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 775 Stress-Cycle Factor for Pitting Resistance CL(ZNT) 2 CL= 103≤Wz<10 3.4822W0.0602 104≤Nz≤1010 (15-14 2 103≤nL<104 3.4822m20.0602 104≤nL≤1010 See Fig.15-8 for a graphical presentation of Eqs.(15-14). Stress-Cycle Factor for Bending Strength KL(YNT) 2.7 102≤N2<103 103≤N2<3(10) KL= 6.1514W01182 1.6831W00323 3(10)≤Nz≤1010 general 1.3558W00178 3(10)≤WL≤1010 critical (15-15) 2.7 102≤nL<103 YNT 6.1514n20.1182 103≤nL<3(10) 1.6831n0.0323 3(10)≤nL≤1010 general 1.3558m-0.0323 3(10)≤nL≤1010 critical See Fig.15-9 for a plot of Eqs.(15-15). 5.0 ￥0 3.0 Case carburized 2.0 C2=3.4822N-m Zr=3.4822n2 1.0 e 103 10 10 10 107 103 10 10 Number of load cycles.N (n) Figure 15-8 Contact stress cycle factor for pitting resistance CZNr)for carburized case-hardened steel bevel gears. (Source:ANSI/AGMA 2003-B97.)
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears 772 © The McGraw−Hill Companies, 2008 Bevel and Worm Gears 775 Stress cycle factor, CL (ZNT) Number of load cycles, NL (nL ) 104 103 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 105 106 107 108 109 1010 Case carburized CL = 3.4822 NL –0.0602 ZNT = 3.4822 nL –0.0602 Figure 15–8 Contact stress cycle factor for pitting resistance CL (ZNT) for carburized case-hardened steel bevel gears. (Source: ANSI/AGMA 2003-B97.) Stress-Cycle Factor for Pitting Resistance CL (ZNT) CL = 2 103 ≤ NL < 104 3.4822N −0.0602 L 104 ≤ NL ≤ 1010 ZN T = 2 103 ≤ nL < 104 3.4822n−0.0602 L 104 ≤ nL ≤ 1010 (15–14) See Fig. 15–8 for a graphical presentation of Eqs. (15–14). Stress-Cycle Factor for Bending Strength KL (YNT) KL = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ 2.7 102 ≤ NL < 103 6.1514N −0.1182 L 103 ≤ NL < 3(106) 1.6831N −0.0323 L 3(106) ≤ NL ≤ 1010 general 1.3558N −0.0178 L 3(106) ≤ NL ≤ 1010 critical (15–15) YN T = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ 2.7 102 ≤ nL < 103 6.1514n−0.1182 L 103 ≤ nL < 3(106) 1.6831n−0.0323 L 3(106) ≤ nL ≤ 1010 general 1.3558n−0.0323 L 3(106) ≤ nL ≤ 1010 critical See Fig. 15–9 for a plot of Eqs. (15–15)
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 15.Bevel and Worm Gears ©The McGraw-Hil 3 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 776 IMechanical Engineering Design 35 3.0 NOTE:The choice of K (Yr)is influenced by: Case carburized Pitch-line velocity Gear material cleanliness Residual stress Material ductility and fracture toughness K=6.1514N,-a12 Vr=6.1514n2a12 1.5 KL=1.3558N2-00178 r=1.3558-0017 1.0 1.0 0.9 0.9 0.8 K=1.683N-0s3 0.8 0.7 Xr=1.683m,0.au 0.7 0.6 0.6 0 103 103 10 10 10 10 103 10' Number of load cycles.N (np) Figure 15-9 Stress cycle factor for bending strength K(YN)for carburized case-hardened steel bevel gears. (Source:ANSI/AGMA 2003-B97.] Hardness-Ratio Factor CH(Zw) CH=1+B1(N/n-1)B1=0.00898(HBP/HBG)-0.00829 (15-16) Zw=1+B1(31/22-1) B1=0.00898(HB1/HB2)-0.00829 The preceding equations are valid when1.2≤Hgr/HaG≤l.7(l.2≤Hg1/Ha2≤ 1.7).Figure 15-10 graphically displays Eqs.(15-16).When a surface-hardened pinion (48 HRC or harder)is run with a through-hardened gear(180 Hg <400),a work- hardening effect occurs.The Cu(Zw)factor varies with pinion surface roughness fp(Ra)and the mating-gear hardness: CH=1+B2(450-HBG) B2=0.00075exp(-0.0122fp) (15-17刀 Zw=1+B2(450-HB2) B2=0.00075exp(-0.52fp) where fp(R)=pinion surface hardness uin (um) HBG(H82)=minimum Brinell hardness See Fig.15-11 for carburized steel gear pairs of approximately equal hardness CH= Zw=1. Temperature Factor KT(Ke) 1 32F≤t≤250°F KT= (460+t)/710 t>250°℉ (15-18) 0°C≤0≤120°C Ka= (273+0)/393 0>120C
