UNIT 3 POWER TRANSMISSION DEVICES

Transcription

1 UNI 3 POWER RANSMISSION DEVIES Power ransmission Devices Srucure 3. Inroducion Objecives 3. Power ransmission Devices 3.. Bels 3.. hain 3..3 Gears 3.3 ransmission Screw 3.4 Power ransmission by Bels 3.4. Law of Beling 3.4. Lengh of he Bel one Pulleys Raio of ensions Power ransmied by Bel Drive ension due o enrifugal Forces Iniial ension Maximum Power ransmied 3.5 Kinemaics of hain Drive 3.6 lassificaion of Gears 3.6. Parallel Shafs 3.6. Inersecing Shafs Skew Shafs 3.7 Gear erminology 3.8 Gear rain 3.8. Simple Gear rain 3.8. ompound Gear rain Power ransmied by Simple Spur Gear 3.9 Summary 3.0 Key Words 3. Answers o SAQs 3. INRODUION he power is ransmied from one shaf o he oher by means of bels, chains and gears. he bels and ropes are flexible members which are used where disance beween he wo shafs is large. he chains also have flexibiliy bu hey are preferred for inermediae disances. he gears are used when he shafs are very close wih each oher. his ype of drive is also called posiive drive because here is no slip. If he disance is slighly larger, chain drive can be used for making i a posiive drive. Bels and ropes ransmi power due o he fricion beween he bel or rope and he pulley. here is a possibiliy of slip and creep and ha is why, his drive is no a posiive drive. A gear rain is a combinaion of gears which are used for ransmiing moion from one shaf o anoher. 79

2 heory of Machines Objecives Afer sudying his uni, you should be able o undersand power ransmission derives, undersand law of beling, deermine power ransmied by bel drive and gear, deermine dimensions of bel for given power o be ransmied, undersand kinemaics of chain drive, deermine gear raio for differen ype of gear rains, classify gears, and undersand gear erminology. 3. POWER RANSMISSION DEVIES Power ransmission devices are very commonly used o ransmi power from one shaf o anoher. Bels, chains and gears are used for his purpose. When he disance beween he shafs is large, bels or ropes are used and for inermediae disance chains can be used. For bel drive disance can be maximum bu his should no be more han en meres for good resuls. Gear drive is used for shor disances. 3.. Bels In case of bels, fricion beween he bel and pulley is used o ransmi power. In pracice, here is always some amoun of slip beween bel and pulleys, herefore, exac velociy raio canno be obained. ha is why, bel drive is no a posiive drive. herefore, he bel drive is used where exac velociy raio is no required. he following ypes of bels shown in Figure 3. are mos commonly used : 80 (a) Fla Bel and Pulley (b) V-bel and Pulley (c) ircular Bel or Rope Pulley Figure 3. : ypes of Bel and Pulley he fla bel is recangular in cross-secion as shown in Figure 3.(a). he pulley for his bel is slighly crowned o preven slip of he bel o one side. I uilises he fricion beween he fla surface of he bel and pulley. he V-bel is rapezoidal in secion as shown in Figure 3.(b). I uilizes he force of fricion beween he inclined sides of he bel and pulley. hey are preferred when disance is comparaive shorer. Several V-bels can also be used ogeher if power ransmied is more. he circular bel or rope is circular in secion as shown in Figure 8.(c). Several ropes also can be used ogeher o ransmi more power. he bel drives are of he following ypes : (a) (b) Open Bel Drive open bel drive, and cross bel drive. Open bel drive is used when sense of roaion of boh he pulleys is same. I is desirable o keep he igh side of he bel on he lower side and slack side a he

3 op o increase he angle of conac on he pulleys. his ype of drive is shown in Figure 3.. Power ransmission Devices Slack Side hickness Driving Pulley Driving Pulley Effecive Radius igh Side Neural Secion ross Bel Drive Figure 3. : Open Bel Derive In case of cross bel drive, he pulleys roae in he opposie direcion. he angle of conac of bel on boh he pulleys is equal. his drive is shown in Figure 3.3. As shown in he figure, he bel has o bend in wo differen planes. As a resul of his, bel wears very fas and herefore, his ype of drive is no preferred for power ransmission. his can be used for ransmission of speed a low power. Figure 3.3 : ross Bel Drive Since power ransmied by a bel drive is due o he fricion, bel drive is subjeced o slip and creep. Le d and d be he diameers of driving and driven pulleys, respecively. N and N be he corresponding speeds of driving and driven pulleys, respecively. he velociy of he bel passing over he driver dn V 60 If here is no slip beween he bel and pulley d N V V 60 d N d N N d N d If hickness of he bel is, and i is no negligible in comparison o he diameer, N d N d Le here be oal percenage slip S in he bel drive which can be aken ino accoun as follows : S V V 00 or d N d N S

4 heory of Machines If he hickness of bel is also o be considered or N ( d ) N ( d ) S 00 ( ) N d S N ( d ) 00 he bel moves from he igh side o he slack side and vice-versa, here is some loss of power because he lengh of bel coninuously exends on igh side and conracs on loose side. hus, here is relaive moion beween he bel and pulley due o body slip. his is known as creep. 3.. hain he bel drive is no a posiive drive because of creep and slip. he chain drive is a posiive drive. Like bels, chains can be used for larger cenre disances. hey are made of meal and due o his chain is heavier han he bel bu hey are flexible like bels. I also requires lubricaion from ime o ime. he lubrican prevens chain from rusing and reduces wear. he chain and chain drive are shown in Figure 3.4. he sprockes are used in place of pulleys. he projeced eeh of sprockes fi in he recesses of he chain. he disance beween roller ceners of wo adjacen links is known as pich. he circle passing hrough he pich ceners is called pich circle. Roller Bushing (a) Pin Pich (b) Pich p φ r Sprocke Le (c) Figure 3.4 : hain and hain Drive be he angle made by he pich of he chain, and r be he pich circle radius, hen pich, p r sin (d) 8 r p cosec he power ransmission chains are made of seel and hardened o reduce wear. hese chains are classified ino hree caegories (a) (b) (c) Block chain Roller chain Invered ooh chain (silen chain)

