12/25/ :27 PM. Chapter 14. Spur and Helical Gears. Mohammad Suliman Abuhaiba, Ph.D., PE

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1 Chapter 14 Spur and Helical Gears 1

2 2 The Lewis Bending Equation Equation to estimate bending stress in gear teeth in which tooth form entered into the formulation:

3 3 The Lewis Bending Equation Assume that max stress in a gear tooth occurs at point a

4 4 The Lewis Bending Equation Only bending of tooth is considered Compression due to radial component of force is neglected

5 5 The Lewis Bending Equation Table 14 2 Lewis Form Factor Y (f n = 20, Full- Depth Teeth, Diametral Pitch of Unity in Plane of Rotation)

6 6 The Lewis Bending Equation Dynamic Effects Barth velocity factor (English units)

7 7 The Lewis Bending Equation Dynamic Effects Barth velocity factor (SI units)

8 8 The Lewis Bending Equation Dynamic Effects Introducing velocity factor into Eq. (14 2) gives The metric version of this equation is Spur gears: face width F = 3 to 5 times circular pitch p

9 9 The Lewis Bending Equation Fatigue stress-concentration factor K f Mitchiner & Mabie by l & t = layout - Fig f = pressure angle r f = fillet radius b = dedendum d = pitch diameter

10 10 Example 14 1 A stock spur gear is available having a diametral pitch of 8 teeth/in, a 1.5 face, 16 teeth, and a pressure angle of 20 with fulldepth teeth. The material is AISI 1020 steel in as rolled condition. Use a design factor of n d = 3 to rate the horsepower output of the gear corresponding to a speed of 1200 rpm and moderate applications.

11 Example Table A 20: S ut = 55 kpsi & S y = 30 kpsi. N d = 3, allowable bending stress = 30/3 = 10 kpsi pitch diameter = N/P = 16/8 = 2 in Table 14 2: form factor Y = for 16 teeth

12 Example Estimate the horsepower rating of the gear in the previous example based on obtaining an infinite life in bending. Table 6 3

13 Example k c = k d = k e = 1

14 Example If a material exhibited a Goodman failure locus, Gerber fatigue locus gives mean values of k f = 1.66

15 Example Fig. A 15 6 K t = 1.68: Fig. 6 20, q = 0.62 ; Eq. (6 32)

16 Example

17 Surface Durability Wear: failure of the surfaces of gear teeth Pitting: a surface fatigue failure due to many repetitions of high contact stresses Scoring: a lubrication failure Abrasion: wear due to presence of foreign material

18 Surface Durability Eq. (3 74): contact stress between two cylinders p max = largest surface pressure Figure 3 38

19 Surface Durability For gears, replace F by W t / cos φ, d by 2r, and l by face width F Replacing p max by σ C, surface compressive stress is found from the equation r 1 & r 2 : instantaneous values of radii of curvature on pinion & gear-tooth profiles at point of contact

20 Surface Durability First evidence of wear occurs near the pitch line Radii of curvature of tooth profiles at pitch point: AGMA defines an elastic coefficient C p

21 Surface Durability With this simplification, and the addition of a velocity factor K v, Eq. (14 11) can be written as

22 Example The pinion of Examples 14 1 and 14 2 is to be mated with a 50-tooth gear manufactured of ASTM No. 50 cast iron. Using the tangential load of 382 lbf, estimate the factor of safety of the drive based on the possibility of a surface fatigue failure.

Example 14 3 23 Table A 5: E P = 30 Mpsi, n P = 0.

23 Example Table A 5: E P = 30 Mpsi, n P = 0.292, E G = 14.5 Mpsi, n G = d P = 2 in, d G = 50/8 = 6.25 in

24 Example F = 1.5 in, K v = 1.52 Surface endurance strength of cast iron for 10 8 cycles: Table A 24: H B = 262 for ASTM No. 50 cast iron factor of safety = S C /σ C

25 AGMA Stress Equations Two fundamental stress equations are used in the AGMA methodology: 1. For bending stress 2. For contact stress In AGMA terminology, called stress numbers

26 AGMA Stress Equations - Bending W t = tangential transmitted load, lbf (N) K o = overload factor K v = dynamic factor K s = size factor P d = transverse diametral pitch F (b) = face width of narrower member, in (mm)

