662 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General T©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Gears-General 663 Figure 13-14 Internal gear and pinion. Pressure line Pitch circk Pitch circle Addendum circle Thus the pitch circles of gears really do not come into existence until a pair of gears are brought into mesh. Changing the center distance has no effect on the base circles,because these were used to generate the tooth profiles.Thus the base circle is basic to a gear.Increasing the center distance increases the pressure angle and decreases the length of the line of action,but the teeth are still conjugate,the requirement for uniform motion transmis- sion is still satisfied,and the angular-velocity ratio has not changed. EXAMPLE 13-1 A gearset consists of a 16-tooth pinion driving a 40-tooth gear.The diametral pitch is 2.and the addendum and dedendum are 1/P and 1.25/P,respectively.The gears are cut using a pressure angle of 20. (a)Compute the circular pitch,the center distance,and the radii of the base circles. (b)In mounting these gears,the center distance was incorrectly made in larger. Compute the new values of the pressure angle and the pitch-circle diameters. Solution ππ Answer (a) =2=1.57in p= The pitch diameters of the pinion and gear are,respectively, 16 40 d=2=8ind6=2=20in Therefore the center distance is Answer dp+dc=8+20 2 =14in 2 Since the teeth were cut on the 20 pressure angle,the base-circle radii are found to be, using ro=rcosφ， 8 Answer n(pinion）=2cos20°=3.76in 20 Answer ro (gear)= cos20°-9.40in
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General 662 © The McGraw−Hill Companies, 2008 Gears—General 663 Thus the pitch circles of gears really do not come into existence until a pair of gears are brought into mesh. Changing the center distance has no effect on the base circles, because these were used to generate the tooth profiles. Thus the base circle is basic to a gear. Increasing the center distance increases the pressure angle and decreases the length of the line of action, but the teeth are still conjugate, the requirement for uniform motion transmission is still satisfied, and the angular-velocity ratio has not changed. EXAMPLE 13–1 A gearset consists of a 16-tooth pinion driving a 40-tooth gear. The diametral pitch is 2, and the addendum and dedendum are 1/P and 1.25/P, respectively. The gears are cut using a pressure angle of 20◦. (a) Compute the circular pitch, the center distance, and the radii of the base circles. (b) In mounting these gears, the center distance was incorrectly made 1 4 in larger. Compute the new values of the pressure angle and the pitch-circle diameters. Solution Answer (a) p = π P = π 2 = 1.57 in The pitch diameters of the pinion and gear are, respectively, dP = 16 2 = 8 in dG = 40 2 = 20 in Therefore the center distance is Answer dP + dG 2 = 8 + 20 2 = 14 in Since the teeth were cut on the 20◦ pressure angle, the base-circle radii are found to be, using rb = r cos φ, Answer rb (pinion) = 8 2 cos 20◦ = 3.76 in Answer rb (gear) = 20 2 cos 20◦ = 9.40 in Pitch circle Base circle 2 Base circle Pitch circle Pressure line Dedendum circle Addendum circle 3 2 3 O2 Figure 13–14 Internal gear and pinion.
