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6320 轴承 外圈 冷碾扩 成形 机理 研究

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6320轴承外圈冷碾扩成形机理研究,6320,轴承,外圈,冷碾扩,成形,机理,研究

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第 27 卷 第 8 期2005 年 8 月武 汉 理 工 大 学 学 报JOURNAL OF WUHAN UNIVERSITY OF TECHNOLOGYVol. 27 No.8 Aug .2005圆 锥 滚 子 轴 承 套 圈 成 对 冷 辗 扩 工 艺 设 计 和 试 验毛 华 杰 ,史 宏 江 , 华 林 ,袁 乐 健(武汉理工大学材料科学与工程学院,武汉 430070)摘 要: 圆锥滚子轴承的套圈是重要的承载部件,对轴承的性能有着重要的影响。成对冷辗扩生产不仅能抵消因不对称变形产生的轴向力、 提高制件精度、 延长芯辊寿命,还能提高生产效率。通过对成对冷辗扩过程中圆锥套圈变形过程分析与受力分析,推导出圆锥套圈成对冷辗扩成型时咬入条件、 锻透条件、 冷辗扩力的计算方法,毛坯设计方法及模具设计方法,并对圆锥轴承 329/ 32(ISO355 系列 2BD)圆锥套圈成对冷辗扩进行试验验证。关键词: 圆锥套圈; 轴承; 冷辗扩中图分类号: TG 376.3文献标志码: A文章编号:1671-4431(2005)08-0070-04Technology Design and Experiment of Cold Ring Rolling with TwinTaper Roller Bearing RaceMAO Hua-jie, SHI Hong-jiang, HUA Lin, Y UAN 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 ofthe bearing .Twinning cold ring rolling can not only counterbalance axial force of asymmetry deformation, increase accuracy andprolong life of idle roller, but also be more productive.Through the deformation process analysis and loading analysis, the bitecondition, plastic penetration condition, calculation method of rolling force, stock design method and tolling design method oftwinning 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 .基金项目:国家自然科学基金(50335060)和高校青年教师奖项目(教人司2002383).作者简介:毛华杰(1962-),男,副教授 . E-mail:maohuajie vip . sina . com圆锥滚子轴承具有高承载能力和高可靠性,广泛用于机械、 汽车、 船舶等重载传动轴的支承1。圆锥滚子轴承套圈冷辗扩成形是一种极具发展前景的新技术,生产中还存在一些关键技术问题有待于解决。例如,圆锥滚子轴承外圈冷辗扩生产中由于它的锥形外形会对芯辊产生较大轴向力,使得芯辊寿命降低,同时影响了生产精度2 5。在环件辗扩成形理论和技术基础上6 9,提出 2 个圆锥套圈同时冷辗扩成型的技术方案,可以抵消因不对称变形产生的轴向力,同时由于成型角度增大一倍减小了应力集中,不仅提高了模具寿命,而且由原来的一次成型 1 个环件变成一次成型 2 个环件,使劳动生产率增长一倍。现针对 D56G90 型精密冷辗环机设计圆锥轴承 329 / 32(ISO355 系列 2BD )外圈冷辗扩试验工艺。1 试验环件的选取及冷辗扩工艺方案选择如图 1 所示圆锥滚子轴承 329 / 32(ISO355 系列 2BD )外圈,设计冷辗扩试验工艺。采用特殊的模具结构同时辗扩 2 个套圈,使其对称冷辗扩从而抵消因不对称变形产生的轴向力。这就大大改善了芯辊的受力状态,提高了制造精度。其冷辗扩示意及受力分析如图 2 所示。2 冷辗扩工艺设计2 . 1 力能计算1)咬入条件 要使环件能顺利冷辗扩必须满足咬入条件,由环件冷辗扩区受力平衡方程Fx=T1 x+ P1 x+ P2 x 0 Fy=T1 y+ P1 y+ P2 y= 0得到咬入条件 tan(11+ 22) P1/ P2 cos(11+ 22)(1)满足咬入条件的进给量h hmax=22R1(1 + R1/ R2)2(1 + R1/ R2+ R1/ R - R1/ r)(2)式中, R2= ( R2+ R2) / 2; r = ( r2+ r2) / 2;1 L / R1;2 L / R2。其中, L 为接触弧长L =2h1R1+1R2+1R-1r()- 1 2)锻透条件 根据环件最大壁厚尺寸计算,因为只要环件最厚端能锻透则整个环件都能锻透。则应满足的锻透条件为: 变形区平均厚度ha= ( h0+ h) / 2由滑移线理论知L / ha 1 / 8 . 74(3)由 h0= h +h,得ha= h + h / 2 h =R - r(4) 将式(4)带入式(3)得到满足锻透条件的进给量h 6 . 5510- 3( R - r )21R1+1R2+1R-1r()(5) 3)环件冷辗扩力 孔型开式p = 2 k(1 +h04 L+38mLh0) 孔型闭式p = 2 k(1 +h04 L+38mLh0+34mLB)P = pBL M = pBR1h hmin h hmax式中, m 为摩擦因子; B 为环件宽度; h0为终轧壁厚。则对称冷辗扩时的冷辗扩力为P = 2 pBL(6)2 . 2 模具设计1)主动辊最小半径 R1min=R2( R - r )17 . 5 R2- ( R - r )2)芯辊最小半径 R2min=R1( R - r )17 . 5 R1- ( R - r )17第 27 卷 第 8 期 毛华杰,等:圆锥滚子轴承套圈成对冷辗扩工艺设计和试验 3 毛坯设计计算根据体积不变原理设计毛坯尺寸,圆锥套圈的体积为圆柱体积减去圆台体积,因此环件体积为V环 件= (D2)2H -13H( r21+ r22+ r1r2) =14D2H -13H(34d2+32Htan + H2tan2 ) 同理得到毛坯的体积 V毛 坯=14D20H -13H(34d20+32Htan + H2tan2 )由体积不变得d0=d2- (1 - 1 / K2) D2式中, d0为毛坯的小端内径,为设计所求尺寸。由上面的推导公式可知毛坯的最小内径必须大于芯辊的最大直径才能穿入芯辊辗扩,所以只要知道芯辊的最大直径就可求出环件的最大辗扩比。d0=d2- (1 - 1 / K2) D2Dmax故K DD2- d2+ D2max 毛坯的外径可由零件外径直接给出,辗扩比 K = 1 . 4,由上式验算确定满足最小辗扩比条件,由于环件轴向展宽很小,可先假设轴向尺寸不变,最后再根据试验进行精确确定。有了这些尺寸就能设计出毛坯的尺寸。锻件图、 毛坯图尺寸如图 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 mm4 试验数据计算与校核1)满足咬入条件的最大进给量hmax=22R1(1 + R1/ R2)2(1 + R1/ R2+ R1/ R - R1/ r) = 0 . 32 mm所以要满足轧辊咬入条件,进给必须小于最大进给量h 0 . 32 mm 2)满足锻透条件的进给量hmin= 6 . 5510- 3( R - r )2(1 / R1+ 1 / R2+ 1 / R - 1 / r) = 5 . 44 10- 2mm所以要锻透进给量必须大于锻透所需最小进给量 h 5 . 4410- 2mm 考虑到生产效率进给应取大值,因此既满足咬入条件,又满足锻透条件的进给量h = 0 . 3 mm3)验证环件的极限壁厚H Hmax=17 . 5 R11 + R1/ R2= 22 . 00 mm 环件大端壁厚 R - r = 5 . 78 mm 22 . 00 mm,满足条件,则环件各处壁厚均满足条件。4)环件冷辗扩力的计算摩擦系数 = 0 . 15 ,相应的摩擦因子 m = 0 . 3 ,进给量 h = 0 . 3 mm, h0= 3 . 87 mm,则有27 武 汉 理 工 大 学 学 报 2005 年 8 月L =2h(1R1+1R2+1R-1r)- 1= 2 . 44 mmp = 2 k(1 +h04 L+38mLh0+34mLB) = 3 . 03 kP = pBl = 73 . 9 k M =pBR1h = 946 . 2 k hminh hmax 冷辗扩过程中 2 个套圈同时进行冷辗扩,因此轧辊的冷辗扩力为 2 P,冷辗扩力矩为 2 M。