6320軸承外圈冷碾擴成形機理研究

6320軸承外圈冷碾擴成形機理研究,軸承,外圈,冷碾擴,成形,機理,研究,鉆研 第 27 卷 第 8期 2005年 8 月 武 漢 理 工 大 學(xué) 學(xué) 報 JOURNAL OF WUHAN UNIVERSITY OF TECHNOLOGY Vol .27 No.8 Aug .2005 圓 錐 滾 子 軸 承 套 圈 成 對 冷 輾 擴 工 藝 設(shè) 計 和 試 驗 毛 華 杰 ,史 宏 江 ,華 林 ,袁 樂 健 (武 漢 理 工 大 學(xué) 材 料 科 學(xué) 與 工 程 學(xué) 院 ,武 漢 430070) null 摘 要 : 圓 錐 滾 子 軸 承 的 套 圈 是 重 要 的 承 載 部 件 ,對 軸 承 的 性 能 有 著 重 要 的 影 響 。 成 對 冷 輾 擴 生 產(chǎn) 不 僅 能 抵 消 因 不 對 稱 變 形 產(chǎn) 生 的 軸 向 力 、 提 高 制 件 精 度 、 延 長 芯 輥 壽 命 ,還 能 提 高 生 產(chǎn) 效 率 。 通 過 對 成 對 冷 輾 擴 過 程 中 圓 錐 套 圈 變 形 過 程 分 析 與 受 力 分 析 ,推 導(dǎo) 出 圓 錐 套 圈 成 對 冷 輾 擴 成 型 時 咬 入 條 件 、 鍛 透 條 件 、 冷 輾 擴 力 的 計 算 方 法 ,毛 坯 設(shè) 計 方 法 及 模 具 設(shè) 計 方 法 ,并 對 圓 錐 軸 承 329/32(ISO355 系 列 2BD)圓 錐 套 圈 成 對 冷 輾 擴 進 行 試 驗 驗 證 。 關(guān) 鍵 詞 : 圓 錐 套 圈 ; 軸 承 ; 冷 輾 擴 中 圖 分 類 號 : TG 376.3 文 獻 標(biāo) 志 碼 : A 文 章 編 號 :1671-4431(2005)08-0070-04 Technology Design and Experiment of Cold Ring Rolling with Twin Taper Roller Bearing Race M AO Hua-jie, SHI Hong-jiang, H UA Lin, Y UA N Le-jian (School of Materials Science and Engineering, Wuhan University of Technology, Wuhan 430070, China) Abstract: Taper roller bearing race, a very important load-bearing component, has significant affection on the property of the bearing . Twinning cold ring rolling can not only counterbalance axial force of asymmetry deformation, increase accuracy and prolong life of idle roller, but also be more productive . Through the deformation process analysis and loading analysis, the bite condition, plastic penetration condition, calculation method of rolling force, stock design method and tolling design method of twinning cold ring rolling were deduced .The taper race of bear 329/32(2BD of ISO355 serials) was twinning cold rolled for ex- periment confirmation . Key words: taper roller; bearing; cold ring rolling 收 稿 日 期 :2005-05-11 . 基 金 項 目 :國 家 自 然 科 學(xué) 基 金 (50335060)和 高 校 青 年 教 師 獎 項 目 (教 人 司 2002383). 作 者 簡 介 :毛 華 杰 (1962-),男 ,副 教 授 .E-mail:maohuajie vip .sina .com 圓 錐 滾 子 軸 承 具 有 高 承 載 能 力 和 高 可 靠 性 ,廣 泛 用 于 機 械 、 汽 車 、 船 舶 等 重 載 傳 動 軸 的 支 承 1 。 圓 錐 滾 子 軸 承 套 圈 冷 輾 擴 成 形 是 一 種 極 具 發(fā) 展 前 景 的 新 技 術(shù) ,生 產(chǎn) 中 還 存 在 一 些 關(guān) 鍵 技 術(shù) 問 題 有 待 于 解 決 。 例 如 , 圓 錐 滾 子 軸 承 外 圈 冷 輾 擴 生 產(chǎn) 中 由 于 它 的 錐 形 外 形 會 對 芯 輥 產(chǎn) 生 較 大 軸 向 力 ,使 得 芯 輥 壽 命 降 低 ,同 時 影 響 了 生 產(chǎn) 精 度 2 5。 在 環(huán) 件 輾 擴 成 形 理 論 和 技 術(shù) 基 礎(chǔ) 上 6 9,提 出 2 個 圓 錐 套 圈 同 時 冷 輾 擴 成 型 的 技 術(shù) 方 案 ,可 以 抵 消 因 不 對 稱 變 形 產(chǎn) 生 的 軸 向 力 ,同 時 由 于 成 型 角 度 增 大 一 倍 減 小 了 應(yīng) 力 集 中 ,不 僅 提 高 了 模 具 壽 命 ,而 且 由 原 來 的 一 次 成 型 1 個 環(huán) 件 變 成 一 次 成 型 2 個 環(huán) 件 ,使 勞 動 生 產(chǎn) 率 增 長 一 倍 。 現(xiàn) 針 對 D56G90 型 精 密 冷 輾 環(huán) 機 設(shè) 計 圓 錐 軸 承 329/ 32(ISO355 系 列 2BD )外 圈 冷 輾 擴 試 驗 工 藝 。 1 試 驗 環(huán) 件 的 選 取 及 冷 輾 擴 工 藝 方 案 選 擇 如 圖 1 所 示 圓 錐 滾 子 軸 承 329/ 32(ISO355 系 列 2BD )外 圈 ,設(shè) 計 冷 輾 擴 試 驗 工 藝 。 采 用 特 殊 的 模 具 結(jié) 構(gòu) 同 時 輾 擴 2 個 套 圈 ,使 其 對 稱 冷 輾 擴 從 而 抵 消 因 不 對 稱 變 形 產(chǎn) 生 的 軸 向 力 。 這 就 大 大 改 善 了 芯 輥 的 受 力 狀 態(tài) ,提 高 了 制 造 精 度 。 其 冷 輾 擴 示 意 及 受 力 分 析 如 圖 2 所 示 。 2 冷 輾 擴 工 藝 設(shè) 計 2 .1 力 能 計 算 1)咬 入 條 件 要 使 環(huán) 件 能 順 利 冷 輾 擴 必 須 滿 足 咬 入 條 件 ,由 環(huán) 件 冷 輾 擴 區(qū) 受 力 平 衡 方 程 Fx = T1 x + P1 x + P2 x 0 Fy = T1 y + P1 y + P2 y = 0 得 到 咬 入 條 件 tan( 1 1 + 2 2) P1/ P2 cos( 1 1 + 2 2) (1) 滿 足 咬 入 條 件 的 進 給 量 h hm a x = 2 2 R 1 (1 + R1/ R2)2(1 + R1/ R2 + R1/ R - R1/ r) (2) 式 中 , R2 = ( R 2 + R 2)/ 2; r = ( r 2 + r 2)/ 2; 1 L/ R1; 2 L/ R2。 其 中 , L 為 接 觸 弧 長 L = 2 h 1R 1 + 1R 2 + 1R - 1r( ) -1 2)鍛 透 條 件 根 據(jù) 環(huán) 件 最 大 壁 厚 尺 寸 計 算 ,因 為 只 要 環(huán) 件 最 厚 端 能 鍛 透 則 整 個 環(huán) 件 都 能 鍛 透 。 