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General ©The McGraw-Hil 63 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 664 Mechanical Engineering Design (b)Designating de and d as the new pitch-circle diameters,the -in increase in the center distance requires that d,+d6=14.250 2 () Also,the velocity ratio does not change,and hence 2 Solving Eqs.(1)and(2)simultaneously yields Answer dp=8.143ind6=20.357in Since ro=rcos中，the new pressure angle is Answer =cos-1 (pinion) dp/2 COS-1_ .76 =22.56° .143/2 13-6 Contact Ratio The zone of action of meshing gear teeth is shown in Fig.13-15.We recall that tooth contact begins and ends at the intersections of the two addendum circles with the pres- sure line.In Fig.13-15 initial contact occurs at a and final contact at b.Tooth profiles drawn through these points intersect the pitch circle at A and B,respectively.As shown, the distance AP is called the arc of approach qa.and the distance PB,the arc of recess q.The sum of these is the arc of action q. Now,consider a situation in which the arc of action is exactly equal to the circular pitch,that is.q=p.This means that one tooth and its space will occupy the entire arc AB.In other words,when a tooth is just beginning contact at a,the previous tooth is simultaneously ending its contact at b.Therefore,during the tooth action from a to b, there will be exactly one pair of teeth in contact. Next,consider a situation in which the arc of action is greater than the circular pitch,but not very much greater,say,1.2p.This means that when one pair of teeth is just entering contact at a,another pair,already in contact,will not yet have reached b. Figure 13-15 Arc of Arc of Pressure line Definition of contact ratio. appro 中 Addendum circle Pitch circle Motion
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 663 Companies, 2008 664 Mechanical Engineering Design Lab Motion A a b B Addendum circle Pressure line Pitch circle Addendum circle Arc of approach qa Arc of recess qr P Figure 13–15 Definition of contact ratio. (b) Designating d P and d G as the new pitch-circle diameters, the 1 4 -in increase in the center distance requires that d P + d G 2 = 14.250 (1) Also, the velocity ratio does not change, and hence d P d G = 16 40 (2) Solving Eqs. (1) and (2) simultaneously yields Answer d P = 8.143 in d G = 20.357 in Since rb = r cos φ, the new pressure angle is Answer φ = cos−1 rb (pinion) d P/2 = cos−1 3.76 8.143/2 = 22.56◦ 13–6 Contact Ratio The zone of action of meshing gear teeth is shown in Fig. 13–15. We recall that tooth contact begins and ends at the intersections of the two addendum circles with the pressure line. In Fig. 13–15 initial contact occurs at a and final contact at b. Tooth profiles drawn through these points intersect the pitch circle at A and B, respectively. As shown, the distance AP is called the arc of approach qa , and the distance P B, the arc of recess qr. The sum of these is the arc of action qt . Now, consider a situation in which the arc of action is exactly equal to the circular pitch, that is, qt = p. This means that one tooth and its space will occupy the entire arc AB. In other words, when a tooth is just beginning contact at a, the previous tooth is simultaneously ending its contact at b. Therefore, during the tooth action from a to b, there will be exactly one pair of teeth in contact. Next, consider a situation in which the arc of action is greater than the circular pitch, but not very much greater, say, qt . = 1.2p. This means that when one pair of teeth is just entering contact at a, another pair, already in contact, will not yet have reached b.
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General T©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Gears-General I 665 Thus,for a short period of time,there will be two teeth in contact,one in the vicinity of A and another near B.As the meshing proceeds,the pair near B must cease contact, leaving only a single pair of contacting teeth,until the procedure repeats itself. Because of the nature of this tooth action,either one or two pairs of teeth in con- tact,it is convenient to define the term contact ratio me as me = (13-8) D a number that indicates the average number of pairs of teeth in contact.Note that this ratio is also equal to the length of the path of contact divided by the base pitch.Gears should not generally be designed having contact ratios less than about 1.20,because inaccuracies in mounting might reduce the contact ratio even more,increasing the pos- sibility of impact between the teeth as well as an increase in the noise level. An easier way to obtain the contact ratio is to measure the line of action ab instead of the arc distance AB.Since ab in Fig.13-15 is tangent to the base circle when extended,the base pitch p must be used to calculate me instead of the circular pitch as in Eq.