Advanced Mechanics of Mechanical Systems

Transcription

1 dvacd Mchac of Mchacal Stm Lctu: Pofo k Mkkola, Ph.D., Lappata Uvt of cholog, Flad. ocat Pofo Shaopg Ba, Ph.D., albog Uvt. ocat Pofo Mchal Skpp d, Ph.D., albog Uvt. M. S. d: dvacd Mchac of Mchacal Stm Wlcom to albog Uvt! Vu Bad ad coff wll b vd v mog. Fb. 14, oom 59, Fb. 16, oom 1.111, Cat, No Bu top, Pa plac M. S. d: dvacd Mchac of Mchacal Stm 1

2 M. S. d: dvacd Mchac of Mchacal Stm Pactcal fomato Food: Roll ad coff wll b vd v mog. Cak ad coff wll b vd v aftoo. Luch: Sadwh, hot dh ad alad ca b puchad th U cat. Pa, Pata ad mo at Blla Itala. Socal vt: D at a tauat dow tow thuda th th. Mo fomato wll follow lat. Wl tt: Eduoam U ou ow cdtal. U-1-DY. Nw pawod v da. W wll povd th fo ou. M. S. d: dvacd Mchac of Mchacal Stm

3 Cou outl Da 1: u., 189: Fudamtal of kmatc of gd mult-bod tm M. S. d Fb. 14, Room 59 ad Fb.16, Room Da : Wd., 199: Damc modlg of gd mult-bod tm S. Ba Fb 14, Room 59 Da 3: F., 19 Itoducto to flbl tm damc. Mkkola Fb 14, Room 59 Wokhop: F. 6 Damc modlg wth dam IdéPo Fb 14, oom 59 NB! Bg ou labtop fo th pat of th cou M. S. d: dvacd Mchac of Mchacal Stm Mult-bod mchacal tm mult-bod mchacal tm cot of bod, ot, actuato ad load. h am of th modl to th comput moto o foc. M. S. d: dvacd Mchac of Mchacal Stm 3

4 Dg of fdom Dg of fdom DOF: a th dpdt wa of moto fo a mchacal tm. h umb of DOF gv th mmum umb of coodat qud to full dcb th moto. Coutg th umb of DOF DOF DOF 6 3 bod bod ot ot cotat, 3D cotat, D patal tm plaa tm Cotat that clud moto fomato a calld kmatc Dv dv ad th umb of th, w dot. M. S. d: dvacd Mchac of Mchacal Stm Eampl Dg of fdom Sld-cak D: 3 bod 3 DOF 9 3 volut ot DOF -6 1 talatoal ot DOF - 1 DOF 4-ba lkag D: 4 bod 3 DOF 1 6 volut ot DOF -1 1 talatoal ot DOF DOF O dudat cotat! M. S. d: dvacd Mchac of Mchacal Stm 4

5 DOF Dv Kmatc dtmac : Kmatcall dtmat tm h tp of tm th mot gal damc of mchacal lkag, wh w hav to olv th cod-od dfftal quato of moto. DOF Dv : Kmatcall dtmat tm h tp of tm much a to hadl. h moto compltl pcbd b th dv, mplg that kmatc ad ktc a dcoupld poblm. DOF Dv pcal ca wh, wh th tm tatcall dtmat. DOF Dv : Kmatcall ov - dtmat tm h tp of tm qu that dfomato mut b tak to accout. M. S. d: dvacd Mchac of Mchacal Stm Kmatc aal Kmatc aal: al of th moto of a mchacal tm wthout codato of th foc that cau th moto. Iclud th dcpto of th volvd bod, whth gd o flbl, th cocto ot ad moto. md at computg th poto, vloct ad acclato latohp btw th volvd bod. al of a mchacal tm pul bad o kmatc fomato qu a kmatcall dtmat tm. M. S. d: dvacd Mchac of Mchacal Stm 5

6 Math plma Gomtc vcto, dotd wth a aow: a XI YJ ZK k a vcto pag btw two pot pac th D o 3D. lgbac vcto, dotd wth a udl: a X Y Z a aa of umb, whch do ot cal hav a gomtc tptato. Z Y Skw-mmtc mat of a : a Z X Y X Matc a dotd wth a doubl udl Co-poduct: a b ab Mat-vcto poduct Rfc fam Global fam: h global fam ou ma fc,.. th fam wh w mau th abolut moto. Local fam bod-fd fam: local fam a atv tool that allow u to cod pat of th moto paatl f covt. vcto o pot pd th global fam calld a global vcto o pot. vcto o pot pd th local fam calld a local vcto o pot. 6

