減振器活塞及支承座上雙面活塞孔的加工組合機床畢業(yè)設(shè)計

減振器活塞及支承座上雙面活塞孔的加工組合機床畢業(yè)設(shè)計,減振器,活塞,支承,雙面,加工,組合,機床,畢業(yè)設(shè)計 河南理工大學本科畢業(yè)設(shè)計（英文翻譯） Taguchi 的申請和回應(yīng)表面方法學 為幾何學的錯誤在表面磨的程序 摘要 幾何學的錯誤在于表面的磨程序中主要地被熱的效果和磨的 系統(tǒng).為 將幾何學的錯誤減到最少的堅硬影響,磨的叁數(shù)選擇非常重要。 這張紙呈現(xiàn)了 Taguchi 的一個申請 而且為幾何學的錯誤回應(yīng)表面方法學。 幾何學的錯誤被評估的磨叁數(shù)對～的效果和將幾何學的錯誤減到最少的最適宜磨的情況是堅決的。 一個秒- 次序回應(yīng)模型為幾何學的錯誤被發(fā)展，而且回應(yīng)表面模型的利用率， 表面的粗糙和那材料的限制被評估移動率。 證實實驗在一種最佳的情況被引導(dǎo)而且為觀察被發(fā)展的回應(yīng)準確性表面模型選擇了二種情況。 一. 介紹 輪磨是一個復(fù)雜的機制程序由于許多交談式叁數(shù), 仰賴輪磨產(chǎn)品的類型和需求。 表面的質(zhì)量在表面的輪磨中生產(chǎn)被影響被各種不同的依下列各項被給的叁數(shù) [1]. (i)旋轉(zhuǎn)叁數(shù): 研磨劑，谷粒大小，等級 , 結(jié)構(gòu),縛者，形狀和尺寸等 (ii)細工品叁數(shù): 破碎模態(tài),機械的財產(chǎn) , 和化學藥品作文等 (iii)處理叁數(shù): 輪子速度, 深度減低,桌子加速, 和穿衣情況等 (iv)以機器制造叁數(shù): 靜電和動態(tài)的特性,變細長系統(tǒng) , 和桌子系統(tǒng)等 限制范圍的完全跟據(jù)經(jīng)驗的情況有有效性照慣例被用在練習因為磨的程序包括許多無法控制的叁數(shù)。 的確如此不可靠的或可接受的在任何的特效藥情形中。 達成在一種特定的情形中的必需表面質(zhì)量,程序叁數(shù)能被決定經(jīng)過一系列的實驗奔跑。 但是, 可能是一耗時的和貴的方法和它也不能夠決定正確因為限制的實驗最適宜。 Taguchi 和回應(yīng)表面方法學能方便的最佳化有一些實驗的奔跑磨叁數(shù)好的設(shè)計。 許多研究有被引導(dǎo)為決定最佳的程序叁數(shù) 。Kim[6] 運行了實驗的分析為一圓筒形的研磨程序使用 Taguchi 方法獲得那變數(shù)和最大的百分比的比較效果表面的粗糙進步。Dhavlikar[7] 呈現(xiàn)了Taguchi 和回應(yīng)方法為減到最小限度決定健康的情況由細工品的圓錯誤為那無中心的磨的程序。 Hashmi[8] 有預(yù)知工具生活在那結(jié)束磨程序藉著回應(yīng)表面方法。 那 最適宜銳利的情況是堅決的對一必需的在瞬間用工具工作生活-次序預(yù)言模型。 Suresh[9] 使用過的回應(yīng)表面方法和遺傳基因的運算法則預(yù)知表面的粗糙而且將程序叁數(shù)最佳化。 這項研究一評估 Taguchi 方法的一個幾何學的錯誤上的磨參數(shù)的影響而且發(fā)展回應(yīng)的一個數(shù)學的模型表面強迫預(yù)知幾何學的錯誤方法。 回應(yīng)表面模型用實驗被查證。 二. 文學檢討 2.1 在磨的程序中的幾何學的錯誤 熱的高比在表面的輪磨期間生產(chǎn)程序轉(zhuǎn)移到一個細工品和能引起各種不同的類型對～的熱傷害 ～最后的產(chǎn)品例如那燃燒現(xiàn)象，幾何學的錯誤，剩余壓迫力,結(jié)構(gòu)變形 , 和其他人。 這些是對～的衡量那土地的表面質(zhì)量評估而且應(yīng)該是 在下面限制如某范圍。 最重要的熱的損害考慮是那幾何學的錯誤, 與～有關(guān)最后產(chǎn)品錯誤。幾何學的錯誤只有藉由一個熱的方面被影響不在細工品和磨的輪子之間的連絡(luò)地域但是也藉著磨的系統(tǒng)一個堅硬那因素一個垂直的換置一些程度反對磨的力量。 在熱的方面情況, 熱在輪磨期間程序能在細工品中產(chǎn)生熱的擴充如中凸的形狀表面。 縮減的真正深度是不常數(shù)和比縮減的理想深度深是那簽姓名的首字母意圖。 然而，細工品能被冷卻和在輪磨之後有收縮。 如此，細工品表面是反的改變?nèi)绨嫉男螤睢?數(shù)量那熱的效果幾何學的錯誤能有關(guān)數(shù)十到達依照應(yīng)用的磨情況的測微計。在磨的系統(tǒng)堅硬的情況,那例如 磨的機器紗錠的成份,磨的輪子，細工品和磨的桌子,不是硬的而且沒有一個重的堅硬。 磨的力量演戲在這些成份垂直地制造一個合量換置。在對熱的方面反對派方面, 縮減的真正深度在這個情形是比縮減的理想深度較多的燕子。如圖 1 所示, 合量描繪那幾何學的被熱的方面和堅硬引起的錯誤那磨的系統(tǒng)用～展現(xiàn)各種不同的表格復(fù)雜的依照應(yīng)用的磨情況的作文。如此，最適宜的磨情況應(yīng)該被選擇為將最后的產(chǎn)品幾何學的錯誤減到最少。 2.2 實驗的設(shè)計 2.2.1. Taguchi 設(shè)計 Taguchi 設(shè)計方法是一簡單的和健康的將程序叁數(shù)最佳化的技術(shù)。在這個方法中, 被假定有程序結(jié)果上的影響力的主要部份叁數(shù)位於不同的排在被設(shè)計的直角排列中。 藉由如此的一個安排完全地隨機化實驗?zāi)鼙灰龑?dǎo)。 大體上, 向謠傳 (S/N) 作信號比 (h,分貝) 表現(xiàn)質(zhì)量特性為被觀察的數(shù)據(jù)在 Taguchi 中實驗的設(shè)計。 仰賴那實驗的目的,有一些質(zhì)量特性。 在那幾何學的錯誤和表面的粗糙情形,比較低的價值他們是令人想要的。 在 Taguchi 中的這些 S/ N 比方法被呼叫當做那-比較低的- 比較好的特性和依下列各項被定義。 （1） yi 在 ith 審判是被觀察的數(shù)據(jù)哪里，而且 n 是那審判的數(shù)字。從 S/N 比,有效的叁數(shù)有程序結(jié)果上的影響力能被看到和最佳的組合程序，叁數(shù)可能是堅決的。 2.3 回應(yīng)表面方法學 時常工程實驗者愿找那一個某程序達到的情況那最佳的結(jié)果。 那是, 他們想要決定水平那設(shè)計叁數(shù)在回應(yīng)延伸它的最適宜。 最適宜可以不是最大值就是一設(shè)計叁數(shù)的一個功能的最小量。 一獲得最適宜的方法學是回應(yīng)升至水面技術(shù)。 回應(yīng)表面方法學是一個收集統(tǒng)計的和是有用的數(shù)學方法對靠模切和分析工程問題。 在這技術(shù),主要的目的要將回應(yīng)最佳化被各種不同的程序叁數(shù)影響的表面?