Budynas-Nisbett Shigley's I Design of Mechanical 15.Bevel and Worm Gears T©The McGraw-Hfll Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 15 Bevel and Worm Gears Chapter Outline 15-1 Bevel Gearing-General 766 15-2 Bevel-Gear Stresses and Strengths 768 15-3 AGMA Equation Factors 771 15-4 Straight-Bevel Gear Analysis 783 15-5 Design of a Straight-Bevel Gear Mesh 786 15-6 Worm Gearing-AGMA Equation 789 15-7 Worm-Gear Analysis 793 15-8 Designing a Worm-Gear Mesh 797 15-9 Buckingham Wear Load 800 765
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears 762 © The McGraw−Hill Companies, 2008 15Bevel and Worm Gears Chapter Outline 15–1 Bevel Gearing—General 766 15–2 Bevel-Gear Stresses and Strengths 768 15–3 AGMA Equation Factors 771 15–4 Straight-Bevel Gear Analysis 783 15–5 Design of a Straight-Bevel Gear Mesh 786 15–6 Worm Gearing—AGMA Equation 789 15–7 Worm-Gear Analysis 793 15–8 Designing a Worm-Gear Mesh 797 15–9 Buckingham Wear Load 800 765
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 15.Bevel and Worm Gears ©The McGraw-Hil 753 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 766 Mechanical Engineering Design The American Gear Manufacturers Association (AGMA)has established standards for the analysis and design of the various kinds of bevel and worm gears.Chapter 14 was an introduction to the AGMA methods for spur and helical gears.AGMA has estab- lished similar methods for other types of gearing,which all follow the same general approach. 15-1 Bevel Gearing-General Bevel gears may be classified as follows: ·Straight bevel gears ·Spiral bevel gears ·Zerol bevel gears ·Hypoid gears ·Spiroid gears A straight bevel gear was illustrated in Fig.13-35.These gears are usually used for pitch-line velocities up to 1000 ft/min(5 m/s)when the noise level is not an important consideration.They are available in many stock sizes and are less expensive to produce than other bevel gears,especially in small quantities. A spiral bevel gear is shown in Fig.15-1;the definition of the spiral angle is illus- trated in Fig.15-2.These gears are recommended for higher speeds and where the noise level is an important consideration.Spiral bevel gears are the bevel counterpart of the helical gear;it can be seen in Fig.15-1 that the pitch surfaces and the nature of con- tact are the same as for straight bevel gears except for the differences brought about by the spiral-shaped teeth. The Zerol bevel gear is a patented gear having curved teeth but with a zero spiral angle.The axial thrust loads permissible for Zerol bevel gears are not as large as those for the spiral bevel gear,and so they are often used instead of straight bevel gears.The Zerol bevel gear is generated by the same tool used for regular spiral bevel gears.For design purposes,use the same procedure as for straight bevel gears and then simply substitute a Zerol bevel gear. Figure 15-1 Spiral bevel gears.(Courtesy of Gleason Works,Rochester N.Y)
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears © The McGraw−Hill 763 Companies, 2008 766 Mechanical Engineering Design The American Gear Manufacturers Association (AGMA) has established standards for the analysis and design of the various kinds of bevel and worm gears. Chapter 14 was an introduction to the AGMA methods for spur and helical gears. AGMA has established similar methods for other types of gearing, which all follow the same general approach. 15–1 Bevel Gearing—General Bevel gears may be classified as follows: • Straight bevel gears • Spiral bevel gears • Zerol bevel gears • Hypoid gears • Spiroid gears A straight bevel gear was illustrated in Fig. 13–35. These gears are usually used for pitch-line velocities up to 1000 ft/min (5 m/s) when the noise level is not an important consideration. They are available in many stock sizes and are less expensive to produce than other bevel gears, especially in small quantities. A spiral bevel gear is shown in Fig. 15–1; the definition of the spiral angle is illustrated in Fig. 15–2. These gears are recommended for higher speeds and where the noise level is an important consideration. Spiral bevel gears are the bevel counterpart of the helical gear; it can be seen in Fig. 15–1 that the pitch surfaces and the nature of contact are the same as for straight bevel gears except for the differences brought about by the spiral-shaped teeth. The Zerol bevel gear is a patented gear having curved teeth but with a zero spiral angle. The axial thrust loads permissible for Zerol bevel gears are not as large as those for the spiral bevel gear, and so they are often used instead of straight bevel gears. The Zerol bevel gear is generated by the same tool used for regular spiral bevel gears. For design purposes, use the same procedure as for straight bevel gears and then simply substitute a Zerol bevel gear. Figure 15–1 Spiral bevel gears. (Courtesy of Gleason Works, Rochester, N.Y.)
