生產(chǎn)類(lèi)型 由題目知：年產(chǎn)量為30000件，備品率為3%，廢品率為2%， Q=30000件/年，m=1件/臺(tái)，備品率a%和廢品率b%分別為3%和2%。代入公式： N=Qm(1+a%)(1+b%) 得：????????????????????N=30000×(1+3%)×(1+2%) ?????????????????=31518件/年 該傳動(dòng)軸的重量約為0.58KG，查表可得，傳動(dòng)軸屬輕型零件，生產(chǎn)類(lèi)型為大批量生產(chǎn)。 表1??不同機(jī)械產(chǎn)品的零件質(zhì)量型別表 機(jī)械產(chǎn)品類(lèi)別 加工零件的質(zhì)量/kg 輕型零件 中型零件 重型零件 電子工業(yè)機(jī)械 <4 4～30 >30 機(jī)床 <15 15～50 >50 重型機(jī)械 <100 100～2000 >2000 表2??各種生產(chǎn)類(lèi)型的規(guī)范 生產(chǎn)類(lèi)型 零件的年生產(chǎn)綱領(lǐng)/（臺(tái)/年或件/年） 輕型機(jī)械 中型機(jī)械 重型機(jī)械 單件生產(chǎn) ≤100 ≤20 ≤5 小批生產(chǎn) 100～500 20～200 5～100 中批生產(chǎn) 500～5000 200～500 100～300 大批生產(chǎn) 5000～50000 500～5000 300～1000 大量生產(chǎn) >500000 >5000 >1000 3．傳動(dòng)軸零件分析 3.1零件用途分析 軸是組成機(jī)器零件的主要零件之一。一切做回轉(zhuǎn)運(yùn)動(dòng)的傳動(dòng)零件（例如：齒輪，蝸輪等）都必須安裝在軸上才能進(jìn)行運(yùn)動(dòng)及動(dòng)力的傳遞。因此，軸的主要用途是支撐回轉(zhuǎn)零件和傳遞運(yùn)動(dòng)和動(dòng)力。跟據(jù)軸線形狀的不同可分為直軸、曲軸、撓性軸。按照承受載荷的不同又可分為轉(zhuǎn)軸，心軸，和傳動(dòng)軸三類(lèi)。工作中，既承受彎矩又承受轉(zhuǎn)矩的稱為轉(zhuǎn)軸，這類(lèi)軸在各種機(jī)器中最為常見(jiàn)，只承受彎矩而不承受轉(zhuǎn)矩的稱為心軸，心軸又分為回轉(zhuǎn)心軸和靜止心軸兩種，只能承受轉(zhuǎn)矩而不承受彎矩的稱為傳動(dòng)軸。 傳動(dòng)軸在各種機(jī)械和傳動(dòng)系統(tǒng)中廣泛使用，用來(lái)傳遞動(dòng)力。在工作過(guò)程中主要承受交變扭轉(zhuǎn)負(fù)荷或有沖擊，因此該零件應(yīng)具有足夠的強(qiáng)度、剛度和韌性，以適應(yīng)其工作條件。 3.2零件技術(shù)要求 零件圖如下圖1所示，其主要加工表面技術(shù)要求見(jiàn)表3 圖1?傳動(dòng)軸零件圖 ? 表3??傳動(dòng)軸零件技術(shù)要求表 加工表面 尺寸及偏差/mm 公差/mm及精度等級(jí) 表面粗糙度/μm 形位公差/mm 傳動(dòng)軸兩端面 160 無(wú) 無(wú) 無(wú) M軸外圓面 ???????+0 ????φ24?-0.021 IT7 Ra1.6 無(wú) N軸肩左端面 40 無(wú) 無(wú) 無(wú) N軸肩外圓面 ?????+0.023 ??φ25?+0.002 IT7 Ra1.6 無(wú) O軸肩左端面 72 無(wú) 無(wú) 無(wú) O軸外圓面 φ28 無(wú) Ra1.6 無(wú) O軸肩右端面 76 無(wú) 無(wú) 無(wú) P軸外圓面 ?????