B6065刨床推動(dòng)架工藝規(guī)程及夾具設(shè)計(jì)【含6張CAD圖紙+PDF圖】
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南京理工大學(xué)泰州科技學(xué)院學(xué)生畢業(yè)設(shè)計(jì)(論文)中期檢查表學(xué)生姓名周曉明學(xué) 號0501510155指導(dǎo)教師武培軍 吳晟選題情況課題名稱B6065刨床推動(dòng)架工藝規(guī)程及夾具設(shè)計(jì)難易程度偏難適中偏易工作量較大合理較小符合規(guī)范化的要求任務(wù)書有無開題報(bào)告有無外文翻譯質(zhì)量優(yōu)良中差學(xué)習(xí)態(tài)度、出勤情況好一般差工作進(jìn)度快按計(jì)劃進(jìn)行慢中期工作匯報(bào)及解答問題情況優(yōu)良中差中期成績評定:良所在專業(yè)意見:學(xué)習(xí)較主動(dòng)、積極,態(tài)度認(rèn)真,階段成果較明顯。 負(fù)責(zé)人: 年 月 日南京理工大學(xué)泰州科技學(xué)院畢業(yè)設(shè)計(jì)(論文)任務(wù)書系部:機(jī)械工程系專 業(yè):機(jī)械工程及自動(dòng)化學(xué) 生 姓 名:周曉明學(xué) 號:0501510155設(shè)計(jì)(論文)題目:B6065刨床推動(dòng)架工藝規(guī)程及夾具設(shè)計(jì)起 迄 日 期:2009年 3 月09日 6月14日設(shè)計(jì)(論文)地點(diǎn):南京理工大學(xué)泰州科技學(xué)院指 導(dǎo) 教 師:武培軍 吳 晟專業(yè)負(fù)責(zé)人:龔光容發(fā)任務(wù)書日期: 2009年 2 月 26 日任務(wù)書填寫要求1畢業(yè)設(shè)計(jì)(論文)任務(wù)書由指導(dǎo)教師根據(jù)各課題的具體情況填寫,經(jīng)學(xué)生所在專業(yè)的負(fù)責(zé)人審查、系部領(lǐng)導(dǎo)簽字后生效。此任務(wù)書應(yīng)在第七學(xué)期結(jié)束前填好并發(fā)給學(xué)生;2任務(wù)書內(nèi)容必須用黑墨水筆工整書寫或按教務(wù)處統(tǒng)一設(shè)計(jì)的電子文檔標(biāo)準(zhǔn)格式(可從教務(wù)處網(wǎng)頁上下載)打印,不得隨便涂改或潦草書寫,禁止打印在其它紙上后剪貼;3任務(wù)書內(nèi)填寫的內(nèi)容,必須和學(xué)生畢業(yè)設(shè)計(jì)(論文)完成的情況相一致,若有變更,應(yīng)當(dāng)經(jīng)過所在專業(yè)及系部主管領(lǐng)導(dǎo)審批后方可重新填寫;4任務(wù)書內(nèi)有關(guān)“系部”、“專業(yè)”等名稱的填寫,應(yīng)寫中文全稱,不能寫數(shù)字代碼。學(xué)生的“學(xué)號”要寫全號;5任務(wù)書內(nèi)“主要參考文獻(xiàn)”的填寫,應(yīng)按照國標(biāo)GB 77142005文后參考文獻(xiàn)著錄規(guī)則的要求書寫,不能有隨意性;6有關(guān)年月日等日期的填寫,應(yīng)當(dāng)按照國標(biāo)GB/T 74082005數(shù)據(jù)元和交換格式、信息交換、日期和時(shí)間表示法規(guī)定的要求,一律用阿拉伯?dāng)?shù)字書寫。如“2008年3月15日”或“2008-03-15”。畢 業(yè) 設(shè) 計(jì)(論 文)任 務(wù) 書1本畢業(yè)設(shè)計(jì)(論文)課題應(yīng)達(dá)到的目的:通過本設(shè)計(jì),使學(xué)生熟悉工序卡制訂及夾具設(shè)計(jì)的一般過程,培養(yǎng)學(xué)生綜合運(yùn)用所學(xué)基礎(chǔ)理論、專業(yè)知識和各項(xiàng)技能,著重培養(yǎng)設(shè)計(jì)、計(jì)算、分析問題和解決問題的能力,進(jìn)而總結(jié)、歸納和獲得合理結(jié)論,進(jìn)行較為系統(tǒng)的工程訓(xùn)練,初步鍛煉科研能力,提高論文撰寫和技術(shù)表述能力,為實(shí)際工作奠定基礎(chǔ),達(dá)到人才培養(yǎng)的目的和要求。2本畢業(yè)設(shè)計(jì)(論文)課題任務(wù)的內(nèi)容和要求(包括原始數(shù)據(jù)、技術(shù)要求、工作要求等):本設(shè)計(jì)要求學(xué)生根據(jù)推動(dòng)架零件的結(jié)構(gòu)特點(diǎn)、機(jī)械制造加工工藝規(guī)程的編制原則,編制出推動(dòng)架零件的加工工藝規(guī)程,并設(shè)計(jì)出特定工序的專用鉆夾具,繪出零件圖和裝配圖,學(xué)習(xí)AutoCAD、Pro/Engineer等軟件在機(jī)械設(shè)計(jì)中的應(yīng)用,具體要求如下:1、查閱相關(guān)資料(不少于15篇),翻譯一定量的外文資料(不少于3000漢字),撰寫開題報(bào)告及文獻(xiàn)綜述(不少于2000字);2、完成推動(dòng)架零件加工工藝規(guī)程的編制;3、選擇合適的工件定位方式;4、確定恰當(dāng)?shù)膴A緊方案;5、進(jìn)行專用夾具總圖及零件圖的繪制;6、撰寫設(shè)計(jì)說明書。相關(guān)的技術(shù)要求:1、被加工工件為灰鑄鐵;2、被加工工件具有較高強(qiáng)度,耐熱性;3、表面無氣孔、縮孔、夾砂。畢 業(yè) 設(shè) 計(jì)(論 文)任 務(wù) 書3對本畢業(yè)設(shè)計(jì)(論文)課題成果的要求包括畢業(yè)設(shè)計(jì)論文、圖表、實(shí)物樣品等:1、相關(guān)資料的英文翻譯與文獻(xiàn)綜述; 2、推動(dòng)架零件的機(jī)械加工工序卡片;3、特定工序?qū)S勉@夾具的裝配圖、零件圖;4、畢業(yè)設(shè)計(jì)論文。畢業(yè)設(shè)計(jì)成果以設(shè)計(jì)圖樣和說明書形式提交,要求圖樣規(guī)范,符合國家標(biāo)準(zhǔn);說明書層次分明、論據(jù)可靠、計(jì)算正確、圖標(biāo)規(guī)范、語句通順。4主要參考文獻(xiàn):1 徐灝,邱懷宣,菜春源,汪愷,余俊. 機(jī)械設(shè)計(jì)手冊M. 北京: 機(jī)械工業(yè)出版社,1989.2 王先逵. 機(jī)械制造工藝學(xué) M. 北京:機(jī)械工業(yè)出版社,2004. 3 張興輝. 實(shí)用機(jī)械加工測量技巧M. 北京:化學(xué)工業(yè)出版社,2008.4 薛源順. 機(jī)床夾具設(shè)計(jì)(第二版)M. 北京:機(jī)械工業(yè)出版社,2003. 5 周海寶. 提高機(jī)床夾具精度的實(shí)用方法 J. 機(jī)床與液壓,2000,4: 9697.6 劉文劍,莫天河,趙繼媛. 夾具工程師手冊M. 黑龍江科學(xué)技術(shù)出版社1987. 畢 業(yè) 設(shè) 計(jì)(論 文)任 務(wù) 書5本畢業(yè)設(shè)計(jì)(論文)課題工作進(jìn)度計(jì)劃:起 迄 日 期工 作 內(nèi) 容2009年 3 月 9 日 3月 23 日 3 月 24 日 4月 6 日4 月 7 日 5月 11日5 月 12 日 5月 31 日6 月 1 日 6月 9 日6月 10日 6月 14 日熟悉課題,準(zhǔn)備相關(guān)資料,完成資料翻譯完成文獻(xiàn)綜述,撰寫開題報(bào)告,熟悉AutoCAD、Pro/Engineer等繪圖軟件掌握機(jī)械制造加工工藝編制的要點(diǎn),分析推動(dòng)架零件的特點(diǎn),完成推動(dòng)架件制造加工工藝規(guī)程的編制掌握機(jī)床夾具設(shè)計(jì)特點(diǎn),完成鉆夾具的結(jié)構(gòu)設(shè)計(jì),畫出相關(guān)零件圖和裝配圖撰寫并打印設(shè)計(jì)說明書,整理相關(guān)資料準(zhǔn)備論文答辯所在專業(yè)審查意見:負(fù)責(zé)人: 2009年 月 日系部意見:系部主任: 2009年 月 日南京理工大學(xué)泰州科技學(xué)院畢業(yè)設(shè)計(jì)(論文)前期工作材料學(xué)生姓名:周曉明學(xué) 號:0501510155系部:機(jī)械工程系專 業(yè):機(jī)械工程及自動(dòng)化設(shè)計(jì)(論文)題目:B6065刨床推動(dòng)架工藝規(guī)程及夾具設(shè)計(jì)指導(dǎo)教師:武培軍高工吳晟助教 材 料 目 錄序號名 稱數(shù)量備 注1畢業(yè)設(shè)計(jì)(論文)選題、審題表12畢業(yè)設(shè)計(jì)(論文)任務(wù)書13畢業(yè)設(shè)計(jì)(論文)開題報(bào)告含文獻(xiàn)綜述14畢業(yè)設(shè)計(jì)(論文)外文資料翻譯含原文15畢業(yè)設(shè)計(jì)(論文)中期檢查表12009年5月夾具夾緊力的優(yōu)化及對工件定位精度的影響B(tài).Li 和 S.N.Mellkote布什伍德拉夫機(jī)械工程學(xué)院,佐治亞理工學(xué)院,格魯吉亞,美國研究所由于夾緊和加工,在工件和夾具的接觸部位會(huì)產(chǎn)生局部彈性變形,使工件尺寸發(fā)生變化,進(jìn)而影響工件的最終加工質(zhì)量。這種效應(yīng)可通過最小化夾具設(shè)計(jì)優(yōu)化,夾緊力是一個(gè)重要的設(shè)計(jì)變量,可以得到優(yōu)化,以減少工件的位移。本文提出了一種確定多夾緊夾具受到準(zhǔn)靜態(tài)加工部位的最佳夾緊力的新方法。該方法采用彈性接觸力學(xué)模型代表夾具與工件接觸,并涉及制定和解決方案的多目標(biāo)優(yōu)化模型的約束。夾緊力的最優(yōu)化對工件定位精度的影響通過3-2-1式銑夾具的例子進(jìn)行了分析。關(guān)鍵詞:彈性 接觸 模型 夾具 夾緊力 優(yōu)化 前言 定位和夾緊的工件加工中的兩個(gè)關(guān)鍵因素。要實(shí)現(xiàn)夾具的這些功能,需將工件定位到一個(gè)合適的基準(zhǔn)上并夾緊,采用的夾緊力必須足夠大,以抑制工件在加工過程中產(chǎn)生的移動(dòng)。然而,過度的夾緊力可誘導(dǎo)工件產(chǎn)生更大的彈性變形 ,這會(huì)影響它的位置精度,并反過來影響零件質(zhì)量。所以有必要確定最佳夾緊力,來減小由于彈性變形對工件的定位誤差,同時(shí)滿足加工的要求。在夾具分析和綜合領(lǐng)域上的研究人員使用了有限元模型的方法或剛體模型的方法。大量的工作都以有限元方法為基礎(chǔ)被報(bào)道參考文獻(xiàn)1-8。隨著得墨忒耳8,這種方法的限制是需要較大的模型和計(jì)算成本。同時(shí),多數(shù)的有限元基礎(chǔ)研究人員一直重點(diǎn)關(guān)注的夾具布局優(yōu)化和夾緊力的優(yōu)化還沒有得到充分討論,也有少數(shù)的研究人員通過對剛性模型9-11對夾緊力進(jìn)行了優(yōu)化,剛型模型幾乎被近似為一個(gè)規(guī)則完整的形狀。