1、12FTM12AGMA Technical PaperManufacturing Methodof Pinion Member ofLarge-Sized SkewBevel Gears UsingMulti-Axis Control andMulti-Tasking MachineToolBy I. Tsuji, Iwasa Tech Co., Ltd.,K. Kawasaki, Niigata University,and H. Gunbara, Matsue NationalCollege of TechnologyManufacturing Method of Pinion Membe
2、r of Large-SizedSkew Bevel Gears Using Multi-Axis Control and Multi-TaskingMachine ToolIsamu Tsuji, Iwasa Tech Co., Ltd., Kazumasa Kawasaki, Niigata University,and Hiroshi Gunbara, Matsue National College of TechnologyThe statements and opinions contained herein are those of the author and should no
3、t be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractInthispaper,amanufacturingmethodofthepinionmemberoflarge-sizedskewbevelgearsusingamulti-axiscontrol and multi-tasking machine tool respecting an existing gear member is proposed. First, the toothsur
4、face forms of skew bevel gears are modeled. Next, the real tooth surfaces of the gear member aremeasured using a coordinate measuring machine and the deviations between the real and theoretical toothsurfaceformsareformalizedusingpolynomialequations. Itispossibletoanalyzethetoothcontactpatternofthe s
5、kew bevel gears with respect to the deviations of the real and theoretical tooth surface forms byexpressing the deviations as polynomial equations. Further, the deviations of the tooth surface form of thegear member are reflected in the analysis of the tooth contact pattern and transmission errors,
6、and the toothsurface form of the pinion member that has good performance mating with the existing gear member isdetermined. Finally, the pinion member is manufactured by swarf cutting using a multi-axis control andmulti-tasking machine tool. Afterward, the real tooth surfaces of the manufactured pin
7、ion member aremeasuredusingacoordinatemeasuringmachineandthetoothsurfaceformerrorsaredetected. Inaddition,the tooth contact pattern of the manufactured pinion member and existing gear member is compared withthose of tooth contact analysis. The results show that there is good agreement.Copyright 2012
8、American Gear Manufacturers Association1001 N. Fairfax Street, Suite 500Alexandria, Virginia 22314October 2012ISBN: 978-1-61481-043-83 12FTM12Manufacturing Method of Pinion Member of Large-Sized Skew Bevel Gears UsingMulti-Axis Control and Multi-Tasking Machine ToolIsamu Tsuji, Iwasa Tech Co., Ltd.,
9、 Kazumasa Kawasaki, Niigata University,and Hiroshi Gunbara, Matsue National College of TechnologyIntroductionBevel gears are used to transmit power and motion between the intersecting axes of the two shafts, and aremostoftenmountedonshaftsthatare90degreesapart. Theymayhavestraight,Zerol,spiral,andsk
10、ewteeth1 2 3 4, and occupy an important place in gear transmissions 5.Thetransmissionofstraightbevelgearsisregardedasaparticularcaseofskewbevelgears6. Thecontactratio of skew gears is larger than that of straight bevel gears because skew bevel gears have oblique teeth.Such skew bevel gears are used
11、at the power generation plants when the gears have large size. In recentyears, the renovation of these plants has been active due to the age of the plants. At the same time, it hasbecomenecessarytoreplacetheskewbevelgearsintheplants. Inthissituation,there arecases whereonlythe pinion member is chang
12、ed. It then becomes necessary to manufacture a pinion member that has goodperformance mating with the existing gear member.It is now possible to machine the complicated tooth surface due to the development of multi-axis control andmulti-taskingmachinetools78. Therefore,highprecisionmachiningoflarge-
13、sizedskewbevelgearshasbeen expected.In this paper, a manufacturing method of the pinion member of the large-sized skew bevel gears usingmulti-axiscontrolandmulti-taskingmachinetoolrespectinganexistinggearmemberisproposed. Theman-ufacturingmethodhastheadvantages ofarbitrary modificationof thetooth su
14、rfaceand machiningof thepartwithout the tooth surface 9.First, the tooth surface forms of skew bevel gears are modeled mathematically. Next, thereal toothsurfacesofthegearmemberaremeasuredusingacoordinatemeasuringmachine(CMM)andthedeviationsbetweenthe real and theoretical tooth surface forms are for
15、malized using the measured coordinates. It is possible toanalyze the tooth contact pattern and transmission errors of the skew bevel gears with consideration of thedeviations of the real and theoretical tooth surface forms by expressing the deviations as polynomial equa-tions. The components of the
16、deviations of tooth surface forms which correspond to the distortions of heattreatment and lapping, etc., are used because the motion concept may be implemented on the multi-taskingmachine. Further, the deviations of the tooth surface forms of the gear member can be reflected in theanalysisofthetoot
