Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General T©The McGraw-Hill G67 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 668 Mechanical Engineering Design Gear teeth may be machined by milling,shaping,or hobbing.They may be finished by shaving,burnishing,grinding,or lapping. Gears made of thermoplastics such as nylon,polycarbonate,acetal are quite popular and are easily manufactured by injection molding.These gears are of low to moderate precision,low in cost for high production quantities,and capable of light loads,and can run without lubrication. Milling Gear teeth may be cut with a form milling cutter shaped to conform to the tooth space. With this method it is theoretically necessary to use a different cutter for each gear, because a gear having 25 teeth,for example,will have a different-shaped tooth space from one having,say,24 teeth.Actually,the change in space is not too great,and it has been found that eight cutters may be used to cut with reasonable accuracy any gear in the range of 12 teeth to a rack.A separate set of cutters is,of course,required for each pitch. Shaping Teeth may be generated with either a pinion cutter or a rack cutter.The pinion cutter (Fig.13-17)reciprocates along the vertical axis and is slowly fed into the gear blank to the required depth.When the pitch circles are tangent,both the cutter and the blank rotate slightly after each cutting stroke.Since each tooth of the cutter is a cutting tool, the teeth are all cut after the blank has completed one rotation.The sides of an involute rack tooth are straight.For this reason,a rack-generating tool provides an accurate method of cutting gear teeth.This is also a shaping operation and is illustrated by the drawing of Fig.13-18.In operation,the cutter reciprocates and is first fed into the gear blank until the pitch circles are tangent.Then,after each cutting stroke,the gear blank Figure 13-17 Generating a spur gear with a pinion cutter.(Courtesy of Boston Gear Works,Inc.)
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 667 Companies, 2008 668 Mechanical Engineering Design Figure 13–17 Generating a spur gear with a pinion cutter. (Courtesy of Boston Gear Works, Inc.) Gear teeth may be machined by milling, shaping, or hobbing. They may be finished by shaving, burnishing, grinding, or lapping. Gears made of thermoplastics such as nylon, polycarbonate, acetal are quite popular and are easily manufactured by injection molding. These gears are of low to moderate precision, low in cost for high production quantities, and capable of light loads, and can run without lubrication. Milling Gear teeth may be cut with a form milling cutter shaped to conform to the tooth space. With this method it is theoretically necessary to use a different cutter for each gear, because a gear having 25 teeth, for example, will have a different-shaped tooth space from one having, say, 24 teeth. Actually, the change in space is not too great, and it has been found that eight cutters may be used to cut with reasonable accuracy any gear in the range of 12 teeth to a rack. A separate set of cutters is, of course, required for each pitch. Shaping Teeth may be generated with either a pinion cutter or a rack cutter. The pinion cutter (Fig. 13–17) reciprocates along the vertical axis and is slowly fed into the gear blank to the required depth. When the pitch circles are tangent, both the cutter and the blank rotate slightly after each cutting stroke. Since each tooth of the cutter is a cutting tool, the teeth are all cut after the blank has completed one rotation. The sides of an involute rack tooth are straight. For this reason, a rack-generating tool provides an accurate method of cutting gear teeth. This is also a shaping operation and is illustrated by the drawing of Fig. 13–18. In operation, the cutter reciprocates and is first fed into the gear blank until the pitch circles are tangent. Then, after each cutting stroke, the gear blank
668 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 669 Figure 13-18 ★ Shaping teeth with a rack. [This is a drawing-board figure that J.E.Shigley executed Gear blank rotates over 35 years ago in in this direction平 response to a question from a student at the University of Michigan.] perpendicular to this page Figure 13-19 Hobbing a worm gear. [Courtesy of Boston Gear Works,Inc.] and cutter roll slightly on their pitch circles.When the blank and cutter have rolled a distance equal to the circular pitch,the cutter is returned to the starting point,and the process is continued until all the teeth have been cut. Hobbing The hobbing process is illustrated in Fig.13-19.The hob is simply a cutting tool that is shaped like a worm.The teeth have straight sides,as in a rack,but the hob axis must be turned through the lead angle in order to cut spur-gear teeth.For this reason,the teeth generated by a hob have a slightly different shape from those generated by a rack cutter. Both the hob and the blank must be rotated at the proper angular-velocity ratio.The hob is then fed slowly across the face of the blank until all the teeth have been cut
