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1 MECHANICAL DRIVES: They are Two groups,. Drives ha ransmi power by means of fricion: eg: bel rives an rope rives.. Drives ha ransmi power by means of engagemen: eg: chain rives an gear rives. However, he selecion of a proper mechanical rive for a given applicaion epens upon number of facors such as cenre isance, velociy raio, shifing arrangemen, Mainenance an cos. GEAR DRIVES Gears are efine as oohe wheels, which ransmi power an moion from one shaf o anoher by means of successive engagemen of eeh. The cenre isance beween he shafs is relaively small.. I can ransmi very large power. I is a posiive, an he velociy raio remains consan. 4. I can ransmi moion a a very low velociy. our groups: CLASSIICATION O GEARS: ) Spur Gears ) Helical gears ) Bevel gears an 4) Worm Gears Spur Gear Helical Gear

2 Bevel Gear Worm Gear Se NOMEN CLATURE Spur gears are use o ransmi roary moion beween parallel shafs. They are usually cylinrical is shape an he eeh are sraigh an parallel o he axis of roaion. In a pair of gears, he larger is ofen calle he GEAR an, he smaller one is calle he PINION Nomenclaure of Spur Gear. Pich Surface: The pich surfaces of he gears are imaginary planes, cyliners or cones ha roll ogeher wih ou slipping.. Pich circle: I is a heoreical circle upon which all calculaions are usually base. I is an imaginary circle ha rolls wih ou slipping wih he pich circle of a maing gear. urher, pich circles of a maing gears are angen o each oher.

3 . Pich circle iameer: The pich circle iameer is he iameer of pich circle. Normally, he size of he gear is usually specifie by pich circle iameer. This is enoe by 4. Top lan: The op lan is he surface of he op of he gear ooh 5. Base circle: The base circle is an imaginary circle from which he involue curve of he ooh profile is generae (he base circles of wo maing gears are angen o he pressure line) 6. Aenum: The Aenum is he raial isance beween he pich an aenum circles. Aenum inicaes he heigh of ooh above he pich circle. 7. Deenum: The eenum is he raial isance beween pich an he eenum circles. Deenum inicaes he eph of he ooh below he pich circle. 8. Whole Deph: The whole eph is he oal eph of he ooh space ha is he sum of aenum an Deenum. 9. Working eph: The working eph is he eph of engagemen of wo gear eeh ha is he sum of heir aenums 0. Clearance: The clearance is he amoun by which he Deenum of a given gear excees he aenum of i s maing ooh.. ace: The surface of he gear ooh beween he pich cyliner an he aenum cyliner is calle face of he ooh.. lank: The surface of he gear ooh beween he pich cyliner an he roo cyliner is calle flank of he ooh.. ace Wih: is he wih of he ooh measure parallel o he axis. 4. ille raius: The raius ha connecs he roo circle o he profile of he ooh is calle fille raius. 5. Circular pich: is he isance measure on he pich circle, from a poin on one ooh o a corresponing poin on an ajacen ooh. 6. Circular ooh hickness: The lengh of he arc on pich circle subening a single gear ooh is calle circular ooh hickness. Theoreically circular ooh hickness is half of circular pich. 7. Wih of space: (ooh space) The wih of he space beween wo ajacen eeh measure along he pich circle. Theoreically, ooh space is equal o circular ooh hickness or half of circular pich 8. Working eph: The working eph is he eph of engagemen of wo gear eeh, ha is he sum of heir aenums 9. Whole eph: The whole eph is he oal eph of he ooh space, ha is he sum of aenum an eenum an (his is also equal o whole eph + clearance) 0. Cenre isance: i is he isance beween cenres of pich circles of maing gears. (i is also equal o he isance beween cenres of base circles of maing gears). Line of acion: The line of acion is he common angen o he base circles of maing gears. The conac beween he involue surfaces of maing eeh mus be on his line o give smooh operaion. The force is ransmie from he riving gear o he riven gear on his line.. Pressure angle: I is he angle ha he line of acion makes wih he common angen o he pich circles.

4 4. Arc of conac: Is he arc of he pich circle hrough which a ooh moves from he beginning o he en of conac wih maing ooh. 4. Arc of approach: i is he arc of he pich circle hrough which a ooh moves from is beginning of conac unil he poin of conac arrives a he pich poin. 5. Arc of recess: I is he arc of he pich circle hrough wich a ooh moves from he conac a he pich poin unil he conac ens. 6. Conac Raio? Velociy raio: if he raio of angular velociy of he riving gear o he angular velociy of riven gear. I is also calle he spee raio. 7. Moule: I is he raio of pich circle iameer in mih meers o he number of eeh. i is usually enoe by m Mahemaically m D Z 8. Back lash: I is he ifference beween he ooh space an he ooh hickness as measure on he pich circle. 9. Velociy Raio: Is he raio of angular velociy of he riving gear o he angular velociy of riven gear. I is also calle he spee raio.

