机械原理Chapter 6Gear Mechanisms-3.

1、Chapter 6 Gear Mechanisms 6.7 Manufacturing Methods of Involute Profiles 6.8 Addendum Modification on Involute Gears Cutting of Tooth Profiles There are various methods for manufacturing tooth profiles , such as die casting , precision forging, powder process , moulding , cutting and so on . Among t

2、hem , cutting is the most widely used . There are generally two types of cutting methods , i.e . form cutting and generating cutting . Form Cutting In form cutting, the desired tooth profile is produced by passing a cutter through the blank . Form milling with a disk milling cutter is shown in Fig.

3、The cutter rotates and the blank is fed axially . Form Cutting End milling cutters are used for large modules and double helical gears . The section of the cutter has the shape of the tooth space of the gear to be cut . Form Cutting Indexing is needed after a tooth space is cut so the accuracy of th

4、e gear depends on the accuracy of the index plate as well as the section shape of the cutter . Form Cutting As the shape of an involute depends on its base radius , a different cutter is required for each number of teeth even for the same module and same pressure angle . rb=r cos=(mz/2)cos Form Cutt

5、ing This is divorced from reality : in practice , only 8 to 26 cutters are available for each module and pressure angle . Each cutter is used for several different numbers of teeth and an error is therefore introduced . Form Cutting The cutting process is interrupted by the indexing and the producti

6、on rate is low in form milling . Its main advantage is that it can be accomplished on commonly available milling machines . Broach(拉刀)-Form Cutting Generating Cutting In a generating cutting method , the edges of a cutter take the form of a gear ( or rack ) with the same module and pressure angle as

7、 the gear to be cut . There is a generating motion between the cutter and the blank as if the cutter engages as a gear with the gear to be cut . By adding the cutting motion between the cutter and the blank , the profile of the teeth is formed gradually by a series of cuts . Fig. 6-22 Generating Cut

8、ting If z0 is the number of teeth of the cutter and z the number of teeth of the gear to be cut , their transmission ratio should be Fig. 6-22 which defines the generating motion to be provided by the kinematic chain in the cutting machine. Generating Cutting The cutting motion is the reciprocation

9、of the cutter while the feed is the movement of the cutter toward the blank . Fig. 6-22 Generating Cutting External gears can be cut with a rack-shaped shaper cutter. The edges of the cutter are now straight lines which can be made accurately . That is why involute gears are easy to manufacture . Fi

10、g.6-23 Generating Cutting The cutting method shown in Fig is not continuous in the shaping of a gear . In mass production , hobbing is used instead of shaping . Fig.6-23 Generating Cutting The shape of a hob is like a screw with some axial slots to form the cut ting edges . The cutting edge takes th

11、e shape of a rack and the hob is therefore a rack-shaped cutter . Generating Cutting The rotation of the hob provides the cutting motion and is also equivalent to the continuous translation of the rack-shaped cutter . At the same time , the blank is driven at a certain transmission ratio to the cutt

12、er by the kinematic chain in the cutting machine and this gives the required generating motion . Generating Cutting Meanwhile , the hob is fed along the axial direction of the blank to cut the teeth for the whole facewidth . Cutting is continuous and the production rate is therefore improved . Gener

13、ating Cutting In the generating method , one cutter is enough for each module and pressure angle, in spite of the number of teeth of the gear to be cut . It is therefore the most widely used method to cut gears . Cutting a Standard Gear with Standard Rack-shaped Cutter The profile of a standard rack

14、-shaped cutter is similar to that of a standard rack but the tip line is c * m higher than the addendum line for cutting the profile at the root to provide the bottom clearance . ha * m c 2 *m m m a h* =0.38m =0.38m * c m * hamah*m * c 2m 2m 2m =20 =20 dedendum line dedendum line reference line adde

15、ndum line cutter tip line reference line addendum line Cutting a Standard Gear with Standard Rack-shaped Cutter The cutting edge between the tip line and the addendum line is not a straight line and the corresponding gear profile cut is not involute . ha * m c 2 *m m m a h* =0.38m =0.38m * c m * ham

16、ah*m * c 2m 2m 2m =20 =20 dedendum line dedendum line reference line addendum line cutter tip line reference line addendum line Cutting a Standard Gear with Standard Rack-shaped Cutter Therefore, in discussing the generating of an involute profile , we simply regard the standard rack-shaped cutter a

