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沈陽(yáng)理工大學(xué)學(xué)士學(xué)位論文
附錄二 ：中文翻譯
通過(guò)夾具布局設(shè)計(jì)和夾緊力的優(yōu)化控制變形
摘 要
工件變形必須控制在數(shù)值控制機(jī)械加工過(guò)程之中。夾具布局和夾緊力是影響加工變形程度和分布的兩個(gè)主要方面。在本文提出了一種多目標(biāo)模型的建立，以減低變形的程度和增加均勻變形分布。有限元方法應(yīng)用于分析變形。遺傳算法發(fā)展是為了解決優(yōu)化模型。最后舉了一個(gè)例子說(shuō)明，一個(gè)令人滿意的結(jié)果被求得， 這是遠(yuǎn)優(yōu)于經(jīng)驗(yàn)之一的。多目標(biāo)模型可以減少加工變形有效地改善分布狀況。
關(guān)鍵詞：夾具布局；夾緊力； 遺傳算法；有限元方法
1 引言
夾具設(shè)計(jì)在制造工程中是一項(xiàng)重要的程序。這對(duì)于加工精度是至關(guān)重要。一個(gè)工件應(yīng)約束在一個(gè)帶有夾具元件，如定位元件，夾緊裝置，以及支撐元件的夾具中加工。定位的位置和夾具的支力，應(yīng)該從戰(zhàn)略的設(shè)計(jì)，并且適當(dāng)?shù)膴A緊力應(yīng)適用。該夾具元件可以放在工件表面的任何可選位置。夾緊力必須大到足以進(jìn)行工件加工。通常情況下，它在很大程度上取決于設(shè)計(jì)師的經(jīng)驗(yàn)，選擇該夾具元件的方案，并確定夾緊力。因此，不能保證由此產(chǎn)生的解決方案是某一特定的工件的最優(yōu)或接近最優(yōu)的方案。因此，夾具布局和夾緊力優(yōu)化成為夾具設(shè)計(jì)方案的兩個(gè)主要方面。 定位和夾緊裝置和夾緊力的值都應(yīng)適當(dāng)?shù)倪x擇和計(jì)算，使由于夾緊力和切削力產(chǎn)生的工件變形盡量減少和非正式化。
夾具設(shè)計(jì)的目的是要找到夾具元件關(guān)于工件和最優(yōu)的夾緊力的一個(gè)最優(yōu)布局或方案。在這篇論文里， 多目標(biāo)優(yōu)化方法是代表了夾具布局設(shè)計(jì)和夾緊力的優(yōu)化的方法。 這個(gè)觀點(diǎn)是具有兩面性的。一，是盡量減少加工表面最大的彈性變形； 另一個(gè)是盡量均勻變形。 ANSYS軟件包是用來(lái)計(jì)算工件由于夾緊力和切削力下產(chǎn)生的變形。遺傳算法是MATLAB的發(fā)達(dá)且直接的搜索工具箱，并且被應(yīng)用于解決優(yōu)化問(wèn)題。最后還給出了一個(gè)案例的研究，以闡述對(duì)所提算法的應(yīng)用。
2 文獻(xiàn)回顧
隨著優(yōu)化方法在工業(yè)中的廣泛運(yùn)用，近幾年夾具設(shè)計(jì)優(yōu)化已獲得了更多的利益。夾具設(shè)計(jì)優(yōu)化包括夾具布局優(yōu)化和夾緊力優(yōu)化。King 和 Hutter提出了一種使用剛體模型的夾具-工件系統(tǒng)來(lái)優(yōu)化夾具布局設(shè)計(jì)的方法。DeMeter也用了一個(gè)剛性體模型，為最優(yōu)夾具布局和最低的夾緊力進(jìn)行分析和綜合。他提出了基于支持布局優(yōu)化的程序與計(jì)算質(zhì)量的有限元計(jì)算法。李和melkote用了一個(gè)非線性編程方法和一個(gè)聯(lián)絡(luò)彈性模型解決布局優(yōu)化問(wèn)題。兩年后， 他們提交了一份確定關(guān)于多鉗夾具受到準(zhǔn)靜態(tài)加工力的夾緊力優(yōu)化的方法。他們還提出了一關(guān)于夾具布置和夾緊力的最優(yōu)的合成方法，認(rèn)為工件在加工過(guò)程中處于動(dòng)態(tài)。相結(jié)合的夾具布局和夾緊力優(yōu)化程序被提出，其他研究人員用有限元法進(jìn)行夾具設(shè)計(jì)與分析。蔡等對(duì)menassa和devries包括合成的夾具布局的金屬板材大會(huì)的理論進(jìn)行了拓展。秦等人建立了一個(gè)與夾具和工件之間彈性接觸的模型作為參考物來(lái)優(yōu)化夾緊力與，以盡量減少工件的位置誤差。Deng和melkote 提交了一份基于模型的框架以確定所需的最低限度夾緊力，保證了被夾緊工件在加工的動(dòng)態(tài)穩(wěn)定。
大部分的上述研究使用的是非線性規(guī)劃方法，很少有全面的或近全面的最優(yōu)解決辦法。所有的夾具布局優(yōu)化程序必須從一個(gè)可行布局開(kāi)始。此外，還得到了對(duì)這些模型都非常敏感的初步可行夾具布局的解決方案。夾具優(yōu)化設(shè)計(jì)的問(wèn)題是非線性的，因?yàn)槟繕?biāo)的功能和設(shè)計(jì)變量之間沒(méi)有直接分析的關(guān)系。例如加工表面誤差和夾具的參數(shù)之間（定位、夾具和夾緊力）。
以前的研究表明，遺傳算法（ GA ）在解決這類(lèi)優(yōu)化問(wèn)題中是一種有用的技術(shù)。吳和陳用遺傳算法確定最穩(wěn)定的靜態(tài)夾具布局。石川和青山應(yīng)用遺傳算法確定最佳夾緊條件彈性工件。vallapuzha在基于優(yōu)化夾具布局的遺傳算法中使用空間坐標(biāo)編碼。他們還提出了針對(duì)主要競(jìng)爭(zhēng)夾具優(yōu)化方法相對(duì)有效性的廣泛調(diào)查的方法和結(jié)果。這表明連續(xù)遺傳算法取得最優(yōu)質(zhì)的解決方案。krishnakumar和melkote 發(fā)展了一個(gè)夾具布局優(yōu)化技術(shù)，用遺傳算法找到夾具布局，盡量減少由于在整個(gè)刀具路徑的夾緊和切削力造成的加工表面的變形。定位器和夾具位置被節(jié)點(diǎn)號(hào)碼所指定。krishnakumar等人還提出了一種迭代算法，盡量減少工件在整個(gè)切削過(guò)程之中由不同的夾具布局和夾緊力造成的彈性變形。Lai等人建成了一個(gè)分析模型，認(rèn)為定位和夾緊裝置為同一夾具布局的要素靈活的一部分。Hamedi 討論了混合學(xué)習(xí)系統(tǒng)用來(lái)非線性有限元分析與支持相結(jié)合的人工神經(jīng)網(wǎng)絡(luò)（ ANN ）和GA。人工神經(jīng)網(wǎng)絡(luò)被用來(lái)計(jì)算工件的最大彈性變形，遺傳算法被用來(lái)確定最佳鎖模力。Kumar建議將迭代算法和人工神經(jīng)網(wǎng)絡(luò)結(jié)合起來(lái)發(fā)展夾具設(shè)計(jì)系統(tǒng)。Kaya用迭代算法和有限元分析，在二維工件中找到最佳定位和夾緊位置，并且把碎片的效果考慮進(jìn)去。周等人。提出了基于遺傳算法的方法，認(rèn)為優(yōu)化夾具布局和夾緊力的同時(shí)，一些研究沒(méi)有考慮為整個(gè)刀具路徑優(yōu)化布局。一些研究使用節(jié)點(diǎn)數(shù)目作為設(shè)計(jì)參數(shù)。一些研究解決夾具布局或夾緊力優(yōu)化方法，但不能兩者都同時(shí)進(jìn)行。 有幾項(xiàng)研究摩擦和碎片考慮進(jìn)去了。
碎片的移動(dòng)和摩擦接觸的影響對(duì)于實(shí)現(xiàn)更為現(xiàn)實(shí)和準(zhǔn)確的工件夾具布局校核分析來(lái)說(shuō)是不可忽視的。因此將碎片的去除效果和摩擦考慮在內(nèi)以實(shí)現(xiàn)更好的加工精度是必須的。
