ME 141 Engineeing Mechnics Lecue 13: Kinemics of igid bodies hmd Shhedi Shkil Lecue, ep. of Mechnicl Engg, UET E-mil: sshkil@me.bue.c.bd, shkil6791@gmil.com Websie: eche.bue.c.bd/sshkil Couesy: Veco Mechnics fo Enginees, ee nd Johnson
Inoducion Kinemics of igid bodies: elions beween ime nd he posiions, elociies, nd cceleions of he picles foming igid body. Clssificion of igid body moions: - nslion: eciline nslion cuiline nslion - oion bou fixed xis - genel plne moion - moion bou fixed poin - genel moion
Tnslion Conside igid body in nslion: - diecion of ny sigh line inside he body is consn, - ll picles foming he body moe in pllel lines. Fo ny wo picles in he body, iffeeniing wih espec o ime, ll picles he he sme elociy. iffeeniing wih espec o ime gin, ll picles he he sme cceleion.
Roion bou Fixed xis. Velociy Conside oion of igid body bou fixed xis Velociy eco d d of he picle P is ngen o he ph wih mgniude ds d s P sin ds d lim 0 sin sin The sme esul is obined fom d d k k ngul elociy
Roion bou Fixed xis. cceleion iffeeniing o deemine he cceleion, d d d d d d d d d d k k k ngul cceleion d d cceleion componen dil cceleion componen ngenil cceleion of P is combinion of wo ecos,
Roion bou Fixed xis. Repesenie Slb Conside he moion of epesenie slb in plne pependicul o he xis of oion. Velociy of ny poin P of he slb, k cceleion of ny poin P of he slb, k Resoling he cceleion ino ngenil nd noml componens, k n n
Equions efining he Roion of Rigid ody bou Fixed xis Moion of igid body oing ound fixed xis is ofen specified by he ype of ngul cceleion. d d d d d d d d d d o Recll Unifom Roion, = 0: 0 Unifomly cceleed Roion, = consn: 0 0 1 0 0 0
Smple Poblem 5.1 Cble C hs consn cceleion of 9 in/s nd n iniil elociy of 1 in/s, boh dieced o he igh. eemine () he numbe of eoluions of he pulley in s, (b) he elociy nd chnge in posiion of he lod fe s, nd (c) he cceleion of he poin on he im of he inne pulley = 0. SOLUTION: ue o he cion of he cble, he ngenil elociy nd cceleion of e equl o he elociy nd cceleion of C. Clcule he iniil ngul elociy nd cceleion. pply he elions fo unifomly cceleed oion o deemine he elociy nd ngul posiion of he pulley fe s. Elue he iniil ngenil nd noml cceleion componens of.
Smple Poblem 5.1 SOLUTION: The ngenil elociy nd cceleion of e equl o he elociy nd cceleion of C. 1in. s 0 C 0 0 0 0 0 1 3 4d s C 9in. s 9 3d 3 pply he elions fo unifomly cceleed oion o deemine elociy nd ngul posiion of pulley fe s. 4d s 3d s s 10 d s 0 0 1 14 d 1e N 14 d numbe of d y 5 in. 10 d s 5 in. 14 d 4d s s 1 3d s s es N.3e y 70 in. s 50in. s
Smple Poblem 5.1 Elue he iniil ngenil nd noml cceleion componens of. 9in. s C 3 in. 4d s 48in n 0 s in. s 48in. s 9 n Mgniude nd diecion of he ol cceleion, 9 48 n 48.8in. s n 48 9 n 79. 4
Genel Plne Moion Genel plne moion is neihe nslion no oion. Genel plne moion cn be consideed s he sum of nslion nd oion. isplcemen of picles nd o nd cn be diided ino wo ps: - nslion o nd 1 - oion of bou o 1
bsolue nd Relie Velociy in Plne Moion ny plne moion cn be eplced by nslion of n biy efeence poin nd simulneous oion bou. k k
bsolue nd Relie Velociy in Plne Moion ssuming h he elociy of end is known, wish o deemine he elociy of end nd he ngul elociy in ems of, l, nd. The diecion of nd / e known. Complee he elociy digm. n n l l cos cos
bsolue nd Relie Velociy in Plne Moion Selecing poin s he efeence poin nd soling fo he elociy of end nd he ngul elociy leds o n equilen elociy ingle. / hs he sme mgniude bu opposie sense of /. The sense of he elie elociy is dependen on he choice of efeence poin. ngul elociy of he od in is oion bou is he sme s is oion bou. ngul elociy is no dependen on he choice of efeence poin.
