ACTIVE VIBRATION CONTROL OF A CANTILEVER BEAM

Transcription

1 ACIVE VIBAIO COOL O A CAILEVE BEAM J. ůma Abtact: he pape eal wth aal of a catleve beam euppe b the actve vbato cotol tem. he catleve beam a a cotuum appoxmate b a lumpe-paamete tem. he lumpepaamete moel eable to eve the tafe fucto elatg foce to placemet to eteme the appopate cotolle tpe a to eteme ue what coto the tem wll wo. ewo: catleve beam lumpe-paamete moel actve vbato cotol cotolle tpe.. Itoucto Cotol theo bae o the moel wth oe (gle) put a oe output (SISO) o moel wth multple put a multple output (MIMO). hee moel ae ecbe b tem of oa ffeetal euato. he ecpto of a cotuum o the cota ue patal ffeetal euato. o the aal of the cotuum utable appoxmato of tem wth a lumpe paamete moel mut be ue (Peumot & Seto 8). he compute tafe fucto of the lumpe-paamete tem allow to eteme the appopate cotolle tpe a to eteme ue what coto the tem wll wo (Geta 9).. Lumpe paamete moel A catleve beam of the legth L a a cotuum ca be ve to cete elemet of the ame legth ΔL that ae moele ug g-bo amc. ow to ceate the lumpe paamete moel of the catleve beam of the ectagula co ecto a to aocate th multbo tem wth the Catea cooate x z how g.. he catleve beam clampe at the x-plae a t cetele paallel to the z-ax. It aume ol plaa moto of the catleve beam the zplae. Let be the umbe of flexble l the moel. Lg of a pa of ajacet beam elemet coee the metoe plae a fee wth a too pg. he cooate of the multbo tem ae uuall aocate wth the gavt cete. Such cooate tem eue the atoal et of cota fo lg of the vual beam elemet oe pot. Atoal euato ae ot eee f the cooate tem choe uch a wa that ecbe moto of the meetg pot of two ajacet elemeta beam. he vetcal cooate of thee pot ae mae b. he agle of otato wth epect to the hozotal ax ca be mae b δ δ δ. he eflecto Δ of the beam ma be expee a a fucto of the beam legth a the ffeece Δδ of the ajacet elemet δ L. he beg tffe δ of the elemeta catleve beam elate the apple beg momet M to the eultg elatve otato Δδ of the elemeta beam M 3EI x δ () δ L whee E Youg moulu of the beam mateal [/m 3 ] I x bh the aea momet of eta of the beam co-ecto [m 4 ] about the hozotal x-ax b the beam wth a h the beam heght. Pof. Ig. Jří ůma CSc.: VSB echcal Uvet of Otava acult of Mechacal Egeeg 7. ltopau Otava Czech epublc e-mal: j.tuma@vb.cz

2 ΔL δ δ δ δ δ x z b h Co-ecto Clampg g. : Cooate a elemet of a catleve beam. he cooate of the beam eutat pot the Catea cooate a the epeet geealze cooate fo Lagaga euato of moto ae etcal. o futhe evato t mae ee ol moto the ecto of the -ax. Becaue the ae aume mall efomato the hft of the meetg pot the ecto of the z-ax ae eglecte. If all agle ae mall eough the the meaue aa gve b the fomula ( ). L δ δ δ δ () he cooate of the gavt cete of the elemeta beam ae a follow ( ) Y Y (3) he potetal V a etc eeg of the catleve beam the hozotal poto a a cotuum eplace b t lumpe paamete moel a follow ( ) δ δ δ x t J t Y m mgy V (4) whee J x the momet of eta [gm ] about the hozotal x-ax a pepecula to the cetele of the elemeta beam. o a ol cubo of heght h a legth ΔL t ( ) h L m J x. he catleve beam coee a a coevatve tem. Lagage' euato of moto of uch a tem ae a follow V t... (5) Afte toucto mbol M fo a ma uae matx a fo a tffe uae matx a G fo a gavt foce colum vecto a fo a cooate colum vecto to the matx euato of moto we obta G M (6) whee δ mg L B A A B A A B G M (7) L h m B L h m A (8)