在這篇論文中，將摩擦和碎片移除考慮在內(nèi)，以達(dá)到加工表面在夾緊和切削力下最低程度的變形。一多目標(biāo)優(yōu)化模型被建立了。一個(gè)優(yōu)化的過(guò)程中基于GA和有限元法提交找到最佳的布局和夾具夾緊力。最后，結(jié)果多目標(biāo)優(yōu)化模型對(duì)低剛度工件而言是比較單一的目標(biāo)優(yōu)化方法、經(jīng)驗(yàn)和方法。
3 多目標(biāo)優(yōu)化模型夾具設(shè)計(jì)
一個(gè)可行的夾具布局必須滿足三限制。首先，定位和夾緊裝置不能將拉伸勢(shì)力應(yīng)用到工件；第二，庫(kù)侖摩擦約束必須施加在所有夾具-工件的接觸點(diǎn)。夾具元件-工件接觸點(diǎn)的位置必須在候選位置。為一個(gè)問(wèn)題涉及夾具元件-工件接觸和加工負(fù)荷步驟，優(yōu)化問(wèn)題可以在數(shù)學(xué)上仿照如下：
這里的△表示加工區(qū)域在加工當(dāng)中j次步驟的最高彈性變形。
其中
是△的平均值；
是正常力在i次的接觸點(diǎn)；
μ是靜態(tài)摩擦系數(shù)；
fhi是切向力在i次的接觸點(diǎn)；
pos(i)是i次的接觸點(diǎn)；
是可選區(qū)域的i次接觸點(diǎn)；
整體過(guò)程如圖1所示，一要設(shè)計(jì)一套可行的夾具布局和優(yōu)化的夾緊力。最大切削力在切削模型和切削力發(fā)送到有限元分析模型中被計(jì)算出來(lái)。優(yōu)化程序造成一些夾具布局和夾緊力，同時(shí)也是被發(fā)送到有限元模型中。在有限元分析座內(nèi)，加工變形下，切削力和夾緊力的計(jì)算方法采用有限元方法。根據(jù)某夾具布局和變形，然后發(fā)送給優(yōu)化程序，以搜索為一優(yōu)化夾具方案。
圖1 夾具布局和夾緊力優(yōu)化過(guò)程
4 夾具布局設(shè)計(jì)和夾緊力的優(yōu)化
4.1 遺傳算法
遺傳算法（ GA ）是基于生物再生產(chǎn)過(guò)程的強(qiáng)勁，隨機(jī)和啟發(fā)式的優(yōu)化方法?；舅悸繁澈蟮倪z傳算法是模擬“生存的優(yōu)勝劣汰“的現(xiàn)象。每一個(gè)人口中的候選個(gè)體指派一個(gè)健身的價(jià)值，通過(guò)一個(gè)功能的調(diào)整，以適應(yīng)特定的問(wèn)題。遺傳算法，然后進(jìn)行復(fù)制，交叉和變異過(guò)程消除不適宜的個(gè)人和人口的演進(jìn)給下一代。人口足夠數(shù)目的演變基于這些經(jīng)營(yíng)者引起全球健身人口的增加和優(yōu)勝個(gè)體代表全最好的方法。
遺傳算法程序在優(yōu)化夾具設(shè)計(jì)時(shí)需夾具布局和夾緊力作為設(shè)計(jì)變量，以生成字符串代表不同的布置。字符串相比染色體的自然演變，以及字符串，它和遺傳算法尋找最優(yōu)，是映射到最優(yōu)的夾具設(shè)計(jì)計(jì)劃。在這項(xiàng)研究里，遺傳算法和MATLAB的直接搜索工具箱是被運(yùn)用的。
收斂性遺傳算法是被人口大小、交叉的概率和概率突變所控制的 。只有當(dāng)在一個(gè)人口中功能最薄弱功能的最優(yōu)值沒(méi)有變化時(shí)，nchg達(dá)到一個(gè)預(yù)先定義的價(jià)值ncmax ，或有多少幾代氮，到達(dá)演化的指定數(shù)量上限nmax， 沒(méi)有遺傳算法停止。有五個(gè)主要因素，遺傳算法，編碼，健身功能，遺傳算子，控制參數(shù)和制約因素。 在這篇論文中，這些因素都被選出如表1所列。
表1 遺傳算法參數(shù)的選擇
由于遺傳算法可能產(chǎn)生夾具設(shè)計(jì)字符串，當(dāng)受到加工負(fù)荷時(shí)不完全限制夾具。這些解決方案被認(rèn)為是不可行的，且被罰的方法是用來(lái)驅(qū)動(dòng)遺傳算法，以實(shí)現(xiàn)一個(gè)可行的解決辦法。1夾具設(shè)計(jì)的計(jì)劃被認(rèn)為是不可行的或無(wú)約束，如果反應(yīng)在定位是否定的。在換句話說(shuō)，它不符合方程（2）和（3）的限制。罰的方法基本上包含指定計(jì)劃的高目標(biāo)函數(shù)值時(shí)不可行的。因此，驅(qū)動(dòng)它在連續(xù)迭代算法中的可行區(qū)域。對(duì)于約束（4），當(dāng)遺傳算子產(chǎn)生新個(gè)體或此個(gè)體已經(jīng)產(chǎn)生，檢查它們是否符合條件是必要的。真正的候選區(qū)域是那些不包括無(wú)效的區(qū)域。在為了簡(jiǎn)化檢查，多邊形是用來(lái)代表候選區(qū)域和無(wú)效區(qū)域的。多邊形的頂點(diǎn)是用于檢查。“inpolygon ”在MATLAB的功能可被用來(lái)幫助檢查。
4.2 有限元分析
ANSYS軟件包是用于在這方面的研究有限元分析計(jì)算。有限元模型是一個(gè)考慮摩擦效應(yīng)的半彈性接觸模型，如果材料是假定線彈性。如圖2所示，每個(gè)位置或支持，是代表三個(gè)正交彈簧提供的制約。
圖2 考慮到摩擦的半彈性接觸模型
在x ， y和z 方向和每個(gè)夾具類(lèi)似，但定位夾緊力在正常的方向。彈力在自然的方向即所謂自然彈力，其余兩個(gè)彈力即為所謂的切向彈力。接觸彈簧剛度可以根據(jù)向赫茲接觸理論計(jì)算如下：
隨著夾緊力和夾具布局的變化，接觸剛度也不同，一個(gè)合理的線性逼近的接觸剛度可以從適合上述方程的最小二乘法得到。連續(xù)插值，這是用來(lái)申請(qǐng)工件的有限元分析模型的邊界條件。在圖3中說(shuō)明了夾具元件的位置，顯示為黑色界線。每個(gè)元素的位置被其它四或六最接近的鄰近節(jié)點(diǎn)所包圍。
圖3 連續(xù)插值
這系列節(jié)點(diǎn)，如黑色正方形所示，是（37，38，31和30 ），（9，10 ，11 ， 18，17號(hào)和16號(hào)）和（ 26，27 ，34 ， 41，40和33 ）。這一系列彈簧單元，與這些每一個(gè)節(jié)點(diǎn)相關(guān)聯(lián)。對(duì)任何一套節(jié)點(diǎn)，彈簧常數(shù)是：
這里，
kij 是彈簧剛度在的j -次節(jié)點(diǎn)周?chē)鷌次夾具元件，
Dij 是i次夾具元件和的J -次節(jié)點(diǎn)周?chē)g的距離，
ki是彈簧剛度在一次夾具元件位置,
ηi 是周?chē)膇次夾具元素周?chē)墓?jié)點(diǎn)數(shù)量
為每個(gè)加工負(fù)荷的一步，適當(dāng)?shù)倪吔鐥l件將適用于工件的有限元模型。在這個(gè)工作里，正常的彈簧約束在這三個(gè)方向（X ， Y ， Z ）的和在切方向切向彈簧約束，（X ， Y ）。夾緊力是適用于正常方向（Z）的夾緊點(diǎn)。整個(gè)刀具路徑是模擬為每個(gè)夾具設(shè)計(jì)計(jì)劃所產(chǎn)生的遺傳算法應(yīng)用的高峰期的X ，Y ，z切削力順序到元曲面，其中刀具通行證。在這工作中，從刀具路徑中歐盟和去除碎片已經(jīng)被考慮進(jìn)去。在機(jī)床改變幾何數(shù)值過(guò)程中，材料被去除，工件的結(jié)構(gòu)剛度也改變。