；貞?yīng)表面方法學也定量關(guān)系在可管理的輸入叁數(shù)之間和那獲得了回應(yīng)表面。 回應(yīng)的設(shè)計程序升至水面方法學依下列各項是 [10]: (i)為興趣的回應(yīng)適當?shù)暮涂煽康臏y量一系列的 實驗的設(shè)計。 (ii)用～發(fā)展第二個次序回應(yīng)表面的一個數(shù)學的模型最好的配件。 (iii)發(fā)現(xiàn)生產(chǎn)最大值或最小量回應(yīng)的價值實驗的叁數(shù)最佳的組合。 (iv)經(jīng)過二和三個空間的情節(jié)表現(xiàn)程序叁數(shù)的直接和交談式效果。 如果所有的變數(shù)被假定可測量,回應(yīng)表面能被表示成追從。 （2 目標要將回應(yīng)變數(shù) y 最佳化。 它是假定獨立變數(shù)是連續(xù)的和 可管理的藉著有可以忽略的錯誤實驗。 它是需要找一個適當?shù)慕浦禐槟钦鎸嵉? 功能的關(guān)系在獨立變數(shù)之間和回應(yīng)表面。 通常一個秒- 次序模型是回應(yīng)表面中利用。 （3） 那里 3 是一個任意的錯誤。 系數(shù), 應(yīng)該在秒- 次序模型中被決定, 被獲得最沒有正直的方法。 大體上情緒商數(shù)。 (3)能被以母體形式寫 Y=bX+E (4) 在 Y 被定義是量過的價值一個母體的地方, X 到是獨立變數(shù)的一個母體。 母體 b 和 E由～所組成系數(shù)和錯誤,分別地。 解決情緒商數(shù)。 (4)能被母體方式獲得 (5) 在哪里那母體 X 和 (X) K1 調(diào)換是那母體X 的相反。 三. 實驗的細節(jié) 一系列的 實驗有被引導(dǎo)評估哪些磨的叁數(shù)影響幾何學的錯誤在磨的表面。 例如 谷粒的四個磨的叁數(shù)大小,輪子速度, 深度減低而且桌子速度被選擇因為實驗。 磨的輪子采用鋁在磨的輪子中的有 vitrified 束縛的氧化物研磨劑 用來磨擦高速度工具鋼。 (SKH51) 一化學藥品細工品的作文在表 1 中被列出。一個幾何學的描繪考試人 (Mahr,OMS-600) 習慣於測量幾何學的錯誤。 幾何學的錯誤價值是定義到一種高度不同在最大的點之間和土地的表面最小的點在總長度里面一個量過的細工品。 表 2 列出了可管理的因素 (磨的叁數(shù))而且他們的水平在這項研究中考慮。 輪磨叁數(shù)是旋轉(zhuǎn)速度 (V) ，桌子速度 (S), 深度減少 (D) 和谷粒大小.(M) 每個因素有了三個水平(程序排列). 例如 研磨劑的類型另一個因素,細工品，冷凍劑和火花在外是不變的。 那冷凍劑不被供應(yīng)和出自途徑火花是不實行 。 被選擇的 L27 直角的排列有27 排在表 3 中被顯示。 交互作用在因素不被考慮。 自由的程度為 實驗是 26 。 四. 實驗的結(jié)果和討論 量過的幾何學的錯誤一個例子是在圖 2 中顯示。 依照縱觀的方向土地的細工品, 一般看到那重要的幾何學的錯誤數(shù)量是量過的。 價值被觀察的數(shù)據(jù)為幾何學的錯誤被列出在表 4 圖 3 呈現(xiàn)了四的有計畫的 S/ N 比在幾何學的錯誤上的因素依照那每個水平。 在最小量之間的那比較高地不同而且在每個因素中的最大 S/ N 比是,愈比較高幾何學的錯誤上的效果是。 如圖 3 所示,深度減低是一個占優(yōu)勢的叁數(shù)為那幾何學的錯誤和下一個是谷粒大小。輪子速度和桌子速度有了比較低的效果在幾何學的錯誤上。 以及因為那-降低-比較好的特性,最高的 S/在每個因素中的 N 比是令人想要的獲得最小的幾何學的錯誤。 在那深度的情形減低何時最低的深度減低當做10 公厘的價值是應(yīng)用的, 幾何學的錯誤可以被減到最少。 它是由於低度的深度減低了對熱的方面是有利潤的實在。 低度的谷粒大小,( 反的平均谷粒的直徑)哪一在磨的程序期間減少了熱世代,可以減少幾何學的錯誤。 和深度相反減低而且谷粒估計,最大的 S/N 比, 是最小的幾何學的錯誤價值,在中央的水平被獲得,這些結(jié)果是由於作文在熱的方面和堅硬之間那磨的系統(tǒng)。 最小的幾何學的錯誤將會是在低度的深度減低達成和谷粒大小聯(lián)合由于中央水平的輪子速度和桌子速度。 如此,最適宜的情況為幾何學的錯誤能是建立在: 輪子速度:(V) 1800 轉(zhuǎn)/每分, 桌子速度:(S) 10.0 m/最小, 深度減低:(D) 10 公厘 谷粒大小:(M) 46 網(wǎng)孔。 表 5 呈現(xiàn)了變化的分析結(jié)果(ANOVA)為幾何學的錯誤 S/ N 比。因素的 F- 比被計算而且與～相較F- 比的統(tǒng)計價值為特定水平的信心。 F- 比的統(tǒng)計價值是3.55而且 6.01 以 95 和 99%個信心消除,分別地[11]. 因為有計畫的價值那F- 比對如表 5 所顯示的所有因素沒有超過 95% 信心水平，全部的效果因素可以被認為適當?shù)摹?那有計畫的百分比分配, 可以展現(xiàn)如何很多的影響力是為幾何學的錯誤,是 在圖 4 中顯示。 4.2.1 秒- 次序幾何學的錯誤模型 秒- 次序回應(yīng)表面的表現(xiàn)那幾何學的錯誤 (Y,公厘) 能被表示成一個功能，例如 輪子速度 (V) 的磨叁數(shù),桌子速度(S), 深度減低 (D) 和谷粒大小.(M) 關(guān)系在幾何學的錯誤和磨的叁數(shù)之間是表示成追從。 從被觀察的數(shù)據(jù)為幾何學的錯誤列出在表 4 和情緒商數(shù)。 (5),秒- 次序回應(yīng)功能是在下面決定。 ANOVA 的一個結(jié)果為回應(yīng)升至水面功能幾何學的錯誤在表 6 中被顯示。 藉由比較由于那有計畫的和統(tǒng)計的 F-比,它被看到秒- 次序回應(yīng)功能相當適當。圖 5 表演 3D 反應(yīng)為幾何學的錯誤升至水面在二的情形改變?nèi)?shù)。 4.2.2. 次序幾何學的錯誤模型的利用從情緒商數(shù)。 (7), 幾何學的錯誤依照各種不同的磨的情況可以被預(yù)知而且減到最少容易地。 但是選擇磨的叁數(shù)從那 第二個次序的幾何學的錯誤模型,在練習中,應(yīng)該是藉由改良限制升至水面粗糙和材料移動率。 平均的粗糙完全跟據(jù)經(jīng)驗的價值,Ra(公厘),可以與～一起呈現(xiàn)下列各項公式[12]. 如情緒商數(shù)所看到。 (8), 減低增加深度和那桌子速度產(chǎn)生平均的表面比較高的價值粗糙而且如此表面惡化。 但是增加那谷粒大小作一個好的表面。 物質(zhì)的移動率在表面的輪磨中， Z(mm3/最小), 被呈現(xiàn)當做那在如下式; 在 B 是磨的輪子寬度 (公厘) 的地方。 情緒商數(shù)。 (8)和(9) 能被利用評估幾何學的錯誤在一特效藥表面粗糙和物質(zhì)的移動率。 