17、s having the same profile as the standard rack . ha * m c 2 *m m m a h* =0.38m =0.38m * c m * hamah*m * c 2m 2m 2m =20 =20 dedendum line dedendum line reference line addendum line cutter tip line reference line addendum line Cutting a Standard Gear with Standard Rack-shaped Cutter To cut a standard

18、gear , the transmission ratio of the cutter and the blank in generating motion is the same as that of the rack and the gear. 1 r ra1 P V m a h* = = 1 1 = = 1 r 2 b1 r 1 N 2 N 211 vr Cutting a Standard Gear with Standard Rack-shaped Cutter Furthermore , the reference line of the cutter should be tang

19、ent to the reference circle of the gear after the feeding has finished. 1 r ra1 P V m a h* = = 1 1 = = 1 r 2 b1 r 1 N 2 N Cutting a Standard Gear with Standard Rack-shaped Cutter Thus the reference line of the cutter rolls with the reference circle of the gear without slipping. 1 r ra1 P V m a h* =

20、= 1 1 = = 1 r 2 b1 r 1 N 2 N Cutting a Standard Gear with Standard Rack-shaped Cutter Therefore , the pitch , module , pressure angle , tooth thickness and spacewidth of the gear on the reference circle are the same as those on the reference line . 1 r ra1 P V m a h* = = 1 1 = = 1 r 2 b1 r 1 N 2 N C

21、utting a Standard Gear with Standard Rack-shaped Cutter As the tooth thickness and spacewidth of the cutter on the reference line are equal to each other , the gear is cut with 1 r ra1 P V m a h* = = 1 1 = = 1 r 2 b1 r 1 N 2 N Cutting a Standard Gear with Standard Rack-shaped Cutter Since the distan

22、ce between the reference line and the tip line of the cutter is ( ha* + c * ) m and the dedendum circle of the gear is cut by the tip line of the cutter , the radius of dedendum circle of the gear 1 r ra1 P V m a h* = = 1 1 = = 1 r 2 b1 r 1 N 2 N rf1 = r1 - ( ha* + c * ) m Cutting a Standard Gear wi

23、th Standard Rack-shaped Cutter Note : the addendum circle of the gear is not cut by the dedendum line of the cutter ! To get a standard gear , the blank must be cut on a lathe with radius ra1 = r1 + ha* m . 1 r ra1 P V m a h* = = 1 1 = = 1 r 2 b1 r 1 N 2 N Cutter Interference In a generating process

24、 , it is sometimes found that the top of the cutter enters the profile of the gear and some part of the involute profile near the root portion is removed. r r N2 r m a h* a1 * c m 1 B 1 = = 1 1 N f1 r 2 P V rb1 = = 1 Cutter Interference This is called cutter interference and will reduce the contact

25、ratio as well as the strength of the tooth . Obviously , cutter interference should be avoided or minimized . Cutter Interference In a generating process , r r N2 r m a h* a1 * c m 1 B 1 = = 1 1 N f1 r 2 P V rb1 = = 1 Cutter Interference For a rack-shaped cutter , it can be proved that cutter interf

26、erence will occur if the addendum line of the cutter passes the limit point N1 of the line of action. r r N2 r m a h* a1 * c m 1 B 1 = = 1 1 N f1 r 2 P V rb1 = = 1 O 1 2 rb B P N ha * m reference circle reference line B r 1 1 1 base circle r 1 = = addendum line pitch circle= = = Cutter Interference

27、In cutting a standard gear , the reference line of the cutter is tangent to the reference circle of the gear and the addendum line of the cutter intersects the line of action at point B2 Cutter Interference To prevent cutter interference , the point B2 should not pass point N1 . 21 PBPN sin sin 2 1

28、* mh mz a 2 * sin 2 a h z Introduction of Addendum Modification Standard gears enjoy interchangeability and are widely used in many kind of machines . However, they also have some disadvantages . The number of teeth should not be less than zmin to prevent cutter interference . More compact construct

29、ion can not be achieved with larger z values . The working centre distance a should be equal to the reference centre distance a to meet the two design requirements . The curvature radius of the tooth profile and the tooth thickness of the pinion at the root are less than those of the gear . Therefor

30、e the strength of the pinion is much lower than that of the gear . The strength of the standard gear mechanism is thus lower even if the strength of the gear is adequate . Introduction of Addendum Modification To improve the performance of gears , addendum modification is employed . Gears with adden