5 Ou of hese hree caegories roller chain shown in Figure 3.4(b) is mos commonly used. he consrucion of his ype of chain is shown in he figure. he roller is made of seel and hen hardened o reduce he wear. A good roller chain is quier in operaion as compared o he block chain and i has lesser wear. he block chain is shown in Figure 3.4(a). I is used for low speed drive. he invered ooh chain is shown in Figures 3.4(c) and (d). I is also called as silen chain because i runs very quiely even a higher speeds Gears Gears are also used for power ransmission. his is accomplished by he successive engagemen of eeh. he wo gears ransmi moion by he direc conac like chain drive. Gears also provide posiive drive. he drive beween he wo gears can be represened by using plain cylinders or discs and having diameers equal o heir pich circles as shown in Figure 3.5. he poin of conac of he wo pich surfaces shell have velociy along he common angen. Because here is no slip, definie moion of gear can be ransmied o gear or vice-versa. Power ransmission Devices he angenial velociy V p = r = r where r and r are pich circle radii of gears and, respecively. V P N N N N r r N r N r N r N r Figure 3.5 : Gear Drive Since, pich circle radius of a gear is proporional o is number of eeh (). N N where and are he number of eeh on gears and, respecively. SAQ In which ype of drive cenre disance beween he shafs is lowes? Give reason for his? 3.3 RANSMISSION SREW In a screw, eeh are cu around is circular periphery which form helical pah. A nu has similar inernal helix in is bore. When nu is urned on he screw wih a force applied angenially, screw moves forward. For one urn, movemen is equal o one lead. In case of lead screw, screw roaes and nu moves along he axis over which ool pos is mouned. 83

6 heory of Machines Le dm be he mean diameer of he screw, be angle of fricion, and p be he pich. If one helix is unwound, i will be similar o an inclined plane for which he angle of inclinaion is given by (Figure 3.6) For single sar L = p an L dm an p dm If force acing along he axis of he screw is W, effor applied angenial o he screw (as discussed in Uni ) for moion agains force. P W an ( ) Also P W an ( ) for moion in direcion of force. P L = p W dm Figure 3.6 : ransmission Screw 3.3. Power ransmied orque acing on he screw dm W dm P an ( ) If speed is N rpm Power ransmied N wa 60 W dm an ( ) N kw POWER RANSMISSION BY BELS 84 In his secion, we shall discuss how power is ransmied by a bel drive. he bels are used o ransmi very small power o he high amoun of power. In some cases magniude of he power is negligible bu he ransmission of speed only may be imporan. In such cases he axes of he wo shafs may no be parallel. In some cases o increase he angle

7 of lap on he smaller pulley, he idler pulley is used. he angle of lap may be defined as he angle of conac beween he bel and he pulley. Wih he increase in angle of lap, he bel drive can ransmi more power. Along wih he increase in angle of lap, he idler pulley also does no allow reducion in he iniial ension in he bel. he use of idler pulley is shown in Figure 3.7. Power ransmission Devices Idler Pulley SAQ (a) (b) Figure 3.7 : Use of Idler in Bel Drive Wha is he main advanage of idler pulley? A prime mover drives a dc generaor by bel drive. he speeds of prime mover and generaor are 300 rpm and 500 rpm, respecively. he diameer of he driver pulley is 600 mm. he slip in he drive is 3%. Deermine diameer of he generaor pulley if bel is 6 mm hick Law of Beling he law of beling saes ha he cenre line of he bel as i approaches he pulley, mus lie in plane perpendicular o he axis of he pulley in he mid plane of he pulley oherwise he bel will run off he pulley. However, he poin a which he bel leaves he oher pulley mus lie in he plane of a pulley. he Figure 3.8 below shows he bel drive in which wo pulleys are a righ angle o each oher. I can be seen ha he cenre line of he bel approaching larger or smaller pulley lies in is plane. he poin a which he bel leaves is conained in he plane of he oher pulley. If moion of he bel is reversed, he law of he beling will be violaed. herefore, moion is possible in one direcion in case of non-parallel shafs as shown in Figure 3.8. Figure 3.8 : Law of Beling 85

8 heory of Machines 3.4. Lengh of he Bel For any ype of he bel drive i is always desirable o know he lengh of bel required. I will be required in he selecion of he bel. he lengh can be deermined by he geomeric consideraions. However, acual lengh is slighly shorer han he heoreically deermined value. Open Bel Drive he open bel drive is shown in Figure 3.9. Le O and O be he pulley ceners and AB and D be he common angens on he circles represening he wo pulleys. he oal lengh of he bel L is given by Le L = AB + Arc BHD + D + Arc GA r be he radius of he smaller pulley, R be he radius of he larger pulley, be he cenre disance beween he pulleys, and be he angle subended by he angens AB and D wih O O. D K J β G β = r β N O O R H A B Figure 3.9 : Open Bel Drive Draw O N parallel o D o mee O D a N. By geomery, For small value of ; O O, N = O J = D O K= Arc BHD = ( + ) R, Arc GA = ( ) r AB = D = O N = O O cos = cos R sin ( R r) sin cos sin sin r L ( ) R ( ) r sin ( R r), he approximae lenghs ( R r) R r L ( R r) ( R r) 86 ( R r) R r ( R r) his provides approximae lengh because of he approximaion aken earlier.

