ON THE MOTION OF PLANAR BARS SYSTEMS WITH CLEARANCES IN JOINTS

Transcription

1 ON THE MOTION OF PLANAR BARS SYSTEMS WITH CLEARANCES IN JOINTS Şl uv dr g Ja-Crsta GRIGORE, Uverstatea d Pteşt, strtîrgu dvale Nr Prof uv dr g Ncolae PANDREA, Uverstatea d Pteşt, strtîrgu dvale Nr Cof uv dr g Mara PANDREA, Uverstatea d Pteşt, strtîrgu dvale Nr Abstract: O cosders the kematc cha composed by artculated bars wth clearaces wth permaetly cotact The goal s to apply the multbody methods ad t eed to determe the costrats matrx whch for jots wth clearaces has a certa form I the fal part s preseted a umercal applcato Keywords: matrx, dfferetal equato, dagrams Itroducere The clearaces jots fluece the moto of a mechacal system by produced chocks ad modfyg the movemet elemets laws I the most cases the chocks avodg ca be realzed usg the elastc supplemetary lks to compesate the clearaces I these cases the cotacts betwee elemets exst ad the mechacal model s to troduce a supplemetary elemet wth legth equal wth half of clearace ad ertaless I the followg o cosders a kematcs lkage wth elemets wth clearace jots By applyg the multbody methods, t obtas the dfferetal matrx equato ad a umercal applcato s doe for the case of vbratory moto The costrats matrx The case of jot wthout clearace The costrat equatos of a joted system are obtaed from equalty of the pot O coordates (fg) whch belogs to the elemet wth pot O co-ordates whch belogs to the elemet Fg Fg Accordg to the multbody methods [], t choces as geeralzed agles ad X, Y beg the coordates of the elemets ceters of gravty ad f the legth of the elemet s equal wth l, o obtas the equalty: X l cos cos Y l s Y l s By dervato of the relatos () by tme ad usg the otatos: X l () 74 Fabltate s Durabltate - Fablty & Durablty r/9 Edtura Academca Brâcuş, Târgu Ju, ISSN X

2 l s B ; B o obtas the followg equalty: l s () l cos l cos X B Y B X Y I the case whe the jot O s fxed, the matrx B elemet The total costrats matrx s obtaed by cocateatg of the matrx from relato () Cosderg the ope kematcs lkage from fg3a ad closed kematcs lkage from fgure (3a) o obtas the followg costrats matrx s zero matrx because do t exsts the (3) a) b) Fg 3 B B B B 3 (4) B B3 B B B 3 3 B 3 3 (5) 3 B B B3 where 3 represets the zero matrx wth m rows ad colums The case of jot wthout clearace I the case of the clearace the jot o accepts the model from [], where a fctve elemet O O ertaless ad wth the legth equal half of clearace s troduced (fg) 75 Fabltate s Durabltate - Fablty & Durablty r/9 Edtura Academca Brâcuş, Târgu Ju, ISSN X

3 Usg the otato from fg the followg relato ca be wrtte: t results X l cos X l cos (6) D cos r Y l s Y l s (7) D s r D D (8) By dervato of the relatos (8) by tme ad usg the relatos: D s D cos l ; D4 D s D D D D D D D D l D3 cos (9) o obtas the expresso D 3 ; 4 () X D Y D Y () I the case whe the jot O s fxed, the relato () t appears oly D matrx because X Y ; Thus for the kematcs lkage from fg3a o obtas the total costrats matrx for the case whe the jots O,O3 are clearaces X B B D D 3 () D D3 3 The matrx dfferetal equato of moto Correspodg to the geeralzed co-ordates X, Y,,, where represets the elemets umber ad deotg by m, J,,, masses respectvely cetral erta momets o obtas [] the ketc eergy expresso T Ec q M q (3) where q X X T Y Y (4) 76 Fabltate s Durabltate - Fablty & Durablty r/9 Edtura Academca Brâcuş, Târgu Ju, ISSN X

4 M m m J m m J (5) Deotg by F, F, the exteror forces projectos o the axes OX, OY ad deotg X Y by M the resultat momet versus C of the exteror forces whch acts o the elemet o obtas [], [4] the matrx dfferetal equato M B T B O j,3 q F B q (6) where: - j s matrx B row umber -,, j are Lagrage multplers F F F M F F M T (7) X Y X j Y (8) R x y I the case of jot wthout clearace the Lagrage multplers from jot are the reactos H, V ad for the case of the jot wth clearace the multpler s the axal force N alog the fctve elemet To equlbrum the followg expresso are obtaed F z B T (9) The equatos (9) together wth equatos () or (8) compose the system whch gves q ~ for the equlbrum estate Makg the substtuto the vector q q ~ () o obtas the matrx dfferetal equato of vbratos versus the précsed equlbrum posto (7) 77 Fabltate s Durabltate - Fablty & Durablty r/9 Edtura Academca Brâcuş, Târgu Ju, ISSN X

5 teta teta 4 Aplcato For the bars system from (fg4) acted oly by ow elemets weghts the goal s to determe the tme varato of the parameters,, kowg that the jot O s wth clearace ad the tal codto are gve below: t ; ; ; X l; Y ; X l r l cos ; Y l s ;, X Numercal applcato: l l 5 m; m m Kg; J J Kg m ; 3 rad; r m The umercal calculus s made wth the help of a route draw up by the algorthm descrbed the paper ad t leads to the results plotted the dagrams from fg5 a, b, c, d 5 Coclusos Fg5c) represets the drecto varato of the fctve elemet ad t detects that the ampltude s dd ot exceed the 5 radas value () Fg4 5 teta(t) 3 teta(t) tmp tmp a) b) 78 Fabltate s Durabltate - Fablty & Durablty r/9 Edtura Academca Brâcuş, Târgu Ju, ISSN X

6 alfa lambda3 5 alfa(t) 5 lambda3(t) tmp tmp c) d) Fg5 Also t detects that the agle ampltude dd ot exceed 5rad value, the agle ampltude dd ot exceed the 3rad value ad the axal force varato (fg5d) s mostly7 N Refereces [] Amroche, F, Fudametals of multbody dyamcs, Brkhäser, Bosto, Berl, 4 [] Grgore, J-C, Cotrbuţ la studul dyamc al mecasmelor cu jocur, Teză de doctorat Uverstatea d Pteşt, 8 [3] Padrea N, Popa D, Mecasme Edtura Tehcă, Bucureşt, [4] Shabaa, A, A, Dyamc of multbody systems, Cambrdge Uversty Press, Cambrdge, 5 79 Fabltate s Durabltate - Fablty & Durablty r/9 Edtura Academca Brâcuş, Târgu Ju, ISSN X