Lecture 5.pptx hoyfsghfdoswyfjhjzkucusasufhj

ywgl

1. Lecture Contents : Kinematics Dynamics Kinematic link or element Types of links Structure Kinematic Pair Classification of Kinematic Pair Types of Joints Degree of Freedom

2. I n t r oduction • If a numb e r of bodies a r e assembled in such a w a y th a t the m o ti o n of one c auses c on s t r ained and p r e d ic t able mo t ion t o the othe r s, i t is kn o wn as a mec h a n ism. A mecha n ism tran s mits a nd m o di f ies a moti o n. A machine is a mecha n ism or a c om b ination of mecha n isms which, a p art f r om imp a rting de f ini t e m o tions t o the p a rts, also tran s mits a nd m o di f ies the a v aila b le mec h a n i c al ene r gy i n t o s o me kind of de s ired work. T h u s , a m ec h a n ism is a f u n d am e n t al unit a nd o n e h a s t o s t art wi t h its s tud y . T h e s tudy of a mecha n ism i n v ol v es its anal y sis as w ell as s y n thesis. Analy s is is the s tudy of m o tions a n d f o r c es c o n c er n ing di f f er e n t p a rts of an e xi s ting mech a nis m , wher e as s y n thesis i n v ol v e s the de s ign o f its dif f ere n t parts. In a mech a nis m , the v a rio u s p a rts are so pr o p o rti o ned a n d rela t ed that the m o tion of o n e imp a rts re q uisi t e motio n s t o the oth e rs a n d the p a rts are able t o wi t h s t a n d the f o r c es impr e s s ed u p on them. • • • • • • 2 / 5 / 2 16 3

3. Ki n emati c s: • It d e als with the r e lative moti o ns o f dif f e r e n t pa r ts of a mechani s m with o ut t aking i nt o c o n sid e r a tion t h e f o r c es pr o ducing the moti o ns. Thus, it is the s tud y , from a g e ometric poi n t of vi e w , t o know the displ a c em e n t, velocity and a c c el e r a tion of a pa r t of a mecha n ism. • Dynamics: • It i n v olves the c alculati o ns o f f o r c es impr e ss e d up o n dif f e r e n t pa r ts of a mecha n ism. The f o r c es c an be either s t atic or dynam i c. Dynamics is fu r th e r sub d ivided i n t o kineti c s and s t atics. • • • Kineti c s is the s tudy of f o r c es when the b o dy is in moti o n s t atics d e als with f o r c es wh e n the b o dy is s t ati o na r y . wh e r e as 2 / 5 / 2 16 4

4. Kine m a tic Link or Elem e n t • E ach part of a ma c hine, which m o v es r el a ti v e t o s o me other part, is kn o wn as a kin e matic link ( o r simply li n k) or el e me n t. A link may c o n si s t of sever a l pa r ts, which are rigi d ly f a st en e d t o g e the r , so th a t th e y do n o t m o v e r el a ti v e t o o n e anot h e r . F or e x ampl e , in a r e c ip r o c a ting st e a m engin e , as sh o wn in below Fig., pi st on, pi st on r od and c r o s shead comprised one link ; c onnecting r od with big and s mall end b e arings comprised a se c ond link ; c r ank, c r ank • • sh a ft main and fl y wheel a thi r d link and the cylinde r , engine f r ame and be a rings a f ourth link. 2 / 5 / 2 16 5

5. A link or element need not to be a rigid body, but it must be a resistant body . A body is said to be a resistant body if it is capable of transmitting the required forces with negligible deformation. Thus a link should have the following two characteristics : 1 . It should have relative motion, and 2. It must be a resistant body.