9 rossed-bel Drive he crossed-bel drive is shown in Figure 3.0. Draw O N parallel o he line D which mees exended O D a N. By geomery O J DO K O O N L Arc AG AB Arc BKD D Arc AG r ( ), and Arc BKD ( ) R Power ransmission Devices For small value of R r ( R r) sin or sin R r ( R r) cos sin sin L r ( ) cos R ( ) ( ) ( R r) cos B G J β R β O O r A β SAQ 3 For approximae lengh N Figure 3.0 : ross Bel Drive ( R r) ( R r) L ( R r) ( R r) ( R r) D K Which ype of drive requires longer lengh for same cenre disance and size of pulleys? one Pulleys Someimes he driving shaf is driven by he moor which roaes a consan speed bu he driven shaf is designed o be driven a differen speeds. his can be easily done by using sepped or cone pulleys as shown in Figure 3.. he cone pulley has differen ses of radii and hey are seleced such ha he same bel can be used a differen ses of he cone pulleys. 87

10 heory of Machines r 3 R 3 Le Figure 3. : one Pulleys N d be he speed of he driving shaf which is consan. N n be he speed of he driven shaf when he bel is on nh sep. r n be he radius of he nh sep of driving pulley. R n be he radius of he nh sep of he driven pulley. where n is an ineger,,,... he speed raio is inversely proporional o he pulley radii N N d r... (3.) R For his firs sep radii r and R can be chosen convenienly. N r Nn rn For second pair, and similarly. N R N R d In order o use same bel on all he seps, he lengh of he bel should be same d i.e. L L... Ln... (3.) hus, wo equaions are available one provided by he speed raio and oher provided by he lengh relaion and for seleced speed raio, he wo radii can be calculaed. Also i has o be kep in mind ha he wo pulleys are same. I is desirable ha he speed raios should be in geomeric progression. Le k be he raio of progression of speed. N N3 Nn... k N N Nn n 88 and 3 N k N N k N r Nn k N k N R n n d r r r r k and k R R R R 3 3 Since, boh he pulleys are made similar.

11 rn R r R or k R r R r n R r n n k... (3.3) If radii R and r have been chosen, he above equaions provides value of k or viceversa. SAQ 4 How he speed raios are seleced for cone pulleys? Power ransmission Devices Raio of ensions he bel drive is used o ransmi power from one shaf o he anoher. Due o he fricion beween he pulley and he bel one side of he bel becomes igh side and oher becomes slack side. We have o firs deermine raio of ensions. Fla Bel Le ension on he igh side be and he ension on he slack side be. Le be he angle of lap and le be he coefficien of fricion beween he bel and he pulley. onsider an infiniesimal lengh of he bel PQ which subend an angle a he cenre of he pulley. Le R be he reacion beween he elemen and he pulley. Le be ension on he slack side of he elemen, i.e. a poin P and le ( + ) be he ension on he igh side of he elemen. he ensions and ( + ) shall be acing angenial o he pulley and hereby normal o he radii OP and OQ. he fricion force shall be equal o R and is acion will be o preven slipping of he bel. he fricion force will ac angenially o he pulley a he poin S. R δ θ R P S Q δ θ δ θ + S O θ Figure 3. : Raio of ensions in Fla Bel onsidering equilibrium of he elemen a S and equaing i o zero. Resolving all he forces in he angenial direcion R cos ( ) cos 0 R cos... (3.4) 89

12 heory of Machines Resolving all he forces in he radial direcion a S and equaing i o zero. R sin ( ) sin 0 R ( ) sin Since is very small, aking limis cos and sin R ( ) Neglecing he produc of he wo infiniesimal quaniies negligible in comparison o oher quaniies : R which is Subsiuing he value of R and cos aking limis on boh sides as 0 d d Inegraing beween limis, i becomes d 0 ln V-bel or Rope d in Eq. (3.4), we ge e... (3.5) he V-bel or rope makes conac on he wo sides of he groove as shown in Figure 3.3. μ R n P δ θ/ S R n sinα Q δ θ/ θ + δ α α O R n α R n 90 (a) Figure 3.3 : Raio of ension in V-Bel (b)

13 Le he reacion be R n on each of he wo sides of he groove. he resulan reacion will be R n sin a poin S. Resolving all he forces angenially in he Figure 3.3(b), we ge Rn cos ( ) cos 0 Rn cos... (3.6) Power ransmission Devices Resolving all he forces radially, we ge Since is very small Rn sin sin ( ) sin sin ( ) sin Rn sin ( ) Neglecing he produc of he wo infiniesimal quaniies Rn sin R n sin Subsiuing he value of R n and using he approximaion cos, in Eq. (3.6), we ge sin sin aking he limis and inegraing beween limis, we ge d 0 ln sin sin d SAQ 5 (a) (b) e sin... (3.7) If a rope makes wo full urn and one quarer urn how much will be angle of lap? If smaller pulley has coefficien of fricion 0.3 and larger pulley has coefficien of fricion 0.. he angle of lap on smaller and larger pulleys are 60 o and 00 o which value of () should be used for raio of ensions? 9