764 Budynas-Nisbett:Shigley's Ill Design of Mechanical 15.Bevel and Worm Gears The McGraw-Hill Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Bevel and Worm Gears 767 Figure 15-2 Circular pitch Cutting spiral-gear teeth on the Face advance basic crown rock. Mean radius of crown rack Spira angle Basic crown rack Figure 15-3 Hypoid gears.(Courtesy of Gleason Works,Rochester, N.YJ It is frequently desirable,as in the case of automotive differential applications,to have gearing similar to bevel gears but with the shafts offset.Such gears are called hypoid gears, because their pitch surfaces are hyperboloids of revolution.The tooth action between such gears is a combination of rolling and sliding along a straight line and has much in common with that of worm gears.Figure 15-3 shows a pair of hypoid gears in mesh. Figure 15-4 is included to assist in the classification of spiral bevel gearing.It is seen that the hypoid gear has a relatively small shaft offset.For larger offsets,the pinion begins to resemble a tapered worm and the set is then called spiroid gearing
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears 764 © The McGraw−Hill Companies, 2008 Bevel and Worm Gears 767 Basic crown rack Cutter radius Spiral angle Mean radius of crown rack Circular pitch Face advance Figure 15–2 Cutting spiral-gear teeth on the basic crown rack. Figure 15–3 Hypoid gears. (Courtesy of Gleason Works, Rochester, N.Y.) It is frequently desirable, as in the case of automotive differential applications, to have gearing similar to bevel gears but with the shafts offset. Such gears are called hypoid gears, because their pitch surfaces are hyperboloids of revolution. The tooth action between such gears is a combination of rolling and sliding along a straight line and has much in common with that of worm gears. Figure 15–3 shows a pair of hypoid gears in mesh. Figure 15–4 is included to assist in the classification of spiral bevel gearing. It is seen that the hypoid gear has a relatively small shaft offset. For larger offsets, the pinion begins to resemble a tapered worm and the set is then called spiroid gearing.
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 15.Bevel and Worm Gears T©The McGraw-Hil 765 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 768 Mechanical Engineering Design Figure 15-4 Comparison of intersecting and offset-shoft beveHtype gearings.(From Gear Handbook by Darle W. Dudley,1962,p.2-24.] Spiroid Ring gear Hypoid Spiral bevel 15-2 Bevel-Gear Stresses and Strengths In a typical bevel-gear mounting,Fig.13-36.for example,one of the gears is often mounted outboard of the bearings.This means that the shaft deflections can be more pronounced and can have a greater effect on the nature of the tooth contact.Another dif- ficulty that occurs in predicting the stress in bevel-gear teeth is the fact that the teeth are tapered.Thus,to achieve perfect line contact passing through the cone center,the teeth ought to bend more at the large end than at the small end.To obtain this condition requires that the load be proportionately greater at the large end.Because of this vary- ing load across the face of the tooth,it is desirable to have a fairly short face width. Because of the complexity of bevel,spiral bevel,Zerol bevel,hypoid,and spiroid gears,as well as the limitations of space,only a portion of the applicable standards that refer to straight-bevel gears is presented here.Table 15-1 gives the symbols used in ANSI/AGMA 2003-B97. Fundamental Contact Stress Equation W 1/2 Se=0e=Cp :KoKKmCsCxe (U.S.customary units) (15-1) 1000W 1/2 OH ZE bdZy KAK KHBZxZxe (SI units) The first term in each equation is the AGMA symbol,whereas;oc,our normal notation, is directly equivalent. Figures 15-5 to 15-13 and Tables 15-1 to 15-7 have been extracted from ANSI/AGMA 2003-B97,Rating the Pitting Resistance and Bending Strength of Generated Straight Bevel.Zerol Bevel and Spiral Bevel Gear Teeth with the permission of the publisher,the American Gear Manufacturers Association, 500 Montgomery Street,Suite 350.Alexandria,VA,22314-1560