因此，這是需要考慮碎片移除的影響。有限元分析模型，分析與重點(diǎn)的工具運(yùn)動(dòng)和碎片移除使用的元素死亡技術(shù)。在為了計(jì)算健身價(jià)值，對(duì)于給定夾具設(shè)計(jì)方案，位移存儲(chǔ)為每個(gè)負(fù)載的一步。那么，最大位移是選定為夾具設(shè)計(jì)計(jì)劃的健身價(jià)值。
遺傳算法的程序和ANSYS之間的互動(dòng)實(shí)施如下。定位和夾具的位置以及夾緊力這些參數(shù)寫(xiě)入到一個(gè)文本文件。那個(gè)輸入批處理文件ANSYS軟件可以讀取這些參數(shù)和計(jì)算加工表面的變形。 因此， 健身價(jià)值觀，在遺傳算法程序，也可以寫(xiě)到當(dāng)前夾具設(shè)計(jì)計(jì)劃的一個(gè)文本文件。
當(dāng)有大量的節(jié)點(diǎn)在一個(gè)有限元模型時(shí)，計(jì)算健身價(jià)值是很昂貴的。因此，有必要加快計(jì)算遺傳算法程序。作為這一代的推移，染色體在人口中取得類(lèi)似情況。在這項(xiàng)工作中，計(jì)算健身價(jià)值和染色體存放在一個(gè)SQL Server數(shù)據(jù)庫(kù)。遺傳算法的程序，如果目前的染色體的健身價(jià)值已計(jì)算之前，先檢查；如果不，夾具設(shè)計(jì)計(jì)劃發(fā)送到ANSYS，否則健身價(jià)值觀是直接從數(shù)據(jù)庫(kù)中取出。嚙合的工件有限元模型，在每一個(gè)計(jì)算時(shí)間保持不變。每計(jì)算模型間的差異是邊界條件，因此，網(wǎng)狀工件的有限元模型可以用來(lái)反復(fù)“恢復(fù)”ANSYS 命令。
5 案例研究
一個(gè)關(guān)于低剛度工件的銑削夾具設(shè)計(jì)優(yōu)化問(wèn)題是被顯示在前面的論文中，并在以下各節(jié)加以表述。
5.1 工件的幾何形狀和性能
工件的幾何形狀和特點(diǎn)顯示在圖4中，空心工件的材料是鋁390與泊松比0.3和71Gpa的楊氏模量。外廓尺寸152.4mm×127mm*76.2mm.該工件頂端內(nèi)壁的三分之一是經(jīng)銑削及其刀具軌跡，如圖4 所示。夾具元件中應(yīng)用到的材料泊松比0.3和楊氏模量的220的合金鋼。
圖4 空心工件
5.2 模擬和加工的運(yùn)作
舉例將工件進(jìn)行周邊銑削，加工參數(shù)在表2中給出。基于這些參數(shù)，切削力的最高值被作為工件內(nèi)壁受到的表面載荷而被計(jì)算和應(yīng)用，當(dāng)工件處于330.94 n（切）、398.11 N （下徑向）和22.84 N （下軸） 的切削位置時(shí)。整個(gè)刀具路徑被26個(gè)工步所分開(kāi)，切削力的方向被刀具位置所確定
表2加工參數(shù)和條件
。
5.3 夾具設(shè)計(jì)方案
夾具在加工過(guò)程中夾緊工件的規(guī)劃如圖5所示。
圖5 定位和夾緊裝置的可選區(qū)域
一般來(lái)說(shuō)， 3-2-1定位原則是夾具設(shè)計(jì)中常用的。夾具底板限制三個(gè)自由度，在側(cè)邊控制兩個(gè)自由度。這里，在Y=0mm截面上使用了4個(gè)定點(diǎn)（L1，L2 ， L3和14 ），以定位工件并限制2自由度；并且在Y=127mm的相反面上，兩個(gè)壓板（C1,C2）夾緊工件。在正交面上，需要一個(gè)定位元件限制其余的一個(gè)自由度，這在優(yōu)化模型中是被忽略的。在表3中給出了定位加緊點(diǎn)的坐標(biāo)范圍。
表3 設(shè)計(jì)變量的約束
由于沒(méi)有一個(gè)簡(jiǎn)單的一體化程序確定夾緊力，夾緊力很大部分（6673.2N）在初始階段被假設(shè)為每一個(gè)夾板上作用的力。且從符合例5的最小二乘法，分別由4.43×107 N/m 和5.47×107 N/m得到了正常切向剛度。
5.4 遺傳控制參數(shù)和懲罰函數(shù)
在這個(gè)例子中，用到了下列參數(shù)值：Ps=30, Pc=0.85, Pm=0.01, Nmax=100和Ncmax=20.關(guān)于f1和σ的懲罰函數(shù)是
這里fv可以被F1或σ代表。當(dāng)nchg達(dá)到6時(shí)，交叉和變異的概率將分別改變成0.6和0.1.
5.5 優(yōu)化結(jié)果
連續(xù)優(yōu)化的收斂過(guò)程如圖6所示。且收斂過(guò)程的相應(yīng)功能（1）和（2）如圖7、圖8所示。優(yōu)化設(shè)計(jì)方案在表4中給出。
圖6 夾具布局和夾緊力優(yōu)化程序的收斂性遺傳算法 圖7 第一個(gè)函數(shù)值的收斂
圖8第二個(gè)函數(shù)值的收斂性
表4 多目標(biāo)優(yōu)化模型的結(jié)果 表5 各種夾具設(shè)計(jì)方案結(jié)果進(jìn)行比較，
5.6 結(jié)果的比較
從單一目標(biāo)優(yōu)化和經(jīng)驗(yàn)設(shè)計(jì)中得到的夾具設(shè)計(jì)的設(shè)計(jì)變量和目標(biāo)函數(shù)值，如表5所示。單一目標(biāo)優(yōu)化的結(jié)果，在論文中引做比較。在例子中，與經(jīng)驗(yàn)設(shè)計(jì)相比較，單一目標(biāo)優(yōu)化方法有其優(yōu)勢(shì)。最高變形減少了57.5 ％，均勻變形增強(qiáng)了60.4 ％。最高夾緊力的值也減少了49.4 ％ 。從多目標(biāo)優(yōu)化方法和單目標(biāo)優(yōu)化方法的比較中可以得出什么呢？最大變形減少了50.2％ ，均勻變形量增加了52.9 ％，最高夾緊力的值減少了69.6 ％ 。加工表面沿刀具軌跡的變形分布如圖9所示。很明顯，在三種方法中，多目標(biāo)優(yōu)化方法產(chǎn)生的變形分布最均勻。
與結(jié)果比較，我們確信運(yùn)用最佳定位點(diǎn)分布和最優(yōu)夾緊力來(lái)減少工件的變形。圖10示出了一實(shí)例夾具的裝配。
圖9沿刀具軌跡的變形分布
圖10 夾具配置實(shí)例
6 結(jié)論
本文介紹了基于GA和有限元的夾具布局設(shè)計(jì)和夾緊力的優(yōu)化程序設(shè)計(jì)。優(yōu)化程序是多目標(biāo)的：最大限度地減少加工表面的最高變形和最大限度地均勻變形。ANSYS軟件包已經(jīng)被用于
健身價(jià)值的有限元計(jì)算。對(duì)于夾具設(shè)計(jì)優(yōu)化的問(wèn)題，GA和有限元分析的結(jié)合被證明是一種很有用的方法。
在這項(xiàng)研究中，摩擦的影響和碎片移動(dòng)都被考慮到了。為了減少計(jì)算的時(shí)間，建立了一個(gè)染色體的健身數(shù)值的數(shù)據(jù)庫(kù)，且網(wǎng)狀工件的有限元模型是優(yōu)化過(guò)程中多次使用的。
傳統(tǒng)的夾具設(shè)計(jì)方法是單一目標(biāo)優(yōu)化方法或經(jīng)驗(yàn)。此研究結(jié)果表明，多目標(biāo)優(yōu)化方法比起其他兩種方法更有效地減少變形和均勻變形。這對(duì)于在數(shù)控加工中控制加工變形是很有意義的。