7 afomato mat a X I Y J Z K Global vcto Popt othoomal mat: 1 I ad Iv latohp: a a X Z I J K Y k k a afomato mat Local vcto h tafomato mat D Dfto of th otatoal coodat D taght fowad. h tafomato mat: co co h tm dvatv of : B wh B co h tm dvatv of B : B co 7

8 u Vloct of th vcto : gula vloct vcto Rotatoal vloct Itataou otato a It ca b pov that th agula vloct vcto a gomtc vcto. I.. fo tac th latv agula vloct ca b computd ug vcto dffc. Rlatohp btw th tafomato mat ad th agula vloct vcto um that a bod otatg a dcbd b th agula vloct vcto,. Hc, ach a otatg. h vloct of th a a: Hb, w ca p th tm-dvatv of th tafomato mat: k k k k k k wh agula vloct vcto local coodat. h w a gog to u wh dvg cotat quato. 8

9 Rotatoal coodat 3D So fa, w hav dcbd th otato of th bod b ma of a otato mat ad th agula vloct. h otato mat: 9 paamt ad 6 othoomalt cotat. Oth opto: Cada agl. Cata otato vcto. Eul paamt. d mo. Rotatoal coodat 3D Idall, a good paamtato of otato mut: Facltat gomtc tptato. B covt fo algbac mapulato. B a la a pobl tm of th otato agl. Pt a mmal 3 umb of paamt. hotcal advatag Computatoal advatag Not lad to gulat: th dfto. th v poblm. th tagt mat. Notc that all otatoal coodat 3D wth ol th paamt wll alwa clud a gulat. 9

10 1 Cada agl Rotato matc fo otato aoud ach a: Combd otato mat: wh l, m, dcat th otato a ad ca b,,. h otato quc dotd l- m-. h pcal ca: -- od calld Eul agl. -- od calld Bat agl. 1 co co, co 1 co, co co 1 m l -- quc -- quc Cada agl fom tafomato mat Rotato quc --: Ft comput th otato agl aoud th -a two pobl oluto: h comput th oth two agl fo ach of th two oluto of : Dtm whch of th two oluto to u. Fo tac ug th oluto that clot to th oluto fom a pvou tm tp o whch clot to o c c c c c c c c c c c c c ac ac co, co acta co, co acta

11 11 Cada agl fom tafomato mat dut th agl b utl th a wth a dd ag fo tac to u that th computd agl a cotuou ov tm. NB! h a gulat fo th patcula otato quc, wh,... 1,, co m-dvatv of Cada agl ad th agula vloct vcto Each compot of th Cada otato quc cotbut towad th total agula vloct. Rlato to th global agula vloct vcto: Computd b tafomg ach vloct compot to th global fc fam. Rlato to th local agula vloct vcto. Obtad b pmultplcato b : to bg th computato to th local fc fam. m l m l m l l m l m m l l l u u u u u u l m m l m m l m m m l l u u u u u u agt mat

12 dvatag: Mmal t of coodat. Nc gomtc tptato. Cada agl Dadvatag: Dpd o otato quc. h otato mat volv umou tgoomc fucto. Sgulat fo th v poblm. Eul thom Eul thom: h gal dplacmt of a bod wth o pot fd a otato about om a. Rotatoal fomula: co u u 1 co u 1

13 u Cata otato vcto 3 paamt Rotato agl Rotato a Rotato mat: 1 co I agt mat: co 1 I 1 gula vloct vcto: Cata otato vcto I 1 co No gulat th dfto: lm lm I Sgulat th v poblm: I f k u caot b dtmd 13

14 Cata otato vcto dvatag: Mmal t of coodat. Ea gomtc tptato. bc of gulat th dfto. Smpl lad po. Dadvatag: Sgulat fo th v poblm. Dfto: co 1 u 3 h otatoal fomula: Eul paamt 1 14

15 15 Eul paamt Sttg outd a backt to obta th tafomato mat: Notc that th Eul paamt a ot dpdt ad mut tatf: lo omtm wtt: wh 1 1 I 1 p p p 3 1 Idtt wth Eul paamt afomato mat: agt mat: gula vloct: G L GL p L G,

16 Eul paamt dvatag: No gulat! Pul algbac quatt,.. o tgoomc fucto. Dadvatag: ta cotat quato. Dffcult to dv. Mo coodat tha tctl dd. Op ad clod loop tm M. S. d: dvacd Mchac of Mchacal Stm 16