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears © The McGraw−Hill 773 Companies, 2008 776 Mechanical Engineering Design Stress cycle factor, KL (YN T) Number of load cycles, NL (nL) 104 102 103 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 3.0 3.5 105 106 107 108 109 1010 KL = 1.3558 NL –0.0178 YNT = 1.3558 nL –0.0178 NOTE: The choice of KL (YNT) is influenced by: Pitch-line velocity Gear material cleanliness Residual stress Material ductility and fracture toughness KL = 1.683 NL –0.0323 YNT = 1.683 nL –0.0323 KL = 6.1514 NL –0.1192 YNT = 6.1514 nL –0.1192 Case carburized Figure 15–9 Stress cycle factor for bending strength KL (YNT) for carburized case-hardened steel bevel gears. (Source: ANSI/AGMA 2003-B97.) Hardness-Ratio Factor CH (ZW) CH = 1 + B1(N/n − 1) B1 = 0.008 98(HB P/HBG) − 0.008 29 ZW = 1 + B1(z1/z2 − 1) B1 = 0.008 98(HB1/HB2) − 0.008 29 (15–16) The preceding equations are valid when 1.2 ≤ HB P/HBG ≤ 1.7 (1.2 ≤ HB1/HB2 ≤ 1.7). Figure 15–10 graphically displays Eqs. (15–16). When a surface-hardened pinion (48 HRC or harder) is run with a through-hardened gear (180 ≤ HB ≤ 400), a workhardening effect occurs. The CH (ZW ) factor varies with pinion surface roughness fP(Ra1) and the mating-gear hardness: CH = 1 + B2(450 − HBG) B2 = 0.000 75 exp(−0.0122 fP) ZW = 1 + B2(450 − HB2) B2 = 0.000 75 exp(−0.52 fP) (15–17) where fP(Ra1) = pinion surface hardness μin (μm) HBG(HB2) = minimum Brinell hardness See Fig. 15–11 for carburized steel gear pairs of approximately equal hardness CH = ZW = 1. Temperature Factor KT (Kθ) KT = 1 32◦F ≤ t ≤ 250◦F (460 + t)/710 t > 250◦F Kθ = 1 0◦C ≤ θ ≤ 120◦C (273 + θ )/393 θ > 120◦C (15–18)
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 15.Bevel and Worm Gears ©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Bevel and Worm Gears 777 Figure 15-10 1.14 Hardness-tatio factor CHZw) for through-hardened pinion 1.12 1.6 and gear. (Source:ANSI/AGMA 2003- B971 1.10 1.5 g 1.4 1.08 1.3 1.06 12 1.04 When 1.02 是会<2 usc CH(亿=1 100 0 6 8 10 12 16 18 20 Reduction gear ratio,NIn ( Figure 15-11 1.20 Hardness-ratio factor CH[Zw) for surface-hardened pinions. 164in (0.44m） Surface roughness of pinion,fp(R) (Source:ANSI/AGMA 2003- 1.15 B97) 32 uin 1.10 (0.8μm) 63 uin (1.64m) 1.05 125μin (324m) 1.0980200 250 300 350 400 Brinell hardness of the gear He Reliability Factors CR(Zz)and Ke(Yz) Table 15-3 displays the reliability factors.Note that Cg=KR and Zz=YZ Logarithmic interpolation equations are 0.50-0.251og(1-R)0.99≤R≤0.999 (15-19) Yz=KR= 0.70-0.151og(1-R) 0.90≤R<0.99 (15-20 The reliability of the stress(fatigue)numbers allowable in Tables 15-4,15-5,15-6,and 15-7is0.99
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears 774 © The McGraw−Hill Companies, 2008 Bevel and Worm Gears 777 Hardness ratio factor, CH (Z W) Reduction gear ratio, N/n (z2/z1) 0 2 4 6 8 10 12 14 16 18 20 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.7 1.6 1.5 1.4 1.3 1.2 Calculated hardness ratio, HBG HBP HB2 HB1 < 1.2 When use CH (ZW) = 1 HBG HBP HB2 HB1 Figure 15–10 Hardness-ratio factor CH (ZW) for through-hardened pinion and gear. (Source: ANSI/AGMA 2003- B97.) Hardness ratio factor CH (Z W) Brinell hardness of the gear HB 180 200 250 300 350 400 1.00 1.05 1.10 1.15 1.20 16 in (0.4 m) Surface roughness of pinion, f P (Ra1) 32 in (0.8 m) 63 in (1.6 m) 125 in (3.2 m) Figure 15–11 Hardness-ratio factor CH (ZW) for surface-hardened pinions. (Source: ANSI/AGMA 2003- B97.) Reliability Factors CR (ZZ) and KR (YZ) Table 15–3 displays the reliability factors. Note that CR = √KR and ZZ = √YZ . Logarithmic interpolation equations are YZ = KR = 0.50 − 0.25 log(1 − R) 0.99 ≤ R ≤ 0.999 0.70 − 0.15 log(1 − R) 0.90 ≤ R < 0.99 (15–19) (15–20) The reliability of the stress (fatigue) numbers allowable in Tables 15–4, 15–5, 15–6, and 15–7 is 0.99.
