SOLID MECHANICS TUTORIAL GEAR SYSTEMS. This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 3.

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1 SOLI MEHNIS TUTORIL GER SYSTEMS This work covers elemens of he syllabus for he Edexcel module 21722P HN/ Mechanical Principles OUTOME 3. On compleion of his shor uorial you should be able o do he following. escribe he differen ypes of gear sysems. escribe a simple gear rain. escribe a compound gear rain. escribe hree ypes of epicyclic gear boxes Solve gear box raios. alculae he inpu and oupus speeds and orques of gear boxes. alculae he holding orque on gear box cases I is assumed ha he suden is already familiar wih he following conceps. ngular moion. Power ransmission by a shaf. ll hese above may be found in he pre-requisie uorials..j.unn 1

2 1. INTROUTION gear box is a device for convering he speed of a shaf from one speed o anoher. In he process he orque T is also changed. This can be done wih pulley and chain drives bu gears have advanages over hese sysem. good example is ha of winch in which a moor wih a high speed and low orque is geared down o urn he drum a a low speed wih a large orque. Similarly, a marine engine may use a reducion gear box o reduce he speed of he engine o ha of he propeller. Oher examples are moor vehicles, lahes, drills and many more. The diagram shows a ypical winch ha has a reducion gear box buil inside he drum. Figure 1 This uorial is no abou he design of gears bu i should be menioned ha here are many ypes of gears, each wih heir own advanages. Here are some examples. Figure 2 Gears are wheels which mesh wih each oher hrough inerlocking eeh. Roaion of one wheel produces roaion of he oher wih no slip beween hem. The shape of he gear eeh is imporan in order o produce a smooh ransfer of he moion. The mos common shape is he INVOLUTE gear form bu i is no our ask o sudy his here. The design of he gear eeh also affecs he relaive posiion of one gear o anoher. For example bevelled gears allow he axis of one gear o be inclined o he axis of anoher. Worm gears conver he moion hrough 90 o and so on. The design also affecs he fricion presen in he ransfer..j.unn 2

3 2. SI GER OX THEORY onsider a simple schemaic of a gear box wih an inpu and oupu shaf. Figure 3 GER OX RTIO INPUT SPEE N1 The raio of he gear box is defined as G.R. OUTPUT SPEE N2 N is usually in rev/min bu he raio is he same whaever unis of speed are used. If angular velociy is used hen INPUT SPEE 1 G.R. OUTPUT SPEE 2 TORQUE N EFFIIENY The power ransmied by a orque T Nm applied o a shaf roaing a N rev/min is 2πNT given by S.P.. In an ideal gear box, he inpu and oupu powers are he 60 same so 2 π N 1 T1 2 N2T2 T2 N1 π N1T1 N2T2 G.R T1 N 2 I follows ha if he speed is reduced, he orque is increased and vice versa. In a real gear box, power is los hrough fricion and he power oupu is smaller han he power inpu. The efficiency is defined as: Power Ou 2π N2T2 x 60 N2T2 η Power In 2π N1T1 x 60 N1T1 ecause he orque in and ou is differen, a gear box has o be clamped in order o sop he case or body roaing. holding orque T 3 mus be applied o he body hrough he clamps. Figure 4 The oal orque mus add up o zero. T 1 + T 2 + T 3 0 If we use a convenion ha ani-clockwise is posiive and clockwise is negaive we can deermine he holding orque. The direcion of roaion of he oupu shaf depends on he design of he gear box..j.unn 3

4 WORKE EXMPLE No.1 1. gear box has an inpu speed of 1500 rev/min clockwise and an oupu speed of 300 rev/min aniclockwise. The inpu power is 20 kw and he efficiency is 70%. eermine he following. i. The gear raio ii. The inpu orque. iii. The oupu power. iv. The oupu orque. v. The holding orque. SOLUTION INPUT SPEE N 1500 G.R. OUTPUT SPEE N πN1T1 60 x Power In Power In T1 60 2πN T T x T Nm (Negaive clockwise) 2π x 1500 Power Ou η 0.7 power Ou 0.7 x Power In 0.7 x kw Power In 2πN 2T2 60 x Power Ou Power ou T2 60 2πN 60 x T Nm (posiive anic clockwise) 2π x 300 T + T + T Nm (niclockwise) J.unn 4