得墨忒耳12,13用螺釘理論解決的最低夾緊力,總的問題是制定一個(gè)線性規(guī)劃,其目的是盡量減少在每個(gè)定位點(diǎn)調(diào)整夾緊力強(qiáng)度的法線接觸力。接觸摩擦力的影響被忽視,因?yàn)樗^法線接觸力相對較小,由于這種方法是基于剛體假設(shè),獨(dú)特的三維夾具可以處理超過6個(gè)自由度的裝夾,復(fù)和倪14也提出迭代搜索方法,通過假設(shè)已知摩擦力的方向來推導(dǎo)計(jì)算最小夾緊力,該剛體分析的主要限制因素是當(dāng)出現(xiàn)六個(gè)以上的接觸力是使其靜力不確定,因此,這種方法無法確定工件移位的唯一性。 這種限制可以通過計(jì)算夾具工件系統(tǒng)15的彈性來克服,對于一個(gè)相對嚴(yán)格的工件,該夾具在機(jī)械加工工件的位置會(huì)受夾具點(diǎn)的局部彈性變形的強(qiáng)烈影響。Hockenberger和得墨忒耳16使用經(jīng)驗(yàn)的接觸力變形的關(guān)系(稱為元功能),解決由于夾緊和準(zhǔn)靜態(tài)加工力工件剛體位移。同一作者還考察了加工工件夾具位移對設(shè)計(jì)參數(shù)的影響17。桂 18 等 通過工件的夾緊力的優(yōu)化定位精度彈性接觸模型對報(bào)告做了改善,然而,他們沒有處理計(jì)算夾具與工件的接觸剛度的方法,此外,其算法的應(yīng)用沒有討論機(jī)械加工刀具路徑負(fù)載有限序列。李和Melkote 19和烏爾塔多和Melkote 20用接觸力學(xué)解決由于在加載夾具夾緊點(diǎn)彈性變形產(chǎn)生的接觸力和工件的位移,他們還使用此方法制定了優(yōu)化方法夾具布局21和夾緊力22。但是,關(guān)于multiclamp系統(tǒng)及其對工件精度影響的夾緊力的優(yōu)化并沒有在這些文件中提到 。本文提出了一種新的算法,確定了multiclamp夾具工件系統(tǒng)受到準(zhǔn)靜態(tài)加載的最佳夾緊力為基礎(chǔ)的彈性方法。該法旨在盡量減少影響由于工件夾緊位移和加工荷載通過系統(tǒng)優(yōu)化夾緊力的一部分定位精度。接觸力學(xué)模型,用于確定接觸力和位移,然后再用做夾緊力優(yōu)化,這個(gè)問題被作為多目標(biāo)約束優(yōu)化問題提出和解決。通過兩個(gè)例子分析工件夾緊力的優(yōu)化對定位精度的影響,例子涉及的銑削夾具3-2-1布局。1 夾具工件聯(lián)系模型 11 模型假設(shè)該加工夾具由L定位器和帶有球形端的c形夾組成。工件和夾具接觸的地方是線性的彈性接觸,其他地方完全剛性。工件夾具系統(tǒng)由于夾緊和加工受到準(zhǔn)靜態(tài)負(fù)載。夾緊力可假定為在加工過程中保持不變,這個(gè)假設(shè)是有效的,在對液壓或氣動(dòng)夾具使用。在實(shí)際中,夾具工件接觸區(qū)域是彈性分布,然而,這種模式的發(fā)展,假設(shè)總觸剛度(見圖1)第i夾具接觸力局部變形如下: (1) 其中(j=x,y,z)表示,在當(dāng)?shù)刈幼鴺?biāo)系切線和法線方向的接觸剛度第 19 頁 共 15 頁圖1 彈簧夾具工件接觸模型。 表示在第i個(gè)接觸處的坐標(biāo)系(j=x,y,z)是對應(yīng)沿著xyz方向的彈性變形,分別 (j= x,y,z)的代表和切向力接觸 ,法線力接觸。12 工件夾具的接觸剛度模型集中遵守一個(gè)球形尖端定位,夾具和工件的接觸并不是線性的,因?yàn)榻佑|半徑與隨法線力呈非線性變化 23。由于法線力接觸變形作用于半徑和平面工件表面之間,這可從封閉赫茲的辦法解決縮進(jìn)一個(gè)球體彈性半空間的問題。對于這個(gè)問題, 是法線的變形,在文獻(xiàn)23 第93頁中給出如下: (2)其中式中 和是工件和夾具的彈性模量,、分別是工件和材料的泊松比。切向變形沿著和切線方向)硅業(yè)切力距有以下形式文獻(xiàn)23第217頁 (3)其中、 分別是工件和夾具剪切模量一個(gè)合理的接觸剛度的線性可以近似從最小二乘獲得適合式 (2),這就產(chǎn)生了以下線性化接觸剛度值:在計(jì)算上述的線性近似, (4) (5)正常的力被假定為從0到1000N,且最小二乘擬合相應(yīng)的R2值認(rèn)定是0.94。2夾緊力優(yōu)化 我們的目標(biāo)是確定最優(yōu)夾緊力,將盡量減少由于工件剛體運(yùn)動(dòng)過程中,局部的夾緊和加工負(fù)荷引起的彈性變形,同時(shí)保持在準(zhǔn)靜態(tài)加工過程中夾具工件系統(tǒng)平衡,工件的位移減少,從而減少定位誤差。實(shí)現(xiàn)這個(gè)目標(biāo)是通過制定一個(gè)多目標(biāo)約束優(yōu)化問題的問題,如下描述。2.1 目標(biāo)函數(shù)配方工件旋轉(zhuǎn),由于部隊(duì)輪換往往是相當(dāng)小17的工件定位誤差假設(shè)為確定其剛體翻譯基本上,其中 、和 是 沿,和三個(gè)正交組件(見圖2)。圖2 工件剛體平移和旋轉(zhuǎn)工件的定位誤差歸于裝夾力,然后可以在該剛體位移的范數(shù)計(jì)算如下: (6)其中表示一個(gè)向量二級標(biāo)準(zhǔn)。 但是作用在工件的夾緊力會(huì)影響定位誤差。當(dāng)多個(gè)夾緊力作用于工件,由此產(chǎn)生的夾緊力為,有如下形式: (7)其中夾緊力是矢量,夾緊力的方向矩陣,是夾緊力是矢量的方向余弦,、和 是第i個(gè)夾緊點(diǎn)夾緊力在、和方向上的向量角度(i=1、2、3.,C)。在這個(gè)文件中,由于接觸區(qū)變形造成的工件的定位誤差,被假定為受的作用力是法線的,接觸的摩擦力相對較小,并在進(jìn)行分析時(shí)忽略了加緊力對工件的定位誤差的影響。意指正常接觸剛度比,是通過(i=1,2L)和最小的所有定位器正常剛度相乘,并假設(shè)工件、取決于、的方向,各自的等效接觸剛度可有下式計(jì)算得出(見圖3),工件剛體運(yùn)動(dòng),歸于夾緊行動(dòng)現(xiàn)在可以寫成: (8)工件有位移,因此,定位誤差的減小可以通過盡量減少產(chǎn)生的夾緊力向量 范數(shù)。因此,第一個(gè)目標(biāo)函數(shù)可以寫為:最小化 (9)要注意,加權(quán)因素是與等效接觸剛度成正比的在、和 方向上。通過使用最低總能量互補(bǔ)參考文獻(xiàn)15,23的原則求解彈性力學(xué)接觸問題得出A的組成部分是唯一確定的,這保證了夾緊力和相應(yīng)的定位反應(yīng)是“真正的”解決方案,對接觸問題和產(chǎn)生的“真正”剛體位移,而且工件保持在靜態(tài)平衡,通過夾緊力的隨時(shí)調(diào)整。因此,總能量最小化的形式為補(bǔ)充的夾緊力優(yōu)化的第二個(gè)目標(biāo)函數(shù),并給出:最小化 (10)其中代表機(jī)構(gòu)的彈性變形應(yīng)變能互補(bǔ),代表由外部力量和力矩配合完成,是遵守對角矩陣的, 和是所有接觸力的載體。如圖3 加權(quán)系數(shù)計(jì)算確定的基礎(chǔ)內(nèi)蒙古科技大學(xué)本科生畢業(yè)設(shè)計(jì)(外文翻譯)2.2 摩擦和靜態(tài)平衡約束在(10)式優(yōu)化的目標(biāo)受到一定的限制和約束,他們中最重要的是在每個(gè)接觸處的靜摩擦力約束。庫侖摩擦力的法律規(guī)定(是靜態(tài)摩擦系數(shù)),這方面的一個(gè)非線性約束和線性化版本可以使用,并且19有: (11)假設(shè)準(zhǔn)靜態(tài)載荷,工件的靜力平衡由下列力和力矩平衡方程確保(向量形式): (12)其中包括在法線和切線方向的力和力矩的機(jī)械加工力和工件重量。2.3界接觸力由于夾具工件接觸是單側(cè)面的,法線的接觸力只能被壓縮。這通過以下的的約束表(i=1,2,L+C) (13)它假設(shè)在工件上的法線力是確定的,此外,在一個(gè)法線的接觸壓力不能超過壓工件材料的屈服強(qiáng)度()。這個(gè)約束可寫為: (i=1,2,,L+C) (14) 如果是在第i個(gè)工件夾具的接觸處的接觸面積,完整的夾緊力優(yōu)化模型,可以寫成:最小化 (15)3模型算法求解式(15)多目標(biāo)優(yōu)化問題可以通過求解約束24。這種方法將確定的目標(biāo)作為首要職能之一,并將其轉(zhuǎn)換成一個(gè)約束對。該補(bǔ)充()的主要目的是處理功能,并由此得到夾緊力()作為約束的加權(quán)范數(shù)最小化。對為主要目標(biāo)的選擇,確保選中一套獨(dú)特可行的夾緊力,因此,工件夾具系統(tǒng)驅(qū)動(dòng)到一個(gè)穩(wěn)定的狀態(tài)(即最低能量狀態(tài)),此狀態(tài)也表示有最小的夾緊力下的加權(quán)范數(shù)。 的約束轉(zhuǎn)換涉及到一個(gè)指定的加權(quán)范數(shù)小于或等于,其中是 的約束,假設(shè)最初所有夾緊力不明確,要確定一個(gè)合適的。在定位和夾緊點(diǎn)的接觸力的計(jì)算只考慮第一個(gè)目標(biāo)函數(shù)(即)。雖然有這樣的接觸力,并不一定產(chǎn)生最低的夾緊力,這是一個(gè)“真正的”可行的解決彈性力學(xué)問題辦法,可完全抑制工件在夾具中的位置。這些夾緊力的加權(quán)系數(shù),通過計(jì)算并作為初始值與比較,因此,夾緊力式(15)的優(yōu)化問題可改寫為: 最小化 (16)由: (11)(14) 得。類似的算法尋找一個(gè)方程根的二分法來確定最低的上的約束, 通過盡可能降低上限,由此產(chǎn)生的最小夾緊力的加權(quán)范數(shù)。 迭代次數(shù)K,終止搜索取決于所需的預(yù)測精度和,有參考文獻(xiàn)15: (17)其中表示上限的功能,完整的算法在如圖4中給出。 圖4 夾緊力的優(yōu)化算法(在示例1中使用)。圖5 該算法在示例2使用4 加工過程中的夾緊力的優(yōu)化及測定上一節(jié)介紹的算法可用于確定單負(fù)載作用于工件的載體的最佳夾緊力,然而,刀具路徑隨磨削量和切割點(diǎn)的不斷變化而變化。因此,相應(yīng)的夾緊力和最佳的加工負(fù)荷獲得將由圖4算法獲得,這大大增加了計(jì)算負(fù)擔(dān),并要求為選擇的夾緊力提供標(biāo)準(zhǔn), 將獲得滿意和適宜的整個(gè)刀具軌跡 ,用保守的辦法來解決下面將被討論的問題,考慮一個(gè)有限的數(shù)目(例如m)沿相應(yīng)的刀具路徑設(shè)置的產(chǎn)生m個(gè)最佳夾緊力,選擇記為, , ,在每個(gè)采樣點(diǎn),考慮以下四個(gè)最壞加工負(fù)荷向量: (18)、和表示在、和方向上的最大值,、和上的數(shù)字1,2,3分別代替對應(yīng)的和另外兩個(gè)正交切削分力,而且有:雖然4個(gè)最壞情況加工負(fù)荷向量不會(huì)在工件加工的同一時(shí)刻出現(xiàn),但在每次常規(guī)的進(jìn)給速度中,刀具旋轉(zhuǎn)一次出現(xiàn)一次,負(fù)載向量引入的誤差可忽略。