上海交通大学：《Design and Manufacturing》课程教学资源（参考文献）Shigley's Mechanical Engineering Design ch13 gears-generals.pdf_大学文库

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 653 Companies, 2008 654 Mechanical Engineering Design Figure 13–1 Spur gears are used to transmit rotary motion between parallel shafts. Figure 13–2 Helical gears are used to transmit motion between parallel or nonparallel shafts. This chapter addresses gear geometry, the kinematic relations, and the forces transmitted by the four principal types of gears: spur, helical, bevel, and worm gears. The forces transmitted between meshing gears supply torsional moments to shafts for motion and power transmission and create forces and moments that affect the shaft and its bearings. The next two chapters will address stress, strength, safety, and reliability of the four types of gears. 13–1 Types of Gears Spur gears, illustrated in Fig. 13–1, have teeth parallel to the axis of rotation and are used to transmit motion from one shaft to another, parallel, shaft. Of all types, the spur gear is the simplest and, for this reason, will be used to develop the primary kinematic relationships of the tooth form. Helical gears,shown in Fig. 13–2, have teeth inclined to the axis of rotation. Helical gears can be used for the same applications as spur gears and, when so used, are not as noisy, because of the more gradual engagement of the teeth during meshing. The inclined tooth also develops thrust loads and bending couples, which are not present with spur gearing. Sometimes helical gears are used to transmit motion between nonparallel shafts. Bevel gears, shown in Fig. 13–3, have teeth formed on conical surfaces and are used mostly for transmitting motion between intersecting shafts. The figure actually illustrates straight-tooth bevel gears. Spiral bevel gears are cut so the tooth is no longer straight, but forms a circular arc. Hypoid gears are quite similar to spiral bevel gears except that the shafts are offset and nonintersecting

656 Budynas-Nisbett Shigley's IIl Design of Mechanical 13.Gears-General T©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Gears-General 657 13-3 Conjugate Action The following discussion assumes the teeth to be perfectly formed,perfectly smooth, and absolutely rigid.Such an assumption is,of course,unrealistic,because the applica- tion of forces will cause deflections. Mating gear teeth acting against each other to produce rotary motion are similar to cams.When the tooth profiles,or cams,are designed so as to produce a constant angular- velocity ratio during meshing,these are said to have conjugate action.In theory,at least, it is possible arbitrarily to select any profile for one tooth and then to find a profile for the meshing tooth that will give conjugate action.One of these solutions is the imolute profile, which,with few exceptions,is in universal use for gear teeth and is the only one with which we should be concerned. When one curved surface pushes against another(Fig.13-6),the point of contact occurs where the two surfaces are tangent to each other(point c),and the forces at any instant are directed along the common normal ab to the two curves.The line ab,rep- resenting the direction of action of the forces,is called the line of action.The line of action will intersect the line of centers O-0 at some point P.The angular-velocity ratio between the two arms is inversely proportional to their radii to the point P.Circles drawn through point P from each center are called pitch circles,and the radius of each circle is called the pitch radius.Point P is called the pitch point. Figure 13-6 is useful in making another observation.A pair of gears is really pairs of cams that act through a small arc and,before running off the involute contour, are replaced by another identical pair of cams.The cams can run in either direction and are configured to transmit a constant angular-velocity ratio.If involute curves are used, the gears tolerate changes in center-to-center distance with no variation in constant angular-velocity ratio.Furthermore,the rack profiles are straight-fanked,making pri- mary tooling simpler. To transmit motion at a constant angular-velocity ratio,the pitch point must remain fixed;that is,all the lines of action for every instantaneous point of contact must pass through the same point P.In the case of the involute profile,it will be shown that all points of contact occur on the same straight line ab,that all normals to the tooth profiles at the point of contact coincide with the line ab,and,thus,that these profiles transmit uniform rotary motion. Figure 13-6 Cam A and follower B in contact.When the contacting surfaces are involute profiles the ensuing conjugate action produces a constant angularvelocity ratio