5 SELF SSESSMENT EXERISE No.1 1. gear box has an inpu speed of 2000 rev/min clockwise and an oupu speed of 500 rev/min aniclockwise. The inpu power is 50 kw and he efficiency is 60%. eermine he following. i. The inpu orque. (238.7 Nm) ii. The oupu power. (30 kw) iii. The oupu orque. (573 Nm) iv. The holding orque. (334.3 Nm clockwise) 2. gear box mus produce an oupu power and orque of 40 kw and 60 Nm when he inpu shaf roaes a 1000 rev/min. eermine he following. i. The gear raio. (0.1571) ii. The inpu power assuming an efficiency of 70% (57.14 kw).j.unn 5

6 3. TYPES OF GER TRINS The meshing of wo gears may be idealised as wo smooh discs wih heir edges ouching and no slip beween hem. This ideal diameer is called he Pich ircle iameer (P) of he gear. 3.1 SIMPLE GER TRIN. Figure 5 These are ypically spur gears as shown in diagram 1. The direcion of roaion is reversed from one gear o anoher. The only funcion of he idler gear is o change he direcion of roaion. I has no affec on he gear raio. The eeh on he gears mus all be he same size so if gear advances one ooh, so does and. number of eeh on he gear. Pich circle diameer. m modem / and his mus be he same for all gears oherwise hey would no mesh. m / / / m m m angular velociy. v linear velociy on he circle. v /2 The velociy v of any poin on he circle mus be he same for all he gears, oherwise hey would be slipping. I follows ha m m or in erms of rev/min N N N The gear raio is defined as GR Inpu speed/oupu speed m If gear is he inpu and gear he oupu, GR N / N /.J.unn 6

7 WORKE EXMPLE No.2 simple rain has 3 gears. Gear is he inpu and has 50 eeh. Gear is he oupu and has 150 eeh. Gear roaes a 1500 rev/min aniclockwise. alculae he gear raio and he oupu speed. The inpu orques on is 12 Nm and he efficiency is 75%. alculae he oupu power and he holding orque. SOLUTION GR N / N / 150/50 3 N / N 3 N N /3 1500/3 500 rev/min (aniclockwise) T 12 Nm P (inpu) 2πN T /60 2π x 1500 x 12/ W P (oupu) P (Inpu) x η 1885 x W 60P(oupu) 60 x T 2πN 2π x 500 T + T + T hold 0 27 Nm T hold 0 T hold - 39 Nm (clockwise) SELF SSESSMENT EXERISE No.2 simple gear rain has 2 spur gears. The inpu gear has 20 eeh and he oupu gear has 100 eeh. The inpu roaes a 2000 rev/min clockwise. alculae he gear raio and he oupu speed. (5 and 400 rev/min aniclockwise) The inpu orque is 15 Nm and he efficiency is 65%. alculae he oupu power and he holding orque. (2 042 W and Nm clockwise).j.unn 7

8 .J.unn OMPOUN GERS ompound gears are simply a chain of simple gear rains wih he inpu of he second being he oupu of he firs. chain of wo pairs is shown below. Gear is he oupu of he firs pair and gear is he inpu of he second pair. Gears and are locked o he same shaf and revolve a he same speed. Figure 6 The velociy of each ooh on and are he same so as hey are simple gears. Likewise for and,. G.R. Since gears and are on he same shaf, x x and Since 2πN hen he gear raio may be wrien as ( ) ( ) G.R. N OUT N IN Gears and are he driven gears. Gears and are he driver gears. I follows ha: Gear raio produc of driven eeh/produc of driving eeh This rule applies regardless of how many pairs of gears here are.

9 WORKE EXMPLE No.3 alculae he gear raio for he compound chain shown below. If he inpu gear roaes clockwise, in which direcion does he oupu roae? Gear has 20 eeh Gear has 100 eeh Gear has 40 eeh Gear has 100 eeh Gear E has 10 eeh Gear F has 100 eeh Figure 7 SOLUTION The driving eeh are, and E. The driven eeh are, and F Gear raio produc of driven eeh/produc of driving eeh Gear raio (100 x 100 x 100)/ (20 x 40 x 10) 125 lernaively we can say here are hree simple gear rains and work o he raio for each. Firs chain GR 100/20 5 Second chain GR 100/ Third chain GR 100/10 10 The overall raio 5 x 2.5 x Each chain reverses he direcion of roaion so if is clockwise, and roae aniclockwise so and E roae clockwise. The oupu gear F hence roaes aniclockwise..j.unn 9

