Plane Kinetics of Rigid Bodies 동역학 및 응용
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1 Ple Kietics of igid odies 동역학 및 응용
2 EQUTONS O PLNE OTON esultt of the pplied etel foces : esultt foce (pss though ss cete) + ouple oets of the etel foces gul otio z ouple : sste of foces with esultt oet but o esultt foce.
3 EQUTONS O PLNE OTON d d
4 EQUTONS O PLNE OTON otio of of igid bod : ss cete of igid bod oves (tsltes) s though the igid bod wee sigle pticle of ss subjected to the esultt foce. otio of ost igid bod : supepositio of Tsltio poduced b the esultt foce ottio poduced b the oet of the esultt foce whe its lie of ctio dose ot pss though z z
5 oet bout : f d Newto s d Lw : f d d d ω ω α d d d d ω ω α d d d Displceet of eleet of ss d - w..t. the ovig poit of z-coodite sste tht is ttched to the bod t : - w..t. the oigi O of XYZ-coodite sste : Displceet of poit w..t. the oigi O of XYZ-coodite sste : EQUTONS O PLNE OTON
6 EQUTONS O PLNE OTON H d el H el α d ω ω H el d d d d d d d d d α ω ω d el H d α ω ω d d
7 EQUTONS O PLNE OTON Ple otio : otio i which ll eleets of the bod ove i pllel ples Ple of otio : The pllel ple tht psses though - ple Ple of otio (Poit is ssued to be i the ple of otio; The z-coodite sste is chose so tht the otio is pllel to the -ple) z d α d ω ω i j ω zk k, α zk k, i jzk d
8 EQUTONS O PLNE OTON i i j zk j, ω k k, α k k, d α d ω ω d zd zd zd zd zd zd z z d d d - Ple otio - Poit is i the ple of otio. z
9 EQUTONS O PLNE OTON ist oet : z d z z, d, zd z Poduct of ieti (eti popeties of igid bod) : z d z, zd oet of ieti (eti popeties of igid bod) : d z z z z z 0 is i the ple of otio : eltioship betwee oets of the etel foces d gul velocities d eti popeties
10 EQUTONS O PLNE OTON od is setic bout the ple of otio zd zd z z 0 0 h h z 0 z z z z z z z z ple of otio
11 EQUTONS O PLNE OTON Seticit + cceletio of is zeo (e: Poit is o fied is of ottio of bod) z z z 0, 0 z 0 z z z z z z Seticit + Poit coicides with z z z 0, 0 z 0
12 OENTS ND PODUTS O NET oet of eti The oet of ieti d of d bout the is OO d d 0 d z z d d d z d d z d The oet of ieti of etie bod The oet of ieti w..t z-coodite sste
13 OENTS ND PODUTS O NET fo sste of pticles 1 1 fo coposite bod z d z di i i
14 OENTS ND PODUTS O NET dius of tio : k, k The distce fo the is to the poit whee the totl ss ust be cocetted to poduce the se oet of ieti with espect to the is s does the ctul (distibuted) ss. d se oet of ieti k O k d O
15 z-coodite sste : Oigi is pllel z -coodite sste : Oigi is O Pllel-is Theoe fo oets of eti z z z ', ', ' Distce b/w - d - is z d oet of ieti bout - is zd z d z d d d z d z z d ' ' OENTS ND PODUTS O NET
16 OENTS ND PODUTS O NET z d d d z d z ' zd z d, d 0, zd 0 d, ' ' z z d d ' ' z' z d z z fo dii of tio k ' k k d, k k d, k k d ' ' z ' z z k d d
17 OENTS ND PODUTS O NET Poduct of eti poduct of the ss of sll eleet d the coodite distces fo pi of othogol coodite ples (Positive o Negtive o Zeo) z z d zd zd Pllel is theoe ' ' ' z' z' ' z z : poduct of ieti w..t. the z- d z-ples z z
18 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES eel Ple otio Oigi of z-coodite sste : z z z z z z Oigi : z z z z z z
19 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES Tsltio No gul otio ω α 0 eel Equtio 0 z 0 0
20 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES 300-kg hge doo is suppoted b two olles s show. foce of 1000 N is pplied to ope the doo. Deteie the cceletio of the doo d the suppot foces eeted o the doo b the two olles. Neglect fictiol foces d the ss of the olles N
21 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES se 1 : Suig oet bout N 1.5 z z :1000 : : g s, N, N