6. T y p es of Lin k s In o r der t o t r an s mit motion, the dri v er and the f o ll o w er m a y be c o nne c t ed b y the f o ll o wing th r ee types o f lin k s : 1. Rigid link. A rigid link is one which does not und e rgo a n y de f or m ation while tran s mi t ti n g motio n . Stri c t ly speakin g , rigid lin k s do not e xi s t . Ho w e v e r , as the d e f o rm a t ion o f a c o nne c ting r o d, c r a n k e t c . o f a r ec i p r o c a ting s t e a m engine is not app r ec i able, th e y c an be c o n s ide r ed a s rigid lin k s . 2 . Fl e xible li n k. A f l e xi b le link is one which is part l y de f ormed in a mann e r not t o a f f e c t t r an s mis s ion o f motio n . F o r e x amp l e, belt s , r o pes, chains and wi r es fl e xible lin k s and t r an s mit t ensile f o r ces o nl y . 3 . Fluid lin k . the a r e A fluid link is one which is f ormed b y h av i ng a fluid in motion is t r a nsmi t t ed th r o ugh the fluid b y p r es s u r e or in the c a s e o f h y d r aulic p r es s es, j a c k s and b r a k e s . a re c e p t acle and the c o mp r es s ion o nl y , as 2 / 5 / 2 16 6

7. Link or ele m e n t types as per connections : It is the name gi v en t o a n y body which has moti o n r el a ti v e t o anothe r . All m a t e r ials h a v e some el a s ticit y . A rigid link is o ne, whose d e f orm a tions a r e so s mall th a t th e y c an be n e gle c t ed in d e t e r mining the motion pa r ame t e r s of the link. • Bi n a r y li n k: Link which is c onnec t ed t o other lin k s a t t w o poi n ts. ( Fig.a) T erna r y li n k: Link which is c onnec t ed t o other lin k s a t th r ee p o i n ts. ( Fig. b ) • • Q u a t erna r y li n k: Link which is c onnec t ed t o other lin k s a t f our p o i n ts. ( F ig. c) 2 / 5 / 2 16 7

8. Structu r e • It is an as s embl a g e of a numb e r of r esi s t a n t bodies ( kn o wn as me m be r s) h a ving no r el a ti v e motion b e t w e e n them and m e a n t f or c ar r ying loads h a ving s t r aining ac t ion. A r ail w a y brid g e, a r o o f truss, ma c hine f r am e s e t c., a r e the e x ampl e s of a s tru c tu r e. Dif f e r ence B e t w een a Ma c hine and a Stru c tu r e The f oll o wing di f f e r en c es b e t w e e n a machine and a s tru c tu r e a r e impor t a n t f r om the subje c t poi n t o f v iew : 1. T he parts of a machine m o v e r el a ti v e t o o n e anot h e r , whe r eas the me m be r s of a s tru c tu r e do not m o v e r el a ti v e t o one anothe r . 2. A ma c hine t r an s f orms the a v ailable en e r gy i nt o s o me us e ful w ork, whe r e a s in a s tru c tu r e no en e r gy is t r an s f ormed i n t o us e ful w ork. 3. T he lin k s of a machine m a y t r ansmit b o th p ow er and m o ti o n, w h ile the m e mb e r s of a s tru c tu r e t r ansmit f o r c e s o nl y . 2 / 5 / 2 16 8

9. Kinem a tic P air The t w o l i n k s or e l e m e n ts of a m a chine, w hen in c o nt act w ith each othe r , a r e s a id t o f orm a p ai r . If the r e l a ti v e mot i on b e t w een them is c omp l e t e l y or suc c es s fully c on s t r ained ( i.e. in a defini t e di r ection ) , the pair i s kn o wn as kinematic p a i r . T ypes of C on s t r ained Motions F ol l o w i ng a r e the th r ee ty p es of c on s t r ained mot i ons : 1. Compl e t ely c on s trai n ed motion When the motion b e t w een a pair is l i m i t ed t o a defin i t e di r ecti o n i r r e s pecti v e of the d i r ection of f o r ce ap p l i ed, then the motion is sa i d t o b e a co m pl e t e l y c on s t r ained mot i on. F or e x amp l e, the pi s t on and c y l i n d er (in a s t eam en g ine) f orm a p air a n d the mot i on of the p i s t on is l i m i t ed t o a d e fini t e di r ection ( i.e. it w i l l only r ec i pro c a t e) re l ati v e t o the c y l i n d er i r r especti v e of the d i r ection of mot i on of the c r ank. The mot i on of a squa r e bar in a squa r e hole, as sh o wn in Fig. 2, and the mot i on of a sh a ft with co l la r s a t ea c h e nd in a ci r c u lar hole, as sh o wn in Fig. 3, a r e al s o e x amp l es of co m pl e t e l y c on s t r ained mot i on. 2 / 5 / 2 16 9