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General 656 © The McGraw−Hill Companies, 2008 Gears—General 657 O B rB rA b c a A O P Figure 13–6 Cam A and follower B in contact. When the contacting surfaces are involute profiles, the ensuing conjugate action produces a constant angular-velocity ratio. 13–3 Conjugate Action The following discussion assumes the teeth to be perfectly formed, perfectly smooth, and absolutely rigid. Such an assumption is, of course, unrealistic, because the application of forces will cause deflections. Mating gear teeth acting against each other to produce rotary motion are similar to cams. When the tooth profiles, or cams, are designed so as to produce a constant angularvelocity ratio during meshing, these are said to have conjugate action. In theory, at least, it is possible arbitrarily to select any profile for one tooth and then to find a profile for the meshing tooth that will give conjugate action. One of these solutions is the involute profile, which, with few exceptions, is in universal use for gear teeth and is the only one with which we should be concerned. When one curved surface pushes against another (Fig. 13–6), the point of contact occurs where the two surfaces are tangent to each other (point c), and the forces at any instant are directed along the common normal ab to the two curves. The line ab, representing the direction of action of the forces, is called the line of action. The line of action will intersect the line of centers O-O at some point P. The angular-velocity ratio between the two arms is inversely proportional to their radii to the point P. Circles drawn through point P from each center are called pitch circles, and the radius of each circle is called the pitch radius. Point P is called the pitch point. Figure 13–6 is useful in making another observation. A pair of gears is really pairs of cams that act through a small arc and, before running off the involute contour, are replaced by another identical pair of cams. The cams can run in either direction and are configured to transmit a constant angular-velocity ratio. If involute curves are used, the gears tolerate changes in center-to-center distance with no variation in constant angular-velocity ratio. Furthermore, the rack profiles are straight-flanked, making primary tooling simpler. To transmit motion at a constant angular-velocity ratio, the pitch point must remain fixed; that is, all the lines of action for every instantaneous point of contact must pass through the same point P. In the case of the involute profile, it will be shown that all points of contact occur on the same straight line ab, that all normals to the tooth profiles at the point of contact coincide with the line ab, and, thus, that these profiles transmit uniform rotary motion

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General T©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 658 Mechanical Engineering Design Figure 13-7 Base circle Pitch circle (a)Generation of an involute; (b)involute action. Gear I Pitch circle Base cirele Gear 2 102 (a) (b) 13-4 Involute Properties An involute curve may be generated as shown in Fig.13-7a.A partial flange B is attached to the cylinder A,around which is wrapped a cord def,which is held tight.Point b on the cord represents the tracing point.and as the cord is wrapped and unwrapped about the cylinder,point b will trace out the involute curve ac.The radius of the curva- ture of the involute varies continuously,being zero at point a and a maximum at point c. At point b the radius is equal to the distance be,since point b is instantaneously rotating about point e.Thus the generating line de is normal to the involute at all points of inter- section and,at the same time,is always tangent to the cylinder A.The circle on which the involute is generated is called the base circle. Let us now examine the involute profile to see how it satisfies the requirement for the transmission of uniform motion.In Fig.13-7b,two gear blanks with fixed centers at O1 and O2 are shown having base circles whose respective radii are Oja and O2b. We now imagine that a cord is wound clockwise around the base circle of gear 1,pulled tight between points a and b,and wound counterclockwise around the base circle of gear 2.If,now,the base circles are rotated in different directions so as to keep the cord tight,a point g on the cord will trace out the involutes cd on gear 1 and ef on gear 2. The involutes are thus generated simultaneously by the tracing point.The tracing point, therefore,represents the point of contact,while the portion of the cord ab is the gener- ating line.The point of contact moves along the generating line;the generating line does not change position,because it is always tangent to the base circles;and since the generating line is always normal to the involutes at the point of contact,the requirement for uniform motion is satisfied. 13-5 Fundamentals Among other things,it is necessary that you actually be able to draw the teeth on a pair of meshing gears.You should understand,however,that you are not doing this for man- ufacturing or shop purposes.Rather,we make drawings of gear teeth to obtain an under- standing of the problems involved in the meshing of the mating teeth

