K09-轉(zhuǎn)向器殼體鉆孔夾具設(shè)計(jì)

資源目錄里展示的全都有，所見(jiàn)即所得。下載后全都有,請(qǐng)放心下載。原稿可自行編輯修改=【QQ：401339828 或11970985 有疑問(wèn)可加】 轉(zhuǎn)向器殼體鉆孔夾具設(shè)計(jì) 摘 要 方向盤(pán)也被稱(chēng)為轉(zhuǎn)向，轉(zhuǎn)向器，轉(zhuǎn)向系統(tǒng)，這是最重要的部分。它的主要作用是增加傳遞到轉(zhuǎn)向力，并在沿垂直于輸送臂的變化的轉(zhuǎn)向力。該結(jié)構(gòu)的轉(zhuǎn)向形式可分為幾種類(lèi)型。歷史上曾有過(guò)許多形式的轉(zhuǎn)向，現(xiàn)在比較常用的齒輪齒條，蝸桿曲柄指銷(xiāo)式，循環(huán)球-齒條齒扇式，循環(huán)球式曲柄手指，蝸輪式等。 本文轉(zhuǎn)向器殼體鉆井工藝和夾具設(shè)計(jì)。分水器外殼的孔中，形狀精度和位置精度要求的尺寸精度非常高，并且反過(guò)來(lái)殼體相交的重合氣缸布置成當(dāng)在組合處理中，需要更準(zhǔn)確和合理定位的殼體。并逐漸整個(gè)殼體的部件的固定和準(zhǔn)確定位，實(shí)現(xiàn)了鉆孔的定位精度所要求的定位精度。 關(guān)鍵詞：轉(zhuǎn)向器 殼體 定位精度 夾具設(shè)計(jì) Steering shell drilling fixture design Summary Steering is also called steering system, steering system, which is the most important part of the steering. Its role is to: increase the propagation direction of the steering wheel to change the power transmission and power steering linkage. Form of the steering structure can be divided into several types. Historically, there have been many forms of steering, there are currently more commonly used rack and pinion, worm crank the pin, recirculating ball - Fan-rack gear, recirculating ball crank pin and worm-type and so on. The main purpose of this system is to study the recirculating ball steering gear. This article discusses the recirculating ball steering gear housing bore and fixture design. Steering shell hole dimensional accuracy, shape accuracy and location accuracy requirements are very high, and the shell is a combination of two overlapping cross cylinder, and the arrangement process, the need for housing is more accurate, reasonable position. The housing member and the step of the overall behavior of the exact positioning, positioning accuracy of the correct positioning step to achieve the desired