因此,在這項(xiàng)工作中,四個(gè)載體負(fù)載適用于同一位置,(但不是同時(shí))對工件進(jìn)行的采樣 ,夾緊力的優(yōu)化算法圖4,對應(yīng)于每個(gè)采樣點(diǎn)計(jì)算最佳的夾緊力。夾緊力的最佳形式有: (i=1,2,m) (j=x,y z,r) (19)其中是最佳夾緊力的四個(gè)情況下的加工負(fù)荷載體,(C=1,2,C)是每個(gè)相應(yīng)的夾具在第i個(gè)樣本點(diǎn)和第j負(fù)荷情況下力的大小。是計(jì)算每個(gè)負(fù)載點(diǎn)之后的結(jié)果,一套簡單的“最佳”夾緊力必須從所有的樣本點(diǎn)和裝載條件里發(fā)現(xiàn),并在所有的最佳夾緊力中選擇。這是通過在所有負(fù)載情況和采樣點(diǎn)排序,并選擇夾緊點(diǎn)的最高值的最佳的夾緊力,見于式 (20): (k=1,2,C) (20)只要這些具備,就得到一套優(yōu)化的夾緊力,驗(yàn)證這些力,以確保工件夾具系統(tǒng)的靜態(tài)平衡。否則,會(huì)出現(xiàn)更多采樣點(diǎn)和重復(fù)上述程序。在這種方式中,可為整個(gè)刀具路徑確定“最佳”夾緊力 ,圖5總結(jié)了剛才所描述的算法。請注意,雖然這種方法是保守的,它提供了一個(gè)確定的夾緊力,最大限度地減少工件的定位誤差的一套系統(tǒng)方法。5影響工件的定位精度它的興趣在于最早提出了評價(jià)夾緊力的算法對工件的定位精度的影響。工件首先放在與夾具接觸的基板上,然后夾緊力使工件接觸到夾具,因此,局部變形發(fā)生在每個(gè)工件夾具接觸處,使工件在夾具上移位和旋轉(zhuǎn)。隨后,準(zhǔn)靜態(tài)加工負(fù)荷應(yīng)用造成工件在夾具的移位。工件剛體運(yùn)動(dòng)的定義是由它在、和方向上的移位和自轉(zhuǎn)(見圖2),如前所述,工件剛體位移產(chǎn)生于在每個(gè)夾緊處的局部變形,假設(shè)為相對于工件的質(zhì)量中心的第i個(gè)位置矢量定位點(diǎn),坐標(biāo)變換定理可以用來表達(dá)在工件的位移,以及工件自轉(zhuǎn)如下: (21)其中表示旋轉(zhuǎn)矩陣,描述當(dāng)?shù)卦诘趇幀相聯(lián)系的全球坐標(biāo)系和是一個(gè)旋轉(zhuǎn)矩陣確定工件相對于全球的坐標(biāo)系的定位坐標(biāo)系。假設(shè)夾具夾緊工件旋轉(zhuǎn),由于旋轉(zhuǎn)很小,故也可近似為: (22) 方程(21)現(xiàn)在可以改寫為: (23)其中是經(jīng)方程(21)重新編排后變換得到的矩陣式,是夾緊和加工導(dǎo)致的工件剛體運(yùn)動(dòng)矢量。工件與夾具單方面接觸性質(zhì)意味著工件與夾具接觸處沒有拉力的可能。因此,在第i裝夾點(diǎn)接觸力可能與的關(guān)系如下: (24)其中是在第i個(gè)接觸點(diǎn)由于夾緊和加工負(fù)荷造成的變形,意味著凈壓縮變形,而負(fù)數(shù)則代表拉伸變形; 是表示在本地坐標(biāo)系第i個(gè)接觸剛度矩陣,是單位向量. 在這項(xiàng)研究中假定液壓/氣動(dòng)夾具,根據(jù)對外加工負(fù)荷,故在法線方向的夾緊力的強(qiáng)度保持不變,因此,必須對方程(24)的夾緊點(diǎn)進(jìn)行修改為: (25)其中是在第i個(gè)夾緊點(diǎn)的夾緊力,讓表示一個(gè)對外加工力量和載體的61矢量。并結(jié)合方程(23)(25)與靜態(tài)平衡方程,得到下面的方程組: (26)其中,其中表示相乘。由于夾緊和加工工件剛體移動(dòng),q可通過求解式(26)得到。工件的定位誤差向量, (見圖6),現(xiàn)在可以計(jì)算如下: (27) 其中是考慮工件中心加工點(diǎn)的位置向量,且 6模擬工作 較早前提出的算法是用來確定最佳夾緊力及其對兩例工件精度的影響例如:1適用于工件單點(diǎn)力。2應(yīng)用于工件負(fù)載準(zhǔn)靜態(tài)銑削序列 如左圖7 工件夾具配置中使用的模擬研究 工件夾具定位聯(lián)系; 、和全球坐標(biāo)系。 3-2-1夾具圖7所示,是用來定位并控制7075 - T6鋁合金(127毫米127毫米38.1毫米)的柱狀塊。假定為球形布局傾斜硬鋼定位器/夾具在表1中給出。工件夾具材料的摩擦靜電對系數(shù)為0.25。使用伊利諾伊大學(xué)開發(fā)EMSIM程序參考文獻(xiàn)26 對加工瞬時(shí)銑削力條件進(jìn)行了計(jì)算,如表2給出例(1),應(yīng)用工件在點(diǎn)(109.2毫米,25.4毫米,34.3毫米)瞬時(shí)加工力,圖4中表3和表4列出了初級夾緊力和最佳夾緊力的算法 。該算法如圖5所示 ,一個(gè)25.4毫米銑槽使用EMSIM進(jìn)行了數(shù)值模擬,以減少起步(0.0毫米,25.4毫米,34.3毫米)和結(jié)束時(shí)(127.0毫米,25.4毫米,34.3毫米)四種情況下加工負(fù)荷載體,(見圖8)。模擬計(jì)算銑削力數(shù)據(jù)在表5中給出。圖8最終銑削過程模擬例如2。表6中5個(gè)坐標(biāo)列出了為模擬抽樣調(diào)查點(diǎn)。最佳夾緊力是用前面討論過的排序算法計(jì)算每個(gè)采樣點(diǎn)和負(fù)載載體最后的夾緊力和負(fù)載。7結(jié)果與討論例如算法1的繪制最佳夾緊力收斂圖9,圖9對于固定夾緊裝置在圖示例假設(shè)(見圖7),由此得到的夾緊力加權(quán)范數(shù)有如下形式:.結(jié)果表明,最佳夾緊力所述加工條件下有比初步夾緊力強(qiáng)度低得多的加權(quán)范數(shù),最初的夾緊力是通過減少工件的夾具系統(tǒng)補(bǔ)充能量算法獲得。由于夾緊力和負(fù)載造成的工件的定位誤差,如表7。結(jié)果表明工件旋轉(zhuǎn)小,加工點(diǎn)減少錯(cuò)誤從13.1到14.6不等。在這種情況下,所有加工條件改善不是很大,因?yàn)閺淖畛跬ㄟ^互補(bǔ)勢能確定的最小化的夾緊力值已接近最佳夾緊力。圖5算法是用第二例在一個(gè)序列應(yīng)用于銑削負(fù)載到工件,他應(yīng)用于工件銑削負(fù)載一個(gè)序列。最佳的夾緊力,對應(yīng)列表6每個(gè)樣本點(diǎn),隨著最后的最佳夾緊力,在每個(gè)采樣點(diǎn)的加權(quán)范數(shù)和最優(yōu)的初始夾緊力繪圖10,在每個(gè)采樣點(diǎn)的加權(quán)范數(shù)的,和繪制。結(jié)果表明,由于每個(gè)組成部分是各相應(yīng)的最大夾緊力,它具有最高的加權(quán)范數(shù)。如圖10所示,如果在每個(gè)夾緊點(diǎn)最大組成部分是用于確定初步夾緊力,則夾緊力需相應(yīng)設(shè)置,有比相當(dāng)大的加權(quán)范數(shù)。故是一個(gè)完整的刀具路徑改進(jìn)方案。上述模擬結(jié)果表明,該方法可用于優(yōu)化夾緊力相對于初始夾緊力的強(qiáng)度,這種做法將減少所造成的夾緊力的加權(quán)范數(shù),因此將提高工件的定位精度。圖108結(jié)論該文件提出了關(guān)于確定多鉗夾具,工件受準(zhǔn)靜態(tài)加載系統(tǒng)的優(yōu)化加工夾緊力的新方法。夾緊力的優(yōu)化算法是基于接觸力學(xué)的夾具與工件系統(tǒng)模型,并尋求盡量減少應(yīng)用到所造成的工件夾緊力的加權(quán)范數(shù),得出工件的定位誤差。該整體模型,制定一個(gè)雙目標(biāo)約束優(yōu)化問題,使用-約束的方法解決。該算法通過兩個(gè)模擬表明,涉及3-2-1型,二夾銑夾具的例子。今后的工作將解決在動(dòng)態(tài)負(fù)載存在夾具與工件在系統(tǒng)的優(yōu)化,其中慣性,剛度和阻尼效應(yīng)在確定工件夾具系統(tǒng)的響應(yīng)特性具有重要作用。9參考資料:1、J. 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DeMeter.加工夾具的性能的最小最大負(fù)荷標(biāo)準(zhǔn) 美國ASME,工業(yè)工程雜志 :199413、E. C. DeMeter .加工夾具最大負(fù)荷的性能優(yōu)化模型 美國ASME,工業(yè)工程雜志 1995。14、JH復(fù)和AYC倪.“核查和工件夾持的夾具設(shè)計(jì)”方案優(yōu)化,設(shè)計(jì)和制造,4,碩士論文: 307-318,1994。15、T. H. Richards、埃利斯 霍伍德.1977,應(yīng)力能量方法分析,1977。16、M. J. Hockenberger and E. C. DeMeter. 對工件準(zhǔn)靜態(tài)分析功能位移在加工夾具的應(yīng)用程序,制造科學(xué)雜志與工程: 325331頁, 1996。 南京理工大學(xué)泰州科技學(xué)院畢業(yè)設(shè)計(jì)(論文)外文資料翻譯系部: 機(jī)械工程系 專 業(yè): 機(jī)械工程及自動(dòng)化 姓 名: 周曉明 學(xué) 號: 0501510155 外文出處:96-8000 rev F, VF SERIES OperatorsManualS. Haas Automation Inc. 2005:93-96. 附 件: 1.外文資料翻譯譯文;2.外文原文。 指導(dǎo)教師評語: 簽名: 年 月 日注:請將該封面與附件裝訂成冊。附件1:外文資料翻譯譯文數(shù)控銑床4軸與5軸編程在VR-11銑床和HAASTRT210上的軸運(yùn)動(dòng)創(chuàng)建5軸程序大多數(shù)的五軸程序是非常復(fù)雜的,可以用CAD/CAM編寫。決定機(jī)床樞軸長度和標(biāo)準(zhǔn)長度非常有必要,將它們輸入到程序中。每一個(gè)機(jī)床都有一個(gè)特定的樞軸長度。它指的是從主軸頭旋轉(zhuǎn)的中心到裝夾刀具器的底面。在設(shè)置116中可以找到樞軸長度,也雕刻在裝有5軸機(jī)床的刀具器上。當(dāng)設(shè)定好程序后,決定每個(gè)刀具的標(biāo)準(zhǔn)長度是非常有必要的。對于每個(gè)刀具,標(biāo)準(zhǔn)長度是從刀具裝置器的底部到刀尖的距離。這個(gè)距離可以通過在工作臺(tái)上放置一個(gè)磁力表計(jì)算,指示刀具裝置器的底面,在控制器中設(shè)著這點(diǎn)為Z0。然后裝載刀具,計(jì)算從刀尖到Z0的距離,這就是標(biāo)準(zhǔn)長度??傞L度是從刀尖到旋轉(zhuǎn)主軸頭的距離。它是標(biāo)準(zhǔn)長度和樞軸長度總和。這個(gè)數(shù)據(jù)將被輸入到CAD/CAM程序中用于計(jì)算。偏置按下PageUp按鈕,在offset顯示中將出現(xiàn)work-offset(工件偏置)。從這里顯示和人工進(jìn)入work-offseet。G54到G59或G129偏置可以用PartZeroSet按鈕設(shè)置。定位軸到工件的工作地點(diǎn)。用光標(biāo)選擇適當(dāng)?shù)妮S和工件數(shù)。按PartZeroSe按鈕,當(dāng)前機(jī)床的位置將會(huì)自動(dòng)儲(chǔ)存在這些地址里。