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 657 Companies, 2008 658 Mechanical Engineering Design + + + Base circle Pitch circle O1 O2 c a P e d g f b d B b c e a A O f Pitch circle Gear 1 Gear 2 Base circle (a) (b) Figure 13–7 (a) Generation of an involute; (b) involute action. 13–4 Involute Properties An involute curve may be generated as shown in Fig. 13–7a. A partial flange B is attached to the cylinder A, around which is wrapped a cord def, which is held tight. Point b on the cord represents the tracing point, and as the cord is wrapped and unwrapped about the cylinder, point b will trace out the involute curve ac. The radius of the curvature of the involute varies continuously, being zero at point a and a maximum at point c. At point b the radius is equal to the distance be, since point b is instantaneously rotating about point e. Thus the generating line de is normal to the involute at all points of intersection and, at the same time, is always tangent to the cylinder A. The circle on which the involute is generated is called the base circle. Let us now examine the involute profile to see how it satisfies the requirement for the transmission of uniform motion. In Fig. 13–7b, two gear blanks with fixed centers at O1 and O2 are shown having base circles whose respective radii are O1a and O2b. We now imagine that a cord is wound clockwise around the base circle of gear 1, pulled tight between points a and b, and wound counterclockwise around the base circle of gear 2. If, now, the base circles are rotated in different directions so as to keep the cord tight, a point g on the cord will trace out the involutes cd on gear 1 and ef on gear 2. The involutes are thus generated simultaneously by the tracing point. The tracing point, therefore, represents the point of contact, while the portion of the cord ab is the generating line. The point of contact moves along the generating line; the generating line does not change position, because it is always tangent to the base circles; and since the generating line is always normal to the involutes at the point of contact, the requirement for uniform motion is satisfied. 13–5 Fundamentals Among other things, it is necessary that you actually be able to draw the teeth on a pair of meshing gears. You should understand, however, that you are not doing this for manufacturing or shop purposes. Rather, we make drawings of gear teeth to obtain an understanding of the problems involved in the meshing of the mating teeth

58 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 659 Figure 13-8 Base circle Construction of an involute Involute curve. 0- First,it is necessary to learn how to construct an involute curve.As shown in Fig.13-8,divide the base circle into a number of equal parts,and construct radial lines OAo.OA1,OA2,etc.Beginning at A1,construct perpendiculars A BI,A2B2,A3 B3. etc.Then along A B lay off the distance A Ao,along A2B2 lay off twice the distance A Ao,etc.,producing points through which the involute curve can be constructed. To investigate the fundamentals of tooth action,let us proceed step by step through the process of constructing the teeth on a pair of gears. When two gears are in mesh,their pitch circles roll on one another without slip- ping.Designate the pitch radii as r and r2 and the angular velocities as and 2, respectively.Then the pitch-line velocity is V Ir1oil Ir2o2l Thus the relation between the radii on the angular velocities is (13-5) r Suppose now we wish to design a speed reducer such that the input speed is 1800 rev/min and the output speed is 1200 rev/min.This is a ratio of 3:2;the gear pitch diam- eters would be in the same ratio,for example,a 4-in pinion driving a 6-in gear.The various dimensions found in gearing are always based on the pitch circles. Suppose we specify that an 18-tooth pinion is to mesh with a 30-tooth gear and that the diametral pitch of the gearset is to be 2 teeth per inch.Then,from Eq.(13-1).the pitch diameters of the pinion and gear are,respectively, N118 N230 d= p=2=9in d2= p=2=15im The first step in drawing teeth on a pair of mating gears is shown in Fig.13-9.The cen- ter distance is the sum of the pitch radii,in this case 12 in.So locate the pinion and gear centers O and 02,12 in apart.Then construct the pitch circles of radii r and r2.These are tangent at P,the pitch point.Next draw line ab,the common tangent,through the pitch point.We now designate gear 1 as the driver,and since it is rotating counter- clockwise,we draw a line cd through point P at an angle to the common tangent ab. The line cd has three names,all of which are in general use.It is called the pressure line,the generating line,and the line of action.It represents the direction in which the resultant force acts between the gears.The angle is called the pressure angle,and it usually has values of20or25°，thoughI4°was once used