5 5 NOTATION ENGLISH SYMBOLS A o B a o r m r a r b r o R R g Z α σ σ b σ H σ HB σ HE σ HL τ ω Suffix Suffix Cenre isance ace wih Aenem circle iameer Pich circke iameer Roo circle iameer Moule Aenem circle raius Base circle raius Pich circle raius Raius of curvaure of ooh profile Gear raio Number of eeh Preasure angle Sress value Bening sress Herz conac sress Conac sress a he beginning of he engagemen Conac sress a he en of he engagemen Piing limi sress Shear sress Angle velociy Pinion Gear Nomenclaure of Spur Gear ailure Map of Involue Gears

6 6 ailure Map of Involue Gears Gear Se

7 7 Differen Phases of Gear Tooh Conac

8 8 Expressions for he Calculaion of Equivalen Raii of Curvaure a Various Phases of Conac Design consieraion for a Gear rive In he esign of gear rive, he following aa is usually given i. The power o be ransmie ii. The spee of he riving gear iii. The spee of he riven gear or velociy raio iv. The cenre isance The following requiremens mus be me in he esign of a gear rive (a) The gear eeh shoul have sufficien srengh so ha hey will no fail uner saic loaing or ynamic loaing uring normal running coniions (b) The gear eeh shoul have wear characerisics so ha heir life is saisfacory. (c) The use of space an maerial shoul be recommene () The alignmen of he gears an eflecions of he shaf mus be consiere because hey effec on he performance of he gears (e) The lubricaion of he gears mus be saisfacory

9 9 Selecion of Gears: The firs sep in he esign of he gear rive is selecion of a proper ype of gear for a given applicaion. The facors o be consiere for eciing he ype of he gear are General layou of shafs Spee raio Power o be ransmie Inpu spee an Cos. Spur & Helical Gears When he shaf are parallel. Bevel Gears When he shafs inersec a righ angles, an,. Worm & Worm Gears When he axes of he shaf are perpenicular an no inersecing. As a special case, when he axes of he wo shafs are neiher inersecing nor perpenicular crosse helical gears are employe. The spee reucion or velociy raio for a single pair of spur or helical gears is normally aken as 6:. On rare occasions his can be raise o 0:. When he velociy raio increases, he size of he gear wheel increases. This resuls in an increase in he size of he gear box an he maerial cos increases. or high spee reucion wo sage or hree sage consrucion are use. The normal velociy raio for a pair of ben gears is : which can be increase o : uner cerain circumsances. or high-spee reucion worm gears offers he bes choice. The velociy raio in heir case is 60:, which can be increase o 00:. They are wiely use in maerials hanling equipmen ue o his avanage. urher, spur gears generae noise in high-spee applicaions ue o suen conac over he enire face wih beween wo meeing eeh. Where as, in helical gears he conac beween he wo meshing eeh begans wih a poin an graually exens along he ooh, resuling in guie operaions. rom consieraions spurgears are he cheapes. They are no only easy o manufacure bu here exiss a number of mehos o manufacure hem. The manufacuring of helical, bevel an worm gears is a specialize an cosly operaion. Low of Gearing: The funamenal law of gearing saes The common normal o he boh profile a he poin of conac shoul always pass hrough a fixe poin calle he pich poin, in orer o obain a consan velociy raio. MODULE: The moule specifies he size of gear ooh. igure shows he acual sizes of gear ooh wih four ifferen moules. I is observe ha as he moules increases, he size of he gear ooh also increases. I can be sai ha moule is he inex of he size of gear ooh.

10 0 Sanar values of moule are as shown. The moule given uner choice, is always preferre. If ha is no possible uner cerain circumsances moule uner choice, can be selece. Sanar proporions of gear ooh in erms of moule m, for 0º full eph sysem. Aenum m Deenum.5 m Cl earance (c) 0.5 m Working eph m Whole eph.5 m Tooh hickness.5708 m Tooh space.5708 m ille raius 0.4 m π z mz π.5708m z

11 Sanar Tooh proporions of involue spur gear Selecion of Maerial : The loa carrying capaciy of he gear ooh epens upon he ulimae ensile srengh or yiel srengh of he maerial. When he gear ooh is subjece o flucuaing forces, he enurance srengh of he ooh is he eciing facor. The gear maerial shoul have sufficien srengh o resis failure ue o breakage of e ooh. In many cases, i is wear raing raher han srengh raing which ecies he imensions of gear ooh. The resisance o wear epens upon alloying elemens, grawn size, percenage of carbon an surface harness. The gear maerial shoul have sufficien surface enurance srengh o avoi failure ue o esrucive piing. or high-spee power ransmission, he sliing velociies are very high an he maerial shoul have a low co-efficien of fricion o avoi failure ue o scoring. The amoun of hermal isorion or wrapping uring he hea reamen process is a major problem on gear applicaion. Due o wraping he loa ges concenrae a one corner of he gear ooh. Alloy seels are superior o plain carbon seel in his respec (Thermal isorion)

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13 orce analysis Spur gearing. We know ha, he reacion beween he maing eeh occur along he pressure line, an he power is ransmie by means of a force exere by he ooh of he riving gear on he meshing ooh of he riven gear. (i.e. riving pinion exering force P N on he ooh of riven gear). Accoring o funamenal law of gear his resulan force P N always acs along he pressure line. This resulan force P N, can be resolve ino wo componens angenial componen P an raial componens P r a he pich poin.