10 SELF SSESSMENT EXERISE No.3 Gear is he inpu and revolves a 1200 rev/min clockwise viewed from he lef end. The inpu orque is 30 Nm and he efficiency is 70%. Gear has 50 eeh Gear has 150 eeh Gear has 30 eeh Gear has 60 eeh alculae he following. Figure 8 i. The oupu speed and is direcion. (200 rev/min clockwise) ii. The oupu power. (2639 W) iii. The fixing orque. (156 Nm aniclockwise).j.unn 10

11 3.3 EPIYLI GERS Epicyclic means one gear revolving upon and around anoher. The design involves plane and sun gears as one orbis he oher like a plane around he sun. Here is a picure of a ypical gear box. Figure 9 This design can produce large gear raios in a small space and are used on a wide range of applicaions from marine gear boxes o elecric screwdrivers. SI THEORY The diagram shows a gear on he end of an arm. Gear meshes wih gear and revolves around i when he arm is roaed. is called he plane gear and he sun. Figure 10 Firs consider wha happens when he plane gear orbis he sun gear. Figure 11.J.unn 11

12 Observe poin p and you will see ha gear also revolves once on is own axis. ny objec orbiing around a cenre mus roae once. Now consider ha is free o roae on is shaf and meshes wih. Suppose he arm is held saionary and gear is roaed once. spins abou is own cenre and he number of revoluions i makes is he raio /. will roae by his number for every complee revoluion of. Now consider ha is unable o roae and he arm is revolved once. Gear will revolve / + 1 because of he orbi. I is his exra roaion ha causes confusion. One way o ge round his is o imagine ha he whole sysem is revolved once. Then idenify he gear ha is fixed and revolve i back one revoluion. Work ou he revoluions of he oher gears and add hem up. The following abular mehod makes i easy. Suppose gear is fixed and he arm makes one revoluion. eermine how many revoluions he plane gear makes. Sep 1 is o revolve everyhing once abou he cenre. Sep 2 idenify ha should be fixed and roae i backwards one revoluion keeping he arm fixed as i should only do one revoluion in oal. Work ou he revoluions of. Sep 3 is simply add hem up and we find he oal revs of is zero and for he arm is 1. Sep cion 1 Revolve all once Revolve by -1 rev 0 + / -1 3 dd 1 1+ / 0 The number of revoluions made by is (1 + /). Noe ha if revolves -1, hen he direcion of is opposie so + / WORKE EXMPLE No.4 simple epicyclic gear has a fixed sun gear wih 100 eeh and a plane gear wih 50 eeh. If he arm is revolved once, how many imes does he plane gear revolve? SOLUTION Sep cion 1 Revolve all once Revolve / dd Gear makes 3 revoluions for every one of he arm. The design so far considered has no idenifiable inpu and oupu. We need a design ha pus an inpu and oupu shaf on he same axis. This can be done several ways..j.unn 12

13 METHO 1 Figure 12 The arm is he inpu and gear is he oupu. Gear is a fixed inernal gear and is normally par of he ouer casing of he gear box. There are normally four plane gears and he arm akes he form of a cage carrying he shafs of he plane gears. Noe ha he plane gear and inernal gear boh roae in he same direcion. Figure 13.J.unn 13

14 WORKE EXMPLE No.5 n epicyclic gear box has a fixed ouer gear wih 240 eeh. The plane gears have 20 eeh. The inpu is he arm/cage and he oupu is he sun gear. alculae he number of eeh on he sun gear and he raio of he gear box. SOLUTION The P of he ouer gear mus he sum of P of he sun plus wice he P of he planes so i follows ha he number of eeh are relaed as follows x Idenify ha gear is fixed and he arm mus do one revoluion so i mus be roaed back one revoluion holding he inpu saionary. Sep cion 1 Revolve all once Revolve / /200 3 dd The raio / is hen 1: 2.2 and his is he gear raio..j.unn 14

15 METHO 2 In his case he sun gear is fixed and he inernal gear is made ino he oupu. Figure 14 WORKE EXMPLE No.6 n epicyclic gear box has a fixed sun gear and he inernal gear is he oupu wih 300 eeh. The plane gears have 30 eeh. The inpu is he arm/cage. alculae he number of eeh on he sun gear and he raio of he gear box. SOLUTION x Idenify ha gear is fixed and he arm mus do one revoluion so i mus be ha is roaed back one revoluion holding he arm saionary. Sep cion 1 Revolve all once Revolve /30 240/ dd The raio / is hen 1: 1.8 and his is he gear raio. Noe ha he soluion would be he same if he inpu and oupu are reversed bu he raio would be 1.8..J.unn 15

