1、09FTM05AGMA Technical PaperHypoloidtGears withSmall Shaft Angles andZero to Large OffsetsBy Dr. H.J. Stadtfeld, TheGleason WorksHypoloidtGears with Small Shaft Angles and Zero to LargeOffsetsDr. Hermann J. Stadtfeld, The Gleason WorksThe statements and opinions contained herein are those of the auth
2、or and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractBeveloid gears are used to accommodate a small shaft angle. The manufacturing technology used forbeveloid gearing is a special set up of cylindrical gear cutting and grinding machine
3、s.A new development, the so called Hypoloid gearing addresses the desire of gear manufacturers for morefreedoms. Hypoloid gear sets can realize shaft angles between zero and 20 and at the same time allow asecond shaft angle (or an offset) in space which provides the freedom to connect two points in
4、space.ThefirstapplicationofHypoloidsisfoundinallwheeldrivenvehiclesthatusetraditionallyatransfercasewithapinion/idler/geararrangementorachain. Inthosecases,theexitofthetransfercaseneedstobeconnectedwiththefrontaxle. Theobstaclehereisthefactthatthepropellershaftbetweenthetransfercaseandthefrontaxle w
5、ill require two CV joints, because the front axle input point has a vertical offset and is shifted sidewayswithrespecttothetransfercaseexit. Thetightpackagingofmodernvehiclesrequiresthepossibilitytooffsetthetwoconnectingpointsofafrontpropellershaft. HoweverthepenaltyforsuchadesignisthecostoftwoCVjoi
6、ntsaswellasthereducedefficiencyofthefrontdrivetrain,whichcouldrangebetween0.5%and2%ofthepower that flows through the front axle.Beveloidscanrealizeanangleinoneplane,whichinmostcasesisnotsufficienttoconnectthetwopointsinquestion without the additional requirement of two CV joints. Only the newly deve
7、loped Hypoloids canconnect those two points due to the shaft angle and the additional offset and obsolete the use of CV joints.TheHypoloidtechnologynotonlyreducescostandincreasesefficiency,italsohasanenhancedperformancecomparedwithbeveloidswithstraightteeth. ThecurvedHypoloidteethenhancetheNVHcharac
8、teristicandshow less contact displacement under load. Flank form generation, tooth contact analysis, Ease-Offcalculation and coordinate measurement with corrective feed back are already possible with todays cuttingand grinding machines.The Hypoloid technology does not apply only to automotive drive
9、trains. All of the mentioned advantagesapply also to aircraft as well as general gearbox manufacturing.Copyright 2009American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314September 2009ISBN: 978-1-55589-958-53HypoloidtGears with Small Shaft Angles and Zero
10、 to Large OffsetsDr. Hermann J. Stadtfeld, The Gleason WorksIntroductionIf two shafts are neither parallel nor perpendicular,but include a small angle in the plane that isdefinedby the axis of rotation, then two possible gearingsolutions are known to accomplish a motiontransmission.Onepossiblesoluti
11、oniscalledbeveloids. Beveloidsaremanufacturedlikecylindricalgearsusing forex-ampleahobbingprocessforsoftmanufacturinganda threaded wheel grinding for hard finishing. Shaftangles between 0 and 15 canbe realizedaccord-ingtotheBeveloid methodwhich resultsdependingon the ratio in gear pitch angles betwe
12、en 0 and7.5, or in case of the combination of one conicalgear with one conventional cylindrical gear, themaximal required pitch angle might be as high as15 1,2.The second possibility is the application of angularspiral bevel gears. The ratio in most real applica-tions is close to miter which results
13、 in pitch anglesbetween 0 and 7.5.The described gearsets are generally used inautomotive transfer cases to transmit rotation andtorquefromtheoutputshaftof atransmission tothefront axle of an all wheel driven vehicle.The mechanical function of both tapered cylindricalgears (Beveloids) and spiral beve
14、l gears is toprovide an angle between the shafts in the planethat their two axes define. In most cases concern-ing all-wheel drive vehicles, this will still requiretwoconstant velocity joints or two universal joints (oneon each end of the drive shaft) in order to connectthe output shaft of the gear
15、box with the input shaftof the front axle, which commonly have differentvertical locations.Inordertoconnect2pointsinspace,likeinthecaseof a propeller shaft between the output of a transfercaseandafrontaxleinput,itisnecessarytoprovideone angle and a linear offset or two angles inperpendicularplanes.