14 4 The angenial componen P is a useful componen (loa) because i eermines he magniue of he orque an consequenly he power, which is ransmie. The raial componen P r services no useful purpose (i is a separaing force) an i is always irece owars he cenre of he gear. The orque ransmie by he gear is given by P 60 M N m π N Where, M Torque ransmie gears (N- m) PkW Power ransmie by gears N Spee of roaion (rev / mn) The angenial componen acs a he pich circle raius. M OR M Where, M Torque ransmie gears N- mm Pich Circle iameer, mm urher, we know, Power ransmie by gears π N M ( kw) 60 Where r an anα resulan force, N Cosα

15 5 The above analysis of gear ooh force is base on he following assumpions. i) As he poin of conac moves he magniue of resulan force P N changes. This effec is neglece. ii) iii) I is assume ha only one pair of eeh akes he enire loa. A imes, here are wo pairs ha are simulaneously in conac an share he loa. This aspecs is also neglece. This analysis is vali uner saic coniions for example, when he gears are running a very low velociies. In pracice here are ynamic forces in aiion o force ue o power ransmission. or gear ooh forces, I is always require o fin ou he magniue an irecion of wo componens. The magniues are eermine by using equaions P 60 M π N M urher, he irecion of wo componens an r are ecie by consrucing he free boy iagram.? How Minimum Number of Teeh: The minimum number of eeh on pinion o avoi inerference is given by Z min sin α or 0 full eph involue sysem, i is always safe o assume he number of eeh as 8 or 0 Once he number of eeh on he pinion is ecie, he number of eeh on he gear is calculae by he velociy raio i Z Z ace Wih: In esigning gears, i is require o express he face wih in erms of moule.

16 6 In pracice, he opimum range of face wih is 9.5m b. 5m Generally, face wih is assume as en imes moule b. 5m The LEWIS Bening Equaion: Wilfre Lewis inrouce an equaion for esimaing he bening sress in gear eeh. This equaion announce in 89 sill remains he basis for mos gear esign oay. In he lewis analysis, he gear ooh is reae as a canilever beam an he angenial componen ( ) causes he bening momen abou he base of he ooh. GEAR TOOTH AS CANTILEVER The Lewis equaion is base on he following assumpion. (i) (ii) (iii) (iv) The effec of raial componen ( r ) which inuces compressive sresses is neglece. I is assume ha he angenial componen ( ) is uniformly isribue over he face wih of he gear (his is possible when he gears are rigi an accuraely machine.) The effec of sress concenraion is neglece. I is assume ha a any ime only one pair of eeh is in conac an Takes oal loa

17 7 I is observe ha he cross secion of he ooh varies from free en o fixe en. Therefore, a parabola is consruce wih in he ooh profile an shown in oe lines. Gear ooh as parabolic beam The avanage of parabolic ou lines is ha i is a beam of uniform srengh, an he sress a any cross secion is uniform. We know M b h I b an y M b y I M M b b σ b (Z Secion moulus) ( I / y) Z b / b 6 σ Permissible bening sress (N/mm ) b 6 b b h b σ b 6 h Muliplying he numeraor an enominaor of he righ han sie by m, (mmoule) mb σ b 6hm The brackee quaniy epens on he from of he ooh an is erme as lewis from sress facor Y Le y 6hm Then he equaion can be rewrien as This y is calle as lewis form facor mbσ y b

18 8 When he sress reaches he permissible magniue of bening sresses, he corresponing force ( ) is calle he beam srengh (S b ) S b mb σ b y Where, S b beam srengh of gear ooh (N) σ b Permissible bening sress (N/mm ) The above equaion is known as LEWIS EQUATION The values of he lewis from facor y is given in able below,

19 9 In orer o avoi he breakage of gear ooh ue o bening, he beam srengh shoul be more han he effecive force beween he meshing eeh In esign of gears, I is require o ecie he weaker beween pinion an gear. When he same maerial is use for pinion an gear, he pinion is always weaker han he gear Why? We know ha S b mbσ y b I can be observe ha m an b are same for pinion an as well as for gear in a gear pair, When ifferen maerials are use, he prouc pinion an gear The lewis form facor y is always less for pinion compare o gear σ. b y ecies he weaker beween he Therefore, when he same maerial is use for pinion an gear, he pinion is always weaker han he gear. Effecive loa-calculaion Earlier we have seen how o eermine he angenial componen of he resulan force beween wo meshing eeh. This componen can be calculae by using I. P 60 M π N An II. M The value of he angenial componen, epens upon rae power an rae spee. In gear esign, he maximum force (ue o maximum orque) is he crierion. This is accoune by means of a facor calle service facor (C s ) This service facor (C s ) is efine as C s Maximum Torque Rae Torque

20 0 C s ( M ) ( ) M max max Where, ( ) is he angenial force ue o rae orque (M ) ( ) Cs max The values of service facors are given in able We know, ha σ b is permissible saic bening sress which is moifie o C vσ b where, C v is he velociy facor use for aking ino accoun he faigue loaing This velociy facor C v evelope by. Carl. G. Barh, expresse in erms of pich line velociy. The values of velociy facor are as below (i) C v +V, for orinary an commercially cu gears (mae wih form cuers) an V<0 m / Sec (ii) 6 C v 6 + V, or accuraely hobbe an generae gears an V < 0 m/sec.