accuracy of drilling. Keywords：steering gear case positioning accuracy fixture design目錄 目錄 3 第一章 緒論 4 1.1 夾具的工作原理 4 1.2夾具的分類(lèi) 4 1.3機(jī)床夾具的發(fā)展趨勢(shì) 4 1.4 本章小結(jié) 5 第二章 轉(zhuǎn)向器殼體鉆夾具的設(shè)計(jì)選擇 5 2.1 轉(zhuǎn)向器殼體的分析 6 2.2 機(jī)床的選擇 6 2.3 鉆床夾具的選擇 7 2.4 選擇鉆模板 7 2.5 鉆套的選擇 8 2.6 鉆套導(dǎo)引孔尺寸和公差的確定 9 2.7 鉆套高度選擇和鉆套與工件距離 9 第三章 工件在夾具中的定位 10 3.1 工件定位的基本原理 10 3.2 確定定位方案 10 3.3 定位元件的選擇與設(shè)計(jì) 11 4.4 導(dǎo)向元件的選擇 12 第四章 定位誤差的分析與計(jì)算 14 4.1 定位誤差 14 4.2 定位誤差的組成及計(jì)算方法 14 第五章 工件的夾緊 15 5.2 夾緊力的計(jì)算 15 參考文獻(xiàn) 17 致謝 18 第一章 緒論 1.1 夾具的工作原理 1 工件加工對(duì)準(zhǔn)夾具的準(zhǔn)確位置。它是由位于相應(yīng)的面部表面與夾具接觸工件的定位元件的定位，有或取向來(lái)實(shí)現(xiàn)。 2 夾具在機(jī)床上應(yīng)確保適當(dāng)和準(zhǔn)確的位置和方向固定結(jié)構(gòu)，并確保相對(duì)于夾具和機(jī)床連接表面之間，從而確保相對(duì)夾具表面切割機(jī)移動(dòng)到的面部的準(zhǔn)確位置的工件的定位元件形成一個(gè)精確的表面幾何形狀，它達(dá)到了面向定位基準(zhǔn)線(xiàn)所需的各等精密加工作業(yè)時(shí)的位置。 3 使刀具定位元件相關(guān)的確切位置，從而確保加工的工件定位基準(zhǔn)位置大小的表面上的刀具相對(duì)定位面的調(diào)整。 1.2夾具的分類(lèi) 根據(jù)不同類(lèi)型的機(jī)床，夾具和固定裝置可分為車(chē)床夾具，銑床夾具，鉆夾具。通過(guò)不同的電源裝置采用，可分為手動(dòng)夾具，氣動(dòng)夾具等。 1.3機(jī)床夾具的發(fā)展趨勢(shì) 1．高精確度 與加工精度的提高，為了降低定位誤差，提高加工精度，制造精密的夾具苛求。 ± 5μm的精密定位夾具高音調(diào)的準(zhǔn)確性，夾具支撐面達(dá)到0.01mm/300mm ，高平行度0.01mm/500mm 。德國(guó)demmeler （戴美樂(lè)）制造的長(zhǎng)4米，寬2米的孔系列組合焊接夾具平臺(tái)， 5μm以下，它的高錯(cuò)誤的定位是± 0.03毫米􀌔平行度和垂直度老虎鉗鉗安裝重復(fù)精度高達(dá)± 5μm的：瑞士EROWA重復(fù)柔性?shī)A具定位精度可達(dá)2 ? 5μm的。精密夾具及固定裝置已提高到微米級(jí)，世界知名的夾具制造公司都是精密機(jī)械制造企業(yè)。事實(shí)上，為了滿(mǎn)足不同行業(yè)和經(jīng)濟(jì)發(fā)展的需要，與文件夾不同型號(hào)和不同檔次的精度標(biāo)準(zhǔn)供選擇。 2高效 為了提高機(jī)器的生產(chǎn)效率。雙面，四片夾緊裝置和越來(lái)越多的產(chǎn)品。為了減少作業(yè)的安裝時(shí)間，各種自動(dòng)定心夾緊臺(tái)鉗，杠桿夾緊，凸輪夾緊，氣動(dòng)和液壓夾緊，以便快速夾緊功能前移。新的電控永磁夾具，加緊和只有1到2秒松開(kāi)工件，夾具結(jié)構(gòu)簡(jiǎn)化為多臺(tái)機(jī)器，多方位，多部分的過(guò)程創(chuàng)造了條件。為了縮短機(jī)器上安裝和調(diào)整夾具的時(shí)間，瑞典3R夾具在短短一分鐘內(nèi)完成安裝和校準(zhǔn)電火花加工夾具。