+0.023 ??φ25?+0.002 IT7 Ra1.6 無(wú) P軸肩右端面 ?115 無(wú) 無(wú) 無(wú) Q軸外圓面 ???????+0 ????φ24?-0.021 IT7 Ra1.6 無(wú) ? 傳動(dòng)軸零件形狀為較簡(jiǎn)單的階梯軸，由圖可知其各個(gè)外圓表面精度等級(jí)和表面粗糙度求不高，并且傳動(dòng)軸沒(méi)有形位誤差要求。為了在工作過(guò)程中承受交變扭轉(zhuǎn)負(fù)荷和沖擊，傳動(dòng)軸需要有良好的力學(xué)綜合性能，一般要對(duì)其進(jìn)行調(diào)質(zhì)處理。 3.3熱處理分析 因以鍛件為毛坯，為改善鍛造過(guò)程中所造成的粗大晶粒，消除加工硬化，殘余應(yīng)力和改善切削加工性能，故毛坯在車(chē)削加工前需要先正火。待粗加工完成后，零件再經(jīng)調(diào)質(zhì)200～230HBS來(lái)滿足設(shè)計(jì)要求（將淬火和高溫回火相結(jié)合的熱處理稱為調(diào)質(zhì)）。 3.4零件表面加工方法的選擇 傳動(dòng)軸的加工面有外圓、端面、鍵槽、退刀槽。從傳動(dòng)軸技術(shù)要求分析，本傳動(dòng)軸僅外圓面有精度要求和表面粗糙度要求，且要求不高。 常用外圓表面加工方法如下表4所示。 表4?外圓表面加工方法 ? 加工方法 經(jīng)濟(jì)加工精度等級(jí)（IT） 表面粗糙度Ra/μm 粗車(chē) 11~12 50~12.5 半精車(chē) 8~10 6.3~3.2 精車(chē) 6~7 1.6~0.8 磨削 6~7 0.8~0.4 常用鍵槽表面加工方法如下表5所示。 表5鍵槽加工方法 ? 加工方案 經(jīng)濟(jì)加工精度等級(jí)（IT） 表面粗糙度Ra 銑 11~12 12.5 因此本傳動(dòng)軸采用車(chē)削加工即可滿足加工精度等級(jí)IT7及表面粗糙度要求。 4.?選擇毛坯 4.1傳動(dòng)軸毛坯材料 題目已選定材料為45，45鋼是軸類(lèi)零件的常用材料，它價(jià)格便宜且經(jīng)過(guò)調(diào)質(zhì)（或正火）后可以得到較好的切削加工性能，而且能獲得較高的強(qiáng)度、剛度和韌性等綜合機(jī)械性能。 4.2毛坯制造方法 由于該傳動(dòng)軸在工作過(guò)程中要承受交變負(fù)荷和沖擊，為增強(qiáng)其強(qiáng)度和沖擊韌度，獲得纖維組織，故毛坯宜選用鍛件，鍛件的形狀和尺寸與零件相近，可以節(jié)約材料，減少切削加工的勞動(dòng)量，降低生產(chǎn)成本。由于生產(chǎn)類(lèi)型為大批量生產(chǎn)，故采用模鍛的方法來(lái)制造毛坯。 4.3毛坯設(shè)計(jì) 如下圖2為傳動(dòng)軸毛坯圖。機(jī)械加工前模鍛毛坯的質(zhì)量為0.86KG。 圖2?傳動(dòng)軸毛坯圖 4.4毛坯加工余量分析 毛坯加工信息量如下表4所示。 表6??傳動(dòng)軸毛坯加工余量表 加工表面 工序名稱 加工余量 工序尺寸 尺寸公差 表面粗糙度（μm） φ24 精車(chē) 0.1 24 h7 Ra1.6μm ? 半精車(chē) 0.4 24.2 h9(+0/-0.052) Ra6.3μm ? 粗車(chē) 2 25 ? Ra12.5μm ? 毛坯 ? 29 ? ? φ25 精車(chē) 0.1 25 k7 Ra1.6μm ? 半精車(chē) 0.4 25.2 k9(+0.052/-0) Ra6.3μm ? 粗車(chē) 1.5 26 ? Ra12.5μm ? 