上海交通大学：《Design and Manufacturing》课程教学资源（参考文献）Shigley's Mechanical Engineering Design ch14 spur and helical gears.pdf_P11-P15

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 14.Spur and Helical Gears T©The McGraw-Hil 721 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Spur and Helical Gears 723 In Ex.14-2 our resources (Fig.A-15-6)did not directly address stress concentra- tion in gear teeth.A photoelastic investigation by Dolan and Broghamer reported in 1942 constitutes a primary source of information on stress concentration.3 Mitchiner and Mabie*interpret the results in term of fatigue stress-concentration factor K as Ky=H+ (14-91 where H=0.34-0.4583662φ L=0.316-0.4583662中 M=0.290+0.4583662φ (b-rt)2 (d/2)+b-rr In these equations and t are from the layout in Fig.14-1,is the pressure angle,rf is the fillet radius,b is the dedendum,and d is the pitch diameter.It is left as an exercise for the reader to compare Kr from Eq.(14-9)with the results of using the approxima- tion of Fig.A-15-6 in Ex.14-2. 14-2 Surface Durability In this section we are interested in the failure of the surfaces of gear teeth,which is generally called wear.Pitting.as explained in Sec.6-16,is a surface fatigue failure due to many repetitions of high contact stresses.Other surface failures are scoring.which is a lubri- cation failure,and abrasion,which is wear due to the presence of foreign material. To obtain an expression for the surface-contact stress,we shall employ the Hertz theory.In Eq.(3-74)it was shown that the contact stress between two cylinders may be computed from the equation 2F Pmax (a) πbl where Pmax =largest surface pressure F=force pressing the two cylinders together I=length of cylinders and half-width b is obtained from Eq.(3-73): b= 2F[-/E]+[1-/E]p (14-10) πl (1/d)+(1/d2) where vi,v2,E1,and E2 are the elastic constants and di and d2 are the diameters, respectively,of the two contacting cylinders. To adapt these relations to the notation used in gearing,we replace F by W/cos, d by 2r,and I by the face width F.With these changes,we can substitute the value of b T.J.Dolan and E.I.Broghamer,A Photoelastic Study of the Stresses in Gear Tooth Fillets,Bulletin 335 Univ.Ill.Exp.Sta.,March 1942,See also W.D.Pilkey,Peterson's Stress Concentration Factors,2nd ed. John Wiley Sons,New York,1997.pp.383-385,412-415. 4R.G.Mitchiner and H.H.Mabie."Determination of the Lewis Form Factor and the AGMA Geometry Factor J of Extemal Spur Gear Teeth,"J.Mech.Des.Vol.104,No.1,Jan.1982.pp.148-158

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears © The McGraw−Hill 721 Companies, 2008 Spur and Helical Gears 723 3 T. J. Dolan and E. I. Broghamer, A Photoelastic Study of the Stresses in Gear Tooth Fillets, Bulletin 335, Univ. Ill. Exp. Sta., March 1942, See also W. D. Pilkey, Peterson’s Stress Concentration Factors, 2nd ed., John Wiley & Sons, New York, 1997, pp. 383–385, 412–415. 4 R. G. Mitchiner and H. H. Mabie, “Determination of the Lewis Form Factor and the AGMA Geometry Factor J of External Spur Gear Teeth,” J. Mech. Des., Vol. 104, No. 1, Jan. 1982, pp. 148–158. In Ex. 14–2 our resources (Fig. A–15–6) did not directly address stress concentration in gear teeth. A photoelastic investigation by Dolan and Broghamer reported in 1942 constitutes a primary source of information on stress concentration.3 Mitchiner and Mabie4 interpret the results in term of fatigue stress-concentration factor Kf as Kf = H + t r L t l M (14–9) where H = 0.34 − 0.458 366 2φ L = 0.316 − 0.458 366 2φ M = 0.290 + 0.458 366 2φ r = (b − rf )2 (d/2) + b − rf In these equations l and t are from the layout in Fig. 14–1, φ is the pressure angle, rf is the fillet radius, b is the dedendum, and d is the pitch diameter. It is left as an exercise for the reader to compare Kf from Eq. (14–9) with the results of using the approximation of Fig. A–15–6 in Ex. 14–2. 