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夾具夾緊力的優(yōu)化及對工件定位精度的影響
B.Li 和 S.N.Mellkote
布什伍德拉夫機械工程學院，佐治亞理工學院，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產(chǎn)生局部彈性變形，使工件尺寸發(fā)生變化，進而影響工件的最終加工質(zhì)量。這種效應可通過最小化夾具設計優(yōu)化，夾緊力是一個重要的設計變量，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式銑夾具的例子進行了分析。
關鍵詞：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關鍵因素。要實現(xiàn)夾具的這些功能，需將工件定位到一個合適的基準上并夾緊，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。然而，過度的夾緊力可誘導工件產(chǎn)生更大的彈性變形 ，這會影響它的位置精度，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，同時滿足加工的要求。在夾具分析和綜合領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被報道[參考文獻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算成本。同時，多數(shù)的有限元基礎研究人員一直重點關注的夾具布局優(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]對夾緊力進行了優(yōu)化，剛型模型幾乎被近似為一個規(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線性規(guī)劃，其目的是盡量減少在每個定位點調(diào)整夾緊力強度的法線接觸力。接觸摩擦力的影響被忽視，因為它較法線接觸力相對較小，由于這種方法是基于剛體假設，獨特的三維夾具可以處理超過6個自由度的裝夾，復和倪[14]也提出迭代搜索方法，通過假設已知摩擦力的方向來推導計算最小夾緊力，該剛體分析的主要限制因素是當出現(xiàn)六個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算夾具——工件系統(tǒng)[15]的彈性來克服，對于一個相對嚴格的工件，該夾具在機械加工工件的位置會受夾具點的局部彈性變形的強烈影響。Hockenberger和得墨忒耳[16]使用經(jīng)驗的接觸力變形的關系（稱為元功能），解決由于夾緊和準靜態(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計參數(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告做了改善，然而，他們沒有處理計算夾具與工件的接觸剛度的方法，此外，其算法的應用沒有討論機械加工刀具路徑負載有限序列。李和Melkote [19]和烏爾塔多和Melkote [20]用接觸力學解決由于在加載夾具夾緊點彈性變形產(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)受到準靜態(tài)加載的最佳夾緊力為基礎的彈性方法。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，這個問題被作為多目標約束優(yōu)化問題提出和解決。通過兩個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
1．1 模型假設
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靜態(tài)負載。夾緊力可假定為在加工過程中保持不變，這個假設是有效的，在對液壓或氣動夾具使用。在實際中，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?shù)刈幼鴺讼登芯€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
接觸處的坐標系
（j=x，y，z）是對應沿著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。
1．2 工件——夾具的接觸剛度模型
集中遵守一個球形尖端定位，夾具和工件的接觸并不是線性的，因為接觸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一個球體彈性半空間的問題。對于這個問題， 是法線的變形，在[文獻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23第217頁]
（3）
其中、 分別是工件和夾具剪切模量
一個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的R2值認定是0.94。
2．夾緊力優(yōu)化
我們的目標是確定最優(yōu)夾緊力，將盡量減少由于工件剛體運動過程中，局部的夾緊和加工負荷引起的彈性變形，同時保持在準靜態(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。實現(xiàn)這個目標是通過制定一個多目標約束優(yōu)化問題的問題，如下描述。
2.1 目標函數(shù)配方