14 heory of Machines Power ransmied by Bel Drive he power ransmied by he bel depends on he ension on he wo sides and he bel speed. Le be he ension on he igh side in N be he ension on he slack side in N, and V be he speed of he bel in m/sec. hen power ransmied by he bel is given by P Power P ( ) V Wa ( ) V kw... (3.8) 000 V kw 000 If bel is on he poin of slipping. e ( e ) V P kw... (3.9) 000 he maximum ension depends on he capaciy of he bel o wihsand force. If allowable sress in he bel is in Pa, i.e. N/m, hen ( b) N... (3.0) where is hickness of he bel in m and b is widh of he bel also in m. he above equaions can also be used o deermine b for given power and speed ension due o enrifugal Forces he bel has mass and as i roaes along wih he pulley i is subjeced o cenrifugal forces. If we assume ha no power is being ransmied and pulleys are roaing, he cenrifugal force will end o pull he bel as shown in Figure 3.4(b) and, hereby, a ension in he bel called cenrifugal ension will be inroduced. δ θ/ r F δ θ/ δ θ (a) (b) Figure 3.4 : ension due o enrifugal Foces Le be he cenrifugal ension due o cenrifugal force. Le us consider a small elemen which subends an angle a he cenre of he pulley. 9 Le m be he mass of he bel per uni lengh of he bel in kg/m.

15 he cenrifugal force F c on he elemen will be given by V F ( r m) r where V is speed of he bel in m/sec. and r is he radius of pulley in m. Resolving he forces on he elemen normal o he angen Power ransmission Devices F sin 0 Since is very small. sin F 0 F Subsiuing for F mv r r m V... (3.) herefore, considering he effec of he cenrifugal ension, he bel ension on he igh side when power is ransmied is given by ension of igh side and ension on he slack side s. he cenrifugal ension has an effec on he power ransmied because maximum ension can be only which is SAQ 6 b b m V Wha will be he cenrifugal ension if mass of bel is zero? Iniial ension When a bel is mouned on he pulley some amoun of iniial ension say 0 is provided in he bel, oherwise power ransmission is no possible because a loose bel canno susain difference in he ension and no power can be ransmied. When he drive is saionary he oal ension on boh sides will be 0. When bel drive is ransmiing power he oal ension on boh sides will be ( + ), where is ension on igh side, and is ension on he slack side. If effec of cenrifugal ension is negleced. 0 93

16 heory of Machines 0 If effec of cenrifugal ension is considered, hen 0 s 0... (3.) Maximum Power ransmied he power ransmied depends on he ension, angle of lap, coefficien of fricion and bel speed V. For a given bel drive, he maximum ension ( ), angle of lap and coefficien of fricion shall remain consan provided ha (a) (b) he ension on igh side, i.e. maximum ension should be equal o he maximum permissible value for he bel, and he bel should be on he poin of slipping. herefore, Power P ( e) V Since, c P ( ) ( e ) V c P ( m V ) ( e ) V For maximum power ransmied dp dv ( 3 m V ) ( e ) 3m V c c 3 mv 3 Also, V... (3.3) 3m A he bel speed given by he Eq. (3.3) he power ransmied by he bel drive shall be maximum. SAQ 7 Wha is he value of cenrifugal ension corresponding o he maximum power ransmied? 94

17 3.5 KINEMAIS OF HAIN DRIVE Power ransmission Devices he chain is wrapped round he sprocke as shown in Figure 3.4(d). he chain in moion is shown in Figure 3.5. I may be observed ha he posiion of axial line changes beween he wo posiion as shown by he doed line and full line. he doed line mees a poin B when exended wih he line of ceners. he firm line mees he line of ceners a poin A when exended. he speed of he driving sprocke say shall be consan bu he velociy of chain will vary beween O and O D. herefore, OA O B Figure 3.5 : Kinemaics of hain Drive he variaion in he chain speed causes he variaion in he angular speed of he driven sprocke. he angular speed of he driven sprocke will vary beween O B O A O A and OB his variaion can be reduced by increasing number of eeh on he sprocke. 3.6 LASSIFIAION OF GEARS here are differen ypes of arrangemen of shafs which are used in pracice. According o he relaive posiions of shaf axes, differen ypes of gears are used Parallel Shafs In his arrangemen, he shaf axes lie in parallel planes and remain parallel o one anoher. he following ype of gears are used on hese shafs : Spur Gears ώ D hese gears have sraigh eeh wih heir alignmen parallel o he axes. hese gears are shown in mesh in Figures 3.6(a) and (b). he conac beween he wo meshing eeh is along a line whose lengh is equal o enire lengh of eeh. I may be observed ha in exernal meshing, he wo shafs roae opposie o each oher whereas in inernal meshing he shafs roae in he same sense. ώ o o A B Line onac Line onac (a) Exernal Meshing Figure 3.6 : Spur Gears (b) Inernal Meshing If he gears mesh exernally and diameer of one gear becomes infinie, he arrangemen becomes Spur Rack and Pinion. his is shown in Figure 3.7. I convers roary moion ino ranslaory moion, or vice-versa. 95

18 heory of Machines Helical Gears or Helical Spur Gears Line onac Figure 3.7 : Spur Rack and Pinion In helical gears, he eeh make an angle wih he axes of he gears which is called helix angle. he wo meshing gears have same helix angle bu is layou is in opposie sense as shown in Figure 3.8. hrus Drivern hrus Figure 3.8 : Helical Gears he conac beween wo eeh occurs a a poin of he leading edge. he poin moves along a diagonal line across he eeh. his resuls in gradual ransfer of load and reducion in impac load and hereby reducion in noise. Unlike spur gears he helical gears inroduce hrus along he axis of he shaf which is o be borne by hrus bearings. Double-Helical or Herringbone Gears Driver A double-helical gear is equivalen o a pair of helical gears having equal helix angle secured ogeher, one having a righ-hand helix and he oher a lef-hand helix. he eeh of wo rows are separaed by a groove which is required for ool run ou. he axial hrus which occurs in case of single-helical gears is eliminaed in double helical gears. If he lef and righ inclinaions of a double helical gear mee a a common apex and groove is eliminaed in i, he gear is known as herringbone gear as shown in Figure Figure 3.9 : Herringbone Gears