22 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES se : Suig oet bout z z 0 z z z 0 z N 1.5,, g z 4 : : : g N, s, N
23 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES The 1400-kg utoobile show hs wheel bse of 3. ts cete of ss is locted 1.30 behid the fot le d 0.5 bove the sufce of the pveet. f the utoobile hs e-wheel dive d the sttic coefficiet of fictio betwee the ties d the pveet is 0.80, deteie the iu cceletio tht the vehicle c develop while tvelig up the 15 iclie kg 0 0.8
24 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES z z : 0.8 : : si15 cos N N 1.51 s
25 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES The pltfo d leve sste show is used to tsfe boes betwee floos i fcto. the positio show, leve is ottig clockwise with gul velocit of 0.5 d/s d the gul velocit is decesig t te of 1.5 d/s. f the ss of the bo is 500 kg, deteie the veticl copoets of the foces eeted o the bo t suppots d D d the iiu coefficiet of fictio equied to pevet slippig of the bo t this istt. D α α 1.5k, ω 0. 5k ω ω 500 kg 1.5 d 0.5 d s s 3 si 60i cos60j 1.5 k 3 si 60i cos60j 0.5k 0.5k 3 si 60i cos60j 1.600i 4. 7j
26 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES 1.600i 4. 7j D z D z : : : D 500g D D g D D 984 N 1985 N 1 1 g 1 D s D D D i D 500 i
27 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES 60-kg cbiet is beig oved with 10-kg skid. The coefficiets of sttic d kietic fictio e 0.6 d 0.5, espectivel, t ll sufces of cotct. Deteie the lgest foce P tht c be pplied to the skid without cusig the cbiet to ove eltive to the skid. 60 kg s 0.6, k kg
28 pedig otio : Slippig s f s f, pedig otio : Tippig, 0 f s f 0 α ω TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES
29 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES uess tht ipedig otio is b slippig : f f c 60 f f b b N d N f f s c N e d, e N, N cot be egtive The guess tht the cbiet slips befoe tippig is icoect.
30 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES uess tht ipedig otio is b tippig : f s, 0 f f f c 60 f f b b N cf s N
31 : 0 10 : 10 : k f z z k f D D g D D P TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES 3.79 N N, N D D N 687 P
32 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES ied-is ottio Poit : tesectio of the is of ottio d ple of otio Equtio of otio z z z z z Setic w..t ple of otio z 0 z Velocit d cceletio of ss cete v v v 0 e 0 e t t z e
33 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES The uifo b show is log, hs ss of 10 kg, d is iitill t est i veticl positio. f the b is distubed d stts to fll to the ight, deteie the gul velocit of the b d the hoizotl d veticl copoets of the foce eeted o the b b the fictioless pivot t whe the b is hoizotl. z : z : : g g 1 si 4 3
34 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES sii cosj k k k cos i si j si i cos j cos si i si cos j cos si g si 0.75g si cos
35 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES d d d d 0.75g si dt d dt d 1.5g cos 1.5g 1 cos g, 1. 5g g g 147. N g 0.75g 4.53 N
36 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES gibl pedestl suppots plod i the spce shuttle d deplos it whe the doos of the cgo b e opeed i obit. The plod is odeled s hoogeeous ectgul block with ss of 6000 kg. The toque o the gibl is O-O is 30 N supplied b d-c bushless oto. With the shuttle obitig i weightless coditio, deteie the tie t equied to big the plod fo its stowed positio t = 0 to its deploed positio t = 90 if the toque is pplied fo the fist 45 of the tvel d the evesed fo the eiig 45 to big the plod to stop.