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ORIGINAL ARTICLE Deformation control through fixture layout design and clamping force optimization Weifang Chen Δ 2 jj; :::; Δ j C12 C12 C12 C12 ; :::; Δ n jj C0C1 t s ; j ? 1; 2; :::; n e1T Subject to m F ni jjC21 ?????????????????? F 2 ti t F 2 hi q e2T F ni C21 0 e3T pos ieT2VieT; i ? 1; 2; :::; p e4T where Δ j refers to the maximum elastic deformation at a machining region in the j-th step of the machining operation, σ? ?????????????????????????????????????? X n j?1 Δj C0Δ C0 C16C17 2 C30 n v u u t Δ is the average of Δ j F ni is the normal force at the i-th contact point μ is the static coefficient of friction F ti ; F hi are the tangential forces at the i-th contact point pos(i) is the i-th contact point V(i) is the candidate region of the i-th contact point. The overall process is illustrated in Fig. 1 to design a feasible fixture layout and to optimize the clamping force. The maximal cutting force is calculated in cutting model and the force is sent to finite element analysis (FEA) model. Optimization procedure creates some fixture layout and clamping force which are sent to the FEA model too. In FEA block, machining deformation under the cutting force and the clamping force is calculated using finite element method under a certain fixture layout, and the deformation is then sent to optimization procedure to search for an optimal fixture scheme. 4 Fixture layout design and clamping force optimization 4.1 A genetic algorithm Genetic algorithms (GA) are robust, stochastic and heuristic optimization methods based on biological reproduction processes. The basic idea behind GA is to simulate “survival of the fittest” phenomena. Each individual candidate in the population is assigned a fitness value through a fitness function tailored to the specific problem. The GA then conducts reproduction, crossover and mutation processes to eliminate unfit individuals and the population evolves to the next generation. Sufficient number of evolutions of the population based on these operators lead to an increase in the global fitness of the population and the fittest individual represents the best solution. The GA procedure to optimize fixture design takes fixture layout and clamping force as design variables to generate strings which represent different layouts. The strings are compared to the chromosomes of natural evolution, and the string, which GA find optimal, is mapped to the optimal fixture design scheme. In this study, the genetic algorithm and direct search toolbox of MATLAB are employed. The convergence of GA is controlled by the population size (P s ), the probability of crossover (P c )andthe probability of mutations (P m ). Only when no change in the best value of fitness function in a population, N chg , reaches a pre-defined value NC max , or the number of generations, N, reaches the specified maximum number of evolutions, N max ., did the GA stop. There are five main factors in GA, encoding, fitness function, genetic operators, control parameters and con- straints. In this paper, these factors are selected as what is listed in Table 1. Since GA is likely to generate fixture design strings that do not completely restrain the fixture when subjected to machining loads. These solutions are considered infeasible and the penalty method is used to drive the GA to a feasible solution. A fixture design