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 F<0.5in C= 0.125F+0.4375 0.5≤F≤4.5in (U.S.customary units) 1 F4.5in (15-91 0.5 b<12.7mm Zx= 0.00492b+0.4375 12.7≤b≤114.3mm (SI units) 1 b>114.3mm 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 < 0.5 in 0.125F + 0.4375 0.5 ≤ F ≤ 4.5 in 1 F > 4.5 in (U.S. customary units) Zx = ⎧ ⎨ ⎩ 0.5 b < 12.7 mm 0.004 92b + 0.4375 12.7 ≤ b ≤ 114.3 mm 1 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.)