Chapter 14: General Gear Theory; Spur Gears

Transcription

1 Chapter 14: General Gear Theory; Spur Gears We are what we repeatedly do. Excellence, therefore, is not an act, but a habit. Aristotle An assortment of gears. Source: Courtesy of Quality Transmission Components.

2 Spur Gears Face view Edge view (a) (b) Figure 14.1: Spur gear drive. (a) Schematic illustration of meshing spur gears; (b) a collection of spur gears. Source: Courtesy of Boston Gear Works, Inc.

3 Helical Gears (a) (b) Figure 14.2: Helical gear drive. (a) Schematic illustration of meshing helical gears; (b) a collection of helical gears. Source: Courtesy of Boston Gear Works, Inc.

4 Bevel Gears (a) (b) Figure 14.3: Bevel gear drive. (a) Schematic illustration of meshing bevel gears with straight teeth; (b) a collection of bevel gears. Source: Courtesy of Boston Gear Works, Inc.

5 Worm Gears (a) (b) (c) Figure 14.4: Worm gear drive. (a) Cylindrical teeth; (b) double enveloping; (c) a collection of worm gears. Source: Courtesy of Boston Gear Works, Inc. Fundamentals of Machine Elements, 3rd ed.

6 Spur Gear Line of action Outside diameter, d op r bp r p r op Pinion Geometry Base circle Pitch circle Pressure angle, Working depth, h k Clearance, c r Base diameter, d bg Root diameter Tooth profile Pitch circle Whole depth, h t Dedendum, b Addendum, a Root (tooth) Fillet Top land Center distance, c d Circular tooth thickness Chordal tooth thickness r bg r og r g Circular pitch, p c Pitch diameter, d g Gear Figure 14.5: Basic spur gear geometry.

7 Gear Tooth Nomenclature Addendum Dedendum Clearance Face width Tooth thickness Top land Circular pitch Fillet radius Width of space Dedendum circle Outside circle Pitch circle Clearance circle Face Flank Bottom land Figure 14.6: Nomenclature of gear teeth.

8 Tooth Size Figure 14.7: Standard diametral pitches compared with tooth size. Full size is assumed.

9 Preferred Pitches Diametral pitch, Class p d, in 1 Coarse 1/2, 1, 2, 4, 6, 8, 10 Medium coarse 12, 14, 16, 18 Fine 20, 24, 32, 48, 64 72, 80, 96, 120, 128 e 150, 180, 200 Table 14.1: Preferred diametral pitches for four tooth classes.

10 Power Capacity Power transmitted, kw pitch; d = 8.00 in. (m = 8; d = 200 mm) 6 pitch; d = 4.00 in. (m = 4; d = 100 mm) 12 pitch; d = 2.00 in. (m = 2; d = 50 mm) 24 pitch; d = 1.00 in. (m = 1; d = 25 mm) Pinion speed, rpm Figure 14.8: TransmiWed power as a function of pinion speed for a number of diametral pitches. For all cases shown, g r =4, N p =24, K o =1.0, ϕ=20. Source: From MoW [2003].

11 Pitch and Base Circles; Line of Action Base circle Pitch circle Pitch point, p p Pitch circle Pinion r p r bp A p g Line of action Base circle (pinion) p Pitch point A L 1 Outside circle (pinion) Outside circle (gear) B Base circle r bg r g Gear B L 2 Base circle (gear) Line of centers L ab (Length of action) g (a) (b) Figure 14.9: (a) Pitch and base circles for pinion and gear as well as line of action and pressure angle. (b) Detail of active profile, showing detail of line of action and length of action, L ab.

12 Addendum, Dedendum & Clearance Metric Coarse pitch Fine pitch module Parameter Symbol (p d < 20 in. 1 ) ( p d 20 in. 1 ) system Addendum a 1/p d 1/p d 1.00 m Dedendum b 1.25/p d 1.200/p d m Clearance c 0.25/p d 0.200/p d m Table 14.2: Formulas for addendum, dedendum and clearance (ϕ=20 ; full- depth involute).