scheme is considered infeasible or unconstrained if the reactions at the locators are negative, in other words, it does not satisfy the constraints in equations (2)and(3). The penalty method essentially involves Machining Process Model FEA Optimization procedure cutting forces fitness Optimization result Fixture layout and clamping force Fig. 1 Fixture layout and clamp- ing force optimization process Table 1 Selection of GA’s parameters Factors Description Encoding Real Scaling Rank Selection Remainder Crossover Intermediate Mutation Uniform Control parameter Self-adapting Int J Adv Manuf Technol assigning a high objective function value to the scheme that is infeasible, thus driving it to the feasible region in successive iterations of GA. For constraint (4), when new individuals are generated by genetic operators or the initial generation is generated, it is necessary to check up whether they satisfy the conditions. The genuine candidate regions are those excluding invalid regions. In order to simplify the checking, polygons are used to represent the candidate regions and invalid regions. The vertex of the polygons are used for the checking. The “inpolygon” function in MATLAB could be used to help the checking. 4.2 Finite element analysis The software package of ANSYS is used for FEA calculations in this study. The finite element model is a semi-elastic contact model considering friction effect, where the materials are assumed linearly elastic. As shown in Fig. 2, each locator or support is represented by three orthogonal springs that provide restrains in the X, Y and Z directions and each clamp is similar to locator but clamping force in normal direction. The spring in normal direction is called normal spring and the other two springs are called tangential springs. The contact spring stiffness can be calculated according to the Herz contact theory [8] as follows k iz ? 16R C3 i E C32 i 9 C16C171 3 f iz 1 3 k iz ? k iy ? 6 E C3 i 2C0v fi G fi t 2C0v wi G wi C16C17 C01 C1 k iz 8 > : e5T where k iz , k ix , k iy are the tangential and normal contact stiffness, 1 R C3 i ? 1 R wi t 1 R fi is the nominal contact radius, 1 E C3 i ? 1C0 V 2 wi E wi t 1C0 V 2 fi E fi is the nominal contact elastic modulus, R wi , R fi are radius of the i-th workpiece and fixture element, E wi , E fi are Young’s moduli for the i-th workpiece and fixture materials, ν wi , ν fi are Poisson ratios for the i-th workpiece and fixture materials, G wi , G fi are shear moduli for the i-th workpiece and fixture materials and f iz is the reaction force at the i-th contact point in the Z direction. Contact stiffness varies with the change of clamping force and fixture layout. A reasonable linear approximation of the contact stiffness can be obtained from a least-squares fit to the above equation. The continuous interpolation, which is used to apply boundary conditions to the workpiece FEA model, is Fig. 2 Semi-elastic contact model taking friction into account Spring position Fixture element position 1234567 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Fig. 3 Continuous interpolation Fig. 4 A hollow workpiece Table 2 Machining parameters and conditions Parameter Description Type of operation End milling Cutter diameter 25.4 mm Number of flutes 4 Cutter RPM 500 Feed 0.1016 mm/tooth Radial depth