37 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES kg 1 : d s O O d kg O t t 39.8 s t O t t t s 1 O O 1.5
38 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES eel equtios fo ple otio : t Oz t z z oet equtio bout the ottio is O : t z Oz z z O t z O ee-od Dig Kietic Dig Pllel is theoe : Oz z
39 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES ottio of igid bod bout fied is though its ss cete : 0 z z z
40 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES t ovig to pllel positio though poit Q o lie O. obie the esultt-foce copoet d esultt couple t z loctio : Q q z Oz q z z k O Q : ete of Pecussio q k O 0 q k t Q Oz z O O q k O Q O
41 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES The esultt of ll foces pplied to the bod ust pss though Q. Su of the oets of ll foces bout Q is lws zeo. Q Q Q z O t q O t O t t t z q q t k k O z q k O
42 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES device fo ipct testig cosists of 34-kg pedulu with ss cete t d with dius gtio bout O of 60. The distce b fo the pedulu is selected so tht the foce o the beig t O hs the lest possible vlue duig ipct with specie t the botto of the swig. Deteie b d clculte the gitude of the totl foce o the beig O istt fte elese fo est t = kg k O 60
43 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES The ipct occus t cete of pecussio. q k O b 0.6 b b si Q 0; 34 t t q t N 0; N cos60 0 t t N g
44 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES toque of 300 N is pplied to pulle of the belt dive show. Pulle is solid cicul disk with ss of 15 kg. Pulle d the cble du hve cobied ss of 75 kg d dius of gtio with espect to the is of ottio of 150. The ss of block is 150 kg. f the sste stts fo est, deteie. The tesio T i the cble suppotig block. b. The cceletio of the block. c. The tie t 10 equied to ise block distce of kg k 15 kg z z 300 N 150 kg
45 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES z z 15 : T T T T 0.1T 750. : : T 150g T T T 0.1, T 1 T 1 0.5T T 1 T 1 75g T T T T T 340 N v d s, 1.985t d s, c 6.493t d s t s 300N 15g 150g
46 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES The speed of ottig sste is cotolled with bke s show. The ottig pts of the sste hve ss of 300 kg d dius of gtio with espect to the is of ottio of 00. The kietic coefficiet of fictio betwee the bke pd d the bke du is Whe foce P of 500 N is beig pplied to the bke leve, deteie the hoizotl d veticl copoets of the ectio t suppot of the bke leve d the tie t equied to educe the speed of the sste fo 1000 ev/i to est. 0 z z k kg, k N : 1000 p 0
47 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES 0 : 0 : N 0 : 0.16N N N N, N, N N N : d s t d s 1000 p 300g k 0. t 0. s
48 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES eel Ple otio of Setic odies i z z z z z z z z z ee-od Dig Kietic Dig
49 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES ied is ottio of Setic odies t t z z z z O Oz Oz z z t z Oz Oz z Oz O t z O ee-od Dig Kietic Dig
50 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES eel Ple otio of Setic odies o bod of bit shpe o the geel ple otio of igid bod setic w..t. the ple of otio z z z z z z z z z z z z z 0 z z z
51 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES eel otio of Setic igid od () = (b) = (c) z () ee-bod dig z (b) Equivlet focecouple t z W b c z (c) Kietic dig
52 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES oet bout : b b b bw e b c e c i b j b i c j t c b, b c b c c c b z c b c b b, c c z z z e e t z
53 j j i j k k j k ω ω α k ω k α, j i j k k j k ω ω α TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES i j i j j i j i j
