LECTURE NOTES ENT345 MECHANICAL COMPONENTS DESIGN Lecture 6, 7 29/10/2015 SPUR AND HELICAL GEARS

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1 LECTURE NOTES ENT345 MECHANICAL COMPONENTS DESIGN Lecture 6, 7 29/10/2015 SPUR AND HELICAL GEARS Dr. HAFTIRMAN MECHANICAL ENGINEEERING PROGRAM SCHOOL OF MECHATRONIC ENGINEERING UniMAP COPYRIGHT RESERVED 2015 ENT345 Mechanical Components Design Sem /2016 Dr. Haftirman School of Mechatronic Engineering 1

2 AGMA The American Gear Manufacturers Association (AGMA) has for many years been the responsible authority for the dissemination of knowledge pertaining to the design analysis of gearing. ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 2

3 The Lewis Bending Equation Wilfred Lewis introduced an equation for estimating the bending stress in gear teeth in which the tooth form entered into the formula. The equation, announced in 1892, still remains the basis for most gear design today. ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 3

4 The Lewis Bending Equation To derive the basic Lewis equation refer to Figure, which shows a cantilever of cross-sectional dimensions F and t, having a length l and a load W t, uniformly distributed across the face width F. The section modulus: I/c = Ft 2 /6 The bending stress (σ ). ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 4

ENT345 Mechanical Componets Design Sem 1-2015/2016 Dr.

5 ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 5 The Lewis bending equations (12) 2 / Ft W l Ft t Ft c I c I M t l t x t l x t 4 2 / 2 / / / x x l t F W l t F W Ft l W t t t

6 The Lewis bending equations t W p 2 F xp 3 t W p Fpy y 2x 3p y is The Lewis form factor W r W t P, Y p y t W P FY Y 2xP 3 Y means that only the bending of the tooth is considered and that the compression due to the radial component of the force is neglected. ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 6

7 Values of the Lewis form factor Y ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 7

8 Dynamic effects When a pair of gears is driven at moderate or high speed and noise is generated, it is certain that dynamic effects are present. If a pair of gears failed at 500 lbf tangential load at zero velocity and at 250 lbf at velocity V 1, then a velocity factor, designated Kv, of 2 was specified for the gears at velocity V 1. ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 8

9 Dynamic effects K K K K v v v v K v = the velocity factor V = the pitch-line velocity in ft/min 600 V ( cast iron, cast profile )) V ( cut or mild profile ) V V 78 ( hobbed or shaped ( shaved or ground profile ) profile ) K K K K v v v v SI units 3.56 V V ( cast iron, cast profile )) V ( cut or mild profile ) V 5.56 K W v FY t ( hobbed or shaped P ( shaved or ground K v W FmY profile ) profile ) t ENT345 Mechanical Componets Design Sem /2016 The metric versions Dr. School of Mechatronic Engineering 9

10 Example 14 1

11 Example 14 1 X X

12 Example 14-1 A stock spur gear is available having a module of 4 mm, a 44 mm face width, 18 teeth, and a pressure angle of 20 with full-depth teeth. The material is AISI 1020 steel in asrolled condition. Use a design factor of n d = 3 to rate the power output of the gear corresponding to a speed of n = 25 rev/s and moderate applications. Solution The term moderate applications seems to imply that the gear can be rated by using the yield strength as a criterion of failure. AISI 1020 steel in as-rolled, from Table A-18, S ut = 380 MPa and S y = 210 MPa. ENT345 Mechanical Components Design Sem /2015 Dr. Haftirman School of Mechatronic Engineering 12

13 Example 14-1 Solution A design factor of 3 means that the allowable bending stress is S y = 210 = 60 MPa n d 3.5 The pitch diameter of d = Nm = 18 (4) = 72 mm. The pitch-line velocity is V = πdn= π(0.072) 25 = m/s The velocity factor (Eq 14-6b): K v = 6.1+V = = ENT345 Mechanical Components Design Sem /2015 Dr. Haftirman School of Mechatronic Engineering 13

14 Table 14-2, Y=0.309 for 18 teeth. The tangential component of load W t W t = FYσ all K v P σ all = S y = 210 n d 3.5 = 60 MPa P = N d N d = 1 m W t = FYσ all K v P = mfyσ all K v 4mm 44mm N/mm2 = = N ( ) The power that can be transmitted is Hp=W t V= ( N)( m/s)= W ENT345 Mechanical Components Design Sem /2015 Dr. Haftirman School of Mechatronic Engineering 14

15 Example 14 2 m = 4 mm P = 25.4 m = = 6.35 m = 3 mm P = 25.4 m = = 8.47

16 Example 14 2

17 Example 14 2

18 Example 14 2

19 Example 14 2

20 Example 14 2

21 Fatigue Stress-Concentration Factor A photoelastic investigation gives an estimate of fatigue stressconcentration factor as