這將工作在選擇的工件零點(diǎn)偏置顯示中。注意輸入非零補(bǔ)償將會(huì)妨礙輸入刀具長度偏置的自動(dòng)操作。工件坐標(biāo)數(shù)字通常以正數(shù)的形式輸入。工件坐標(biāo)只以數(shù)字的形式輸入工作臺(tái)。將X2.00輸入到G54,選中X欄,輸入2.0。五軸編程注意在CAD/CAM中用嚴(yán)密同步切削幾何分解面能夠形成光滑的流線輪廓,得到更精確的工作。定位機(jī)床到一個(gè)向量途徑,只可以在工件上方安全距離或者是弓箭邊上可以做到。在高速模式下,軸在不同的時(shí)期到達(dá)編程位置;離目標(biāo)最近的軸將先達(dá)到目標(biāo),最遠(yuǎn)的后達(dá)到。高進(jìn)給速度將會(huì)使軸在同一時(shí)間到達(dá)編輯的位置避免碰撞。G代碼選擇英制(G20)或公制(G21)將不會(huì)影響第五軸編程,A和B軸通常用度來編輯。在同時(shí)發(fā)生五軸運(yùn)動(dòng)的事G93反時(shí)必須生效。在G93模式下,最大進(jìn)給速度將包括一個(gè)代碼塊中國所有運(yùn)動(dòng)的總和??刂破髟O(shè)置了限制,查看編碼器在塊中為所有軸編輯的程序步驟。如果可能,限制您的后置處理器(CAD/CAM);在G93模式的最大速度是32每分鐘。這將導(dǎo)致更平滑的運(yùn)動(dòng),當(dāng)沿傾斜的面展開的時(shí)候是需要它的。M代碼重要!高度推薦當(dāng)沒有5-軸運(yùn)動(dòng)的時(shí)候A/B制動(dòng)接合。解除制動(dòng)下切削能引起齒輪裝置的過度磨損。M10/M11接合/不接合A軸制動(dòng)控制M12/M13接合、不接合B軸制動(dòng)裝置在4或5軸切削的時(shí)候,機(jī)床在塊中將暫停,這個(gè)暫停是由于A和B制動(dòng)軸制動(dòng)裝置松開造成的。為了避免這個(gè)暫停,得到平穩(wěn)的程序執(zhí)行,在G93之前編輯M11或M13.M代碼將不會(huì)接合制動(dòng)裝置,會(huì)導(dǎo)致平穩(wěn)運(yùn)動(dòng)和不中斷流程。記住,如果制動(dòng)裝置從未重新接合,它們將無限處于不解和狀態(tài)。設(shè)置在5-軸切削中設(shè)置85被設(shè)置為0.500.小于0.500的設(shè)置將會(huì)移動(dòng)機(jī)床更接近于精確停止,引起不均勻運(yùn)動(dòng)。G187應(yīng)用于程序中使軸速減慢。小心!如果刀具長度偏置(H-code)沒有取消,當(dāng)切削在5-軸模式下,粗劣定位和過行程將會(huì)發(fā)生。在換刀后,第一個(gè)塊里用G90G40,H00和G49避免發(fā)生這種情況。當(dāng)混用3-軸和5-軸程序的時(shí)候發(fā)生這種情況;重新開始這個(gè)程序或者當(dāng)開始是一個(gè)工作,刀具長度偏置仍然有效。進(jìn)給速度在4或5軸代碼的每一行必須調(diào)用進(jìn)給速度。在鉆孔的時(shí)候限制進(jìn)給速度小于751PM。推薦3軸精加工的速度不能超過50到60IPM,在完成加工的時(shí)候,至少有0.500到0.750的余量。不允許快速移動(dòng);快速移動(dòng),進(jìn)入和離開孔(完全撤出灼鉆循環(huán))是不支持的。當(dāng)編輯同時(shí)5-軸運(yùn)動(dòng),需要少的資料供應(yīng)和允許較高的進(jìn)給速度。依靠于完成定量材料,切削長度,高速進(jìn)給都不可能。例如,當(dāng)切削一個(gè)模型線的時(shí)候或者是長流線輪廓的時(shí)候,進(jìn)給速度可以超過100IPM。手動(dòng)控制第4軸和第5軸手動(dòng)控制第五軸的操作與其它軸的操作是相同的。不同的是在A軸和B軸間選擇進(jìn)給。按“+A”或“-A”減將選擇A進(jìn)給。按shift鍵然后按“+A”或“-A”鍵選擇B軸進(jìn)給。設(shè)置程序中的會(huì)用一定數(shù)量的設(shè)置編輯第四軸和第五軸。見第四軸設(shè)置30.34和38,第五軸設(shè)置78和80.碰撞恢復(fù)程序當(dāng)機(jī)床在切削一個(gè)五軸工件發(fā)生碰撞的時(shí)候,由于包含有監(jiān)督很難清楚工件導(dǎo)軌。不要立刻按下ToolChanger Restore鍵或關(guān)閉電源。從撞擊處恢復(fù),也就是主軸已經(jīng)停止,然而刀具仍在切削,用Vector Jog特征縮回去主軸。按下面板上的字母V,按下“Handle Jog”,用手輪移動(dòng)軸。這個(gè)特征允許運(yùn)動(dòng)哪個(gè)方向的移動(dòng)軸是由A和B軸決定的。Vector Jog模式下不提供G28,當(dāng)選擇單個(gè)軸的時(shí)候,它只提供于X,Y,Z,A和B。如果在切削矢量進(jìn)給中電量不足,他將不會(huì)到達(dá)控制器所要求的參考位置。將會(huì)需要從工件上清理切削刀具的其他方法。當(dāng)發(fā)生碰撞的時(shí)候,刀具沒有切削,按Tool Changer Restore按鈕,回答屏幕上顯示的問題。當(dāng)過按下Tool Changer Restore后,主軸頭將同時(shí)移動(dòng)A,B和Z軸,為了縮回刀具。如果刀具以一個(gè)角度切削,當(dāng)按下這個(gè)鍵的時(shí)候,刀具將發(fā)生碰撞。安裝可選擇第四軸當(dāng)在HAAS銑床中加入旋轉(zhuǎn)工作臺(tái)的時(shí)候,對于指定的工作臺(tái)改變設(shè)置30和34,以及當(dāng)前應(yīng)用的工件直徑。警告:對于銑床安裝的有刷或無刷旋轉(zhuǎn)設(shè)置的錯(cuò)誤配置將導(dǎo)致電動(dòng)機(jī)危險(xiǎn)。在設(shè)置中的B指的是無刷旋轉(zhuǎn)工件。無刷分度器有倆個(gè)來自工作臺(tái)的電纜線。和倆個(gè)來自銑床控制器的連接器。參數(shù)在很少的情況下,用戶需要修改一些參數(shù)獲得分度器以外的特殊執(zhí)行。沒有改變參數(shù)列表的時(shí)候不要左這個(gè)。(如果在您的分度器里沒有得到特定參數(shù)的列表,那么您就不需要改變默認(rèn)參數(shù)-不要改變,否則將不在在保修范圍內(nèi)。)改變第四軸或者第五軸分度器中的參數(shù),執(zhí)行以下的操作:按下E-stop開關(guān),保持按下狀態(tài)。你必須關(guān)閉參數(shù)鎖(設(shè)置7)。按下Setting按鈕跳到設(shè)置頁面。輸入“7”,按向下箭頭,選擇Off按Write鍵。參數(shù)鎖現(xiàn)在是關(guān)閉狀態(tài)。轉(zhuǎn)到參數(shù)頁,輸入改變參數(shù),按向下箭頭。為新的參數(shù)輸入新值,按Write按鈕。如果需要,改變其他的參數(shù)?;氐皆O(shè)置7,開啟這個(gè)設(shè)置。重置E-stop鍵?;氐椒侄绕?,按Handle Jog和“A”按鈕,確信操作正確。Uong手輪進(jìn)給A軸,分度器將移動(dòng)。檢查標(biāo)記在工作臺(tái)上的正確速率,一在位置頁所顯示的旋轉(zhuǎn)360,檢查標(biāo)記是否在相同的位置。如果它關(guān)閉了(在10內(nèi)),速率是正確的。原始啟動(dòng)開啟銑床(和伺服控制,如果應(yīng)用),歸位分度器。從前視圖方向看,所有的HAAS分度器都順時(shí)針方向歸位。如果分度器逆時(shí)針歸位,那E-stop,與HAAS服務(wù)部門聯(lián)系。安裝一個(gè)可選擇第五軸第五軸安裝與第四軸的安裝方式相同。設(shè)置78和79支配第五軸,用B地址進(jìn)給和調(diào)用軸。輔助器控制器除了直接控制這五個(gè)軸,它還可以控制最多四個(gè)附加的外部定位軸。這些軸可以用軸代碼C,U,V和W從程序中直接調(diào)用。只有在G00或G01塊中允許調(diào)用這些這些軸。通過一個(gè)或更多HAAS單個(gè)軸控制器的第二RS-232端口鏈接這些軸。再Setting頁中,設(shè)置38用來選擇輔助軸的數(shù)目。機(jī)床中的定位顯示將顯示這些軸的當(dāng)前位置。一次只能移動(dòng)一個(gè)輔助軸。如果編輯了進(jìn)給(G01),在CNC編輯中的進(jìn)給速度將被發(fā)送到輔助控制器中去。一個(gè)輔助軸G00運(yùn)動(dòng)將會(huì)以它的最大進(jìn)給速度移動(dòng)。當(dāng)CNC控制器等待輔助軸運(yùn)動(dòng)完成的時(shí)候,屏幕的底端將顯示“CFIN”。RS-232端口有關(guān)輔助軸的傳輸錯(cuò)誤將會(huì)顯示無線中斷。Reset鍵將中斷任何“掛斷”輔助軸傳輸。Emergency Stop或者開啟Single Block是停止輔助軸的唯一方法,F(xiàn)eed Hold或Reset將不停止軸。如果用戶加入一個(gè)輔助軸,設(shè)置38,然后輔助軸將會(huì)被指定為C。如果加入倆個(gè)輔助器,它們將被指定為C和U等。參數(shù)當(dāng)訪問一個(gè)輔助器的時(shí)候,HAAS單個(gè)伺服控制器必須根據(jù)下表設(shè)置的參數(shù)21。在CNC中命名: 參數(shù)21: 所選擇: C 6 Z U 1 U V 2 V W 3 W多個(gè)輔助軸必須通過第二RS-232端口采集,這和操作手冊中的輔助軸的描述是一樣的。用進(jìn)給手輪可以從CNC前面板進(jìn)給輔助軸。沒有對這些軸做共建設(shè)置,所以所有的指令在機(jī)床的操作坐標(biāo)系統(tǒng)中。但是如果一個(gè)替代零點(diǎn)位置已經(jīng)輸入到HAAS伺服控制中去,那個(gè)位置將被定位零點(diǎn)。當(dāng)CNC通電時(shí)。輔助軸將會(huì)初始化,零點(diǎn)將會(huì)被設(shè)置在但控制器中的值代替。設(shè)置一個(gè)替代零點(diǎn),用戶必須進(jìn)給但控制軸到另一個(gè)新的零點(diǎn)位置,然后按下控制軸但控制中的Clear按鈕。輔助軸傳輸通常有7個(gè)數(shù)據(jù)位,偶同位,倆個(gè)停止位。數(shù)據(jù)傳輸速率是CNC設(shè)置54,將被設(shè)置為4800。CNC設(shè)置50必須設(shè)置為XON/XOFF。在單個(gè)軸控制器中的參數(shù)26,對于4800位每分鐘必須設(shè)置為5.對于XON/XOFF,參數(shù)33必須為1。在單個(gè)軸控制器中的參數(shù)12總是被設(shè)置為3或4阻止循環(huán)包圍。連接CNC和單軸控制器的電纜線必須是DB-25電纜(倆個(gè)末端是陽性導(dǎo)線,來自HassCNC(低)串行端口的1,2,3和7針。使軸失效如果用戶有第四軸旋轉(zhuǎn)的工作臺(tái)或者是5C分度器,或者是第五軸,它將在Setting顯示中生效,并從機(jī)床中移出。當(dāng)控制器開啟的時(shí)候不要連續(xù)或不連接任何電纜。如果它在沒有連接的時(shí)候不能使旋轉(zhuǎn)軸失效,將會(huì)得到警告。