13 Involute Curve Base circle Involute A 4 A 3 A 2 C 4 C 3 C 2 A 1 C 1 B 3 B 4 A o B 1 B 2 0 Figure 14.10: Construction of involute curve.

14 Arc of Approach Arc of approach a Arc of recess r Line of action a P Outside circle Outside circle Motion b Pitch circle L ab Figure 14.11: Illustration of arc of approach, arc of recess and length of action.

15 Backlash Pitch circle Base Backlash 0 Base Pitch Backlash Pressure line Diametral pitch Center distance, c d, in. p d, in Figure 14.12: Illustration of backlash in gears. Table 14.3: Recommended minimum backlash for coarse- pitched gears.

16 Meshing Gears Gear 1 (N 1 teeth) Gear 2 (N 2 teeth) 1 2 r 2 Gear 1 (N 1 ) Gear 2 (N 2 ) r 1 r 2 r 1 1 (+) ( ) 2 Figure 14.13: Externally meshing spur gears. Figure 14.14: Internally meshing spur gears.

17 Gear Trains N 2 Figure 14.15: Simple gear train. N 1 Shaft 2 Shaft 3 N 1 N 2 N 5 N 6 Shaft 4 Shaft 1 N 3 N4 N 7 N 8 Shaft 5 Figure 14.16: Compound gear train.

18 Example 14.6 Input Shaft 1 N A = 20 A B N B = 70 A B Only pitch circles of gears shown Shaft 2 N C = 18 C C N D = 22 D D E Shaft 3 N E = 54 E Shaft 4 Output Figure 14.17: Gear train used in Example 14.6.

19 Planetary Gear Train R P Ring Planet Arm Sun R P A S S P P (a) (b) Figure 14.18: Illustration of planetary gear train. (a) With three planets; (b) with one planet (for analysis only).

20 Gear Cost vs. Quality Dimensional tolerance (in.) Special methods Relative cost 10 Shaving Production grinding Shaping & Hobbing Powder Metal Figure 14.19: Gear cost as a function of gear quality. Note that the powder metallurgy approaches of pressing and sintering and metal injection molding can produce gears up to a quality index of 8 without additional machining. Recent research has suggested that similar quality levels can be achieved from cold forging as well AGMA quality index 6 4

21 Quality Index Application Quality index, Q v Cement mixer drum driver 3-5 Cement kiln 5-6 Steel mill drives 5-6 Corn pickers 5-7 Punch press Mining conveyor Clothes washing machine 8-10 Printing press 9-11 Automotive transmission Marine propulsion drive Aircraft engine drive Gyroscope Pitch velocity Quality index, Q v ft/min m/s > 4000 > Table 14.4: Quality index Q v for various applications. Source: Courtesy of the American Gear Manufacturers Association.

22 Form CuWing of Gears Form cutter Gear blank (a) (b) (c) Figure 14.20: Form cuwing of gear teeth. (a) A form cuwer. Notice that the tooth profile is defined by the cuwer profile; (b) schematic illustration of the form cuwing process; (c) form cuwing of teeth on a bevel gear. Source: (a) and (b) From Kalpakjian and Schmid [2010]; (c) Courtesy Schafer Gear Works, Inc. Fundamentals of Machine Elements, 3rd ed.

23 Shaping Figure 14.21: Production of gear teeth with a pinion- shaped cuwer. (a) Schematic illustration of the process. Source: From Kalpakjian and Schmid [2010]; (b) photograph of the process with gear and cuwer motions indicated. Source: Courtesy Schafer Gear Works, Inc. Fundamentals of Machine Elements, 3rd ed.

24 Hobbing Top view Gear blank (b) Helical gear Hob Hob rotation Hob Gear blank (b) (a) Figure 14.22: Production of gears through the hobbing process. (a) A hob, along with a schematic illustration of the process. Source: From Kalpakjian and Schmid [2010]; (b) production of a worm gear through hobbing. Source: Courtesy Schafer Gear Works, Inc. Fundamentals of Machine Elements, 3rd ed.

