Budynas-Nisbett Shigleys I Design of Mechanical 14.Spur and Helical Gears T©The McGraw-Hil m Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 4 Spur and Helical Gears Chapter Outline 14-1 The Lewis Bending Equation 714 14-2 Surface Durability 723 14-3 AGMA Stress Equations 725 14-4 AGMA Strength Equations 727 14-5 Geometry Factors I and J(Z and Y1 731 14-6 The Elastic Coefficient Cp (Z 736 14-7 Dynamic Factor K,736 14-8 Overload Factor K。738 14-9 Surface Condition Factor Cr (Zel 738 14-10 Size Factor K 739 14-11 Load-Distribution Factor Km(KH 739 14-12 Hardness-Ratio Factor CH 741 14-13 Stress Cycle Life Factors YN and ZN 742 14-14 Reliability Factor Ke(Y 743 14-15 Temperature Factor KT (Ya 744 14-16 Rim-Thickness Factor K8 744 14-17 Safety Factors Se and SH 745 14-18 Analysis 745 14-19 Design of a Gear Mesh 755 713
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears © The McGraw−Hill 711 Companies, 2008 Spur and Helical Gears Chapter Outline 14–1 The Lewis Bending Equation 714 14–2 Surface Durability 723 14–3 AGMA Stress Equations 725 14–4 AGMA Strength Equations 727 14–5 Geometry Factors I and J (ZI and YJ) 731 14–6 The Elastic Coefficient Cp (ZE) 736 14–7 Dynamic Factor Kv 736 14–8 Overload Factor Ko 738 14–9 Surface Condition Factor Cf (ZR) 738 14–10 Size Factor Ks 739 14–11 Load-Distribution Factor Km (KH) 739 14–12 Hardness-Ratio Factor CH 741 14–13 Stress Cycle Life Factors YN and ZN 742 14–14 Reliability Factor KR (YZ) 743 14–15 Temperature Factor KT (Yθ) 744 14–16 Rim-Thickness Factor KB 744 14–17 Safety Factors SF and SH 745 14–18 Analysis 745 14–19 Design of a Gear Mesh 755 14 713
12 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 14.Spur and Helical Gears T©The McGraw-Hill Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 714 Mechanical Engineering Design This chapter is devoted primarily to analysis and design of spur and helical gears to resist bending failure of the teeth as well as pitting failure of tooth surfaces.Failure by bend- ing will occur when the significant tooth stress equals or exceeds either the yield strength or the bending endurance strength.A surface failure occurs when the significant contact stress equals or exceeds the surface endurance strength.The first two sections present a little of the history of the analyses from which current methodology developed. The American Gear Manufacturers Association'(AGMA)has for many years been the responsible authority for the dissemination of knowledge pertaining to the design and analysis of gearing.The methods this organization presents are in general use in the United States when strength and wear are primary considerations.In view of this fact it is important that the AGMA approach to the subject be presented here. The general AGMA approach requires a great many charts and graphs-too many for a single chapter in this book.We have omitted many of these here by choosing a single pressure angle and by using only full-depth teeth.This simplification reduces the complexity but does not prevent the development of a basic understanding of the approach.Furthermore,the simplification makes possible a better development of the fundamentals and hence should constitute an ideal introduction to the use of the general AGMA method.2 Sections 14-1 and 14-2 are elementary and serve as an examination of the foundations of the AGMA method.Table 14-1 is largely AGMA nomenclature. 14-1 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 formulation.The equation,announced in 1892, still remains the basis for most gear design today To derive the basic Lewis equation,refer to Fig.14-1a,which shows a cantilever of cross-sectional dimensions F and t,having a length and a load W,uniformly dis- tributed across the face width F.The section modulus //c is Ft-/6,and therefore the bending stress is M 6W'I (a) Gear designers denote the components of gear-tooth forces as W,Wr,Wa or W,W, Wa interchangeably.The latter notation leaves room for post-subscripts essential to free- body diagrams.For instance,for gears 2 and 3 in mesh,W is the transmitted force of 500 Montgomery Street.Suite 350,Alexandria.VA 22314-1560. 