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears © The McGraw−Hill 765 Companies, 2008 768 Mechanical Engineering Design Worm Spiroid Hypoid Spiral bevel Ring gear Figure 15–4 Comparison of intersectingand offset-shaft bevel-type gearings. (From Gear Handbook by Darle W. Dudley, 1962, p. 2–24.) 15–2 Bevel-Gear Stresses and Strengths In a typical bevel-gear mounting, Fig. 13–36, for example, one of the gears is often mounted outboard of the bearings. This means that the shaft deflections can be more pronounced and can have a greater effect on the nature of the tooth contact. Another dif- ficulty that occurs in predicting the stress in bevel-gear teeth is the fact that the teeth are tapered. Thus, to achieve perfect line contact passing through the cone center, the teeth ought to bend more at the large end than at the small end. To obtain this condition requires that the load be proportionately greater at the large end. Because of this varying load across the face of the tooth, it is desirable to have a fairly short face width. Because of the complexity of bevel, spiral bevel, Zerol bevel, hypoid, and spiroid gears, as well as the limitations of space, only a portion of the applicable standards that refer to straight-bevel gears is presented here.1 Table 15–1 gives the symbols used in ANSI/AGMA 2003-B97. Fundamental Contact Stress Equation sc = σc = Cp Wt FdP I KoKvKmCsCxc1/2 (U.S. customary units) σH = Z E 1000Wt bd Z1 KAKvK Hβ Zx Zxc1/2 (SI units) (15–1) The first term in each equation is the AGMA symbol, whereas; σc, our normal notation, is directly equivalent. 1 Figures 15–5 to 15–13 and Tables 15–1 to 15–7 have been extracted from ANSI/AGMA 2003-B97, Rating the Pitting Resistance and Bending Strength of Generated Straight Bevel, Zerol Bevel and Spiral Bevel Gear Teeth with the permission of the publisher, the American Gear Manufacturers Association, 500 Montgomery Street, Suite 350, Alexandria, VA, 22314-1560
766 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 15.Bevel and Worm Gears ©The McGraw-Hill Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Bevel and Worm Gears 769 Table 15-1 Symbols Used in Bevel Gear Rating Equations,ANSI/AGMA 2003-B97 Standard Source:ANSI/AGMA 2003-B97 AGMA ISO Symbol Symbol Description Units Mean cone distance in (mm) Outer cone distance in (mm) Hardness ratio factor for pitting resistance Inertia factor for pitting resistance Stress cycle factor for pitting resistance Elastic coefficient [bf/in2]0.5 N/mm2°.) 3 Reliability factor for pitting Service factor for pitting resistance Size factor for pitting resistance Crowning factor for pitting resistance d de2,del Outer pitch diameters of gear and pinion,respectively in (mm) Ep E2,E1 Young's modulus of elasticity for materials of gear and pinion,respectively Ibf/in2 (N/mm2) Base of natural (Napierian)logarithms Net face width in (mm) 8 b2 b1 Effective face widths of gear and pinion,respectively in (mm) ⊙ Pinion surface roughness uin (um] Minimum Brinell hardness number for gear material HB HB1 Minimum Brinell hardness number for pinion material HB Eht min Minimum total case depth at tooth middepth in (mm) 云 Minimum effective case depth in (mm lim Suggested maximum effective case depth limit at tooth middepth in (mm) Geometry factor for pitting