19 3.6. Inersecing Shafs he moion beween wo inersecing shafs is equivalen o rolling of wo conical frusums from kinemaical poin of view. Sraigh Bevel Gears hese gears have sraigh eeh which are radial o he poin of inersecion of he shaf axes. heir eeh vary in cross secion hrough ou heir lengh. Generally, hey are used o connec shafs a righ angles. hese gears are shown in Figure 3.0. he eeh make line conac like spur gears. Power ransmission Devices Figure 3.0 : Sraigh Bevel Gears As a special case, gears of he same size and connecing wo shafs a righ angle o each oher are known as mire gears. Spiral Bevel Gears When he eeh of a bevel gear are inclined a an angle o he face of he bevel, hese gears are known as spiral bevel gears or helical bevel gears. A gear of his ype is shown in Figure 3.(a). hey run quier in acion and have poin conac. If spiral bevel gear has curved eeh bu wih zero degree spiral angle, i is known as zerol bevel gear Skew Shafs (a) Spiral Bevel Gear Figure 3. : Spiral Bevel Gears (b) Zerol Bevel Gear hese shafs are non-parallel and non-inersecing. he moion of he wo maing gears is equivalen o moion of wo hyperboloids in conac as shown in Figure 3.. he angle beween he wo shafs is equal o he sum of he angles of he wo hyperboloids. ha is he minimum perpendicular disance beween he wo shafs is equal o he sum of he hroa radii. Line of conac A B Ψ θ Ψ Figure 3. : Hyperboloids in onac 97

20 heory of Machines rossed-helical Gears or Spiral Gears hey can be used for any wo shafs a any angle as shown in Figure 3.3 by a suiable choice of helix angle. hese gears are used o drive feed mechanisms on machine ool. Worm Gears Figure 3.3 : Spiral Gears in onac I is a special case of spiral gears in which angle beween he wo axes is generally righ angle. he smaller of he wo gears is called worm which has large spiral angle. hese are shown in Figure 3.4. (a) (b) Hypoid Gears (c) Figure 3.4 : Worm Gears hese gears are approximaions of hyperboloids hough look like spiral bevel gears. he hypoid pinion is larger and sronger han a spiral bevel pinion. hey have qui and smooh acion and have larger number of eeh is conac as compared o sraigh bevel gears. hese gears are used in final drive of vehicles. hey are shown in Figure 3.5. (d) 98 Figure 3.5 : Hypoid Gears

21 3.7 GEAR ERMINOLOGY Power ransmission Devices Before considering kinemaics of gears we shall define he erms used for describing he shape, size and geomery of a gear ooh. he definiions given here are wih respec o a sraigh spur gear. Pich ircle or Pich urve I is he heoreical curve along which he gear rolls wihou slipping on he corresponding pich curve of oher gear for ransmiing equivalen moion. Pich Poin Pinion Rack I is he poin of conac of wo pich circles. I is he smaller of he wo maing gears. I is usually he driving gear. I is ype of he gear which has infinie pich circle diameer. ircular Pich I is he disance along he pich circle circumference beween he corresponding poins on he consecuive eeh. I is shown in Figure 3.6. Face Widh Addendum ircle Pich ircle op Land ircular Pich Space ooh Widh hicknes s Face Flank Addendu mm Working Deph Boom Land Dedendum learance Dedendum (Roo) ircle Figure 3.6 : Gear erminology If d is diameer of he pich circle and be number of eeh, he circular pich (p c ) is given by Diamenal Pich p c d... (3.4) I is defined as he number of eeh per uni pich circle diameer. herefore, diamenal pich (p d ) can be expressed as From Eqs. (3.4) and (3.5) p d... (3.5) d p c d d p d pc pd... (3.6) 99

22 heory of Machines 00 Module I is he raio of he pich circle diameer o he number of eeh. herefore, he module (m) can be expressed as From Eqs. (8.4) d m... (3.7) pc Addendum ircle and Addendum m... (3.8) I is he circle passing hrough he ips of gear eeh and addendum is he radial disance beween pich circle and he addendum circle. Dedendum ircle and Dedendum I is he circle passing hrough he roos of he eeh and he dedendum is he radial disance beween roo circle and pich circle. Full Deph of eeh and Working Deph Full deph is sum of addendum and dedendum and working deph is sum of addendums of he wo gears which are in mesh. ooh hickness and Space Widh ooh hickness is he hickness of ooh measured along he pich circle and space widh is he space beween wo consecuive eeh measured along he pich circle. hey are equal o each oher and measure half of circular pich. op Land and Boom Land op land is he op surface of he ooh and he boom land is he boom surface beween he adjacen filles. Face and Flank ooh surface beween he pich surface and he op land is called face whereas flank is ooh surface beween pich surface and he boom land. Pressure Line and Pressure Angle he driving ooh exers a force on he driven ooh along he common normal. his line is called pressure line. he angle beween he pressure line and he common angen o he pich circles is known as pressure angle. Pah of onac he pah of conac is he locus of a poin of conac of wo maing eeh from he beginning of engagemen o he end of engagemen. Arc of Approach and Arc of Recess Arc of approach is he locus of a poin on he pich circle from he beginning of engagemen o he pich poin. he arc of recess is he locus of a poin from pich poin upo he end of engagemen of wo maing gears. Arc of onac I is he locus of a poin on he pich circle from he beginning of engagemen o he end of engagemen of wo maing gears. Angle of Acion Arc of onac = Arc of Approach + Arc of Recess I is he angle urned by a gear from beginning of engagemen o he end of engagemen of a pair of eeh. Angle of acion = Angle urned during arc of approach + Angle urned during arc of recess