2The standards ANSI/AGMA 2001-D04 (revised AGMA 2001-C95)and ANSI/AGMA 2101-D04 (metric edition of ANSUAGMA 2001-D04).Fundamental Rating Factors and Calculation Methods for Imvolute Spur and Helical Gear Teeth,are used in this chapter.The use of American National Standards is completely voluntary:their existence does not in any respect preclude people,whether they have approved the standards or not,from manufacturing,marketing.purchasing,or using products.processes.or procedures not conforming to the standards. The American National Standards Institute does not develop standards and will in no circumstances give an interpretation of any American National Standard.Requests for interpretation of these standards should be addressed to the American Gear Manufacturers Association.[Tables or other self-supporting sections may be quoted or extracted in their entirety.Credit line should read:"Extracted from ANSI/AGMA Standard 2001-D04 or 2101-D04 Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth"with the permission of the publisher.American Gear Manufacturers Association, 500 Montgomery Street.Suite 350.Alexandria,Virginia 22314-1560.]The foregoing is adapted in part from the ANSI foreword to these standards
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears 712 © The McGraw−Hill Companies, 2008 714 Mechanical Engineering Design 1 500 Montgomery Street, Suite 350, Alexandria, VA 22314-1560. 2 The standards ANSI/AGMA 2001-D04 (revised AGMA 2001-C95) and ANSI/AGMA 2101-D04 (metric edition of ANSI/AGMA 2001-D04), Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth, are used in this chapter. The use of American National Standards is completely voluntary; their existence does not in any respect preclude people, whether they have approved the standards or not, from manufacturing, marketing, purchasing, or using products, processes, or procedures not conforming to the standards. The American National Standards Institute does not develop standards and will in no circumstances give an interpretation of any American National Standard. Requests for interpretation of these standards should be addressed to the American Gear Manufacturers Association. [Tables or other self-supporting sections may be quoted or extracted in their entirety. Credit line should read: “Extracted from ANSI/AGMA Standard 2001-D04 or 2101-D04 Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth” with the permission of the publisher, American Gear Manufacturers Association, 500 Montgomery Street, Suite 350, Alexandria, Virginia 22314-1560.] The foregoing is adapted in part from the ANSI foreword to these standards. This chapter is devoted primarily to analysis and design of spur and helical gears to resist bending failure of the teeth as well as pitting failure of tooth surfaces. Failure by bending will occur when the significant tooth stress equals or exceeds either the yield strength or the bending endurance strength. A surface failure occurs when the significant contact stress equals or exceeds the surface endurance strength. The first two sections present a little of the history of the analyses from which current methodology developed. The American Gear Manufacturers Association1 (AGMA) has for many years been the responsible authority for the dissemination of knowledge pertaining to the design and analysis of gearing. The methods this organization presents are in general use in the United States when strength and wear are primary considerations. In view of this fact it is important that the AGMA approach to the