圖 6 表演回應(yīng)表面為幾何學的錯誤有關(guān)於桌子速度和縮減的深度在一常數(shù)輪子 1800 轉(zhuǎn)/每分的速度和常數(shù)谷粒大小120 網(wǎng)孔。 圖 7 表現(xiàn)了表面的等高線為幾何學的錯誤回應(yīng)。 表面的粗糙等高線0.35,0.40 和 0.45 公厘和物質(zhì)的2500,3200 和 4000 mm3/ 最小的等高線也被增加。使用過的 B 的價值是 22 公厘。 表面的粗糙可以與～一起改良減低減退所有的深度和桌子速度。 增加這些叁數(shù)引導(dǎo)到一壞的 表面以及一個重的幾何學的錯誤。 但是不管 相同的表面粗糙在點 A,B而且 C,幾何學的錯誤和那材料的價值移動率不同於彼此。 增加那有一個不變的表面粗糙的物質(zhì)移動率增加幾何學的錯誤。 如此,適當范圍的輪磨叁數(shù)應(yīng)該被藉由犧牲一選擇是不重要的。 五. 證實實驗 在秒- 次序回應(yīng)表面的證實中模型 , 確認測試被引導(dǎo)在那最佳的被 Taguchi 方法決定的情況 (測試 1) 和二選擇了情況 ( 測試 2 和 3) 那不是在表 3 中實行 . 在測試 2 中和 3,那12 網(wǎng)孔的 15 公厘和谷粒大小的縮減相同的深度被用。 輪子速度和桌子速度是2100 轉(zhuǎn)/每分和 15 m/ 最小和 1500 轉(zhuǎn)/每分和 12.5 m/最小。圖 8 呈現(xiàn)了測試結(jié)果。 如圖 8 所示,在被預(yù)知的幾何學的錯誤之間的不同被那秒- 次序回應(yīng)表面和尺寸結(jié)果被實驗很小。 如此,秒-次序回應(yīng)模型對預(yù)知非常有用那幾何學的錯誤。 六. 結(jié)論 在表面的磨程序中的幾何學的錯誤是測量依照實驗的直角排列。 被那實驗的和分析的結(jié)果, 那獲得結(jié)論是依下列各項。 1. 磨的叁數(shù)對～的效果幾何學的錯誤用 來自 Taguchi 方法的幫忙被評估。 那深度減低是一個占優(yōu)勢的叁數(shù)為幾何學的錯誤和下一個是谷粒大小。 最佳的磨 以將減到最少為條件幾何學的錯誤是決定 2. 一個秒- 次序回應(yīng)表面模型為那幾何學的錯誤從被觀察的數(shù)據(jù)被發(fā)展?；貞?yīng)的利用升至水面模型是評估用～選擇適當?shù)哪デ闆r表面的粗糙和那材料的限制移動率。 3. 回應(yīng)表面模型的證實實驗在一種最佳的情況被引導(dǎo)和二選擇情況。 一般顯示被發(fā)展的回應(yīng)表面模型對預(yù)知非常有用那幾何學的錯誤。 承認 這個工作部份地被腦韓國 21 支援在 2004 年的計畫. Application of Taguchi and response surface methodologiesfor geometric error in surface grinding process Jae-Seob Kwak* School of Mechanical Engineering, Pukyong National University, San 100, Yongdang-Dong, Nam-Ku, Busan 608-739, South Korea Received 24 February 2004; accepted 3 August 2004 Available online 16 September 2004 Abstract The geometric error in the surface grinding process is mainly affected by the thermal effect and the stiffness of the grinding system. For minimizing the geometric error, the selection of grinding parameters is very important. This paper presented an application of Taguchi and response surface methodologies for the geometric error. The effect of grinding parameters on the geometric error was evaluated and optimum grinding conditions for minimizing the geometric error were determined. A second-order response model for the geometric error was developed and the utilization of the response surface model was evaluated with constraints of the surface roughness and the material removal rate. Confirmation experiments were conducted at an optimal condition and selected two conditions for observing accuracy of the developed response surface model. q 2004 Elsevier Ltd. All rights reserved. Keywords: Geometric error; Grinding process; Taguchi method; Response surface; Optimal conditions。 1. Introduction Grinding is a complex machining process with a lot of interactive parameters, which depend upon the grinding type and requirements of products. The surface quality produced in surface grinding is influenced by various parameters given as follows [1]. (i) Wheel parameters: abrasives, grain size, grade, structure,binder, shape and dimension, etc. (ii) Workpiece parameters: fracture mode, mechanical properties, and chemical