54 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES solid hoogeeous clide with ss of 100 kg ests o iclied sufce s show. The sttic coefficiet of fictio betwee the clide d the iclied sufce is cble wpped oud shllow goove i the clide coects it to block, which hs ss of 75 kg. The pulle tht the cble psses ove hs ss of 10 kg. f the sste is elesed fo est i the positio show, deteie the cceletio of the ss cete of the block, the cceletio of the ss cete of the clide, d the tesios i the two pts of the cble. 100 kg s kg 75 kg
55 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES T1 T The diffeece b/w the tesios povides the oet which cuses the ss of the pulle to otte with N s lock lide : T zc z z : T : 1 N : 0.4 b 981si cos30 0 N T1 zc c 0.4 8kg c N
56 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES Pulle Kietics : z z : 0. b b c p p 0.8 c c T 1 T zp p p zp kg
57 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES Equtios T T T1 10b T T b 50 b b 0 Solutios T N, b s, c 5.181d c T s 4.66 N, N, p d s s
58 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES heck vil N N s f > vil, slippig will occu 0.8 N b c k c c c c b b c 0.4 c b
59 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES sste cosists of flwheel, coectig od, d pisto s show. The coectig od hs uifo coss sectio d ss of 10 kg. The ss of the pisto is 15 kg. couple T ottes the flwheel couteclockwise t costt gul velocit of 500 ev/i. Deteie the veticl d hoizotl copoets of the foces eeted o od b the pis t d whe = 60. Neglect fictio betwee the clide wll d the pisto. 10 kg 15 kg 500 ev i
60 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES Pisto 15 : od z z z : : : L 10 10g z kg () (b) (c)
61 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES v v 6.80i 3.97j s si 1 v v v α v 5.36k 0.15 cos60i si 60j ω ω ω 5.36k 0.15cos60i si 60j 05.6i s 5.36k j i v ω 6.80i 3.97j k 0.75cosi sij i j ω v v 5.316k d 7.493i s s
62 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES i α ω ω k 0.75cosi sij 5.316k 5.316k 0.75cosi sij 05.6i j 0.75 sii cosj cosi sij i j α α i k d i j ω k ω s s cosi sij 5.316k 5.316k 0.375cosi sij s N 466. N N N
63 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES 00--diete clide is outed o 50--diete le s show i the figue. The ss of the clide d le is 50 kg d the dius of gtio with espect to the is of the le is 70. leible cods wpped oud the le o both sides of the clide e coected to 100-kg block tht slides o hoizotl sufce. The kietic coefficiet of fictio betwee the hoizotl sufce d the block is 0.5. f the clide olls without slippig, deteie the cceletio of the block d the tesios i the two cods. 100 kg k 0.5 k 50 kg 70
64 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES t 0 : 50 : 0.05 N 50g cos60 0 : T 50g si T 0.1 (1) () (3) T : N : T 100g (4) (5) t 50g N 0.1, (1) N 45.5 N (5) 981 N N k N 45.5 N () T (3) T (4) T d kg N T N s s s T 100g N
65 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES t 50 : 0.05 : T 50g si T 0.1 () (3) T T () T (3) (4) T t 50g N 5.14 d s T N N s s
66 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES 1--diete hoogeeous clide with ss of 100 kg ests o the bed of fltbed tuck s show. The tuck cceletes fo est t te of 1. /s fo 10 s d the oves with costt velocit. f the clide olls without slippig o the bed of the tuck, deteie the distce tveled b the tuck befoe the clide olls off the tuck. d 1 100kg t 1. s 3
67 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES t 1. s, v t 1.t, t 0.6t fo 0 t 10 s 100g 0 : N : 100g : (1) () (3) 0.5 i 0.5 j k 0.5jk k 0. j 5 t t 0.5 (4) N (1),(4) (3) t t t 100
68 TNSLTON, OTTON, ND ENEL PLNE OTON O D ODES d t s (4) 0.4 s v 0.4t c t 0.t c 4 0.t c4 The clide will be t the bck of the tuck whe = t. 0.t 3 0.6t t 0.6t t 7.5, t.7386 s 0.t 3 t t
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