subject be presented here. The general AGMA approach requires a great many charts and graphs—too many for a single chapter in this book. We have omitted many of these here by choosing a single pressure angle and by using only full-depth teeth. This simplification reduces the complexity but does not prevent the development of a basic understanding of the approach. Furthermore, the simplification makes possible a better development of the fundamentals and hence should constitute an ideal introduction to the use of the general AGMA method.2 Sections 14–1 and 14–2 are elementary and serve as an examination of the foundations of the AGMA method. Table 14–1 is largely AGMA nomenclature. 14–1 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 formulation. The equation, announced in 1892, still remains the basis for most gear design today. To derive the basic Lewis equation, refer to Fig. 14–1a, which shows a cantilever of cross-sectional dimensions F and t, having a length l and a load Wt , uniformly distributed across the face width F. The section modulus I/c is Ft 2/6, and therefore the bending stress is σ = M I/c = 6Wt l Ft 2 (a) Gear designers denote the components of gear-tooth forces as Wt , Wr, Wa or Wt , Wr, Wa interchangeably. The latter notation leaves room for post-subscripts essential to freebody diagrams. For instance, for gears 2 and 3 in mesh, Wt 23 is the transmitted force of
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 14.Spur and Helical Gears ©The McGraw-Hil 13 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Spur and Helical Gears 715 Table 14-1 Symbol Name Where Found Symbols,Their Names, b Net width of face of narrowest member Eq.(14-161 and Locations* Ce Mesh alignment correction factor Eq.(14-351 C Surface condition factor Eq.14-16) CH Hardness-ratio factor Eq.(14-18) Cma Mesh alignment factor Eq.14-34 Cme Load correction factor Eq.(14-31) Cmf Face load-distribution factor Eq.(14-30) Cp Elastic coefficient Eq.14-13 Cpf Pinion proportion factor Eq.(14-32 Cpm Pinion proportion modifier Eq.(14-33) d Operating pitch diameter of pinion Ex.(14-1) dp Pitch diameter,pinion Eq.(14-22) Pitch diameter,gear Eq.(14-22) E Modulus of elasticity Eq.14-10 F Net face width of narrowest member Eq.I14-15 Pinion surface finish Fig.14-13 H Power Fig.14-17 Ha Brinell hardness Ex.14-3 HBG Brinell hardness of gear Sec.14-12 Hap Brinell hardness of pinion Sec.14-12 hp Horsepower Ex.14-] hr Gear-tooth whole depth Sec.14-16 Geometry factor of pitting resistance Eq.(14-16) Geometry factor for bending strength Eq.14-15) K Contact load factor for pitting resistance Eq.6-651 K Rim-thickness factor Eq.(14-401 K Fatigue stress-concentration factor Eq.(14-91 Load-distribution factor Eq.(14-301 to Overload factor Eq.14-15) Reliability factor Eq.(14-17) K Size factor Sec.14-10 K Temperature factor Eq.(14-17刀 K Dynamic factor Eq.(14-27 m Metric module Eq.(14-15) mg Backup ratio Eq.(14-391 mG Gear ratio (never less than 1) Eq.14-22) mN Load-sharing ratio Eq.14-21 N Number of stress cycles Fig.14-14 NG Number of teeth on gear Eq.(14-22) Np Number of teeth on pinion Eq.I14-22) n Speed E×.14-1 [Continued)
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears © The McGraw−Hill 713 Companies, 2008 Spur and Helical Gears 715 Symbol Name Where Found b Net width of face of narrowest member Eq. (14–16) Ce Mesh alignment correction factor Eq. (14–35) Cf Surface condition factor Eq. (14–16) CH Hardness-ratio factor Eq. (14–18) Cma Mesh alignment factor Eq. (14–34) Cmc Load correction factor Eq. (14–31) Cmf Face load-distribution factor Eq. (14–30) Cp Elastic coefficient Eq. (14–13) Cpf Pinion proportion factor Eq. (14–32) Cpm Pinion proportion modifier Eq. (14–33) d Operating pitch diameter of pinion Ex. (14–1) dP Pitch diameter, pinion Eq. (14–22) dG Pitch diameter, gear Eq. (14–22) E Modulus of