10. 2 . In c ompl e t ely c o n s tr a ined mot i on Wh e n the moti o n between a pair c an t a k e pl a c e in mo r e than one di r e c tion, then the moti o n is c all e d an in c ompl e t ely c on s t r ained moti o n. T h e chan g e in the d i r e c ti o n of imp r essed f o r ce m a y al t er the di r e c tion of r el a ti v e motion b e t w e e n the pai r . A ci r cul a r bar or sh a ft in a ci r cul a r hole, as s h o wn in Fig . , is an e x ample of an in c ompl e t ely c on s t r ained moti o n as it m a y eith e r r o t a t e or slide in a h o le. T hese b o th m o ti o ns h a v e no r el a ti o nship with the othe r . 2 / 5 / 2 16 10

11. 3 . Su c c e s sf u l l y c on s trained m o tion When the motion b e tween the el e me n ts, f o r ming a pai r , is such th a t the c on s t r ained moti o n is n o t c ompl e t ed b y itsel f , but b y s o me other me a ns, then the moti o n is s aid t o be suc c es s fully c on s t r ained moti o n. Consider a sh a ft in a f o o t - st ep be a ring as sh o wn in Fig. T h e sh a ft m a y r o t a t e in a bearing o r it m a y m o v e u p w a r ds. This is a c ase of in c ompl e t ely c on s t r ained moti o n. But if the load is pla c ed on the sh a ft t o p r e v e n t a xi a l u p w a r d m o v em e n t o f the sh a ft, then the motion of the pair is s aid t o be succ e s s fully c on s t r ained moti o n. T h e moti o n of an I . C. engi n e v al v e and the p i st on r e c ip r o c a ting inside an engine c y linder a r e also the e x ampl e s of succ e s s fully c on s t r ained moti o n. 2 / 5 / 2 16 11

12. C l assif i c a ti o n of K i nem a tic P ai r s 1. A c c o r ding t o t h e ty p e of r e lative mo t i o n b e twe e n t h e eleme n ts. ( a) Sl i ding p a i r . Wh e n the two e lem e n ts of a pair are c o n n e c t ed in such a way that one c an o n ly slide r el a ti v e t o the o the r , the pair is k n o wn as a slidi n g pai r . The pi st on and c y linde r , c r o s s - he a d and guides o f a r e c ip r o c a ting st e a m engin e , r am and its guides in s hap e r , t ail st ock on the l a the bed e t c. a r e the e x amples of a sli d ing pai r . A li t tle c o n side r a ti o n will sh o w , th a t a sliding pair has a c ompl e t ely c on s t r ained moti o n. 2 / 5 / 2 16 12

13. ( b) T u r n i ng p a i r . Wh e n the two e lem e n ts of a pair are c o n n e c t ed in such a way that o n e c an only turn or r e v ol v e about a fi x ed a xis of another link, the pair is kn o wn as turning pai r . ( c) Sphe r i c al p a i r . Wh e n the two e lem e n ts of a pair are c o n n e c t ed in such a way that o n e el e me n t (with spheri c al shape) turns or s wi v els about the other fi x ed e le m e n t, the pair f ormed is c all e d a spheri c al pai r . The ball and soc k et joi n t, a t t ach m e n t of a c ar mir r o r , pen s t and e t c., a r e the e x amples of a spheri c al pai r . 2 / 5 / 2 16 13