則 應(yīng) 滿 足 的 鍛 透 條 件 為 : 變 形 區(qū) 平 均 厚 度 ha = ( h0 + h)/ 2 由 滑 移 線 理 論 知 L/ ha 1/ 8 .74 (3) 由 h0 = h + h,得 ha = h + h/ 2 h = R - r (4) 將 式 (4)帶 入 式 (3)得 到 滿 足 鍛 透 條 件 的 進 給 量 h 6 .55 10-3( R - r )2 1R 1 + 1R 2 + 1R - 1r( ) (5) 3)環(huán) 件 冷 輾 擴 力 孔 型 開 式 p = 2 k(1 + h04 L + 38 m Lh 0 ) 孔 型 閉 式 p = 2 k(1 + h04 L + 38 m Lh 0 + 34 m LB ) P = pB L M = pBR1 h hmin h hm a x 式 中 , m 為 摩 擦 因 子 ; B 為 環(huán) 件 寬 度 ; h0 為 終 軋 壁 厚 。 則 對 稱 冷 輾 擴 時 的 冷 輾 擴 力 為 P = 2 pBL (6) 2 .2 模 具 設(shè) 計 1)主 動 輥 最 小 半 徑 R1min = R2( R - r )17 .5 R 2 - ( R - r ) 2)芯 輥 最 小 半 徑 R2min = R1( R - r )17 .5 R 1 - ( R - r ) 17第 27 卷 第 8 期 毛 華 杰 ,等 :圓 錐 滾 子 軸 承 套 圈 成 對 冷 輾 擴 工 藝 設(shè) 計 和 試 驗 3 毛 坯 設(shè) 計 計 算 根 據(jù) 體 積 不 變 原 理 設(shè) 計 毛 坯 尺 寸 ,圓 錐 套 圈 的 體 積 為 圓 柱 體 積 減 去 圓 臺 體 積 ,因 此 環(huán) 件 體 積 為 V環(huán) 件 = null ( D2 )2 H - 13 H( r21 + r22 + r1 r2) = 1 4 D 2 H - 1 3 H( 3 4 d 2 + 3 2 Htan + H 2 tan2 ) 同 理 得 到 毛 坯 的 體 積 V毛 坯 = 14 D20 H - 13 H( 34 d20 + 32 Htan + H2 tan2 ) 由 體 積 不 變 得 d0 = d2 - (1 - 1/ K2) D2 式 中 , d0 為 毛 坯 的 小 端 內(nèi) 徑 ,為 設(shè) 計 所 求 尺 寸 。 由 上 面 的 推 導(dǎo) 公 式 可 知 毛 坯 的 最 小 內(nèi) 徑 必 須 大 于 芯 輥 的 最 大 直 徑 才 能 穿 入 芯 輥 輾 擴 ,所 以 只 要 知 道 芯 輥 的 最 大 直 徑 就 可 求 出 環(huán) 件 的 最 大 輾 擴 比 。 d0 = d2 - (1 - 1/ K2) D2 Dm a x 故 K D D2 - d2 + D2m a x 毛 坯 的 外 徑 可 由 零 件 外 徑 直 接 給 出 ,輾 擴 比 K = 1 .4,由 上 式 驗 算 確 定 滿 足 最 小 輾 擴 比 條 件 ,由 于 環(huán) 件 軸 向 展 寬 很 小 ,可 先 假 設(shè) 軸 向 尺 寸 不 變 ,最 后 再 根 據(jù) 試 驗 進 行 精 確 確 定 。 有 了 這 些 尺 寸 就 能 設(shè) 計 出 毛 坯 的 尺 寸 。 鍛 件 圖 、 毛 坯 圖 尺 寸 如 圖 5 所 示 。 毛 坯 的 尺 寸 R = 18 .55 mm, r = 12 .77 mm, r = 14 .90 mm, r = ( r + r )/ 2 = 13 .83 mm 4 試 驗 數(shù) 據(jù) 計 算 與 校 核 1)滿 足 咬 入 條 件 的 最 大 進 給 量 hma x = 2 2 R 1 (1 + R1/ R2)2(1 + R1/ R2 + R1/ R - R1/ r) = 0 .32 mm 所 以 要 滿 足 軋 輥 咬 入 條 件 ,進 給 必 須 小 于 最 大 進 給 量 h 0 .32 mm 2)滿 足 鍛 透 條 件 的 進 給 量 hmin = 6 .55 10- 3( R - r )2(1/ R1 + 1/ R2 + 1/ R - 1/ r) = 5 .44 10-2 mm 所 以 要 鍛 透 進 給 量 必 須 大 于 鍛 透 所 需 最 小 進 給 量 h 5 .44 10- 2 mm 考 慮 到 生 產(chǎn) 效 率 進 給 應(yīng) 取 大 值 ,因 此 既 滿 足 咬 入 條 件 ,又 滿 足 鍛 透 條 件 的 進 給 量 h = 0 .3 mm 3)驗 證 環(huán) 件 的 極 限 壁 厚 H Hm ax = 17 .5 R11 + R 1/ R2 = 22 .00 mm 環(huán) 件 大 端 壁 厚 R - r = 5 .78 mm 22 .00 mm,滿 足 條 件 ,則 環(huán) 件 各 處 壁 厚 均 滿 足 條 件 。 4)環(huán) 件 冷 輾 擴 力 的 計 算 摩 擦 系 數(shù) = 0 .15 ,相 應(yīng) 的 摩 擦 因 子 m = 0 .3 ,進 給 量 h = 0 .3 mm, h0 = 3 .87 mm,則 有 27 武 漢 理 工 大 學(xué) 學(xué) 報 2005年 8 月 L = 2 h( 1R 1 + 1R 2 + 1R - 1r )-1 = 2 .44 mm p = 2 k(1 + h04 L + 38 m Lh 0 + 34 m LB ) = 3 .03 k P = pBl = 73 .9 k M = pBR1 h = 946 .2 k hmin h hma x 冷 輾 擴 過 程 中 2 個 套 圈 同 時 進 行 冷 輾 擴 ,因 此 軋 輥 的 冷 輾 擴 力 為 2 P,冷 輾 擴 力 矩 為 2 M。 對 于 此 次 試 驗 采 用 不 同 材 料 的 毛 坯 進 行 試 驗 ,分 別 為 :軸 承 鋼 、 45 鋼 、 純 鋁 。 試 驗 計 算 參 數(shù) 見 表 2。 表 2 不 同 材 料 輾 擴 力 、 力 矩 計 算 結(jié) 果 材 料 屈 服 強 度 s/ MPa 剪 切 屈 服 強 度 k/ MPa 軋 輥 壓 力 2 nullP/ N 冷 輾 擴 力 矩 2 nullM/(N m) 導(dǎo) 向 輥 壓 力 / N GCr15 P400 w231 null34 null141 437 114 4 null969 45 null# 360 w207 null30 null597 391 726 4 null453 Al 200 w115 null16 null997 217 626 2 null474 5 結(jié) 論 a .根 據(jù) D56G90 型 精 密 冷 輾 環(huán) 機 ,設(shè) 備 參 數(shù) 最 大 冷 輾 擴 壓 力 為 100 kN,可 知 試 驗 所 需 壓 力 遠(yuǎn) 遠(yuǎn) 小 于 額 定 冷 輾 擴 力 ,因 此 得 出 結(jié) 論 ,試 驗 從 力 能 角 度 考 慮 圓 錐 滾 子 軸 承 329/ 32(ISO355 系 列 2BD )外 圈 對 稱 冷 輾 擴 成 型 的 方 案 可 行 。 b .當(dāng) 進 給 量 控 制 在 h = 0 .3 mm 時 ,既 滿 足 咬 入 條 件 ,又 滿 足 鍛 透 條 件 ;按 上 述 方 法 設(shè) 計 出 了 毛 坯 形 狀 和 尺 寸 。 通 過 試 驗 證 明 ,該 工 藝 可 生 產(chǎn) 高 質(zhì) 量 圓 錐 滾 子 軸 承 外 圈 。 參 考 文 獻 1 w史 宏 江 ,毛 華 杰 ,袁 樂 健 ,等 .冷 輾 擴 技 術(shù) 在 新 時 期 的 發(fā) 展 J.機 械 制 造 ,2004,(10):23 24. 2 時 大 方 .