POLITECNICO DI TORINO Faculty of Automotive Engineering Masters of Automotive Engineering Masters Thesis Academic Year 2019-2020 GENERATION OF TOOTH PROFILE STARTING FROM CREATOR SHAPE Supervisor: Candidate: Professor Carlo Rosso Jean Marc Moujabber Co-supervisors: Mr. Fabio Bruzzone Mr. Tomasso Maggi Mr. Claudio Marcellini 1 2 Table of Contents Figures Index ................................................................................................................................................. 5 Tables Index .................................................................................................................................................. 7 1. Introduction .......................................................................................................................................... 8 2. Nomenclature ..................................................................................................................................... 11 3. Conjugate Action ................................................................................................................................. 13 4. Involute Properties ............................................................................................................................. 14 5. Construction of a gear tooth profile ................................................................................................... 15 6. Gear Forming ...................................................................................................................................... 17 6.1 Milling ......................................................................................................................................... 17 6.2 Hobbing ....................................................................................................................................... 18 6.3 Shaping ........................................................................................................................................ 19 7. Contact Ratio ....................................................................................................................................... 20 8. Tooth Systems ..................................................................................................................................... 21 9. Coordinate Transformation for externally tangent spur gears ........................................................... 22 10. Applications for external spur gears ................................................................................................. 24 10.1 Application to a generic shape .................................................................................................... 24 10.2 Refined application ..................................................................................................................... 25 10.2.1 Data given ........................................................................................................................... 25 10.2.2 Results ................................................................................................................................. 26 11. Coordinate transformation for internally tangent spur gears......................................................... 27 12. Application for internal spur gears .................................................................................................. 29 12.1 Data given .................................................................................................................................... 29 12.2 Results ......................................................................................................................................... 29 13. Involute helical gears with parallel axes .......................................................................................... 31 13.1 Design procedure ........................................................................................................................ 33 13.2 Virtual number of teeth .............................................................................................................. 34 14. Coordinate transformation for externally tangent gears ............................................................... 35 15. Application for parallel helical gears ............................................................................................... 37 15.1 Data given ................................................................................................................................... 37 15.2 Results ......................................................................................................................................... 39 16. Alternative coordinate transformation for externally tangent helical gears .................................. 40 17. Alternative application for parallel helical gears ............................................................................ 42 3 17.1 Data given ................................................................................................................................... 42 17.2 Results ......................................................................................................................................... 43 18. Face Gears ....................................................................................................................................... 45 19. Coordinate transformation for face-gears ..................................................................................... 49 20. Application for face-gears ............................................................................................................... 51 20.1 Data given ................................................................................................................................... 51 20.2 Results ......................................................................................................................................... 51 21. Envelope of Internal Spur Gears ..................................................................................................... 53 22. Conclusion ....................................................................................................................................... 61 23. References ...................................................................................................................................... 62 4 Figures Index Figure 2.1: Nomenclature of spur-gear teeth 11 Figure 3.1: Cam A and follower B 13 Figure 4.1: Generation of an involute 14 Figure 4.2: Construction of an involute 14 Figure 5.1: Circles of a gear layout 15 Figure 5.2: Base circle construction 15 Figure 5.3: Tooth profile construction 16 Figure 6.1: Gear forming via milling 17 Figure 6.2: Gear forming via hobbing 18 Figure 6.3: Standard reference profile for hob cutter 18 Figure 6.4: Gear forming via shaping 19 Figure 6.5: Connection between gear and rack 19 Figure 7.1: Definition of contact ratio 20 Figure 9.1: Centrodes in rotational motions in opposite directions 22 Figure 10.1: Generic tooth shape 23 Figure 10.2: Generated tooth profile 24 Figure 10.3: 2 teeth section of the shaper gear cutter 25 Figure 10.4: Shaper gear cutter vane profile 25 Figure 10.5: Generated tooth profile on the work gear 26 Figure 11.1: Centrodes in rotational motions of the same direction 27 Figure 12.1: Shaper cutter gear vane profile 29 Figure 12.2: Generated tooth shape pre-correction 30 Figure 12.3: Generated tooth shape post-correction 30 Figure 13.1: Helical gear tooth 31 Figure 13.2: Contact lines on tooth surface of a helical gear (a) and a spur gear (b) 31 Figure 13.3: Nomenclature of helical gears 32 Figure 13.4: A cylinder cut by an oblique plane 34 Figure 14.1: Centrodes in translation-rotation motions 35 5 Figure 15.1: Rack tangential tooth profile 37 Figure 15.2: Rack vane in 3D 38 Figure 15.3: Generated tooth profile on the work gear 39 Figure 17.1: Shaper cutter vane profile 42 Figure 17.2: Shaper cutter vane (3D) 42 Figure 17.3: Generated helical tooth 43 Figure 17.4: Section A-A 44 Figure 17.5: Section B-B 44 Figure 18.1: Face-gear drive 45 Figure 18.2: Structure of face-gear tooth 45 Figure 18.3: Axodes and pitch cones 46 Figure 18.4: Tangency of pinion and shaper tooth profiles 47 Figure 18.5: Instantaneous axes of rotation 48 Figure 19.1: Coordinate system for generation of face-gear surface: (a) illustration of installation; (b) derivation of coordinate transformation 49 Figure 20.1: Shaper cutter gear vane profile 51 Figure 20.2: Generated face-gear tooth profiles 51 Figure 21.1: Rack geometry (vane) 53 Figure 21.2: Relative motion 53 Figure 21.3: Envelope of the generated spur gear tooth 54 Figure 21.4: Normals of the cutter profile 54 Figure 21.5: Vane profile 55 Figure 21.6: Relative motion for the internal gear shaping 55 Figure 21.7: Internal gear tooth profile 56 Figure 21.8: Normals at the angular tip 56 Figure 21.9: Uncorrected envelope 57 Figure 21.10: First correction of the envelope 57 Figure 21.11: Second correction of the envelope 58 Figure 21.12: Final Result, relative motion and generated envelope 58 6 Figure 21.13: Module = 2 59 Figure 21.14: Number of teeth = 30 59 Figure 21.15: Gear ratio = 3 59 Figure 21.16: Root fillet coefficient = 0.01 59 Figure 21.17: Pressure angle = 14.5o 60 Figure 21.18: Pressure angle = 22o 60 Figure 21.19: Addendum = 1.75m 60 Figure 21.20: Addendum = 1m 60 Figure 21.21: Dedendum = 1.1m 60 Figure 21.22: Dedendum = 0.8m