resistance Geometry factor for bending strength Geometry factor for bending strength for gear and pinion,respectively Stress correction and concentration factor Inertia factor for bending strength Y Stress cycle factor for bending strength KHB Load distribution factor Overload factor Reliability factor for bending strength Size factor for bending strength Service factor for bending strength Temperature factor Dynamic factor Lengthwise curvature factor for bending strength met Outer transverse module (mm) mmt Mean transverse module (mm) mn Mean normal module (mm) mN Load sharing ratio,pitting EN Load sharing ratio,bending Number of gear teeth Number of load cycles 21 Number of pinion teeth Pinion speed ev/min Continued
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears 766 © The McGraw−Hill Companies, 2008 Bevel and Worm Gears 769 Table 15–1 Symbols Used in Bevel Gear Rating Equations, ANSI/AGMA 2003-B97 Standard Source: ANSI/AGMA 2003-B97. AGMA ISO Symbol Symbol Description Units Am Rm Mean cone distance in (mm) A0 Re Outer cone distance in (mm) CH ZW Hardness ratio factor for pitting resistance Ci Zi Inertia factor for pitting resistance CL ZNT Stress cycle factor for pitting resistance Cp ZE Elastic coefficient [lbf/in2] 0.5 ([N/mm2] 0.5) CR ZZ Reliability factor for pitting CSF Service factor for pitting resistance CS Zx Size factor for pitting resistance Cxc Zxc Crowning factor for pitting resistance D, d de2, de1 Outer pitch diameters of gear and pinion, respectively in (mm) EG, EP E2, E1 Young’s modulus of elasticity for materials of gear and pinion, respectively lbf/in2 (N/mm2) e e Base of natural (Napierian) logarithms F b Net face width in (mm) FeG, FeP b 2, b 1 Effective face widths of gear and pinion, respectively in (mm) fP Ra1 Pinion surface roughness μin (μm) HBG HB2 Minimum Brinell hardness number for gear material HB HBP HB1 Minimum Brinell hardness number for pinion material HB hc Eht min Minimum total case depth at tooth middepth in (mm) he h c Minimum effective case depth in (mm) he lim h c lim Suggested maximum effective case depth limit at tooth middepth in (mm) I ZI Geometry factor for pitting resistance J YJ Geometry factor for bending strength JG, JP YJ2, YJ1 Geometry factor for bending strength for gear and pinion, respectively KF YF Stress correction and concentration factor Ki Yi Inertia factor for bending strength KL YNT Stress cycle factor for bending strength Km KHβ Load distribution factor Ko KA Overload factor KR Yz Reliability factor for bending strength KS YX Size factor for bending strength KSF Service factor for bending strength KT Kθ Temperature factor Kv Kv Dynamic factor Kx Yβ Lengthwise curvature factor for bending strength met Outer transverse module (mm) mmt Mean transverse module (mm) mmn Mean normal module (mm) mNI εNI Load sharing ratio, pitting mNJ εNJ Load sharing ratio, bending N z2 Number of gear teeth NL nL Number of load cycles n z1 Number of pinion teeth nP n1 Pinion speed rev/min (Continued)
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 15.Bevel and Worm Gears ©The McGraw-Hil 767 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 770 Mechanical Engineering Design Table 15-1 Symbols Used in Gear Rating Equations,ANSI/AGMA 2003-B97 Standard (Continued) AGMA 15O Symbol Symbol Description Units P P Design power through gear pair hp (kW) Pa Allowable transmitted power hp (kW) Allowable transmitted power for pitting resistance hp (kW) Allowable transmitted power for pitting resistance at unity service factor hp (kw) Allowable transmitted power for bending strength hp (kW) Payu Allowable transmitted power for bending strength at unity service factor hp (kw) Pa