圓 錐 套 圈 精 密 成 型 工 藝 分 析 J .軸 承 ,1998,(1):14 16 . 3 時 大 方 .錐 形 環(huán) 形 件 套 鍛 輾 擴 工 藝 分 析 J .礦 山 機 械 ,2003,(5):69 70 . 4 田 民 ,王 衛(wèi) 榮 ,吳 宗 彥 .圓 錐 滾 子 軸 承 套 圈 精 密 輾 擴 J .軸 承 ,2003,(7):13 16 . 5 曹 淑 明 ,李 法 泉 .圓 錐 內(nèi) 圈 輾 擴 時 內(nèi) 徑 凹 心 分 析 J .軸 承 ,1995,(3):45 46 . 6 Yea Y, Ko Y, Kim N, et al .Prediction of Spread, Pressure Distribution and Roll Force in Ring Rolling Process Using Rigid- plastic Finite Element MethodJ .Journal of Materials Processing Technology, 2003, 140(1): 478 486 . 7 Hua Lin, Zhao Zhongzhi . 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Parvizi,K. Abrinia.A two dimensional upper bound analysis of the ring rolling process with experimental and FEM vericationsJ.International Journal of Mechanical Sciences, 2014(79):17618122 M.R. Forouzan,M.S alimi,M.S. Gadala,A.A.Aljawi. Guide roll simulation in FE analysis of ring rollingJ. Journal of Materials Processing Technology,2003(142): 213223 Journal of Materials Processing Technology 142 (2003) 213223 Guide roll simulation in FE analysis of ring rolling M.R. Forouzan a , M. Salimi a , M.S. Gadala b, , A.A. Aljawi c a Mechanical Engineering Department, IUT, Esfahan, Iran b Mechanical Engineering Department, University of British Columbia, 2324 Main Mall, Vancouver, BC, Canada V6T 1Z4 c Mechanical Engineering Department, KAAU, Jeddah, Saudi Arabia Received 6 September 2001; received in revised form 17 April 2002; accepted 26 February 2003 Abstract A new method (thermal spokes) has been proposed to simulate the guide roll effect in FE analysis of the ring rolling process. The method is simple to use with existing FE formulations, does not introduce further nonlinearities, and results in more stable and faster solution to ring rolling simulations. The method has been successfully employed in a 2D FE simulation of rolling flat rings. A special feature of the method is its ability to take into account the stiffness of the adjustment mechanism of the guide rolls. On the other hand, a simple modification has been introduced on the lever arm principle, which might be used to estimate guide roll forces in elementary analysis. Incorporating the thermal spokes method in the ring rolling simulation showed important effects of the guide rolls on the ringtool contact region, roll force and drive torque. Finite element results are in good agreement with experimental results and confirm the validity of the thermal spokes approach as well as the proposed modified lever arm principle. 2003 Elsevier Science B.V. All rights reserved. Keywords: Ring rolling; Guide roll; Finite element; Thermal spoke 1. Introduction Ring rolling is used to manufacture a wide variety of seamless ring shaped components for small parts such as bearing races to large rings for rocket space crafts and tur- bines. In this process an annular blank is formed between two rolls. The inner roll (mandrel or pressure roll) is idle and approaches the outer (main roll or driven roll), which is driven with a constant angular velocity and causes the ring to rotate. Decreasing the workpiece thickness leads to an in- crease of the ring radius/height. Ring height is controlled in a closed-pass radial rolling or by using a pair of identical conical rolls. Pair of guide rolls, which is usually controlled by a linkage mechanism, keeps the ring central and maintain circularity. Fig. 1 shows the main components of a typical radialaxial ring rolling mill. In addition to classical elementary analysis based