of cut 2.54 mm Axial depth of cut 25.4 mm Radial rake angle 10 Helix angle 30 Projection length 92.07 mm Int J Adv Manuf Technol illustrated in Fig. 3. Three fixture element locations are shown as black circles. Each element location is surrounded by its four or six nearest neighboring nodes. These sets of nodes, which are illustrated by black squares, are {37, 38, 31 and 30}, {9, 10, 11, 18, 17 and 16} and {26, 27, 34, 41, 40 and 33}. A set of spring elements are attached to each of these nodes. For any set of nodes, the spring constant is k ij ? d ij P k2h i d ik k i e6T where k ij is the spring stiffness at the j-th node surrounding the i-th fixture element, d ij is the distance between the i-th fixture element and the j-th node surrounding it, k i is the spring stiffness at the i-th fixture element location. η i is the number of nodes surrounding the i-th fixture element location. For each machining load step, appropriate boundary conditions have to be applied to the finite element model of the workpiece. In this work, the normal springs are constrained in the three directions (X, Y, Z)andthe tangential springs are constrained in the tangential direc- tions (X, Y). Clamping forces are applied in the normal direction (Z) at the clamp nodes. The entire tool path is simulated for each fixture design scheme generated by the GA by applying the peak X, Y, Z cutting forces sequentially to the element surfaces over which the cutter passes [23]. In this work, chip removal from the tool path is taken into account. The removal of the material during machining alters the geometry, so does the structural stiffness of the workpiece. Thus, it is necessary to consider chip removal affects. The FEA model is analyzed with respect to tool movement and chip removal using the element death technique. In order to calculate the fitness value for a given fixture design scheme, displacements are stored for each load step. Then the maximum displacement is selected as fitness value for this fixture design scheme. The interaction between GA procedure and ANSYS is implemented as follows. Both the positions of locators and clamps, and the clamping force are extracted from real strings. These parameters are written to a text file. The input batch file of ANSYS could read these parameters and calculate the deformation of machined surfaces. Thus the fitness values in GA procedure can also be written to a text file for current fixture design scheme. It is costly to compute the fitness value when there are a largenumberofnodesinanFEMmodel.Thusitisnecessary to speed up the computation for GA procedure. As the generation goes by, chromosomes in the population are getting similar. In this work, calculated fitness values are stored in a SQL Server database with the chromosomes and fitness values. GA procedure first checks if current chromosome’s fitness value has been calculated before, if not, fixture design scheme are sent to ANSYS, otherwise fitness values are directly taken from the database. The meshing of workpiece FEA model keeps same in every calculating time. The difference among every calculating model is the boundary conditions. Thus, the meshed workpiece FEA model could be used repeatedly by the “resume” command in ANSYS. 