23 onac Raio I is equal o he number of eeh in conac and i is he raio of arc of conac o he circular pich. I is also equal o he raio of angle of acion o pich angle. Pich ircle Dedendum ircle Power ransmission Devices B F D Pah of onac Angle of Acion P Drivers A E Dedendum ircle Base ircle Pich ircle Ψ Pressure Angle Figure 3.7 : Gear erminology 3.8 GEAR RAIN A gear rain is combinaion of gears ha is used for ransmiing moion from one shaf o anoher. here are several ypes of gear rains. In some cases, he axes of roaion of he gears are fixed in space. In one case, gears revolve abou axes which are no fixed in space Simple Gear rain In his gear rain, here are series of gears which are capable of receiving and ransmiing moion from one gear o anoher. hey may mesh exernally or inernally. Each gear roaes abou separae axis fixed o he frame. Figure 3.8 shows wo gears in exernal meshing and inernal meshing. Le, be number of eeh on gears and. + P + (a) Exernal Meshing Figure 3.8 : Simple Gear rain (b) Inernal Meshing 0

24 heory of Machines Le N, N be speed in rpm for gears and. he velociy of P, V P N d N d N d N d Referring Figure 3.8, he wo meshing gears in exernal meshing roae in opposie sense whereas in inernal meshing hey roae in same sense. In simple gear rain, here can be more han wo gears also as shown in Figure Figure 3.9 : Gear rain Le N, N, N 3,... be speed in rpm of gears,, 3,... ec., and,, 3,... be number of eeh of respecive gears,, 3,..., ec. In his gear rain, gear is inpu gear, gear 4 is oupu gear and gears, 3 are inermediae gears. he gear raio of he gear rain is give by N N N N3 Gear Raio N N N N N N N ; and N N N N 3 herefore, N his expression indicaes ha he inermediae gears have no effec on gear raio. hese inermediae gears fill he space beween inpu and oupu gears and have effec on he sense of roaion of oupu gear. SAQ 8 (a) (b) here are six gears meshing exernally and inpu gear is roaing in clockwise sense. Deermine sense of roaion of he oupu gear. Deermine sense of roaion of oupu gear in relaion o inpu gear if a simple gear rain has four gears in which gears and 3 mesh inernally whereas oher gears have exernal meshing ompound Gear rain In his ype of gear rain, a leas wo gears are mouned on he same shaf and hey roae a he same speed. his gear rain is shown in Figure 3.30 where gears and 3 are mouned on same shaf and hey roae a he same speed, i.e. N N 3

25 3 4 Power ransmission Devices Figure 3.30 : ompound Gear rain Le N, N, N 3,... be speed in rpm of gears,, 3,..., ec. and,, 3,..., ec. be number of eeh of respecive gears,, 3,..., ec. N N N N N3 Gear Raio N N N N N herefore, unlike simple gear rain he gear raio is conribued by all he gears. his gear rain is used in convenional auomobile gear box. onvenional Auomobile Gear Box A convenional gear box of an auomobile uses compound gear rain. For differen gear engagemen, i may use sliding mesh arrangemen, consan mesh arrangemen or synchromesh arrangemen. Discussion of hese arrangemens is beyond he scope of his course. We shall resric ourselves o he gear rain. I can be explained beer wih he help of an example. Example 3. Soluion A sliding mesh ype gear box wih four forward speeds has following gear raios : op gear = hird gear =.38 Second gear =.4 Firs gear = 4 Deermine number of eeh on various gears. he minimum number of eeh on he pinion should no be less han 8. he gear box should have minimum size and variaion in he raios should be as small as possible. he gears in he gear box are shown in Figure 3.3 below : Engine Shaf Inpu Shaf Dog luch Main Splined Shaf (Oupu Shaf) A E G D F H Lay Shaf B Figure 3.3 : onvenional Gear Box 03

26 heory of Machines For providing firs gear raio, gear A meshes wih gear B and gear H meshes wih gear G. Speed of engine shaf Firs gear raio = Speed of oupu shaf N N N N N N N N N N A A H A H G H G B G [i.e. N B = N H ] B For smalles size of gear box A G H B B G A H If A = 0 eeh H = 0 A B = 0 = 40 eeh and G = 0 = 40 eeh Since cenre disance should be same A B D E F H G G H 04 E (3.9) D (3.0) F For second gear, gear A meshes wih gear B and gear E meshes wih gear F. N A.4 N G NA NF.4 N N B E.4 F.4 From Eqs. (0.) and (0.3) A E F B G F E (3.). 60 F F F E F E Since number of eeh have o be in full number. herefore, F can be eiher 8 or 9 and E can be eiher 3 or 3. If F = 8 and E = Second gear raio A E B F

27 If F = 9 and E = 3. Second gear raio From hese wo values of gear raios,.86 is closer o.4 han.38. For hird gear, gear A meshes wih gear B and gear D meshes wih gear. N A.38 N Power ransmission Devices NA ND.38 N N B.38 A B D D.38 From Eqs. (3.9) and (3.0) D D (3.) = 0.69 D D D = 60 D Eiher = 4 and D = 36 or = 5 and D = 35. If = 5 and D = 35. hird gear raio If = 4 and D = 36 hird gear raio D B A D Since.333 is closer o.38 as compared o.486. herefore, = 4 and D = 36 he op gear requires direc connecion beween inpu shaf and oupu shaf Power ransmied by Simple Spur Gear When power is bring ransmied by a spur gear, ooh load F n acs normal o he profile. I can be resolved ino wo componens F n cos and F n sin. F n cos acs angenially o he pich circle and i is responsible for ransmission of power Power ransmied (P) = F n cos. V where V is pich line velociy. 05