(13-8).If the length of the line of action is Lab,the contact ratio is Lab mc= (13-91 pcos中 in which Eq.(13-7)was used for the base pitch. 13-7 Interference The contact of portions of tooth profiles that are not conjugate is called interference. Consider Fig.13-16.Illustrated are two 16-tooth gears that have been cut to the now obsolete 14 pressure angle.The driver,gear 2,turns clockwise.The initial and final points of contact are designated A and B,respectively,and are located on the pressure line.Now notice that the points of tangency of the pressure line with the base circles C and D are located inside of points A and B.Interference is present. The interference is explained as follows.Contact begins when the tip of the driven tooth contacts the flank of the driving tooth.In this case the flank of the driving tooth first makes contact with the driven tooth at point A,and this occurs before the involute portion of the driving tooth comes within range.In other words,contact is occurring below the base circle of gear 2 on the nonimolute portion of the flank.The actual effect is that the involute tip or face of the driven gear tends to dig out the noninvolute flank of the driver. In this example the same effect occurs again as the teeth leave contact.Contact should end at point D or before.Since it does not end until point B.the effect is for the tip of the driving tooth to dig out.or interfere with,the flank of the driven tooth. When gear teeth are produced by a generation process,interference is automati- cally eliminated because the cutting tool removes the interfering portion of the flank. This effect is called undercutting:if undercutting is at all pronounced,the undercut tooth is considerably weakened.Thus the effect of eliminating interference by a gener- ation process is merely to substitute another problem for the original one. The smallest number of teeth on a spur pinion and gear,one-to-one gear ratio,which can exist without interference is Np.This number of teeth for spur gears is Robert Lipp."Avoiding Tooth Interference in Gears."Machine Design,Vol.54.No.1.1982.pp.122-124
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General 664 © The McGraw−Hill Companies, 2008 Gears—General 665 Thus, for a short period of time, there will be two teeth in contact, one in the vicinity of A and another near B. As the meshing proceeds, the pair near B must cease contact, leaving only a single pair of contacting teeth, until the procedure repeats itself. Because of the nature of this tooth action, either one or two pairs of teeth in contact, it is convenient to define the term contact ratio mc as mc = qt p (13–8) a number that indicates the average number of pairs of teeth in contact. Note that this ratio is also equal to the length of the path of contact divided by the base pitch. Gears should not generally be designed having contact ratios less than about 1.20, because inaccuracies in mounting might reduce the contact ratio even more, increasing the possibility of impact between the teeth as well as an increase in the noise level. An easier way to obtain the contact ratio is to measure the line of action ab instead of the arc distance AB. Since ab in Fig. 13–15 is tangent to the base circle when extended, the base pitch pb must be used to calculate mc instead of the circular pitch as in Eq. (13–8). If the length of the line of action is Lab, the contact ratio is mc = Lab p cos φ (13–9) in which Eq. (13–7) was used for the base pitch. 13–7 Interference The contact of portions of tooth profiles that are not conjugate is called interference. Consider Fig. 13–16. Illustrated are two 16-tooth gears that have been cut to the now obsolete 141 2 ◦ pressure angle. The driver, gear 2, turns clockwise. The initial and final points of contact are designated A and B, respectively, and are located on the pressure line. Now notice that the points of tangency of the pressure line with the base circles C and D are located inside of points A and B. Interference is present. The interference is explained as follows. Contact begins when the tip of the driven tooth contacts the flank of the driving tooth. In this case the flank of the driving tooth first makes contact with the driven tooth at point A, and this occurs before the involute portion of the driving tooth comes within range. In other words, contact is occurring