14–2 Surface Durability In this section we are interested in the failure of the surfaces of gear teeth, which is generally called wear. Pitting, as explained in Sec. 6–16, is a surface fatigue failure due to many repetitions of high contact stresses. Other surface failures are scoring, which is a lubrication failure, and abrasion, which is wear due to the presence of foreign material. To obtain an expression for the surface-contact stress, we shall employ the Hertz theory. In Eq. (3–74) it was shown that the contact stress between two cylinders may be computed from the equation pmax = 2F πbl (a) where pmax = largest surface pressure F = force pressing the two cylinders together l = length of cylinders and half-width b is obtained from Eq. (3–73): b = 2F πl 1 − ν2 1 E1 + 1 − ν2 2 E2 (1/d1) + (1/d2) 1/2 (14–10) where ν1, ν2, E1, and E2 are the elastic constants and d1 and d2 are the diameters, respectively, of the two contacting cylinders. To adapt these relations to the notation used in gearing, we replace F by Wt /cos φ, d by 2r, and l by the face width F. With these changes, we can substitute the value of b

722 Budynas-Nisbett:Shigley's IIL Design of Mechanical 14.Spur and Helical Gears T©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 724 Mechanical Engineering Design as given by Eq.(14-10)in Eq.(a).Replacing Pmax by oc,the surface compressive stress(Hertzian stress)is found from the equation (1/r)+(1/r2) 2= (14-11) -πFcos中[1-)/E1]+[(1-)/E2] where r and r2 are the instantaneous values of the radii of curvature on the pinion-and gear-tooth profiles,respectively,at the point of contact.By accounting for load sharing in the value of W used,Eg.(14-11)can be solved for the Hertzian stress for any or all points from the beginning to the end of tooth contact.Of course,pure rolling exists only at the pitch point.Elsewhere the motion is a mixture of rolling and sliding. Equation(14-11)does not account for any sliding action in the evaluation of stress.We note that AGMA uses u for Poisson's ratio instead of v as is used here. We have already noted that the first evidence of wear occurs near the pitch line.The radii of curvature of the tooth profiles at the pitch point are r=desing 2 r2=da sind (14-12) 2 where o is the pressure angle and dp and dc are the pitch diameters of the pinion and gear,respectively. Note,in Eq.(14-11),that the denominator of the second group of terms contains four elastic constants,two for the pinion and two for the gear.As a simple means of com- bining and tabulating the results for various combinations of pinion and gear materials, AGMA defines an elastic coefficient Cp by the equation 1/2 (14-13) EP EG With this simplification,and the addition of a velocity factor K,Eq.(14-11)can be written as c-c[(+】 (14-14) where the sign is negative because oc is a compressive stress. EXAMPLE 14-3 The pinion of Examples 14-1 and 14-2 is to be mated with a 50-tooth gear manufac- tured of ASTM No.50 cast iron.Using the tangential load of 382 Ibf,estimate the factor of safety of the drive based on the possibility of a surface fatigue failure. Solution From Table A-5 we find the elastic constants to be Ep =30 Mpsi,vp =0.292,EG 14.5 Mpsi,v =0.211.We substitute these in Eq.(14-13)to get the elastic coefficient as 1/2 1 C =1817 「1-(0.292)2,1-(0.211)2 30(106) 14.5(10)

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears 722 © The McGraw−Hill Companies, 2008 724 Mechanical Engineering Design as given by Eq. (14–10) in Eq. (a). Replacing pmax by σC , the surface compressive stress (Hertzian stress) is found from the equation σ2 C = Wt π F cos φ (1/r1) + (1/r2) 1 − ν2 1 E1 + 1 − ν2 2 E2 (14–11) where r1 and r2 are the instantaneous values of the radii of curvature on the pinion- and gear-tooth profiles, respectively, at the point of contact. By accounting for load sharing in the value of Wt used, Eq. (14–11) can be solved for the Hertzian stress for any or all points from the beginning to