22 Surface durability Wear is the failure of the surfaces of gear teeth. Pitting is a surface fatigue failure due to many repetitions of high contact stresses. Scoring is a lubrication failure, and abrasion, which is wear due to the presence of foreign material. ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 22

23 Surface durability To obtain an expression for the surface-contact stress, we shall employ the Hertz theory. The contact stress between two cylinders may computed from the equation; p max =largest surface pressure. F= force pressing the two cylinders together. l = length of cylinders. ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 23

24 Cylindrical Contact Stress Two right circular cylinders with length l and diameters d 1 and d 2 Area of contact is a narrow rectangle of width 2b and length l Pressure distribution is elliptical Half-width b Maximum pressure Fig. 3 38

25 Surface durability Half-width b is obtained from The surface compressive stress (Hertzian stress) ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 25

26 Surface durability The radii of curvature of the tooth profiles at the pitch point are An elastic coefficient C p ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 26

27 Example 14 3

28 Example 14 3

29 SPUR GEAR BENDING Based on ANSI/AGMA 2001-D04 (US. Customary units) Fig

30 SPUR GEAR WEAR Based on ANSI/AGMA 2001-D04 (US. Customary units) Fig

31 AGMA equations Two fundamental stress equations are used in the AGMA methodology, one for bending stress and another for pitting resistance (contact stress). In AGMSA terminology, these are called stress numbers, as contrasted with actual applied stress (σ). ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 31

32 AGMA Bending Stress equations ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 32

33 AGMA Contact Stress equations ENT345 Mechanical Componets Design Sem /2016 Dr. School of Mechatronic Engineering 33

34 AGMA Strengths AGMA uses allowable stress numbers rather than strengths. We will refer to them as strengths for consistency within the textbook. The gear strength values are only for use with the AGMA stress values, and should not be compared with other true material strengths. Representative values of typically available bending strengths are given in Table 14 3 for steel gears and Table 14 4 for iron and bronze gears. Figs. 14 2, 14 3, and 14 4 are used as indicated in the tables. Tables assume repeatedly applied loads at 10 7 cycles and 0.99 reliability.

35 Bending Strengths for Steel Gears

36 Bending Strengths for Iron and Bronze Gears

37 Bending Strengths for Through-hardened Steel Gears Fig. 14 2

38 Bending Strengths for Nitrided Through-hardened Steel Gears Fig. 14 3

39 Bending Strengths for Nitriding Steel Gears Fig. 14 4

40 Allowable Bending Stress

41 Allowable Contact Stress

42 Nominal Temperature Used in Nitriding and Hardness Obtained Table 14 5

43 Contact Strength for Steel Gears

44 Contact Strength for Iron and Bronze Gears

45 Contact Strength for Through-hardened Steel Gears Fig. 14 5

46 Geometry Factor J (Y J in metric) Accounts for shape of tooth in bending stress equation Includes A modification of the Lewis form factor Y Fatigue stress-concentration factor K f Tooth load-sharing ratio m N AGMA equation for geometry factor is Values for Y and Z are found in the AGMA standards. For most common case of spur gear with 20º pressure angle, J can be read directly from Fig For helical gears with 20º normal pressure angle, use Figs and 14 8.

47 Spur-Gear Geometry Factor J Fig. 14 6

48 Helical-Gear Geometry Factor J Get J' from Fig. 14 7, which assumes the mating gear has 75 teeth Get multiplier from Fig for mating gear with other than 75 teeth Obtain J by applying multiplier to J' Fig. 14 7

49 Modifying Factor for J Fig. 14 8

50 Surface Strength Geometry Factor I (Z I in metric) Called pitting resistance geometry factor by AGMA

51 Elastic Coefficient C P (Z E ) Obtained from Eq. (14 13) or from Table 14 8.

52 Elastic Coefficient

53 Dynamic Factor K v Accounts for increased forces with increased speed Affected by manufacturing quality of gears A set of quality numbers Q v define tolerances for gears manufactured to a specified accuracy. Quality numbers 3 to 7 include most commercial-quality gears. Quality numbers 8 to 12 are of precision quality. The AGMA transmission accuracy-level number A v is basically the same as the quality number.