一分鐘之內(nèi)，美國(guó)杰根斯（詹金斯）的球鎖裝夾系統(tǒng)將能夠找到并鎖定在機(jī)床工作臺(tái)上的夾具，球鎖裝夾系統(tǒng)用于柔性生產(chǎn)線(xiàn)更換夾具，起到停機(jī)時(shí)間提高生產(chǎn)效率的效果。 3模塊，所述組合 組合夾具元件是一個(gè)組合的基礎(chǔ)。省工，省時(shí)􀌑物質(zhì)，能量，體現(xiàn)在創(chuàng)新的夾具系統(tǒng)中先進(jìn)的。模擬演練切割工藝，既為用戶(hù)提供正確，合理的固定裝置及零部件配套項(xiàng)目，也是積累經(jīng)驗(yàn)，了解市場(chǎng)需求，不斷改進(jìn)和完善夾具系統(tǒng)。 4 適用 ，經(jīng)濟(jì) 夾具的通用性直接影響經(jīng)濟(jì)。模塊化，組合夾具系統(tǒng)，一次性投資比較大，只有夾具系統(tǒng)的可重構(gòu)性，可重構(gòu)性和可擴(kuò)展性的特點(diǎn)，以及應(yīng)用范圍廣，通用性好，夾具利用率，更快的投資回報(bào)率，以反映經(jīng)濟(jì)性好。強(qiáng)大的功能元素，使得夾具通用性好，簡(jiǎn)潔的組件，配套成本低，經(jīng)濟(jì)實(shí)用的應(yīng)用程序具有推廣價(jià)值。 1.4 本章小結(jié) 本章的重點(diǎn)是夾具及固定裝置的制造機(jī)器，裝置及工作原理的作用及其發(fā)展趨勢(shì)的份額的重要作用。 第二章 轉(zhuǎn)向器殼體鉆夾具的設(shè)計(jì)選擇 2.1 轉(zhuǎn)向器殼體的分析 為了安裝需要鉆孔在所述殼體的4 Φ13 HFC方向側(cè)的表面上。下面工件。由于沒(méi)有現(xiàn)成的裝置可用，這樣流程來(lái)設(shè)計(jì)專(zhuān)用夾具。 2.2 機(jī)床的選擇 根據(jù)所須加工孔的，四孔的φ13基本尺寸。該機(jī)有三種可用的機(jī)床，數(shù)據(jù)如下： 型 號(hào) 最大鉆削直徑 主軸端面至工作臺(tái)最大距離 工作臺(tái)面積 主 軸 孔莫氏錐度號(hào) 最大行 程 轉(zhuǎn) 速 Z518 18 600 350×350 2 145 330-3040 ZQ4015 15 475 250×300 2 100 480-4100 Z535 35 750 450×500 4 225 68-1100 考慮到裝卸和拆卸，和表面積通過(guò)對(duì)各方面工件的比較，決定采用Z535立式鉆床。 2.3 鉆床夾具的選擇 許多類(lèi)型的鉆孔夾具，可根據(jù)孔的分布可以被加工成： a 固定式鉆模： 在使用過(guò)程中，例如一夾具，固定在鉆探工作表面，但一般只用于垂直鉆孔加工時(shí)間。 b 回轉(zhuǎn)式鉆模： 用于處理分布式對(duì)在工件上的同一圓周上的相同或平行的孔的徑向孔的周?chē)? c 翻轉(zhuǎn)式鉆模： 沒(méi)有這樣的夾具軸和索引方法。在此過(guò)程中需要使用一個(gè)手翻轉(zhuǎn)的?？字械闹魈幚淼男×慵讉€(gè)方向分布。 d 蓋板式鉆模： 這樣的夾具沒(méi)有特定的文件夾中，除了鉆孔模板鉆套，以及所述定位元件和夾緊裝置。在加工過(guò)程中，鉆模板為掩護(hù)覆蓋同一工件上。 e 滑柱式鉆模： 這是一種常見(jiàn)的可調(diào)夾具與升降鉆模板。 根據(jù)工件的特殊結(jié)構(gòu)，決定采用蓋板夾具。它的特點(diǎn)是簡(jiǎn)單，重量輕，易于清除切屑。對(duì)于大型和重型加工工件孔的大小，使用最合適的蓋板夾具。對(duì)于小批量，在那里鉆鉸鏈后，立即倒角，而其他進(jìn)程，使用蓋鉆取模式也很方便。鉆模式每次需要從工件處理，比較麻煩，覆蓋時(shí)間，所以不適合大批量生產(chǎn)。 2.4 選擇鉆模板 鉆模板是供安裝鉆套用的。它有一定的剛度和強(qiáng)度，以防止套管和導(dǎo)向精度的影響，位置精度的變形。 常用的有以下幾種： a固定式鉆模板： 直接固定到所述上主體，并且是不可移動(dòng)的。