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General 668 © The McGraw−Hill Companies, 2008 Gears—General 669 Figure 13–19 Hobbing a worm gear. (Courtesy of Boston Gear Works, Inc.) and cutter roll slightly on their pitch circles. When the blank and cutter have rolled a distance equal to the circular pitch, the cutter is returned to the starting point, and the process is continued until all the teeth have been cut. Hobbing The hobbing process is illustrated in Fig. 13–19. The hob is simply a cutting tool that is shaped like a worm. The teeth have straight sides, as in a rack, but the hob axis must be turned through the lead angle in order to cut spur-gear teeth. For this reason, the teeth generated by a hob have a slightly different shape from those generated by a rack cutter. Both the hob and the blank must be rotated at the proper angular-velocity ratio. The hob is then fed slowly across the face of the blank until all the teeth have been cut. Gear blank rotates in this direction Rack cutter reciprocates in a direction perpendicular to this page Figure 13–18 Shaping teeth with a rack. (This is a drawing-board figure that J. E. Shigley executed over 35 years ago in response to a question from a student at the University of Michigan.)
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General T©The McGraw-Hill 669 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 670 Mechanical Engineering Design Finishing Gears that run at high speeds and transmit large forces may be subjected to additional dynamic forces if there are errors in tooth profiles.Errors may be diminished somewhat by finishing the tooth profiles.The teeth may be finished,after cutting,by either shav- ing or burnishing.Several shaving machines are available that cut off a minute amount of metal,bringing the accuracy of the tooth profile within the limits of 250 uin. Burnishing,like shaving,is used with gears that have been cut but not heat-treated. In burnishing,hardened gears with slightly oversize teeth are run in mesh with the gear until the surfaces become smooth. Grinding and lapping are used for hardened gear teeth after heat treatment.The grinding operation employs the generating principle and produces very accurate teeth. In lapping,the teeth of the gear and lap slide axially so that the whole surface of the teeth is abraded equally. 13-9 Straight Bevel Gears When gears are used to transmit motion between intersecting shafts,some form of bevel gear is required.A bevel gearset is shown in Fig.13-20.Although bevel gears are usu- ally made for a shaft angle of 90,they may be produced for almost any angle.The teeth may be cast,milled,or generated.Only the generated teeth may be classed as accurate. The terminology of bevel gears is illustrated in Fig.13-20.The pitch of bevel gears is measured at the large end of the tooth,and both the circular pitch and the pitch diam- eter are calculated in the same manner as for spur gears.It should be noted that the clear- ance is uniform.The pitch angles are defined by the pitch cones meeting at the apex,as shown in the figure.They are related to the tooth numbers as follows: Np tany NG tanr= NG Np (13-14 Figure 13-20 Pitch angle Terminology of bevel gears. Cone distance A. Face Uniform Pitch angle clearance -Pitch diameter D Back-cone Back radius.r con
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 669 Companies, 2008 670 Mechanical Engineering Design Back-cone radius, r b F Cone distance Ao Face Pitch angle Uniform clearance Pitch diameter DG Back cone Γ Pitch angle Figure 13–20 Terminology of bevel gears. Finishing Gears that run at high speeds and transmit large forces may be subjected to additional dynamic forces if there are errors in tooth profiles. Errors may be diminished somewhat by finishing the tooth profiles. The teeth may be finished, after cutting, by either shaving or burnishing. Several shaving machines are available that cut off a minute amount of metal, bringing the accuracy of the tooth profile within the limits of 250 μin. Burnishing, like shaving, is used with gears that have been cut but not heat-treated. In burnishing, hardened gears with slightly oversize teeth are run in mesh with the gear until the surfaces become smooth. Grinding and lapping are used for hardened gear teeth after heat treatment. The grinding operation employs the generating principle and produces very accurate teeth. In lapping, the teeth of the gear and lap slide axially so that the whole surface of the teeth is abraded equally. 