3 . Stea tate efomato hape of the catleve beam Stea tate efomato of the beam the hozotal poto ue to the gavt foce eultg fom oluto of the euato G. If the catleve beam vetcal poto the. o tetg a beam wth the followg paamete pepae: L.5 [m] b.4 [m] h.5 [m]. he beam ve to elemet. Deflecto of the beam' ow elf-weght how g. A.. ee vbato g. : A) Deflecto of the catleve beam' ow elf-weght B) he ft 5 of moal hape of the catleve beam. o vetcal poto of the catleve beam the goveg euato of fee vbato a follow M (9) he oluto of th euato of the homogeou tpe aume to be the fom of u exp( jωt) whee u a -meoal vecto of the ocllato ampltue a ω a agula feuec. Afte ubttuto to (9) we obta ( ω M) u () Gve that the ma matx mmetc a potve efte the ubttuto λ ω a multplcato of the pevou euato b the vee ma matx o the left e eult ( A λi) u A M () whee I a ut matx. Gve that the ma a tffe matce ae mmetc the the matx A mmetc a well. o the ozeo vecto u the etemat of the matx ( A λ I) ha to be zeo. Becaue the etemat et ( A λ I) a -egee chaactetc polomal of λ the umbe of oot λ calle the egevalue eual to the egee of polomal. he coepog ozeo oluto of the homogeou euato calle a egevecto. It pove the lea algeba theo that all the egevalue of the mmetc matx ae eal a a abta pa of the egevecto coepog to ffeet egevalue othogoal. We fom a pectal matx Λ ag( λ λ... λ ) a a U u u.... egevecto matx [ ] u Becaue the beam ve to elemet t poble to calculate moal feuece a moal hape of vbato. Ol 5 out of the moal hape etfable b the umbe of oe ae how g. B. he moal feuece ae ummaze ab.. ab. : Moal feuece fo. Moe e z Excte vbato he excte vbato of the catleve beam the vetcal poto ecbe the euato of moto wth the exteal foce p p p aemble to a vecto p o the ght e a actg at the gavt cete of the beam elemet

4 M p () he peece of vcou ampg uch a a patve foce exte the left e of the euato of moto b a atoal tem whch popotoal to veloct M C p C α M β (3) whee the matx of popotoalt C fo alegh ampg a lea combato of the ma a tffe matce a α β ae cotat of popotoalt. he elatohp to the ampg ato ξ ca ξ π α f βf whee f the feuec hetz be ee ug the fomula ( ) 3. afe ucto Vbato of mechacal tuctue ampee ve lghtl. It' ol a few pecet of ctcal ampg. he pupoe of actve vbato cotol ceae ablt of tuctue to abob vbato b ag a atfcal electoc feebac. o aalze the effect of actve vbato ampg t ftl aume that the tem ot ampe at all. It aume that a p ae complex hamoc fucto of tme ( exp ( ωt) ) a Y a P ae complex ampltue epectvel. he tafe fucto the fom of a uae matx elatg the placemet... of the lumpe mae to the foce p... actg at thee mae efe b the followg fomula ( & u ) ( λm) P P Y A moal tafom V the ba fo the evato of the tafe fucto. he cooate ae tafome to geealze cooate b ug the matx V. he elatohp of the tafe fucto to the moal popete of the tuctue ca be efe f the moal tafomato matx V ha the followg popet V M V I. It ca be pove that the othoomal egevecto v aage the matx V ae gve b (3) v u u Mu... (4) he tafe fucto matx calle the eceptace a a fucto of egevecto a egevalue accog to the fomula D ( λm ) VDV ag ( λ λ) ( λ λ)... ( λ λ) ( ) λ ω epe o the whee D a agoal matx. he matx elate the foce actg at the -th lumpe ma to the placemet of the -th lumpe ma whee meaue. he Laplace tafom of the vual elemet of the matx a follow vv... (6) ω whee v... the -th elemet of the -th omalze egevecto. he pole of the tafe fucto le o the maga ax of the complex plae. he tem o the tablt mag ot table a multaeoul ot utable. ab. : Value of the pouct v v fo v v he pouct v v of the elemet of the matx V gve ab.. he aumpto (6) about the tablt mag of the catleve beam a a amc tem the wot oe becaue the ampltue of vbato alwa eca afte ome tme. (5)

5 4. Actve vbato cotol he pupoe of the tem fo the actve vbato cotol (AVC) to compeate the effect of a tubg exteal foce o the vbato of the beam. It eable to elocate the pole of the tafe fucto of the cotolle tem fom the maga ax the left half-plae of the complex plae. he catleve beam coee a a MIMO tem compoe of the lumpe mae whoe cout eual to. Vbato of all thee mae ca be cotolle b foce actg at all of them a t how g. 3. Actuato Seo eebac Cotolle g. 3: Actve vbato cotol. We aume that the tem of the SISO tpe wth a cotolle wth a tafe fucto (). hee ae two poble oluto the collocate a o-collocate actve vbato cotol. o the collocate tem the coectg foce p act a the epoe Y meaue the ame cete gavt of the beam elemet. o the o-collocate tem t aume that the coectg foce p act at the lumpe ma exe b a the vbato ae ee at the lumpe ma exe b. A example of the o-collocate tem how g. 4. Vbato of the fee e elemet of th catleve beam ee at a the coectg foce act at the elemet jut ext to the clampe e theefoe. A bloc agam of the cloe loop tem how g. 5. Actuato Seo eebac Cotolle g. 4: o-collocate tem of actve vbato cotol. Cotolle f Beam - g. 5: Cloe loop tem of AVC ~ hee a tafe fucto of the cloe-loop tem elatg the placemet of -th lumpe ma to the et pot x SP a the fucto foce p actg at -th lumpe ma SP elatg the placemet to the feebac