工件旋轉(zhuǎn)，由于部隊輪換往往是相當小[17]的工件定位誤差假設為確定其剛體翻譯基本上，其中 、、和 是 沿，和三個正交組件（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)計算如下：
（6）
其中表示一個向量二級標準。
但是作用在工件的夾緊力會影響定位誤差。當多個夾緊力作用于工件，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個夾緊點夾緊力在、和方向上的向量角度（i=1、2、3...,C）。
在這個文件中，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，并在進行分析時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=1，2…L）和最小的所有定位器正常剛度相乘，并假設工件、、取決于、、的方向，各自的等效接觸剛度可有下式計算得出（見圖3），工件剛體運動，歸于夾緊行動現(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，第一個目標函數(shù)可以寫為：
最小化 （9）
要注意，加權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參考文獻[15，23]的原則求解彈性力學接觸問題得出A的組成部分是唯一確定的，這保證了夾緊力和相應的定位反應是“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，通過夾緊力的隨時調(diào)整。因此，總能量最小化的形式為補充的夾緊力優(yōu)化的第二個目標函數(shù)，并給出：
最小化 （10）
其中代表機構的彈性變形應變能互補，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 加權系數(shù)計算確定的基礎
內(nèi)蒙古科技大學本科生畢業(yè)設計（外文翻譯）
2.2 摩擦和靜態(tài)平衡約束
在（10）式優(yōu)化的目標受到一定的限制和約束，他們中最重要的是在每個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)）,這方面的一個非線性約束和線性化版本可以使用，并且[19]有：
（11）
假設準靜態(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）
它假設在工件上的法線力是確定的，此外，在一個法線的接觸壓力不能超過壓工件材料的屈服強度（）。這個約束可寫為：
（i=1，2，…,L+C） (14)
如果是在第i個工件——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）多目標優(yōu)化問題可以通過求解約束[24]。這種方法將確定的目標作為首要職能之一，并將其轉(zhuǎn)換成一個約束對。該補充（）的主要目的是處理功能，并由此得到夾緊力（）作為約束的加權范數(shù)最小化。對為主要目標的選擇，確保選中一套獨特可行的夾緊力，因此，工件——夾具系統(tǒng)驅(qū)動到一個穩(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范數(shù)。 的約束轉(zhuǎn)換涉及到一個指定的加權范數(shù)小于或等于，其中是 的約束，假設最初所有夾緊力不明確，要確定一個合適的。在定位和夾緊點的接觸力的計算只考慮第一個目標函數(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“真正的”可行的解決彈性力學問題辦法，可完全抑制工件在夾具中的位置。這些夾緊力的加權系數(shù)，通過計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范數(shù)。 迭代次數(shù)K，終止搜索取決于所需的預測精度和，有參考文獻[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的不斷變化而變化。因此，相應的夾緊力和最佳的加工負荷獲得將由圖4算法獲得，這大大增加了計算負擔，并要求為選擇的夾緊力提供標準， 將獲得滿意和適宜的整個刀具軌跡 ，用保守的辦法來解決下面將被討論的問題，考慮一個有限的數(shù)目（例如m）沿相應的刀具路徑設置的產(chǎn)生m個最佳夾緊力，選擇記為， ， …,在每個采樣點，考慮以下四個最壞加工負荷向量：
(18)、和表示在、和方向上的最大值，、和上的數(shù)字1，2，3分別代替對應的和另外兩個正交切削分力，而且有：
雖然4個最壞情況加工負荷向量不會在工件加工的同一時刻出現(xiàn)，但在每次常規(guī)的進給速度中，刀具旋轉(zhuǎn)一次出現(xiàn)一次，負載向量引入的誤差可忽略。因此，在這項工作中，四個載體負載適用于同一位置，（但不是同時）對工件進行的采樣 ，夾緊力的優(yōu)化算法圖4，對應于每個采樣點計算最佳的夾緊力。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情況下的加工負荷載體，(C=1，2，…C)是每個相應的夾具在第i個樣本點和第j負荷情況下力的大小。是計算每個負載點之后的結果，一套簡單的“最佳”夾緊力必須從所有的樣本點和裝載條件里發(fā)現(xiàn)，并在所有的最佳夾緊力中選擇。這是通過在所有負載情況和采樣點排序，并選擇夾緊點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，驗證這些力，以確保工件夾具系統(tǒng)的靜態(tài)平衡。否則，會出現(xiàn)更多采樣點和重復上述程序。在這種方式中，可為整個刀具路徑確定“最佳”夾緊力 ，圖5總結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)生在每個工件夾具接觸處，使工件在夾具上移位和旋轉(zhuǎn)。隨后，準靜態(tài)加工負荷應用造成工件在夾具的移位。工件剛體運動的定義是由它在、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)生于在每個夾緊處的局部變形，假設為相對于工件的質(zhì)量中心的第i個位置矢量定位點，坐標變換定理可以用來表達在工件的位移，以及工件自轉(zhuǎn)如下: (21)
