TOOTH CURVES AND ENTIRE CONTACT AREA IN PROCESS OF SPLINE COLD ROLLING

CHINESE JOURNAL OF MECHANICAL ENGINEERING●94.Vol. 21, No. 6, 2008DO: 10.3901/CIME 2008.06 094, available online at www.cjmenet.com; www.cjmenel.com.cnTOOTH CURVES AND ENTIRECONTACT AREA IN PROCESSZHANG DaweiSchool of Matenals Science andOF SPLINE COLD ROLLING*Engineeing,Northwestem Polytechnical UniversityXian 710072, ChinaAbstract: In spline rolling process, the contact area between roller and workpiece plays a importantU Yongtangrole in calculating rolling-force and rolling-moment. For the purpose of studying the contact area,contact state between roller and workpiece in process of spline cold rolling based upon cross rolling isanalyzed. According to the suitable hypothesis, the mathematic model of oller-tooth-curve inFU Jianhuaoptional position of rolling process is established. Combing the theory of conjugate curves with theSchool of Matrials Science andtheory of envelope curve, the corresponding mathematic model of workpiece-tootb-curve isEngineeing.establisbed. By utilizing establishing mathematic models, the algorithm of entire contact area inTajyuanUnivvrsity of Sciencerolling process is created. On the basis of the algorithm, calculation-program is compiled underand Technology,MATLAB program language environment. The calculation-program actualizes quantitative analysisand quantiative calculation of contact areas. Utilizing the calculation-program, the influence ofTaiyuan 030024 Chinaparameters on contact area is analyzed, and the tendency is consistent with the manufacturingexperience. In consideration of rolling- force optimization, the primary process parameters may beselected according to results of calculation. The result of the present study may provide basis forresearch on rolling- force and rolling-moment.Key words: Spline Cold rllig Contact area Mathematic model MATLABof workpiece-tooth-curve will be also established. So the0 INTRODUCTIONmathematic model of contact area in process of extemnal splineWith the rapid growth in manufacture, spline parts of highercold rlling may approach the true contact area in process.precision and better mechanical property are in great demand. 1 MATHEMATIC MODELS OFCold rolling is getting to be increasingly, utilized in manufacturingROLLER-TOOTH-CURVE ANDhigh strength and high precision splinet. Up to now, however, theWORKPIECE-TOOTH CURVEtheoretical research on extemal spline cold rolling is fewer,besides_ some papers about the mechanical analysis of theprocess(2]. And other related papers are calculation of blank 1,1 Coordinate and bypothesisdiameter and generalizing of manufacturing experienceAccording to the feature of extermal spline cold rlling, anySome metals have plasticity in cold (room-temperature) state, time in processf rolling, contact curve in cross-section isso the spline matched the die can be formed 4. Usually there are invariable in axial direction. If the contact curve is solved, thetwo shapes of die as fllows:①cylindrical die;②flat die. The contact area can be calculated as fllowspresent study is concermed chiefly with the study of rolling withA=S.L1)cylindrical die. Tbe principle of cold rolling process of extemalspline cold rlling with cylindrical die is as fllows 20:a couple where A-Entire contact areaof rllrs with identical-parameters, are paral!l assembled on two-Contact curve in cross-sectiondrive shafts, which are the rlling die, two shafts rotate in theL---Axial length of workpiecesynchronization and the same direction. At the same time, underThe rectangular coordinate systems on cross-section arethe action of the horizontal force that provided by bhydraulicestablished: Oxy andO,xyz (shown in Fig. 1). 02x2 is coupledsystem, two drive shafts infeed synchronized in radial direction,with roller. The coordinate angle日is the angle between yz axistwo drive shafts keep a constant ifeeding speed and continualand x axis (clockwise is negative and anticlockwise is positive).radial-infeed rolling until the extermpal spline is formed.Rolling-force and rolling-moment are prime processs of cold rollinAnd they are principal elements whenParamneterscalculate the fluid pressure of special spline cold rlling machineand the power of the motor driving the roller. Calculation andanalysis on the contact area in process of rolling are importantlink in calculation of rlling-force and rolling -moment. ZHANG,d-xet alt", studied the mathematic model of contact area betweenaddendum of the roller and workpiece. But the change of|芗workpiece-tooth-curve in rolling process wasn't described, andthe contact area of between the flank of roller and the workpieceFig. 