14. ( d) R o l li n g p a i r . Wh e n the two e lem e n ts of a pair are c o n n e c t ed in such a way that o n e r olls o v er another fi x ed link, the pair is kn o wn as r olling pai r . Ball and r oller be a rings a r e e x amples of r oll i ng pai r . ( e) Sc r e w pai r . Wh e n the two e lem e n ts of a pair are c o n n e c t ed in such a way that o n e e lem e n t c an turn about the other b y sc r ew th r e a ds, the pair is kn o wn as s c r ew pai r . The le a d sc r ew of a l a the with nut, and bolt with a nut a r e e x ampl e s of a sc r ew pai r . 2 / 5 / 2 16 14

15. 2 . A cc o r ding t o t h e ty p e of c o n t act b e twe e n t h e eleme n ts. (a) L o wer pai r . Wh e n the two e lem e n ts of a pair have a su r f a c e c o n t act wh e n r e lative moti o n t a k es pla c e and the sur f ace o f o ne e le m e n t slides o v er the sur f ace o f the othe r , the pair f ormed is kn o wn as l o w er pai r . It will be se e n th a t sliding pai r s, turning pai r s and s c r ew pai r s f orm l o w er pai r s. ( b) H igh e r pai r . Wh e n the two e lem e n ts of a p a ir h a ve a line or p o i n t c o n t act wh e n r e lative moti o n t a k es pla c e and the moti o n b e t w e e n the t w o el e me n ts is partly turning and partly slidin g , then the pair is kn o wn higher pai r . A pair of friction discs, t o o thed g e a rin g , belt and r ope dri v es, ball and r oller be a rings and c am and f oll o w er a r e the as e x ampl e s of higher pai r s. 2 / 5 / 2 16 15

16. 3 . A c c ording t o the type of closur e . (a) S e lf closed pai r . When the two e l eme n ts o f a p a ir are c o n nec t ed t o g e ther mechani c a l l y in s u c h a w a y th a t o n l y r equi r ed kind of r e l a t i v e m otion o c c u r s, i t i s t hen k n o wn a s se l f closed pa i r . The l o w er pai r s a r e se l f closed F or c e - closed pai r . pai r . ( b) Wh e n t h e two eleme n ts of in a pair a re not c o nnec t ed mechani c a l l y but a r e k e p t c o n t a c t by t he a c tion of e x t ernal f o r ces, the pair i s said t o b e a f o r c e - clo s ed pai r . The c am and f ol lo w er i s a n e x a mp l e o f f o r ce i t i s k e p t i n c o n t a c t b y t he f o r ces e x er t ed g r a v i t y . clo s ed pai r , as b y s pring a n d 2 / 5 / 2 16 16

17. K i nem a tic Ch a in • When the kine m a t i c pai r s a r e c oupl e d in such a w a y th a t t h e la s t link is joined t o the fi r s t link t o t r ansmit d e fini t e mot i on ( i . e . c o m ple t ely succ e s s fully c on s t r ained moti o n), it is c all e d a kin e matic chai n . I n oth e r w o r ds, a kinem a tic ch a in m a y be d e fin e d as a c omb i n a ti o n or • of of is kine m a t ic p ai r s, joined in such a w a y th a t ea c h link f orms a part t w o pai r s a nd the r el a ti v e moti o n b e t w e e n the lin k s or el e me n ts c ompl e t ely or succ e s s fully c on s t r aine d . 2 / 5 / 2 16 17

18. Types of Joints The f oll o wing types Bina r y j o i n t. T e r na r y J oi n t Qu a t e r na r y J o i n t of j oi n ts a r e usually f ound in a ch a in : Bi n a r y jo i n t: • When t w o lin k s a r e joined a t the same c o n nection, the joi n t is k n o wn as bina r y joi n t 2 / 5 / 2 16 18

19. T ernary joi n t : • Wh e n th r ee l i n k s a r e jo i ned a t the same c onnectio n , the joi n t is known as t ernary joi n t. • It is equ i v ale n t t o t w o bi n a r y joi n ts as one of the th r ee l i n k s joined c ar r y the p i n f or the other t w o li n k s. 2 / 5 / 2 16 19