其中表示旋轉(zhuǎn)矩陣,描述當?shù)卦诘趇幀相聯(lián)系的全球坐標系和是一個旋轉(zhuǎn)矩陣確定工件相對于全球的坐標系的定位坐標系。假設夾具夾緊工件旋轉(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，是夾緊和加工導致的工件剛體運動矢量。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裝夾點接觸力可能與的關系如下：
（24）
其中是在第i個接觸點由于夾緊和加工負荷造成的變形，意味著凈壓縮變形，而負數(shù)則代表拉伸變形; 是表示在本地坐標系第i個接觸剛度矩陣，是單位向量. 在這項研究中假定液壓/氣動夾具，根據(jù)對外加工負荷，故在法線方向的夾緊力的強度保持不變，因此，必須對方程（24）的夾緊點進行修改為：
（25）
其中是在第i個夾緊點的夾緊力，讓表示一個對外加工力量和載體的6×1矢量。并結合方程（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，q可通過求解式（26）得到。工件的定位誤差向量， （見圖6），
現(xiàn)在可以計算如下： （27）
其中是考慮工件中心加工點的位置向量，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．適用于工件單點力。
2．應用于工件負載準靜態(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。使用伊利諾伊大學開發(fā)EMSIM程序[參考文獻26] 對加工瞬時銑削力條件進行了計算，如表2給出例（1），應用工件在點（109.2毫米，25.4毫米，34.3毫米）瞬時加工力，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25.4毫米銑槽使用EMSIM進行了數(shù)值模擬，以減少起步（0.0毫米，25.4毫米，34.3毫米）和結束時（127.0毫米，25.4毫米，34.3毫米）四種情況下加工負荷載體，
（見圖8）。模擬計算銑削力數(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個坐標列出了為模擬抽樣調(diào)查點。最佳夾緊力是用前面討論過的排序算法計算每個采樣點和負載載體最后的夾緊力和負載。
7．結果與討論
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（見圖7），由此得到的夾緊力加權范數(shù)有如下形式：.結果表明，最佳夾緊力所述加工條件下有比初步夾緊力強度低得多的加權范數(shù)，最初的夾緊力是通過減少工件的夾具系統(tǒng)補充能量算法獲得。由于夾緊力和負載造成的工件的定位誤差，如表7。結果表明工件旋轉(zhuǎn)小，加工點減少錯誤從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，因為從最初通過互補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序列應用于銑削負載到工件，他應用于工件銑削負載一個序列。最佳的夾緊力，，對應列表6每個樣本點，隨著最后的最佳夾緊力，在每個采樣點的加權范數(shù)和最優(yōu)的初始夾緊力繪圖10，在每個采樣點的加權范數(shù)的，，和繪制。
結果表明，由于每個組成部分是各相應的最大夾緊力，它具有最高的加權范數(shù)。如圖10所示，如果在每個夾緊點最大組成部分是用于確定初步夾緊力，則夾緊力需相應設置，有比相當大的加權范數(shù)。故是一個完整的刀具路徑改進方案。上述模擬結果表明，該方法可用于優(yōu)化夾緊力相對于初始夾緊力的強度，這種做法將減少所造成的夾緊力的加權范數(shù)，因此將提高工件的定位精度。
圖10
8．結論
該文件提出了關于確定多鉗夾具，工件受準靜態(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的夾具與工件系統(tǒng)模型，并尋求盡量減少應用到所造成的工件夾緊力的加權范數(shù)，得出工件的定位誤差。該整體模型，制定一個雙目標約束優(yōu)化問題，使用-約束的方法解決。該算法通過兩個模擬表明，涉及3-2-1型，二夾銑夾具的例子。今后的工作將解決在動態(tài)負載存在夾具與工件在系統(tǒng)的優(yōu)化，其中慣性，剛度和阻尼效應在確定工件夾具系統(tǒng)的響應特性具有重要作用。
9．參考資料：
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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 =