25 Gear Gear Grinding wheel Finishing Worm gear Single-ribbed grinding wheel (a) Multiribbed grinding wheel Two grinding wheels Grinding wheels Gear 15 or 20 position 0 position Figure 14.23: Finishing gears by grinding: (a) form grinding with shaped grinding wheels; (b) grinding by generating, using two wheels. Source: From Kalpakjian and Schmid [2010]. (b)

26 Hardness Effects on Bending Strength Bending strength, S b, MPa Through-hardened Through-hardened Grade 2 Nitrided Nitrided Grade Brinell hardness, HB ksi Bending strength, S b, MPa Grade 2 - Nitralloy Grade 1 - Nitralloy Grade 3-2.5% Chrome Grade 2-2.5% Chrome Grade 1-2.5% Chrome Brinell hardness, HB ksi Figure 14.24: Effect of Brinell hardness on allowable bending stress for steel gears. (a) Through- hardened steels; (b) Flame or induction hardened nitriding steels. Note that Brinell hardness refers to case hardness for these gears. Source: ANSI/AGMA Standard D04 [2004].

27 Hardness Effects on Contact Strength Contact strength, S c, MPa Grade 1: S 2.41 HB +237 (MPa) c = { HB (ksi) Grade 2: S 2.22 HB +200 (MPa) c = { HB (ksi) Grade 2 Grade ksi Brinell hardness, HB Figure 14.25: Effect of Brinell hardness on allowable contact stress number for two grades of through- hardened steel. Source: ANSI/AGMA Standard D04 [2004]

28 Strength of Gear Materials Material designation Grade Typical Bending strength, S b Contact strength, S c Hardness a lb/in. 2 MPa ksi MPa Steel Through-hardened b HB HB HB HB HB HB HB HB HB HB Carburized & hardened HRC HRC 65.0 c 450 c HRC Nitrided and through HR15N HB HB , hardened b HB HB , Nitralloy 135M and HR15N HB HB , Nitralloy N, nitrided b HR15N HB HB , % Chrome, nitrided b HR15N HB HB , HR15N HB HB , HR15N HB HB , Cast Iron ASTM A48 gray cast Class iron, as-cast Class HB Class HB ASTM A536 ductile HB (nodular) iron HB HB HB Bronze S ut > 40, 000 psi (S ut > 275 GPa) S ut > 90, 000 psi (S ut > 620 GPa) Powder Metal FL-4405, = 7.30 g/cm 3 80 HRB FLN2-4405, = 7.35 g/cm 3 90 HRB FLC-4608, = 7.30 g/cm 3 65 HRB FN-0205, = 7.10 g/cm 3 69 HRB a Hardness refers to case hardness unless through-hardened. b See Figs and/or c 70,000 psi (485 MPa) may be used if bainite and microcracks are limited to grade 3levels. Table 14.5: Bending and contact strength for selected gear materials. Source: Adapted from ANSI/AGMA D04 [2004] and MPIF Standard 35 [2009].

29 Gear Strength Modification Factors and Reliability Factor Allowable bending stress: σ b,all = S b n s Allowable contact stress: Y N K t K r σ c,all = S c n s Z N C H K t K r Probability of survival, percent Reliability factor a K r 0.70 b 0.85 b a Based on surface pitting. If tooth breakage is considered a greater hazard, a larger value may be required. b At this value, plastic flow may occur rather than pitting. Table 14.6: Reliability factor, K r. Source: From ANSI/AGMA D04 [2004].

30 Bending Stress Cycle Factor HB: Y N = N Case Carb.: Y N = N Stress cycle factor, Y n HB: Y N = N Nitrided: Y N = N HB: Y N = N Y N = N Y N = N Number of load cycles, N Figure 14.26: Stress cycle factor. (a) Bending strength stress cycle factor Y N ; Source: ANSI/AGMA Standard D04 [2004].

31 PiWing Resistance Cycle Factor 2.0 Grade 1 & 2 Nitrided 1.5 Z n = 1.47, N < 10 4 Hydrodynamic: Z n = N Stress cycle factor, Z n Z n = 1.10, N < 10 4 Z n = N Z n = N Mixed: Z n = N , N > Boundary: Z n = N , N > Number of load cycles, N Figure 14.26: Stress cycle factor. (b) PiWing resistance stress cycle factor Z N. Source: ANSI/AGMA Standard D04 [2004].