27 AGMA Stress Equations - Bending K m (K H ) = load-distribution factor K B = rim-thickness factor J (Y J ) = geometry factor for bending strength (includes root fillet stress-concentration factor K f ) m t = transverse metric module

28 AGMA Stress Equations Fundamental equation for contact stress C p (Z E ) = elastic coefficient, lbf/in 2 ( N/mm 2 ) C f (Z R ) = surface condition factor d P (d w1 ) = pitch diameter of pinion, in (mm) I (Z I ) = geometry factor for pitting resistance

29 AGMA Strength Equations Uppercase letter S = strength Lowercase Greek letters σ and τ = stress Gear strength = allowable stress numbers as used by AGMA Values for gear bending strength, S t = Figs. 14 2, 14 3, & 14 4, and Tables 14 3 & 14 4

30 AGMA Strength Equations S t = 0.533HB MPa, grade 1 S t = 0.703HB MPa, grade 2 Figure 14 2: Allowable bending stress number for through-hardened steels

31 AGMA Strength Equations Figure 14 3: Allowable bending stress number for nitrided through hardened steel gears (AISI 4140, 4340) S t = 0.568HB MPa, grade 1 S t = 0.749HB MPa, grade 2

32 AGMA Strength Equations Figure 14 4: Allowable bending stress numbers for nitriding steel gears

33 AGMA Strength Equations Figure 14 4: Allowable bending stress numbers for nitriding steel gears The SI equations are: 1. Nitralloy grade 1 S t = 0.594HB Mpa 2. Nitralloy grade 2 S t = 0.784HB Mpa % chrome, grade 1 S t = HB Mpa % chrome, grade 2 S t = HB Mpa % chrome, grade 3 S t = HB Mpa

34 14 4 AGMA Strength Equations Table 14 3: Repeatedly Applied

34 AGMA Strength Equations Table 14 3: Repeatedly Applied Bending Strength S t at 10 7 Cycles & 0.99 Reliability for Steel Gears

35 14 4 AGMA Strength Equations Table 14 4: Repeatedly Applied Bending

35 AGMA Strength Equations Table 14 4: Repeatedly Applied Bending Strength S t for Iron & Bronze Gears at 10 7 Cycles & 0.99 Reliability for Steel Gears

36 AGMA Strength Equations The equation for the allowable bending stress is S t = allowable bending stress, lbf/in 2 (N/mm 2 ) Y N = stress cycle factor for bending stress K T (Y θ ) = temperature factors K R (Y Z ) = reliability factors S F = AGMA factor of safety, a stress ratio

37 AGMA Strength Equations The equation for allowable contact stress σ c,all is S c = allowable contact stress, lbf/in 2 (N/mm 2 ) Z N = stress cycle life factor C H (Z W ) = hardness ratio factors for pitting resistance K T (Y θ ) = temperature factors K R (Y Z ) = reliability factors S H = AGMA factor of safety, a stress ratio

38 AGMA Strength Equations Allowable contact stress, S c : Fig and Tables 14 5, 14 6, and 14 7 AGMA allowable stress numbers (strengths) for bending and contact stress are for Unidirectional loading 10 million stress cycles 99 % reliability

39 AGMA Strength Equations Figure 14 5: Contact-fatigue strength S c at 10 7 cycles and 0.99 reliability for through-hardened steel gears S c = 2.22HB MPa, grade 1 S c = 2.41HB MPa, grade 2

40 14 4 AGMA Strength Equations Table 14 5: Nominal Temperature Used in Nitriding & Hardnesses

40 AGMA Strength Equations Table 14 5: Nominal Temperature Used in Nitriding & Hardnesses Obtained Source: Darle W. Dudley, Handbook of Practical Gear Design, rev. ed., McGraw-Hill, New York, 1984.

41 14 4 AGMA Strength Equations Table 14 6: Repeatedly Applied Contact Strength S

41 AGMA Strength Equations Table 14 6: Repeatedly Applied Contact Strength S c at 10 7 Cycles and 0.99 Reliability for Steel Gears Source: ANSI/AGMA 2001-D04.