Outer transverse diametral pitch in-1 Mean transverse diametral pitch in-1 Mean normal diametral pitch in-1 Q Q Transmission accuracy number Exponent used in formula for lengthwise curvature factor R,r Impr2,Imprl Mean transverse pitch radii for gear and pinion,respectively in (mm) R 「myo2,fmyol Mean transverse radii to point of load application for gear in (mm) and pinion,respectively Te0 Cutter radius used for producing Zerol bevel and spiral bevel gears in (mm) ge Length of the instantaneous line of contact between mating tooth surfaces in (mm) Allowable contact stress number Ibf/in2 (N/mm) Bending stress number (allowable) lbf/in2 (N/mm2) aH Calculated contact stress number Ibf/in2 (N/mm2) SF 收 Bending safety factor SH Contact safety factor Calculated bending stress number Ibf/in2 (N/mm2) Swc CHP Permissible contact stress number Ibf/in2 (N/mm2) Swt app Permissible bending stress number Ibf/in2 (N/mm2) Te Operating pinion torque Ibf in (Nm) Tr 0r Operating gear blank temperature FC) o Sai Normal tooth top land thickness at narrowest point in (mm) Core hardness coefficient for nitrided gear Ibf/in2 (N/mm2) Uh UH Hardening process factor for steel lbf/in2 (N/mm2) Pitch-ine velocity at outer pitch circle #/min (m/s) YkG,Y知 YK2,YKI Tooth form factors including stress-concentration factor for gear and pinion,respectively HG.Hp 2,1 Poisson's ratio for materials of gear and pinion,respectively po Pyo Relative radius of profile curvature at point of maximum contact stress in (mm) between mating tooth surfaces an Normal pressure angle at pitch surface g Owt Transverse pressure angle at pitch point Bm Mean spiral angle at pitch surface % b Mean base spiral angle
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears © The McGraw−Hill 767 Companies, 2008 770 Mechanical Engineering Design AGMA ISO Symbol Symbol Description Units P P Design power through gear pair hp (kW) Pa Pa Allowable transmitted power hp (kW) Pac Paz Allowable transmitted power for pitting resistance hp (kW) Pacu Pazu Allowable transmitted power for pitting resistance at unity service factor hp (kW) Pat Pay Allowable transmitted power for bending strength hp (kW) Patu Payu Allowable transmitted power for bending strength at unity service factor hp (kW) Pd Outer transverse diametral pitch in−1 Pm Mean transverse diametral pitch in−1 Pmn Mean normal diametral pitch in−1 Qv Qv Transmission accuracy number q q Exponent used in formula for lengthwise curvature factor R, r rmpt2, rmpt1 Mean transverse pitch radii for gear and pinion, respectively in (mm) Rt , rt rmyo2, rmyo1 Mean transverse radii to point of load application for gear in (mm) and pinion, respectively rc rc0 Cutter radius used for producing Zerol bevel and spiral bevel gears in (mm) s gc Length of the instantaneous line of contact between mating tooth surfaces in (mm) sac σH lim Allowable contact stress number lbf/in2 (N/mm2) sat σF lim Bending stress number (allowable) lbf/in2 (N/mm2) sc σH Calculated contact stress number lbf/in2 (N/mm2) sF sF Bending safety factor sH sH Contact safety factor st σF Calculated bending stress number lbf/in2 (N/mm2) swc σHP Permissible contact stress number lbf/in2 (N/mm2) swt σFP Permissible bending stress number lbf/in2 (N/mm2) TP T1 Operating pinion torque lbf in (Nm) TT θ T Operating gear blank temperature °F(°C) t0 sai Normal tooth top land thickness at narrowest point in (mm) Uc Uc Core hardness coefficient for nitrided gear lbf/in2 (N/mm2) UH UH Hardening process factor for steel lbf/in2 (N/mm2) vt vet Pitch-line velocity at outer pitch circle ft/min (m/s) YKG, YKP YK2, YK1 Tooth form factors including stress-concentration factor for gear and pinion, respectively μG, μp ν2, ν1 Poisson’s ratio for materials of gear and pinion, respectively ρ0 ρyo Relative radius of profile curvature at point of maximum contact stress in (mm) between mating tooth surfaces φ αn Normal pressure angle at pitch surface φt αwt Transverse pressure angle at pitch point ψ βm Mean spiral angle at pitch surface ψb βmb Mean base spiral angle Table 15–1 Symbols Used in Gear Rating Equations, ANSI/AGMA 2003-B97 Standard (Continued)
