Unit 2 - Helical gear & Bevel gears.pptx

Design of helical gear

1. Helical and Bevel Gears Types of helical and Bevel gears, Terminology, Virtual number of teeth, and force analysis of Helical and Straight Bevel Gear. Design of Helical and Straight Bevel Gear based on Beam Strength, Wear strength and estimation of effective load based on Velocity factor (Barth factor) and Buckingham’s equation. Mountings of Bevel Gear. (No numerical on force analysis of helical & Bevel Gear) By S M Gujrathi

2. Helical Gear By S M Gujrathi

3. Helix angle: It is defined as the angle between the axis of the shaft and the centre line of the tooth taken on the pitch plane( ψ ) 1.XX is the plane of rotation, 2.YY is a plane perpendicular to the tooth elements. transverse circular pitch=A1A2 Normal circular pitch( pn )=B1B2=A1C Formative or Virtual Gear Concept By S M Gujrathi

4. where Pn and P are normal and transverse diametral pitches respectively. Substituting ( P = 1/m) in the above expression, we have m n =m* cos ( ψ ) mn = normal module (mm) m = transverse module (mm) By S M Gujrathi

5. By S M Gujrathi

6. VIRTUAL NUMBER OF TEETH Semi major(a)= semi-minor(b)= Radius of curvature r’ at the point B is given by, Substituting the values of a and b in the expression for r’, By S M Gujrathi

7. In the design of helical gears, an imaginary spur gear is considered in the plane A–A with centre at O’ having a pitch circle radius of r’ and module mn . It is called a ‘ formative’ or ‘virtual’ spur gear. The pitch circle diameter d’ of the virtual gear is given by, By S M Gujrathi

8. The number of teeth z’ on this imaginary spur gear is called the virtual number of teeth where z is the actual number of teeth. By S M Gujrathi

9. TOOTH PROPORTIONS In helical gears, the normal module m n should be selected from standard values. The first preference values of the normal module are m n (in mm) = 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8 and 10 The standard proportions of the addendum and the dedendum are, addendum ( h a ) = m n dedendum ( h f ) = 1.25 m n clearance ( c) = 0.25 m n The addendum circle diameter d a is given by By S M Gujrathi

10. Similarly, the dedendum circle diameter df is given by The normal pressure angle an is always 20°. The helix angle varies from 15 to 25°. A portion of the top view of a helical gear is shown in Fig By S M Gujrathi

11. For this rotation, the point A1 will be the fi rst point to come in contact with its meshing tooth on the other gear. It is called the ‘leading’ edge of the tooth. Also, the point A2 will be the last point to come in contact with its meshing tooth on the other gear. It is called the ‘trailing’ edge of the tooth. In order that the contact on the face of the tooth shall always contain at least one point, the leading edge of the tooth should be advanced ahead of the trailing end by a distance greater than the circular pitch. Or, x ≥ p By S M Gujrathi

12. FORCE ANALYSIS The resultant force P acting on the tooth of a helical gear is resolved into three components, Pt, Pr and Pa as shown in Fig. By S M Gujrathi

13. How to find force magnitude? How to find directions of forces? First find: ( i ) Find driven and driving element (ii) Direction of rotation of pinion(CW/CCW) (iii)Right hand or left hand helix ( i ) Tangential Component (Pt) (a) The direction of tangential component for a driving gear is opposite to the direction of rotation. (b) The direction of tangential component for a driven gear is same as the direction of rotation. (ii) Radial Component (Pr ) (a) The radial component on the pinion acts towards the centre of the pinion. (b) The radial component on the gear acts towards the centre of the gear. (iii) Thrust Component (Pa) (a) Select the driving gear from the pair. (b) Use right hand for RH-helix and left hand for LH-helix. (c) Keep the fingers in the direction of rotation of the gear and the thumb will indicate the direction of the thrust component for the driving gear. (d) The direction of the thrust component for the driven gear will be opposite to that for the driving gear. By S M Gujrathi

14. A pair of parallel helical gears is shown in Fig. A 5 kW power at 720 rpm is supplied to the pinion A through its shaft. The normal module is 5 mm and the normal pressure angle is20°. The pinion has right-hand teeth, while the gear has left-hand teeth. The helix angle is 30°. The pinion rotates in the clockwise direction when seen from the left side of the page. Determine the components of the tooth force and draw a free-body diagram showing the forces acting on the pinion and the gear . By S M Gujrathi

15. BEAM STRENGTH OF HELICAL GEARS In order to determine beam strength, the helical gear is considered to be equivalent to a formative spur gear. EFFECTIVE LOAD ON GEAR TOOTH By S M Gujrathi

16. By S M Gujrathi

17. By S M Gujrathi

18. By S M Gujrathi

19. By S M Gujrathi

20. By S M Gujrathi

21. By S M Gujrathi

22. By S M Gujrathi

23. By S M Gujrathi

24. By S M Gujrathi

25. By S M Gujrathi

26. By S M Gujrathi

27. By S M Gujrathi

28. By S M Gujrathi

29. By S M Gujrathi

30. By S M Gujrathi

31. By S M Gujrathi

32. By S M Gujrathi

33. Bevel Gears Bevel gears are used to transmit power between two intersecting shafts. Types of Bevel Gears: (a) Straight Bevel Gear (b) Spiral Bevel Gear Three types of bevel gears that are based on pitch angle are as follows: ( i ) When the pitch angle is less than 90°, it is called external bevel gear. (ii) When the pitch angle is equal to 90°, it is called crown bevel gear. (iii) When the pitch angle is more than 90°, it is called internal bevel gear. ( i ) Miter Gears When two identical bevel gears are mounted on shafts, which are intersecting at right angles, they are called ‘ miter’ gears. By S M Gujrathi