21 (iii) 5. 6 C v v, or precision gears wih shaving grining an lapping an V > 0 m/sec Where, v pich line Velociy (m/sec) πn 60 0, mm n, rev/min (The velociy facor is an empirical relaionship evelope by pas experience). Dynamic effecs (Dynamic Tooh Loa) When gears roae a very low spee, he ransmie loa P can be consiere o be he acual force presen beween wo meshing eeh However in mos of he cases he gears roae a appreciable spee an i becomes necessary o consier he ynamic force resuling from impac beween maing eeh. The Dynamic force is inuce ue o he following facors. Inaccuracies of he ooh profile. Errors in ooh spacings. Misalignmen beween bearings 4. Elasiciy of pars, an 5. Ineria of roaing masses. There are wo mehos o accoun for Dynamics loa. I. Approxinae esimaion by he velociy facor in he preliminary sages of gear esign II. Precise esimaion by Bucking Hams equaion in he final sages of gear esign. Noe: Approximae esimaion, Using velociy facor (C v ) evelope by Barh iscusse earlier. In he final sages of gear esing when gear imensions are known errors specifie an qualiy of gears eermine, he Dynamic loa is calculae by equaion erive by Earle Buckingham Where, Dynamic loa + i Where, Tangenial ooh loa i Inevemen loa ue o ynamic acion

22 kv + k V + ( Cb + ) c b + Where, V Pich line Velociy (m/sec) C Dynamic facor (N/mm ) epening upon machining errors e measure error in acion beween gears in mm b face wih of ooh (mm) angenial force ue o rae orque (N) K 0.67 in SI unis The Dynamic facor C, epens upon moulus of elasiciy of maerials for pinion an gear an he form ooh or pressure angle an i is given by Where C K E e + E K Consan epening upon he form of ooh (ake from DDH) E Moulus of elasiciy of pinion maerial (N/mm ) E Moulus of elasiciy of gear maerial (N/mm ) The Values of K, for various ooh forms are given as. The error, e, in he ynamic loa equaion is measure error in acion beween gears in mm This error epens upon he qualiy of gear an manufacuring mehos.

23 WEAR TOOTH LOAD WEAR: or gears wear is efine as loss of maerial from conacing surfaces of eeh. I is furher classifie as Normal wear Moerae wear Desrucive wear Abrasive wear Scraching an ec. Generally, normal wear (Polishinging in) oes no consiue failure because i involves loss of meal a a rae oo slow o affec performance Moerae wear refers o loss of meal more rapi han normal wear. This nee no necessarily be esrucive an may evelop on heavily loae gear eeh. Desrucive wear usually resuls from loaing ha is excessive for he lubrican employe. The effec of esrucive wear on he ooh profile of an involue gear is epice in he figure. PITTING Piing is he principal moe of failure of rolling surfaces. The eails of he process vary wih he maerial an operaing coniions, bu in all cases i manifess iself by he iniiaion an propagaion of cracks in he near surface layer unil microscopic pieces each hemselves o form a pi or a spall. In spur gears surface piing has long been recognise as one of he failure moes. This is ofen referee o as Pich line Piing The main facors affecing piing ype of failure, Conac sress. Maerial pouring an harness. Surface finish an lubricaion

24 4 Conac sress was originally conceive By HERTZ (896) in whose name i is ofen referre o as Herz Conac Sress. The failure of he gear ooh ue o piing occurs when he conac sress beween wo meshing eeh excees he surface enurance srengh he maerial In orer o avoi his ype of failure, he proporions of he gear ooh an surface properies such as surface harness shoul be selece in such a way ha he wear srengh of he gear ooh is more han he effecive loa beween he meshing eeh. The Herz sress is base on he assumpions of elecric an isomeric maerial behaviours, loa is compressive an normal o he conacing surfaces which are saionary an he size of conacing area whose imensions are relaively smaller compare wih he curvaure raius of he conacing boies The above figure, Illusraes he conac are an corresponing sress isribuion beween wo cyliners. Here he area of conac sress which is heoreically recangular wih one imension being he cyliner lengh L. (i.e. corresponing o face wih of he gear) The isribuion of pressure is represene by a semi ellipical prism an he maximum conac pressure Po exiss on he loa axis, The curren gear esign pracice is o esimae he conac sress a he pich poin of he eeh by assuming line conac beween wo cyliners whose rail of conac epens on he gear geomery a he pich poin. The analysis of wear srengh was one by Earle Buckingham an was accepe by AGMA (American Gear Manufacuring Associaion) in 96. This Buckingham equaion gives he wear srengh of he gear ooh base on Herz heory of conac sress. Hence, he maximum ooh loa from wear consieraion as evaluae from Herz conac sress equaion applie for pich poin conac is given by bqk

25 5 Where, Pich circle iameer of pinion in mm. b ace wih of he pinion in mm. Q Raio facor VR VR + Z Z + Z an K Loa sress facor (also known as maerial combinaion facor in N / mm ) This loa sress facor epens upon he maximum faigue limi of compressive sress, he pressure angle, an he moulus of elasiciy of he maerials of he gear. Accoring o Buckingham, his loa sress facor is given by ( σ ) es Sin α K +.4 E E Where, σ es Surface enurance limi of a gear pair. (N / mm ) [.75 (BHN) -70] Where BHN he average Brinall harness number of gear an pinion for he seels Design proceure for spur gears: (i) in esign angenial ooh loa, from power ransmie an pich line velociy T 000 p C V s. (ii) Apply lewis relaionship i.e. σ. P. b y σ C bπ m y C. v a) This lewis equaion is applie only o he weaker of he wo wheels b) When boh he pinion an gear are mae of he same maerial, hen pinion is weaker. c) When he pinion an gear are mae of ifferen maerials hen he prouc of (σ o x y) is he eciing facor. The lewis equaion is use o ha wheel for which (σ o x y) is less. ) The prouc of [(σ o. C v ).y] is calle as srengh facor of he gear e) The face wih may be aken as 9.5m o.5m for cu eeh an 6.5m o 9.5 m for cas eeh. (iii) Calculae he ynamic loa ( ) on he ooh by using Backingham equaion, i.e.,

