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Geometrical aspects of double enveloping worm gear driveL.V. Mohan, M.S. Shunmugam*Manufacturing Engineering Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, Indiaa r t i c l ei n f oArticle history:Received 10 November 2008Received in revised form 16 May 2009Accepted 21 May 2009Available online 13 June 2009Keywords:Double enveloping wormContact patternFly-cutting toola b s t r a c tDouble enveloping worm gearing is expected to have contact over larger number of teethand higher load carrying capacity compared to single enveloping worm gearing. In thispaper, contact in this gearing is analysed by geometrical simulation of worm gear toothgeneration using intersection profiles of different axial sections of worm representingthe hob tooth profile with transverse plane of worm gear. The analysis reveals that inthe engaging zone a straight line contact always exists in the median plane and intermit-tent contact exists at the extreme end sections of worm. This has lead to the idea of usingtwo fly cutters positioned at the location identical to the extreme end sections of worm togenerate full worm gear tooth thereby eliminating the need for hobs of complex geometry.For a given worm, a mating worm gear is machined in a gear hobbing machine using flytool in two settings and nature of contact with the worm is checked by a blue test.? 2009 Elsevier Ltd. All rights reserved.1. IntroductionWorm gear drives are used to transmit motion and power between two mutually perpendicular non-intersecting axeswith large reduction in a single step. Cylindrical worms with corresponding single enveloping worm gears are quite commonin usage due to their simplicity in manufacturing and assembly. In heavy-duty applications, double enveloping (DE) wormgear drives are recommended owing to their large load carrying capacity compared to single enveloping worm gear drives1. In double enveloping worm gear drive, greater number of teeth is in contact at any instant compared to single envelopingcylindrical worm gear drive. This drive is also named after its inventor as Hindley Hour-glass worm gear drive. They are usedin sugar mills and coal mines due to high resistance to tooth breakage and better lubrication conditions. However, they poseproblems in manufacture and require precise assembly.Cylindrical worm having straight sided profiles in axial section (referred to as ZA-type) is cut by a trapezoidal tool that isset in an axial plane and moved parallel to the axis, as in a thread-chasing operation on a lathe. Mating worm gear is ma-chined by a hob having appropriate cutting elements lying on the cylindrical worm surface. When manufacture of a hobis not justified in view its cost, a fly tool can be used for cutting of cylindrical worm gear. However, a hobbing machine witha tangential feed is required to produce accurate worm gear. Also, it is a very slow process and can be used for machiningworm gears, few in numbers. Without a tangential feed, fly cutter simply performs form cutting. For cutting a double envel-oping worm, a trapezoidal tool set in axial plane is used with its edge always tangential to base circle while moving in a cir-cular path concentric to gear axis. The mating double enveloping worm gear requires a hob resembling the doubleenveloping worm with cutting elements lying on the worm surface. Hobs with such complex geometry are difficult tomanufacture.It is presumed that due to the enveloping nature of worm, the contact in this gearing is over much larger area and greaternumber of teeth is in contact compared to single enveloping worm gear set. Earlier work reveals that in the engaging zone,0094-114X/$ - see front matter ? 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2009.05.008* Corresponding author. Tel.: +91 44 22574677; fax: +91 44 22570509.E-mail address: shuniitm.ac.in (M.S. Shunmugam).Mechanism and Machine Theory 44 (2009) 20532065Contents lists available at ScienceDirectMechanism and Machine Theoryjournal homepage: contact is established as straight line in median plane of the worm gear irrespective of the rotation angle and intermit-tent contact appears on transverse end planes and central plane of the worm. Further it shows that the meshing action in thisgearing is like cam action with more of sliding 2. Review of literature shows that differential geometry approach is widelyused to analyze this gearing 39. However, it is also reported that an undercutting is performed by the extreme edge of thehob 10. As differential geometry approach does not address the problem fully when interference or undercutting phenom-enon exits in machining, intersection profile method as reported by Buckingham 2 is chosen. The nature of contact andundercutting phenomenon in generation of double enveloping worm gear is analysed by geometrical simulation of geartooth profile generation using the different axial section profiles of the worm representing