13–9 Straight Bevel Gears When gears are used to transmit motion between intersecting shafts, some form of bevel gear is required. A bevel gearset is shown in Fig. 13–20. Although bevel gears are usually made for a shaft angle of 90◦, they may be produced for almost any angle. The teeth may be cast, milled, or generated. Only the generated teeth may be classed as accurate. The terminology of bevel gears is illustrated in Fig. 13–20. The pitch of bevel gears is measured at the large end of the tooth, and both the circular pitch and the pitch diameter are calculated in the same manner as for spur gears. It should be noted that the clearance is uniform. The pitch angles are defined by the pitch cones meeting at the apex, as shown in the figure. They are related to the tooth numbers as follows: tan γ = NP NG tan = NG NP (13–14)
670 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 671 where the subscripts P and G refer to the pinion and gear,respectively,and where y and I are,respectively,the pitch angles of the pinion and gear. Figure 13-20 shows that the shape of the teeth,when projected on the back cone, is the same as in a spur gear having a radius equal to the back-cone distance r.This is called Tredgold's approximation.The number of teeth in this imaginary gear is W'=2% (13-15) where N'is the virtual number of teeth and p is the circular pitch measured at the large end of the teeth.Standard straight-tooth bevel gears are cut by using a 20 pressure angle,unequal addenda and dedenda,and full-depth teeth.This increases the contact ratio,avoids undercut,and increases the strength of the pinion. 13-10 Parallel Helical Gears Helical gears,used to transmit motion between parallel shafts,are shown in Fig.13-2. The helix angle is the same on each gear,but one gear must have a right-hand helix and the other a left-hand helix.The shape of the tooth is an involute helicoid and is illus- trated in Fig.13-21.If a piece of paper cut in the shape of a parallelogram is wrapped around a cylinder,the angular edge of the paper becomes a helix.If we unwind this paper,each point on the angular edge generates an involute curve.This surface obtained when every point on the edge generates an involute is called an involute helicoid. The initial contact of spur-gear teeth is a line extending all the way across the face of the tooth.The initial contact of helical-gear teeth is a point that extends into a line as the teeth come into more engagement.In spur gears the line of contact is parallel to the axis of rotation;in helical gears the line is diagonal across the face of the tooth.It is this gradual engagement of the teeth and the smooth transfer of load from one tooth to another that gives helical gears the ability to transmit heavy loads at high speeds. Because of the nature of contact between helical gears,the contact ratio is of only minor importance,and it is the contact area,which is proportional to the face width of the gear,that becomes significant. Helical gears subject the shaft bearings to both radial and thrust loads.When the thrust loads become high or are objectionable for other reasons,it may be desirable to use double helical gears.A double helical gear(herringbone)is equivalent to two helical gears of opposite hand,mounted side by side on the same shaft.They develop opposite thrust reactions and thus cancel out the thrust load. When two or more single helical gears are mounted on the same shaft,the hand of the gears should be selected so as to produce the minimum thrust load. Figure 13-21 An involute helicoid. Edge of paper Base helix angle Base cylinder
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General 670 © The McGraw−Hill Companies, 2008 Gears—General 671 Involute Base cylinder Edge of paper Base helix angle Figure 13–21 An involute helicoid. where the subscripts P and G refer to the pinion and gear, respectively, and where γ and are, respectively, the pitch angles of the pinion and gear. Figure 13–20 shows that the shape of the teeth, when projected on the back cone, is the same as in a spur gear having a radius equal to the back-cone distance rb. This is called Tredgold’s approximation. The number of teeth in this imaginary gear is N = 2πrb p (13–15) where N is the virtual number of teeth and p is the circular pitch measured at the large end of the teeth. Standard straight-tooth bevel gears are cut by using a 20◦ pressure angle, unequal addenda and dedenda, and full-depth teeth. This increases the contact ratio, avoids undercut, and increases the strength of the pinion. 