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General 658 © The McGraw−Hill Companies, 2008 Gears—General 659 O Base circle Involute A4 A3 A2 A1 A0 B1 B2 B3 B4 Figure 13–8 Construction of an involute curve. First, it is necessary to learn how to construct an involute curve. As shown in Fig. 13–8, divide the base circle into a number of equal parts, and construct radial lines O A0, O A1, O A2, etc. Beginning at A1, construct perpendiculars A1B1, A2B2, A3B3, etc. Then along A1B1 lay off the distance A1A0, along A2B2 lay off twice the distance A1A0, etc., producing points through which the involute curve can be constructed. To investigate the fundamentals of tooth action, let us proceed step by step through the process of constructing the teeth on a pair of gears. When two gears are in mesh, their pitch circles roll on one another without slipping. Designate the pitch radii as r1 and r2 and the angular velocities as ω1 and ω2, respectively. Then the pitch-line velocity is V = |r1ω1| = |r2ω2| Thus the relation between the radii on the angular velocities is ω1 ω2 = r2 r1 (13–5) Suppose now we wish to design a speed reducer such that the input speed is 1800 rev/min and the output speed is 1200 rev/min. This is a ratio of 3:2; the gear pitch diameters would be in the same ratio, for example, a 4-in pinion driving a 6-in gear. The various dimensions found in gearing are always based on the pitch circles. Suppose we specify that an 18-tooth pinion is to mesh with a 30-tooth gear and that the diametral pitch of the gearset is to be 2 teeth per inch. Then, from Eq. (13–1), the pitch diameters of the pinion and gear are, respectively, d1 = N1 P = 18 2 = 9 in d2 = N2 P = 30 2 = 15 in The first step in drawing teeth on a pair of mating gears is shown in Fig. 13–9. The center distance is the sum of the pitch radii, in this case 12 in. So locate the pinion and gear centers O1 and O2, 12 in apart. Then construct the pitch circles of radii r1 and r2. These are tangent at P, the pitch point. Next draw line ab, the common tangent, through the pitch point. We now designate gear 1 as the driver, and since it is rotating counterclockwise, we draw a line cd through point P at an angle φ to the common tangent ab. The line cd has three names, all of which are in general use. It is called the pressure line, the generating line, and the line of action. It represents the direction in which the resultant force acts between the gears. The angle φ is called the pressure angle, and it usually has values of 20 or 25◦, though 141 2 ◦ was once used.

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 659 Companies, 2008 660 Mechanical Engineering Design O r P Pitch circle Pressure line Base circle rB Figure 13–10 Base circle radius can be related to the pressure angle φ and the pitch circle radius by r b = r cos φ. Next, on each gear draw a circle tangent to the pressure line. These circles are the base circles. Since they are tangent to the pressure line, the pressure angle determines their size. As shown in Fig. 13–10, the radius of the base circle is rb = r cos φ (13–6) where r is the pitch radius. Now generate an involute on each base circle as previously described and as shown in Fig. 13–9. This involute is to be used for one side of a gear tooth. It is not necessary to draw another curve in the reverse direction for the other side of the tooth, because we are going to use a template which can be turned over to obtain the other side. The addendum and dedendum distances for standard interchangeable teeth are, as we shall learn later, 1/P and 1.25/P, respectively. Therefore, for the pair of gears we are constructing, a = 1 P = 1 2 = 0.500 in b = 1.25 P = 1.25 2 = 0.625 in Using these distances, draw the addendum and dedendum circles on the pinion and on the gear as shown in Fig. 13–9. Next, using heavy drawing paper, or preferably, a sheet of 0.015- to 0.020-in clear plastic, cut a template for each involute, being careful to locate the gear centers properly with respect to each involute. Figure 13–11 is a reproduction of the template used to create some of the illustrations for this book. Note that only one side of the tooth pro- file is formed on the template. To get the other side, turn the template over. For some problems you might wish to construct a template for the entire tooth. Base circle + + Dedendum circle Pitch circle Base circle Involute Addendum circles Pitch circle b d a c P O1 O2 r1 r2 Dedendum circle Involute 1 2 Figure 13–9 Circles of a gear layout.