hob cutting edge geometry. Atrace of intersection points of an axial section profile of worm with a transverse plane of worm gear on its generation is ob-tained with reference to fixed gear coordinate frame. This trace is called the intersection profile of particular axial sectionprofile on a transverse plane under consideration as reported by Buckingham and Niemann in their books 2,11. The innerenvelope of such intersection profiles of different axial section profiles of worm obtained for a particular transverse plane ofworm gear constitutes the worm gear tooth profile at the corresponding transverse plane. The analysis of generation of toothprofile at different transverse planes of worm gear shows that extreme end-sections of worm determine the worm gear sur-face. This has lead to the idea that fly cutters representing end tooth of hob are sufficient to machine the full worm geartooth. In the present work, a single fly cutter is used to machine both the right and left flanks of the worm gear teeth intwo settings. The contact pattern is also checked by a blue test with a mating worm in meshing.2. Surface geometry2.1. Worm surfaceFig. 1 shows the details of the double enveloping worm gear drive. Fig. 2 shows the coordinate frame used for analysis ofdouble enveloping worm gear drive. Fig. 3 shows the coordinate systems used for obtaining worm surface. St(Xt, Yt, Zt) andSw(Xw, Yw, Zw) are reference frames corresponding to tool and worm respectively. Coordinate frames S1(X1, Y1, Z1) and S2(X2,Y2, Z2) are rigidly connected to tool and worm respectively. A trapezoidal tool with basic rack profile, rotating about the gearaxis produces the helicoidal surface on a kinematically linked worm blank rotating about its own axis. A right-hand single-start worm is considered in this study. Eq. (1) gives the right-side tool profile with parameter u anda. The reference (pitch)circle radius of the gear at throat is given by rpand half-width of tool at the reference circle is given by bp1b usina?rp ucosa0126664377751Nomenclaturebhalf width of worm cutting tool at pitch circleCcenter distanceknumber of axial sectionsmmodulengear ratiorbbase circle radiusrppitch circle radius of worm gear at throatSg(Xg, Yg, Zg) reference coordinate frame of worm gearSw(Xw, Yw, Zw) reference coordinate frame of wormSt(Xt, Yt, Zt) tool coordinate frameS1(X1, Y1, Z1) fixed coordinate frame on worm gearS2(X2, Y2, Z2) fixed coordinate frame on wormtdistance of a plane from median planeZwnumber of start on wormZgnumber of teeth on worm gearutool edge parameteracutting edge/pressure angle/1tool rotation parameter/2, h2,w2worm rotation parametersh1,w1worm gear rotation parameterwangular disposition of axial plane2054L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 20532065For left side tool profile, the values ofaand b are taken as negative. The gear ratio n is defined as the ratio of number of teethon the worm gear to number of start on the worm. This gives /2= n/1,where /1and /2are rotation parameters of tool andworm respectively. The worm surface is obtained using coordinate transformation matrices as given by Eq. (2)p2 M2w?Mwt?Mt1?p12Fig. 1. Double enveloping worm gear drive.Fig. 2. Coordinate frame for double enveloping worm gear drive.L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 205320652055whereMt1? cos/1?sin/100sin/1cos/100001000012666437775;Mwt? 0010010C?100000012666437775;M2w? cos/2sin/200?sin/2cos/200001000012666437775Eq. (3) is the explicit form of representation of worm surfacep2x2y2z2264375 sin/2fbsin/1? rpcos/1 ucosa? /1 Cgcos/2fbsin/1? rpcos/1 ucosa? /1 Cg?bcos/1? rpsin/1? usina? /1264375 Asin/2Acos/2B2643753where A = b sin /1? rpcos /1+ u cos (a? /1) + C and B = ?b cos /1? rpsin /1? u sin (a? /1).2.2. Worm gear surfaceWorm gear surface is considered as a conjugate surface generated as an envelope of series of worm surfaces placed on theworm gear on its kinematic motion. From the principles of differential geometry, at any instant both envelope and the gen-erating surface contact each other on a line and curve called characteristic or contact line 3. Series of contact lines at dif-ferent instances generate the worm gear surface. If a family of generating surfaces is represented by implicit form asF(x, y, z, h) with h as parameter of motion, then contact line can be determined by solving the Eq. (4) given belowFx;y;z;h 0;Fx;y;z;hh 04Fig. 4 shows the coordinate systems used for generation of worm gear surface. Reference frames Sg(Xg, Yg, Zg) andSw(Xw, Yw, Zw) correspond to worm gear and worm respectively. Coordinate frames S1(X1, Y1, Z1) and S2(X2, Y2, Z2) are rigidlyconnected to worm gear and worm respectively. Eq. (5) gives the surface coordinates of family of worm surfaces transformedto worm gear coordinate frame with kinematic parameters h1and h2. As the gear ratio is n, h2is represented in terms of h1ash2= nh1Fig. 3. Coordinate system to arrive at worm surface.2056L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 