on ex- periments and the slip line field method 15, ring rolling has been analyzed using the upper bound method 6,7, the energy method 8 and Hills general method 9. Ring rolling is different from the standard rolling pro- cess in the fact that the ring rolling process is non-steady Corresponding author. Tel.: +1-604-822-2777; fax: +1-604-822-2403. E-mail address: gadalamech.ubc.ca (M.S. Gadala). state throughout and a large number of ring rotations are required to finish the product. Numerical simulation of the process normally necessitates several times the number of increments required for a regular metal forming application. Although the finite element method might simulate this pro- cess accurately, it suffers from excessive computation time and convergence problems due to the highly nonlinear na- ture of the problem and the existence of complicated contact conditions. In order to reduce computational requirements several as- sumptions have been employed, e.g., simulation in the vicin- ity of the roll gap 1012, pseudo plane strain simulations 13, and the axisymmetric forging approach method 14. Decreasing the mesh density out of the roll gap zone might greatly reduce the computational requirements. Us- ing this idea, Kim et al. 15,Huetal.16 and Lim et al. 17 have introduced two different mesh systems. A com- putational mesh, constructed with a fine mesh in the main deformation zone, and a material mesh, constructed with a uniform fine density. As the computational mesh deforms and changes its geometry, new deformation state variables are calculated and transferred to the material mesh at the end of each increment. The process is similar in nature to the re-meshing technique and it suffers from the usual prob- lems of interpolation between the two meshes. Xie et al. 18 have used Lagrangian formulation, whereby the entire mesh 0924-0136/$ see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00600-9 214 M.R. Forouzan et al. / Journal of Materials Processing Technology 142 (2003) 213223 Fig. 1. Schematic illustration of radialaxial ring rolling mill. rotates with the material. A fine mesh is constructed in the main deformation zone between the gap of the driving roll and the mandrel. It is assumed that every section of the ring deforms under the same conditions during one ring revolu- tion and, therefore, the deformation of the fine sector is ob- tained and generalized to the whole ring. The program then updates the ring geometry and re-meshes the workpiece for the new deformed geometry such that the fine mesh is again located in the desired vicinity of the roll gap zone. As an approximation this method assumes steady state condition in each revolution. A promising technique for analyzing ring rolling is the ALE formulation in which mesh distortion is independent of material flow; therefore, a fine mesh may always be main- tained in the roll gap zone. The ALE formulation has been used for plain strain 2D FE simulation of plane ring rolling by Hu and Liu 19 but they used constant mesh density over the entire ring body. By combining the ALE formula- tion, the idea of reducing mesh density out of the deforma- tion zone and the SPCGM technique, Davey and Ward 20 have proposed an efficient method for a complete 3D sim- ulation of the process. The ALE simulation to ring rolling suffers, however, from the difficulty in specifying a general mesh motion scheme for the application. Unlike the regu- lar rolling