对于此次试验采用不同材料的毛坯进行试验,分别为:轴承钢、 45 钢、 纯铝。试验计算参数见表 2。表 2 不同材料辗扩力、 力矩计算结果材料屈服强度 s/ MPa剪切屈服强度 k / MPa轧辊压力 2 P / N冷辗扩力矩 2 M/ (N m)导向辊压力 / NGCr15P400w23134 1414371144 96945#360w20730 5973917264 453Al200w11516 9972176262 4745 结 论a .根据 D56G90 型精密冷辗环机,设备参数最大冷辗扩压力为 100 kN,可知试验所需压力远远小于额定冷辗扩力,因此得出结论,试验从力能角度考虑圆锥滚子轴承 329 / 32(ISO355 系列 2BD )外圈对称冷辗扩成型的方案可行。b .当进给量控制在h = 0 . 3 mm 时,既满足咬入条件,又满足锻透条件;按上述方法设计出了毛坯形状和尺寸。通过试验证明,该工艺可生产高质量圆锥滚子轴承外圈。参考文献1 w史宏江,毛华杰,袁乐健,等 .冷辗扩技术在新时期的发展J .机械制造,2004,(10):2324 .2 时大方 . 圆锥套圈精密成型工艺分析J .轴承,1998,(1):1416 .3 时大方 . 锥形环形件套锻辗扩工艺分析J . 矿山机械,2003,(5):6970 .4 田 民,王卫荣,吴宗彦 . 圆锥滚子轴承套圈精密辗扩J .轴承,2003,(7):1316 .5 曹淑明,李法泉 . 圆锥内圈辗扩时内径凹心分析J .轴承,1995,(3):4546 .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): 478486 .7 Hua Lin, Zhao Zhongzhi . The Extreme Parameters in Ring RollingJ . Journal of Materials Processing Technology, 1997,69(13):273276 .8 Hua Lin, Mei Xuesong, Wu Xutang .Vibration and Control in Ring Rolling ProcessJ .Transactions of Nonferrous Metals So-ciety of China, 1999, 9(2): 213217.9 华 林,黄兴高,朱春东 . 环件轧制理论和技术M . 北京:机械工业出版社,2001.37第 27 卷 第 8 期 毛华杰,等:圆锥滚子轴承套圈成对冷辗扩工艺设计和试验 1 漆良涛.轴承套圈冷辗扩成形关键技术研究D.宁波：宁波大学，2011 2 田亮.双沟轴承外圈冷辗扩成型理论及工艺研究D.湖北：武汉理工大学，20093 赵秀婷,王雅红.外球面轴承外圈冷辗扩有限元分析J.轴承, 2009(09):39-42 4 张利斌,张洛平,彭晓南.双沟道球轴承外圈冷辗扩变形规律有限元分析J.轴承,2008(10):9-125 毛华杰,田亮.双沟球轴承外圈冷辗扩数值模拟与试验研究J.轴承,2010(02):25-286 赵伟,贺红霞,张洛平.梯形轴承外圈冷辗扩的有限元模拟J.轴承,2008(03):18-217 朱春东,程孟彪.深沟球轴承外圈冷辗扩破坏分析J.热加工工艺,2010(07):185-1898 何杰,束学道,彭文飞,孙宝寿.高铁轴承外圈的冷辗扩变形机理研究J.热加工工艺,2014,43(21):101-103 9 毛华杰,史宏江,华 林,袁乐健.圆锥滚子轴承套圈成对冷辗扩工艺设计和试验J.自然科学,2015,27(08):70-7310 漆良涛,肖旻,孙宝寿,束学道.异型截面环件径向轧制的数值模拟研究J.轻工机械,2011,29(02):48-52 11 高乃杰.辗扩技术在轴承锻造中的应用和发展J.金属加工(热加工),2008(07):22-23 12 赵伟敏.轴承套圈冷辗扩加工技术J.新技术新工艺,2001(05):24-26 13 张夕凤,管延锦.滚珠轴承套圈冷辗扩成形过程的数值模拟研究J.锻压装备与制造技术,2008(02):75-80 14 华林,赵仲治,王华昌.环件轧制原理和设计方法J.湖北工学院学报,1995,(10):2-5 15 许思广,连家创.环件轧制的有限元分析J.锻压技术,1989(06):21-27 16 罗洲,华林,王志慧,周勇强.环件轧制过程的显式有限元模拟分析J.金属成形工艺 2003(12):12-16 17 许思广,连家创.环件连续轧制过程的有限元分析J.东北重型机械学院学报,1993,17(01):1-7 18 华林,左治江,兰箭,钱东升.沟球断面环件冷辗扩三维有限元模拟与工艺设计J.机械工程学报,2008,44(10):202-205 19 左治江.环件冷辗扩变形规律和工艺模拟研究D.湖北：武汉理工大学,2006 20 左治江.沟球环件冷辗扩过程中截面变化规律J.塑性工程学,2009,16(2):210-21321 A. 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 A two dimensional upper bound analysis of the ring rolling processwith experimental and FEM verificationsA. Parvizia,n, K. Abriniab,1aDepartment of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Hesarak, Tehran 14778-93855, I.R. IranbSchool of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Irana r t i c l e i n f oArticle history:Received 30 July 2013Received in revised form23 November 2013Accepted 18 December 2013Available online 28 December 2013Keywords:Ring rollingUpper bound methodAdmissible velocity fieldExperimentFEMa b s t r a c tIn this paper, an upper bound solution is used to determine the ring rolling power and force.An admissible velocity field and strain rates are derived from the parametric definition of streamlinesin the deforming zone. Minimizing the upper bound power with respect to neutral point position, theneutral point situation and the rolling force are determined. The present method is validated usingexperimental results extracted from ring rolling mill. Moreover, using ABAQUS/Explicit software, the ringrolling process was simulated in every respect. Comparison of the present upper bound results withthose from experimental study, finite element simulation and our previous slab analysis, have proved theaccuracy of the present analysis. As compared with the FEM simulation, this method is very muchquicker and less expensive and could be used as an engineering tool in the ring rolling industrialapplications.