Budynas-Nisbett:Shigley's lll.Design of Mechanical 15.Bevel and Worm Gears T©The McGraw-Hill Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Bevel and Worm Gears 771 Permissible Contact Stress Number(Strength)Equation SaeCLCH Swc (c)all SHKTCR (U.S.customary units) (15-2) OH limZNT ZW OHP= (SI units) SH KeZz Bending Stress W KsKm S1= Pakak.K:J (U.S.customary units) (15-3) OF= 1000W KAK Yx KHB (SI units) b met YBYJ Permissible Bending Stress Equation SatKL Swt= (U.S.customary units) SEKTKR (15-4 OFlimYNT OFP= (SI units） SEKaY, 15-3 AGMA Equation Factors Overload Factor Ko(KA) The overload factor makes allowance for any externally applied loads in excess of the nominal transmitted load.Table 15-2,from Appendix A of 2003-B97,is included for your guidance. Safety Factors SH and Se The factors of safety Su and Se as defined in 2003-B97 are adjustments to strength,not load,and consequently cannot be used as is to assess (by comparison)whether the threat is from wear fatigue or bending fatigue.Since W is the same for the pinion and gear,the comparison ofS to Sr allows direct comparison. Dynamic Factor Ky In 2003-C87 AGMA changed the definition of K to its reciprocal but used the same symbol.Other standards have yet to follow this move.The dynamic factor K makes Table 15-2 Character of Character of Load on Driven Machine ○verload Factors K(KA) Prime Mover Uniform Light Shock Medium Shock Heavy Shock Source:ANSI/AGMA Uniform 1.00 1.25 1.50 1.75 or higher 2003B97. Light shock 110 1.35 1.60 1.85 or higher Medium shock 1.25 1.50 1.75 2.00 or higher Heavy shock 1.50 1.75 2.00 2.25 or higher Note:This table is for speed-decreasing drives.For speed-increasing drives,odd 0.01(N/n)or0.01(z2/)to the above factors
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears 768 © The McGraw−Hill Companies, 2008 Bevel and Worm Gears 771 Permissible Contact Stress Number (Strength) Equation swc = (σc)all = sacCLCH SH KTCR (U.S. customary units) σH P = σH lim ZN T ZW SH Kθ ZZ (SI units) (15–2) Bending Stress st = Wt F PdKoKv KsKm Kx J (U.S. customary units) σF = 1000Wt b KAKv met Yx K Hβ YβYJ (SI units) (15–3) Permissible Bending Stress Equation swt = sat KL SF KT KR (U.S. customary units) σF P = σF limYN T SF KθYz (SI units) (15–4) 15–3 AGMA Equation Factors Overload Factor Ko (KA) The overload factor makes allowance for any externally applied loads in excess of the nominal transmitted load. Table 15–2, from Appendix A of 2003-B97, is included for your guidance. Safety Factors SH and SF The factors of safety SH and SF as defined in 2003-B97 are adjustments to strength, not load, and consequently cannot be used as is to assess (by comparison) whether the threat is from wear fatigue or bending fatigue. Since Wt is the same for the pinion and gear, the comparison of √SH to SF allows direct comparison. Dynamic Factor Kv In 2003-C87 AGMA changed the definition of Kv to its reciprocal but used the same symbol. Other standards have yet to follow this move. The dynamic factor Kv makes Table 15–2 Overload Factors Ko (KA) Source: ANSI/AGMA 2003-B97. Character of Character of Load on Driven Machine Prime Mover Uniform Light Shock Medium Shock Heavy Shock Uniform 1.00 1.25 1.50 1.75 or higher Light shock 1.10 1.35 1.60 1.85 or higher Medium shock 1.25 1.50 1.75 2.00 or higher Heavy shock 1.50 1.75 2.00 2.25 or higher Note: This table is for speed-decreasing drives. For speed-increasing drives, add 0.01(N/n)2 or 0.01(z2 /z1) 2 to the above factors