process, the fine mesh between the driving roll and the mandrel is moving radially and material is flowing through it in the tangential direction. This introduces an ex- tra set of unknowns in the process of generalizing the mesh motion scheme. Simulation and studying the effect of guide rolls on the process parameters has not been carefully discussed in ring rolling studies. This may be attributed to the complexity of modeling the guide rolls. The addition of contact sur- faces, the need to increase the mesh density in the guide roll contact area, and the unknown position and forces of the guide roll are just few factors contributing to the complex- ity of the simulation. In addition, the absence of the guide rolls normally does not lead to instability of the process simulation. In this paper, a new and simple procedure, the thermal spokes method, is introduced to accurately simulate the guide roll effects. This method does not introduce additional non- linearities to the model and has a minimal effect on the com- putational cost. The method may be used with Lagrangian or ALE formulation. The results of the new method show that guide rolls do affect the process parameters considerably. Good agree- ments between experimental reports and FE results have been achieved. The paper also presents some modifications to the elementary lever arm method, which confirm the ef- ficiency and efficiency of the thermal spokes method and might be used for the calculation of the guide roll forces in the classical analysis. Fig. 2. Guide roll adjustment mechanism. M.R. Forouzan et al. / Journal of Materials Processing Technology 142 (2003) 213223 215 2. Guide rolls adjustment mechanism Synchronized arms, which the guide rolls are attached to, exert stabilizing forces to keep the ring in central position and maintain circularity. Fig. 2 shows how a four bar linkage assemblage paves the way for the symmetry of the guide roll passes. The adjustment mechanism of the guide rolls keeps the guide rolls position symmetrical about a vertical axis passing through the centers of the ring, the driven roll and the mandrel. The pressure on the actuator piston of the linkage assembly always forces at least one of the guide rolls to be in contact with the ring. Whenever the ring tends to tilt left or right, the contact force between the ring and proper guide roll increases and prevents the ring from tilting. The guide roll force has been reported to be very small under ordinary conditions 4. However, sometimes eccen- tricity in the initial blank may run the ring back and forth Fig. 3. Lever arm method. with small guide roll force 21. In industry, the guide roll force is preferred, however, to be set at a higher level at the start of the rolling operation. In the final stages, the centering rolls must not press the ring oval when it loses its rigidity due to decreasing wall thickness and increasing diameter. In this case, a special ge- ometry design on the adjustment arms may reduce the ne- cessity of additional control on hydraulic actuator pressure. 