& 2013 Elsevier Ltd. All rights reserved.1. IntroductionRing rolling is a specialized form of rolling process for manu-facturing seamless rings in a wide geometrical variety with outerdiameters ranging from 100 mm all the way up to 8 m with cold orhot workpieces. Large rings like those used in power generationplants, aircraft engines and rolling stock wheels are usuallymanufactured by hot ring rolling and small ones such as innerand outer bearing races by cold ring rolling. Compared to a ringforging process which is different from a rolling process, advan-tages of the ring rolling process include that the working speed israpid, the temperature can be maintained, the production yieldcan be enhanced, and so on. Particularly, in the case of a rolled ringproduct that is manufactured by a ring rolling process, the grainflow line is continuously formed in the circumferential direction ofthe product, thus providing superior characteristics 1.The upper-bound method predicts an upper bound on thedeformation energy. The first step in applying the upper boundmethod is the identification of kinematically admissible velocityfields through an assumption of the shape of the streamlines inthe deformation zone. Based on this velocity field, the total powerdissipation for the metal forming process is calculated. Sincethis predicted power is inherently higher than the actual valuerequired for the process, the shape of streamlines and metal flowdistribution can be determined by minimizing the power dissipa-tion with respect to unknown variables of the velocity field.Over the years, some analytical, physical and numerical model-ing methods including experimental formulae 2, slip line fieldsmethod 3,4, slab method 5 and FE analysis 6 are usedby many researchers to predict the final results of ring rollingprocesses. But, because of some difficulties in the ring rollingprocess including the unsteady state nature of the process (beingtime dependent),the continuously varyingdeformingzone,unsymmetrical nature of the geometry, complexity of the bound-ary conditions and the irregular behavior of friction force at thedeforming zone, the upper bound analysis of this process has beenremained so limited. In other words, there is hardly any upperbound analysis of ring rolling process except that given by Ryooet al. 7 and Yang and Ryoo 8. Taking into consideration the ringwith rectangular cross section, Ryoo et al. 7 derived the dual twodimensional velocity-field and developed an upper-bound solu-tion for the ring rolling process. By using the same velocity field,Yang and Ryoo 8 proposed a concept of equivalent coefficient offriction and derived a relationship based on the process para-meters. But, their proposed velocity field has some errors whichwill be discussed in details later.Using the upper bound elemental technique (UBET), Hahn andYang 9 proposed an analytical method for the simulation of rolltorque and profile formation during profile ring rolling. The plasticflow of material during rolling was described by dividing thewhole deforming region at the roll gap into simple elementsContents lists available at ScienceDirectjournal homepage: /locate/ijmecsciInternational Journal of Mechanical Sciences0020-7403/$-see front matter & 2013 Elsevier Ltd. All rights reserved./10.1016/j.ijmecsci.2013.12.012nCorresponding author. Tel./fax: 98 21 44865239.E-mail addresses: parvizisrbiau.ac.ir (A. Parvizi), Cabriniaut.ac.ir (K. Abrinia).1Tel.: 98 21 82084026; fax: 98 21 88013029.International Journal of Mechanical Sciences 79 (2014) 176181of rectangular cross-section. Considering a new element with acurvilinear side, Hahn and Yang 10 carried out an upper-boundanalysis for rolling rings with a circular groove or protrusionhaving a round corner fillet. The upper bound elemental techniquewas used by Alfozan and Gunasekera 11 to determine theoptimum intermediate shape for profile ring rolling using back-ward simulation. The ring was divided into features, whichprovided an approximated profile consisting of a number ofrectangular elements. Later on, Alfozan and Gunasekera 12 testedthe two simulations (forward and backward) with