因此，為了獲得高的定位精度，但不是很方便的裝卸工作件。 b鉸鏈?zhǔn)姐@模板： 鉆孔模板及固定裝置鉸鏈連接。因?yàn)楸仨氂性阢q鏈的間隙，所以該鉆孔加工精度的位置不是固定模更低。 c可卸式鉆模板： 當(dāng)工件裝卸必須鉆去除模板，你應(yīng)該可以采用可拆卸鉆模板。然而，由于處理鉆孔模板進(jìn)行比較費(fèi)時(shí)，而較少精密鉆孔的位置，所以一般使用時(shí)，其它類(lèi)型的鉆孔模板是不容易的使用工件進(jìn)行安裝。 d懸掛式鉆模板： 該鉆模板適合于大量生產(chǎn)的大量的鉆在相同的方向平行的孔，并與多軸驅(qū)動(dòng)器或組合使用的垂直鉆孔機(jī)上。 應(yīng)為工件結(jié)構(gòu)有點(diǎn)獨(dú)特，要加工孔的位置精度不高，而且是小量生產(chǎn)。它決定采用可拆卸鉆模板。 2.5 鉆套的選擇 在鉆孔夾具通常是與鉆具對(duì)齊設(shè)置來(lái)實(shí)現(xiàn)，處理只需要對(duì)齊鉆頭夾持器，定位精度可以通過(guò)鉆孔工藝要求來(lái)實(shí)現(xiàn)。當(dāng)然，也有增強(qiáng)的工具襯套剛度的影響。 因?yàn)檫@些孔的大小相同于在同一平面上處理不同的位置，它僅需要使用一個(gè)襯套。 鉆套的四種形式： a固定式鉆套： 一鉆套入固定式無(wú)肩，有肩。肩部主要用于鉆孔模板更薄要保持鉆頭導(dǎo)向套長(zhǎng)度時(shí)。圓筒形軸套，以H7/r6或H7/n6壓力直接與此特定鉆頭或鉆模板集的缺點(diǎn)，該文件夾是不容易的更換磨損，它主要用于生產(chǎn)小批量加工鉆孔夾具或螺距和球場(chǎng)高精度小孔。為了防止切屑進(jìn)入鉆孔設(shè)置鉆頭套筒的上端和下端應(yīng)稍突出鉆孔模板為宜，通常不大于鉆孔模板更小。。 b可換鉆套： 可互換的襯套的實(shí)際功能和一個(gè)固定的軸襯，如在較大數(shù)量時(shí)，它可以被磨損后立即更換。避免穿鉆模板，鉆套不直接安裝在鉆模板或一個(gè)特定的文件夾但H6/G5 H7/G6或與襯套孔，具有防旋轉(zhuǎn)螺釘以防止摩擦處理當(dāng)用工具解除工具，芯片和鉆孔的鉆套護(hù)套產(chǎn)生旋轉(zhuǎn)作用或回縮，并夾在襯套或鉆模板與特定的使用H7/n6或H7/r6 。 c 快換鉆套： 快速變化的位持有人受到相同的孔中使用的多個(gè)加工步驟。因?yàn)樵谶@個(gè)過(guò)程中，這反過(guò)來(lái)需要更換，取下鉆套，以適應(yīng)不同的切削工具的需求，宜采用快換鉆套。 d 特殊鉆套： 根據(jù)在順序的具體情況而設(shè)計(jì)，以補(bǔ)充組標(biāo)準(zhǔn)鉆頭性能不足。 通過(guò)比較，結(jié)合工件結(jié)構(gòu)，決定采用固定鉆套的A型無(wú)肩鉆套。（數(shù)量為四個(gè)） 2.6 鉆套導(dǎo)引孔尺寸和公差的確定 設(shè)置的選擇標(biāo)準(zhǔn)鉆孔結(jié)構(gòu)時(shí)，引導(dǎo)襯套孔的尺寸和公差需要根據(jù)以下原則確定： 1）導(dǎo)套基本尺寸鉆孔直徑應(yīng)等于最大極限尺寸指南工具，以防止卡住而死亡 2）并與導(dǎo)向孔刀具的鉆頭套筒，軸應(yīng)根據(jù)選擇系統(tǒng)中，因?yàn)檫@些工具的結(jié)構(gòu)和尺寸已經(jīng)標(biāo)準(zhǔn)化。 3）刀具和導(dǎo)向套筒之間的鉆孔應(yīng)確保有一定的配合間隙，以防止堵塞。根據(jù)所選刀具的類(lèi)型和精度要求導(dǎo)孔公差引導(dǎo)，鉆孔和擴(kuò)孔選F7，F(xiàn)8 ;原油相比G7選舉;精度被選擇比G6 。 4）標(biāo)準(zhǔn)鉆孔的最大尺寸被處理的基本尺寸，鉆導(dǎo)向孔的基本尺寸和孔的加工公差的相同的基本尺寸取F7 。 5）如果工具不開(kāi)機(jī)的切削部分，但與導(dǎo)柱部分，可以按與H7/f7 ， H7/g6 ，或H6/g5選擇相應(yīng)的孔基礎(chǔ)。 