Fundamental principles

Transcription

1 JU 07/HL Dnacs and contol of echancal sstes Date Da (0/08) Da (03/08) Da 3 (05/08) Da 4 (07/08) Da 5 (09/08) Da 6 (/08) Content Reve of the bascs of echancs. Kneatcs of gd bodes coodnate tansfoaton, angula veloct vecto, descpton of veloct and acceleaton n elatvel ovng faes. Eule angles, Reve of ethods of oentu and angula oentu of sste of patcles, neta tenso of gd bod. Dnacs of gd bodes Eules equaton, applcaton to oton of setc tops and goscopes and pobles of sste of bodes. Knetc eneg of a gd bod, vtual dsplaceent and classfcaton of constants. D Alebet s pncple. ntoducton to genealed coodnates, devaton of Laganges equaton fo D Alebet s pncple. Sall oscllatons, at foulaton, Egen value poble and nuecal solutons. Modellng echancal sstes, ntoducton to MALAB, copute geneaton and soluton of equatons of oton. ntoducton to cople analtc functons, Laplace and Foue tansfo. PD contolles, Phase lag and Phase lead copensaton. Analss of Contol sstes n state space, pole placeent, copute sulaton though MALAB. DMS60 Dnacs and Contol of Mechancal Sstes Pupose: Knetc eneg Fundaental pncples Focus on 4 Knetc eneg of gd bodes 4 Concept of vtual ok and vtual dspalceent 4 Classfcaton of constants 4 DAlebets Pncple 4 Eaples JU 07/HL DMS60 Dnacs and Contol of Mechancal Sstes

2 JU 07/HL 3 Knetc eng of a gd bod DMS60 Dnacs and Contol of Mechancal Sstes Knetc eneg of a gd bod can be sepaated nto: the knetc eneg of oton of the ass cente C and the knetc eneg of otaton of the bod about C. Let s consde a gd bod of ass n plane oton. #"! #"! #$"$! Rot v v v v v ans ; ø è ω ( ) Rot ω v JU 07/HL 4 Knetc eng of a gd bod f aes of coodnates concde th the pncpal aes of the bod DMS60 Dnacs and Contol of Mechancal Sstes v ( ) O v o ø è!" $!# Fo a gd bod s otatng about a fed as though O. n pncpal coodnates

3 Consevaton of Eneg Eaple: Let s epesse the ok of consevatve foces as a change n potental eneg, the pncple of ok and eneg becoes V V Consde the slende od of ass. 0, V 0 v ( ) ( l l l ) V Wl snq gl snq Moent of neta of od of length l 3 JU 07/HL ass eleased th eo veloct detene at q V V l 0 ω 3 3g ω sn θ è l ø gl sn θ KE 4 Eaple A heel s feel otatng about a hoontal as O and an deal stng s apped on t has a sall ass attached at ts end as shon. he heel has a oent of neta and adus R. Fnd the epesson fo the speed of the ass afte t has fallen though a dstance h. O ω v/r Sol.: Usng eneg theoe V V JU 07/HL 0 gh v ω 0è v DMS60 Dnacs and Contol of Mechancal Sstes 6 ghr R (,R )

4 D Alebet s Pncple 4 Equlbu equaton n statcs: F 0 è can be used to solve fo thee unknons n 3D space 4 Equlbu equaton of oton n dnacs: F.a hee F s the su of the etenal foces actng on the patcle, s the patcle ass (constant), and a s the acc. of the patcle elatve to an netal ef. fae. Retng the equaton of oton: F.a 0 Etenal foces neta foces 0 Conveson of the sste nto a dnacs equllbu è D Alebet s pncple Fo a bod n otatonal oton:.a and.a 0 JU 07/HL hee s the su of etenal toques actng on the bod, s the ass oent of neta of the bod t the otatng as, and a s the angula acceleaton of the bod. he pncple s appled to solve pobles fo a bod sultaneousl undegong tanslaton and otaton. t geatl splfes coplcated dnac pobles n echancs. DMS60 Dnacs and Contol of Mechancal Sstes 7 Vtual ok and vtual dsplaceent 4 Fo a sste of N patcles Let the poston of each patcle be gven b,, 3N Let s consde that thee ae 3N foces F, F, F 3N actng n the decton of each coodnate Let the sste at a gven nstant s subjected to sall dsplaceents d, d, d 3N n the decton of each coodnate Wok done b the foces: dw F d Vtual ok Vtual dsplaceent JU 07/HL OR, n vecto notaton dw F d Note: hee s no passage of te fo the dsplaceent d he foces ean constant. DMS60 Dnacs and Contol of Mechancal Sstes 8