composition, etc. (iii) Process parameters: wheel speed, depth of cut, table speed, and dressing condition, etc. (iv) Machine parameters: static and dynamic characteristics,spindle system, and table system, etc. The empirical conditions having restricted range of validity are conventionally used in practice because grinding process involves many uncontrollable parameters. * Tel.: C82 51 620 1622; fax: C82 51 620 1531. E-mail address: jskwak5@pknu.ac.kr. 0890-6955/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2004.08.007 So the ground surface quality with these conditions is not reliable or acceptable in any specific situation. To achieve the required surface quality in a specific situation, process parameters can be determined through a series of experimental runs. But, that may be a time-consuming and expensive method and also it cannot determine the exact optimum because of restricted experiments. Taguchi and response surface methodologies can conveniently optimize the grinding parameters with several experimental runs well designed. A lot of research has been conducted for determining optimal process parameters [2–5]. Kim [6] performed experimental analysis for a cylindrical lapping process using the Taguchi method to obtain the relative effects of variables and the largest percentage improvement of surface roughness. Dhavlikar [7] presented the Taguchi and response method to determine the robust condition for minimization of out of roundness error of workpieces for the centerless grinding process. Hashmi [8] has predicted a tool life in the end milling process by response surface method. The optimum cutting conditions were determined for a required tool life by a second-order prediction model. Suresh [9] used the response surface method and genetic algorithm forpredicting the surface roughness and optimizing process parameters. This study forced on evaluating the grinding parameters’ effect on a geometric error by the Taguchi method and developing a mathematical model by the response surface method for predicting the geometric error. The response surface model was verified with experiments. 2. Literature review 2.1. Geometric error in grinding process A high ratio of heats produced during surface grinding process transfers to a workpiece and can cause various types of thermal damage to the final product such as the burn phenomenon, geometric error, residual stress, structure transformation, and others. These are a measure of quality evaluation in the ground surface and should limited as a certain range below. The most important consideration of thermal damages is the geometric error, which is related to inaccuracy of the final product.The geometric error is affected not only by a thermal aspect of the contact zone between workpiece and grinding wheel but also by a stiffness of the grinding system that causes some degree of a vertical displacement against grinding force. In the case of thermal aspect, the heats during