42 14 4 AGMA Strength Equations Table 14 7: Repeatedly Applied Contact

42 AGMA Strength Equations Table 14 7: Repeatedly Applied Contact Strength S c at 10 7 Cycles and 0.99 Reliability for Iron and Bronze Gears

43 AGMA Strength Equations When two-way (reversed) loading occurs, AGMA recommends using 70 % of S t values.

44 Geometry Factors I & J (Z I & Y J ) Y is used in the Lewis equation to introduce the effect of tooth form into the stress equation. AGMA factors I & J: accomplish same purpose in a more involved manner Face-contact ratio m F p x = axial pitch F = face width For spur gears, m F = 0

45 Geometry Factors I & J (Z I & Y J ) Bending-Strength Geometry Factor J (Y J ) Equation for J for spur and helical gears is Factor Y in Eq is not Lewis factor at all Value of Y here is obtained from calculations within AGMA 908-B89, and is often based on the highest point of single-tooth contact.

46 Geometry Factors I & J (Z I & Y J ) Bending-Strength Geometry Factor J (Y J ) Factor K f in Eq. (14 20): stress-correction factor by AGMA. based on a formula deduced from a photo-elastic investigation of stress concentration in gear teeth.

47 Geometry Factors I & J (Z I & Y J ) Bending-Strength Geometry Factor J (Y J ) Load-sharing ratio m N = face width / min total length of lines of contact This factor depends on: Transverse contact ratio m p Face-contact ratio m F Effects of any profile modifications, and tooth deflection For spur gears, m N = 1.0

48 Geometry Factors I & J (Z I & Y J ) Bending-Strength Geometry Factor J (Y J ) For helical gears having a face-contact ratio m F > 2.0, a conservative approximation is given by p N = normal base pitch Z = length of line of action in the transverse plane = distance L ab in Fig

49 Geometry Factors I & J (Z I & Y J ) Bending-Strength Geometry Factor J (Y J ) Figure 13 15

50 14 5 Geometry Factors I & J (Z I & Y J ) Bending-Strength Geometry Factor J (Y J ) Figure 14 6: geometry

50 Geometry Factors I & J (Z I & Y J ) Bending-Strength Geometry Factor J (Y J ) Figure 14 6: geometry factor J for spur gears having a 20 pressure angle and full-depth teeth Number of teeth for which factor is desired

51 Geometry Factors I & J (Z I & Y J ) Bending-Strength Geometry Factor J (Y J ) Figure 14 7: Helical-gear geometry factors J

52 Geometry Factors I & J (Z I & Y J ) Bending-Strength Geometry Factor J (Y J ) Figure 14 8: J -factor multipliers for use with Fig to find J. Modifying factor can be applied to J factor when other than 75 teeth are used in the mating element

53 Geometry Factors I & J (Z I & Y J ) Surface-Strength Geometry Factor I (Z I ) Pitting-resistance geometry factor by AGMA

54 Geometry Factors I & J (Z I & Y J ) Surface-Strength Geometry Factor I (ZI) Pitting-resistance geometry factor by AGMA m N = 1 for spur gears p N = normal base pitch Z = length of line of action in the transverse plane (L ab in Fig )

55 Geometry Factors I & J (Z I & Y J ) Bending-Strength Geometry Factor J (Y J ) r P & r G = pitch radii r bp & r bg = base-circle radii of pinion & gear

56 The Elastic Coefficient C p (Z E ) Eq Table 14 8

57 Dynamic Factor K v Dynamic factor = account for inaccuracies in manufacture & meshing of gear teeth in action Transmission error = departure from uniform angular velocity of the gear pair

58 Dynamic Factor K v Effects that produce transmission error are: Inaccuracies produced in generation of tooth profile Vibration of tooth during meshing due to tooth stiffness Magnitude of pitch-line velocity Dynamic unbalance of rotating members

59 Dynamic Factor K v Effects that produce transmission error are: Wear & permanent deformation of contacting portions of teeth Gear shaft misalignment and linear & angular deflection of shaft Tooth friction

60 Dynamic Factor K v AGMA has defined a set of quality numbers These numbers define tolerances for gears of various sizes manufactured to a specified accuracy Quality numbers 3 to 7 = most commercial quality gears Quality numbers 8 to 12 = precision quality AGMA transmission accuracy level number Q v = quality number

