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)