1 Contact state in cos-sectionwasnconsidered.So this calculation of contact area isimperfection. In this study, the mathematic model coming fromThe following bhypotheses are proposed, when analysis on thethe Ref. [7] will be improved, and the entire contact area between change of contact area in process and establishing of mathematicroller and workpiece is able to consider well. Mathematic modelmodels中国煤化工and workpice isn'rThis project is supported by National Naturnl Science Foundation ofconsideChina (No. 50675145)。 Shanxi Province Key Project of Science andMYHc N M H Gner and wortpice isTechnology of China (No. 2006031147), Shanxi Province Innovationconstant in establishment of decrement. The decrement (ds) isProject for Graduate Students of China (No. 20061027) and the ShanxiProvince Key Project for Studied-abroad Reumee of China. Receivedresult that latter root radius of workpiecesubtracted from formerJanuary 22, 2008; received in revised form October 6, 2008; aceptedroot radius of workpiece in process of one rolling.October 28, 2008CHINESE JOURNAL OF MECHANICAL ENGINEERING(3) The rise per tooth is invariable in establishment of envelop curve of 1。9. So the chief problem is obtaining thedecrement.eavelop curve. Suppose the rotation angle of roller isφ, the1.2 Contact boundary of starting and separatingcoordinate conversion fom 1, to corresponding eavelop curveThe rllretooth-urve (4,) is composed oflwR, laa and (1。 ) is shown as fllowsshml. ZHANG, et al7, gave the parametric equations in natural(x(,)。cosip sinip)( cosP sinφsystem of coordinate, and they founded the formula of knuckle(y(,o)-sinip cosip八- - sinφ coso(6radius of roller-addendum (r.).In the starting age, l. R and the corresponding involute flank(52)+(( acosipi -asinip)of workpiece arc tangent, and the tangent point is K.. Accordingwhere i= ra2/r , is base radius of spline.to the theory of gear. engagemente), the point K, is cossing pointAccording to the theory of eovelop!9, the equations of。isbetween the former addendum circle of workpiece in one rllingcomposed of Eq. (6) and the fllowing equationand engaging line. So the coordinates (x,y%,)of K, in 0xy .ay(,o) ax(t,p)_ ay(t,o) ax(t,),can be solved. Then polar radius (Pn, ) and pressure angle (axn)a9 i=0(7of K, in wr may be expressed as followsThe individual expressions are put into Eq. (7), ie.|Px =(x,-a)Y+或sin(p-x)= +)(x(sin+t :(C0s2)(8)(2ia]a, = arcsPrwhereh= artan dx()/d" dy,()drwhere r- Base radius of rlleAccording to Eq. (8), there isAccording to geometrical relationship, in initial contactage can be expressed as follows中= arcsin(9arctan+β- iuva,)(3In 8 position, the workpice-toh-curve (L。) is composedof Eqs. (6) and (9). The points of 。may be obtained aboutwhere β一Angle between y2 axis and polar axis of involutedifferent t. In different curve, the parameter t and the expressionIn the same way, the point K. of contact point in separatingof x() and y2(t) are dference.age can be obtained. Then can be expressed as follows2 ENTIRE CONTACT CURVE IN CROSSINGb =arctans --β+ invax,(4)SECTIONAccording to the mathematic model based in sections 1.3 and1.4， tooth curves of roller and workpiece may be plotted by13 Roller-tooth-curveMATLAB (Fig. 2). In Fig. 2, the solid line, the dashed line and theIn[ur,8,], the parametric equations of the rllertoothodashed-dotted line represent the roller-tooth-curve, thcurve (4,) in Oxy can be expressed as fllowssworkpiece-tooth-curve and the addendum circle of workpiece,respectively. The parameters are a =37.5" , m=1.0 mm, f=0.5 mm(x=-rgsecancos(β-inva+6)+a(5a)in Fig. 2. It can be seen form Fig. 2 that lm contacts the的:{y= 6asecacsin(β- inva+e)workpiece firstly, afterwards lr and Love contacts the[a e[Qw,a]workpiece, respectively in scquential time. And in intermediate[x=rosin(θ-r)-(ra -r)cos0+asection of variation range of O(around 0=0), the contact area will的n: {y=r,.sin(θ-y)+(口-r,)sinθ(5b)reach the maximum.r∈[tanCu-β,π- tanGμ+B]0.[x =-rasecancos(inva- B+ e)+a0.0-LwL: y=fna.secasin(inva-β+日)(5c)(a e[aw,Qa].5-where a一Central distance between roller and workpiece-2.0-a,y一 -Parameters of equations-Pressure angle in tangent point between involute-2517.017.5 18.018.5 19.0 195 -0517.0175 18.0 18.5 19.0 19.sx/mmflank of roller and kouckle of rllraddendum(a) Afer contacting(b) Before separatingacas一Pressure angle in rller-dedendum circle wbenFig. 2 Tooth shapes of rller and workpieceengagement has not the headspaceTbe [0r,0r] is discrete to N points: B(0,) , 62 ,"1.4 Workpiece tooth .curveIn the process of splie cold rllig, the relative movement中国煤化工0 orolel in poesbetween roller and workpiece is similar to generative movement.of one rooth-curve is expressedIn θ position, the rllerooth-curve isILg, and it is better to takeYHC NMH Gred asla. In otherl instead of a andγ . The conjugatc curve of lgis tbewords,1g. is the workpiecetooth-curve. Then, the contact curveworkpiece-tootb-curve in θ position, and the conjugate curve isbetweenlg. and is entire contact curve in crossing-section.