20. Qu at ernary joi n t : • When f our l i n k s a r e joined a t t he same c onne c t i on, the joi n t is c alled a qu a t ernary joi n t . It is equi v ale n t t o th r ee bi n ary joi n t s . • • In g ene r al, when l number of l i n k s a r e joined a t the t o ( l same c onnect i on, the joi n t is equi v ale n t - 1) bi n ary joi n t s . Qu a t ernary joi n t: 2 / 5 / 2 16 20

21. M e chanism • When one of the lin k s of a kine m a tic ch a in is fi x ed, the c hain is kn o wn as mechani s m. It may be used f or t r ansmi t ting or t r an s f orming moti o n A me c hanism with f o u r lin k s is kn o wn as simple mechanism, and mechani s m with more than f our lin k s is kn o wn as c omp o und mechani s m. Wh e n a mechani s m is r e qui r ed t o tr a nsmit power or t o do s o me particular type of w ork, it then be c omes a machine. • • the • • In such c as e s, the vari o us lin k s or el e me n ts h a v e t o be d e signed t o with st and the f o r c e s (both st a tic and kinetic) s a f el y . A li t tle c on s ide r a tion will sh o w th a t a me c hanism m a y be r e g a r ded a machine in w h ich ea c h part is r educ e d t o the simple s t f orm t o t r ansmit the r equi r ed motion. • as 2 / 5 / 2 16 21

22. Mechanisms a n d S imple M a chines • Mac h in e : an ass e mblage of parts that transmit f or c e s , motion and e nergy in a pr e d e t ermin e d manne r . The t erm me c h a nism is applied t o the c o mbin a tion o f g e o m e tri c al b o dies which comprised a machine o r part of a machin e . A mech an ism m a y the r e f o r e be d e fined as a c o mbin a tion o f rigid o r r esi s t a n t bodie s , f o rmed and c o nne c t ed so t h a t th e y m o v e with d e fini t e r el a ti v e motions with r espe c t t o one another . The simila r ity b e t w een machin e s and mechan i sms is th a t – th e y a r e b o th c o mbin a tions o f rigid bodies – the r el a ti v e motion am o ng the rigid bodies a r e d e fini t e. Th e dif f eren c e b e t w een mach i ne and me c han i sm is t h a t ma c hin e s t r an s f o rm ene r gy t o do w o rk, while mechanisms do n o t necess a rily per f o rm this fun c tion. All m achines a r e m ec h ani s ms. But all me c hani s ms a r e n o t machines. • • • • • • 2 / 5 / 2 16 22

23. Mechanisms Degree of Freedom: DOF (Also called as movability ) is defined as the number of input parameters (usually pair variables) which must be independently controlled in order to bring the mechanism into useful engineering purpose. Four bar chain, as shown in figure (a) shows that only one variable θ is needed to define the relative positions of all links. Five bar chain, as shown in figure (b) shows that two variables θ 1 and θ 2 are needed to define completely the relative positions of all the links.

24. Kutzbach Criterion Kutzbach equation to calculate the movability of a mechanism having plane motion is n = 3(l-1) - 2j – h If there are no two degree freedom pairs ( i.e higher pairs), then h = 0. n = 3(l-1) – 2j - h

25. Kutzbach Criterion For binary joint n = 0 For four bar mechanism n = 1 For five bar mechanism n = 2 For five bar mechanism n = 0 For six bar mechanism n = -1 (redundant constraints forming statically indeterminate structure)

26. Kutzbach Criterion For higher pair or two degree of freedom joints is shown as below. n = 3 (3-1) – 2 * 2 – 1 = 1 n = 3 (4-1) – 2 * 3 – 1 = 2 Grubler’s Condition:

27. • P l anar m e chanism s : When all the lin k s of a m e ch a nism h a v e plane moti o n, it is c alled as a planar me c hanism. All the lin k s in a planar me c hanism m o v e in plan e s pa r all e l t o the r e f e r en c e plan e . 2 / 5 / 2 16 23