below the base circle of gear 2 on the noninvolute portion of the flank. The actual effect is that the involute tip or face of the driven gear tends to dig out the noninvolute flank of the driver. In this example the same effect occurs again as the teeth leave contact. Contact should end at point D or before. Since it does not end until point B, the effect is for the tip of the driving tooth to dig out, or interfere with, the flank of the driven tooth. When gear teeth are produced by a generation process, interference is automatically eliminated because the cutting tool removes the interfering portion of the flank. This effect is called undercutting; if undercutting is at all pronounced, the undercut tooth is considerably weakened. Thus the effect of eliminating interference by a generation process is merely to substitute another problem for the original one. The smallest number of teeth on a spur pinion and gear,1 one-to-one gear ratio, which can exist without interference is NP . This number of teeth for spur gears is 1 Robert Lipp, “Avoiding Tooth Interference in Gears,” Machine Design, Vol. 54, No. 1, 1982, pp. 122–124
Budynas-Nisbett:Shigley's I Design of Mechanical 13.Gears-General I©The McGraw-Hil 65 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 666 Mechanical Engineering Design Figure 13-16 Interference in the action of gear teeth. Driven gear 3 Base circle This portion of profile Is not an involute Addendum Pressure li circles Interference is on flank of driver during approach Base circle This portion of profile is not an involute Driving gear 2 given by 2k Np=3sin2φ 1+V1+3sin2Φ (13-101 where k=1 for full-depth teeth,0.8 for stub teeth and=pressure angle. For a 20 pressure angle,with k =1, Np= 3sim20(1+V1+3sin220）=12.3=13teth 2(1) Thus 13 teeth on pinion and gear are interference-free.Realize that 12.3 teeth is possi- ble in meshing arcs.but for fully rotating gears,13 teeth represents the least number. For a 14 pressure angle,Ne=23 teeth,so one can appreciate why few 14-tooth systems are used,as the higher pressure angles can produce a smaller pinion with accompanying smaller center-to-center distances. If the mating gear has more teeth than the pinion,that is,mG =NG/Np=m is more than one,then the smallest number of teeth on the pinion without interference is given by 2k Np= (1+2m)sin2φ m+Vm2+(1+2m)sin2φ (13-11)
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 665 Companies, 2008 666 Mechanical Engineering Design given by NP = 2k 3 sin2 φ 1 + 1 + 3 sin2 φ (13–10) where k = 1 for full-depth teeth, 0.8 for stub teeth and φ = pressure angle. For a 20◦ pressure angle, with k = 1, NP = 2(1) 3 sin2 20◦ 1 + 1 + 3 sin2 20◦ = 12.3 = 13 teeth Thus 13 teeth on pinion and gear are interference-free. Realize that 12.3 teeth is possible in meshing arcs, but for fully rotating gears, 13 teeth represents the least number. For a 141 2 ◦ pressure angle, NP = 23 teeth, so one can appreciate why few 141 2 ◦ -tooth systems are used, as the higher pressure angles can produce a smaller pinion with accompanying smaller center-to-center distances. If the mating gear has more teeth than the pinion, that is, mG = NG/NP = m is more than one, then the smallest number of teeth on the pinion without interference is given by NP = 2k (1 + 2m)sin2 φ m + m2 + (1 + 2m)sin2 φ (13–11) Driving gear 2 Driven gear 3 Base circle Base circle O2 O3 2 3 Interference is on flank of driver during approach This portion of profile is not an involute This portion of profile is not an involute Addendum circles Pressure line A C D B Figure 13–16 Interference in the action of gear teeth.
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General T©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Gears-General 667 For example,ifm=4,中=20°， 2(1) Np=1+2(41sim220 4+√42+[1+2(4]sim220° =15.4=16 teeth Thus a 16-tooth pinion will mesh with a 64-tooth gear without interference. The largest gear with a specified pinion that is interference-free is N3sin2中-4k2 NG= (13-12) 4k-2 Ve sin2Φ For example,for a 13-tooth pinion with a pressure angleφof2O°， 132sin220°-4(1)2 Ng= 4(1)）-2(13)sin220° ,=16.45=16 teeth For a 13-tooth spur pinion,the maximum number of gear teeth possible without inter- ference is 16. The smallest spur pinion that will operate with a rack without interference is 2(k) Np = sin (13-13) For a 20 pressure angle full-depth tooth the smallest number of pinion teeth to mesh with a rack is 2(1) Np= =17.1=18 teeth sin2 200 Since gear-shaping tools amount to contact with a rack,and the gear-hobbing process is similar,the minimum number of teeth to prevent interference to