16、Hypoidgearsrepresentsuchageneralvalidsolutionofinput/outputshaftorienta-tion in three dimensional space. However, thefeaturesoftodayshypoidgeardesignsdonotcoverthe case of low shaft angle and high offset. Thedifferent hypoid theories applied today do not evenallow gear engineers to design low shaft
17、anglegearswithanyoffset. Thecommon hypoidtheoriesrely on a flat or conical generating gear as the basisfor basic setting and tool parameter calculation 3.Shaft angles close to and including 90 combinedwith ratios of 2.5 and higher lead to gear pitch coneangles of 68 and higher, and pinion pitch cone
18、angles of 22and lower. This leads to a typical ringgear whose cone is close to a plane, with a tangentplanetothepitchcone,whichiscloseenoughtothepitch cone in the neighborhood of the contacting li-ne. This allows to apply certain amounts of hypoidoffset, derived in the pitchcone tangentplane inthetr
19、aditionalhypoidtheory. Thetraditional theoryfailsincasesofhighhypoidoffsets(closeorequaltohalfthe ring gear diameter). The traditional hypoidtheory also fails in cases of low ratio (close to 1.0).In cases of high ratios, worm gear drives can beusedtorealizea90shaftangleandanoffsetofhalfthe gear diam
20、eter plus half the worm diameter (likecenterdistanceincylindricalgearing). Inthecaseoflow ratios, crossed helical gears can be used toachieve any desired shaft angle combined with anoffset equal to the center distance of those crossedhelical gears.The freedom of any small shaft angle (e.g., 0 to20)c
21、ombinedwithanyoffsetbetweenzeroandthesum of half the mean pitch diameters of the twomembers will for the first time be possible byapplying the Hypoloid system.The basics of hypoloidsThe generating principle is applied between thedrivingpinionandthedrivengear. Although inmostcases, pinion and gear mi
22、ght have the same num-ber of teeth, the gear is used as a generating gear.The new method even goes one step further anduses a non-generated gear with straight or curvedtooth profile as generating gear for the pinion. Inbevelandhypoidgearsthenon-generatedprincipleis generally only used in cases when
23、the ring gear4pitch cone angle is 68 and higher. The Hypoloidmethod derives, in a first step, bevel gear machinebasic settings for a non-generated gear member.Thosesettings areused toderive, ina secondstep,thebasicsettingsforabevelgeargeneratorinorderto manufacture the pinion. The cutter head for th
24、epinion cutting (positioned by the basic settings androtatedaroundthecradleaxis)representsonetoothof the non-generated gear member on which thepinion rolls during the generating roll process.The principle of applying a non-generated gearmemberinordertogeneratethematingpinionistheonly technique that
25、delivers a precise conjugaterelationship between pinion and gear, even if theaxes of the two members are not in one plane. Aconjugate basic geometry also requires the pitchcone to be parallel to the root cone. It has beenobserved that in case oflow shaftangle spiralbevelgearsets,thetoothdepthwascalc
26、ulatedtobetallerat the toe (reverse tooth taper) in order to fulfill therequirements of completing (matching tooththickness and opposite member slot width). OneelementoftheHypoloidgeometryisaparalleldepthtooth design, which will lead to more optimal toothproportions than a reverse taper and also ful
27、fill therequirement of parallelism between pitch line androot line. If the axes of the two members are in twoparallel planes, the distance between the planes isdefined as offset. In the case of conical pitchelements of the two members, this offset iscommonly called hypoid offset.One member is define
28、d as a pinion (which is thegenerated member) and one member is defined asa gear (which is the non-generated member). Inspiteofthetraditionaldefinition,Hypoloidgearsandpinions can have a similar number of teeth. It isevenpossiblethatthepinionhasahighernumberofteeth than the gear.The conjugacy between
29、 the two members is onlythe basis for the generating principle. In order tomake the gearset insensitive to tolerances inmanufacturing and assembly, a located contact isachieved using flank surface crowning in thedirectionofthetoothprofile,theleadandthepathofcontact.If the non-generating process of t
30、he gear memberis performed with straight cutting blades, then thegeneration of a pinion tooth will cause additionalprofile curvature versus an involute (or moreprecisely defined as spherical involute or octoid).The additional pinion profile curvature can causeundercut in the pinion root area and a p
31、ointedtopland. To reduce the additional profile curvatureinthegeneratedpinion,itispossibletomanufacturethenon-generatedgearteethwithcurvedblades. Ifthegearcutterbladesareformedliketheinvoluteofa similar generated gear, then the pinion toothprofiles will be regular involutes without additionalprofile
32、 curvature, without additional undercut in theroot area and with no pointed topland versus astandard profile. It is also possible to approximatethe involute function of the gear blade profile withcircular or parabolic shape functions. This willachieve a similar effect and reduce the complexityof bla