differentprofiles of disks. Furthermore, the simulation of complex ringprofiles using an improved Upper Bound Elemental Techniquewas presented by Ranatunga et al. 13. The profile of the ringwas approximated with a collection of triangular-prismatic andrectangular-brick elements. Ranatunga and Gunasekera 14 alsopresented the UBET-based preform design tool as a process anddie design tool for multistage forging processes.In this paper, a modified solution based on upper boundanalysis of Ryoo et al. 7 and Yang and Ryoo 8 is presented tosolve the ring rolling process. The ring with rectangular geome-trical cross section is assumed to be isotropic, incompressible andfollows a rigidperfect plastic behavior. The elastic deformationsare assumed to be negligible. An admissible velocity field andstrain rates are derived from the parametric definition of stream-lines in the deforming zone. Using the proposed velocity field, theinternal power of deformation, the shear power, the frictionalpower as well as the upper bound on power for the ring rolling areobtained, parametrically. Minimizing the upper bound power withrespect to neutral point position, the neutral point situation andthe rolling force are determined. In addition, the measuring andmonitoring systems for the ring rolling machine are designed andset up. The present method is validated using an experimental ringrolling mill equipped in the Department of Mechanical Engineer-ing at Tehran University. Also, the ring rolling process is simulatedin the ABAQUS/Explicit software.2. Upper bound analysis2.1. Ryoo et al. 7 and Yang and Ryoo 8 velocity fieldInvestigating the literature of ring rolling process reveals thatthe almost only admissible velocity field for carrying out the upperbound analysis has been given by Ryoo et al. 7 and Yang andRyoo 8. Considering a ring with rectangular cross section, theyderived the dual two dimensional velocity-field based on therotational velocity of the main roll and the linear velocity of themandrel. According to this dual velocity field, they developed anupper-bound solution for the ring rolling process. Studying theirproposed velocity field, the following inaccuracies are noted:?The velocity component in the x direction (Vx) at the entryboundary plane was chosen constant (V0 Const:), whereas,the circumferential velocity component (Vx) in the ring withthe rotational movement should change in the radial direction.?In the resulting velocity field, the component in the x direction(Vx) at the deforming zone is constant at any radial plane anddoes not change in the radial directin (y), while, in order tohold the circular shape of the ring, the material velocity in the xdirection (Vx) at the outer radius of the ring should be higherthan those at the inner radius.?At the neutral point (x Ln), the circumferential velocitycomponent based on the rotational velocity of the main rollwas considered to be equal to the circumferential velocity ofthe main roll (V1xjx Ln Vr). Also, the component based onthe linear velocity of the mandrel was supposed to be zero(V2xjx Ln 0). But at the neutral point, the total x-componentof velocity in the deforming zone should be assumed to beequaltothecircumferentialvelocityofthemainroll(Vxjx Ln V1xV2xjx Ln Vr).Considering the above points, a two dimensional velocity fieldwas derived for expressing the material plastic flow in thedeforming zone of the ring rolling process and an upper boundsolution was attained.2.2. Deformation zone geometryA ring of rectangular cross section and a uniform unit thicknessis rolled through a pair of cylindrical rolls. Fig. 1 shows thecoordinates and notations. The coordinate axes in the rolling andthickness (radial) directions are the x and y, respectively. Theorigin of the Cartesian coordinate used is taken to be at the middleof the exit boundary plane. hiand hodenote the thickness of thering at entry