the end of tooth contact. Of course, pure rolling exists only at the pitch point. Elsewhere the motion is a mixture of rolling and sliding. Equation (14–11) does not account for any sliding action in the evaluation of stress. We note that AGMA uses μ for Poisson’s ratio instead of ν as is used here. We have already noted that the first evidence of wear occurs near the pitch line. The radii of curvature of the tooth profiles at the pitch point are r1 = dP sin φ 2 r2 = dG sin φ 2 (14–12) where φ is the pressure angle and dP and dG are the pitch diameters of the pinion and gear, respectively. Note, in Eq. (14–11), that the denominator of the second group of terms contains four elastic constants, two for the pinion and two for the gear. As a simple means of combining and tabulating the results for various combinations of pinion and gear materials, AGMA defines an elastic coefficient Cp by the equation Cp = ⎡ ⎢ ⎢ ⎢ ⎣ 1 π 1 − ν2 P EP + 1 − ν2 G EG ⎤ ⎥ ⎥ ⎥ ⎦ 1/2 (14–13) With this simplification, and the addition of a velocity factor Kv , Eq. (14–11) can be written as σC = −Cp KvWt F cos φ 1 r1 + 1 r2 1/2 (14–14) where the sign is negative because σC is a compressive stress. EXAMPLE 14–3 The pinion of Examples 14–1 and 14–2 is to be mated with a 50-tooth gear manufactured of ASTM No. 50 cast iron. Using the tangential load of 382 lbf, estimate the factor of safety of the drive based on the possibility of a surface fatigue failure. Solution From Table A–5 we find the elastic constants to be EP = 30 Mpsi, νP = 0.292, EG = 14.5 Mpsi, νG = 0.211. We substitute these in Eq. (14–13) to get the elastic coefficient as Cp = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ 1 π 1 − (0.292)2 30(106) + 1 − (0.211)2 14.5(106) ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ 1/2 = 1817

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 14.Spur and Helical Gears I©The McGraw-Hil 728 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Spur and Helical Gears 725 From Example 14-1,the pinion pitch diameter is de =2 in.The value for the gear is dg =50/8=6.25 in.Then Eg.(14-12)is used to obtain the radii of curvature at the pitch points.Thus 2sin20=0.342inn= 6.25sin20° i= 2 2 =1.069in The face width is given as F=1.5 in.Use K=1.52 from Example 14-1.Substituting all these values in Eq.(14-14)with=20 gives the contact stress as 1 1 0c=-1817 1.52(380) （(0.342+1.069月 =-72400psi 1.5 cos 200 The surface endurance strength of cast iron can be estimated from Sc =0.32Hg kpsi for 108 cycles,where Sc is in kpsi.Table A-24 gives Hg =262 for ASTM No.50 cast iron.Therefore Sc 0.32(262)=83.8 kpsi.Contact stress is not linear with transmit- ted load [see Eq.(14-14)].If the factor of safety is defined as the loss-of-function load divided by the imposed load,then the ratio of loads is the ratio of stresses squared.In other words. nloss-of-function load 83.8 2 =1.34 imposed load 0 72.4 One is free to define factor of safety as Sc/oc.Awkwardness comes when one com- pares the factor of safety in bending fatigue with the factor of safety in surface fatigue for a particular gear.Suppose the factor of safety of this gear in bending fatigue is 1.20 and the factor of safety in surface fatigue is 1.34 as above.The threat,since 1.34 is greater than 1.20,is in bending fatigue since both numbers are based on load ratios.If the factor of safety in surface fatigue is based on Sc/oc =v1.34=1.16,then 1.20 is greater than 1.16,but the threat is not from surface fatigue.The surface fatigue factor of safety can be defined either way.One way has the burden of requiring a squared number before numbers that instinctively seem comparable can be compared. In addition to the dynamic factor K,already introduced,there are transmitted load excursions,nonuniform distribution of the transmitted load over the tooth contact,and the influence of rim thickness on bending stress.Tabulated strength values can be means, ASTM minimums,or of unknown heritage.In surface fatigue there are no endurance lim- its.Endurance strengths have to be qualified as to corresponding cycle count,and the slope of the S-N curve needs to be known.In bending fatigue there is a definite change in slope of the S-N curve near 10 cycles,but some evidence indicates that an endurance limit does not exist.Gearing experience leads to cycle counts of 101 or more.Evidence of dimin- ishing endurance strengths in bending have been included in AGMA methodology. 