26 6 D + i by (iv) in he saic ooh loa (i.e.,. Beam srengh or he enurance srengh of he ooh) by using he relaion, s σ e b p c y for safey agains breakage σ e b π m y S > (v) inally fin he wear ooh loa by using he relaion W b Q K The wear loa ( W ) shoul no be less han he ynamic loa ( D ) Design a pair of spur gears o ransmi 0kW of power while operaing for 8 0 hrs/ay susaining meium shock, from shaf roaing a 000 rev/min o a parallel shaf which is o roae a 0 rev/min. Assume he number of eeh on pinion o be an 0 full eph involue ooh profile. if loa facor C N/mm an also for wear loa aking loa sress facor, K 0.79 N/mm. Sugges suiable harness. Bohhe pinio gears are mae of cas seel 0.% corbon (unreae) whose σ 7.4 N/mm check he esign for ynamic loa if Given: P 0kW, 0x0 W, Z, Z 00, V. R :.5, 0 ull eph. N 000 rev/min, N 0 rev/min Maerial: Cas seel 0.% C, (unreae) σ 7.4 N/mm Type of loa: Meium skock, wih 8-0hrs/ay. C ynamic facors epening up on machining errors N/mm K loa sress facor (wear) 0.79 N/mm Soluion: σ Allowable saic sress 07.0 N/mm(Pinion) σ 8.N/mm(Gear).05 Le C v (assume).05+v V pich lin velociy V N π 60

27 7 π m Z N 60 m 000 π 6.5m 60 V.6m mm/sec mm / Sec or, Meium shock, wih 08-0 hrs/ay he service facor C s, for gear, C s.5 The angenial ooh loa 000 p. C s P, in kw, V V, m/ Sec m N m Now C v m. W.K.T, Tooh form facor for he pinion, (for 0 full eph) Z an Tooh from facor, for he gear Z ( Q Z 00)

28 8 σ x y 7.4 x σ x y 7.4 x Since (σ.y )is less han (σ.y ), here fore PINION is WEAKER Now, by using he lewis equaion o he pinion, We have, σ bπ m y C v m m π m + m m m m By hi an rial meh, m LHS RHS Le m M 04.5 Hence, m moule 4.5 is OK. Bu he sanar moule is 5.0 mm Le us ake m 5.0 mm ace wih b 0m (assume) 0 x 5 50mm Pich circle iameer of i) Pinion, mz 5 x 55mm ii)gear, mz 5 x mm m 5 N s

29 9 V.6m.6 x m/sec Checking he gear for ynamic an wear loas We know, ha, he Dynamic loa on gear ooh + i i K v ( Cb + ) K v + Cb ( ) (980) N s Assuming: σ en 59.0 N /mm Sae ooh loa or enurance srengh of he ooh en σ en.bπ en 59 x 50 x π x 5 x N or meium shock aking en.5.5 x en i.e., Design is safe

30 0 Wear loa W.K.T, VR Q Raio facor. VR w, b Q K 55 x 50 x.8 x N Is he esign is safe from he poin of wear? fin new k k.08 Hea reae for 50 BHN w > esign is safe A pair of carefully cu spur gears wih 0 sub involue profile is use o ransmi a maximum power.5 kw a 00 rev/min. The velociy raio is :. The maerial use for boh pinion an gear is meium cas iron, whose allowable, saic sress may be aken as 60 Mpa. The approximae cener isance may be aken as 600 mm, eermine moule an face wih of he spur pinion an gear. Check he gear pair for ynamic an wear loas The ynamic facor or eformaions facor in Bucking hams ynamic loa equaion may be aken as 80, an maerial combinaion/loa sress facor for he wear may be aken as.4 Given: VR, N 00 rev/min, N 00 rev/ min, P Power ransmie,.5 kw Cener isance L 600mmσ σ 60Mpa, C 80, K.4 Assumpion: i) b face wih 0m ii) Seay loa coniion an 8 0 hrs/ay C s.0

31 Boh he gear an pinion are moe of he same maerial. Therefore pinion is weaker an he esign will be base on pinion. W.K.T, Cenre isance beween he shafs (L) 600mm +. an mm 400mm 0.4 m 800mm 0.8m V Pich line velociy of pinion N π V π. 4.m / sec 60 Since V i pich line velociy is less han m/sec he velociy facor C v, may be aken as v v Now, Z. m m y ooh form facor (for 0 sub sysems) Z m m W.K.T, Design angenial ooh loa P 0 C v s N

32 W.K.T, σ. C v bπ m x y 60 x 0.4 x 0m x πm x ( m) Solving for m, we ge m 6.5 m 8.0 (sanar) ace wih b 0m 0 x 8 80mm Z Z m 400. m m Checking wo gears for ynamic an wear loa W.K.T (i) Dynamic loa T + i g ( ) T ( ) N W.K.T, y Tooh form facor for pinion m [ x 8] Le flexural enurance limi (σe) for cas iron may be aken as 85 Mpa ( 85 N/mm) en σ en. b π my π N or seay loas en.5 f..