20532065p1 M1w?Mgw?Mw2?p25whereM1g? cosh1sinh100?sinh1cosh100001000012666437775;Mgw? 00?10010?C100000012666437775;Mw2? cosh2?sinh200sinh2cosh200001000012666437775The explicit form of the above equation is given asx1y1z1264375 Asin/2sinh1sinh2 cos/2cosh2sinh1 ? Bcosh1? C sinh1Asin/2sinh2cosh1 cos/2cosh1cosh2 Bsinh1? C cosh1Acosh2sin/2? sinh2cos/22643756Eliminating A, B, /1and /2from Eq. (6), the implicit form of the equation is obtainedFh1 x1sinh1 y1cosh1 C2 z21hi12? C?sina? /1 ? x1cosh1? y1sinh1cosa? /1 bcosa rpsina 07where/1 h1?1ncos?1x1sinh1 y1cosh1 Cx1sinh1 y1cosh1 C2 z21hi128:9=;8Differentiating Eqs. (7) and (8) with respect to h1, we get Eqs. (9) and (10)Fh1h1 x1sinh1 y1cosh1 C2 z21?12cosa? /1 ?d/1dh1?x1sinh1 y1cosh1 Cx1cosh1? y1sinh1x1sinh1 y1cosh1 C2 z21?12sina? /1 x1cosh1? y1sinh1sina? /1 ?d/1dh1? cosa? /1x1sinh1 y1cosh1 09Fig. 4. Coordinate system to arrive at worm gear surface.L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 205320652057d/1dh1 1 1nz1x1cosh1? y1sinh1x1sinh1 y1cosh1 C2 z2110To obtain the surface coordinates of the worm gear, Eqs. (7)(10) have to be solved for different values of z1and rotationparameter h1.The nature of contact at the median plane is obtained by substituting z1= 0 in Eqs. (7)(10). Substituting z1= 0 in Eqs. (8)and (10), h1= /1andd/1dh1 1.Substituting above values in Eq. (9), a? h1 p2.Substituting the above equality in Eq. (7), Eq. (11) is obtained. This represents an equation of a straight line correspondingto tool edge and is independent of the rotation parameter h1x1? by1 rp tana11This shows that there exits a straight line contact always on the median plane irrespective of the rotational position of thegear. Though it can be solved for median plane with z1= 0, for other planes these equations cannot be solved in a straight-forward way. Hence, to understand the generation of gear tooth profiles at different transverse planes, geometrical simula-tion of gear tooth profile generation is carried out, using intersection profiles of different axial section profiles of wormgroove at different angular positions. The procedure is explained briefly in next section.3. Geometrical simulation of gear tooth generationDifferent axial section profiles of worm groove representing the hob tooth profile and angularly disposed by anglewin auniform manner are considered for the simulation. Intersection points of an axial section profile when crossing a transverseplane of worm gear on its kinematic motion are obtained with reference to the fixed gear coordinate frame. Plotting thesepoints gives the intersection profile of the particular axial section profile of worm groove. Similarly the intersection profilesof different axial sections of worm groove are obtained and the inner envelope of all these intersection profiles gives the gen-erated gear tooth profile.The axial section profile coordinates of the worm surface are obtained for different values of rotation parameter using Eq.(3) by substituting the value ofwfor h2. Value of h1is obtained from the relation h2= nh1. If k axial sections are considered,then the value ofwwill be integer multiple of 360/k degree. The intersection profiles of these axial sections are obtainedusing homogenous coordinate transformation matrices. Considering an axial section at angle ofwfrom the median planeof worm gear, the axial section is brought to the median plane by rotation of the worm through an anglewin the anti-clock-wise direction. Fig. 5 shows that a point p2(x2, y2, z2) on this axial section is moved to point P(x, y, z) on the median plane byrotation of this axial section. The coordinate of point P on the right-flank of worm groove is obtained byp xyz264375 coswsinw0?sinwcosw0001264375p212When this point moves to pointP0(x0, y0, z0) on a plane at a distance t from the median plane by rotation of worm through anangle h2, the worm gear also rotates through an angle h2/n anticlockwise for a right-hand worm. The coordinates of the ro-tated point in the worm reference frame Sware given byP0 x0y0z0264375 ?tycosh2z264375 where h2 sin?1ty? ?13The point P0can be transferred to worm gear coordinate system as P00by coordinate transformation matrices as given by Eq.(14) with values of h11nh2 wP00 M1g?Mgw?P014whereMgw? 