1. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.1 (p. 591) Primitive gears.

2. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Introduction-Spur Gears Function: To transmit power, motion and position. Advantage: High power transmission efficiency, 98%, compact, high speed, precise timing. Disadvantage: Gears are more costly than belts and chains. Gear manufacturing costs increase sharply with increased precision including high speeds, heavy loads, and low noise.

3. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.2 (p. 592) Spur gears.

4. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Conjugate Gear-Tooth Action The basic requirement of gear-tooth geometry is the ability to transmit motion in a constant angular velocity ratio at all times. For example, the angular velocity ratio between a 20-tooth and a 40-tooth gear must be precisely 2 in every position. The action of a pair of gear teeth satisfying this requirement is termed conjugate gear tooth action, see fig. 15.3. The basic law of conjugate gear-tooth action is: As the gear rotate, the common normal to the surfaces at the point of contact must always intersect the line of centers at the same point P, called the pitch point.

5. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.3 (p. 593) Conjugate gear-tooth action.

6. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Involute Profile The law of conjugate gear-tooth action can be satisfied by various tooth shapes, but the most important one is the involute of the circle. An involute of the circle is the curve generated by any point on a taut thread as it unwinds from a circle, see Fig. 15.4. Correspondingly, involutes generated by unwinding a thread wrapped counterclockwise around the base circle would form the outer portion of the left sides of the teeth. Note that at every point, the involute is perpendicular to the taut thread. An involute cannot exist inside its base circle.

7. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.4 (p. 593) Generation of an involute from its base circle.

8. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.5 (p. 594) Friction gears of diameter d rotating at angular velocity .

9. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.6 (p. 594) Belt drive added to friction gears.

10. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.7 (p. 595) Belt cut at c to generate conjugate involute profiles.

11. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.8 (p. 596) Further development and nomenclature of involute gear teeth. Note: The diagram shows the special case of maximum possible gear addendum without interference; pinion addendum is far short of the theoretical limit.

12. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Fig. 15.5 shows two pitch circles. If there is no slippage, rotation of one cylinder will cause rotation of the other at an angular velocity ratio inversely proportional to their diameters. The smaller is called pinion and the larger one the gear. We have,  p /  g = - d g /d p (15.1) Where  is the angular velocity, d is the pitch diameter, and the minus sign indicates that the two cylinders rotate in opposite directions. The center distance is c = (d p + d g )/2 = r p + r g (15.1a) where r is the pitch circle radius.

13. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.9 (p. 597) Nomenclature of gear teeth.

14. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Circular pitch, p, measured in inches or mm. If N is the number of teeth and d is the pitch diameter, then p =  d/N, p =  d p /N p, p =  d g /N g (15.2)

15. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Diametral pitch, P, is defined as the number of teeth per inch of pitch diameter (used only with English units): P = N/d, P = Np/dp, P = Ng/dg(15.3) Module m, which is essentially the reciprocal of P, is defined as the pitch diameter in millimeters divided by the number of teeth: m = d/N, m = dp/Np, m = dg/Ng(15.4) It can be seen that, pP =  (p in inches and P in teeth per inch) (15.5) and p/m =  (p in millimeters and m in millimeters per tooth)(15.6) m = 25.4/P(15.7) In English units the “pitch” means diametral pitch P, a “12-pitch gear” refers to a gear with 12 teeth per inch of pitch diameter, whereas in SI units “pitch” means circular pitch p, a “gear of pitch = 3.14 mm” refers to a gear having a circular pitch of 3.14 mm.

16. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Gears are commonly made to an integral value of diametral pitch P (English units) or standard value of module m (SI units). Fig. 15.10 shows the actual size of gear teeth of several standard diametral pitches. With SI units, commonly used standard values of module are: 0.2 to 1.0 by increments of 0.1 1.0 to 4.0 by increments of 0.25 4.0 to 5.0 by increments of 0.5 The most commonly used pressure angle, , with both English and SI units is 20 0. For all systems, the standard addendum is a = 1/P, in inches, or a = m, in millimeters, and the standard dedendum is 1.25 * a. The fillet radius at the base of the tooth, is 0.35/P (English units) or m/3 (SI units). Face width, b is generally, 9/P < b < 14/P or 9m < b < 14m Gears made to standard systems are interchangeable and are usually available in stock.

17. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.10 (p. 598) Actual sizes of gear teeth of various diametral pitches. Note: In general, fine- pitch gears have P  20; coarse-pitch gears have P < 20. (Courtesy Bourn & Koch Machine Tool Company.)

18. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.11 (p. 599) Involute pinion and rack.

19. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.12 (p. 600) Involute pinion and internal gear. Note that both rotate in the same direction.

20. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.13 (p. 600) Generating a gear with a shaping machine suitable for external and internal gears. (For additional information see http://www.liebher.com/gt/en/). (Courtesy Cleason-Pfauter Maschinenfabik GmbH.) http://www.liebher.com/gt/en/

21. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.14 (p. 601) Shaping teeth with a rack cutter.

22. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.15 (p. 602) Interference of spur gears (eliminated by removing the shaded tooth tips).

23. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Interference of spur gears and Contact Ratio From Fig. 15.15, ra = r + a where ra = addendum circle radius, r = pitch circle radius, a = addendum. The maximum possible addendum circle radius without interference can be obtained from right triangle O1ab or O2ab, (15.8) where ra = maximum noninterfering addendum circle radius of pinion or gear rb = base circle radius, c = center distance,  = pressure angle The average number of teeth in contact as the gears rotate together is the contact ratio (CR), (15.9)

24. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.16 (p. 603) Spur gears for Sample Problem 15.1D.

25. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.17 (p. 605) Gear-tooth force F, shown resolved at pitch point. The driving pinion and driven gear are shown separately.

26. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Gear Force Analysis 1. Tangential component Ft, which when multiplied by the pitch line velocity, accounts for the power transmitted. 2. Radial component Fr, which does not work but tends to push the gear apart. Fig. 15. 17 illustrates that Fr = F t tan  (15.12) Gear pitch line velocity V, in feet per minute, V =  dn/12(15.13) where d is the pitch diameter in inch, gear rotating in n rpm. The transmitted power in horse power hp is W’ = F t V/33,000(15.14) where Ft is in pound and V is in feet per minute. In SI units V =  dn/60,000(15.13a) where d is in mm, n is in rpm, and V is in meters per second. Transmitted power in watts, is in Newton W’ = F t V(15.14a)

27. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.18 (p. 607) Gear forces in Sample Problem 15.2. (a) Gear layout. (b) Forces acting on idler b.

28. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.19 (p. 608) Photoelastic pattern of stresses in a spur gear tooth. (From T.J. Dolan and E.L. Broghammer, A Study of Stresses in Gear Tooth Fillets, Proc. 14 th Eastern Photoelasticity Conf., PE December 1941.)

29. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.20 (p. 609) Bending stresses in a spur gear tooth (comparison with a constant-stress parabola).

30. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Basic Analysis of Gear-Tooth-Bending Stress (Lewis Equation) By using English unit system, the bending stress is: where F t : Tangential force; P: Diametral pitch; b: Face width; Y: Lewis form factor (see Fig. 15.21). When using SI units, we have (15.16a) (15.16)

31. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.21 (p. 611) Values of Lewis form factor Y for standard spur gears (load applied at tip of the tooth).

32. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Refined Analysis of Gear-Tooth-Bending Strength: Basic Concepts The important strength property is usually the bending fatigue strength, as represented by the endurance limit. From Eqa. 8.1 Sn = Sn’ C L C G C S C L = load factor = 1.0 for bending loads C G = gradient factor = 1.0 for P > 5, and 0.85 for P <= 5 C S = Surface factor from Fig. 8.13 For steel members S = (0.55Su) C L C G C S

33. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. –Refined Analysis of Gear-Tooth-Bending Strength: Recommended Procedure In the absence of more specific information, the factors affecting gear-tooth-bending stress can be taken into account by embellishing the Lewis equation to the following form: (15.17) J = spur gear geometry factor from Fig. 15.23 Kv = velocity or dynamic factor Ko = overload factor Km = mounting factor

34. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.23b (cont.)

35. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.24 (p. 615) Velocity factor K v. (Note: This figure, in a very rough way, is intended to account for the effects of tooth spacing and profile errors, tooth stiffness and the velocity, inertia, and stiffness of the rotating parts.)

36. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Table 15.1 (p. 615) Overload Correction Factor K O.

37. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Table 15.2 (p. 616) Mounting Correction Factor K m

38. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Table 15.3 (p. 616) Reliability Correction Factor k r, from Figure 6.19 with Assumed Standard Deviation of 8 Percent.

39. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.25 (p. 617) Date for Sample Problem 15.3.

40. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.23a (p. 614) Geometry factor J for standard spur gears (based on tooth fillet radius of 0.35/P). (From AGMA Information Sheet 225.01; also see AGMA 908-B89.) (Continued on next slide.)

41. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.26 (p. 620) Gear-tooth sliding velocity.

42. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. The effective fatigue stress from Equa. 15.17 must be compared with the corresponding fatigue strength. For infinite life the appropriate endurance limit is : Sn = Sn’ C L C G C S k r k t k ms (15.18) where Sn’ = standard R.R. Moore endurance limit C L = load factor = 1.0 for bending loads C G = gradient factor = 1.0 for P > 5, and 0.85 for P <= 5 C S = Surface factor from Fig. 8.13 k r = reliability factor from Fig. 6.19 or Table 15.3. k t = temperature factor. For steel gears k t = 1.0 if temp. 160 F, k t = 620/(460 + T). k ms = mean stress factor. Use 1.0 for idler gears, use 1.4 for input and output gears

43. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Gear-Tooth Surface Fatigue Analysis- Recommended Procedure Surface fatigue stress (15.24) where elastic coefficient: Geometry factor: Table 15.4a (p. 621)

44. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Table 15.4a (p. 621)

45. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Table 15.4b (p. 621)

46. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Table 15.5 (p. 624) Surface Fatigue Strength S fe, for Use with Metallic Spur Gears (107-Cycle Life, 99 Percent Reliability, Temperature < 250  F)

47. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.27 (p. 624) Values of C Li for steel gears (general shape of surface fatigue S-N curve).

48. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Table 15.6 (p. 625) Reliability Factor C R.

49. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.28 (p. 629) Single-reduction spur gear train.......

50. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Gear Trains The speed ratio (or gear ratio of a single pair of external spur gear is: (15.26) where  and n are rotating speed in radians per second and rpm, respectively, d represents pitch diameter, and N is the number of teeth. The minus mean opposite directions. If it is internal teeth the sign will be positive and is in same direction. Pinion is usually driver and the gear the driven, which provide a reduction ratio, but increase in torque.

51. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.29 (p. 633) Double-reduction gear train.

52. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Gear Trains Fig. 15.29 shows a double reduction gear train involving countershaft b as well as input shaft a and output shaft c. The overall speed ratio is: (15.26) This equation can be extended to three, four or any number of gear pairs.

53. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.30 (p. 633) Typical planetary gear train.

54. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.31 (p. 634) Torque ratio (1 divided by the speed ratio) determined by free-body diagrams.

55. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.32 (p. 635) Speed ratio determined by velocity vector diagram.

56. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure 15.33 (p. 636) Geometric study of two versus four equally spaced planets for a 20-tooth sun and a 70-tooth ring.

57. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure P15.23 (p. 639)

58. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure P15.25 (p. 640)

59. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure P15.28 (p. 640)

60. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure P15.29 (p. 641)

61. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure P15.47 (p. 644)

62. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure P15.49 (p. 645)

63. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure P15.52 (p. 645)

64. Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Figure P15.53 (p. 646)