毛坯 ? 29 ? ? φ28 半精車(chē) 0.5 28 ? Ra6.3μm ? 粗車(chē) 1.5 29 ? Ra12.5μm ? 毛坯 ? 32 ? ? 5.工藝規(guī)程設(shè)計(jì) 5.1定位基準(zhǔn)的選擇 定位基準(zhǔn)有粗基準(zhǔn)和精基準(zhǔn)之分，通常先確定精基準(zhǔn)，然后再確定粗基準(zhǔn)。 5.1.1精基準(zhǔn)的選擇 根據(jù)傳動(dòng)軸零件圖的設(shè)計(jì)意圖和精基準(zhǔn)的選擇原則要求定位基準(zhǔn)與設(shè)計(jì)基準(zhǔn)相重合，這里選擇傳動(dòng)軸的兩端面中心孔作為定位基準(zhǔn)，這樣可以方便的加工各個(gè)軸肩端面和各外圓表面，而且能保證加工軸肩面相對(duì)于中心軸線的圓跳動(dòng)誤差，保證加工軸線相對(duì)于中心軸線的同軸度誤差。在加工鍵槽時(shí)需改變定位基準(zhǔn)，根據(jù)設(shè)計(jì)要求選擇左右兩個(gè)端面作為定位基準(zhǔn)加工鍵槽。 ?5.1.2粗基準(zhǔn)的選擇 一般先選擇外圓表面作為粗基準(zhǔn)，先加工出一個(gè)端面和端面的中心孔，然后再以加工出的端面定位加工另一個(gè)端面和其中心孔，而不是用外圓表面定位把兩個(gè)端面同時(shí)加工出來(lái)，這樣加工可以保證兩端面中心線的同軸度，并為后續(xù)的精加工做好準(zhǔn)備。5.2零件表面加工方法的選擇 根據(jù)傳動(dòng)軸零件上各加工表面的尺寸精度和表面粗糙度，確定加工件各表面的加工方法如下表： ? 表7???傳動(dòng)軸各表面加工方案 加工表面 尺寸精度等級(jí) 表面粗糙度/μm 加工方法 傳動(dòng)軸兩端面 無(wú) 無(wú) 粗車(chē) M軸外圓面 IT7 Ra1.6 粗車(chē)-半精車(chē)-精車(chē) N軸肩左端面 無(wú) 無(wú) 粗車(chē) N軸肩外圓面 IT7 Ra1.6 粗車(chē)-半精車(chē)-精車(chē) O軸肩左端面 無(wú) 無(wú) 粗車(chē) O軸外圓面 無(wú) Ra1.6 粗車(chē)-半精車(chē)-精車(chē) O軸肩右端面 無(wú) 無(wú) 粗車(chē) P軸外圓面 IT7 Ra1.6 粗車(chē)-半精車(chē)-精車(chē) P軸肩右端面 無(wú) 無(wú) 粗車(chē) Q軸外圓面 IT7 Ra1.6 粗車(chē)-半精車(chē)-精車(chē) ? 5.3加工順序的安排 ?工工藝路線為：下料→鍛造→正火→粗加工→調(diào)質(zhì)→半精加工→淬火→精加工→銑槽→去毛刺→清洗→檢驗(yàn)。 5.4工藝路線 該軸為階梯軸，為了提高毛胚生產(chǎn)效率，需要對(duì)毛胚進(jìn)行簡(jiǎn)化，但是這會(huì)使后面的切削加工余量增大，使車(chē)削過(guò)程中產(chǎn)生大量的切削熱，從而引起殘余應(yīng)力重新分布而變形。因此，安排工序時(shí)需要將加工過(guò)程分為以下階段。 5.4.1粗加工階段 粗加工階段主要是去除各加工表面的余量，并做出精基準(zhǔn)。包括粗車(chē)外圓，鉆中心孔。 5.4.1.1粗車(chē)兩端面，鉆中心孔為精基面作好準(zhǔn)備，使后續(xù)工序定位精準(zhǔn)，從而保證其他加工表面的形狀和位置要求。 5.4.1.2粗車(chē)階梯軸外圓，使此時(shí)毛坯的形狀接近工件的最終形狀和尺寸，只留下適當(dāng)?shù)募庸び嗔俊? 5.4.1.3切出退刀槽。 