●96●ZHANG Dawei et al: Tooth curves and entire contact area in process of spline cold rllingWhen the distance between a couple of corresponding pointsmany, but their degree of contribution are different. Thecoming foml, andlg, , respectively is less than s(ε is small changing-curves of unit-contact. area about dfferent parameter arepositive number), the couple of points are considered to contactshown in Fig. 4, and the different paraneters of the eight curves inwith each other. One by one, the distance between couple ofFig. 4 are shown in Table. When tbe pressure angle is different,corresponding points may be calculated through numericaladdendum cofficient and dedendum coeficient are different, andtheir values are given by Ref. [10]. The reason of different contactother. Then the endpoints of contact curve may be found.area about parameters is that changing parameters will cbange theTherefore the s can be obtained. Based on this algoribm, the contat state between rller and wokpiee. Fig 5 shows the:urve length calculated program of entire contact curve incontact state about diferent parameters intuitionally. In Fig. 5,crossing-section may be compiled. The program flowchart isθ=0 and f=0.5 mm.shown in Fig 3.2.000∠Input x271.0801.040-1.020中00-0.080L lete ]0.040S.=0NI Are the endpoints0.020(A and B) existent--0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.02Coordinate angle 0/ra<运有公NAmend Aand司Fig 4 Uni-contact-arca about dfferent paraneterTable Parsmeters In changlng-curves of contact areasin Flg.4sn=BCurvePressureModuleNumber of Number of Amount ofDecrementnglespline teeth roller teet infeedNal(m/mm2f/mmds/mm37.51.0200).30.05.53600.0).s0.0:( SIop400Fig. 3 Program flowchart of curve leagth calculated progrum30.0:00).5.057.50.03 ANALYSIS OF EFFECTING PARAMETERS.,5.03ON UNIT-CONTACT-AREA45.0Unit-contact-area is the contact area in unit axial length.2.0[According to program flowchart (Fig. 3) and Eq. (1), the main1.0-calculationam and supplementary subroutines (add up to 19program and Sprograms) are compiled under MATLAB program languagea.otenovironment. The changing tendency of unit-contact-area in0.5-process of one rolling may be observed. If the1.s-calculation-program is utilized to calculate about differentparameters, quantitative analysis on unit- contact-area can be done.-97.0 175 18.0 18.5 19.0 26.0 27.0 28.0 29.0Because roller-tooth-curve is composed of three curves, the(a)a=375,m=-1.0 mm(b)=37.5,m=1.5 mmcontact area can be calculated respectively, then add them up toentire contact area in process of one rolling. The maximal contactarea|etween flank of roller and workpiece is only more than halfof the maximal contact area between addendum of roller andwprkpiecce. But in initial contact age and in separating age, thecontact area between flank of roller and workpiece plays animportant role in rolling-force. The contact area between-1.5-addendum of roller and wprkpiecce appears primarily inintermediate section of variation range of θ. And its shape is中国煤化工$ 190almost symnetry abouty; axis. So the force on it isn't the chieffactor for rlling-moment, and the force detrminingMHC N M H Gmt prnmcersrolling-moment is the force on the contact area between fank ofroller and workpiece.Around 0=0, the contact area between山and workpice isThe process parameters efecting on entire contact area are more than two thirds of the whole contact area. Comparing Fig. 5aCHINESE JOURNAL OF MECHANICAL ENGINEERING.97.and Fig. 5b indicates that the module bigger the contact areapressure angle may change the contact area sharply. So they arebetween flank of roller and workpiece smaller around θ =0. The prime parameters.contact area between addendum of roller and wprkpiccce is(2) The contact area is increasing function of amount ofchanged sharply when the pressure angle and module are infed, decrement, number of the rller teeth and wokpiece teth.changedT. So it can be found from Fig. 4 that:①contact area in But the influences of decrement and umber of roller teeth areprocess of one rolling is on the increase when the pressure anglesmallin dividing circle is on the decrease, and this regularity changeis(3) When the number of workpiece teeth reaches criticalexhibited evidently in maximal value of contact area; ②value (Z=40-50), the contact area isn't changed along with theproportional relation between contact area and module is nearoumber changed.linearity.ReferencesIf the number of spline teeth is on the increase, the contactarea is on the increase. But a series of quantitative calculations[1] KLEPIKOV V v, BODROV A N. Precise sbaping of splied shafs inindicate that the cootact area is not changed when the number ofautomobile maoufacturing[J]。 