bsolue nd Relie Velociy in Plne Moion ssuming h he elociy of end is known, wish o deemine he elociy of end nd he ngul elociy in ems of, l, nd. The diecion of nd / e known. Complee he elociy digm. n n l l cos cos
bsolue nd Relie Velociy in Plne Moion Selecing poin s he efeence poin nd soling fo he elociy of end nd he ngul elociy leds o n equilen elociy ingle. / hs he sme mgniude bu opposie sense of /. The sense of he elie elociy is dependen on he choice of efeence poin. ngul elociy of he od in is oion bou is he sme s is oion bou. ngul elociy is no dependen on he choice of efeence poin.
Smple Poblem 15.3 SOLUTION: Will deemine he bsolue elociy of poin wih The elociy is obined fom he gien cnk oion d. The cnk hs consn clockwise ngul elociy of 000 pm. Fo he cnk posiion indiced, deemine () he ngul elociy of he connecing od, nd (b) he elociy of he pison P. The diecions of he bsolue elociy nd he elie elociy e deemined fom he poblem geomey. The unknowns in he eco expession e he elociy mgniudes nd which my be deemined fom he coesponding eco ingle. The ngul elociy of he connecing od is clculed fom.
Smple Poblem 15.3 SOLUTION: Will deemine he bsolue elociy of poin wih The elociy 000 is obined fom he cnk oion d. e min min 60s d 09.4 d e 3in. 09.4 d s The elociy diecion is s shown. The diecion of he bsolue elociy is hoizonl. The diecion of he elie elociy is pependicul o. Compue he ngle beween he hoizonl nd he connecing od fom he lw of sines. sin 40 8in. sin 3in. 13.95 s
Smple Poblem 15.3 eemine he elociy mgniudes fom he eco ingle. 68.3in. s sin 53.95 sin 50 sin76.05 nd 53.4in. s 43.6f 495.9in. s s P 43.6f s l 495.9in. l 8 in. 6.0 d s s 6.0 d sk
Pob # 15.63 Knowing h he insn shown he ngul elociy of od is 15 d/s clockwise, deemine () he ngul elociy of od, (b) he elociy of he midpoin of od.
Pob# 15.71 The 80-mm-dius wheel shown olls o he lef wih elociy of 900 mm/s. Knowing h he disnce is 50 mm, deemine he elociy of he coll nd he ngul elociy of od when () b = 0, (b) b = 90.
Insnneous Cene of Roion in Plne Moion Plne moion of ll picles in slb cn lwys be eplced by he nslion of n biy poin nd oion bou wih n ngul elociy h is independen of he choice of. The sme nslionl nd oionl elociies e obined by llowing he slb o oe wih he sme ngul elociy bou he poin C on pependicul o he elociy. The elociy of ll ohe picles in he slb e he sme s oiginlly defined since he ngul elociy nd nslionl elociy e equilen. s f s he elociies e concened, he slb seems o oe bou he insnneous cene of oion C.
Insnneous Cene of Roion in Plne Moion If he elociy wo poins nd e known, he insnneous cene of oion lies he inesecion of he pependiculs o he elociy ecos hough nd. If he elociy ecos e pllel, he insnneous cene of oion is infiniy nd he ngul elociy is zeo. If he elociy ecos nd e pependicul o he line, he insnneous cene of oion lies he inesecion of he line wih he line joining he exemiies of he elociy ecos nd. If he elociy mgniudes e equl, he insnneous cene of oion is infiniy nd he ngul elociy is zeo.