00?10010?C100000012666437775;M1g? cosh1sinh110?sinh1cosh100001000012666437775In a similar way, transforming all the points for a particular axial section to the gear coordinate frame and plotting the trans-formed points, the intersection profile of the particular axial section is obtained. Intersection profiles of other axial sectionsof worm are also obtained following the above procedure. The inner envelope of all these intersection profiles represents the2058L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 20532065gear tooth profile at that plane. Computer code is written in C for simulation of gear tooth profile generation and plot of theintersection profiles are made using MATLAB software.3.1. Simulation case studyA case study of worm gear tooth generation using four axial sections of worm at 90? apart is considered in this study. Thedesign details of worm and worm gear are given in Table 1. Two turns of the worm thread is considered with a total of nineaxial section profiles as shown in Fig. 6. Initial position of axial section of worm groove with its flanks equally displaced onboth sides of the central plane is numbered as A0and other teeth in the clockwise direction have positive subscript withincreasing order as A1, A2, A3and A4. Similarly for anti-clockwise direction the teeth are given negative subscript as A?1,A?2, A?3and A?4. Intersection profiles of these nine axial sections at different locations are obtained for median plane andshown in Fig. 7. Intersection profiles of planes away form the median plane by a distance of 10 mm on both sides of the med-ian plane are shown in Fig. 8.It is seen from Fig. 7 that the intersection profiles of different axial sections of worm at the median plane of gear are merg-ing into one profile. This gives the inference that in the median plane straight line contact exists irrespective of the rotationalFig. 5. Determination of intersection profiles.Table 1Design details of double enveloping worm and worm gear.Design parameterValueModule (m)2.5 mmNo. of start of worm (Zw)1No. of teeth on worm gear (Zg)40Cutting edge angle (a)20?Half tool width at pitch circle1.9635 mmOuter diameter of worm at throat47.5 mmReference diameter of worm at throat42.5 mmRoot diameter of worm at throat36.5 mmAddendum1 moduleDedendum1.2 moduleCentre distance71.25 mmWorm length39.9 mmL.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 205320652059position of the worm. In off-median planes, as seen from Fig. 8 the nature of intersection profiles and the resultant gear toothprofile reveals that there is no conjugate action. It is also seen from Fig. 8 that the resultant gear tooth profile is generatedonly by the intersection profiles of end axial sections A?4and A4. This shows that cutting is taking place by teeth representedby end axial sections A?4and A4. For planes oppositely displaced by equal distance from median plane, the gear tooth profileis found to be mirror image of the other. As these axial sections represent the hob teeth profile geometry with zero rake an-gle, it leads to an inference that one end-tooth of a hob is sufficient and both sides of the gear tooth can be obtained in twosettings.4. Contact between worm and gearIn order to study the contact for different rotational positions of worm, a worm gear generated by end axial-section (A4) ofworm is considered. The surface coordinates of the worm gear are obtained by substituting /2= 2pin Eq. (6). Eq. (15) givesthe surface coordinates of the right-flank of the worm gear toothx1ry1rz1r12666437775Asinh1cosh2? Bcosh1? C sinh1Acosh1cosh2 Bsinh1? C cosh1?Asinh21266643777515Fig. 6. Axial sections of worm.Fig. 7. Intersection profiles at median plane (double enveloping worm gear) (m = 2.5 mm, Zg= 40, Zw= 1, C = 71.25 mm,a= 20?).2060L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 20532065If the worm is rotated to an anglew2, then the worm gear is rotated through an anglew1given by the relationw2= nw1. Thesurface coordinates of right-flank of the worm groove atw2position for the axial section A4with /2= 2pin gear referencecoordinate frame Sgis given by Eq. (16)xwrywrzwr1266643777500?10010?C100000012666437775cosw2?sinw200sinw2cosw2000010000126664377750AB12666437775?BAcosw2? C?Asinw21266643777516Fig. 8. Intersection profiles at off-median planes (double enveloping worm gear) (m = 2.5 mm, Zg= 40, Zw= 1, C = 71.25 mm,a= 20?).L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 205320652061The surface coordinates of the worm gear in the gear reference coordinate frame Sgfor the rotated positionw1is given by Eq.(16)xgrygrzgr1266664377775cosw1?sinw100sinw1cosw1000010001266664377775x1ry1rz1r1266664377775cosw1Asinh1cosh2? Bcosh1? C sinh1 ? sinw1Acosh1cosh2 Bsinh1? C cosh1sinw1Asinh1cosh2? Bcosh1? C sinh1 cosw1Acosh1cosh2 Bsinh1? C cosh1?Asinh2126666437777517It is easily found that Eqs. (16) and (17) are same, ifw2= h2andw1= h1. The value of h2assumes a value within the face widthof the gear. This means that for any valuew2within the range of h2, this contact exists as a sweeping intermittent contact onthe right flank of the worm gear by the end axial section A4of worm. Besides this, contact always exists at the median planeirrespective of the rotation anglew2. Atw2= 0, the contact is only at the median plane in engaging teeth. With face widthangle of 60?, for any other value ofw2between ?30? and 30? an intermittent contact lines starts from the top of the facewidth of the gear and sweeps the face width and disappears. As the left flank of axial section A?4with /2= ?2pgeneratesthe left flank of the worm gear, similar intermittent contact exists on the left flank of worm gear tooth at t