33 N W.K.T, Q Raio facor VR VR + +. w b Q K 400 x 80 x. x N Since boh en an w are greaer han, he esign is safe

34 4 Helical Gears: A Helical gear has eeh in he form of helix aroun he gear. Two such gears may be use o connec wo parallel shafs in place of spur gears. The helixes may be righ hane on one gear an lef hane on he oher. The pich surfaces are cylinrical as in spurgearing, bu he eeh insea of being parallel o he axis, win aroun he cyliners helically like screw hreas. The eeh of helical gears wih parallel axis have line conacs as in spurgearing. This provies graual engagemen an coninuous conac of he engaging eeh. Hence helical gears give smooh rive wih high efficiency of ransmission. The helical gears may be single helical ype or ouble helical ype. In case of single helical ype here is some axial hrus beween he eeh which is a isavanage. In orer o eliminae his axial rus ouble helical gears (i.e., herning bone gears) are use. I is equivalen o wo single helical gears, In which equal an opposie hruss are provie on each gear an he resuling axial hrus is zero.

35 5 Terms use: Helix angle: I is consan angle mae by he helices wih he axis of roaion Axial pich: I is he isance parallel o he axis beween similar faces of ajacen eeh. I is same as circular pich an is herefore enoe by P C. Normal pich: I is he isance beween similar faces of ajacen eeh along a helix on he pich cyliners normal o he eeh. I is enoe by P N. again P N P C cos β an N an cosβ N Normal pressure angle Pr. angle ace wih: In orer o have more han one pair of eeh in conac, he ooh isplacemen (i.e., he avancemen of one en of ooh over he oher en) or over lap shoul be aleas equal o he axial pich such ha, over lap P C b an β (i) The normal ooh loa ( N ) has wo componens, one is angenial componen ( ) an he oher axial componen ( A ) as shown in fig The axial or en hrus is given by A N sin β an β (ii) rom he above equaion (i), we see ha as he helix angle increases hen he ooh over lop increases. Bu a he same ime he en hrus as given by he equaion (ii) also increases which is no esirable. I is usually recommene ha he over lop shoul be 5% of he circular pich. Over lop b an β. P C

36 6. P an β Q b minimum face wih m Moule, Noe:. The maximum face wih may be aken as.5 o 0.0m. In case of ouble helical or herring bone gears he minimum face wih is given by b ( ) c p c π ª m.pc b b an β. π m an β. π m sin β. In a single helical gears, he helix angle ranges from 0 o 5, while for ouble helical gears i may be mae up o 45 b.5 m n To 0.m n. ormaive or equivalen number of eeh for helical gear: The formaive or equivalen number of eeh for a helical gear may be efine as he number of eeh ha can be generae on he surface of a cyliner having a raius equal o he raius of curvaure a a poin a he ip of he minor axis of an ellipse obaine by aking a secion of he gear in he normal plane. Mahemaically, formaive or equivalen number of eeh an a helical gear Z E Z / cos. β Z Acual number of eeh on a helical gear an β helix angle. Proporion of Helical Gears: AGMA Recommenaions. Pressure angle in he plane of roaion 5 o 5 Helix angle, β 0-45 Aenum 0.8 m (maximum) Deenum.0 m Minimum oal eph.8 m (maximum) Minimum clearance 0. m Thickness of ooh.5708 m STRENGTH O HELICAL GEARS: (P96 K/G) In helical gears, he conac beween maing eeh is graual, saring a one en an moving along he eeh so ha a any insan he line of conac runs iagonally across he eeh. Therefore, in orer o fin he srengh of helical gear, a moifie lewis equaion is use. I is given by, T σ o. C V b π m y. Where (i) T, σ o, C V, b, π, m, as usual, wih same meanings,

37 7 An y Tooh from facor or lewis facor corresponing o he ORMATIVE OR VIRTUAL OR EQUIVALENT NUMBER O TEETH. The values of C V, velociy facor, from equaion, (D.D.H) Iem (a) or low-angle helical gears when v is less han 5 m/s (b) or all helical an herringbone gears when v is 5 o 0 m/ (c) or gears when v is 0 o 0 m/s (Barh s formula) () or precision gear wih v greaer han 0 m/s (e) or non meallic gears 4.58 C v C v 6. + v 5.5 C v v 5.55 C v C v v v v Equaion (ii) The ynamic ooh loa, + i Where i K K v ( cb cos β + ) cos + ( cb cos β + ) / v K S 0.67 in SI unis 6.60 in meric unis, (iii) The saic ooh loa or enurance srengh of he ooh is given by β S σ e b π my The maximum or limiing wear ooh loa for helical gears is given by, b Q K w cos β Where. b, Q an K have usual meanings as iscusse in spur gears In his case, Where K The loa sress facor K ( σ ) es sin.4 α N E + E