31、dum modification are also called corrected gears . The gear-cutting machines and the cutters are the same as those for standard gears . In adition , the transmission ratio between the cutter and the blank remains unchanged . O 1 rb P N ha * m reference circle reference line r 1 1 1 base circle r 1 =

32、 = addendum line pitch circle= pitch line Introduction of Addendum Modification The difference from cutting standard gears is that the reference line of the cutter may not be tangent to the reference circle of the gear . r1= r1 for rack and pinion gearing . The line tangent to and rolling without sl

33、ipping on the reference circle of the gear is the pitch line of the rack-shaped cutter . O 1 rb P N ha * m reference circle reference line r 1 1 1 base circle r 1 = = addendum line pitch circle= pitch line Introduction of Addendum Modification the pressure angle and the pitch of a rack are the same

34、on any line . Therefore , parameters z , m , r = m z/ 2, , and rb = rcos of the corrected gear remain the same as those of standard gears . This means that different portions of the same involute are employed for the profiles of the standard gear and the corrected gear . O 1 rb P N ha * m reference

35、circle reference line r 1 1 1 base circle r 1 = = addendum line pitch circle= pitch line Geometric Dimensions of Corrected Gears In cutting the corrected gear , the rackshaped cutter is located a distance xm from the position used for cutting the standard gear , i.e . the distance between the pitch

36、line and the reference line is x m . Here , m is the module and x is the modification coefficient . reference line reference circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m ha xm m 2 pitch circle= Geometric Dimensions of Corrected Gears-轮齿的切制 If the cutter is placed further away from th

37、e position for cutting a standard gear , x is positive and the modification is considered positive . Otherwise , if the cutter is placed towards the axis of the blank , x and the modification are negative. reference line reference circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m ha xm m

38、2 pitch circle= Geometric Dimensions of Corrected Gears To prevent cutter interference , the modification distance xm of the cutter should be sufficiently large reference line reference circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m ha xm m 2 pitch circle= GNxmmha 1 * 2/sin 2 1 * 1 * m

39、zmhGNmhxm aa 2 sin 2 * z hx a 2 * min sin 2 a h z * min min min () a h zz x z - = Geometric Dimensions of Corrected Gears if ha*=1 and =20, zmin=17 reference line reference circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m ha xm m 2 pitch circle= * min min min () a h zz x z - = min 17 17

40、Z x - = Geometric Dimensions of Corrected Gears Obviously , if z 0. When z zmin , the gear can be negatively modified , as xmin 0 . reference line reference circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m ha xm m 2 pitch circle= * min min min () a h zz x z - = Geometric Dimensions of Co

41、rrected Gears Since the dedendum circle of the gear is cut by the tip line of the cutter and the cutter has retreated xm , the radius of the dedendum circle rf for the corrected gear is now given by reference line reference circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m ha xm m 2 pitch

42、 circle= Geometric Dimensions of Corrected Gears the tooth thickness and the spacewidth of a corrected gear on the reference circle are respectively reference line reference circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m ha xm m 2 pitch circle= The tooth thickness of a positively modif

43、ied gear is larger than that of a standard gear . The bending strength of a positively modified gear is therefore improved . Gearing of a Corrected Gear Pair To keep zero backlash for a corrected gear pair , the following relations should hold , as in the case of standard gears reference line refere

44、nce circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m ha xm m 2 pitch circle= O 1 2 r 1 r 2 s 1e 1 s 2e 2 e 1 s 1 2 e 2 s Gearing of a Corrected Gear Pair Then the pitch on pitch circles of both gears is reference line reference circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m

45、 ha xm m 2 pitch circle= O 1 2 r 1 r 2 s 1e 1 s 2e 2 e 1 s 1 2 e 2 s This is called the gearing equation without backlash . The following formula can be derived Gearing of a Corrected Gear Pair This equation can be used to calculate the working pressure angle according to the modification coefficien

46、ts x1 and x2 . Then the working centre distance a can be calculated from acos= acos . reference line reference circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m ha xm m 2 pitch circle= Gearing of a Corrected Gear Pair For a given specific working centre distance a, the working pressure angle should be calculated first from acos= acos . Then the sum of x1 and x2 can be calculated. reference line reference circle pitch line base circle O1 PG N1 m 2 K I J xm xmtan *m ha xm m 2 p