elasticity Eq. (14–10) F Net face width of narrowest member Eq. (14–15) fP Pinion surface finish Fig. 14–13 H Power Fig. 14–17 HB Brinell hardness Ex. 14–3 HBG Brinell hardness of gear Sec. 14–12 HBP Brinell hardness of pinion Sec. 14–12 hp Horsepower Ex. 14–1 ht Gear-tooth whole depth Sec. 14–16 I Geometry factor of pitting resistance Eq. (14–16) J Geometry factor for bending strength Eq. (14–15) K Contact load factor for pitting resistance Eq. (6–65) KB Rim-thickness factor Eq. (14–40) Kf Fatigue stress-concentration factor Eq. (14–9) Km Load-distribution factor Eq. (14–30) Ko Overload factor Eq. (14–15) KR Reliability factor Eq. (14–17) Ks Size factor Sec. 14–10 KT Temperature factor Eq. (14–17) Kv Dynamic factor Eq. (14–27) m Metric module Eq. (14–15) mB Backup ratio Eq. (14–39) mG Gear ratio (never less than 1) Eq. (14–22) mN Load-sharing ratio Eq. (14–21) N Number of stress cycles Fig. 14–14 NG Number of teeth on gear Eq. (14–22) NP Number of teeth on pinion Eq. (14–22) n Speed Ex. 14–1 Table 14–1 Symbols, Their Names, and Locations∗ (Continued)
14 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 14.Spur and Helical Gears ©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 716 Mechanical Engineering Design Table 14-1 Symbol Name Where Found Symbols,Their Names, np Pinion speed Ex.14-4 and Locations' P Diametral pitch Eq.14-21 [Continued) Pd Diametral pitch of pinion Eq.(14-151 PN Normal base pitch Eq.(14-24) Pn Normal circular pitch Eq.(14-241 Px Axial pitch Eq.(14-191 Q, Transmission accuracy level number Eq.(14-291 R Reliability Eq.(14-381 R Root-mean-squared roughness Fig.14-13 Tooth fillet radius Fig.14-1 rG Pitch-circle radius,gear In standard p Pitch-circle radius,pinion In standard Tbp Pinion base-circle radius Eq.(14-251 fbG Gear base-circle radius Eq.(14-251 Sc Buckingham surface endurance strength E×.14-3 Se AGMA surface endurance strength Eq.(14-181 S AGMA bending strength Eq.14-17 Bearing span Fig.14-10 S1 Pinion offset from center span Fig.14-10 Se Safety factor-bending Eq.(14-41) SH Safely factor-pitting Eq.(14-42) Wor W! Transmitted load Fig.14-1 YN Stress cycle factor for bending strength Fig.14-14 ZN Stress cycle factor for pitting resistance Fig.14-15 B Exponent Eq.(14-44 0 Bending stress Eq.(14-2) oc Contact stress from Hertzian relationships Eq.(14-14) e Contact stress from AGMA relationships Eq.(14-161 Call Allowable bending stress Eq.14-17 Oc,all Allowable contact stress,AGMA Eq.(14-181 必 Pressure angle Eq.(14-12) Transverse pressure angle Eq.(14-23) Helix angle at standard pitch diameter Ex.14-5 Because ANSI/AGMA 2001-95inuedsignificntmo ofnew nomendatre,nd cnind in ANSI/AGMA 21-04 this summary and references are provided for use until the reader's vocabulary has grown. See preferenie following E.()Se.14-1. body 2 on body 3,and W is the transmitted force of body 3 on body 2.When working with double-or triple-reduction speed reducers,this notation is compact and essential to clear thinking.Since gear-force components rarely take exponents,this causes no com- plication.Pythagorean combinations,if necessary,can be treated with parentheses or avoided by expressing the relations trigonometrically
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears 714 © The McGraw−Hill Companies, 2008 716 Mechanical Engineering Design Symbol Name Where Found nP Pinion speed Ex. 14–4 P Diametral pitch Eq. (14–2) Pd Diametral pitch of pinion Eq. (14–15) pN Normal base pitch Eq. (14–24) pn Normal circular pitch Eq. (14–24) px Axial pitch Eq. (14–19) Qv Transmission accuracy level number Eq. (14–29) R Reliability Eq. (14–38) Ra Root-mean-squared roughness Fig. 14–13 rf Tooth fillet radius Fig. 14–1 rG Pitch-circle radius, gear In standard rP Pitch-circle radius, pinion In standard rbP Pinion base-circle radius Eq. (14–25) rbG Gear base-circle radius Eq. (14–25) SC