61 Dynamic Factor K v Equations for dynamic factor are based on Q v numbers:

62 Dynamic Factor K v Maximum velocity, representing the end point of the Q v curve, is given by

63 Dynamic Factor K v Figure 14 9: Dynamic factor K v

64 Overload Factor K o Intended to make allowance for all externally applied loads in excess of nominal tangential load W t in a particular application An extensive list of service factors appears in: Howard B. Schwerdlin, Couplings, Chap. 16 Joseph E. Shigley, Charles R. Mischke, and Thomas H. Brown, Jr. (eds.), Standard Handbook of Machine Design, 3rd ed., McGraw-Hill, New York, 2004

65 65 Surface Condition Factor C f (Z R ) C f or Z R = used only in pitting resistance Eq It depends on Surface finish as affected by cutting, shaving, lapping, grinding, shot peening Residual stress Plastic effects (work hardening) AGMA specifies a value of C f greater than unity

66 Size Factor K s Size factor reflects non-uniformity of material properties due to size and it depends upon: Tooth size Diameter of part Ratio of tooth size to diameter of part Face width Area of stress pattern Ratio of case depth to tooth size Hardenability and heat treatment

67 Size Factor K s K s is given by If K s in the preceding Eq is less than 1, use K s = 1.

68 Load-Distribution Factor K m (K H ) Reflects non-uniform distribution of load across the line of contact The ideal is to locate the gear mid-span between two bearings at the zero slope place when the load is applied.

69 Load-Distribution Factor K m (K H ) The following procedure is applicable to: Net face width to pinion pitch diameter ratio F/d 2 Gear elements mounted between the bearings Face widths up to 40 in Contact, when loaded, across the full width of the narrowest member

70 Load-Distribution Factor K m (K H ) The load-distribution factor is given by for values of F/(10d) < 0.05, F/(10d) = 0.05 is used

71 Load-Distribution Factor K m (K H )

72 Load-Distribution Factor K m (K H ) Figure 14 10: Definition of distances S and S 1 used in evaluating C pm, Eq

73 14 11 Load-Distribution Factor K m (K H ) Table 14 9: Empirical Constants

73 Load-Distribution Factor K m (K H ) Table 14 9: Empirical Constants A, B, and C for Eq. (14 34), Face Width F in Inches Source: ANSI/AGMA 2001-D04

74 Load-Distribution Factor K m (K H ) Figure 14 11: Mesh alignment factor C ma. Curve-fit equations in Table 14 9

75 Hardness-Ratio Factor C H Pinion generally has a smaller number of teeth than gear and consequently is subjected to more cycles of contact stress. If both pinion & gear are through-hardened, then a uniform surface strength can be obtained by making pinion harder than gear. A similar effect can be obtained when a surface-hardened pinion is mated with a through hardened gear.

76 Hardness-Ratio Factor C H Hardness-ratio factor C H is used only for the gear Its purpose is to adjust surface strengths for this effect.

77 14 12 Hardness-Ratio Factor C H Figure 14 12

77 Hardness-Ratio Factor C H Figure Hardness-ratio factor CH (throughhardened steel)

78 Hardness-Ratio Factor C H When surface-hardened pinions with Rockwell C48 or harder are run with through-hardened gears ( Brinell), a work hardening occurs. C H factor is a function of pinion surface finish f P and mating gear hardness. Figure displays the relationships:

79 Hardness-Ratio Factor C H Figure Hardness-ratio factor C H (surfacehardened steel pinion)

80 Stress-Cycle Factors Y N & Z N AGMA strengths as given in Figs through 14 4, in Tables 14 3 and 14 4 for bending fatigue, and in Fig and Tables 14 5 and 14 6 for contact-stress fatigue are based on 10 7 load cycles applied. Load cycle factors Y N & Z N modify gear strength for lives other than 10 7 cycles.

81 Stress-Cycle Factors Y N & Z N Values for these factors: Figs & For life goals slightly higher than 10 7 cycles, mating gear may be experiencing fewer than 10 7 cycles and the equations for (Y N ) P and (Y N ) G can be different.