17 17 M. S. d: dvacd Mchac of Mchacal Stm Op loop tm Op-loop tm wth latv coodat mmal t: Op-loop tm ca b hadld fal a b dfg coodat aocatd wth th DOF of ach ot. h coodat copod dctl to th DOF of th tm ad a kmatc quatt ca b computd dctl fom th. M. S. d: dvacd Mchac of Mchacal Stm Op loop tm Poto aal: Rlatv talato, ot coodat ad tafomato mat. Vloct aal: cclato aal:

18 Clod-loop tm Clod-loop tm Clod-loop tm a mo complcatd. pcall, w gt ola quato, whch caot b hadld aaltcall. W hall fomulat th a gal ma that allow u to olv th quato umcall a tmatcal wa. M. S. d: dvacd Mchac of Mchacal Stm Mthod of appdd dvg cotat mbl th cotat quato, cludg dv, to o tm of ola quato. Poto aal: q t, t wh: qt t a th tm coodat. tm. M. S. d: dvacd Mchac of Mchacal Stm 18

19 Sld-cak ampl St up kmatc cotat quato fo th ld-cak mcham ug th mthod of appdd dvg cotat. h kmatc dv quato gv a: θ t Blackboad l 1 l Moto of a pot 3D pac. Cotat quato: Sphcal ot. wo ppdcula vcto. wo paalll vcto. Uval ot. Rvolut ot. Sphcal-phcal ot. Dv cotat. 3D cotat quato 19

20 Moto of a pot 3D pac. Cotat quato: Sphcal ot. wo ppdcula vcto. wo paalll vcto. Uval ot. Rvolut ot. Sphcal-phcal ot. Dv cotat. 3D cotat quato Moto of a pot 3D pac Global coodat of th pot P: P

21 Poto cotat:,3 q, t Sphcal ot cotat P P Poto cotat: wo ppdcula vcto tp I 1,1 h two vcto, ad hav cotat poto latv to th bod. 1

22 wo ppdcula vcto tp II Poto cotat:,1 d d ha cotat poto latv to bod wha d dfd btw two pot o th bod. Poto cotat: p, NB! h povd 3 cotat ath tha two, o o of th th quato mut b obmttd. ltatvl, p two ppdcula vcto: p, wo paalll vcto h cotat a alwa vald, but qu th dfto of two w vcto. t t

23 Poto cotat:,3 p, Rvolut ot cotat Poto cotat:,3 1,1 Uval ot cotat 3

24 4 Uval ot cotat Poto cotat: 1,1,3 P P Sphcal-phcal ot cotat Poto cotat: P P d l d d,1, Ma mo ot cotat ca b foud Nkavh Scto 7..

25 Ud to mpo moto o th modl. Dv cotat h mut b a ma dv cotat a modl DOF fo a kmatcall dtmat tm. Poto cotat: d q, t Eampl: d q, t q t Som popt of th modl plctl pcfd b th kmatc dv. E.g. a ot agl Wok o pat 1 of th c. Ec M. S. d: dvacd Mchac of Mchacal Stm 5

26 Vloct ad acclato aal M. S. d: dvacd Mchac of Mchacal Stm Mthod of appdd dvg cotat Poto aal: q t, t M. S. d: dvacd Mchac of Mchacal Stm 6

27 Mthod of appdd dvg cotat Vloct aal: q, q, t q q t h acoba mat ad : t q q t t th ow th colum M. S. d: dvacd Mchac of Mchacal Stm Mthod of appdd dvg cotat cclato aal: q, q, q, t q q q q q q, q, t q q q q q q qt q q tt qt q tt M. S. d: dvacd Mchac of Mchacal Stm 7

28 Dfft poto, vloct ad acclato coodat It omtm dabl to u dfft coodat fo poto aal tha fo vloct ad acclato aal. Fo tac wth cata coodat:?? v v q Vloct aal: q, v, t ˆ q v cclato aal: q, v, v, t v t ˆ v v v qˆ q ˆ qt ˆ q tt Moto of a pot 3D pac Global coodat of th pot P: P Vloct of th pot P: P cclato of th pot P: P Zo fo gd bod 8

29 9 Sphcal ot cotat Poto cotat: Vloct cotat: cclato cotat:,,3 P P t q,,,3 P P P P v I v I t v q v v,,,,,,,3 t v v q P P P P v I v I t v v q wo ppdcula vcto tp I Poto cotat: 1,1 Vloct cotat: 1,1 cclato cotat: b a ab Ht:,, 1,1 v v q

30 h pocdu th am fo th mag cotat tp. Sld-cak ampl Dv th Jacoba mat ad gamma vcto fo th ld-cak mcham. Blackboad l 1 l 3

31 Wok o pat of th c. Ec M. S. d: dvacd Mchac of Mchacal Stm hak fo ou attto! M. S. d: dvacd Mchac of Mchacal Stm 31