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General T©The McGraw-Hill 661 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 662 Mechanical Engineering Design Referring again to Fig.13-12,the pinion with center at O is the driver and turns counterclockwise.The pressure,or generating,line is the same as the cord used in Fig.13-7a to generate the involute,and contact occurs along this line.The initial con- tact will take place when the flank of the driver comes into contact with the tip of the driven tooth.This occurs at point a in Fig.13-12,where the addendum circle of the dri- ven gear crosses the pressure line.If we now construct tooth profiles through point a and draw radial lines from the intersections of these profiles with the pitch circles to the gear centers,we obtain the angle of approach for each gear. As the teeth go into mesh,the point of contact will slide up the side of the driving tooth so that the tip of the driver will be in contact just before contact ends.The final point of contact will therefore be where the addendum circle of the driver crosses the pressure line.This is point b in Fig.13-12.By drawing another set of tooth profiles through b,we obtain the angle of recess for each gear in a manner similar to that of find- ing the angles of approach.The sum of the angle of approach and the angle of recess for either gear is called the angle of action.The line ab is called the line of action. We may imagine a rack as a spur gear having an infinitely large pitch diameter. Therefore,the rack has an infinite number of teeth and a base circle which is an infinite distance from the pitch point.The sides of involute teeth on a rack are straight lines making an angle to the line of centers equal to the pressure angle.Figure 13-13 shows an involute rack in mesh with a pinion.Corresponding sides on involute teeth are par- allel curves;the base pitch is the constant and fundamental distance between them along a common normal as shown in Fig.13-13.The base pitch is related to the circu- lar pitch by the equation Pb=Pe cos中 (13-7 where p is the base pitch. Figure 13-14 shows a pinion in mesh with an internal,or ring,gear.Note that both of the gears now have their centers of rotation on the same side of the pitch point.Thus the positions of the addendum and dedendum circles with respect to the pitch circle are reversed;the addendum circle of the internal gear lies inside the pitch circle.Note,too, from Fig.13-14,that the base circle of the internal gear lies inside the pitch circle near the addendum circle. Another interesting observation concerns the fact that the operating diameters of the pitch circles of a pair of meshing gears need not be the same as the respective design pitch diameters of the gears,though this is the way they have been constructed in Fig.13-12.If we increase the center distance,we create two new operating pitch circles having larger diameters because they must be tangent to each other at the pitch point. Figure 13-13 Involute-toothed pinion and rack

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 661 Companies, 2008 662 Mechanical Engineering Design Circular pitch Base pitch pc pb Figure 13–13 Involute-toothed pinion and rack. Referring again to Fig. 13–12, the pinion with center at O1 is the driver and turns counterclockwise. The pressure, or generating, line is the same as the cord used in Fig. 13–7a to generate the involute, and contact occurs along this line. The initial contact will take place when the flank of the driver comes into contact with the tip of the driven tooth. This occurs at point a in Fig. 13–12, where the addendum circle of the driven gear crosses the pressure line. If we now construct tooth profiles through point a and draw radial lines from the intersections of these profiles with the pitch circles to the gear centers, we obtain the angle of approach for each gear. As the teeth go into mesh, the point of contact will slide up the side of the driving tooth so that the tip of the driver will be in contact just before contact ends. The final point of contact will therefore be where the addendum circle of the driver crosses the pressure line. This is point b in Fig. 13–12. By drawing another set of tooth profiles through b, we obtain the angle of recess for each gear in a manner similar to that of finding the angles of approach. The sum of the angle of approach and the angle of recess for either gear is called the angle of action. The line ab is called the line of action. We may imagine a rack as a spur gear having an infinitely large pitch diameter. Therefore, the rack has an infinite number of teeth and a base circle which is an infinite distance from the pitch point. The sides of involute teeth on a rack are straight lines making an angle to the line of centers equal to the pressure angle. Figure 13–13 shows an involute rack in mesh with a pinion. Corresponding sides on involute teeth are parallel curves; the base pitch is the constant and fundamental distance between them along a common normal as shown in Fig. 13–13. The base pitch is related to the circular pitch by the equation pb = pc cos φ (13–7) where pb is the base pitch. Figure 13–14 shows a pinion in mesh with an internal, or ring, gear. Note that both of the gears now have their centers of rotation on the same side of the pitch point. Thus the positions of the addendum and dedendum circles with respect to the pitch circle are reversed; the addendum circle of the internal gear lies inside the pitch circle. Note, too, from Fig. 13–14, that the base circle of the internal gear lies inside the pitch circle near the addendum circle. Another interesting observation concerns the fact that the operating diameters of the pitch circles of a pair of meshing gears need not be the same as the respective design pitch diameters of the gears, though this is the way they have been constructed in Fig. 13–12. If we increase the center distance, we create two new operating pitch circles having larger diameters because they must be tangent to each other at the pitch point.