grinding process can generate a thermal expansion in the workpiece surface as a convex shape. The real depth of cuts is not constant and deeper than the ideal depth of cuts that are initial intention. However, the workpiece can be cooled and have shrinkage after grinding. So, the workpiece surface inversely changed as a concave shape. The amount of geometric error by the thermal effect can reach about tens micrometers according to the applied grinding conditions. In the case of stiffness of the grinding system, the components such as the spindle of the grinding machine, grinding wheel, workpiece and grinding table, are not rigid and do not have a heavy stiffness. Grinding forces acting on these components make vertically a resultant displacement.In opposition to the thermal aspect, the real depth of cuts in this case is more swallow than the ideal depth of cuts.As shown in Fig. 1, the resultant profile of the geometric error caused by the thermal aspect and the stiffness of the grinding system shows various forms with complicated compositions according to the applied grinding conditions. So, the optimum grinding condition should be selected forminimizing the geometric error of the final product. 2.2. Design of experiments 2.2.1. Taguchi design The Taguchi design method is a simple and robust technique for optimizing the process parameters. In this method, main parameters which are assumed to haveinfluence on process results are located at different rows in a designed orthogonal array. With such an arrangement completely randomized experiments can be conducted. In general, signal to noise (S/N) ratio (h, dB) represents quality characteristics for the observed data in the Taguchi design of experiments. Depending on the experimental objective, there are several quality characteristics. In the case of geometric error and surface roughness, lower values of them are desirable. These S/N ratios in the Taguchi method are called as the-lower-the better characteristics and are defined as follows. where yi is the observed data at the ith trial and n is the number of trials. From the S/N ratio, the effective parameters having influence on process results can be seen and the optimal sets of process parameters can be determined. 2.3. Response surface methodology Often engineering experimenters wish to find the conditions under which a certain process attains the optimal results. That is, they want to determine the levels of the design parameters at which the response reaches its optimum. The optimum could be either a maximum or a minimum of a function of the design parameters. One of methodologies for obtaining the optimum is response surface technique. Response surface methodology is a collection of statistical and mathematical methods that are useful for the modeling and analyzing engineering problems. In this technique, the main objective is to optimize the response surface that is influenced by various process parameters. Response