54 Dynamic Factor equation Dynamic Factor K v Or can obtain value directly from Fig Maximum recommended velocity for a given quality number,

55 Dynamic Factor K v Fig. 14 9

56 Overload Factor K O To account for likelihood of increase in nominal tangential load due to particular application. Recommended values,

57 Surface Condition Factor C f (Z R ) To account for detrimental surface finish No values currently given by AGMA Use value of 1 for normal commercial gears

58 Size Factor K s Accounts for fatigue size effect, and non-uniformity of material properties for large sizes AGMA has not established size factors Use 1 for normal gear sizes Could apply fatigue size factor method from Ch. 6, where this size factor is the reciprocal of the Marin size factor k b. Applying known geometry information for the gear tooth,

59 Load-Distribution Factor K m (K H ) Accounts for non-uniform distribution of load across the line of contact Depends on mounting and face width Load-distribution factor is currently only defined for Face width to pinion pitch diameter ratio F/d p 2 Gears mounted between bearings Face widths up to 40 in Contact across the full width of the narrowest member

60 Load-Distribution Factor K m (K H ) Face load-distribution factor

61 Load-Distribution Factor K m (K H )

62 Load-Distribution Factor K m (K H ) Fig

63 Load-Distribution Factor K m (K H ) C ma can be obtained from Eq. (14 34) with Table 14 9 Or can read C ma directly from Fig

64 Load-Distribution Factor K m (K H ) Fig

65 Hardness-Ratio Factor C H (Z W ) Since the pinion is subjected to more cycles than the gear, it is often hardened more than the gear. The hardness-ratio factor accounts for the difference in hardness of the pinion and gear. C H is only applied to the gear. That is, C H = 1 for the pinion. For the gear, Eq. (14 36) in graph form is given in Fig

66 Hardness-Ratio Factor C H Fig

67 Hardness-Ratio Factor If the pinion is surface-hardened to 48 Rockwell C or greater, the softer gear can experience work-hardening during operation. In this case, Fig

68 Stress-Cycle Factors Y N and Z N AGMA strengths are for 10 7 cycles Stress-cycle factors account for other design cycles Fig gives Y N for bending Fig gives Z N for contact stress

69 Stress-Cycle Factor Y N Fig

70 Stress-Cycle Factor Z N Fig

Reliability Factor K R (Y Z ) Accounts for statistical distributions of material fatigue failures Does not account for load variation Use Table

71 Reliability Factor K R (Y Z ) Accounts for statistical distributions of material fatigue failures Does not account for load variation Use Table Since reliability is highly nonlinear, if interpolation between table values is needed, use the least-squares regression fit, Table 14 10

72 Temperature Factor K T (Y q ) AGMA has not established values for this factor. For temperatures up to 250ºF (120ºC), K T = 1 is acceptable.

73 Rim-Thickness Factor K B Accounts for bending of rim on a gear that is not solid Fig

74 Safety Factors S F and S H Included as design factors in the strength equations Can be solved for and used as factor of safety Or, can set equal to unity, and solve for traditional factor of safety as n = all /

75 Comparison of Factors of Safety Bending stress is linear with transmitted load. Contact stress is not linear with transmitted load To compare the factors of safety between the different failure modes, to determine which is critical, Compare S F with S H 2 for linear or helical contact Compare S F with S H 3 for spherical contact

76 Summary for Bending of Gear Teeth Fig

77 Summary for Surface Wear of Gear Teeth Fig

78 Example 14 4

79 Example 14 4

80 Example 14 4

81 Example 14 4

82 Example 14 4

83 Example 14 4

84 Example 14 4

85 Example 14 4

86 Example 14 4

87 Example 14 4

88 Example 14 4

89 Example 14 4

90 Example 14 5

91 Example 14 5

92 Example 14 5

93 Example 14 5

94 Example 14 5

95 Example 14 5

96 Example 14 5

97 Example 14 5

98 Comparing Pinion with Gear Comparing the pinion with the gear can provide insight. Equating factors of safety from bending equations for pinion and gear, and cancelling all terms that are equivalent for the two, and solving for the gear strength, we get Substituting in equations for the stress-cycle factor Y N, Normally, m G > 1, and J G > J P, so Eq. (14 44) indicates the gear can be less strong than the pinion for the same safety factor.

99 Comparing Pinion and Gear Repeating the same process for contact stress equations, Neglecting C H which is near unity,

100 Example 14 6

101 Example 14 7

102 Example 14 8

103 Example 14 8

104 Example 14 8

105 Example 14 8

106 Example 14 8

107 Example 14 8

108 Example 14 8

109 Example 14 8

110 Example 14 8

111 Example 14 8

112 Assignment 1 Page Problem:

113 Thank you ENT345 Mechanical Components Design Sem /2016 Dr. Haftirman School of Mechatronic Engineering 113

Lecture Slides. Chapter 14. Spur and Helical Gears

Lecture Slides. Chapter 14. Spur and Helical Gears Lecture Slides Chapter 14 Spur and Helical Gears The McGraw-Hill Companies 2012 Chapter Outline Cantilever Beam Model of Bending Stress in Gear Tooth Fig. 14 1 Lewis Equation Lewis Equation Lewis Form