故鉆套導(dǎo)引孔以便在相同的一套基本的基本尺寸和孔的加工公差的大小取F7。 2.7 鉆套高度選擇和鉆套與工件距離 1．鉆套高度 是由鉆頭套筒的精度，工件材料，孔的加工精度，工具壽命，其表面形狀因子的高度確定。的± 0.25毫米間距或自由公差尺寸精度鉆，鉆套的高度取H=（1.5～2.5）d。鉆套內(nèi)徑采用基軸制F8的公差。所以H=（1.5～2.5）d=（1.5～2.5）×13=19.5～32.5mm 加工IT6～I(xiàn)T7級(jí)精度，孔距在12mm以上的孔或加工工件孔距精度要求在±0.10～±0.15mm時(shí)，鉆套的高度取H=（2.5～3.5）d鉆套內(nèi)徑采用基軸制G7的公差。 H=（2.5～2）d=（2.5～3.5）×13=32.5～45.5mm 由于該零件加工孔精度不怎么高，故鉆套高度取27mm. 2．鉆套與工件的距離 工件和鉆頭套筒留有一定距離h之間，如果h過(guò)大傾斜量增加，使得鉆頭夾持器是不是一個(gè)很好的指導(dǎo)。 h過(guò)小，則切屑排出困難，不僅提高了工件的表面粗糙度，鉆頭有時(shí)會(huì)斷裂。 H值可按下面經(jīng)驗(yàn)公式選?。? 加工鑄鐵、黃銅時(shí)，h=（0.3～0.7）d； 加工鋼件時(shí)，h=（0.7～1.5）d； 由于工件是鑄鐵，故取 h=（0.3～0.7）d=（0.3～0.7）×13=3.9～9.1mm取7.5mm。 第三章 工件在夾具中的定位 3.1 工件定位的基本原理 1．六點(diǎn)定位原理 工件定位性質(zhì)是使工件占據(jù)在夾具中一個(gè)確定的位置。這個(gè)位置可以通過(guò)將支持相應(yīng)的限制，獲得自由的程度來(lái)確定。在空間中的對(duì)象有六個(gè)自由度笛卡爾坐標(biāo)。這種自由沿三個(gè)相互垂直的軸線(xiàn)的運(yùn)動(dòng)，以及圍繞旋轉(zhuǎn)自由度的三個(gè)軸 2．完全定位與不完全定位 如果僅基于所述臺(tái)階部的自由限制了加工要求，而另一度，而不會(huì)限制所述工件的自由而沒(méi)有占據(jù)一個(gè)唯一的位置確定，但是這個(gè)步驟不影響加工要求不完全清楚，此時(shí)定位。應(yīng)該使用或不完全定位完全定位主要是由加工要求的方法測(cè)定。 3．欠定位與過(guò)定位 由于定位裝置的程度的工件的定位限制自由的實(shí)際數(shù)目，該過(guò)程是小于的自由度，通過(guò)限制所述處理要求是必要的數(shù)目。因此，由于結(jié)果的定位將導(dǎo)致自由和應(yīng)該限制限制是并非不合理現(xiàn)象，這并不能保證該程序規(guī)定的處理要求。定位兩個(gè)或多個(gè)點(diǎn)的支持與限制的自由反復(fù)，這種現(xiàn)象稱(chēng)為過(guò)度重復(fù)定位的定位。 4．定位支承點(diǎn)的配置 在六個(gè)定位，定位的支持點(diǎn)的配置，定位精度和穩(wěn)定性很大。 3.2 確定定位方案 根據(jù)上述被處理部和各個(gè)表面的特殊結(jié)構(gòu)。具體方案如下： 工件應(yīng)該限制的自由：X，移動(dòng)在Y方向上的自由， X，Y， Z方向的轉(zhuǎn)動(dòng)自由度 2） a以已加工的底面為基準(zhǔn)面，限制 X ，Y方向的轉(zhuǎn)動(dòng)自由度，Z方向的移動(dòng)自由度。 b 以中間孔套入一根長(zhǎng)芯軸，限制X， Y方向的轉(zhuǎn)動(dòng)和移動(dòng)自由度。 c 以后面的已加工的孔為基準(zhǔn)套入一定位軸，限制Z 方向的自由度。 所以出現(xiàn)了重復(fù)定位的情況。但由于是采用已加工面為基準(zhǔn)，所以對(duì)該定位方案是合理的。 3.3 定位元件的選擇與設(shè)計(jì) 工件在夾具中位置的確定，主要是通過(guò)各種類(lèi)型的定位元件實(shí)現(xiàn)的。一般的定位元件有： 1） 平面定位元件：固定支承、可調(diào)支承、自位支承、輔助支承。 固定支承：支承點(diǎn)的位置固定不變的定位元件。