Russian Engineering Research, 2003,spline teeth more than cnitical value. The contact area is23(12): 37-40increasing function of amount of infeed. So rolling -force will] ZHANG Dawei, LI Yongtang, FU Jianbua, et al, Mechanics analysis onprecise forming process of extemal spline cold olligJ]. Chinese Journalreach the maximal value before end of olling (inishing rolling isof Mechanical Engineering, 2007, 20(3) 54-58.excluded), and the rolling-force at this time is maximal value in3] MASTUNAGA R TAKEMASU T, OZAKI T. Spline rling of depindustrial process.drawn cups [C/]Advanced Technology of Plasticit. Proeeding of theThe contact area is also increasing function of decrement and6th ICTP, Nuretmberg, DE. Sept. 19-24, 999 Bertin: Springer Verlag.oumber of roller teeth, but the degree of change is small. When4] ZHENG Quangang. Cold rlling process of spur somall module invoute1999: 2431-2436.the ratio of variation to oumber of roller teeth and the ratio ofspline D. Automobile Technology and Materia, 1997, 0000716-18.variation to decrement are near equal, tbe variations of contactarea are near equal. In consideration of operation, modifying the[5] WU Xiuyi. Cold rlling process for small module involute spline[].oumber of roller teeth is more difficult than varying decrement.Machinery, 1998, 000(001) 16- 18. (in ChineseAnd the number of roller teetb can't be decreased blindly, because[6] ASM Intermational Handbook Comitee Thread rlling (ASMhandbook machining, Vol. (16) [M/CD]. Materials Pank, Ohio: ASMthe operating life of roller and quality of workpiece are worse inIntemational, 2001.[7] ZHANG Dawei, LI Yongtang, FU Jianbua. Thcoretical calculation andsimulation analysis on rolling cotact area of exrernal spline cold rlling4 CONCLUSIONS闪Journal of Taiyuan University of Science and Technology, 2007, 28(1]):64-68. (inChinse)On the basis of the theory of the conjugate and the envelope,] WU Xutang. Principle of gear engagement[M]. Bejing: China Machinemathematic models of workpiece-tooth-curve and roller-tooth-Press, 1982. (in Chinese)curve in optional position of one rolling process are established.9] Department of Mathematics of Fudan University. Curve and curved faceAccording to the mathematic models, the eatire contact area10] ZHAN Zhaoping, CHANG Baoyin, MING Cuixin. Involute spline(including contact area between fank of roller and workpiece andstandards spplication manual M]. Beijing; Stundards Press of China,contact area between addendum of roller and wprkpiecce) in1997. (in Chinese)process of one rolling is solved. The contact area of between flankBlographical notesof roller and workpiece is considered well in the mathematicmodel, so the modei of contact area of the present study is more ZHANG Dawei, male, bom in 1982, isa PhD candidate in School of Materialsaccurate than mathematic model in Ref. [7]. Thus, in the present Science and Engineering, Norbwestem Plyechical Unvrsity, Xi'an, China.study, the contact area calculated may approach the true contact His reserch. interests include advanced plasic process and oumericalarea of extemal spline cold rolling.simulaion technology.According to identification rule of contacting andTel: +86-29-88460212-806; E-mail: zhangdawei2000@yaboo.com.cnmatbematic model of curves, the algorithm of entire contact areaLI Yongtang is cureatly a professor in School of Materials Science andis created and calculation-program is cormpiled. Utilizing thEngineering, Taiyuao Uaiversity of Science and Techology, China. Hisproggram, tbe entire contact area may be calculated and analyzedresearch interests inelude bydraulic hammer, elacto-hydraulic hammer,modeling and simulation of hydraulic system and automatic contolling ofquantitatively.hen the primary process parameters may beforging equipment, etc.selected. Therefore the tests selecting process parameters may be Tel: +6-351-6998029; E mail: lyogtyast.du.odecreased, and material and time may be saving.Based on the resuts of analyzing efet of parametrs on FU Jianhua is curealy . profeserr in Sctool of Materials Scienee aodEngineering, Taiyuan University of Science and Technology, China. Haunit-contact-area, the following conclusions may be concluded.research interests include Dew processing and tecbnology of material foraing,(1) The contact area is increasing function of module anddecreasing function of pressure angle. The changes of module andTe: +86-351 6963339; E-mail: bua5963@sina.com中国煤化工MYHCNMHG