3. Modified lever arm method 3.1. Lever arm method Fig. 3a shows the typical shape of the pressure distribu- tion exerted by rolls on the ring. Two resultant forces L 1 and L 2 are considered as the mandrel and main roll reactions, 216 M.R. Forouzan et al. / Journal of Materials Processing Technology 142 (2003) 213223 respectively, where L 2 passes through the center of the man- drel, as it is idle. Based on experimental results, small guide roll forces are necessary to maintain the stability of the ring. The lever arm method neglects these forces completely. Con- sequently, to maintain force equilibrium, the main roll resul- tant force L 1 must be collinear with L 2 and thus the required drive torque is given by T 0 = L 1 a 0 (1) where the lever arm a 0 is the perpendicular distance from the center of the main roll onto the line of action of L 1 4. 3.2. Proposed modification In large reduction values, as presented in Fig. 3, it is easy to realize that L 1 and L 2 are not, in general, collinear. In Fig. 3b, L 1 has been moved to its actual position, i.e., on the centroid of the outside pressure. Now L 1 and L 2 are parallel but not collinear and the small distance a 2 defines the stationary moment M 1 , which intends to rotate the ring clockwise: M 1 = L 1 a 2 (2) Although in the real case a 2 is very small, M 1 may be quite considerable because the rolling force L 1 is very large. This proposed modification divides the lever arm a 0 into two parts, a 1 and a 2 , and, therefore: T 0 = L 1 (a 1 +a 2 ) = T 1 +M 1 (3) where T 1 is the modified required drive torque. Under ideal conditions, one of the guide rolls prevents the ring from tilting by exerting a small contact force to the ring while the other guide roll is assumed to be not in contact, as shown in Fig. 3c. Any other case may be treated as the ideal condition as shown in Appendix A. As the guide rolls are mounted freely and friction is as- sumed negligible, the line joining the guide roll center to the ring center gives the direction of the contact force L 3 , which prevents the ring from tilting. On the other hand, a force L 4 of equal magnitude and opposite direction in the roll gap zone might be assumed as its conjugate. The moment equi- librium equation may be used to estimate the contact force between the guide roll and the ring: L 1 a 2 = L 3 a 3 (4) where a 3 is the length of the perpendicular from the roll gap zone to the line that joins the guide roll center to that of the ring. Despite its simplicity, the above modification and analysis results in the following two important points: It may be used to determine the side to which the ring tilts. It shows that guide rolls may decrease the required drive moment of the main roll. Further details and assumptions have been introduced in Appendix A to estimate the guide roll force explicitly. 4. Finite element simulation of the guide rolls The idea behind the thermal spokes method stems from the fact that the guide roll contact forces are very small and, therefore, their local effects recover due to elastic contact conditions. Consequently, the guide rolls just affect the tilt- ing of the ring. The thermal spokes method is proposed to replace the guide rolls by a few imaginary links to prevent the ring from tilting (Fig. 4). The main characteristic and steps of the method is summarized in the following points: a. A node is created in the center of the initial annular blank. b. Elastic truss or spring elements are created to connect the central node to the nodes on the mid-layer of the ring. The mid-layer nodes should already be defined on the mean ring radius. No bending or buckling effect is introduced due to the introduction of the truss elements, which look like bicycle wheel spokes. c. Time dependent thermal body force is prescribed on the spokes in order to adjust the link length. The