Int J Adv Manuf Technol (2001) 17:104113 2001 Springer-Verlag London Limited Fixture Clamping Force Optimisation and its Impact on Workpiece Location Accuracy B. Li and S. N. Melkote George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Georgia, USA Workpiece motion arising from localised elastic deformation at fixtureworkpiece contacts owing to clamping and machining forces is known to affect significantly the workpiece location accuracy and, hence, the final part quality. This effect can be minimised through fixture design optimisation. The clamping force is a critical design variable that can be optimised to reduce the workpiece motion. This paper presents a new method for determining the optimum clamping forces for a multiple clamp fixture subjected to quasi-static machining forces. The method uses elastic contact mechanics models to represent the fixtureworkpiece contact and involves the formulation and solution of a multi-objective constrained optimisation model. The impact of clamping force optimisation on workpiece location accuracy is analysed through examples involving a 32-1 type milling fixture. Keywords: Elastic contact modelling; Fixture clamping force; Optimisation 1. Introduction The location and immobilisation of the workpiece are two critical factors in machining. A machining fixture achieves these functions by locating the workpiece with respect to a suitable datum, and clamping the workpiece against it. The clamping force applied must be large enough to restrain the workpiece motion completely during machining. However, excessive clamping force can induce unacceptable level of workpiece elastic distortion, which will adversely affect its location and, in turn, the part quality. Hence, it is necessary to determine the optimum clamping forces that minimise the workpiece location error due to elastic deformation while satisfying the total restraint requirement. Previous researchers in the fixture analysis and synthesis area have used the finite-element (FE) modelling approach or Correspondence and offprint requests to: Dr S. N. Melkote, George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0405, USA. E-mail: shreyes.melkoteme.gatech.edu the rigid-body modelling approach. Extensive work based on the FE approach has been reported 18. With the exception of DeMeter 8, a common limitation of this approach is the large model size and computation cost. Also, most of the FE- based research has focused on fixture layout optimisation, and clamping force optimisation has not been addressed adequately. Several researchers have addressed fixture clamping force optimisation based on the rigid-body model 911. The rigid body modelling approach treats the fixture-element and work- piece as perfectly rigid solids. DeMeter 12, 13 used screw theory to solve for the minimum clamping force. The overall problem was formulated as a linear program whose objective was to minimise the normal contact force at each locating point by adjusting the clamping force intensity. The effect of the contact friction force was neglected because of its relatively small magnitude compared with the normal contact force. Since this approach is based on the rigid body assumption, it can uniquely only handle 3D fixturing schemes that involve no more than 6 unknowns. Fuh and Nee 14 also presented an iterative search-based method that computes the minimum clamping force by assuming that the friction force directions are known a priori. The primary limitation of the rigid-body analysis is that it is statically indeterminate when more than six contact forces are unknown. As a result, workpiece displace- ments cannot be determined uniquely by this method. This limitation may be overcome by accounting for the elasticity of the fixtureworkpiece system 15. For a relatively rigid workpiece, the location of the workpiece in the machining fixture is strongly influenced by the localised elastic defor- mation at the fixturing points. Hockenberger and DeMeter 16 used empirical contact force-deformation relations (called meta- functions) to solve for the workpiece rigid-body displacements due to clamping and quasi-static machining forces. The same authors also investigated the effect of machining fixture design parameters on workpiece displacement 17. Gui et al 18 reported an elastic contact model for improving workpiece location accuracy through optimisation of the clamping force. However, they did not address methods for calculating the fixtureworkpiece contact stiffness. In addition, the application of their algorithm for a sequence of machining loads rep- resenting a finite tool path was not discussed. Li and Melkote 19 and Hurtado and Melkote 20 used contact mechanics to Fixture Clamping Force Optimisation 105 solve for the contact forces and workpiece displacement pro- duced by the elastic deformation at the fixturing points owing to clamping loads. They also developed methods for optimising the fixture layout 21 and