14-3 AGMA Stress Equations Two fundamental stress equations are used in the AGMA methodology,one for bend- ing stress and another for pitting resistance(contact stress).In AGMA terminology, these are called stress numbers,as contrasted with actual applied stresses,and are

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears © The McGraw−Hill 723 Companies, 2008 Spur and Helical Gears 725 From Example 14–1, the pinion pitch diameter is dP = 2 in. The value for the gear is dG = 50/8 = 6.25 in. Then Eq. (14–12) is used to obtain the radii of curvature at the pitch points. Thus r1 = 2 sin 20◦ 2 = 0.342 in r2 = 6.25 sin 20◦ 2 = 1.069 in The face width is given as F = 1.5 in. Use Kv = 1.52 from Example 14–1. Substituting all these values in Eq. (14–14) with φ = 20◦ gives the contact stress as σC = −1817 1.52(380) 1.5 cos 20◦ 1 0.342 + 1 1.0691/2 = −72 400 psi The surface endurance strength of cast iron can be estimated from SC = 0.32HB kpsi for 108 cycles, where SC is in kpsi. Table A–24 gives HB = 262 for ASTM No. 50 cast iron. Therefore SC = 0.32(262) = 83.8 kpsi. Contact stress is not linear with transmitted load [see Eq. (14–14)]. If the factor of safety is defined as the loss-of-function load divided by the imposed load, then the ratio of loads is the ratio of stresses squared. In other words, n = loss-of-function load imposed load = S2 C σ2 C = 83.8 72.4 2 = 1.34 One is free to define factor of safety as SC/σC . Awkwardness comes when one compares the factor of safety in bending fatigue with the factor of safety in surface fatigue for a particular gear. Suppose the factor of safety of this gear in bending fatigue is 1.20 and the factor of safety in surface fatigue is 1.34 as above. The threat, since 1.34 is greater than 1.20, is in bending fatigue since both numbers are based on load ratios. If the factor of safety in surface fatigue is based on SC/σC = √1.34 = 1.16, then 1.20 is greater than 1.16, but the threat is not from surface fatigue. The surface fatigue factor of safety can be defined either way. One way has the burden of requiring a squared number before numbers that instinctively seem comparable can be compared. In addition to the dynamic factor Kv already introduced, there are transmitted load excursions, nonuniform distribution of the transmitted load over the tooth contact, and the influence of rim thickness on bending stress. Tabulated strength values can be means, ASTM minimums, or of unknown heritage. In surface fatigue there are no endurance limits. Endurance strengths have to be qualified as to corresponding cycle count, and the slope of the S-N curve needs to be known. In bending fatigue there is a definite change in slope of the S-N curve near 106 cycles, but some evidence indicates that an endurance limit does not exist. Gearing experience leads to cycle counts of 1011 or more. Evidence of diminishing endurance strengths in bending have been included in AGMA methodology. 14–3 AGMA Stress Equations Two fundamental stress equations are used in the AGMA methodology, one for bending stress and another for pitting resistance (contact stress). In AGMA terminology, these are called stress numbers, as contrasted with actual applied stresses, and are