82 14 13 Stress-Cycle Factors Y N & Z N Figure 14 14:

82 Stress-Cycle Factors Y N & Z N Figure 14 14: Repeatedly applied bending strength stress-cycle factor Y N

83 14 13 Stress-Cycle Factors Y N & Z N Figure

83 Stress-Cycle Factors Y N & Z N Figure 14 14: Pitting resistance stress-cycle factor Z N

84 Reliability Factor K R (Y Z ) Reliability factor accounts for the effect of statistical distributions of material fatigue failures. Gear strengths S t & S c are based on a reliability of 99% Table 14 10

85 14 14 Reliability Factor K R (Y Z ) Table 14 10:

85 Reliability Factor K R (Y Z ) Table 14 10: Reliability Factors K R (Y Z ) Source: ANSI/AGMA 2001-D04

86 Temperature Factor K T (Y θ ) For oil or gear-blank temperatures up to 120 C, use K T = Y θ = 1.0. For higher temperatures, the factor should be greater than unity.

87 Rim-Thickness Factor K B When rim thickness is not sufficient to provide full support for the tooth root, the location of bending fatigue failure may be through the gear rim rather than at the tooth fillet. Rim-thickness factor K B, adjusts estimated bending stress for thin-rimmed gear. It is a function of the backup ratio m B

88 Rim-Thickness Factor K B t R = rim thickness below the tooth h t = tooth height Figure 14 16

89 Safety Factors S F & S H S F = safety factor guarding against bending fatigue failure S H = safety factor guarding against pitting failure

90 Safety Factors S F & S H Caution is required when comparing S F with S H in an analysis in order to ascertain nature and severity of threat to loss of function. To render S H linear with the transmitted load, W t it could have been defined as with the exponent 2 for linear or helical contact, or 3 for crowned teeth (spherical contact)

91 14 18 Analysis 91 Figure 14 17: Roadmap of gear bending equations based on AGMA standards. (ANSI/AGMA 2001-D04.)

92 14 18 Analysis 92 Figure 14 17: Roadmap of gear bending equations based on AGMA standards. (ANSI/AGMA 2001-D04.)

93 14 18 Analysis 93 Figure 14 17: Roadmap of gear bending equations based on AGMA standards. (ANSI/AGMA 2001-D04.)

94 14 18 Analysis 94 Figure 14 18: Roadmap of gear wear equations based on AGMA standards. (ANSI/AGMA 2001-D04.)

95 14 18 Analysis 95 Figure 14 18: Roadmap of gear wear equations based on AGMA standards. (ANSI/AGMA 2001-D04.)

96 14 18 Analysis 96 Figure 14 18: Roadmap of gear wear equations based on AGMA standards. (ANSI/AGMA 2001-D04.)

97 Example A 17-tooth 20 pressure angle spur pinion rotates at 1800 rpm & transmits 4 hp to a 52-tooth disk gear. The diametral pitch is 10 teeth/in, the face width 1.5 in, and the quality standard is No. 6. The gears are straddle-mounted with bearings immediately adjacent. The pinion is a grade 1 steel with a hardness of 240 Brinell tooth surface and throughhardened core. The gear is steel, through-hardened also, grade 1 material, with a Brinell hardness of 200, tooth surface and core. Poisson s ratio is 0.30, J P = 0.30, J G = 0.40, and Young s modulus is 30(10 6 ) psi. The loading is smooth because of motor and load. Assume a pinion life of 10 8 cycles and a reliability of 0.90, and use Y N = N , Z N = N The tooth profile is uncrowned. This is a commercial enclosed gear unit. a. Find the factor of safety of the gears in bending. b. Find the factor of safety of the gears in wear. c. By examining the factors of safety, identify the threat to each gear and to the mesh.

98 Example Summary: Np= 17, f = 20,, n p = 1800 rpm, Pin = 4 hp, NG = 52, P = 10 teeth/in, F = 1.5 in, Qv = 6, straddle-mounted gears. pinion material= grade 1 steel, 240 HB tooth surface and throughhardened core gear material = steel, through-hardened, grade 1, 200HB, tooth surface and core v = 0.30, J P = 0.30, J G = 0.40, E = 30 Mpsi, smooth loading. pinion life = 10 8 cycles, R = 0.90, Y N = N , Z N = N 0.023, uncrowned tooth profile, commercial enclosed gear unit

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

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