clamping force using this method 22. However, clamping force optimisation for a multiclamp system and its impact on workpiece accuracy were not covered in these papers. This paper presents a new algorithm based on the contact elasticity method for determining the optimum clamping forces for a multiclamp fixtureworkpiece system subjected to quasi- static loads. The method seeks to minimise the impact of workpiece motion due to clamping and machining loads on the part location accuracy by systematically optimising the clamping forces. A contact mechanics model is used to deter- mine a set of contact forces and displacements, which are then used for the clamping force optimisation. The complete prob- lem is formulated and solved as a multi-objective constrained optimisation problem. The impact of clamping force optimis- ation on workpiece location accuracy is analysed via two examples involving a 32-1 fixture layout for a milling oper- ation. 2. FixtureWorkpiece Contact Modelling 2.1 Modelling Assumptions The machining fixture consists of L locators and C clamps with spherical tips. The workpiece and fixture materials are linearly elastic in the contact region, and perfectly rigid else- where. The workpiecefixture system is subjected to quasi- static loads due to clamping and machining. The clamping force is assumed to be constant during machining. This assumption is valid when hydraulic or pneumatic clamps are used. In reality, the elasticity of the fixtureworkpiece contact region is distributed. However, in this model development, lumped contact stiffness is assumed (see Fig. 1). Therefore, the contact force and localised deformation at the ith fixturing point can be related as follows: F i j = k i j d i j (1) where k i j (j = x,y,z) denotes the contact stiffness in the tangential and normal directions of the local x i ,y i ,z i coordinate frame, d i j Fig. 1. A lumped-spring fixtureworkpiece contact model. x i , y i , z i , denote the local coordinate frame at the ith contact. (j = x,y,z) are the corresponding localised elastic deformations along the x i ,y i , and z i axes, respectively, F i j (j = x,j,z) represents the local contact force components with F i x and F i y being the local x i and y i components of the tangential force, and F i z the normal force. 2.2 WorkpieceFixture Contact Stiffness Model The lumped compliance at a spherical tip locator/clamp and workpiece contact is not linear because the contact radius varies nonlinearly with the normal force 23. The contact deformation due to the normal force P i acting between a spherical tipped fixture element of radius R i and a planar workpiece surface can be obtained from the closed-form Hertz- ian solution to the problem of a sphere indenting an elastic half-space. For this problem, the normal deformation D i n is given as 23, p. 93: D i n = S 9(P i ) 2 16R i (E*) 2 D 1/3 (2) where 1 E* = 1 - n 2 w E w + 1 - n 2 f E f E w and E f are Youngs moduli for the workpiece and fixture materials, respectively, and n w and n f are Poisson ratios for the workpiece and fixture materials, respectively. The tangential deformation D i t (= D i tx or D i ty in the local x i and y i tangential directions, respectively) due to a tangential force Q i (= Q i x or Q i y ) has the following form 23, p. 217: D ti t = Q i 8a i S 2 - n f G f + 2 - n w G w D (3) where a i = S 3P i R i 4 S 1 - n f E f + 1 - n w E w DD 1/3 and G w and G f are shear moduli for the workpiece and fixture materials, respectively. A reasonable linear approximation of the contact stiffness can be obtained from a least-squares fit to Eq. (2). This yields the following linearised contact stiffness values: k i z = 8.82 S 16R i (E*) 2 9 D 1/3 (4) k i x = k i y = 4 E* S 2 - n j G f + 2 - n w G w D - 1 k i z (5) In deriving the above linear approximation, the normal force P i was assumed to vary from 0 to 1000 N, and the correspond- ing R 2 value of the least-squares fit was found to be 0.94. 3. Clamping Force Optimisation The goal is to determine the set of optimal clamping forces that will minimise the workpiece rigid-body motion due to 106 B. Li and S. N. Melkote localised elastic deformation induced by the clamping and machining loads, while maintaining the fixtureworkpiece sys- tem in quasi-static equilibrium during machining. Minimisation of the workpiece motion will, in turn, reduce the location error. This goal is achieved by formulating the problem as a multi- objective constrained optimisation problem, as described next. 3.1 Objective Function Formulation Since the workpiece rotation due to fixturing forces is often quite small 17 the workpiece location error is assumed to be determined largely by its rigid-body translation Dd w = DX w DY w DZ w T , where DX w , DY w , and DZ w are the three orthogonal components of Dd w along the X g , Y g , and Z g axes (see Fig. 2). The workpiece location error due to the fixturing forces can then be calculated in terms of the L 2 norm of the rigid-body