如各種固定支承釘。 可調(diào)支承：支承點(diǎn)的位置可調(diào)節(jié)的定位元件。如可擰動(dòng)的螺釘。 自位支承：支承本身在定位過(guò)程中所處的位置，隨工件定位基準(zhǔn)面的變化而自動(dòng)與之適應(yīng)，其作用相當(dāng)于一個(gè)固定支承，只限制一個(gè)自由度。由于增加了與定位基準(zhǔn)面接觸的點(diǎn)數(shù)，故可提高工件的安裝剛性和穩(wěn)定性。適用于工件以粗基準(zhǔn)定位或剛性不足的場(chǎng)合。 輔助支承：這類(lèi)支承只起提高工件支承剛性或起輔助作用的定位元件而不起定位作用。 2） 圓孔表面定位元件 這類(lèi)定位元件常用于圓孔表面。一般定位元件有：定位銷(xiāo)、剛性心軸、錐度心軸等。 3） 外圓表面定位元件 這類(lèi)定位元件常用于外圓表面定位。一般有：定位套、支承板、V型塊等。 定位套對(duì)工件外圓表面主要實(shí)現(xiàn)定心定位；支承板實(shí)現(xiàn)對(duì)外圓表面的支承定位；V型塊則實(shí)現(xiàn)對(duì)外圓表面的定心對(duì)中定位。 4 ）錐面定位元件 主要用于加工軸類(lèi)零件或某些要求精度定心的零件時(shí)，以工件上的錐孔作為定位基準(zhǔn)?？商岣吖ぜS向的定位精度。 根據(jù)工件在工件孔，類(lèi)似于懸臂的向下的力，鉆頭的一部分，扭轉(zhuǎn)力矩和重力的力和作用時(shí)引起的鉆孔的工件的平面結(jié)構(gòu)的特殊性。工件通過(guò)重力下垂的作用，而使從所述定位元件將工件定位基準(zhǔn)。同時(shí)，也使工件的振動(dòng)，從而導(dǎo)致?lián)p害的工具，最終影響加工質(zhì)量。根據(jù)這一特點(diǎn)，決定采用輔助支持。在工件的下部可能需要的孔的輔助支持。在第一預(yù)定位置，然后在主重新定位和所有的接觸元件的準(zhǔn)確定位的夾緊力的作用下。所述支撐構(gòu)件是球形的支柱?？紤]四個(gè)孔被加工，并且孔的位置不是對(duì)稱(chēng)的，該分布是大的，所以使用兩個(gè)這樣的柱子。 這樣的支柱，具體如上圖。 對(duì)于工件的后部，由于采用了一個(gè)定位軸，所以要保證定位軸的定位。決定一端用定位螺釘，一端用壓緊螺釘。 4.4 導(dǎo)向元件的選擇 行動(dòng)導(dǎo)向元件被用來(lái)確定刀具與工件的相對(duì)位置時(shí)，刀具導(dǎo)向作用正確播放。時(shí)也定位元件都可以使用。這些元素包括各種鉆模板，鉆套，鉸鏈蓋和引導(dǎo)支持等 1）鉆模板 我們已經(jīng)選擇了可拆卸鉆模板。但是普通的可拆卸鉆模板還不能滿(mǎn)足我們的加工孔的要求，所以要另外設(shè)計(jì)。因?yàn)榭资莾商撞⑿刑幚砜撞粚?duì)稱(chēng)的。的分析表明，它們與中間孔之間的垂直連接成164°角。為了確保加工的精度，鉆模板可以被設(shè)計(jì)成三角形花鍵連接，花鍵齒的中心線(xiàn)164 °角，只是垂直的設(shè)計(jì)寬度的形式?；ㄦI與拉削或鉆孔加工，熱處理后的方法用于提高定心面的磨削精度。三角形與外花鍵齒銑機(jī)，熱處理后的磨削方法也可用于改善定心表面的精度和側(cè)翼，具體設(shè)計(jì)如下： 三角花鍵 根據(jù)Dg=32 查表6-74內(nèi)花鍵弧齒槽寬和外花鍵弧齒厚偏差取m=2 齒數(shù)：z=14 模數(shù)：m=2 分度圓：df=dg-2(0.6+ξ)m=32-2(0.6+0.1)×2=29.2mm 齒頂圓直徑：dd= df+2(0.4+ξ)m=29.2+2(0.4+0.1)m=31.2mm 齒根圓直徑: dg=df -2(0.6-ξ)m=29.2-2(0.6-0.1)×2=26.6mm 公稱(chēng)圓：D=df +2×(0.4+ξ)m=29.2+2×（0.4+0.1）×2=31.2mm 內(nèi)花鍵齒根圓弧起始點(diǎn)直徑：dq=(z+1.1)m=(14+1.1)2=30.2mm 