NON-LINEAR PARAMETER ESTIMATION USING VOLTERRA SERIES WITH MULTI-TONE EXCITATION

Transcription

1 NON-LINER PRMETER ESTIMTION USING VOLTERR SERIES WIT MULTI-TONE ECITTION imsh Char Dparm of Mchaical Egirig Visvsvaraya Rgioal Collg of Egirig Nagpur INDI-00 Naliash Vyas Dparm of Mchaical Egirig Iia Isiu of Tchology Kapur INDI-006 STRCT Th fucioal form rprsaio of ipu-oupu rlaioship hrough Volrra sris provis a srucur mahmaical plaform for h suy of oliar sysms. I mploys muli-imsioal rls which upo covoluio wih h appli ciaio forcs prss sysm rspos i h form of a powr sris. Frqucy omai rasforms of hs muli-imsioal rls provi highr orr frqucy rspos fucios which ar rla o firs orr rl rasfroms hrough h s of oliar paramrs of h sysm. This rlaioship has b mploy i his papr o vlop a oliar paramr simaio procur. Muli-o ciaio is us hr a rspos harmoic amplius a fuamal as wll as highr orr combiaio os ar masur. Volrra sris prssio for hs rspos amplius provis a s of oliar quaios ivolvig h firs orr rl rasforms a h oliar paramrs. Liar a oliar paramrs ar sima by solvig h s of oliar quaios a by saar curv fiig procurs. NOMENCLTURE f Eciaio forc h h orr Volrra rl r No-imsioal frqucy h orr rspos compo h orr Volrra rl rasform or ighr orr FRFs m m Rspos harmoic ampliu of a combiaio o m m Zr No-imsioal harmoic ampliu η No-imsioal rspos ς Dampig raio λ No-imsioal cubic oliar paramr µ ciaio ampliu raio INTRODUCTION Sysm aural frqucy No-imsioal im Volrra sris [] provis a srucur rspos rprsaio for o-liar sysms hrough highr orr Volrra rls. For harmoic ciaios h rspos ca b covily prss i rms of h firs a highr orr FRFs. Esiv rsarch wors [-5] hav b o i h ara of o-paramric sysm iificaio hrough masurm of hs highr orr FRFs. owvr rlaivly fw rsarch wors ar availabl i h ara of paramric sysm iificaio hrough Volrra sris. L [6] has sima oliar paramrs from h highr orr FRFs which wr masur from h firs harmoic amplius hrough rspos compo sparaio chiqu. Char a Vyas [] hav suggs simaio of olirar paramrs hrough rcursiv iraio usig h firs a highr orr rspos harmoic amplius. Sigl-o ciaio is mploy i hs procurs a h prim is rpa svral ims wih h ciaio frqucy varyig ovr a wi rag covrig h sysm aural frqucy. Th prs procur is bas o muli-o ciaio which gras a larg umbr of combiaio os i h rspos. armoic amplius a h ciaio frqucis a various combiaio os ar masur a h paramrs ar sima by solvig h s of oliar quaios hrough saar umrical chiqu. Th procur is umrically illusra a i is show ha a accura sima of h oliar paramr ca b obai wih much lss umbr of prims ha h prvious mhos mployig sigl-o ciaios. owvr h ciaio frqucis ar o b slc approprialy for covrgc rquirms. 0

2 VOLTERR SERIES RESPONSE REPRESENTTION Volrra sris rspos rprsaio for a gral physical sysm wih f as ipu ciaio a as oupu rspos is giv by wih f f h h is h h orr Volrra rl a is Fourir rasform provis h h orr Frqucy Rspos Fucios FRFs or Volrra rl rasforms as i i i h For a muli-o harmoic ciaio cosisig of wo probig frqucis giv by f cos cos rspos compos followig quaio bcom couga rms 5a couga rms 5b couga rms 5c a so o For Duffig oscillaor i. sysm wih cubic oliariy rprs by h quaio of moio f c m & & & 6 i ca b show [7] ha all v orr FRFs ruc o zro a h rspos sris coais oly h o orr rspos compos. Th o highr orr FRFs for a Duffig oscillaor ar rla o h firs orr FRFs [9] hrough h oliar paramr a a ypical rlaioship for hir orr FRF is giv by 7 RESPONSE RMONIC MPLITUDES FOR DUFFING OSCILLTOR I ca b s from quaios 5a-c ha a muli-o ciaio gras svral combiaio os i h rspos sris i aiio o h fuamal ciaio frqucis. For waly oliar sysms Volrra sris hibis rapi covrgc [7] a highr orr rspos compos i h sris bcom gligibly small. Trucaig h rspos sris afr h hir rspos compo rspos amplius of various fuamal a combiaio os followig quaios 5ac ca b prss as Usig h hir orr FRF syhsis rlaioship giv by quaio 7 abov s of quaios ca b r-wri i rms of oliar paramr a h firs orr FRFs as -