32 Loads on Gear Tooth W r W W t Pitch circle Tangential load: W t = h p u = 60h p πdn a If h p is in horsepower: W t = 126, 050h p dn a Figure 14.27: Loads acting on an individual gear tooth. Normal load: W = Radial load: W t cos φ W r = W t tan φ

33 Bending of Gear W r W W t Teeth l r f x t (a) W t b w t l Figure 14.28: A crack that has developed at the root of a gear tooth due to excessive bending stresses. Source: Courtesy of the American Gear Manufacturers Association. (b) Figure 14.29: Loads and length dimensions used in determining tooth bending stress. (a) Tooth; (b) cantilevered beam.

34 Lewis Form Factor Lewis Lewis Number form Number form of teeth factor of teeth factor Table 14.7: Lewis form factor for various numbers of teeth (pressure angle, 14.5 ; full- depth involute).

35 Spur Gear Geometry Factors Geometry factor, Y j Number of teeth in mating gear. Load considered applied at highest point of single-tooth contact Load applied at tip of tooth Geometry factor, Y j Number of teeth in 1000 mating gear. 85 Load considered 50 applied at 25 highest point 17 of single-tooth contact Load applied at tip of tooth Number of teeth, N (a) Number of teeth, N (b) Figure 14.30: Spur gear geometry factors for full- depth involute profile. (a) ϕ=20 ; (b) ϕ=25. Source: AGMA Standard 908- B89 [1989].

36 AGMA Bending Stress Equation σ b = W tp d K o K s K m K v K b b w Y j where W t = transmiwed load, N p d = diametral pitch, m - 1 K o = overload factor K s = size factor K m = load distribution factor K v = dynamic factor K b = rim thickness factor

37 Overload and Size Factors Driven Machines Light Moderate Heavy Power source Uniform shock shock shock Uniform Light shock Moderate shock Figure 14.8: Overload factor, K o, as function of driving power source and driven machine. Table 14.9: Recommended values of size factor, K s. Diametral pitch, p d, Module, m, in. 1 mm Size factor, K s

38 Load Distribution Factor Load distribition factor: K m =1.0+C mc (C pf C pm + C ma C e ) where C mc = lead correction factor C pf = pinion proportion factor C pm = pinion proportion modifier C ma = mesh alignment factor C e = mesh alignment correction factor Lead correction factor: 1.0 for uncrowned teeth C mc = 0.8 for crowned teeth

39 Pinion Proportion Factor Pinion proportion factor, C pf b w /d p = 2.00 Face width, b w (in.) For b w /d p < 0.5 use curve for b w /d p = Face width, b w (mm) Figure 14.31: Pinion proportion factor, C pf. Source: ANSI/AGMA Standard D04 [2004]. 40 Pinion proportion factor: If b w < 25 mm, b w C pf = d p For 25 mm < bw < 432 mm, b w C pf = b w 10d p For 432 mm < bw 1020 mm, C pf = b w 10d p b w ( )b 2 w (see text if b w is in inches)

40 Mesh Alignment Factor Face width, b w (in) Mesh alignment factor, C ma Open gearing Commercial enclosed gear units Precision enclosed gear units Extra-precision enclosed gear units Face width, b w (mm) C ma = A + Bb w + Cb 2 w If b w is in inches: Condition A B C Open gearing Commercial enclosed gears Precision enclosed gears Extraprecision enclosed gears If b w is in mm: Condition A B C Open gearing Commercial enclosed gears Precision enclosed gears Extraprecision enclosed gears Figure 14.33: Mesh alignment factor. Source: ANSI/AGMA Standard D04 [2004].

41 Pinion Proportion and Mesh Alignment Modifiers Pinion proportion modifier: C pm = 1.0, (S1 /S) < , (S 1 /S) S 1 S S/2 Mesh alignment correction factor: C e = 0.80 when gearing is adjusted at assembly 0.80 when compatibility between gear teeth is improved by lapping 1.0 for all other conditions Figure 14.32: Evaluation of S and S 1. Source: ANSI/AGMA Standard D04 [2004].