Buckingham surface endurance strength Ex. 14–3 Sc AGMA surface endurance strength Eq. (14–18) St AGMA bending strength Eq. (14–17) S Bearing span Fig. 14–10 S1 Pinion offset from center span Fig. 14–10 SF Safety factor—bending Eq. (14–41) SH Safety factor—pitting Eq. (14–42) Wt or W† t Transmitted load Fig. 14–1 YN Stress cycle factor for bending strength Fig. 14–14 ZN Stress cycle factor for pitting resistance Fig. 14–15 β Exponent Eq. (14–44) σ Bending stress Eq. (14–2) σC Contact stress from Hertzian relationships Eq. (14–14) σc Contact stress from AGMA relationships Eq. (14–16) σall Allowable bending stress Eq. (14–17) σc,all Allowable contact stress, AGMA Eq. (14–18) φ Pressure angle Eq. (14–12) φt Transverse pressure angle Eq. (14–23) ψ Helix angle at standard pitch diameter Ex. 14–5 ∗Because ANSI/AGMA 2001-C95 introduced a significant amount of new nomenclature, and continued in ANSI/AGMA 2001-D04, this summary and references are provided for use until the reader’s vocabulary has grown. † See preference rationale following Eq. (a), Sec. 14–1. Table 14–1 Symbols, Their Names, and Locations∗ (Continued) body 2 on body 3, and Wt 32 is the transmitted force of body 3 on body 2. When working with double- or triple-reduction speed reducers, this notation is compact and essential to clear thinking. Since gear-force components rarely take exponents, this causes no complication. Pythagorean combinations, if necessary, can be treated with parentheses or avoided by expressing the relations trigonometrically
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 14.Spur and Helical Gears T©The McGraw-Hil 715 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Spur and Helical Gears 717 I Figure 14-1 (a) Referring now to Fig.14-16,we assume that the maximum stress in a gear tooth occurs at point a.By similar triangles,you can write 业= x 1/2 (6) By rearranging Eq.(a), 6W1w1W11 0= F=下21a=下24吾 (d) If we now substitute the value ofx from Eq.(b)in Eq.(c)and multiply the numerator and denominator by the circular pitch p,we find Wp G=F③xP (d) Letting y =2x/3p,we have W 0= (14-11 Fpy This completes the development of the original Lewis equation.The factor y is called the Lewis form factor,and it may be obtained by a graphical layout of the gear tooth or by digital computation. In using this equation,most engineers prefer to employ the diametral pitch in determining the stresses.This is done by substituting P=z/p and Y=zy in Eq.(14-1).This gives WP 0= FY (14-2 where r=2tp (14-3) 3 The use of this equation for 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.Values of Y obtained from this equation are tabulated in Table 14-2
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears © The McGraw−Hill 715 Companies, 2008 Spur and Helical Gears 717 Figure 14–1 l F t W t W t W r l t a rf x W (a) (b) Referring now to Fig. 14–1b, we assume that the maximum stress in a gear tooth occurs at point a. By similar triangles, you can write t/2 x = l t/2 or x = t 2 4l (b) By rearranging Eq. (a), σ = 6Wt l Ft 2 = Wt F 1 t 2/6l = Wt F 1 t 2/4l 1 4 6 (c) If we now substitute the value of x from Eq. (b) in Eq. (c) and multiply the numerator and denominator by the circular pitch p, we find σ = Wt p F 2 3 xp (d) Letting y = 2x/3p, we have σ = Wt Fpy (14–1) This completes the development of the original Lewis equation. The factor y is called the Lewis form factor, and it may be obtained by a graphical layout of the gear tooth or by digital computation. In using this equation, most engineers prefer to employ the diametral pitch in determining the stresses. This is done by substituting P = π/p and Y = πy in Eq. (14–1). This gives σ = Wt P FY (14–2) where Y = 2x P 3 (14–3) The use of this equation for 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. Values of Y obtained from this equation are tabulated in Table 14–2.