內(nèi)花鍵齒槽寬：s=(π/2+2ξtanα)=(3.14/2+tan45°×2×0.1)=3.54 內(nèi)花鍵齒槽角：β=90°-203°/z=90°-203°/14=76.5° 外花鍵基圓:dj=df+2×(0.4+ξ)m=29.2+2×(0.4+0.1)×2=31.2mm 內(nèi)花鍵齒根高：h"1=(0.6+ξ)× m=(0.6+0.1)×2=1.4 內(nèi)花鍵齒頂高：h′1=(0.4-ξ)×m=(0.4-0.1)×2=0.6 外花鍵齒根高：h"=(0.4+ξ)m=(0.4+0.1)×2=1 外花鍵齒頂高：h′=(0.6-ξ)m=(0.6-0.1)×2=1 內(nèi)花鍵齒頂圓直徑：Dg=df+2×(0.6+ξ)m=29.2+2(0.6+0.1)×2= 32 分度圓周節(jié)：t=πm=3.14×2=6.28 內(nèi)外表面光潔度的粗糙度必須在3.2～1.6之間。 而芯軸也就設(shè)計(jì)如下圖： 第四章 定位誤差的分析與計(jì)算 4.1 定位誤差 定位誤差是由于一個(gè)過(guò)程取向所造成的步長(zhǎng)（通常指的是加工表面的距離的步長(zhǎng)的基礎(chǔ)上），或者處理錯(cuò)誤的位置的要求是不允許的。的定位方案，通過(guò)分析其可能的定位誤差，只要大于約1/3至1/5的工件的公差的尺寸或位置減計(jì)算。這個(gè)取向的程序，通常被認(rèn)為是能滿(mǎn)足精密加工工藝的要求。 工件在夾具中的位置是由定位件確定的，一旦工件的定位面時(shí)，與定位部件或夾具的整體接觸，該位置將決定工件。然而，對(duì)于一個(gè)數(shù)的工件，由于每個(gè)工件的表面是介于約彼此在尺寸和位置上的差在容許范圍內(nèi)。是鉗和相應(yīng)的定位元件本身之間的大小和位置公差。因此，工件的定位雖然，被定位于特定的表面的每個(gè)元素將具有自己的位置變動(dòng)量，造成了步長(zhǎng)大小，并且處理錯(cuò)誤的位置的要求。 4.2 定位誤差的組成及計(jì)算方法 定位誤差是加工調(diào)整法，該法只允許因定位所造成的進(jìn)程大小或位置的要求，盡最大可能的變化過(guò)程中工件的數(shù)量。該定位誤差由基準(zhǔn)誤差占主導(dǎo)地位，并引用沒(méi)有重合誤差兩部分組成。因此，在夾具的設(shè)計(jì)中，定位方案的任何一個(gè)，由多個(gè)定位工件，直接計(jì)算的參考步驟的最大波動(dòng)范圍的兩個(gè)極限位置，該定位誤差為定位程序。 孔的工件的最大直徑和當(dāng)所述定位銷(xiāo)O1O2, O1相對(duì)于上述圓筒狀工件和工件的尺寸，最小步長(zhǎng)Hmin的最高位置時(shí)的相對(duì)位置OO2針的定位銷(xiāo)的最低條件沿O2工件的幾何形狀，以最大升力的底部位置的上述圓筒狀工件的最大尺寸下，定位誤差的過(guò)程，此時(shí)，我們可以看到的H在工件定位銷(xiāo)孔的直徑的大小和最大的最小條件的直徑中，當(dāng)相對(duì)于定位銷(xiāo)O1O2 O1在最高位置與上述圓筒狀工件小時(shí)的尺寸工件時(shí)，最小步長(zhǎng)Hmin的，當(dāng)相對(duì)于定位銷(xiāo)OO2向下沿O2外側(cè)的工件的底部位置的工件圓的最大大小，則處理大小為最大揚(yáng)程時(shí)，定位誤差在這個(gè)時(shí)間步長(zhǎng)大小H被稱(chēng)為： δ定位 =A1A2=Hmax–Hmin=O1O2+1/2d-1/2(d-Td)=O1O2 +1/2Td 根據(jù)定位誤差產(chǎn)生的原因也可按定位誤差的組成進(jìn)行計(jì)算： δ定位=δ位置+δ不重=O1O2 +1/2Td 在工件的裝夾配合當(dāng)中，由于芯軸與工件采用的是H7/k6過(guò)渡配合，故認(rèn)為O1O2基本為0。所以： δ定位=0.5Td 即為芯軸公差的一半。 第五章 工件的夾緊 5.2 夾緊力的計(jì)算 1）夾緊力的方向 由于在整個(gè)加工過(guò)程是以工件底面為準(zhǔn)，鉆模板壓在上面，所以上面可以用螺母直接夾緊。 