33、de grinding or grinding wheel dressingkinematics.The adjustment of the offset is done in two steps,starting from the spiral bevel non-generated geardesign. Theaxisofthespiral bevelgear setlies inahorizontal plane (Figure 1). The first step to offsetthe pinion relative to the gear is to move the pini
34、onaxis in a vertical direction. A positive offset for pin-ions with left hand spiral direction means a downmovement of the pinion, if the view is directed ontothe front of the pinion and if the pinion is located totheright(geartotheleft). Theruleforpositiveoffsetis identical to hypoid gear sets. The
35、 amount of thedown movement is identicalto theamount ofoffset.However, this vertical movement would increasethe center distance (e.g. at the center of the facewidth) as well as the diameter of the pinion (of thesameamountthenthecenterdistanceincrease). Incontrast to regular hypoid pinions where thed
36、iameter is also increased (in case of a positiveoffset)thediameterincreaseoftheHypoloidpinionswouldbelargerthanrequiredtocompensatefortheincreasingpinionspiralangle. Thisisthereasonforthe correction of the pinion diameter, accomplishedby a second movement of the pinion axis in ahorizontal plane towa
37、rds the gear. The amount ofthis movement is determined such that the originalcenter distance is re-established. A pinion diame-ter increase, because of the spiral angle increasedue to the offset is not required (would lead to asmall addendum modification) but can beintroduced in order to achieve a m
38、inimal impact ofthe hypoid offset onto the pinion profile.5Figure 1. Pinion offset position (see ANSI/AGMA/ISO 23509-A08)Four step approach to develop the blankgeometryIn order to present a possibility to optimize the gen-erated pinion tooth profiles along the face width, apinion diameter increase o
39、r reduction is proposed(leads to a profile shift or addendum modification).The procedure used for this pinion diameter in-crease has to assure that the offset remainsconstant. This is achieved if center distance andoffset angle are calculated as follows:A =m2z2+ m2z22(1) = arcsinaA(2)=m2z2+ m2z1+ Dm
40、2A*=A + DM2(3)*= arcsinaA*(4)whereA is center distance, calculated like incase of cylindrical gears at center facewidth of conical gears; is offset angle;a is hypoid offset;A* is center distance in case of pinion di-ameter increase;DM is pinion diameter increase;m2is face module of gear;z1is number
41、of pinion teeth;z2is number of gear teeth.The horizontal movement in order to re-establishthe pinion diameter is calculated:s = A (1 cos ) (5)or in case of a pinion diameter increase:s*= A* (1 cos *) (6)whereS is horizontal movement of pinion axistowards gear axis;S* ishorizontalmovementofpinionaxis
42、incase of a pinion diameter increase.Initially the pitch cone angles are calculated forspiral bevel gears using the following equation,solved with an iteration process:6sin 2-spiralsin1-spiral=z2z1(7)where1-spiralis pitch angle of spiral bevel pinion be-fore introduction of offset;2-spiralispitchang
43、leofspiralbevelgearbeforeintroduction of offset.The Hypoloid geometry takes into account that theoffsetwillchangetheconeangleofthepinionsignif-icantlyratherthanthespiralangle(ascommonlyex-pected in hypoid gears). In order to keep the coneangles of pinion and gear similar (to the spiral bevelgear des
44、ign) and avoid exotic pinion tooth profilesnamely in the root fillet or undercut area, it is pro-posed to change the gear cone angles dependingon the offset:2hypoid= gearspiralcos() (8)or2-hypoid= gear-spiralcos(*) (9)where2-hypoidis pitch angle of hypoid gear setmember.Establishingthegearblankdimen
45、sionsisexplainedin a four step process.Step 1. the spiral bevel gear version has a pitchangle of 2-spiraland a distance crossing point topitch cone (ZTKR2) of zero (Figure 2).Step2. In case of a pinion diameter increaseDM,the crossing point moves in negative z2axis direc-tionand establishesanewcoord
46、inatesystemorigin(Figure 3):ZTKR2*=DM2(10)ZTKR2*=DM2sin(11)whereZTKR2* is crossing point to pitch apex.Step 3. The changed pitch angle of the gear for ahypoid pair shifts the location of the pitch apex(while the pitch point P at the center of the facewidth remains unchanged) as shown in Figure 4.The
47、valueofZTKR2*isnegative,becausethegearconeapexisnowshiftedinnegativez2directionandliesleftoftheR2axis.ZTKR2 * *=DM2sin+RPOtan 2-spiralRPOtan 2-hypoid(12)2-hypoid= 2-spiralcos* (13)Figure 2. Spiral bevel gear cones7Figure 3. Pinion diameter increaseFigure 4. Gear hypoid pitch angleStep 4. The horizon
48、tal movement (Figure 1) of S(orS*) will move the pinion axis in case of positiveS values towards the gear axis and move thecrossing point in positive Z-axis direction byS/sin(), which will establish the final distance,crossingpointtopitchapex(Figure5). The valueofZTKR2* is negative, because the gear
49、 cone apexis now shifted in negative z2direction and lies left ofthe R3axis.SsinZTKR2* *=DM2sin+ RPOItan 2-spiralItan2-hypoid(14)8Figure 5. Hypoid crossing pointThe pinion cone angles are calculated using thethree dimensional point generating principlebetween two axes in a general three dimensionalarrangement in space (Figure 6). This principlebases on a given gear coneand agiven pinionaxis.A pinion cone (or more correctly a hyperboloid)around the pinion axis is calcul