and exit of the roll gap, respectively. R1, R2and R3iaredesignated to main roll, mandrel and ring inner radii, respectively.The contact length is small as compared with the roll circum-ference and can be obtained as follows 7:LR21?R1R2ho?hi2R21?R222R1R2ho?hi()224351=21If the slip between the main roll and the ring workpiece isneglected, then the circumference of the outer surface of the ringworkpiece in one rotation is equal to the perimeter of the mainroll. Considering this fact, the decrease in ring thickness in anyNomenclatureFMain roll force per unit widthH;hInitial and variable ring thicknesses at roll gapJnUpper bound powerLThe contact lengthR1;R2Radius of the main roll and the mandrelR3i;R3o;RmInner, outer and mean radius of the ringTRolling torqueVx;VyVelocity in the x and y direction at the deforming zoneWWidth of the ringWi;Ws;WfInternal, shear and friction powerf1x;f2x Functions for surfaces definition of the main roll andthe mandrelhi;hoRing thicknesses at entry and exithxFunction for definition of the mid-line at the roll gapgxFunction for definition of the roll gapkMean shear yield stress of the ringmFriction factor at the main roll and the ring interfacenRotational speed of the main rollvFeed speed (linear speed of mandrel)x;yHorizontal and vertical distance from the exit pointand centerline of the ringxn;LnNeutral point at the main roll and ring interfacehDecrease in the ring thicknesssYield stress of the ring_ x; _ yNormal strain rates at the x and y direction_ xyShear strain rateA. Parvizi, K. Abrinia / International Journal of Mechanical Sciences 79 (2014) 176181177rotation is given by Lin and Zhi 15h hi?hovnR3oR12The deforming zone is defined as shown in Fig. 2, bounded bytwo flat surfaces at the entry and exit and the surfaces of the mainroll and the mandrel. The peripheral surfaces of the main roll andthe mandrel are expressed by y f1x and y f2x, respectively.From simple geometrical analysis, the expressions representingthe main roll and the mandrel surfaces are given in the followingforms 8:f1x ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR21?x2qR1h0=2f2x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR22?x2qR2?h0=23For simplicity of formulation, hx and 2gx are defined as thecenter and depth of the roll gap at any position from the x-axis,respectively. Thushx f1xf2x?=2gx f1x?f2x?=24In the development of the mathematical model, it was assumedthat the rolls were rigid and the material being rolled was rigidperfectly plastic. For rings, where the width is much larger thanthe thickness, lateral flow is negligibly small. So, the plasticdeformation was supposed to occur under plane strain conditions.The friction factor multiplied by the shear yield strength ( mk)is used to present interface friction between the main roll and thering. The friction factor between the main roll and the ring isassumed constant along the contact length. Since the mandrel isidle, there is no power transmitted to it except frictional loss atthe mandrel-ring interface and that at the mandrel bearings. Thepower loss at the mandrel is very small and can be practicallyignored. Then the analysis may be simplified by assuming africtionless mandrel (i.e. m0 at the mandrel-ring interface) 16.In passing through the roll gap (Fig. 2), the ring thickness wassteadily reduced and the ring radius progressively increased fromentry to exit. The main roll surface speed was taken to be equal toan intermediate value between the ring entry and exit speeds. Thepoint at which the ring circumferential speed and the main rollsurface speed became equal was called a neutral point. The ringwas advanced by the frictional force acting from entry towards theneutral point and opposed by the frictional forces acting fromneutral point to the exit. Thus, the plastic deformation zone wasdivided into two distinct regions, zone I from the exit to theneutral