prevent under- cutting by the hobbing process is equal to the value of Np when No is infinite. The importance of the problem of teeth that have been weakened by undercutting cannot be overemphasized.Of course,interference can be eliminated by using more teeth on the pinion.However,if the pinion is to transmit a given amount of power,more teeth can be used only by increasing the pitch diameter. Interference can also be reduced by using a larger pressure angle.This results in a smaller base circle,so that more of the tooth profile becomes involute.The demand for smaller pinions with fewer teeth thus favors the use of a 25 pressure angle even though the frictional forces and bearing loads are increased and the contact ratio decreased. 13-8 The Forming of Gear Teeth There are a large number of ways of forming the teeth of gears,such as sand casting. shell molding.investment casting,permanent-mold casting,die casting,and centrifugal casting.Teeth can also be formed by using the powder-metallurgy process;or,by using extrusion,a single bar of aluminum may be formed and then sliced into gears.Gears that carry large loads in comparison with their size are usually made of steel and are cut with either form cutters or generating cutters.In form cutting,the tooth space takes the exact form of the cutter.In generating,a tool having a shape different from the tooth profile is moved relative to the gear blank so as to obtain the proper tooth shape.One of the newest and most promising of the methods of forming teeth is called cold form- ing,or cold rolling,in which dies are rolled against steel blanks to form the teeth.The mechanical properties of the metal are greatly improved by the rolling process,and a high-quality generated profile is obtained at the same time
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General 666 © The McGraw−Hill Companies, 2008 Gears—General 667 For example, if m = 4, φ = 20◦, NP = 2(1) [1 + 2(4)] sin2 20◦ 4 + 42 + [1 + 2(4)] sin2 20◦ = 15.4 = 16 teeth Thus a 16-tooth pinion will mesh with a 64-tooth gear without interference. The largest gear with a specified pinion that is interference-free is NG = N2 P sin2 φ − 4k2 4k − 2NP sin2 φ (13–12) For example, for a 13-tooth pinion with a pressure angle φ of 20◦, NG = 132 sin2 20◦ − 4(1)2 4(1) − 2(13)sin2 20◦ = 16.45 = 16 teeth For a 13-tooth spur pinion, the maximum number of gear teeth possible without interference is 16. The smallest spur pinion that will operate with a rack without interference is NP = 2(k) sin2 φ (13–13) For a 20◦ pressure angle full-depth tooth the smallest number of pinion teeth to mesh with a rack is NP = 2(1) sin2 20◦ = 17.1 = 18 teeth Since gear-shaping tools amount to contact with a rack, and the gear-hobbing process is similar, the minimum number of teeth to prevent interference to prevent undercutting by the hobbing process is equal to the value of NP when NG is infinite. The importance of the problem of teeth that have been weakened by undercutting cannot be overemphasized. Of course, interference can be eliminated by using more teeth on the pinion. However, if the pinion is to transmit a given amount of power, more teeth can be used only by increasing the pitch diameter. Interference can also be reduced by using a larger pressure angle. This results in a smaller base circle, so that more of the tooth profile becomes involute. The demand for smaller pinions with fewer teeth thus favors the use of a 25◦ pressure angle even though the frictional forces and bearing loads are increased and the contact ratio decreased. 13–8 The Forming of Gear Teeth There are a large number of ways of forming the teeth of gears, such as sand casting, shell molding, investment casting, permanent-mold casting, die casting, and centrifugal casting. Teeth can also be formed by using the powder-metallurgy process; or, by using extrusion, a single bar of aluminum may be formed and then sliced into gears. Gears that carry large loads in comparison with their size are usually made of steel and are cut with either form cutters or generating cutters. In form cutting, the tooth space takes the exact form of the cutter. In generating, a tool having a shape different from the tooth profile is moved relative to the gear blank so as to obtain the proper tooth shape. One of the newest and most promising of the methods of forming teeth is called cold forming, or cold rolling, in which dies are rolled against steel blanks to form the teeth. The mechanical properties of the metal are greatly improved by the rolling process, and a high-quality generated profile is obtained at the same time