displacement as follows: iDd w i = (DX w ) 2 + (DY w ) 2 + (DZ w ) 2 ) (6) where ii denotes the L 2 norm of a vector. In particular, the resultant clamping force acting on the workpiece will adversely affect the location error. When mul- tiple clamping forces are applied to the workpiece, the resultant clamping force, P R C = P R X P R y P R Z T , has the form: P R C = R C P C (7) where P C = P L+1 .P L+C T is the clamping force vector, R C = n L+1 .n L+C T is the clamping force direction matrix, n L+i = cosa L+i cosb L+i cosg L+i T is the clamping force direction cosine vector, and a L+i , b L+i , and g L+i are angles made by the clamping force vector at the ith clamping point with respect to the X g , Y g , Z g coordinate axes (i = 1,2,. . .,C). In this paper, the workpiece location error due to contact region deformation is assumed to be influenced only by the normal force acting at the locatorworkpiece contacts. The frictional force at the contacts is relatively small and is neg- lected when analysing the impact of the clamping force on the workpiece location error. Denoting the ratio of the normal contact stiffness, k i z , to the smallest normal stiffness among all locators, k s z ,byj i (i = 1,. . .,L), and assuming that the workpiece rests on N X , N Y , and N Z number of locators oriented in the X g , Fig. 2. Workpiece rigid body translation and rotation. Y g , and Z g directions, the equivalent contact stiffness in the X g , Y g , and Z g directions can be calculated as k s zSO N X i=1 j iD , k s zSO N Y i=1 j iD , and k s zSO N Z i=1 j iD respectively (see Fig. 3). The workpiece rigid-body motion, Dd w , due to clamping action can now be written as: Dd w = 3 P R X k s zSO N X i=1 j iD P R Y k s zSO N Y i=1 j iD P R Z k s z SO N Z i=1 j iD 4 T (8) The workpiece motion, and hence the location error can be reduced by minimising the weighted L 2 norm of the resultant clamping force vector. Therefore, the first objective function can be written as: Minimize iP R C i w = ! 11 P R X O N X i=1 j i 2 2 + 1 P R Y O N Y i=1 j i 2 2 + 1 P R Z O N Z i=1 j i 2 2 2 (9) Note that the weighting factors are proportional to the equival- ent contact stiffnesses in the X g , Y g , and Z g directions. The components of P R C are uniquely determined by solving the contact elasticity problem using the principle of minimum total complementary energy 15, 23. This ensures that the clamping forces and the corresponding locator reactions are “true” solutions to the contact problem and yield “true” rigid- body displacements, and that the workpiece is kept in static equilibrium by the clamping forces at all times. Therefore, the minimisation of the total complementary energy forms the second objective function for the clamping force optimisation and is given by: Minimise (U* - W*) = 1 2 FO L+C i=1 (F i x ) 2 k i x + O L+C i=1 (F i y ) 2 k i y + O L+C i=1 (F i z ) 2 k i z G (10) = .l T Ql Fig. 3. The basis for the determination of the weighting factor for the L 2 norm calculation. Fixture Clamping Force Optimisation 107 where U* represents the complementary strain energy of the elastically deformed bodies, W* represents the complementary work done by the external force and moments, Q = diag c 1 x c 1 y c 1 z .c L+C x c L+C y c L+C z is the diagonal contact compliance matrix, c i j = (k i j ) - 1 , and l = F 1 x F 1 y F 1 z .F L+C x F L+C y F L+C z T is the vector of all contact forces. 3.2 Friction and Static Equilibrium Constraints The optimisation objective in Eq. (10) is subject to certain constraints and bounds. Foremost among them is the static friction constraint at each contact. Coulombs friction law states that (F i x ) 2 +(F i y ) 2 ) #m i s F i z (m i s is the static friction coefficient). A conservative and linearised version of this nonlinear con- straint can be used and is given by 19: uF i x u + uF i y u #m i s F i z (11) Since quasi-static loads are assumed, the static equilibrium of the workpiece is ensured by including the following force and moment equilibrium equations (in vector form): O F = 0 (12) O M = 0 where the forces and moments consist of the machining forces, workpiece weight and the contact forces in the normal and tangential directions. 3.3 Bounds Since the fixtureworkpiece contact is strictly unilateral, the normal contact force, P i , can only be compressive. This is expressed by the following bound on P i : P i $ 0(i = 1, . . ., L + C) (13) where it is assumed that normal forces directed into the workpiece are positive. In addition, the normal compressive stress at a contact cannot exceed the compressive yield strength (S y ) of the workpiece material. This upper bound is written as: P i # S y A i (i = 1, . . .,L+C) (14) where A i is the contact area at the ith workpiecefixture con- tact. The complete clamping force optimisation model can now be written as: Minimize f = H f 1 f 2 J = H .l T Ql iP R C i w J (15) subject to: (11)(14). 