point and zone II from the neutral point to the entry.2.3. Assumed velocity fieldIt is obvious that the plastic flow of ring material in thecircumferential direction has to increase in the ring diameterduring the process. Having higher material velocity in the xdirection (Vx) at the outer radius of the ring than those one atthe inner radius was the necessary condition to hold the cylind-rical shape of the ring. From the nature of the plastic flow in thedeforming zone, the 2nd order polynomial function was consid-ered to define the variation of circumferential velocity in the radialdirection as follows:Vx y2C1xyC2x5where C1x and C2x were two unknown functions that relate Vxto the variation of x parameter. According to Eq. (5), in each radialcross section at the roll gap, the velocity in the horizontal directionwas varied with 2nd order polynomial. Incompressibility conditionof the work material in the plastic zone requires_ x_ yVxxVyy 06Substituting Eq. (5) into Eq. (6) and integrating the subsequentresult with respect to y leads toVy ?dC1xdxy22?dC2xdxyDx7Considering Eqs. (5) and (7), five boundary conditions arerequired to find the unknowns as follows:f1gy hx ) Vy 0f2gy f1x ) Vy ?v=2f3gy f2x ) Vy v=2f4gRf10f20Vxdy Rf1xnf2xnVxdyf5gx xn& y f1x ) Vx ?R1n8In the items 13 of Eq. (8), it was assumed that themagnitude of the radial velocity (Vy) at the main roll and themandrel surfaces are maximum and equal to v=2, then it decreasesgradually till vanishes along the center line of the roll gap, hx.Furthermore, the conditions of the volume constancy and no slip(neutral) point on the main roll and the ring interface are given initems 4 and 5 of Eq. (8), respectively.Fig. 1. Ring rolling geometry.Fig. 2. Geometry of deforming zone.A. Parvizi, K. Abrinia / International Journal of Mechanical Sciences 79 (2014) 176181178Imposing items 13 of Eq. (8) to Eq. (7) and using subse-quent results with items 4 and 5 in Eq. (5), the kinematicallyadmissible velocity field for the plastic deformation zone in ringrolling process were obtained as follows:Vx y2ayv2gxxbVyv2gxy?hx9wherea C?B ?R1n?f21x?v2gxnxn?A?Bf1xnb ?R1n?f21xn?af1xn?v2gxnxn10andA 12f210?f220?f21xn?f22xn?B f10?f20?f1xn?f2xn?C 13f31xn?f32xn?f310?f320?vL2gLf1xn?f2xn?112.4. Power balanceThe upper bound theorem states that among all kinematicallyadmissible velocity fields, the nearly actual one minimizes thefollowing expression 7:Jn2ffiffiffi3psyZVffiffiffiffiffiffiffiffiffiffiffiffi12_ ij_ ijrdVmkZSfjVdjdskZSejVsjds12In the right side of Eq. (12), the first, second and the third termsare designated to internal power of deformation, frictional power atthe roll-material interface and shear power at the plastic bound-aries, respectively. As the working material was assumed to berigid-perfectly plastic and the Mises yield criterion was applied, theinternal power consumption was calculated as_Wisyffiffiffi3pZL0Zf1xf2xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_ 2x_ 2y_ 2xyqdydx13The strain rates associated with the assumed velocity field caneasily be derived using Eq. (9) as follows:_ xVxxv2gx?xg0xg2xhi_ yVyyv2gx_ xyVxxVxx 2ya?v2h0xgx?hx?y?g0xg2xhi14Taking into consideration that jVdj is the velocity differencealong the arc of contact between the main roll and the ring andinterfacial friction is characterized by constant friction factor, m,the frictional power consumption is computed as_Wf msyffiffi3pRSfjVdjds msyffiffi3pRxn0Vxjy f1x?R1ndxRLxn?Vxjy f1xR1ndx?15The entry and exit planes are the velocity discontinuity planesin the ring rolling process. Supposing that the ring materials enterand exit the deforming zone vertically, the only velocity disconti-nuities will be as a result of the radial velocity. Therefore, the shearpower consumption at the velocity discontinuities is calculated as_Wssyffiffi3pRSejVsjdssyffiffi3pRf10f20jVyjx 0jdyRf1Lf2LjVyjx Ljdy?16Substituting velocity fields