1. The McGraw-Hill Companies © 2012
Lecture Slides
Chapter 15
Bevel and Worm Gears
The McGraw-Hill Companies © 2012

2. Chapter Outline
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3. Bevel Gearing - General
Bevel gear classifications
Straight bevel gears
Spiral bevel gears
Zerol bevel gears
Hypoid gears
Spiroid gears
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4. Perpendicular shafts lying in a plane
Straight Bevel Gear
Perpendicular shafts lying in a plane
Usually used for pitch line velocities up to 1000 ft/min (5 m/s)
Fig. 13–3
Fig. 13–35
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5. Recommended for higher speeds Recommended for lower noise levels
Spiral Bevel Gear
Recommended for higher speeds
Recommended for lower noise levels
The bevel counterpart of the helical gear
Fig. 15–1
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6. Cutting spiral-gear teeth
Spiral Bevel Gear
Cutting spiral-gear teeth
Fig. 15–2
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7. Patented gear with curved teeth but with a zero spiral angle
Zerol Bevel Gear
Patented gear with curved teeth but with a zero spiral angle
Axial thrust loads are less than spiral bevel gear
Often used instead of straight bevel gears
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8. Allows for offset in shaft center-lines
Hypoid Gears
Allows for offset in shaft center-lines
Pitch surfaces are hyperboloids of revolution
Fig. 15–3
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9. Greater offset of center-lines than hypoid gears
Spiroid Gears
Greater offset of center-lines than hypoid gears
Hypoid and Spiroid gears are progressions from spiral gear to worm gear
Fig. 15–4
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10. AGMA Straight-Bevel Gear Equations
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11. AGMA Straight-Bevel Gear Equations
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12. Overload Factor KO (KA)
Table 15–2
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13. Dynamic Factor Kv
Fig. 15–5
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14. Dynamic Factor Kv
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15. Size Factor for Pitting Resistance Cs (Zx)
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16. Size Factor for Bending Ks (Yx)
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17. Load-Distribution Factor Km (KHb)
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18. Crowning Factor for Pitting Cxc (Zxc)
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19. Lengthwise Curvature Factor for Bending Strength Kx (Yb)
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20. Pitting Resistance Geometry Factor I (ZI)
Fig. 15–6
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21. Bending Strength Geometry Factor J (YJ)
Fig. 15–7
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22. Stress-Cycle Factor for Pitting Resistance CL (ZNT)
Fig. 15–8
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23. Stress-Cycle Factor for Bending Strength KL (YNT)
Fig. 15–9
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24. Stress-Cycle Factor for Bending Strength KL (YNT)
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25. Hardness-Ratio Factor CH (ZW)
Fig. 15–10
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26. Hardness-Ratio Factor CH (ZW) for Work-Hardened Gear
Fig. 15–11
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27. Temperature Factor KT (Kq)
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28. Reliability Factors CR (ZZ) and KR (YZ)
Table 15–3
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29. Elastic Coefficient for Pitting Resistance Cp (ZE)
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30. Allowable Contact Stress Number for Steel Gears
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31. Allowable Contact Stress Number for Through-Hardened Steel Gears
Fig. 15–12
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32. Allowable Contact Stress Number for Iron Gears
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33. Allowable Bending Stress Number for Steel Gears
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34. Allowable Bending Stress Number for Through-Hardened Steel Gears
Fig. 15–13
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35. Allowable Bending Stress Number for Iron Gears
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36. Summary for Straight-Bevel Gear Wear
Fig. 15–14
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37. Summary for Straight-Bevel Gear Bending
Fig. 15–15
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38. Example 15–1
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39. Example 15–1
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40. Example 15–1
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41. Example 15–1
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42. Example 15–1
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43. Example 15–1
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44. Example 15–1
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45. Example 15–1
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46. Example 15–1
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47. Design of Straight-Bevel Gear Mesh
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48. Recommended Face Width
Bending strength is not linear with face width
Added material is placed at the small end of the teeth
Recommended face width,
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49. Example 15–2
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50. Example 15–2
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51. Example 15–2
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52. Example 15–2
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53. Example 15–2
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54. Example 15–2
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55. Example 15–2
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56. Example 15–2
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57. Example 15–2
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58. Usually perpendicular Relation between shaft angle and helix angles is
Worm Gearing
Used to transmit rotary motion between non- parallel and non-intersecting shafts
Usually perpendicular
Relation between shaft angle and helix angles is
Crossed helical gears can be considered as non-enveloping worm gears
Fig. 15–16
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59. Cylindrical worm dimensions common to both worm and gear,
Worm Gear Dimensions
With center-to-center distance C, good proportions indicate the pitch worm diameter d should be in the range
Cylindrical worm dimensions common to both worm and gear,
Table 15–8
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60. Friction Force
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61. Sliding Velocity and Torque
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62. Worm Gearing Equations for Allowable Tangential Force
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63. Worm Gearing Equations for Allowable Tangential Force
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64. Worm Gearing Equations for Allowable Tangential Force
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65. Coefficient of Friction f
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66. Worm-Gear Geometry
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67. Face Width
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68. Heat Loss Rate From Worm-Gear Case
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69. Heat loss rate from worm-gear case in ft·lbf/min,
Energy Issues
Heat loss rate from worm-gear case in ft·lbf/min,
Overall coefficient for combined convective and radiative heat transfer from the worm-gear case,
With case lateral area A, the oil sump temperature,
AGMA recommended minimum lateral area in in2
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70. Buckingham Stress Equation
Worm teeth are inherently much stronger than worm-gear teeth
Worm-gear teeth are short and thick on the edges of the face
Midplane they are thinner as well as curved
Buckingham adapted the Lewis equation for this case,
y is the Lewis form factor
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71. Mechanical efficiency with worm driving,
Worm-Gear Analysis
Mechanical efficiency with worm driving,
Mechanical efficiency with gear driving,
To ensure worm gear will drive the worm,
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72. Relation of tangential worm force and tangential gear force,
Worm-Gear Analysis
Relation of tangential worm force and tangential gear force,
Due to low efficiency of worm gearing, output power is not considered equivalent to input power
Relating tangential gear force to output power and efficiency,
Power for worm and gear,
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73. Sliding velocity of worm at pitch cylinder,
Worm-Gear Analysis
Friction force,
Sliding velocity of worm at pitch cylinder,
Friction power,
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74. Maximum Lead Angle for Worm Gearing
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75. Example 15–3
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76. Example 15–3
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77. Example 15–3
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78. Example 15–3
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79. Example 15–3
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80. Example 15–3
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81. Example 15–3
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82. Example 15–3
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83. Recommended Minimum Number of Worm-Gear Teeth
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84. Example 15–4
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85. Example 15–4
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86. Example 15–4
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87. Example 15–4
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88. Example 15–4
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89. Example 15–4
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90. Example 15–4
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91. Buckingham Wear Load
Buckingham showed that the allowable gear-tooth loading for wear can be estimated from
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92. Wear Factor Kw for Worm Gearing
Table 15–11
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93. Example 15–5
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