17、hcontactpatternandtransmissionerrors,andthetoothsurfaceformofthepinionmemberthathasgoodperformancematingwiththeexistinggearmemberisdetermined. Finally,thepinionmemberismanufactured by a swarf cutting that is machined using the side of the end mill using amulti-axis controlandmulti-tasking machine to
18、ol. Afterward, the real tooth surfaces of the manufactured pinion member weremeasured using a CMM and the tooth surface form errors were detected. Although the tooth surface formerrorswerelargerelativelyonthecoastside,thoseweresmallonthedriveside. Inaddition,thetoothcontactpattern of the manufacture
19、d pinion member and the provided original gear member was compared with theresults from tooth contact analysis and there was good agreement.Tooth surfaces of skew bevel gearsIn this section, the tooth surface forms of skew bevel gears are modeled mathematically. In general, thegeometry of the skew b
20、evel gears is achieved by considering the complementary crown gear as thetheoretical generating tool. Therefore, first the tooth surface form of the complementary crown gear isconsidered.4 12FTM12The number of teeth of the complementary crown gear is represented by:zc=zpsin p0=zgsin g0(1)wherezcis n
21、umber of teeth of complementary crown gear;zpis number of teeth of the pinion;zgis number of teeth of the gear;p0is pitch cone angles of the pinion;g0is pitch cone angles of the gear.Figure 1 shows the tooth surface formof thecomplementary crowngear assumingto bestraight bevelgearswith depth-wise to
22、oth taper. O-xyz is the coordinate system fixed to the crown gear and z axis is the crowngearaxisofrotation. PointPisareferencepointatwhichtoothsurfacesmeshwitheachotherandisdefinedinthecenteroftoothsurface. Thecirculararcswithlargeradiiofcurvaturesaredefinedbothinxzandxyplanes.xzand xyplanes corres
23、pond to the sections of the tooth profile and tooth trace of the tooth surface, respect-ively. This curved surface is defined as the tooth surface of the complementary crown gear. The followingequationsyieldconsideringtherelationsbetweenc,c,andMninxz,andbetweens,s,andbinxyplanes,respectively 10.Sinc
24、eskewbevelgearshaveteeththatarestraightandoblique,theskewbevelgearshavetheskewangleasdescribedinFigure 2. Therefore,thecomplementarycrowngearalsohastheskewanglethatisdefinedas.The tooth surface of the complementary crown gear is expressed in O-xyz using cand s:s=s2+b242 s(2)c=c2+ Mncos22 cwherecis r
25、adius of the curvature of the circular arcs in the xz plane, and has influence on c;c is amount of tooth profile modification;Mnis the normal module; is pressure angle;sis radius of curvature of the circular arcs in the xy plane, and influence on s;s is amount of tooth profile crowning;b is face wid
26、th.Figure 1. Tooth surface form of complementary crown gear5 12FTM12Figure 2. Skew angle of complementary crown gearX (u,) = c(cos cos) s(1 cosu) + ssinu tanssinu + Rmc(sin sin)(3)whereX is position vector of tooth surface of complementary crown gear in O-xyz;u is parameter which represents curved l
27、ines; is parameter which represents curved lines;Rmis mean cone distance.The unit normal of X is expressed by N.X expresses the equation of the tooth surface of the complementary crown gear. The complementary crowngear is rotated about z axis by angle and generates the tooth surface of the skew beve
28、l gear. This rotationangle, , of the crown gear, is thegenerating angle. Whenthe generatingangle is, XandNare rewrittenasXandNinO-xsyszsassumingthatthecoordinatesystemO-xyzisrotatedaboutzaxisbyinthecoordinatesystem O-xsyszsfixed in space. When is zero, O-xsyszscoincides with O-xyz.Assuming the relat
29、ive velocity W (X) between the crown gear and the generated gear at the moment whengenerating angle is , the equation of meshing between the two gears is as follows 11 12:Nu, ; Wu, ; = 0(4)whereNis unit normal vector of Xin O-xsyszs;Xis position vector of tooth surface of complementary crown gear in
30、 O-xsyszs; is parameter representing rotation angle of complementary crown gear about z axis;Fromequation4,wehave=(u,). Substituting(u,)intoXandN,anypointonthetoothsurfaceofthecrowngearand itsunit normalare definedby acombination of(u, ),respectively. When thetooth surfaceofthe complementary crown g
31、ear in O-xsyszsis transformed into the coordinate system fixed to the generatedgear, the tooth surface of the skew bevel gear is expressed. The tooth surfaces of the pinion and gear areexpressed as xpand xg, respectively. Moreover, the unit normals of xpand xgare expressed as npand ng,respectively.