surface methodology also quantifies the relationshipbetween the controllable input parameters and the obtained response surfaces. The design procedure of response surface methodologyis as follows [10]: (i) Designing of a series of experiments for adequate and reliable measurement of the response of interest. (ii) Developing a mathematical model of the second order response surface with the best fittings. (iii) Finding the optimal set of experimental parameters that produce a maximum or minimum value of response. (iv) Representing the direct and interactive effects of process parameters through two and three dimensional plots. If all variables are assumed to be measurable, the response surface can be expressed as follows. The goal is to optimize the response variable y. It is assumed that the independent variables are continuous and controllable by experiments with negligible errors. It is required to find a suitable approximation for the true functional relationship between independent variables and the response surface. Usually a second-order model is utilized in response surface methodology. where 3 is a random error. The b coefficients, which should be determined in the second-order model, are obtained by the least square method. In general Eq. (3) can be written in matrix form. where Y is defined to be a matrix of measured values, X to be a matrix of independent variables. The matrixes b and E consist of coefficients and errors, respectively. The solution of Eq. (4) can be obtained by the matrix approach. where XT is the transpose of the matrix X and (XTX)K1 is the inverse of the matrix XTX. 3. Experimental details A series of experiments have been conducted to evaluate which grinding parameters affect the geometric error in surface grinding. Four grinding parameters such as grain size, wheel speed, depth of cut and table speed were selected for experimentation. Grinding wheel adopting aluminum oxide abrasives with vitrified bond in the grinding wheel was used to grind a high-speed tool steel (SKH51). A chemical composition of the workpiece is listed in Table 1.A geometric profile tester (Mahr, OMS-600) was used to measure the geometric error. Values of geometric error were defined to a height difference between a maximum point and a minimum point of the ground surface within a total length of a measured workpiece. Table 2 listed controllable factors (grinding parameters) and their levels considered in this study. The grinding parameters were wheel speed (V), table speed (S), depth of cut (D) and grain size (M). Each factor had three levels (process ranges). The other factors such as type of abrasive, workpiece, coolant, and spark out were constant. The coolant was not supplied and the spark out pass was not carried out. The selected L27 orthogonal array having 27 rows is shown in Table 3. The interaction between factors was not considered. Degree of freedom for experiments is 26. Table 2 Levels of independent factors 4. Experimental results and discussion An example of the measured geometric error was shown in Fig. 2. According to the longitudinal direction of the ground workpiece, it was seen that the significant amount of geometric error was measured. The values of the observed data for geometric error were listed in Table 4. 