5.4.2半精加工階段 半精加工階段的任務(wù)是減小粗加工留下的誤差，使加工表面達(dá)到一定的精度，為精加工做好準(zhǔn)備。包括主軸各處外圓和修研中心孔。 5.4.3精加工階段 精加工階段的任務(wù)是確保達(dá)到圖紙規(guī)定的精度要求和表面粗糙度要求。它包括對(duì)表面粗糙度要求較高的外圓面M、N、P和Q面的精車(chē)加工，然后銑槽。 5.5?機(jī)床設(shè)備的選用 由于該軸生產(chǎn)類(lèi)型為大批量生產(chǎn)，為提高生產(chǎn)效率及確保品質(zhì)的穩(wěn)定性，故采用沈陽(yáng)一機(jī)數(shù)控車(chē)床CAK6140來(lái)加工外圓及各端，該機(jī)床最大功率7.5KW，主軸轉(zhuǎn)速范圍150-2400rpm，加工精度IT6-IT7。 圖3數(shù)控車(chē)床CAK6140 ? 鍵槽的加工采用臺(tái)灣邁鑫立式加工中心CNC VH-610，其最大功率7.5KW，定位精密：±0.005mm／300mm，重復(fù)精度：±0.003mm，主軸轉(zhuǎn)速8000rpm，F(xiàn)ANUC數(shù)控系統(tǒng)。 圖4立式加工中心VH-610 5.6?刀具的選擇 選用車(chē)刀時(shí)，車(chē)端面選用硬質(zhì)合金YT5主偏角45°端面車(chē)刀，粗車(chē)、半精車(chē)及精車(chē)外圓時(shí)選用YT15主偏角90°偏刀，退刀槽選用YT5切槽刀，銑鍵槽選用YT15圓柱鍵槽銑刀。 5.7?傳動(dòng)軸工藝過(guò)程卡片 跟據(jù)上述加工工藝過(guò)程，其加工工藝過(guò)程卡片如表8： 表8?加工工藝過(guò)程卡片 機(jī)械加工工藝過(guò)程卡片 軸類(lèi) 階梯軸 零件名稱 傳動(dòng)軸 材料牌號(hào) 45 毛坯種類(lèi) 鍛造 毛坯外形尺寸 164mm×φ32?mm 工序號(hào) 工序名稱 工序內(nèi)容 設(shè)備 刀具 量具 1 鍛造毛坯 鍛造毛坯 模鍛錘 / / 2 熱處理 正火處理 熱處理爐 / / 3 粗車(chē)、鉆孔 粗車(chē)傳動(dòng)軸兩端面并在兩端面鉆中心孔 CAK6140 45°刀 鉆頭 游標(biāo)卡尺 4 粗車(chē) 粗車(chē)各軸肩端面及傳動(dòng)軸各外圓表面 CAK6140 90°刀 游標(biāo)卡尺 5 熱處理 調(diào)質(zhì)處理?200—230HBS 熱處理爐 / 布氏硬度計(jì) 6 研修 研修中心孔 鉗工臺(tái) 麻花鉆 游標(biāo)卡尺 7 半精車(chē) 半精車(chē)M、N、O、P、Q軸外圓面及各軸肩端面 CAK6140 90°刀 游標(biāo)卡尺、外圓千分尺 8 車(chē)退刀槽 在O左右端面車(chē)出退刀槽 CAK6140 切槽刀 游標(biāo)卡尺 9 精車(chē) 精車(chē)M、N、P、Q外圓表面至尺寸 CAK6140 90°刀 游標(biāo)卡尺、外圓千分尺 10 銑槽 在M、Q面上銑槽至規(guī)定尺寸 VH-610 鍵槽銑刀 游標(biāo)卡尺 11 去毛刺 去除各銳利毛邊毛刺 鉗工臺(tái) 平銼 / 12 清洗 去除零件加工過(guò)程中的污物 清洗機(jī) / / 13 終檢 檢查傳動(dòng)軸零件外觀及尺寸 檢驗(yàn)臺(tái) / 游標(biāo)卡尺、外圓千分尺 重慶大學(xué)網(wǎng)絡(luò)教育 學(xué)院 設(shè)計(jì)（日期） 校對(duì)（日期） 審核（日期） 標(biāo)準(zhǔn)化（日期） 張小波 ? ? ? ? ? 5.8?