16 METHO 3 In his design a compound gear and is inroduced. Gear is fixed and gears roae upon i and around i. Gears are rigidly aached o gears and hey all roae a he same speed. Gears mesh wih he oupu gear E. Figure 15 WORKE EXMPLE No.7 n epicyclic gear box is as shown above. Gear has 100 eeh, has 50, has 50 and E has 100. alculae he raio of he gear box. SOLUTION Idenify ha gear is fixed and ha mus do one revoluion so i mus be ha is roaed back one revoluion holding saionary. Sep cion / E 1 Revolve all once Revolve ½ -¼ 3 dd ½ ¾ The raio /E is hen ¾ :1 or 3:4 Noe ha he inpu and oupu may be reversed bu he soluion would be he same wih a raio of 4:3 insead of 3:4.J.unn 16

17 SELF SSESSMENT EXERISE No.4 1. n epicyclic gear box is designed as shown. The inpu roaes a 200 rev/min clockwise viewed from he lef wih a orque of 40 Nm. The efficiency is 75%. alculae he following. i. The gear box raio. (16:1) ii. The oupu speed and is direcion. (12.5 rev/min clockwise) iii. The power oupu. (628.3 W) iv. The holding orque. (520 Nm aniclockwise) Figure n epicyclic gear box is designed as shown. The inpu roaes a 100 rev/min clockwise viewed from he righ wih a orque of 20 Nm. The efficiency is 65%. alculae he following. i. The gear box raio. (1:5) ii. The oupu speed and is direcion. (500 rev/min clockwise) iii. The power oupu. (136.1 W) iv. The holding orque. (22.6 Nm aniclockwise) Figure 17.J.unn 17

18 3. n epicyclic gear box as shown has a fixed sun gear and he inernal gear is he oupu wih 400 eeh. The plane gears have 150 eeh. The inpu is he arm/cage. The oupu mus deliver 5 kw of power a 900 rev/min. The inpu power is 7 kw. alculae he following. i. The gear box raio. (1:5) ii. The inpu speed and is direcion. (180 rev/min same direcion as inpu) iii. The efficiency. (71.4%) iv. The holding orque. (424.4 Nm) Figure n epicyclic gear box as shown has a fixed sun gear wih 150 eeh. Gear has 30 eeh and i is compounded wih which has 130 eeh. The inpu shaf delivers 200 W a 2400 rev/min. The gear box efficiency is 55%. alculae he following. i. The number of eeh on E. (50) ii. The gear box raio. (12:1) iii. The oupu speed and is direcion. (200 rev/min in reverse direcion) iv. The holding orque. (4.46 Nm) Figure 19.J.unn 18

19 3.4 MORE VNE EPIYLI OXES Someimes he case of he epicyclic gear box is allowed o roae. In his case he soluion should be along hese lines. Figure 20 Keep saionary give 1 rev 1 / 0 - / Muliply by x (revs of ) x x / 0 -x / Lock he gears and roae all y imes x + y y -x / + y x is he revoluions of and y he revoluions of Given he speed of any wo gears, he speed of he oher may be deduced. WORKE EXMPLE No.8 n epicyclic gear box is as shown above. Gear has 28 eeh and has 64. Shaf roaes a 200 rev/min and he case is allowed o roae a 100 rev/min in he same direcion. alculae he speed of shaf. alculae he raio of he gear box. SOLUTION The eeh on mus be Keep saionary give 1 rev 1 120/ /64 Muliply by x (revs of ) x 120x/ x/64 Lock he gears and roae all y imes x + y 120x/28+y y -120x/64+ y The speed of is 100 rev/min so y 100 The speed of is 200 rev/min so -120x/64+ y x/ x rev/min (opposie direcion).j.unn 19

20 SELF SSESSMENT EXERISE No.5 1. n epicyclic gear box is designed as shown. The number of eeh on is 80 and on is 40. The inpu roaes a 300 rev/min clockwise viewed from he lef and he case is allowed o roae a 150 rev/min in he opposie direcion. alculae he speed of shaf. (-75 rev/min).j.unn 20