temperature should keep the free stress length of the spokes equal to the ring mid-layer radius during the rolling process. d. Displacement boundary condition is prescribed to keep the central node on the symmetry plane of the mill and prevent the ring from tilting, i.e., u x (N) = 0, where N is the node number of the central node. In a plane strain case, the required temperature over time may be easily calculated based on volume consistency. Knowing the process feed rate will enable the calculation of an average ring mean radius as function of time. The tem- perature load to be specified on the spokes should maintain compatibility between the ring mean radius and the spokes Fig. 4. Thermal spokes model. M.R. Forouzan et al. / Journal of Materials Processing Technology 142 (2003) 213223 217 length. The flexibility of the adjusting mechanism of the guide rolls may be simulated using a very small stiffness for the links, thus small errors in the calculated tempera- tures will not affect the process parameters noticeably. In addition, a large number of truss elements and symmetric positioning helps the process in tolerating imperfect tem- peratures. Numerical experiments show that the simulation is not sensitive to small errors in temperature calculations. In three-dimensional analysis, fish tailing and incomplete filling may affect the calculation of an accurate mean radius for the ring. In such case, a preliminary run with a coarse mesh will help in calculating a more accurate function of the ring mean radius. 5. Ring rolling FE model A 2D FE model, with 8-noded plane strain rectangular solid elements, has been developed to simulate cold rolling of plane aluminum rings. A total number of 360 divisions in the hoop direction and six divisions in the radial direction yield 2160 elements and 7200 nodes in the workpiece body. In addition, two target elements simulate the mandrel and the main roll as two rigid circles. Two sets of contact elements, including 360 contact elements each, are used on the surface of the inner and the outer elements of the ring. The first model is constructed with no thermal spokes and no guide roll simulation is provided. The second model contains 360 truss elements to model guide rolls. Each truss element starts from a corner node of the mid-layer of the ring elements and connects to the central node of the ring. Material property and geometry of the mill have been chosen according to Ref. 4 and are listed in Table 1.A friction coefficient of = 0.15 for the main roll surface Fig. 5. Element distortion versus state of reduction. Table 1 Rolling properties Material Aluminum-alloy HE 30 (0.8% Mg, 1% Si, 0.7% Mn) Main roll rotational speed 31 rev/min Mandrel feed rat 0.4826 mm/rev (0.019 in./rev) Main roll diameter 228.6 mm (9 in.) Mandrel diameter 69.85 mm (2.75 in.) Outside diameter of initial blank 127 mm (5 in.) Inside diameter of initial blank 76.2 mm (3 in.) Initial ring thickness 25.4 mm (1 in.) Initial ring height 25.4 mm (1 in.) has been assumed. For simplicity the mandrel surface is assumed to be frictionless 10,16. Mechanical properties of truss elements may be chosen arbitrarily. A numerical assessment of truss element stiffness is provided below. Static Lagrangian formulation with about 25,000 time- steps has been used to simulate near 80% reduction. For this purpose the commercial software ANSYS 22 was used. Fine meshes in the hoop direction as well as the augmented-Lagrange