機(jī)械加工工序卡片 跟據(jù)上述加工工藝過(guò)程，其機(jī)械加工工序卡片如表9： 表9??機(jī)械加工工序卡片 ? ? 重慶大學(xué)網(wǎng)絡(luò)教育 學(xué)院 產(chǎn)品名稱 / 零件圖號(hào) / ? 零件名稱 傳動(dòng)軸 車(chē)間 機(jī)加 ? 機(jī)械加工工序卡片 ? 夾具編號(hào) / 使用設(shè)備 / ? 工步號(hào) 工步內(nèi)容 工藝設(shè)備 刀具 主軸轉(zhuǎn)速 切削速度 進(jìn)給量 背吃刀量 工裝 ? r/min m/min mm/r mm ? 1 粗車(chē)兩端面并在兩端面鉆中心孔 CAK6140 45°端面車(chē)刀 500 45.53 0.2 / 三抓卡盤(pán) ? 2 粗車(chē)軸肩端面及各外圓表面 CAK6140 90°偏刀 600 54.64 0.3 / 一夾一頂 ? 3 半精車(chē)M、N、O、P、Q軸外圓面及各軸肩端面 CAK6140 90°偏刀 700 / 0.2 / 一夾一頂 ? 4 車(chē)退刀槽 CAK6140 切槽刀 500 39.25 0.1 / 一夾一頂 ? 5 精車(chē)M、N、P、Q外圓表面至尺寸 CAK6140 90°偏刀 1000 / 0.1 0.1 一夾一頂 ? 6 在M、Q面上銑槽至規(guī)定尺寸 VH-610 鍵槽銑刀 1200 30 0.08 / 專用夾具 ? 備注：部分參數(shù)因加工對(duì)像而沒(méi)能統(tǒng)一填入此表格，請(qǐng)參考下面各工步計(jì)算過(guò)程。 ? ? 各工步加工時(shí)間計(jì)算 工步1?車(chē)兩端面 毛坯兩端外圓直徑φ29，需要切削深度是直徑的1/2，為14.5mm。 進(jìn)給量f=0.2mm/r 進(jìn)給速度Vf=fn=0.2x500=100mm/min 加工時(shí)間Tm=L/vf=14.5/100=0.145min 則車(chē)左右端面共用時(shí)間為2x0.145=0.29min ? 工步2?粗車(chē)各軸肩端面及傳動(dòng)軸各外圓表面 粗車(chē)φ24左右兩端總長(zhǎng)L=40+45=85mm的加工時(shí)間，從毛坯φ29加工到φ25，背吃刀量2mm 切削速度：Vc=πdn/1000=3.14X29X600/1000=54.64m/min 進(jìn)給量取f=0.3mm/r 進(jìn)給速度Vf=fn=0.3x600=180mm/min 加工時(shí)間tm=L/Vf=85/180=0.472min ? 粗車(chē)φ25左右兩端總長(zhǎng)L=32+39=71 mm的加工時(shí)間，從毛坯φ29加工到φ26，背吃刀量1.5mm 切削速度：Vc=πdn/1000=3.14X29X600/1000=54.64m/min 進(jìn)給量取f=0.3mm/r 進(jìn)給速度Vf=fn=0.3x600=180mm/min 加工時(shí)間tm=L/Vf=71/180=0.394min ? 粗車(chē)φ28總長(zhǎng)L=4mm的加工時(shí)間，從毛坯φ32加工到φ29，背吃刀量1.5mm 切削速度：Vc=πdn/1000=3.14X32X600/1000=60.29m/min 進(jìn)給量取f=0.3mm/r 進(jìn)給速度Vf=fn=0.3x600=180mm/min 加工時(shí)間Tm=L/Vf=4/180=0.022min ? 工步3?