42 Dynamic Factor Dynamic factor, K v Pitch line velocity, ft/min Q v = 5 Q v = Q v = 7 Q v = Pitch line velocity, m/s 7500 Q v = 9 "Very accurate" gearing Q v = 10 Figure 14.34: Dynamic factor as a function of pitch- line velocity and transmission accuracy level number. Source: ANSI/AGMA Standard D04 [2004]. Q v = 11 Dynamic factor: A + C vt K v = A Where B A = (1.0 B) B =0.25(12 Q v ) C = 1 for v t in ft/min C = 200 = for v t in m/s Maximum recommended pitch line velocity: v t,max = 1 C 2 [A +(Q v 3)] 2

43 Design Procedure 14.2: Methods to Increase Bending Performance of Gears If a gear does not produce a satisfactory design based on bending requirements, a design alteration may be needed. This is not always straightforward, since such alterations may help in one area and hurt in another, and may affect associated machine elements such as bearings. However, some factors that improve bending performance are the following: 1. Reduction in the load, such as by increasing contact ratio, or altering other aspects of the system. 2. Increase the center distance. 3. Apply gears with a coarser pitch. 4. Use a higher pressure angle. 5. Use a helical gear instead of a spur gear. 6. Use a carburized material. 7. Improve the gear accuracy through manufacturing process selection. 8. Select a bewer (stronger) gear material. 9. Use a wider effective face width. 10. Apply shot peening to the teeth.

44 PiWing Figure 14.35: A gear showing extreme piwing or spalling. Source: Courtesy of the American Gear Manufacturing Association.

45 AGMA Contact Stress Equation σ c = E W K o K s K m K v 2π Maximum Hern pressure: Where p H = E W 1 2 = ph (K o K s K m K v ) 1 2 8W b = R x π Ko, Ks, Km, and Kv are defined as with the AGMA Bending Stress Equation. 2π 1 2 E 2 = effective elastic modulus = 1 νa ν2 b E a E b W = dimensionless load = w W E = R x E R x b w 1 1 = R x r p r g sin φ = d p d g sin φ 1/2

46 Design Procedure 14.3: Methods to Increase PiWing Resistance If a gear does not produce a satisfactory design based on surface piwing requirements, a design alteration may be needed. This is not always straightforward, since such alterations may help in one area and hurt in another, and may affect associated machine elements such as bearings. However, some factors that improve piwing performance are the following: 1. Reduction in the load, such as by increasing contact ratio, or altering other aspects of the system. 2. Increase the center distance. 3. Apply gears with a finer pitch. 4. Use a higher pressure angle. 5. Use a helical gear instead of a spur gear. 6. Use a carburized material. 7. Increase the surface hardness by material selection, or by performing a surface hardening operation. 8. Improve the gear accuracy through manufacturing process selection. 9. Use a wider effective face width. 10. Increase the lubricant film thickness.

47 Design Procedure 14.4: Lubricant Film Thickness It will be assumed that the power transmiwed, gear and pinion angular velocity, number of teeth in pinion and gear, gear materials, lubricant properties, and geometry of the pinion and gear are known or can be determined from design constraints. Of these, the lubricant properties, especially the pressure exponent of viscosity, are most likely to be unknown or will have the largest uncertainty. Regardless, these properties should be awainable from the lubricant supplier. The normal force acting on the gear teeth can be obtained from Eq. (14.50) as: W = cos φ The lubrication velocity for the pinion and gears is obtained from Eqs. (14.29) and (14.30) as ũ = u p + u g = ω pr p sin φ + ω g r g sin φ 2 2 The effective radius is obtained from Eq. (14.74) as 1 R x = 1 W t r p + 1 r g 1 sin φ

48 Design Procedure 14.4 (concluded) 5. The effective modulus of elasticity is obtained from Eq. (8.16): 6. For steel- on- steel, E = 227.5$ GPa (32.97 Mpsi). 7. Equations (13.68), (13.69), and (13.72) yield the dimensionless load, speed, and materials parameters as: W = E = 2 (1 νa) 2 + (1 ν2 b ) E a E b W b w E ; U = η oũ R z E ; R x G = ξe 8. The lubricant film thickness is then obtained from the Hamrock- Dowson equation for rectangular contacts given by Eq.~(13.71): H min = h min R x =1.714(W ) U G 0.568