from Eq. (9) into Eqs. (13), (15) and(16) and substituting the subsequent results in Eq. (12), the upperbound power for the ring rolling process is obtained as a function ofneutral point position (xn). Therefore, the quantity xnis regarded asa pseudo independent process parameter and is subjected to thepower optimization. When values of xnvary from the exit plane(x 0) to the entry plane (x L), the resulting power (Jn) initiallydecreases, passes through a minimum and then increases. It can besaid that the almost the actual value of xnis the one whichminimizes the power Jn, and the correspondent power is nearlythe actual one.The external power (Jn) is the sum of the power imposed by thedriving torque of the main roll and the power imposed by thepressing load of the mandrel. Since the upper bound on power ishigher than the real power, the following relation is concluded:JnZTnFv17Substituting the driving torque from Eq. (17) of our previousstudy 5 into Eq. (17), the ring rolling force per unit width of thering based on the upper bound solution is obtained as follows:F Jn?msyffiffiffi3p L?2xnR1n?=v183. ExperimentsIn order to accomplish experimental remarks and verify thepresent theory, a measuring and monitoring system for the ringrolling machine was designed and set up. This machine had beendesigned and built in the Department of Mechanical Engineeringat Tehran University.3.1. Ring rolling machineIn order to measure the experimental data, the measuring andmonitoring system for the ring rolling machine was designed andset up. The ring rolling machine is shown in Fig. 3(a). The machineconsists of a main roll, a mandrel, two radial and two axial guiderolls, an electric motor, a hydraulic power pack, four sensors and amonitoring system. The main roll and the mandrel were made ofsteel CK60 and the guide rolls were made of steel CK45. Thediameter of main roll, mandrel, radial and axial guide rolls are 400,90, 150 and 120 mm, respectively. The power of electric motor is6.7 hp. Two pressure sensors were adopted to measure themandrel cylinder pressure and the hydraulic motor pressure. Aninductive sensor with 12 magnetic particles is used to measure therotational velocity of the ring. Also, using a digital ruler, the instantposition of the mandrel is measured and its linear velocity iscalculated. Programming the PLC type step7-200, the sensorsignals became ready for monitoring and the graphical resultsare shown in WinCC software.3.2. Ring rolling testThe material used for the billet is lead having a yield strengthof 15 MPa which was obtained using a standard compression test.The primary ring billet was manufactured by casting. Afterassembly of the ring, the machine started working with electricmotor and hydraulic units while the resulting parameters weremeasured and graphically shown in WinCC software. The finalstate of test process are shown in the Fig. 3(b). Furthermore, inTable 1, the primary and final geometry of the ring are given. Fortytimes of ring rotation were done in this test and the mandrelvelocity was equal to 0.4 mm/s.4. Finite element simulationUsing ABAQUS/Explicit, the experimental machine presented inthe previous section was modeled and the ring rolling process wasA. Parvizi, K. Abrinia / International Journal of Mechanical Sciences 79 (2014) 176181179simulated (Fig. 4). The 3D elements used for meshing the work-piece were eight nodes C3D8R. All the rolls considered in theexperiment section are modeled and the simulation showedthat the ring thickness was steadily reduced and the ring radiusprogressively increased by doing the process. The numbers ofelements at the deforming zone were considered to verify themesh sensitivity. This part of the simulation is vital for completingthe applicable analytical part of the work. However, the obtainedresults from the finite element analysis were used to evaluate thering rolling force