半精車(chē)各外圓表面 半精車(chē)φ24左右兩端總長(zhǎng)L=40+45=85 mm的加工時(shí)間，從粗車(chē)后的尺寸φ25加工到φ24.2，背吃刀量0.4mm 切削速度：Vc=πdn/1000=3.14X25X700/1000=54.95m/min 進(jìn)給量取f=0.2mm/r 進(jìn)給速度Vf=fn=0.2x700=140mm/min 加工時(shí)間tm=L/Vf=85/140=0.607min ? 半精車(chē)φ25左右兩端總長(zhǎng)L=32+39=71 mm的加工時(shí)間，從粗車(chē)后的尺寸φ26加工到φ25.2，背吃刀量0.4mm 切削速度：Vc=πdn/1000=3.14X26X700/1000=57.148m/min 進(jìn)給量取f=0.2mm/r 進(jìn)給速度Vf=fn=0.2x700=140mm/min 加工時(shí)間tm=L/Vf=71/140=0.507min ? 半精車(chē)φ28總長(zhǎng)L=4mm的加工時(shí)間，從粗車(chē)后的尺寸φ29加工到φ28，背吃刀量0.5mm 切削速度：Vc=πdn/1000=3.14X29X700/1000=63.742m/min 進(jìn)給量取f=0.2mm/r 進(jìn)給速度Vf=fn=0.2x700=140mm/min 加工時(shí)間tm=L/Vf=4/140=0.029min ? 工步4?車(chē)退刀槽 半精加工后的外圓直徑為φ25，切槽1mm深。 進(jìn)給量f=0.1mm/r 進(jìn)給速度Vf=fn=0.1x500=50mm/min 加工時(shí)間Tm=L/vf=1/50=0.02min 則切左右2個(gè)退刀槽共用時(shí)間為2x0.02=0.04min ? 工步5?精車(chē)外圓 精車(chē)φ24左右兩端總長(zhǎng)L=40+45=85 mm的加工時(shí)間，從半精車(chē)后的尺寸φ24.2加工到φ24，背吃刀量0.1mm 切削速度：Vc=πdn/1000=3.14X24.2X1000/1000=75.988m/min 進(jìn)給量取f=0.1mm/r 進(jìn)給速度Vf=fn=0.1x1000=100mm/min 加工時(shí)間tm=L/Vf=85/100=0.85min ? 精車(chē)φ25左右兩端總長(zhǎng)L=32+39=71 mm的加工時(shí)間，從半精車(chē)后的尺寸φ25.2加工到φ25，背吃刀量0.1mm 切削速度：Vc=πdn/1000=3.14X25.2X1000/1000=79.128m/min 進(jìn)給量取f=0.1mm/r 進(jìn)給速度Vf=fn=0.1x1000=100mm/min 則加工時(shí)間tm=L/Vf=71/100=0.71min ? 工步6?銑鍵槽 軸M、Q外圓面上鍵槽尺寸為8×4，加工余量4mm，故一次走刀完成。左右2個(gè)鍵槽總長(zhǎng)度尺寸L=26+31=57 mm 刀具選YT15銑刀，d=8mm ,z=2，參考機(jī)床切削范圍Vc取30m/min Vc=πdn/1000 算得機(jī)床轉(zhuǎn)速：n=1000*30/3.14*8=1194(r/min)，n取1200 r/min, fz取0.08mm/z Vf=fn=fzzn 每分鐘進(jìn)給量：Vf=0.08*2*1200=192(mm/min)。 加工時(shí)間：tm=57/192=0.297(min)。 ? 機(jī)械加工時(shí)間總共為： tm=0.29+0.472+0.394+0.022+0.607+0.507+0.029+0.04+0.85+0.71+0.297=3.92min 6.?鍵槽加工用夾具設(shè)計(jì) 6.1夾具的目的和作用 工件如何準(zhǔn)確的裝夾在機(jī)床上？