3 9 Equaios 9 provi a s of igh oliar quaios ivolvig h oliar paramr a igh firs orr FRF valus a. For a fasibl soluio of hs i uows o mor quaio is rquir which ca b ~ obai hrough masurig h rspos ampliu for a sigl-o ciaio a which givs ~ 0 Th s of hs i oliar quaios is solv by saar algorihms such as Nwo-Raphso mho o obai h sima of h oliar paramr a h simas of firs orr FRF a igh frqucy pois. Th fuamal frqucis a ar slc approprialy so ha h igh frqucy pois corrspoig o h combiaio os covr a wi rag icluig h aural frqucy. curv fiig algorihm usig h simas of igh FRF valus h givs h simas of liar paramrs. is solv by Rug-Kua umrical igraio algorihm a h o-imsioal rspos η is filr o obai h harmoic amplius a h igh frqucy pois corrspoig o h fuamal a combiaio os. Noliar paramr λ a h liar paramrs ar h sima usig h procur scrib abov. Th o-imsioal ciaio frqucis r a r ar slc such ha o combiaio o is form clos o h aural frqucy. Th raio of succssiv rms o h prvious rms i h rspos ampliu sris is of h orr of λ r a h sris covrgc bcom slow or i may ivrg also whvr a combiaio o is gra clos o h sysm aural frqucy [7]. Th squc of h combiaio os i icrasig orr ca b wri as r r < r < r < r r < < r < r r < r r < r which iicas ha h os which ar closr o h aural frqucy ar r r a r. If i is rsric ha o combiaio o shoul occur clos o h aural frqucy giv by h rag h h ciaio frqucis shoul saisfy h cririo r.0 ; r r.0 This provis a basis for slcig h ciaio frqucis. Numrical simulaio is carri ou for wo combiaios of ciaio frqucis as giv blow Cas I: r 0.6; r 0. for 0. Cas II: r 0.55; r 0. 5 for 0. 5 Combiaio os i cas I ar i closr proimiy of h aural frqucy ha i cas II. Two valus of oimsioal oliar paramr λ a λ 0. 0 ar cosir i h simulaio for ach cas. Dampig is cosir as ς 0. 0 a h ciaio amplius of boh h frqucis ar p sam i.. µ. CSE I Figur shows h spcrum of h wo-o ciaio wih frqucis a r 0.6; r 0. 0 NUMERICL SIMULTION For a wo-o ciaio giv by quaio h quaio of moio of a Duffig oscillaor quaio 6 ca b wri i a o-imsioal form as η ςη η λη cos r µ cos r whr ri i / / m ς c / m η / λ / µ / Eciaio 0 0 Numrical simulaio is carri ou for iffr valus of h o-imsioal oliar paramrλ a for iffr combiaio of ciaio frqucis r a r. Equaio No-imsioal frqucy Figur : Eciaio spcrum wih r 0.6; r 0.

4 Rspos spcra for λ a λ 0. 0 ar rspcivly show i Figurs a a b 0 0 s fi curv Esima FRF valus Rspos Firs orr FRF No-imsioal frqucy Figur a: Rspos spcrum for λ No-imsioal frqucy Figur a: Firs orr FRF simas a bs fi curv for λ s fi curv Esima FRF valus 0 0 Rspos 0 - Firs orr FRF No-imsioal frqucy Figur b: Rspos spcrum for λ 0. 0 I ca b s ha h combiaio os ar i h rag r a r.-.. Th sigal srgh of h combiaio os ar rlaivly lss compar o hos of fuamal os a r a r. Th rlaiv sigal srgh howvr icrass wih icrasig valus of λ. Th simas of firs orr FRF valus a h igh frqucy pois alog wih h bs fi curv ar show i Figurs a a b rspcivly for λ a λ Th fial simas of h oliar a liar paramrs ar giv i Tabl blow Figur b: Firs orr FRF simas a bs fi curv for λ CSE II Figur shows h spcrum of h wo-o ciaio wih frqucis a r 0.55; r No-imsioal frqucy Tabl : Esimas of liar a oliar paramrs i Cas I Paramr simas For λ For λ 0. 0 Eciaio 0 0 Noliar paramr Mass Siffss Dampig No-imsioal frqucy Figur : Eciaio spcrum wih r 0.55; r 0. 5