32、Henceforth, the subscripts “p” and “g” indicate that each is related to the pinion and gear,respectively.Measurement of gear memberManufacturing errors occur in bevel gear cutting. Whether the mathematical model as mentioned earlier fitsthe real tooth surface of the existing gear member or not is no
33、t obvious. Therefore, the tooth surfaces of thegear member are measured using a CMM and the deviations between the real and theoretical tooth surfaceforms are formalized.Coordinate measurement of real tooth surfaceThetheoreticaltoothsurfacesofthegearmemberisexpressedasxg(ug,g)asmentionedearlier. Agr
34、idofnlinesandmcolumnsis definedand apoint calledthe referencepoint isspecified onthe toothsurfaces ofboth6 12FTM12drive and coast sides. The reference point is usually chosen in the center of the grid. The position vectorxg(x, y,z),namely,u,andaredeterminedforthesolutionofsimultaneousequationsconsid
35、eringonepointon the grid of the tooth surface and the unit normal (nx, ny, nz) of the corresponding surface point is alsodetermined since u, , and are determined 13.For measurement, the gear member is set up arbitrarily on a CMM whose coordinate system is defined asOm- xmymzm. We can make origin Oma
36、nd axis zmcoincide with the origin and the axis of the gear member,respectively. The whole grid of surface points together with the theoretical tooth surfaces is rotated about zmaxis so that ymis equal to zero at the reference point. Therefore, the position vector of the point and its unitnormal are
37、 transformed into the coordinate system Om- xmymzmand are represented by:ni=nix, niy, nizT(5)xi=xi, yi, ziTi = 1, 2,2nmwherex(i)is position vector of i-th point of tooth surface in Om- xmymzm;n(i)is unit normal vector of x(i);The real tooth surface of the gear member was measured using a CMM (Sigma
38、M and M3000 developed byGleason Works). When the real tooth surface is measured accordingto theprovided grid,the i-thmeasuredtooth surface coordinates are obtained and is numerically expressed as the position vector 13 14:xim=x(i)m, yim, zimTi = 1, 2,2nm (6)wherexm(i)is position vector of i-th measu
39、red tooth surface coordinates in Om- xmymzm.When the deviation between the measured coordinates and nominal data of theoretical tooth surfaces foreach point on the grid is defined toward the direction of normal of theoretical tooth surface, i-th can bedetermined by:i=xim xinii = 1, 2,2nm (7)where(i)
40、is deviation between measured coordinates and nominal data of tooth surface for each point on gridtoward direction of normal of tooth surface;isequaltozeroatthereferencepoint. Thefundamentalcomponentsofthedeviationsoftoothsurfaceformswhichcorrespondtothedistortionsofheattreatmentand lapping,etc., ar
41、eused becausethe motionconceptmay be implemented on a multi-tasking machine.Formalization of deviations of tooth surface formBasedonthemethodmentionedearlier,thedeviationforeachpointonthegridiscalculatedwhenthepointson the tooth surface are measured 15. However, it is difficult to fit to the theoret
42、ical tooth surface wellbecausevariesateachpointonthegrid. Therefore,wedefine(X,Y)whoseXandYaretowardthedirectionsof the tooth profile and tooth trace, respectively, and form the following polynomial expression: = 11+ 12+ 21+ 22+ 31+ 32+ 41(8)where11is a parameter defining deviation;12is a parameter
43、defining deviation;21is a parameter defining deviation;22is a parameter defining deviation;31is a parameter defining deviation;32is a parameter defining deviation;41is a parameter defining deviation.7 12FTM12Figure 3 shows the procedure formalizing the relation between the fundamental components of
44、polynomialexpression and the deviation of tooth surface form. First, the tooth trace deviation 11and tooth profiledeviation12are expressed as the following first order equations of X and Y using fundamental componentsa11and a12, respectively (see Figure 3a):a12=110.5 T(9)12= a12Ya11=110.5 H11= a11Xw
45、herea11is the fundamental component of polynomial expression;H is the range of the evaluation of the tooth surface in X directions;a12is the fundamental component of polynomial expression;T is the range of the evaluation of the tooth surface in Y directionsThe tooth trace deviation 21and tooth profi
46、le deviation 22are expressed as the following second orderequations of both X and Y using fundamental components a21and a22, respectively (see Figure 3b):a22=220.5 T2=4 22T2(10)22= a21Y2a21=210.5 H2=4 21H221= a21X2wherea21is the fundamental component of polynomial expression;a22is the fundamental co
47、mponent of polynomial expression.Figure 3. Procedure formalizing relation between fundamental components of polynomialexpression and deviation of tooth surface form8 12FTM12Further,thedeviations31and32inthedirectionsofthebias-inandbias-outare expressedas thefollowingsecond order equations of both X
48、and Y using fundamental components a31and a32, respectively (seeFigure 3c):a32=320.5 L02=4 31L20(11)a31=310.5 L02=4 31L2031= a31X cos1 Y sin121= tan1TH, L0=Hcos132= a32X cos1+ Y sin12wherea31is the fundamental component of polynomial expression;a32is the fundamental component of polynomial expressio
49、n;Thetoothtracedeviation41isexpressedasthefollowingthirdorderequations ofX andYusingfundamentalcomponents b1, b2, and b3, respectively (Figure 3d):41= b3X3+ b2X2+ b1X(12)b1,b2,andb3aredeterminedfromthefollowingconditions: isequaltozerowhenX=-0.5HandX=0.5H.Inaddition, is equal to 41when X =0.25H. Reflecting