28. The Grubler’s criterion applies to mechanisms with only single degree of freedom joints where the overall movability of the mechanism is unity. Substituting n = 1 and h = 0 in Kutzbach equation , we have 1 = 3 ( l – 1) – 2 j or 3l – 2j – 4 = 0 Gr u b l e r ’ s equ a t i on: Num b er of d e g r e e s of f r e e dom of a me c h a nism is g i v en b y W here n = to t al deg r ees of fre e dom in t he m echan i sm l = nu m ber of l inks ( i nclud i ng the f ra m e) j = Nu m ber h = nu m ber o f equiv a lent bina r y jo i nts of higher pai r s ( t wo degrees of f reedo m ) 2 / 5 / 2 16 26

29. 1. The m e c han i s m , as sh o wn in Fi g . ( a ) , has t h ree li n ks and th r ee binary jo i nt s , i.e. l= 3 and j = 3. n = 3 ( 3 -1)- 2 x 3=0 2. The m e c han i s m , as sh o wn in Fi g . ( b ) , has f o ur li n ks and f o ur bi n ary j o i n ts, i.e. l= 4 and j = 4 . N = 3 ( 4 -1 ) - 2 x 4 = 1 ( c), 3. The m e c han i s m , as sh o wn in Fi g . h as fi v e li n ks and fi v e b i nary j o i n ts, i.e. l = 5 , and j = 5. n = 3 (5 - l )- 2 x 5 = 2 2 / 5 / 2 16 27

30. N u m ber of l i n k s, l = 11 N u m ber of Eq u iva l ent binary j oints, J b = 15 N u m ber of Hig h er pairs, h do f = 3( l - 1) – 2 J b – h = 3 ( 1 1 - 1) – 2x1 5 – 0 = 30 – 3 – 0 = = N u m ber of l i n k s, l = 4 N u m ber of Eq u iva l ent binary j oints, J b = 3 N u m ber of Hig h er pairs, h = 1 do f = 3( l - 1) – 2 J b – h = 3 (4 - 1) – 2x 3 – 1 = 9 – 6 – 1 = 2

31. 2 / 5 / 2 16

32. 2 / 5 / 2 16

33. 2 / 5 / 2 16 N u m ber Nu m ber Nu m ber of of of l i nks, l = 7 Equiva l ent bina r y jo i nts, J b = 8 High e r pai r s, h = dof = 3 ( l - 1) – 2 J b –h = = = 3( 7- 1) – 2x8 – 18 – 2 16 Nu m ber Nu m ber N u m ber of of of l inks, l = 9 Equ i val e nt bina r y jo i nts, J b = 10 Higher pai r s, h = dof = 3 ( l - 1) – 2J b –h = 3( 9- 1) – 2x10 – 0 = 24 – 20 = 4 28

34. Nu m ber Nu m ber Nu m ber of of of l inks, l = 4 Equ i val e nt bina r y jo i nts, J b = 4 High e r pai r s, h = 1 dof = 3 ( l - 1) – 2J b –h = = = 3( 4- 1) – 2x4 – 9 – 8 – 1 1 Nu m ber Nu m ber Nu m ber of of of l inks, l = 8 Equ i val e nt bina r y jo i nts, J b = 10 High e r pai r s, h = 1 dof = 3 ( l - 1) – 2J b –h = = = 3(8 - 1) – 2x10 – 0 21 – 20 – 0 1 2 / 5 / 2 16 29

35. (a) N u m ber of l i n k s, l = N u m ber of Eq u ival e nt bi n ary j oi n ts, J b = N u m ber of Hig h er pairs, h = do f = - 1 T ry th e se ( b ) N u m ber of l i n k s, l = N u m ber of Eq u ival e nt bi n ary j oi n ts, J b = N u m ber of Hig h er pairs, h = do f = 1 (c) N u m ber of l i n k s, l = N u m ber of Eq u ival e nt bi n ary j oi n ts, J b = N u m ber of Hig h er pairs, h = do f =