# KINEMATICS OF RIGID BODIES

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1 KINEMTICS OF RIGID ODIES

2 In igid body kinemaics, we use he elaionships govening he displacemen, velociy and acceleaion, bu mus also accoun fo he oaional moion of he body.

3 Descipion of he moion of igid bodies is impoan fo wo easons: 1) To geneae, ansmi o conol moions by using cams, geas and linkages of vaious ypes and analyze he displacemen, velociy and acceleaion of he moion o deemine he design geomey of he mechanical pas. ) To deemine he moion of a igid body caused by he foces applied o i. Calculaion of he moion of a ocke unde he influence of is hus and gaviaional aacion is an example of such a poblem.

4 igid body is a sysem of paicles fo which he disances beween he paicles and he angle beween he lines emain unchanged. Of couse his is an idealizaion since all solid maeials change shape when foces ae applied o hem. Neveheless, if he movemens associaed wih he changes in shape ae vey small compaed wih he movemens of he body as a whole, hen his assumpion is usually accepable.

5 ll pas of he body move in paallel planes. The plane moion of a igid body is divided ino seveal caegoies: 1. Tanslaion. Roaion 3. Geneal Moion

6 1. TRNSLTION I is any moion in which evey line in he body emains paallel o is oiginal posiion a all imes. In anslaion, hee is no oaion of any line in he body. 1. Recilinea Tanslaion: ll poins in he body move in paallel saigh lines. Rocke es sled

7 . Cuvilinea Tanslaion: ll poins move on conguen cuves. In each of he wo cases of anslaion, he moion of he body is compleely specified by he moion of any poin in he body, since all he poins have he same moion.

8 . Fixed xis Roaion Roaion abou a fixed axis is he angula moion abou he axis. ll paicles in a igid body move in cicula pahs abou he axis of oaion and all lines in he body oae hough he same angle a he same ime. C C

9 3. Geneal Plane Moion I is he combinaion of anslaion and oaion.

10 Cank (Kank) (Roaion) Pison (Tanslaion) O Connecing od (Geneal Moion) hinge

11 The oaion of a igid body is descibed by is angula moion. The figue shows a igid body which is oaing. The angula posiions of any wo lines 1 and aached o he body ae specified by θ 1 and θ measued fom any convenien fixed efeence diecion. ecause he angle β is invaian, he elaion θ θ 1 + β upon diffeeniaion wih espec o ime gives θ θ 1 and θ θ 1 duing a finie ineval, θ θ 1. ll lines on a igid body have he same angula displacemen, he same angula velociy and he same angula acceleaion.

12 The angula velociy and angula acceleaion α of a igid body in plane oaion ae, especively, he fis and second ime deivaives of he angula posiion coodinae θ of any line in he plane of moion of he body. These definiions give dθ θ d d αdθ o d α d θd θ θdθ o α d θ θ d Fo oaion wih consan angula acceleaion, he elaionships become α 0 + α ( θ θ ) θ θ + α

13 Roaion bou a Fixed xis When a igid body oaes abou a fixed axis, all poins move in concenic cicles abou he fixed axis. Thus, fo he igid body in he figue oaing abou a fixed axis hough poin O, any poin such as moves in a cicle of adius. So he velociy and he acceleaion of poin can be wien as v a a n α v v

14 These quaniies may be expessed using he coss poduc elaionship of veco noaion, v k k α α, ( ) ( ) ( ) n a a v d d d d d d v d d a + + α α

15 PROLEMS 1. The cicula disk oaes abou is cene O. Fo he insan epesened, he velociy of is 00 j mm s and he angenial acceleaion of is v ( a ) 150i mm s. Wie he veco expessions fo he angula velociy and angula acceleaion α of he disk. lso deemine he acceleaion of poin C.

16 PROLEMS. The angula velociy of a gea is conolled accoding o 1 3, whee in ads and is he ime in seconds. Find he ne angula displacemen θ fom he ime 0 o 3 s. lso find he oal numbe of evoluions N hough which he gea uns duing he hee seconds. SOLUTION dθ dθ d d θ 0 dθ 3 0 ( ) d, θ 1 1( 3) θ 9 ad ad

17 SOLUTION Does he gea sop beween 0 and 3 seconds? s ( i sops a s) θ θ dθ dθ ( ) d θ 1 1( ) ( 1 3 ) ad 0 3 d θ ad 3 16 ad 1evoluion π ad N evoluions 3 ad N 3.66 evoluions

18 PROLEMS 3. The bel-diven pulley and aached disk ae oaing wih inceasing angula velociy. a ceain insan he speed v of he bel is 1.5 ms, and he oal acceleaion of poin is 75 ms. Fo his insan deemine (a) he angula acceleaion a of he pulley and disk, (b) he oal acceleaion of poin, and (c) he acceleaion of poin C on he bel.

19 C ? a c)? a b)? a) s m a s m a s m a s m a s ad R a s m a s m R a s ad v s m a s m v C C C n n + α α α α SOLUTION

20 PROLEMS 4. The design chaaceisics of a gea-educion uni ae unde eview. Gea is oaing clockwise (cw) wih a speed of 300 evmin when a oque is applied o gea a ime s o give gea a couneclockwise (ccw) acceleaion a which vaies wih ime fo a duaion of 4 seconds as shown. Deemine he speed N of gea when 6 s.