根據(jù)材料的磨損和耐用度標(biāo)準(zhǔn)8-37位查表孔尺寸： 后刀面最大磨損極限：0.5～0.8mm，T=3600 刀具切削速度： V=27.91/51.62=0.52 m/s 表10-3中鉆切削速度的計(jì)算公式： Cv=14.7 Zv=0.25 Yv=0.55 m=0.125 f=0.3mm/r 查表10-4 鉆削時(shí)軸向力，扭距和功率的計(jì)算公式 F=9.81×42.7×13×0.3×1=3471.57N M=9.81×0.021×13×0.3×1 =13.29N·m Pm=2Mv/d=2×13.29/13 =2.04k N F =Fw，簡(jiǎn)化FW朝向中心,芯軸的軸向力Fw和旋轉(zhuǎn)力矩T ,當(dāng)鉆削在如圖的孔時(shí)力矩最大,故計(jì)算這一個(gè)便可: T =FwL=3471.5×157=545000N·mm 預(yù)緊力: Fp= kuF/uc=1.2×3471.57/0.15=27800N 根據(jù)芯軸許用應(yīng)力驗(yàn)算公式芯軸: 表11-6,螺釘,螺柱和螺母的力學(xué)性能等級(jí) = [σ]=σs/s=480/1.5=320 MP D=4×1.3×2.78×10000/3.14×320=12mm 考慮到安全性螺母M14。由于橫向負(fù)荷，在這里不作計(jì)算。 參考文獻(xiàn) 1.機(jī)床夾具零部件 第一機(jī)械工業(yè)部機(jī)床研究所 1966 2.吳克堅(jiān)主編 機(jī)械原理 高等教育出版社 1990 3.吳宗澤主編 機(jī)械設(shè)計(jì)高等教育出版社 1991 4.吳可想主編 機(jī)械設(shè)計(jì)及其自動(dòng)化 2008 5.李明亞主編 機(jī)床夾具設(shè)計(jì) 1993 6.李偉主編 機(jī)械設(shè)計(jì)手冊(cè) 1993 7.機(jī)床夾具設(shè)計(jì)手冊(cè) 上?？茖W(xué)技術(shù)出版社 1990 8.機(jī)床夾具 上海科學(xué)技術(shù)出版社 1991 9.機(jī)床夾具及良具設(shè)計(jì) 白海清主編 2013 致謝 也許是時(shí)間的流逝是客觀的，但時(shí)光流逝純粹是主觀感覺(jué)。當(dāng)我終于從考公務(wù)員，找工作，畢業(yè)設(shè)計(jì)釋放的壓力之下解脫出來(lái)，長(zhǎng)嘆一聲，我突然明白了，原來(lái)已經(jīng)在過(guò)去的四年里，到了告別的時(shí)候。一念至此，居然在恍惚中，所謂的流年，想必就是這么憂(yōu)郁。但之后的失落感，總是有話(huà)說(shuō)。大學(xué)四年，生活其實(shí)很簡(jiǎn)單，但一次又一次的一些閱讀，寫(xiě)作和考試。如果生命作為秀場(chǎng)的這個(gè)單調(diào)的循環(huán)，我只是提供了一個(gè)安靜的演員。雖然沒(méi)有什么精彩的臺(tái)詞，但這個(gè)詞是幕即將告一段落這個(gè)時(shí)期。但不管有多么的差勁的演員，不管有多少觀眾觀眾，哪怕只是說(shuō)給自己聽(tīng)，在他的謝幕也永遠(yuǎn)感謝一些人，這些人幫助他走上舞臺(tái)，成功或不那么成功的表現(xiàn)。我在這里首先要感謝我的論文指導(dǎo)教授 – 樊十全老師。從本論文的開(kāi)題，數(shù)據(jù)查詢(xún)，修改定稿沒(méi)有他的努力，不知道會(huì)以何種面目出現(xiàn)。我很榮幸有這樣的老師，他是值得我感敬和尊重。我感謝工學(xué)院所有的學(xué)生和精彩的度過(guò)大學(xué)四年的人生的2010農(nóng)業(yè)機(jī)械化及其自動(dòng)化類(lèi)的同學(xué)。感謝學(xué)院的所有老師，你讓我終身難忘。感謝所有關(guān)心，鼓勵(lì)和支持我的家人，親戚和朋友。 18 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