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 15.Bevel and Worm Gears T©The McGraw-Hill 769 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 772 Mechanical Engineering Design Figure 15-5 Pitch-line velocity.(m/s) 20 10 20 30 40 50 Dynamic factor Kv. (Source:ANSI/AGMA 2003- 2。=5 1.9 B97.J Q=6 1.7 0=7 1.6 2n=8 1.5 2。=9 1.4 Q。=10 13 1.2 g=11 2000 4000 6000 8000 10000 Pitch-line velocity.(ft/min) allowance for the effect of gear-tooth quality related to speed and load,and the increase in stress that follows.AGMA uses a transmission accuracy number to describe the precision with which tooth profiles are spaced along the pitch circle.Figure 15-5 shows graphically how pitch-line velocity and transmission accuracy number are related to the dynamic factor K.Curve fits are Kp= A+√： A (U.S.customary units) (15-5) K。= A+√200ve: (SI units) A where A=50+56(1-B) (15-6) B=0.25(12-Q)2B and v:(ver)is the pitch-line velocity at outside pitch diameter,expressed in ft/min (m/s): =πdpnr/12 (U.S.customary units) (15-7刀 vau=5.236(10-5)dn1 (SI units) The maximum recommended pitch-line velocity is associated with the abscissa of the terminal points of the curve in Fig.15-5: 4max=[A+(Qw-3)]2 (U.S.customary units) [A+(2。-3)2 (15-8) Vte max (SI units) 200 where vrmax and ver max are in ft/min and m/s,respectively
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears © The McGraw−Hill 769 Companies, 2008 772 Mechanical Engineering DesignDynamic factor, Kv Pitch-line velocity, vt (ft/min) Pitch-line velocity, vet (m/s) 0 2000 4000 6000 8000 10 000 0 10 20 30 40 50 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 Qv = 5 Qv = 7 Qv = 6 Qv = 8 Qv = 9 Qv = 10 Qv = 11 Figure 15–5 Dynamic factor Kv. (Source: ANSI/AGMA 2003- B97.) allowance for the effect of gear-tooth quality related to speed and load, and the increase in stress that follows. AGMA uses a transmission accuracy number Qv to describe the precision with which tooth profiles are spaced along the pitch circle. Figure 15–5 shows graphically how pitch-line velocity and transmission accuracy number are related to the dynamic factor Kv . Curve fits are Kv = A + √vt A B (U.S. customary units) Kv = A + √200vet A B (SI units) (15–5) where A = 50 + 56(1 − B) B = 0.25(12 − Qv) 2/3 (15–6) and vt(vet) is the pitch-line velocity at outside pitch diameter, expressed in ft/min (m/s): vt = πdPnP/12 (U.S. customary units) vet = 5.236(10−5 )d1n1 (SI units) (15–7) The maximum recommended pitch-line velocity is associated with the abscissa of the terminal points of the curve in Fig. 15–5: vt max = [A + (Qv − 3)] 2 (U.S. customary units) vte max = [A + (Qv − 3)] 2 200 (SI units) (15–8) where vt max and vet max are in ft/min and m/s, respectively.
0 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 15.Bevel and Worm Gears T©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Bevel and Worm Gears 773 Size Factor for Pitting Resistance Cs(Zx) 0.5 F114.3mm Size Factor for Bending Ks (Yx) K,= 0.4867+0.2132/Pa0.5≤P≤16in-1 (U.S.customary units) 0.5 Pa>16 in-1 10.5 mer 1.6 mm (15-101 Yt= 10.4867+0.008339n1.6≤mer≤50mm (SI units) Load-Distribution Factor Km(KHp) Km=Kmb+0.0036F2 (U.S.customary units) (15-111 KHB=Kmb+5.6(10-6)b2 (SI units) where 1.00 both members straddle-mounted Kmb 1.10 one member straddle-mounted 1.25 neither member straddle-mounted Crowning Factor for Pitting Cxe(Zxe) The teeth of most bevel gears are crowned in the lengthwise direction during manufac- ture to accommodate to the deflection of the mountings. 