38 8 Pair of helical gears are o ransmi 5 kw. The eeh are 0 sub in iameral plane an have a helix angle of 45. The pinion runs a 0,000 rev/min an has 80 mm pich iameer. The gear has 0 mm pich iameer. If he gears are mae of cas seel having allowable saic srengh of 00 Mpa. Deermine he suiable moule an face wih from saic srengh consieraions an check he gears for ynamic an wear loas. given σ es 68 MPa Given: P 5kW 5x0 W, 0, β 45, N 0,000 rev/min, 80mm 0.08 m, 0 mm 0. m, σ σ 00 MPa 00 N/mm, σ es 68 MPa 68 N/mm Since, boh he pinion an gear are mae of he same maerial (i.e., cas seel) he pinion is weaker. Thus he esign is base on he pinion. W K T, Torque ransmie by he pinion T P π N π 0, N-m Tangenial ooh loa on he pinion T / / 58 N W.K.T Number of eeh on he pinion Z m 80 m An formaive or equivalen number of eeh for pinion Z E Z cos β 80 cos / m / m 6 ( ) m.. 4 Tooh from facor for pinion for 0 sub eeh y' Z E m 6.4 / m W.K.T V π N ,000 π 4 m / Sec C v Q V is greaer han 0 m/sec V

39 9 C v Since maximum face wih,(b) for helical gear may be aken as.5 m o 0.0 m. Le us ake b.5 m W.K.T angenial ooh loa ( ) 58 (σ. C V ) b π m y (00 x 0.46) x.5m x π x m x ( ) 7m.5m By Trial an hi meho, Soluion for m, m. say.5 mm (sanar) an face wih b.5 m.5 x.5.5 mm say.0 mm cheeking he gear for wear: 0 WKT. V.R 4 80 VR 4 8 Q.6 VR WKT. an α N an cos β an 0 cos α N 4.4 Since, boh he gears are mae of same maerial (i.e., cas seel). Therefore, le E E 00 x 0 N/ mm Loa sress facor K σ. Sin N es α.4 E + E 68 Sin N / mm

40 40 W.K.T, b Q K 80 Cos β W Cos N Since maximum loa for wear is much more han he angenial loa on he ooh. Design is saisfacory for wear consieraion. Seminally ry for ynamic loa + i k v ( Cb cos β ) cos β + k v + C Cos β + b C ynamic facor epening upon machine error 7.0 (for an error of 0.04) (7 Cos (0.67 4) + (7 Cos ) cos ) cos 45 D?

41 4 Bevel gears: The bevel gears are use o ransmi power a a consan velociy raio beween wo shafs whose axes inersec a a cerain angle. The pich surfaces for he bevel gear are frusums of cones. CLASSIICATION O BEVEL GEARS: Classifie epening upon he angles beween he shafs an he pich surfaces. (i) Mier gears: when equal bevel gears (having equal eeh an equal pich angles) connec wo shafs whose axes inersec a righ angles as shown, hen hey are known as mier gear. (ii) Angular bevel gears: when he bevel gears connec wo shafs whose axes inersec a an angle oher han a righ angle, hen hey are known as angular bevel gears. (iii)crown bevel gears: when bevel gears connec wo shafs whose axes inersec a an angle greaer han a righ angle an one of he bevel gears has a pich angle of 90 hen i is known as a crown gear. The crown gear correspons o a rack in spur gearing as shown. (iv) Inernal bevel gears: when he eeh on he bevel gear are cu on he insie of he pich cone hen hey are known ass iner bevel gears. Noe: The bevel gears may have sraigh or spiral eeh. I may be assume, unless oher wise sae ha he bevel gear has sraigh eeh an an he axes of he shafs inersec a righ angle.

42 4 TERMS USED IN BEVEL GEARS: A secional view of wo bevel gears in mesh is as shown. The following erms are imporan from he subjec poin of view. (i) Pich cone: I is a cone conaining he pich elemens of he eeh. (ii) Cone cenre: I is he apex of he pich cone. I may be efine as ha poin where he axes of wo maing gears inersec each oher. (iii) Pich angle: I is he angle mae by he pich line wih he axis of he shaf. I is enoe by (i.e, δ & δ ) (iv) Cone isance: I is he lengh of he pich cone elemen. I is also calle as a pich cone raius. I is enoe by OP Mahemaically cone isance or pich cone raius pich raius D p/ DG/ OP sino sino sino p p p (v) Aenum angle: I is he angle subene by he aenum of he ooh a he cone cenre. I is enoe by θ a. Mahemaically aenum angle. an θ a h a Sin δ h a Sin δ (vi) Deenum angle: I is he angle subene by he Deenum of he ooh a he cone cenre. I is enoe by θ. Mahemaically,

43 4 an θ h f sin δ h sin f δ Where, h a, h a aenum of he pinion an gear respecively, mm h f, h f eenum of pinion an gear respecively, mm (vii) ace angle: I is he angle subene by he face of he ooh a he cone cenre. The face angle is equal o he pich angle plus aenum angle. (viii) Roo angle: I is he angle subene by he roo of he ooh a he cone cenre. I is equal o he pich angle minus eenum angle (ix) Back cone: (Normal cone): I is he imaginary cone perpenicular o he pich cone a he en of he ooh. (x) Crown heigh: I is he isance of he crown poin C, from he cone cenre O, parallel o he axis of he gear. I is he enoe by C (xi) (xii) (xiii) Mouning heigh: I is he isance of he back of he boss from he cone cenre. I is enoe by m Pich iameer: I is he iameer of he larges pich circle. Ousie or aenum cone iameer: I is he maximum iameer of he eeh of he gear. I is equal o he iameer of he blank from which he gear can be cu. Mahemaically ousie ia, O + ha, Cos δ O + ha, Cos δ Proporions of Bevel gears: The proporion for he bevel gear may be aken as (i) Aenum: a.0 m (ii) Deenum:. m (iii) Clearance 0. m (iv) Working eph.0m (v) Tooh hiknes.5708 ormaive or Equivalen number of eeh for Bevel Gears: (Tregol s approximaion) Z e Z/Cosδ