5 Rspos spcra for λ a λ 0. 0 for cas II ar rspcivly show i Figurs 5a a 5b. h simas of firs orr FRF valus ar show i Figurs 6a a 6b rspcivly for λ a λ s fi curv Esima FRF valus 0 0 Rspos Firs orr FRf Figur 5a: Rspos spcrum for λ Rspos No-imsioal frqucy Figur 5b: Rspos spcrum for λ Th combiaio os i his cas ar i h rag r a r Th bs fi curv alog wih Firs orr FRF No-imsioal frqucy No-imsioal frqucy s fi curv Esima FRF valus Figur 6a: Firs orr FRF simas a bs fi curv For λ No-imsioal frqucy Figur 6b: Firs orr FRF simas a bs fi curv For λ Fial simas of liar a oliar paramrs ar giv i Tabl. Tabl : Esimas of liar a oliar paramrs i Cas II Paramr simas For λ For λ 0.0 Noliar paramr Mass Siffss Dampig Followig obsrvaios ca b ma from h abov simulaio rsuls a Errors of simaio i liar a oliar paramrs ar highr i cas I ha i Cas II. This ca b aribu o h closr proimiy of h combiaio os o h sysm aural frqucy i Cas I. b Fairly goo simas ar obai for h oliar paramr maimum rror is.% i Cas I a 0.% i Cas II. c Esimas of mass a siffss paramrs ar fairly goo i Cas II bu h rror is o h highr si i Cas I. Dampig sima is o as accura as ha of ohr paramrs a his ca b plai by h fac ha h masurms ar ma away from h aural frqucy whr FRF is lss ssiiv o ampig. Th procur alhough scrib for a wo-o ciaio ca b asily for a muli-o ciaio wih mor ha wo frqucis. Mor frqucis i ciaio will provi mor FRF valus for Curv fiig. For ampl i cas of a hr-o ciaio firs orr FRF ca b sima a frqucy pois. owvr wih mor ciaio frqucis sris covrgc bcoms slowr

6 a i bcoms mor ifficul o slc h frqucis compaibl wih h icommsuracy cririo which rquirs ha wo iffr combiaio os amog all possibl ss shoul o b qual o ach ohr. 5 CONCLUSION w oliar paramr simaio procur is prs usig muli-o ciaio. Th mho rquirs fwr prims compar o ohr mhos usig siglo ciaios. Th mho givs fairly accura simaio of oliar paramrs. owvr i os o sima ampig wih sam accuracy. CKNOWLEDGEMENT Th auhors wish o prss hir has o h fiacial ai big provi by h Propulsio Pal of roauical Rsarch a Dvlopm oar Miisry of Dfc Govrm of Iia i carryig ou his suy. REFERENCES [] Schz M Th Volrra a Wir Thoris of Noliar Sysms Joh Wily a Sos: Nw Yor 90. [] oy Tag a Chua Masurig Volrra Krls IEEE Tras. o Circuis a Sysms CS 0 pp [] Chua L. O. a Liao Y. Masurig Volrra Krls II I. Joural of Circui Thory a pplicaios Vol. 7 pp [] Giffor S. J. a Tomliso G. R. Rc vacs i h pplicaios of Fucioal Sris o Noliar Srucurs Joural of Sou a Vibraio Vol. 5 pp [5] Giffor S. J. Esimaio of Sco a Thir Orr Frqucy Rspos Fucios Usig Truca Mols Mchaical Sysms a Sigal Procssig Vol. 7 pp [6] L G. M. Esimaio of Noliar Sysm paramrs Usig ighr Orr Frqucy Rspos Fucios Mchaical Sysms a Sigal procssig Vol. pp [7] Char. a Vyas N. S. Covrgc alysis of Volrra Sris Rspos of Noliar Sysms Subc o armoic Eciaio Joural of Sou a Vibraio Vol. 6 pp [] Char. a Vyas N. S. Noliar Paramr Esimaio hrough Volrra Sris Usig Mho of Rcursiv Iraio Commuica o Joural of Sou a Vibraio. [9] rosia E. a Ric S. O. Th oupu Propris of Volrra Sysms Driv by armoic a Gaussia Ipus Procigs of IEEE 59 pp