4. Algorithm for Model Solution The multi-objective optimisation problem in Eq. (15) can be solved by the e-constraint method 24. This method identifies one of the objective functions as primary, and converts the other into a constraint. In this work, the minimisation of the complementary energy (f 1 ) is treated as the primary objective function, and the weighted L 2 norm of the resultant clamping force (f 2 ) is treated as a constraint. The choice of f 1 as the primary objective ensures that a unique set of feasible clamping forces is selected. As a result, the workpiecefixture system is driven to a stable state (i.e. the minimum energy state) that also has the smallest weighted L 2 norm for the resultant clamping force. The conversion of f 2 into a constraint involves specifying the weighted L 2 norm to be less than or equal to e, where e is an upper bound on f 2 . To determine a suitable e,itis initially assumed that all clamping forces are unknown. The contact forces at the locating and clamping points are computed by considering only the first objective function (i.e. f 1 ). While this set of contact forces does not necessarily yield the lowest clamping forces, it is a “true” feasible solution to the contact elasticity problem that can completely restrain the workpiece in the fixture. The weighted L 2 norm of these clamping forces is computed and taken as the initial value of e. Therefore, the clamping force optimisation problem in Eq. (15) can be rewritten as: Minimize f 1 = .l T Ql (16) subject to: iP R C i w $e, (11)(14). An algorithm similar to the bisection method for finding roots of an equation is used to determine the lowest upper bound for iP R C i w . By decreasing the upper bound e as much as possible, the minimum weighted L 2 norm of the resultant clamping force is obtained. The number of iterations, K, needed to terminate the search depends on the required prediction accuracy d and ueu, and is given by 25: K = F log 2 S ueu d DG (17) where I denotes the ceiling function. The complete algorithm is given in Fig. 4. 5. Determination of Optimum Clamping Forces During Machining The algorithm presented in the previous section can be used to determine the optimum clamping force for a single load vector applied to the workpiece. However, during milling the magnitude and point of cutting force application changes continuously along the tool path. Therefore, an infinite set of optimum clamping forces corresponding to the infinite set of machining loads will be obtained with the algorithm of Fig. 4. This substantially increases the computational burden and calls for a criterion/procedure for selecting a single set of clamping forces that will be satisfactory and optimum for the entire tool path. A conservative approach to addressing these issues is discussed next. Consider a finite number (say m) of sample points along the tool path yielding m corresponding sets of optimum clamp- ing forces denoted as P 1 opt , P 2 opt ,.,P m opt . At each sampling 108 B. Li and S. N. Melkote Fig. 4. Clamping force optimisation algorithm (used in example 1). point, the following four worst-case machining load vectors are considered: F X max = F max X F 1 Y F 1 Z T F Y max = F 2 X F max Y F 2 Z T F Z max = F 3 X F 3 Y F max Z T (18) F r max = F 4 X F 4 Y F 4 Z T where F max X , F max Y , and F max Z are the maximum X g , Y g , and Z g components of the machining force, the superscripts 1, 2, 3 of F X , F Y , and F Z stand for the other two orthogonal machining force components corresponding to F max X , F max Y , and F max Z , respectively, and iF r max i = max(F X ) 2 +(F Y ) 2 +(F Z ) 2 ). Although the four worst-case machining load vectors will not act on the workpiece at the same instant, they will occur once per cutter revolution. At conventional feedrates, the error introduced by applying the load vectors at the same point would be negligible. Therefore, in this work, the four load vectors are applied at the same location (but not simultaneously) on the workpiece corresponding to the sam- pling instant. The clamping force optimisation algorithm of Fig. 4 is then used to calculate the optimum clamping forces corresponding to each sampling point. The optimum clamping forces have the form: P i jmax = C i 1j C i 2j .C i Cj T (i = 1, . . .,m)(j = x,y,z,r) (19) where P i jmax is the vector of optimum clamping forces for the four worst-case machining load vectors, and C i kj (k = 1,. . .,C) is the force magnitude at each clamp corresponding to the ith sample point and the jth load scenario. After P i jmax is computed for each load application point, a single set of “optimum” clamping forces must be selected from all of the optimum clamping forces found for each clamp from all the sample points and loading conditions. This is done by sorting the optimum clamping force magnitudes at a clamping point for all load scenarios and sample points and selecting the maximum value, C max k , as given in Eq. (20): C max k # C i kj (k =
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