5 Constants and degees of feedo Classfcaton of constants 4 f a constant can be tten as an equalt functon of the fo f(,. n, t) 0 o f(q, q, q N, t) 0 (hee q ae genealed coodnates), then t s called holonoc constant. A gd bod has such tpes of constants and estence of holonoc constants allo elnaton of soe vaables 4 Constants that can not be epessed n the above fo ae called nonholonoc constants. Fo eaple, oton of a gas n a contane JU 07/HL DMS60 Dnacs and Contol of Mechancal Sstes 9 Vtual ok and vtual dsplaceent Classfcaton of constants 4 Equalt constants nvolve onl genealed coodnates and te (holonoc), hle nonholoonc constants depend on genealed coodnates and hghe devatves (veloctes), as ell as te. 4 nequaltes do not constan the poston n the sae a as equalt constants do, thus the ae nonholonoc. Othe classfcatons 4 A geoetc constant estcts the confguatons that can be acheved dung oton. JU 07/HL 4 A kneatc constant onl estcts the veloctes that can be acqued at a gven poston. he sste can, hoeve, occup an poston desed DMS60 Dnacs and Contol of Mechancal Sstes 0

6 Vtual ok and vtual dsplaceent 4 Constants and degees of feedo (DOF) A DOF s an potant eleent n descbng the dnacs of a sste consstng of ultple luped paaetes 4 Nube of degees of feedo of a sste the nu nube of vaables to copletel specf the poston of eve patcle n the sste Nube of knetatcall ndependent confguaton constants o vaables equed to descbe copletel the oton of a sste NB: A gd bod n (unconstaned) space has 6 DOF (,Ω) JU 07/HL he nube of degees of feedo of a patcle/luped ass gets educed f t s subjected to constants. Fo nstance, a gd bod n a 3D space has 3 DOF constaned, hence ts oton s defned b 3 DOF. DMS60 Dnacs and Contol of Mechancal Sstes Vtual ok and vtual dsplaceent 4 Constants and degees of feedo (DOF) 4 Eaples Rollng contact Pnned jont Fed jont (Geneall onl appled to a gd bod) (Geneall not appled to a gd bod oton) JU 07/HL DMS60 Dnacs and Contol of Mechancal Sstes

7 Vtual ok and vtual dsplaceent 4 Eaple Use the vtual ok ethod and detene the elatonshp beteen the toque M appled to the cank R and the foce F appled to the slde n the shon echans. JU 07/HL Fo sste n statc equlbu dw Fd Mdq Fd 0 hee R cosq L cosf h Rsnq Lsnf Þ snf gonoetc elaton : cosf cos R Þ f sn q è è L ø ø ( sn f) snq DMS60 Dnacs and Contol of Mechancal Sstes R L 3 cos R R q L sn q è è L ø ø R Þ d Rsnqdq L Substtutng and splfng ì Þ M FRsnq í L î snq cosq R sn q è è L ø ø ü Rosq ý R sn q è L ø þ Sua and Questons n ths lectue e focused on KE of a gd bod, a.o. 4 Knetc eneg of gd bodes: anslaton and otaton KE 4 KE of gd bod hen coodnates of oton concde th the pncpal aes of the bod 4 Concept of vtual ok and vtual dspalceent 4 DOF and classfcaton of constants 4 DAlebets Pncple JU 07/HL Net: nto. genealed coodnates, devaton of Laganges equaton fo D Alebet s pncple. DMS60 Dnacs and Contol of Mechancal Sstes 4?