28 heory of Machines N m Since V 60 N m P F n cos 60 where is number of eeh and m is module. F n Pressure angle 06 Example 3. Soluion Figure 3.3 An open fla bel drive is required o ransmi 0 kw. he diameer of one of he pulleys is 50 cm having speed equal o 300 rpm. he minimum angle of conac may be aken as 70 o. he permissible sress in he bel may be aken as 300 N/cm. he coefficien of fricion beween bel and pulley surface is 0.3. Deermine (a) (b) widh of he bel neglecing effec of cenrifugal ension for bel hickness equal o 8 mm. widh of bel considering he effec of cenrifugal ension for he hickness equal o ha in (a). he densiy of he bel maerial is.0 gm/cm 3. Given ha Power ransmied (p) = 0 kw (a) Diameer of pulley (d) = 50 cm =.5 m Speed of he bel (N) = 300 rpm Angle of lap () oefficien of fricion () = 0.3 o radian 80 Permissible sress () = 300 N/cm hickness of he bel () = 8 mm = 0.8 cm Le higher ension be and lower ension be e e.53 he maximum ension is conrolled by he permissible sress.

29 b b b N 0 Here b is in mm 4b herefore, N Velociy of bel V N d m/s b 3.5 Power ransmied p ( ) V 4b kw Power ransmission Devices b 4b Since P = 0 kw 347.3b b 36.4 mm (b) he densiy of he bel maerial = gm/cm 3 Mass of he bel maerial/lengh, m = b mere b b kg/m b 0 kg/m enrifugal ension = m V 3 8b 0 (3.5) = 4.48b N Maximum ension ( max ) = 4b N max 4b 4.48b 9.58b Example 3.3 Power ransmied P V e Also P = 0 kw b b = 45.4 mm b 9.58b he effec of he cenrifugal ension increases he widh of he bel required. An open bel drive is required o ransmi 5 kw from a moor running a 740 rpm. he diameer of he moor pulley is 30 cm. he driven pulley runs a 300 rpm and is mouned on a shaf which is 3 meres away from he driving shaf. Densiy of he leaher bel is 0. gm/cm 3. Allowable sress for he bel maerial is 50 N/cm. If coefficien of fricion beween he bel and pulley is 0.3, deermine widh of he bel required. he hickness of he bel is 9.75 mm. 07

30 heory of Machines Soluion Given daa : Power ransmied (P) = 5 kw Speed of moor pulley (N ) = 740 rpm Diameer of moor pulley (d ) = 30 cm Speed of driven pulley (N ) = 300 rpm Disance beween shaf axes () = 3 m Densiy of he bel maerial () = 0. gm/cm 3 Allowable sress () = 50 N/cm oefficien of fricion () = 0.3 Le he diameer of he driven pulley be d N d = N d d N Nd cm 300 d d sin sin = radian.94 rad Mass of bel m = b one mere lengh 0. b where b is widh of he bel in mm 3 m b kg/m max b b N 0 0 Acive ension = max N d Velociy of bel V V =.6 m/s 3 m V b (.6) = 0.3 b N b 0.3 b 4.43 b 08 Power ransmied P V e

31 Example 3.4 Soluion e e b P b 5 or b 9 mm 00 An open bel drive has wo pulleys having diameers. m and 0.5 m. he pulley shafs are parallel o each oher wih axes 4 m apar. he mass of he bel is kg per mere lengh. he ension is no allowed o exceed 000 N. he larger pulley is driving pulley and i roaes a 00 rpm. Speed of he driven pulley is 450 rpm due o he bel slip. he coefficien of he fricion is 0.3. Deermine (a) (b) (c) Daa given : power ransmied, power los in fricion, and efficiency of he drive. Diameer of driver pulley (d ) =. m Diameer of driven pulley (d ) = 0.5 m enre disance () = 4 m Mass of bel (m) = kg/m Maximum ension ( max ) = 000 N Speed of driver pulley (N ) = 00 rpm Speed of driven pulley (N ) = 450 rpm oefficien of fricion () = 0.3 N 00 (a) 0.93 r/s N r/s Velociy of he bel (V) m/s enrifugal ension ( ) = m V = (.56) = N Power ransmission Devices Acive ension on igh side ( ) = max = = 84.5 N d d. 0.5 sin = 5.05 o o e e.43 09

32 heory of Machines Power ransmied ( P ) kw = 3.67 kw d (b) Power oupu W kw Power los in fricion = =.47 kw (c) Efficiency of he drive Power ransmied or 89%. Power inpu 3.67 Example 3.5 Soluion A leaher bel is mouned on wo pulleys. he larger pulley has diameer equal o. m and roaes a speed equal o 5 rad/s. he angle of lap is 50 o. he maximum permissible ension in he bel is 00 N. he coefficien of fricion beween he bel and pulley is 0.5. Deermine he maximum power which can be ransmied by he bel if iniial ension in he bel lies beween 800 N and 960 N. Given daa : Diameer of larger pulley (d ) =. m Speed of larger pulley = 5 rad/s Speed of smaller pulley = 50 rad/s Angle of lap () = 50 o Iniial ension ( 0 ) = 800 o 960 N Le he effec of cenrifugal ension be negligible. he maximum ension ( ) = 00 N e e N N Maximum power ransmied (P max ) = ( ) V d. Velociy of bel (V) 5 V = 5 m/s Pmax ( ) V ( ) W or kw