Insnneous Cene of Roion in Plne Moion The insnneous cene of oion lies he inesecion of he pependiculs o he elociy ecos hough nd. C l sin C l cos l cos n The elociies of ll picles on he od e s if hey wee oed bou C. The picle he cene of oion hs zeo elociy. The picle coinciding wih he cene of oion chnges wih ime nd he cceleion of he picle he insnneous cene of oion is no zeo. The cceleion of he picles in he slb cnno be deemined s if he slb wee simply oing bou C. The ce of he locus of he cene of oion on he body is he body cenode nd in spce is he spce cenode.
Smple Poblem 15.5 The cnk hs consn clockwise ngul elociy of 000 pm. Fo he cnk posiion indiced, deemine () he ngul elociy of he connecing od, nd (b) he elociy of he pison P. SOLUTION: eemine he elociy fom he gien cnk oion d. The diecion of he elociy ecos nd e known. The insnneous cene of oion is he inesecion of he pependiculs o he elociies hough nd. eemine he ngul elociy bou he cene of oion bsed on he elociy. Clcule he elociy bsed on is oion bou he insnneous cene of oion.
Smple Poblem 15.5 SOLUTION: Fom Smple Poblem 15.3, 403.9i 481.3 j in. s 13.95 68.3in. s The insnneous cene of oion is he inesecion of he pependiculs o he elociies hough nd. 40 53.95 90 76.05 C C 8 in. sin 76.05 sin 53.95 sin50 C 10.14 in. C 8.44 in. eemine he ngul elociy bou he cene of oion bsed on he elociy. C C 68.3in. s 10.14 in. Clcule he elociy bsed on is oion bou he insnneous cene of oion. C 8.44 in. 6.0d s 6.0d s P 53in. s 43.6f s
bsolue nd Relie cceleion in Plne Moion bsolue cceleion of picle of he slb, Relie cceleion ssocied wih oion bou includes ngenil nd noml componens, n k n
bsolue nd Relie cceleion in Plne Moion Gien nd, deemine nd. n Veco esul depends on sense of nd elie mgniudes of n nd he Mus lso know ngul elociy.
bsolue nd Relie cceleion in Plne Moion Wie x componens: 0 l sin l cos y componens: l cos l sin Sole fo nd. in ems of he wo componen equions,
Smple Poblem 15.7 Cnk G of he engine sysem hs consn clockwise ngul elociy of 000 pm. Fo he cnk posiion shown, deemine he ngul cceleion of he connecing od nd he cceleion of poin. SOLUTION: The ngul cceleion of he connecing od nd he cceleion of poin will be deemined fom n The cceleion of is deemined fom he gien oion speed of. The diecions of he cceleions,, nd e n deemined fom he geomey. Componen equions fo cceleion of poin e soled simulneously fo cceleion of nd ngul cceleion of he connecing od.
Smple Poblem 15.7 SOLUTION: The ngul cceleion of he connecing od nd he cceleion of poin will be deemined fom n The cceleion of is deemined fom he gien oion speed of. 000 pm 09.4 d s consn 0 3 f 09.4 d s 10,96 f s 1 10,96 f s cos40i sin j 40
Smple Poblem 15.7 The diecions of he cceleions deemined fom he geomey. i,, nd n Fom Smple Poblem 15.3, = 6.0 d/s, = 13.95 o. 8 f 6.0 d s 563 f s n 1 563 f s cos13.95i sin13. 95 j n 8 f 0. 667 1 The diecion of ( / ) is known bu he sense is no known, 0.667 sin 76.05i cos76. 05 j e
Smple Poblem 15.7 Componen equions fo cceleion of poin e soled simulneously. x componens: n,96 cos40 563 cos13.95 0.667 sin13. 95 y componens: 10 0 10,96 sin 40 563 sin13.95 0.667 cos13. 95 9940 d s k 990 f s i
Pob # 15.131 Knowing h he insn shown b hs consn ngul elociy of 4 d/s clockwise, deemine he ngul cceleion () of b, (b) of b E.
Pob # 15.13 The disk shown hs consn ngul elociy of 500 pm couneclockwise. Knowing h od is 10 in. long, deemine he cceleion of coll when () θ= 90, (b) θ= 180.