5 Case study An example of milling fixture design optimization problem for a low rigidity workpiece displayed in previous research papers [16, 18, 22] is presented in the following sections. Fig. 5 Candidate regions for the locators and clamps Table 3 Bound of design variables Minimum Maximum X /mm Z /mm X /mm Z /mm L 1 0 0 76.2 38.1 L 2 76.2 0 152.4 38.1 L 3 0 38.1 76.2 76.2 L 4 76.2 38.1 152.4 76.2 C 1 0 0 76.2 76.2 C 2 76.2 0 152.4 76.2 F 1 /N 0 6673.2 F 2 /N 0 6673.2 Int J Adv Manuf Technol 5.1 Workpiece geometry and properties The geometry and features of the workpiece are shown in Fig. 4. The material of the hollow workpiece is aluminum 390 with a Poisson ration of 0.3 and Young’s modulus of 71 Gpa. The outline dimensions are 152.4 mm×127 mm× 76.2 mm. The one third top inner wall of the workpiece is undergoing an end-milling process and its cutter path is also shown in Fig. 4. The material of the employed fixture elements is alloy steel with a Poisson ration of 0.3 and Young’s modulus of 220 Gpa. 5.2 Simulating and machining operation A peripheral end milling operation is carried out on the example workpiece. The machining parameters of the operation are given in Table 2. Based on these parameters, the maximum values of cutting forces that are calculated and applied as element surface loads on the inner wall of the workpiece at the cutter position are 330.94 N (tangential), 398.11 N (radial) and 22.84 N (axial). The entire tool path is discretized into 26 load steps and cutting force directions are determined by the cutter position. 5.3 Fixture design plan The fixture plan for holding the workpiece in the machining operation is shown in Fig. 5.Generally,the3–2–1 locator principleisusedinfixturedesign.Thebasecontrols3degrees. One side controls two degrees, and another orthogonal side controlsonedegree.Here,itusesfourlocators(L1,L2,L3and L4) on the Y=0 mm face to locate the workpiece controlling two degrees, and two clamps (C1, C2) on the opposite face where Y=127 mm, to hold it. On the orthogonal side, one locator is needed to control the remaining degree, which is neglectedintheoptimalmodel.Thecoordinateboundsforthe locating/clamping regions are given in Table 3. Since there is no simple rule-of-thumb procedure for determining the clamping force, a large value of the clamping force of 6673.2 N was initially assumed to act at each clamp, and the normal and tangential contact stiffness obtained from a least-squares fit to Eq. (5) are 4.43×10 7 N/m and 5.47×10 7 N/m separately. 5.4 Genetic control parameters and penalty function The control parameters of the GA are determined empiri- cally. For this example, the following parameter values are Fig. 6 Convergence of GA for fixture layout and clamping force optimization procedure Fig. 7 Convergence of the first function values Fig. 8 Convergence of the second function values Table 4 Result of the multi-objective optimization model Multi-objective optimization X /mm Z /mm L 1 17.102 30.641 L 2 108.169 25.855 L 3 21.315 56.948 L 4 127.846 60.202 C 1 22.989 62.659 C 2 117.615 25.360 F 1 /N 167.614 F 2 /N 382.435 f 1 /mm 0.006568 σ/mm 0.002683 Int J Adv Manuf Technol used: P s =30, P c =0.85, P m =0.01, N max =100 and N cmax = 20. The penalty function for f 1 and σ is φ f v eT?f v t 50 Here f v can be represented by f 1 or σ. When N chg reaches 6 the probability of crossover and mutation will be change into 0.6 and 0.1 separately. 