33 Example 3.6 Soluion A shaf carries pulley of 00 cm diameer which roaes a 500 rpm. he ropes drive anoher pulley wih a speed reducion of :. he drive ransmis 90 kw. he groove angle is 40 o. he disance beween pulley ceners is.0 m. he coefficien of fricion beween ropes and pulley is 0.0. he rope weighs 0. kg/m. he allowable sress for he rope is 75 N/cm. he iniial ension in he rope is limied o 800 N. Deermine : (a) (b) Given daa : number of ropes and rope diameer, and lengh of each rope. Diameer of driving pulley (d ) = 00 cm = m Speed of he driving pulley (N ) = 500 rpm Speed of he driven pulley (N ) = 50 rpm Power ransmied (P) = 90 kw Groove angle () = 40 o enre disance () = m oefficien of fricion () = 0. Mass of rope = 0. kg/m Allowable sress () = 75 N/cm Iniial ension ( 0 ) = 800 N d N 500 he velociy of rope 6.8 m/s enrifugal ension ( ) = 0. (6.8) = 8.5 N ( d d) sin 0.5 Power ransmission Devices = sin 0.5 = 4.8 Angle of lap () = = 5 o o sin 0 e 4.67 or.636 radian = 4.67 Iniial ension ( 0 ) N = 53. N = 4.67 = 8.0 N V 6.8 P ( ) ( ) 4.3 kw Numbers of ropes required (n) 7.8 or say 8 ropes, 4.3

34 heory of Machines Maximum ension ( max ) = + = = 64.5 N max d d d = 3.03 cm his is open bel drive, herefore, formula for lengh of rope is given by ( R r) R r L ( R r) d d R m, r 0.5 m ( 0.5) 0.5 L ( 0.5) 3.9 SUMMARY ( ) 8.7 m. he power ransmission devices are bel drive, chain drive and gear drive. he bel drive is used when disance beween he shaf axes is large and here is no effec of slip on power ransmission. hain drive is used for inermediae disance. Gear drive is used for shor cenre disance. he gear drive and chain drive are posiive drives bu hey are comparaively coslier han bel drive. Similarly, bel drive should saisfy law of beling oherwise i will slip o he side and drive canno be performed. When bel drive ransmis power, one side will become igh side and oher side will become loose side. he raio of ension depends on he angle of lap and coefficien of fricion. If coefficien of fricion is same on boh he pulleys smaller angle of lap will be used in he formula. If coefficien of fricion is differen, he minimum value of produc of coefficien of fricion and angle of lap will decide he raio of ension, i.e. power ransmied. Due o he mass of bel, cenrifugal ension acs and reduces power ransmied. For a given bel drive he power ransmied will be maximum a a speed for which cenrifugal ension is one hird of maximum possible ension. he gears can be classified according o he layou of heir shafs. For parallel shafs spur or helical gears are used and bevel gears are uded for inersecing shafs. For skew shafs when angle beween he axes is 90 o worm and worm gears are used. When disance beween he axes of shaf is larger and posiive drive is required, chain drive is used. We can see he use of chain drive in case of anks, moorcycles, ec. 3.0 KEY WORDS Spur Gears Helical Gears : hey have sraigh eeh wih eeh layou is parallel o he axis of shaf. : hey have curved or sraigh eeh and is inclinaion wih shaf axis is called helix angle.

35 Herringbone Gears Bevel Gear Spiral Gears Worm Gears Rack and Pinion Hypoid Gears Pich ylinders Pich Diameer ircular Pich Diameral Pich Module Addendum Dedendum Pressure Angle : I is a double helical gear having lef and righ inclinaions which mee a a common apex and here is no groove in beween hem. : hey have eeh radial o he poin of inersecion of he shaf axes and hey vary in cross-secion hroughou heir lengh. : hey have curved eeh which are inclined o he shaf axis. hey are used for skew shafs. : I is special case of spiral gears where angle beween axes of skew shafs is 90 o. : Rack is special case of a spur gear whose pich circle diameer is infinie and i meshes wih a pinion. : hese gears are approximaions of hyperboloids bu hey look like spiral gears. : A pair of gears in mesh can be replaced by a pair of imaginary fricion cylinders which by pure rolling moion ransmi he same moion as pair of gears. : I is diameer of pich cylinders. : I is he disance beween corresponding poins of he consecuive eeh along pich cylinder. : I is he raio of number of eeh o he diameer of he pich cylinders. : I is he raio of diameer of pich cylinder o he number of eeh. : I is he radial heigh of ooh above pich cylinder. : I is he radial deph of ooh below pich cylinder. : I is he angle beween common angen o he wo pich cylinders and common normal a he poin of conac beween eeh (pressure line). Power ransmission Devices 3. ANSWERS O SAQs SAQ Available in ex. SAQ (a) Available in ex. (b) Available in ex. SAQ 3 (a) Available in ex. (b) Daa given : Speed of prime mover (N ) = 300 rpm Speed of generaor (N ) = 500 rpm Diameer of driver pulley (d ) = 600 mm Slip in he drive (s) = 3% hickness of bel () = 6 mm 3

36 heory of Machines N d If here is no slip. N d If hickness of bel is appreciable and no slip N d N d If hickness of bel is appreciable and slip is S in he drive N d S N d 00 SAQ 4 SAQ 5 SAQ 6 SAQ 7 SAQ d ( d 6) d mm Available in ex. Available in ex. Available in ex. Available in ex. Available in ex. 4