如何保證同一批次的每個(gè)工作被裝夾在相同的位置？夾具的實(shí)質(zhì)是在機(jī)床上對(duì)工件進(jìn)行定位和夾緊，其目的是通過(guò)定位和夾緊使工件在加工過(guò)程中始終保持正確的加工位置。夾具的作用表現(xiàn)在以下幾個(gè)方面。 6.1.1提高生產(chǎn)率? 　??機(jī)床夾具能快速地將工件定位和夾緊，可以減少輔助時(shí)間，提高生產(chǎn)效率。尤其對(duì)于大批量零件的加工制造，優(yōu)良的夾具是提高生產(chǎn)效率的前提和保證。 6.1.2確保并穩(wěn)定加工精度，保證產(chǎn)品質(zhì)量? 　??加工過(guò)程中，工件與刀具的相對(duì)位置容易得到保證，并且裝夾過(guò)程中不受各種主觀因素的影響，因而工件的加工精度穩(wěn)定可靠。? 6.1.3降低對(duì)操作工人的技術(shù)要求和工人的勞動(dòng)強(qiáng)度? ????由于多數(shù)專用夾具的夾緊裝置只需工人操縱扳手即可實(shí)現(xiàn)對(duì)工件的夾緊，這在相當(dāng)大程度上降低了工人的工作強(qiáng)度、同時(shí)夾具的使用可以節(jié)省每次裝夾工件時(shí)找正和調(diào)整工件的時(shí)間與難度。 6.2.夾具的設(shè)計(jì) 6.2.1定位基準(zhǔn)的選擇 由零件工藝分析可知，軸鍵槽與軸外圓表面雖然沒(méi)有對(duì)稱度要求，但我們也應(yīng)盡量提高其制造精度。加工鍵槽時(shí)零件外圓已經(jīng)完成精車(chē)，故以零件外圓柱表面為定位基準(zhǔn)。為了簡(jiǎn)化夾具的結(jié)構(gòu)，方便操作，這里采用手動(dòng)夾緊。 6.2.2夾具定位元件 工件以外圓柱面作為定位基準(zhǔn)時(shí)，根據(jù)外圓柱面的完整程度，加工要求及安裝方式，可以在V形塊，定位套，半圓套及圓錐套中定位。 本夾具是用來(lái)加工傳動(dòng)軸零件的外鍵槽，這里選用V形塊作為定位元件，V形塊對(duì)中性好，可以使工件的定位基準(zhǔn)軸線對(duì)中在V形塊兩斜面的對(duì)稱面上，不會(huì)發(fā)生偏移而且安裝方便，應(yīng)用范圍廣泛，不論定位基準(zhǔn)是否經(jīng)過(guò)加工，不論是否完整，都可以采用V形塊定位。 V形塊上兩斜面間的夾角一般選用60°，90°，和120°。隨著V形塊的夾角的增大，其定位誤差減小，但夾角過(guò)大時(shí)，則工件需要更大的夾緊力，不然會(huì)引起工件定位不穩(wěn)定，綜合兩方面考慮選擇90°的V形塊。 本夾具采用的定位結(jié)構(gòu)如圖5所示，其中視圖A是V形定位結(jié)構(gòu)。 本夾具設(shè)計(jì)有2個(gè)凸起的定位塊，如圖5中視圖B所示。該定位塊用來(lái)將夾具體定位于數(shù)控銑床的T形槽上，然后用6個(gè)M10螺栓緊固于銑床T形臺(tái)上。 ? 圖5夾具底座零件圖 6.2.3夾具導(dǎo)向及防誤操作機(jī)構(gòu) 本夾具設(shè)計(jì)具有如下圖6所示的導(dǎo)向機(jī)構(gòu)及防誤操作機(jī)構(gòu)，裝夾時(shí)先將工件左端頂住V形塊左擋塊，右端用力按壓工件即可保證正確定位。如工件裝夾方向弄錯(cuò)，則工件中間部分最大外圓柱面會(huì)與基座V形塊斜面干涉，導(dǎo)致其不能正確入位，如圖7所示。 圖6夾具導(dǎo)向機(jī)構(gòu) 圖7工件反向裝夾 6.2.4夾具夾緊機(jī)構(gòu) 夾緊機(jī)構(gòu)常用的有斜楔，螺旋，偏心等夾緊機(jī)構(gòu)，它們都是根據(jù)斜面加緊原理來(lái)實(shí)現(xiàn)夾緊工作的。