49 Design Procedure 14.5: Gear Design Synthesis 1. From design requirements, determine the power transfer, total life required, and pinion and gear speed. 2. The gear ratio, g r, must be known from design requirements. If this value is near or in excess of 6:1, a second stage is advisable. 3. If the pinion face- to- diameter ratio is not a design requirement, it can be estimated from the following equation for spur gears: b w d p = Where g r = N g /N p is the gear ratio. 4. Estimate K o from Table g r g r +1,

50 Design Procedure 14.5 (continued) 5. The load distribution factor, K m, cannot be determined until knowledge of the design, manufacturing approach, and mounting is established. A rough approximation based on pinion torque can be made according to K m =1+ b 0.33 w Tp K o d p b w /d p However, if the pitch diameter is known (from design requirements or selected based on experience), then the load distribution factor can be more accurately estimated from K m =1+ b w ( d p ) d p Recall that the torque is related to the power by h p = Tω. 6. The dynamic factor, K v, depends on gear speed and quality. For simplicity, a value of K v = 1.43 can be assumed, which is conservative for most applications.

51 Design Procedure 14.5 (continued) 7. The pressure angle, φ, is generally chosen as 20, since these gears are widely available. However, the pressure angle can be reduced to obtain higher contact ratios, or increased when precision and noise are not issues. Regardless, it is recommended that standard values of 14.5, 17.5, 22.5, or 25 be used. 8. A geometry factor for spur gears is given by: I = sin φ cos φ 2 g r g r ± 1 Where g r is the gear ratio. Note that the plus sign in the denominator of Eq.~ (14.79) applies for external gearsets, the negative sign for internal gearsets. 9. Y j can be conservatively estimated as approximately 0.45 (see Fig ). 10. The life factors Y n and Z n can be estimated from Fig Obtain the bending (σ b ) and contact (σ c ) strengths for the gear material according to Eqs.~(14.45) or (14.46), respectively.

52 Design Procedure 14.5 (continued) 12. An estimate for the preferred number of pinion teeth is then: N p = 1 K b Y j I σ b σc 2 For the special case of steel gears (E = 227 GPa) a pressure angle of φ=20, K b =1, n sc =n sb =n s, and Y j =0.45, the preferred number of pinion teeth becomes N p = ( GPa) g r +1σ b 13. Obtain the piwing resistance constant: E 2π n 2 sc n sb 14. The pinion pitch diameter is then estimated as 1/3 Kc d p = b w /d p g r K c = E h p K o K m K v Iω p σ 2 c n s 2 nsc σ c

53 Design Procedure 14.5 (concluded) 15. The face width can be calculated from the ratio of b w /d p established in step 3 if a specific value was not specified as a design requirement. 16. When the gear profile is selected from this design procedure, it is necessary to analyze the design more closely in accordance with the approaches described in Sections and This is clearly required since a number of approximate values were used. 17. It should also be noted that an essential part of the gear design synthesis is calculating lubricant film thickness as described in Section

54 Upper punch PM Spur Gear Powder Feed Shoe Die Compacted shape (green) Manufacture Lower punch Ejector (a) Upper punch Die PM spur gear (green) Lower punch Core rod (b) Figure 14.36: Production of gears through the powder compaction process. (a) Steps required to produce a part; (b) illustration of tooling required for a simple spur gear. Source: After Kalpakjian and Schmid [2010].

55 Surface Densified PM Gears Subsurface porosity 100 µm Densified surface Figure 14.37: Image of a PM material after roll densification. Note the low porosity near the surface, resulting in high allowable contact stresses. Source: Courtesy Capstan Atlantic Corp. Figure 14.38: A stepped gear produced through the metal injection molding process. Source: Courtesy Perry Tool & Research, Inc. Fundamentals of Machine Elements, 3rd ed. Figure 14.39: A collection of PM gears used in automotive applications. Source: Courtesy Capstan Atlantic Corp.