5.5 Optimization result The convergence behavior for the successive optimization steps is shown in Fig. 6, and the convergence behaviors of corresponding functions (1) and (2) are shown in Fig. 7 and Fig. 8. The optimal design scheme is given in Table 4. 5.6 Comparison of the results The design variables and objective function values of fixture plans obtained from single objective optimization and from that designed by experience are shown in Table 5. The single objective optimization result in the paper [22]is quoted for comparison. The single objective optimization method has its preponderance comparing with that designed by experience in this example case. The maximum deformation has reduced by 57.5%, the uniformity of the deformation has enhanced by 60.4% and the maximum clamping force value has degraded by 49.4%. What could be drawn from the comparison between the multi-objective optimization method and the single objective optimization method is that the maximum deformation has reduced by 50.2%, the uniformity of the deformation has enhanced by 52.9% and the maximum clamping force value has degraded by 69.6%.The deformation distribution of the machined surfaces along cutter path is shown in Fig. 9. Obviously, the deformation from that of multi-objective optimization method distributes most uniformly in the deformations among three methods. With the result of comparison, we are sure to apply the optimal locators distribution and the optimal clamping force to reduce the deformation of workpiece. Figure 10 shows the configuration of a real-case fixture. 6 Conclusions This paper presented a fixture layout design and clamping force optimization procedure based on the GA and FEM. The optimization procedure is multi-objective: minimizing the maximum deformation of the machined surfaces and maximizing the uniformity of the deformation. The ANSYS software package has been used for FEM calculation of fitness values. The combinationof GAand FEM isproven to be a powerful approach for fixture design optimization problems. In this study, both friction effects and chip removal effects are considered. In order to reduce the computation time, a database is established for the chromosomes and fitness values, and the meshed workpiece FEA model is repeatedly used in the optimization process. Table 5 Comparison of the results of various fixture design schemes Experimental optimization Single objective optimization X/mm Z/mm X/mm Z/mm L 1 12.700 12.700 16.720 34.070 L 2 139.7 12.700 145.360 17.070 L 3 12.700 63.500 18.400 57.120 L 4 139.700 63.500 146.260 58.590 C 1 12.700 38.100 5.830 56.010 C 2 139.700 38.100 104.400 22.740 F 1 /N 2482 444.88 F 2 /N 2482 1256.13 f 1 /mm 0.031012 0.013178 σ/mm 0.014377 0.005696 Fig. 9 Distribution of the deformation along cutter path Fig. 10 A real case fixture configuration Int J Adv Manuf Technol Thetraditionalfixturedesignmethodsaresingleobjective optimization method or by experience. The results of this study show that the multi-objective optimization method is more effective in minimizing the deformation and uniform- ing the deformation than other two methods. It is meaningful for machining deformation control in NC machining. References 1. 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