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears 724 © The McGraw−Hill Companies, 2008 726 Mechanical Engineering Design designated by a lowercase letter s instead of the Greek lower case σ we have used in this book (and shall continue to use). The fundamental equations are σ = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ Wt KoKvKs Pd F Km KB J (U.S. customary units) Wt KoKvKs 1 bmt K H KB YJ (SI units) (14–15) where for U.S. customary units (SI units), Wt is the tangential transmitted load, lbf (N) Ko is the overload factor Kv is the dynamic factor Ks is the size factor Pd is the transverse diameteral pitch F (b) is the face width of the narrower member, in (mm) Km (KH) is the load-distribution factor KB is the rim-thickness factor J (YJ ) is the geometry factor for bending strength (which includes root fillet stress-concentration factor Kf ) (mt) is the transverse metric module Before you try to digest the meaning of all these terms in Eq. (14–15), view them as advice concerning items the designer should consider whether he or she follows the voluntary standard or not. These items include issues such as • Transmitted load magnitude • Overload • Dynamic augmentation of transmitted load • Size • Geometry: pitch and face width • Distribution of load across the teeth • Rim support of the tooth • Lewis form factor and root fillet stress concentration The fundamental equation for pitting resistance (contact stress) is σc = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ Cp Wt KoKvKs Km dP F Cf I (U.S. customary units) Z E Wt KoKvKs K H dw1b Z R ZI (SI units) (14–16) where Wt , Ko, Kv , Ks, Km, F, and b are the same terms as defined for Eq. (14–15). For U.S. customary units (SI units), the additional terms are Cp (Z E ) is an elastic coefficient, √ lbf/in2 ( √ N/mm2) Cf (Z R) is the surface condition factor dP (dw1) is the pitch diameter of the pinion, in (mm) I (ZI) is the geometry factor for pitting resistance The evaluation of all these factors is explained in the sections that follow. The development of Eq. (14–16) is clarified in the second part of Sec. 14–5

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 14. Spur and Helical Gears © The McGraw−Hill 725 Companies, 2008 14–4 AGMA Strength Equations Instead of using the term strength, AGMA uses data termed allowable stress numbers and designates these by the symbols sat and sac. It will be less confusing here if we continue the practice in this book of using the uppercase letter S to designate strength and the lowercase Greek letters σ and τ for stress. To make it perfectly clear we shall use the term gear strength as a replacement for the phrase allowable stress numbers as used by AGMA. Following this convention, values for gear bending strength, designated here as St , are to be found in Figs. 14–2, 14–3, and 14–4, and in Tables 14–3 and 14–4. Since gear strengths are not identified with other strengths such as Sut , Se, or Sy as used elsewhere in this book, their use should be restricted to gear problems. In this approach the strengths are modified by various factors that produce limiting values of the bending stress and the contact stress. Spur and Helical Gears 727 Figure 14–2 Allowable bending stress number for through-hardened steels. The SI equations are St = 0.533HB + 88.3 MPa, grade 1, and St = 0.703HB + 113 MPa, grade 2. (Source: ANSI/AGMA 2001-D04 and 2101-D04.) Metallurgical and quality control procedure required 150 200 250 300 350 400 450 10 20 30 40 50 Brinell hardness, HB Allowable bending stress number, St kpsi Grade 1 St = 77.3 HB + 12 800 psi Grade 2 St = 102 HB + 16 400 psi Figure 14–3 Allowable bending stress number for nitrided throughhardened steel gears (i.e., AISI 4140, 4340), St . The SI equations are St = 0.568HB + 83.8 MPa, grade 1, and St = 0.749HB + 110 MPa, grade 2. (Source: ANSI/AGMA 2001-D04 and 2101-D04.) Metallurgical and quality control procedures required 250 275 300 325 350 20 30 40 50 60 70 80 Grade 1 St = 82.3HB + 12 150 psi Grade 2 St = 108.6HB + 15 890 psi Allowable bending stress number, St kpsi Core hardness, HB