C211 2009 Elsevier Ltd. All rights reserved. Cylindrical to their drive, is also high cular path concentric to gear axis. The mating double enveloping worm gear requires a hob resembling the double enveloping worm with cutting elements lying on the worm surface. Hobs with such complex geometry are difficult to manufacture. It is presumed that due to the enveloping nature of worm, the contact in this gearing is over much larger area and greater number 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 C211 2009 Elsevier Ltd. All rights reserved. * Corresponding author. Tel.: +91 44 22574677; fax: +91 44 22570509. E-mail address: shun@iitm.ac.in (M.S. Shunmugam). Mechanism and Machine Theory 44 (2009) 2053–2065 Contents lists available at ScienceDirect Mechanism and Machine Theory doi:10.1016/j.mechmachtheory.2009.05.008 problems 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 is set 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 hob is not justified in view its cost, a fly tool can be used for cutting of cylindrical worm gear. However, a hobbing machine with a tangential feed is required to produce accurate worm gear. Also, it is a very slow process and can be used for machining worm 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- 1. Introduction Worm gear drives are used to transmit with large reduction in a single step. in usage due to their simplicity in manufac gear drives are recommended owing [1]. In double enveloping worm gear cylindrical worm gear drive. This drive in sugar mills and coal mines due to motion and power between two mutually perpendicular non-intersecting axes worms with corresponding single enveloping worm gears are quite common turing and assembly. In heavy-duty applications, double enveloping (DE) worm large load carrying capacity compared to single enveloping worm gear drives greater number of teeth is in contact at any instant compared to single enveloping named after its inventor as Hindley Hour-glass worm gear drive. They are used resistance to tooth breakage and better lubrication conditions. However, they pose Geometrical aspects of double enveloping worm gear drive L.V. Mohan, M.S. Shunmugam * Manufacturing Engineering Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India article info Article history: Received 10 November 2008 Received in revised form 16 May 2009 Accepted 21 May 2009 Available online 13 June 2009 Keywords: Double enveloping worm Contact pattern Fly-cutting tool abstract Double enveloping worm gearing is expected to have contact over larger number of teeth and higher load carrying capacity compared to single enveloping worm gearing. In this paper, contact in this gearing is analysed by geometrical simulation of worm gear tooth generation using intersection profiles of different axial sections of worm representing the hob tooth profile with transverse plane of worm gear. The analysis reveals that in the 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 using two fly cutters positioned at the location identical to the extreme end sections of worm to generate 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 fly tool in two settings and nature of contact with the worm is checked by a blue test. journal homepage: www.elsevier.com/locate/mechmt Nomenc b C Z g number of teeth on worm gear 2054 L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 2053–2065 the 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 this gearing is like cam action with more of sliding [2]. Review of literature shows that differential geometry approach is widely used to analyze this gearing [3–9]. However, it is also reported that an undercutting is performed by the extreme edge of the hob [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 and undercutting phenomenon in generation of double enveloping worm gear is analysed by geometrical simulation of gear tooth profile generation using the different axial section profiles of the worm representing hob cutting edge geometry. A trace of intersection points of an axial section profile of worm with a transverse plane of worm gear on its generation is ob- u tool edge parameter a cutting edge/pressure angle / 1 tool rotation parameter / 2 , h 2 , w 2 worm rotation parameters h 1 , w 1 worm gear rotation parameter w angular disposition of axial plane k number of axial sections m module n gear ratio r b base circle radius r p pitch circle radius of worm gear at throat S g (X g , Y g , Z g ) reference coordinate frame of worm gear S w (X w , Y w , Z w ) reference coordinate frame of worm S t (X t , Y t , Z t ) tool coordinate frame S 1 (X 1 , Y 1 , Z 1 ) fixed coordinate frame on worm gear S 2 (X 2 , Y 2 , Z 2 ) fixed coordinate frame on worm t distance of a plane from median plane Z w number of start on worm tained profile envelope worm profile face. This tooth. two settings. 