1.5 properly crowned teeth Cxc =Zxc= (15-12) 2.0 or larger uncrowned teeth Lengthwise Curvature Factor for Bending Strength Kx(Ys) For straight-bevel gears, K=Y8=1 (15-13) Pitting Resistance Geometry Factor I(Zi) Figure 15-6 shows the geometry factor /(Z)for straight-bevel gears with a 20 pressure angle and 90 shaft angle.Enter the figure ordinate with the number of pinion teeth, move to the number of gear-teeth contour,and read from the abscissa. Bending Strength Geometry Factor J(Yj) Figure 15-7 shows the geometry factor for straight-bevel gears with a 20 pressure angle and 90 shaft angle
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears 770 © The McGraw−Hill Companies, 2008 Bevel and Worm Gears 773 Size Factor for Pitting Resistance Cs (Zx) Cs = ⎧ ⎨ ⎩ 0.5 F 4.5 in (U.S. customary units) Zx = ⎧ ⎨ ⎩ 0.5 b 114.3 mm (SI units) (15–9) Size Factor for Bending Ks (Yx) Ks = 0.4867 + 0.2132/Pd 0.5 ≤ Pd ≤ 16 in−1 0.5 Pd > 16 in−1 (U.S. customary units) Yx = 0.5 met < 1.6 mm 0.4867 + 0.008 339met 1.6 ≤ met ≤ 50 mm (SI units) (15–10) Load-Distribution Factor Km (KHβ) Km = Kmb + 0.0036F2 (U.S. customary units) K Hβ = Kmb + 5.6(10−6)b2 (SI units) (15–11) where Kmb = ⎧ ⎨ ⎩ 1.00 both members straddle-mounted 1.10 one member straddle-mounted 1.25 neither member straddle-mounted Crowning Factor for Pitting Cxc (Zxc) The teeth of most bevel gears are crowned in the lengthwise direction during manufacture to accommodate to the deflection of the mountings. Cxc = Zxc = 1.5 properly crowned teeth 2.0 or larger uncrowned teeth (15–12) Lengthwise Curvature Factor for Bending Strength Kx (Yβ) For straight-bevel gears, Kx = Yβ = 1 (15–13) Pitting Resistance Geometry Factor I (ZI) Figure 15–6 shows the geometry factor I (ZI) for straight-bevel gears with a 20◦ pressure angle and 90◦ shaft angle. Enter the figure ordinate with the number of pinion teeth, move to the number of gear-teeth contour, and read from the abscissa. Bending Strength Geometry Factor J (YJ) Figure 15–7 shows the geometry factor J for straight-bevel gears with a 20◦ pressure angle and 90◦ shaft angle
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 15.Bevel and Worm Gears ©The McGraw-Hil m Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 774 Mechanical Engineering Design Figure 15-6 Number of gear teeth Contact geometry factor I(Z 0 60 80 90 100 50 for coniflex straighbevel gears with a 20 normal pressure angle and a 90 shaft angle. (Source:ANSI/AGMA 2003- B97) 30 30 20 15 0.06 0.07 0.08 0.09 0.10 0.11 Gcometry factor,.I(Z） Figure 15-7 Number of teeth in mate 13 15 20 253035404550 100 Bending factor J[Y)for 100 coniflex straighi-bevel gears with a 20 normal pressure 色 angle and 90 shaft angle (Source:ANSI/AGMA 2003. B97.) 60 18160.180200220.240260280300320340360.480.40 Gcometry factor,J(Y)
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears © The McGraw−Hill 771 Companies, 2008 774 Mechanical Engineering DesignNumber of pinion teeth Geometry factor, I (ZI) 0.05 0.06 0.07 0.08 0.09 Number of gear teeth 0.10 0.11 10 20 30 40 50 15 20 25 30 35 45 50 60 70 80 90 100 40 Figure 15–6 Contact geometry factor I (ZI ) for coniflex straight-bevel gears with a 20◦ normal pressure angle and a 90◦ shaft angle. (Source: ANSI/AGMA 2003- B97.) Number of teeth on gear for which geometry factor is desired Geometry factor, J (YJ) Number of teeth in mate 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.48 0.40 10 20 30 40 50 60 13 15 20 25 30 35 40 45 50 100 90 70 80 90 100 80 70 60 Figure 15–7 Bending factor J (YJ) for coniflex straight-bevel gears with a 20◦ normal pressure angle and 90◦ shaft angle. (Source: ANSI/AGMA 2003- B97.)