13–10 Parallel Helical Gears Helical gears, used to transmit motion between parallel shafts, are shown in Fig. 13–2. The helix angle is the same on each gear, but one gear must have a right-hand helix and the other a left-hand helix. The shape of the tooth is an involute helicoid and is illustrated in Fig. 13–21. If a piece of paper cut in the shape of a parallelogram is wrapped around a cylinder, the angular edge of the paper becomes a helix. If we unwind this paper, each point on the angular edge generates an involute curve. This surface obtained when every point on the edge generates an involute is called an involute helicoid. The initial contact of spur-gear teeth is a line extending all the way across the face of the tooth. The initial contact of helical-gear teeth is a point that extends into a line as the teeth come into more engagement. In spur gears the line of contact is parallel to the axis of rotation; in helical gears the line is diagonal across the face of the tooth. It is this gradual engagement of the teeth and the smooth transfer of load from one tooth to another that gives helical gears the ability to transmit heavy loads at high speeds. Because of the nature of contact between helical gears, the contact ratio is of only minor importance, and it is the contact area, which is proportional to the face width of the gear, that becomes significant. Helical gears subject the shaft bearings to both radial and thrust loads. When the thrust loads become high or are objectionable for other reasons, it may be desirable to use double helical gears. A double helical gear (herringbone) is equivalent to two helical gears of opposite hand, mounted side by side on the same shaft. They develop opposite thrust reactions and thus cancel out the thrust load. When two or more single helical gears are mounted on the same shaft, the hand of the gears should be selected so as to produce the minimum thrust load.
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General I©The McGraw-Hil 671 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 672 Mechanical Engineering Design Figure 13-22 Nomenclature of helical gears. (a) Section B-E (e) SectionA-A Figure 13-22 represents a portion of the top view of a helical rack.Lines ab and cd are the centerlines of two adjacent helical teeth taken on the same pitch plane.The angle is the helix angle.The distance ac is the transverse circular pitch p in the plane of rotation (usually called the circular pitch).The distance ae is the normal circular pitch P and is related to the transverse circular pitch as follows: Pn P:cos (13-16) The distance ad is called the axial pitch p,and is related by the expression A=品 (13-17刀 Since Pn Pn=π，the normal diametral pitch is P Pn=cos (13-18) The pressure anglein the normal direction is different from the pressure anglein the direction of rotation,because of the angularity of the teeth.These angles are related by the equation tann cos= (13-19列 tan Figure 13-23 illustrates a cylinder cut by an oblique plane ab at an angle v to a right section.The oblique plane cuts out an arc having a radius of curvature of R.For the condition that=0,the radius of curvature is R=D/2.If we imagine the angle to be slowly increased from zero to 90,we see that R begins at a value of D/2 and
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 671 Companies, 2008 672 Mechanical Engineering Design t pt n Section B-B b d pn a c e px A B A B Section A-A (a) (b) (c) Figure 13–22 Nomenclature of helical gears. Figure 13–22 represents a portion of the top view of a helical rack. Lines ab and cd are the centerlines of two adjacent helical teeth taken on the same pitch plane. The angle ψ is the helix angle. The distance ac is the transverse circular pitch pt in the plane of rotation (usually called the circular pitch). The distance ae is the normal circular pitch pn and is related to the transverse circular pitch as follows: pn = pt cos ψ (13–16) The distance ad is called the axial pitch px and is related by the expression px = pt tan ψ (13–17) Since pn Pn = π, the normal diametral pitch is Pn = Pt cos ψ (13–18) The pressure angle φn in the normal direction is different from the pressure angle φt in the direction of rotation, because of the angularity of the teeth. These angles are related by the equation cos ψ = tan φn tan φt (13–19) Figure 13–23 illustrates a cylinder cut by an oblique plane ab at an angle ψ to a right section. The oblique plane cuts out an arc having a radius of curvature of R. For the condition that ψ = 0, the radius of curvature is R = D/2. If we imagine the angle ψ to be slowly increased from zero to 90◦, we see that R begins at a value of D/2 and