2. Surface 2.1. Worm Fig. double S w (X w Y 2 , Z 2 ) axis produces start worm circle lature half width of worm cutting tool at pitch circle center distance with reference to fixed gear coordinate frame. This trace is called the intersection profile of particular axial section on a transverse plane under consideration as reported by Buckingham and Niemann in their books [2,11]. The inner of such intersection profiles of different axial section profiles of worm obtained for a particular transverse plane of gear constitutes the worm gear tooth profile at the corresponding transverse plane. The analysis of generation of tooth at different transverse planes of worm gear shows that extreme end-sections of worm determine the worm gear sur- has lead to the idea that fly cutters representing end tooth of hob are sufficient to machine the full worm gear In the present work, a single fly cutter is used to machine both the right and left flanks of the worm gear teeth in The contact pattern is also checked by a blue test with a mating worm in meshing. geometry surface 1 shows the details of the double enveloping worm gear drive. Fig. 2 shows the coordinate frame used for analysis of enveloping worm gear drive. Fig. 3 shows the coordinate systems used for obtaining worm surface. S t (X t , Y t , Z t ) and , Y w , Z w ) are reference frames corresponding to tool and worm respectively. Coordinate frames S 1 (X 1 , Y 1 , Z 1 ) and S 2 (X 2 , are rigidly connected to tool and worm respectively. A trapezoidal tool with basic rack profile, rotating about the gear the helicoidal surface on a kinematically linked worm blank rotating about its own axis. A right-hand single- is considered in this study. Eq. (1) gives the right-side tool profile with parameter u and a. The reference (pitch) radius of the gear at throat is given by r p and half-width of tool at the reference circle is given by b p 1 ¼ b þ usina C0r p þ ucosa 0 1 2 6 6 6 4 3 7 7 7 5 ð1Þ L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 2053–2065 2055 For left on the worm side tool profile, the values of a and b are taken as negative. The gear ratio n is defined as the ratio of number of teeth worm gear to number of start on the worm. This gives / 2 = n/ 1, where / 1 and / 2 are rotation parameters of tool and respectively. The worm surface is obtained using coordinate transformation matrices as given by Eq. (2) p 2 ¼½M 2 w C138½M w t C138½M t 1 C138p 1 ð2Þ Fig. 1. Double enveloping worm gear drive. Fig. 2. Coordinate frame for double enveloping worm gear drive. 2056 L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 2053–2065 Fig. 3. Coordinate system to arrive at worm surface. where Eq. (3) where 2.2. Worm Worm worm erating ferent F(x, y, Fig. 4 S w (X w connected to worm h 2 =nh ½M t 1 C138¼ cos/ 1 C0sin/ 1 00 sin/ 1 cos/ 1 00 0010 0001 2 6 6 6 4 3 7 7 7 5 ; ½M w t C138¼ 0010 010C C01000 0001 2 6 6 6 4 3 7 7 7 5 ; ½M 2 w C138¼ cos/ 2 sin/ 2 00 C0sin/ 2 cos/ 2 00 0010 0001 2 6 6 6 4 3 7 7 7 5 is the explicit form of representation of worm surface p 2 ¼ x 2 y 2 z 2 2 6 4 3 7 5 ¼ sin/ 2 fbsin/ 1 C0 r p cos/ 1 þ ucosðaC0 / 1 ÞþCg cos/ 2 fbsin/ 1 C0 r p cos/ 1 þ ucosðaC0 / 1 ÞþCg C0bcos/ 1 C0 r p sin/ 1 C0 usinðaC0 / 1 Þ 2 6 4 3 7 5 ¼ Asin/ 2 Acos/ 2 B 2 6 4 3 7 5 ð3Þ A = b sin / 1 C0 r p cos / 1 + u cos (a C0 / 1 )+C and B = C0b cos / 1 C0 r p sin / 1 C0 u sin (a C0 / 1 ). gear surface gear surface is considered as a conjugate surface generated as an envelope of series of worm surfaces placed on the gear on its kinematic motion. From the principles of differential geometry, at any instant both envelope and the gen- surface contact each other on a line and curve called characteristic or contact line [3]. Series of contact lines at dif- instances generate the worm gear surface. If a family of generating surfaces is represented by implicit form as z, h) with h as parameter of motion, then contact line can be determined by solving the Eq. (4) given below Fðx;y;z;hÞ¼0; @Fðx;y;z;hÞ @h ¼ 0 ð4Þ shows the coordinate systems used for generation of worm gear surface. Reference frames S g (X g , Y g , Z g ) and , Y w , Z w ) correspond to worm gear and worm respectively. Coordinate frames S 1 (X 1 , Y 1 , Z 1 ) and S 2 (X 2 , Y 2 , Z 2 ) are rigidly to worm gear and worm respectively. Eq. (5) gives the surface coordinates of family of worm surfaces transformed gear coordinate frame with kinematic parameters h 1 and h 2 . As the gear ratio is n, h 2 is represented in terms of h 1 as 1 where cosh 1 sinh 1 00 00C010 cosh 2 C0sinh 2 00 The explicit Eliminating where Differentiat L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 2053–2065 2057 1 C0sinh 1 cosh 1 00 6 6 7 7 g 01 0 C0C 6 6 7 7 w sinh 2 cosh 2 00 6 6 7 7 2 3 2 3 2 3 p 1 ¼½M 1 w C138½M g w C138½M w 2 C138p 2 ð5Þ Fig. 4. Coordinate system to arrive at worm gear surface. ½M g C138¼ 0010 0001 6 4 7 5 ; ½M w C138¼ 10 0 0 00 0 1 6 4 7 5 ; ½M 2 C138¼ 0010 0001 6 4 7 5 form of the above equation is given as x 1 y 1 z 1 2 6 4 3 7 5 ¼ Aðsin/ 2 sinh 1 sinh 2 þ cos/ 2 cosh 2 sinh 1 ÞC0Bcosh 1 C0 C sinh 1 Aðsin/ 2 sinh 2 cosh 1 þ cos/ 2 cosh 1 cosh 2 ÞþBsinh 1 C0 C cosh 1 Aðcosh 2 sin/ 2 C0 sinh 2 cos/ 2 Þ 2 6 4 3 7 5 ð6Þ A, B, / 1 and / 2 from Eq. (6), the implicit form of the equation is obtained Fðh 1 Þ¼ ðx 1 sinh 1 þ y 1 cosh 1 þ CÞ 2 þ z 2 1 hi1 2 C0 C C26C27 sinðaC0 / 1 ÞC0ðx 1 cosh 1 C0 y 1 sinh 1 ÞcosðaC0/ 1 Þþbcosaþ r p sina ¼ 0 ð7Þ / 1 ¼ h 1 C0 1 n cos C01 ðx 1 sinh 1 þ y 1 cosh 1 þ CÞ ðx 1 sinh 1 þ y 1 cosh 1 þ CÞ 2 þ z 2 1 hi1 2 8 : 9 = ; ð8Þ ing Eqs. (7) and (8) with respect to h 1 , we get Eqs. (9) and (10) @Fðh 1 Þ @h 1 ¼½ðx 1 sinh 1 þ y 1 cosh 1 þ CÞ 2 þ z 2 1 C138 1 2 cosðaC0 / 1 ÞC0 d/ 1 dh 1 C26C27 þ ðx 1 sinh 1 þ y 1 cosh 1 þ CÞðx 1 cosh 1 C0 y 1 sinh 1 Þ ½ðx 1 sinh 1 þ y 1 cosh 1 þ CÞ 2 þ z 2 1 C138 1 2 sinðaC0 / 1 Þþðx 1 cosh 1 C0 y 1 sinh 1 ÞsinðaC0 / 1 ÞC0 d/ 1 dh 1 C26C27 þ cosðaC0 / 1 Þðx 1 sinh 1 þ y 1 cosh 1 Þ¼0 ð9Þ To obtain parameter The and (10) Substituting gear. Though forward tion of groove uniform plane points of different (14) with In a similar formed of worm 2058 L.V. Mohan, M.S. Shunmugam/Mechanism and Machine Theory 44 (2009) 2053–2065 way, transforming all the points for a particular axial section to the gear coordinate frame and plotting the trans- points, the intersection profile of the particular axial section is obtained. Intersection profiles of other axial sections are also obtained following the above procedure. The inner envelope of all these intersection profiles represents the where ½M g w C138¼ 00C010 01 0 C0C 10 0 0 00 0 1 2 6 6 6 4 3 7 7 7 5 ; ½M 1 g C138¼ cosh 1 sinh 1 10 C0sinh 1 cosh 1 00 0010 0001 2 6 6 6 4 3 7 7 7 5 values of h 1 ¼ 1 n ðh 2 þ wÞ P 00 ¼½M 1 g C138½M g w C138P 0 ð14Þ 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 of w for h 2 . Value of h 1 is obtained from the relation h 2 =nh 1 .Ifk axial sections are considered, then the value of w will be integer multiple of 360/k degree. The intersection profiles of these axial sections are obtained using homogenous coordinate transformation matrices. Considering an axial section at angle of w from the median plane of worm gear, the axial section is brought to the median plane by rotation of the worm through an angle w in the anti-clock- wise direction. Fig. 5 shows that a point p 2 (x 2 , y 2 , z 2 ) on this axial section is moved to point P(x, y, z) on the median plane by rotation of this axial section. The coordinate of point P on the right-flank of worm groove is obtained by p ¼ x y z 2 6 4 3 7 5 ¼ cosw sinw 0 C0sinw cosw 0 001 2 6 4 3 7 5p 2 ð12Þ When this point moves to pointP 0 (x 0 , y 0 , z 0 ) on a plane at a distance t from the median plane by rotation of worm through an angle h 2 , the worm gear also rotates through an angle h 2 /n anticlockwise for a right-hand worm. The coordinates of the ro- tated point in the worm reference frame S w are given by P 0 ¼ x 0 y 0 z 0 2 6 4 3 7 5 ¼ C0t ycosh 2 z 2 6 4 3 7 5 where h2 ¼ sin C01 t y C18C19 ð13Þ The point P 0 can be transferred to worm gear coordinate system as P 00 by coordinate transformation matrices as given by Eq. manner are considered for the simulation. Intersection points of an axial section profile when crossing a transverse of worm gear on its kinematic motion are obtained with reference to the fixed gear coordinate frame. Plotting these gives the intersection profile of the particular axial section profile of worm groove. Similarly the intersection profiles axial sections of worm groove are obtained and the inner envelope of all these intersection profiles gives the gen- 3. Geometrical simulation of gear tooth generation Different axial section profiles of worm groove representing the hob tooth profile
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