4.1. Effect of grinding parameters Fig. 3 presented the calculated S/N ratios of four factors on the geometric error according to the each level. The higher the difference between the minimum and the maximum S/N ratios in each factor is, the higher the effect on the geometric error is. As shown in Fig. 3, the depth of cut was a dominant parameter for the geometric error and the next was the grain size. The wheel speed and the table speed had lower effects on the geometric error. And also because of the-lower-the better characteristics, the highest S/N ratio in the each factor was desirable to obtain the minimum geometric error. In the case of the depth of cut when the lowest depth of cut as a value of 10 mm was applied, the geometric error could be minimized. It was due to a low level of the depth of cut being profitable to the thermal aspect. A low level of the grain size (inversely average diameter of grain), which reduced heat generation during grinding process,could reduce the geometric error. Contrary to the depth of cut and the grain size, the maximum S/N ratios, which were the values of minimum geometric error, were obtained at the middle levels. These results were due to the composition between the thermal aspect and the stiffness of the grinding system. The minimum geometric error will be achieved at the low levels of the depth of cut and of the grain size combined with the middle levels of the wheel speed and the table speed. So, the optimum conditions for the geometric error can be established at: ? wheel speed (V): 1800 rpm, ? table speed (S): 10.0 m/min, ? depth of cut (D): 10 mm ? grain size (M): 46 mesh. Table 5 presented the result of analysis of variation (ANOVA) for the S/N ratio of the geometric error.F-ratios of factors were calculated and compared with the statistical values of the F-ratio for a specific level of confidence. The statistical values of the F-ratio were 3.55 and 6.01 at 95 and 99% confidence levels, respectively [11]. Because the calculated values of the F-ratio for all factors as shown in Table 5 did not exceed the 95% confidence level, the effects of all factors could be considered adequate. The calculated percentage distributions, which could show how much influence was for the geometric error, were shown in Fig. 4. 4.2. Response surface analysis 4.2.1. Second-order geometric error model The second-order response surface representing the geometric error (Y, mm) can be expressed as a function of grinding parameters such as wheel speed (V), table speed (S), depth of cut (D) and grain size (M). The relationship between the geometric error and grinding parameters