34. Terminologies of bevel gear ( i ) Pitch Cone: Pitch cone is an imaginary cone the surface of which contains the pitch lines of all teeth in the bevel gear. (ii) Cone Centre The apex of the pitch cone is called the cone centre. It is denoted by O. (iii) Cone Distance Cone distance is the length of the pitch-cone element. It is also called pitch-cone radius. It is denoted by A0. (iv) Pitch Angle The angle that the pitch line makes with the axis of the gear, is called the pitch angle. It is denoted by γ . The pitch angle is also called centre angle. (v) Addendum Angle It is the angle subtended by the addendum at the cone centre. It is denoted by α . (vi) Dedendum Angle It is the angle subtended by the dedendum at the cone centre. It is denoted by δ . (vii) Face Angle It is the angle subtended by the face of the tooth at the cone centre. Face angle = pitch angle + addendum angle = γ + α . (viii) Root Angle It is the angle subtended by the root of the tooth at the cone centre. Root angle = pitch angle – dedendum angle = γ – δ (ix) Back Cone The back cone is an imaginary cone and its elements are perpendicular to the elements of the pitch cone. (x) Back Cone Distance It is the length of the back cone element. It is also called back cone radius. It is denoted by r b . By S M Gujrathi

35. Cross section of the tooth decreases in size as it approaches towards the apex point O Pitch circle diameter, module, addendum, and dedendum decreases and there is no single value for these parameters The dimensions of the bevel gear are always specified and measured at the large end of the tooth Observe ha, hf and D in a figure. An imaginary spur gear is considered in a plane perpendicular to the tooth at the large end. r b is the pitch circle radius of this imaginary spur gear(formative gear) and z’ is the number of teeth on this gear. By S M Gujrathi

36. where m is the module at the large end of the tooth. If z is the actual number of teeth on the bevel gear , then A pair of bevel gears is shown in Fig. 19.8. Dp and Dg are the pitch circle diameters of pinion and gear respectively. γ is the pitch angle of the pinion, while Г is the pitch angle of the gear. Line AB is perpendicular to the line OB. Consider the triangle OAB, By S M Gujrathi

37. FORCE ANALYSIS By S M Gujrathi

38. By S M Gujrathi

39. ( i ) Tangential Component (Pt) (a) The direction of tangential component for the driving gear is opposite to the direction of rotation. (b) The direction of tangential component for the driven gear is same as the direction of rotation. (ii) Radial Component (Pr) (a) The radial component on the pinion acts towards the centre of the pinion. (b) The radial component on the gear acts towards the centre of the gear. (iii) Thrust Component (Pa) The following guidelines can be used to determine the direction of the thrust component: (a) The thrust component on the pinion is equal and opposite of the radial component on the gear. (b) The thrust component on the gear is equal and opposite of the radial component on the pinion. By S M Gujrathi

40. A pair of bevel gears transmitting 7.5 kW at 300 rpm is shown in Fig. 19.11(a). The pressure angle is 20°. Determine the components of the resultant gear tooth force and draw a free-body diagram of forces acting on the pinion and the gear. By S M Gujrathi

41. By S M Gujrathi

42. BEAM STRENGTH OF BEVEL GEARS Consider an elemental section of the tooth at a distance x from the apex O and having a width dx . Applying the Lewis equation to a formative spur gear at a distance x from the apex, By S M Gujrathi

43. By S M Gujrathi

44. The face width of the bevel gear is generally taken as 10 m or ( Ao /3), whichever is smaller, i.e., b = 10 m or b = Ao /3 (whichever is smaller) By S M Gujrathi

45. WEAR STRENGTH OF BEVEL GEARS By S M Gujrathi

46. pinion or the gear is generally overhanging. It has been found that to transmit the load, only three quarters of the face width is effective By S M Gujrathi

47. EFFECTIVE LOAD ON GEAR TOOTH By S M Gujrathi

48. Bevel gears are usually made of steel and the deformation factor C is 11 400 N/mm2. In practice, it is necessary to contact the manufacturer and find out the expected error between meshing teeth. In the absence of such information. Table 19.1 may be used to get the values of error e. The classes of gears mentioned in the table indicate the following manufacturing methods, Class-1 Well cut commercial gear teeth Class-2 Gear teeth cut with great care Class-3 Ground and lapped precision gear teeth By S M Gujrathi

49. By S M Gujrathi

50. By S M Gujrathi

51. By S M Gujrathi

52. By S M Gujrathi

53. By S M Gujrathi

54. By S M Gujrathi

55. By S M Gujrathi

56. Reference V.B.Bhandari,Design of Machine Elements,3 rd Edition,Chpter 18 and 19,pp:694-728 By S M Gujrathi