multiplier method with high penalty factor provide the ability to compute the pressure distribu- tion after the roll gap throttle, which may be affected by ring tilting as well as by the spring-back of the elasticplastic workpiece material. Although the total number of DOFs of the first and sec- ond model are almost the same (14,400 DOFs and 14,401 DOFs, respectively), introducing one node, which is con- nected directly to 360 other nodes, increases the computa- tional cost of each iteration by increasing the front width of the global stiffness matrix. A similar optimizing method of the node-numbering scheme has been used for both models in order to minimize the wave front. The final wave front of the second model was found to be just 21 DOFs greater 218 M.R. Forouzan et al. / Journal of Materials Processing Technology 142 (2003) 213223 than the first. However, the total CPU time for the latter is less than that for the former and it is expected that the sim- ulation of the guide rolls helps the stability of the system and, therefore, reduces the CPU time requirements. 6. Results and discussion In the ring rolling process, large deformation and distor- tion of elements occur in the main deformation area between the driven roll and the mandrel. The second order elements with high order shape functions used in modeling the pro- cess help to calculate severe element distortion in the final stages. Fig. 5 shows how the element shape changes over time. It also shows that the mechanism of deformation on the inside and outside layer elements is completely different. Although the contact forces exerted by the guide rolls are very small with respect to the roll force, the thermal spokes method shows that the guide roll simulation do affect the process parameters. Similar to other metal forming processes, the die geometry in ring rolling plays a great role in determining the process parameters. Obviously, the contact region between the ring and the tools might be interpreted as the die shape, which is controlled by the guide rolls as they determine the angle at which the ring elements approach the tool. This indicates that the flexibility of the guide roll controlling mechanism may have a great effect on the simulation. While this is a Fig. 7. Ring shape after 78.8% reduction (plane strain conditions). Fig. 6. Effect of roll tilting on the pressure distribution. simple and automatic procedure through the thermal spokes method, it will greatly add to the complexity of modeling the guide rolls by the normal modeling procedures. As the ring is gripped between the main roll and the mandrel, ring tilting may be assumed as rigid body rotation around the roll gap. In the absence of the guide rolls the ring M.R. Forouzan et al. / Journal of Materials Processing Technology 142 (2003) 213223 219 will tilt until the resultant force L 1 and L 2 become collinear. Fig. 6 illustrates that for the tilted ring, the start and the end points of the contact region between the ring and the mandrel move in the rolling direction and the main rollring contact region moves in the opposite direction. This con- tact region has been found sensitive to spoke elasticity until the spokes become rigid enough. In order to show the ef- fect of the elasticity of the guide rolls arm on the contact region, four different sets of FE simulations have been per- formed. The first model has no spokes; the second contains soft spokes with a stiffness