and compare it with the analytical results.5. Results and discussionThe results obtained from the analytical and numerical (finiteelement method) solutions as well as experimental observationhave been illustrated in Figs. 5 and 6 and Table 3. Since theproposed velocity field is the modified version of velocity fieldgiven by Ryoo et al. 7 and Yang and Ryoo 8, their processparameters are taken in to consideration to do the analysis(Table 2).In Fig. 5, results of the present modified upper bound analysisare compared with the FEM simulation results and the upperbound and experimental studies of Ryoo et al. 7 and Yang andRyoo 8. In this figure, the ring rolling forces are plotted withrespect to the current mean radius of the ring. It could be seenthat the values for force obtained from the present upper boundmethod are in good agreement with experimental and FEMones. Considering the experimental results, the differences in thevariation of the required force during the process in two upperbound method are noticeable. Based on the modified solution,the necessary force for accomplishment of the process increasesslowly with increase in the ring mean radius. The same result wasTable 1The primary and final geometry of the ring (mm).Ring inner radius (R3i)Ring outer radius (R3o)Ring width (W)Primary60.5108.5158Final94124158Fig. 4. The finite element model.10012014016018020022024000.511.522.53Mean Radius of Ring (mm)Load (KN)/y (N/mm2): Experiment 7-8: Upper Bound Method 7-8: Upper Bound Method (Author): FEM (Author)Fig. 5. Ring rolling force with respect to the mean radius of the ring. Comparison ofpresent modified upper bound analysis with FEM simulation results and thetheoretical and the experimental results of 7,8.84868890929420253035404550Mean radius of Ring (mm)Load (KN): Experiment: Upper Bound Method: FEMFig. 6. The ring rolling force with respect to the current mean radius of ring.Comparison of upper bound results with experimental and FEM results.Table 2Primary parameters of ring rolling process 7,8.Main rollRingRadiusR1 275 mmInner radiusR3i 60 mmRotational velocityn 47 rpmOuter radiusR3o 107:5 mmWidthW 52 mmMandrelRadiusR2 45 mmFeed Speedv 1:2 mm=sFig. 3. (a) The ring rolling machine. (b) The final state of the test process.A. Parvizi, K. Abrinia / International Journal of Mechanical Sciences 79 (2014) 176181180also concluded in our previous study 5 based on the slab analysisof the ring rolling process. But, according to upper bound analysisof the Ryoo et al. 7, there is a descending behavior in the forcevariation during the process which is contrary to experimentalevidence. Moreover, in comparison to the experimental results, themodified solution gives upper values for force which verify theaccuracy of the present solution.The results obtained from the experimental test of the ringrolling process and the FEM simulation are shown in Fig. 6 alongwith the modified upper bound results. It could be seen that thereare good conformities between the theoretical, the FEM and theexperimental work. According to this figure, the higher ring rollingforces are required for the ring with bigger diameter. Therefore,this result also confirms the consequences of Fig. 5.Considering the primary parameters of the experiment, theresults for mandrel force at the end of third rotation of the ringfrom upper bound analysis, experimental study, finite elementsimulation and our previous slab analysis of the process 5 aregiven in Table 3. According to this table, the maximum deviation inthe obtained results is 11% and occurs between the results of slabanalysis and FEM simulation. Using the Core 2 Duo CPU 2.4 GHz, itwas taken more than 2 h to simulate only the three rotation of thering, while the upper bound calculation for the same problemtakes less than 5 min to complete which is 4% of the FEM time.6. ConclusionsModifying the velocity field given by Ryoo et