44 44 STRENGTH O BEVEL GEARS: The srengh of a bevel gear ooh is obaine in a similar way as iscusse in he previous aricles. The moifie form of he lewis equaion for he angenial ooh loa is given as follows ( σ C v ) b π m y L L b y lewis form facor base on formaive or equivalen number of eeh L Slan heigh of pich cone (or cone isance) + Where an are he pich circle iameers on he larger iameer of pinion an gears respecively L b (i) The facor i.e, may be calle as bevel facor L (ii) or saisfacory operaion of bevel gears he face wih shoul be from 6m o 0 m. Also raio L/b shoul no excee, (i.e., b L / )for his he number of eeh 48 in he pinion mus no be less han + ( vr) (iii) The ynamic loas for bevel gears may be obaine in he same similar manner as iscusse for spur gears. (iv) The saic ooh loa or enurance srengh of he ooh for bevel gears is given by e σ e b π m y L b L The value of flexural enurance limi (σ e ) may be aken from able (v) The maximum or limiing loa for wear for bevel gears is given by w D b Q e k Cos δ Where, D, b, Q, k, have usual meanings as iscusse in spur gears excep ha Q e is base on formaive or equivalen number of eeh, such ha, Q Ze Ze + Ze

45 45 A pair of bevel gears o connec wo shafs a righ angles an ransmi 9 kw. The allowable saic sress for pinion an gear maerials may be aken respecively as 85 MPa an 55 MPa an brinill harness of 00 an 60. The spee may be assume as 00/40 an number of eeh may be assume as for pinion an 60 for gear. Tooh profile may be aken as 0 full eph involue. Check he esign for ynamic an wear loas. Given: θ s 90, P 9kW 9000W, Z, Z 60, σ 85 MPa, σ 55 MPa, N 00 rev/min, N 40 rev/min, α 0 (full eph involue) in, moule, an check he esign for ynamic an wear loas, Since, he shafs are a righ angles, Therefore, pich angle for pinion, anδ Z Z i δ an δ 90 δ an he pich angle for gear W.K.T, formaive number of eeh for pinion Z e δ 9. Z cos δ δ cos 9. Z Z e. 5 Z Z an e cos δ cos Z e 8. 54

46 46 W.K.T. for 0 full eph involue sysem ooh from facor or pinion y Z e ( ) y an for gear σ y σ y Since he prouc, σ is less han σ herefore Gear is weaker, an hus he esign shoul be base on gear only. W.K.T. y y P 0 v Here v. π. N π mz N m π 60 0m v.0m m / Sec mm/ Sec

47 47 Now P v.0 m m m N Taking velociy facor C v v (aking ino consieraion ha gears are very accuraely cu an groun gears having a pich line velociy from 6 m/ sec o 0 m/sec) C v m W.K.T, Lengh of pich cone elemen L Sin δ m 60 Sin 70.7 m m L.78 m Assuming he face wih b / r of he lengh of he pich cone elemen L, L.78 m b m b 0.60 m

48 48 W.K.T, The angenial ooh loa on gear ( σ C b π m v ) y L L b m m 0.60 m.78 m 0.60m π m m m m m m 09 m Solving his by hi an rial meho we ge, m 4.58 m 5.0 (Sanar) an b 0.60 x m 0.60 x mm b face wih 5.0 mm Thus, m x 60 5 x mm m x 5 x 05 mm & L.78 m.78 x

49 49 Check for ynamic loa W.K.T, Pich line velociy V.0 m m/sec. x 5 v m / Sec an angenial ooh loa on he gear m N N rom able he ooh error in acion for firs class commercial gears having moule 5 mm is e Take K 9.0 for 0 full eph eeh an E 0 x 0 N/mm an E 84 x 0 N/mm C ynamic facor epening upon machining errors e k( / E + / E ) ( )

50 ( ) N / m C ynamic facor 70..N/m W.K.T Dynamic loa on he gear + i k + k v + v ( cb + ) c b ( ) ( ) N

51 5 σ en for gear maerial of BHN 60, is aken as 8.5 N/mm urher, we know saic ooh loa or enurance srengh of he ooh S σ en L b b π my L 8.5 x 5 x π x 5 x x 5 x π x 5 x x S N Since S <, he esign is no saisfacory from he sanpoin of ynamic loa. I is known ha S.5 for seay loas i.e., ynamic loa on gear mus be reuce i.e, by assuming for a saisfacory esign agains ynamic loa, le us ake he precision gears (class III) having ooh error in acion e mm C 00.0 N/mm ( ) D ( )

52 D N rom he above we see ha by aking precision gear, S is greaer han D, herefore, he esign is saisfacory from he sanpoin of ynamic loa s here D (Hence, esign is safe) Check for wear loa or 80 BHN, σ es may be 67.8 N /mm normally seel for pinion an cos iron for gear of 00 & 60, (Hence in he able ake 80 & 80) k loa sress facor ( σ es ) Sinφ.4 E + E ( 67.8) Sin x N / mm k.55 N / mm

53 5 an Q e raio facor Ze Ze + Ze W.K.T Maximum or limiing loa for wear W * b Q e K (*? or pinion, please explain) 05 x 5 x.78 x.55 w N Since, w is greaer han D he esign is saisfacory from he sanpoin of wear also