但本工序采用數(shù)控銑床加工，考慮到切削力的方向變化不斷，加工時(shí)極易產(chǎn)生振動(dòng)等因素，不建議采用偏心夾緊機(jī)構(gòu)，而采用螺栓鉸鏈壓板組合夾緊機(jī)構(gòu)，如圖8所示。 工件需要用壓板來(lái)將其固定在V形塊上，夾緊力過(guò)小會(huì)使夾緊不可靠；過(guò)大會(huì)使夾緊變形增大，因此，有必要確定一個(gè)恰當(dāng)?shù)膴A緊力。由于切削力本身在加工過(guò)程中是受到切削用量、工件材料、刀具等多種因素影響，并且這些影響因素又是變化的，所以?shī)A緊力大小的計(jì)算是一個(gè)很復(fù)雜的問(wèn)題，實(shí)際上常常通過(guò)工藝實(shí)驗(yàn)來(lái)確定夾緊力的大小。 圖8夾具結(jié)構(gòu)3D視圖 本夾具可以同時(shí)裝夾2個(gè)工件，用六角扳手即可方便裝卸零件。若將多套夾具同時(shí)固定在機(jī)床的T形臺(tái)上，則可以一次性加工多個(gè)工件。 6.3夾具裝配圖的繪制 因生產(chǎn)類(lèi)型為大批量生產(chǎn)，故設(shè)計(jì)夾具時(shí)除要考慮保證工件制造精度外，還要考慮如何提高生產(chǎn)效率。其設(shè)計(jì)如圖9所示。 ? 圖9夾具爆炸圖 ? 6.4夾具的對(duì)刀 夾具在機(jī)床上安裝完畢，在進(jìn)行加工之前，還需進(jìn)行夾具的對(duì)刀，使刀具相對(duì)夾具定位元件處于正確位置。對(duì)刀的方法通常有三種：試切法；調(diào)整法；用樣件或?qū)Φ堆b置對(duì)刀。試切法對(duì)刀會(huì)在工件上留下刀痕，并且操作繁瑣；調(diào)整法對(duì)刀雖然不會(huì)留下刀痕，但操作起來(lái)仍然不方便。故這里采用方便的光電尋邊器對(duì)刀。 7夾具的精度核校 工件的外圓柱面在90度V形塊上定位時(shí)，若不考慮V形塊的制造誤差，則工件定位基準(zhǔn)在V形塊的對(duì)稱面上，因此工件中心線在水平方向上的位移為零。在垂直方向上，因工件外圓有制造誤差會(huì)產(chǎn)生基準(zhǔn)位移。工件精加工后外圓直徑為25mm，其上偏差為+0.023mm，下偏差為+0.002mm； 則基準(zhǔn)位移誤差=0.707X0.021=0.015 mm 工序基準(zhǔn)選在尺寸上線或是下線時(shí)，工序基準(zhǔn)與定位基準(zhǔn)不重合，其誤差值為 準(zhǔn)不重合誤差=0.021/2=0.011 mm 工序基準(zhǔn)選在下母線時(shí)，其定位誤差最小。其值為： 定位誤差=基準(zhǔn)位移誤差-基準(zhǔn)不重合誤差=0.015-0.011=0.04mm； 已知鍵槽的深度方向尺寸為20mm，圖上未標(biāo)注公差，說(shuō)明該工件對(duì)鍵槽深度尺寸要求不高，故滿足要求。 參考文獻(xiàn)： 【1】《機(jī)械制造工藝與裝備課程設(shè)計(jì)指導(dǎo)書(shū)》，倪森壽 【2】《機(jī)床夾具圖冊(cè)》，機(jī)械工業(yè)出版社 【3】《現(xiàn)代機(jī)械設(shè)備設(shè)計(jì)手冊(cè)》，機(jī)械工業(yè)出版社 【4】《機(jī)械工程手冊(cè)》，機(jī)械工業(yè)出版社 【5】《機(jī)床夾具設(shè)計(jì)》，機(jī)械工業(yè)出版社 Robotics 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