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 15.Bevel and Worm Gears ©The McGraw-Hil 75 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 778 Mechanical Engineering Design Table 15-3 Reliability Reliability Factors Factors for Steel' Source:ANSI/AGMA Requirements of Application CR (Zz) KR (Yz)t 2003-B97 Fewer than one failure in 10000 1.22 1.50 Fewer than one failure in 1000 1.12 1.25 Fewer than one failure in 100 1.00 1.00 Fewer than one failure in 10 0.92 0.85 Fewer than one failure in 2 0.84 0.70§ *At the present time there are insufficndato conceing the reliobility of bevel gears made from other materials. Tooth breakoge is sometimes considered a greater hazard than pitting.In such cases a greater value of Ke(Yz)is selected for bending. At this vale plasticowmightouher than pitting. Fmtstdati Table 15-4 Allowable Contact Stress Number for Steel Gears,Sac (Hm Source:ANSI/AGMA 2003-B97 Minimum Allowable Contact Stress Number, Material Heat Surface* sac (OH lim)Ibf/in2 (N/mm2) Designation Treatment Hardness Grade 1t Grade 2t Grade 3t Steel Through-hardened Fig.15-12 Fig.15-12 fig.15-12 Flame or induction 50 HRC 175000 190000 hardeneds (12101 (13101 Carburized and 2003-B97 200000 225000 250000 case hardeneds Table 8 1380) (1550) (1720) A1SI4140 Nitrideds 84.5HR15N 145000 (10001 Nitralloy 160000 135M Nitrideds 90.0HR15N (11001 *Hardness to be equivalent to that at the tooth middepth in the center of the foce width. tSee ANSIAGMA 3-Tblesthough 1for metafoforstgde of steegers +These materials must be aneed or nomalized as a minumum. SThellwablestress numbers indiated may be used with the cse depths presibed in21.1,ANSI/AGMA 2003-897. Elastic Coefficient for Pitting Resistance Cp(Zg) Cp= Vπ[I-)/Ep+(I-哈)/Ec] (15-21) ZE=√x0-/61+0-哈)/E阿
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 15. Bevel and Worm Gears © The McGraw−Hill 775 Companies, 2008 778 Mechanical Engineering Design Reliability Factors for Steel* Requirements of Application CR (ZZ) KR (YZ) † Fewer than one failure in 10 000 1.22 1.50 Fewer than one failure in 1000 1.12 1.25 Fewer than one failure in 100 1.00 1.00 Fewer than one failure in 10 0.92 0.85‡ Fewer than one failure in 2 0.84 0.70§ *At the present time there are insufficient data concerning the reliability of bevel gears made from other materials. † Tooth breakage is sometimes considered a greater hazard than pitting. In such cases a greater value of KR (YZ) is selected for bending. ‡ At this value plastic flow might occur rather than pitting. § From test data extrapolation. Table 15–3 Reliability Factors Source: ANSI/AGMA 2003-B97. Elastic Coefficient for Pitting Resistance Cp (ZE) Cp = 1 π 1 − ν2 P EP + 1 − ν2 G EG Z E = 1 π 1 − ν2 1 E1 + 1 − ν2 2 E2 (15–21) Minimum Allowable Contact Stress Number, Material Heat Surface* sac (H lim) lbf/in2 (N/mm2) Designation Treatment Hardness Grade 1† Grade 2† Grade 3† Steel Through-hardened‡ Fig.15–12 Fig.15–12 Fig.15–12 Flame or induction 50 HRC 175 000 190 000 hardened§ (1210) (1310) Carburized and 2003-B97 200 000 225 000 250 000 case hardened§ Table 8 (1380) (1550) (1720) AISI 4140 Nitrided§ 84.5 HR15N 145 000 (1000) Nitralloy 160 000 135M Nitrided§ 90.0 HR15N (1100) *Hardness to be equivalent to that at the tooth middepth in the center of the face width. † See ANSI/AGMA 2003-B97, Tables 8 through 11, for metallurgical factors for each stress grade of steel gears. ‡ These materials must be annealed or normalized as a minumum. § The allowable stress numbers indicated may be used with the case depths prescribed in 21.1, ANSI/AGMA 2003-B97. Table 15–4 Allowable Contact Stress Number for Steel Gears, sac (σH lim) Source: ANSI/AGMA 2003-B97.