21 SOLUTION s ad ev N s min 300 π π The velociies of geas and ae same a he conac poin. ( ) ( ) ( ) ( ) ( ) min ) 6 ( ev N s ad b b s s a s ad d d d d s ad b b v v π α α π π

22 bsolue Moion In his appoach, we make use of he geomeic elaions which define he configuaion of he body involved and hen poceed o ake he ime deivaives of he defining geomeic elaions o obain velociies and acceleaions.

23 PROLEM wheel of adius olls on a fla suface wihou slipping. Deemine he angula moion of he wheel in ems of he linea moion of is cene O. lso deemine he acceleaion of a poin on he im of he wheel as he poin comes ino conac wih he suface on which he wheel olls.

24 The wheel olls o he igh fom he dashed o he full posiion wihou slipping. The linea displacemen of he cene O is s, which is also he ac lengh C' along he im on which he wheel olls. The adial line CO oaes o he new posiion C'O' hough he angle θ, whee θ is measued fom he veical diecion. Since he wheel olls wihou slipping, he linea displacemen of he cene is θ. s θ ( θ is in adians) The velociy of cene O s vo θ The acceleaion of cene O s a v θ α o o

25 If he wheel is slowing down, a o will be dieced opposie o v o and and α will have opposie diecions. When poin C has moved along is cycloidal pah o C, is new coodinaes and hei ime deivaives become x y x s sinθ θ sinθ ( θ sinθ ) y cosθ ( 1 cosθ ) x ( θ θ cosθ ) θ ( 1 cosθ ) y θ sinθ vo sinθ x v vo x x x o ( 1 cosθ ) θ θ cosθ + θ α α cosθ + sinθ a o a o a o ( 1 cosθ ) + sinθ sinθ y y y θ cosθ + θ sinθ cos θ + α sin θ a o cosθ + a o sinθ

26 Fo he desied insan of conac, θ 0 x y 0 v 0 c (he poin of conac has zeo velociy) (INSTNTNEOUS CENTER OF ZERO VELOCITY) a C C v c 0 (IC) j cceleaion a he conac poin: x 0, y

27 Relaive Moion The second appoach o igid body kinemaics uses he pinciples of elaive moion. In kinemaics of paicles fo moion elaive o anslaing axes, we applied he elaive velociy equaion v v + v o he moions of wo paicles and. We now choose wo poins on he same igid body fo ou wo paicles. The consequence of his choice is ha he moion of one poin as seen by an obseve anslaing wih he ohe poin mus be cicula since he adial disance o he obseved poin fom he efeence poin does no change.

28 The figue shows a igid body moving in he plane of he figue fom posiion o duing ime. This movemen may be visualized as occuing in wo pas. Fis, he body anslaes o he paallel posiion wih he displacemen. Second, he body oaes abou hough he angle θ, fom he nonoaing efeence axes x -y aached o he efeence poin, giving ise o he displacemen of wih espec o.

29 Wih as he efeence poin, he oal displacemen of is + Whee has he magniude θ as θ appoaches zeo. Dividing he ime ineval and passing o he limi, we obain he elaive velociy equaion v v + v The disance beween and emains consan.

30 The magniude of he elaive velociy is hus seen o be which, wih becomes v Using o epesen he veco, we may wie he elaive velociy as he veco d d v θ θ 0 0 lim lim θ v Theefoe, he elaive velociy equaion becomes v v +

31 Hee, is he angula velociy veco nomal o he plane of he moion in he sense deemined by he igh hand ule. I should be noed ha he diecion of he elaive velociy will always be pependicula o he line joining he poins and. Inepeaion of he Relaive Velociy Equaion We can bee undesand he elaive velociy equaion by visualizing he anslaion and oaion componens sepaaely. Tanslaion Fixed axid oaion

32 In he figue, poin is chosen as he efeence poin and he velociy of is he veco sum of he anslaional poion v v, plus he oaional poion, which has he magniude v, whee θ, he absolue angula velociy of. The elaive linea velociy is always pependicula o he line joining he wo poins and.

33 Relaive cceleaion Equaion of elaive velociy is y diffeeniaing he equaion wih espec o ime, we obain he elaive acceleaion equaion, which is o v v + v This equaion saes ha he acceleaion of poin equals he veco sum of he acceleaion of poin and he acceleaion which appeas o have o a nonoaing obseve moving wih. v v + v a a + a

34 If poins and ae locaed on he same igid body, he disance beween hem emains consan. ecause he elaive moion is cicula, he elaive acceleaion em will have boh a nomal componen dieced fom owad due o he change of diecion of v and a angenial componen pependicula o due o he change in magniude of. Thus, we may wie, v ( a ) ( ) n + a a a + Whee he magniudes of he elaive acceleaion componens ae n v ( a ) ( a ) v α

35 In veco noaion he acceleaion componens ae ( ) ( ) ( ) a a n α The elaive acceleaion equaion, hus, becomes ( ) a a + + α

36 The figue shows he acceleaion of o be composed of wo pas: he acceleaion of and he acceleaion of wih espec o.

37 Soluion of he Relaive cceleaion Equaion s in he case of he elaive velociy equaion, he elaive acceleaion equaion may be caied ou by scala o veco algeba o by gaphical consucion. ecause he nomal acceleaion componens depend on velociies, i is geneally necessay o solve fo he velociies befoe he acceleaion calculaions can be made.

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