6 ~ SP X X SP v ω v v ω v v v ( ω ) ( ω ) v v ( ω ) (7) he tem the eomato of () a polomal of the uae vaable the coeffcet of the o powe ae eual to zeo. o a table tem the vaable ae to o powe mut be ae D D (8) vv ( ω ) v v ( ω ) whee D... ae the metoe potve coeffcet. he egee of the polomal the umeato of the tafe fucto () geate tha the egee of the polomal the eomato. he feebac cotolle of the evatve tpe o the tablt mag b telf ue to lac of tem wth vaable ae to a o powe. he atoal popotoal pat of the cotolle eue tablt (ůma ). whee a ga a a tme cotat. (9) he tubace foce ca be of the boa o aow feuec pectum. Suppoe that the feuec pectum of tubace affect ol the -th moe of vbato. v v ω ω () he tafe fucto of the cloe loop wth the cotolle ecbe b the tafe fucto (9) a follow ~ SP whee a ga facto X X ( ) ω ω ( ) SP ( ) ( ω ) () v v calle a euum. A t how ab. ome of the ga facto ae egatve. he ampg ato ξ a the eca cotat σ of the tem ecbe b () eual to ξ σ ξ ω () ω he eca cotat σ fo < ξ < a ω > eteme a evelope of ecag vbato exp ( σt). Stctl peag f the eca cotat egatve ce < the the vbato of the coepog feuec oe ot eca a the tem utable. Ol atual ampg of the catleve beam ca compeate th tablt caue b actve vbato cotol. Deg of the cotolle paamete eue tag to accout a umbe of exteal fluece a mal the atual mateal ampg of the beam vbato. he effect of actve vbato cotol ofte emotate o the vbato eca of the beam whch bee to a eflecte poto a uel eleae. I th cae ol the lowet moe of vbato ae excte. Both the cotolle paamete ca be calculate ug the patal pole placemet (Mottehea & eha & am 9) a well. he pa of the complex cojugate pole jω a jω of the tafe

7 fucto ( ) wll be place the complex cojugate pole j Ω σ a j Ω σ of the cloe loop tafe fucto ( ) SP ~. o both the eplace pole the eomato of ( ) SP ~ have to be zeo. We obta two euato wth uow paamete a. (3) Afte eaagg the euato (9) we get. (4) he oluto of thee two euato a follow (5) Ug the cotolle of the popotoal-evatve tpe ol oe pa of pole ca be place to the table pat of the complex plae. 5. Smulato of actve vbato cotol Euato of moto of a mechacal tem wth a electoc feebac ca be wtte the fom p b C M u u (6) whee b a ae colum vecto efe a follow. b (7) he b combg euato (5) we obta the coecte ampg a tffe matce ( ). p b b C M (8) o the catleve beam ecbe above we aume ow that 5. Vbato of the fee e elemet of th catleve beam ee at 5 a the coectg foce act at the elemet jut ext to the clampe e theefoe. he ampg C a tffe matce ae fluece b the feebac the followg wa C b C b (9) Both the matce C a become ammetc feuec [z] [-] alegh expemet g. 6: he epeece of the ampg ato ξ o feuec

8 It aume the vcou ampg foce (3) efe b the matx C α M β whee α.59 β. 4. he ampg ato (ξ) fo the ft two moe of vbato about.4 a the coepog ampg cotat (σ) about.6. he epeece of the ampg ato ξ o feuec how g. 6. Small ccle cate the ampg ato fom the expemetal moal aal. A foce of 9.8 [] actg at the fee e of the catleve beam whch o the oppote e of the clampe e. he foce uel eleae at t. A wea ampg ae to avo tablt of the uampe epoe. he eult of mulato of fee vbato how the left pael of g. 7 (AVC O). he effect of the actve vbato cotol opeato how the ght pael of g. 7 (AVC O) a well. he ga 66.8 a the tme cotat.7 [] wee ege fo the uampe catleve beam. he ampg cotat of the ft vbato moe wa ceae tme b ug a mulate feebac. All the epoe wee calculate ug ewma metho fo tegatg ffeetal euato. g. 7: ee vbato of the catleve beam wth AVC O a O 6. Cocluo he lumpe-paamete moel of the catleve beam wa ege ug the metho bae o the moal aal. It wa pove that the catleve beam ca be actvel ampe ol b a foce whch cotolle b the PD cotolle. he feebac of the D tpe ot uffcet fo ampg uampe tem. h pape focue o the computato of the PD cotolle paamete ug the pole placemet metho. he fomula fo calculato the cotolle ga a tme cotat ae eve. Acowlegemet h eeach ha bee uppote b the Czech Gat Agec poject o. P//5 Actve vbato ampg of oto wth the ue of paametc exctato of joual beag. efeece Peumot A. & Seto. (8) Actve Cotol of Stuctue Wle. Geta G. (9) Vbato Damc a Cotol Spge. J. & u Z-. () Moal Aal Buttewoth eema Oxfo. ůma J. () Aal of actve vbato cotol fo a catleve beam wth the ue of the lumpe paamete moel. I Damc of mache. ebua 7-8 Pague Cech epublc pp Mottehea J. E. & eha M. G. & Y. M. am Y. M. (9) A Itoucto to the eceptace Metho Actve Vbato Cotol. I Poceeg of the IMAC-XXVII ebua 9-9 Olao loa USA.