It is quickly verified that the dynamic response of this system is entirely governed by τ or equivalently the pole s = 1.

Transcription

1 Tim Domai Prforma I orr o aalyz h im omai rforma of ym, w will xami h hararii of h ouu of h ym wh a ariular iu i ali Th iu w will hoo i a ui iu, ha i u ( < Th Lala raform of hi iu i U ( Thi iu i l bau i i iml i rlaivly ay o xrimally imlm o ym a o imlm o ym imulaio a oai a rih frquy o i ha h Fourir raform of h fuio oai all frquy igal Thu h iu i a igal rih iu oially aabl of roviig a walh of iformaio abou h ym Thi ui iu will b ali o fir, o a highr orr ym i orr o rmi h gral bhavior of h y of ro Fir Orr Sym Coir h ym X ( k G( U ( a uig U ( giv k X ( G( U ( ( Coruig a arial fraio xaio of h ouu, X ( whr k X ( k k ( X ( k Thu x( k I i quikly vrifi ha h yami ro of hi ym i irly govr by or quivally h ol

2 W will labl h im oa of hi fir orr ym By valuaig h yami ar of h ro, h rm , w a omu ha Thi ay ha h iiial valu of h ro i zro a wh ha 63% of h yami hav iia Alo by h im 3, 95% of h yami hav ourr Fially, wh 4 ha la, 98% of h yami ar of h ro ha iaar W a alo o ha k xd ( Thrfor h iiial lo of h ro i k a h lo moooially ra a h am ra a h yami fa Evually h lo bom vry mall a aymoially aroah zro Thi imli ha h ym vually rah a ay a I i imora o rogiz ha a h im oa g mallr, h ol loaio, mov mor o h lf o h ral umbr li Tha i, h ol bom mor gaiv Furhr, a g mallr, h ym ro ffivly g far Thu hr i a fuamal oio bw fa ym ro a ym ol ha ar mor o h lf i h omlx la S Ro S ro of wo fir orr ym 8 l iu m A Tim ( Th figur how h ro o a iu of wo fir orr ym wih iffr im oa Th fir a far roig ym ha a im oa of o Th lowr ym ha a im oa of 33 o I i o ha hi ym ro roughly hr im lowr ha h fir ym

3 So Orr Sym Coir h girig ym X ( G( U ( whr k G( whr h variabl i uually all h ym aural frquy a h variabl i of rfrr o a h amig raio Thi aar form a a b u o ovr ially all oibl alraiv or variaio o ariular o orr ym a Th variaio ilu h followig Uram, ovram a riially am ym Th ol of h ym i aar for ar h roo of h hararii quaio whih ar ( ( ± j ± 4, ± From hi i i ha hr ar hr bai oibilii: < Ur am a Hr hr will b wo omlx ojuga ol wih qual ral ar a omlx ar ha ar gaiv ivr of ah ohr Criially am a Hr hr will b wo ol ha ar iial i h irimia i h quarai form i rily zro Th wo ol will b, > Ovr am a Hr hr will b wo ral ol Th ol will boh b o h gaiv ral axi (for abl ym a hy will b ii Thi a a b ra a h aa or ri ombiaio of wo fir orr ym Aiio of a zro a b If h aual ym i G(, hi a b wri a a b G( Th o rm i xaly a r h aar form, giv ha b k, a h fir rm i imly h Lala vrio of h im rivaiv of h aar form (agai wih a orrio for h gai To vrify hi, rall ha akig h rivaiv i h im omai i quival o mulilyig by i h Lala omai Thu hi ym a b brok io wo ar o ha i xaly quival o h aar form of h roblm a o ha i h rivaiv of h aar form i h im omai

4 Thu if h o rm ha a ivr raform ha i a ayig i wav, h h fir rm will b a ayig oi wav wih a iffr amliu bu h am ay ra a h am oillaio frquy 3 Exlii ol form If h aual ym i wri G( ( (, h hi a b xr a G( ( Thu, uig h rlaio,, k h wo form ar o b quival Furhrmor h ol of h ym a b valua a j, a j I i quikly vrifi ha h ol will alway aar a ihr a of ral ol or a a air of omlx ojuga 4 Ral-imagiary ol form If h aual ym i wri G( ( a b, h hi a b xr a G( a a b Thu, uig h rlaio a a b,, k a b a b h aar a hi form ar fou o b quival I hi form, h ol of h ym ar omu o b a jb, a a jb Thrfor w a ha a, b I ohr wor, h ral ar of h ym ol i ± b ± a a h imagiary ar i 5 Pol-zro lo Th ol a zro of a yami ym a b lo o a omlx la From h loaio of h ol a zro, o a a a of h yami rforma of h urlyig hyial ym For xaml, from h iuio of abiliy aalyi, whr i wa how ha for aymoi abiliy of a ym,

5 R( i < i, l, Thi ay ha h ral ar of all of h ym ol mu b gaiv, or li i h lo lf half la of h omlx la Similarly for BIBO abiliy, h ol of h ym mu li i h o lf half la of h omlx la Th iffr big ha ol wih zro ral ar ar allow Pol wih ral ar qual o zro ar of all margially abl ol i hy imly yami ym mo ha ar o h vrg of iabiliy 6 Prforma rgio Naural frquy For a o orr ym wri i aar form wih ol j, a j, h magiu of h ol i M R( Im( Th magiu i h am for boh ol Subiuig i hi aa, M ( ( ( ( Thu ym wih qual valu for (of all h aural frquy of h ym hav qual magiu for h air of omlx ojuga ol I ohr wor, h ol li o irl r a h origi a of magiu M W uually rri iuio o abl ym, or ho wih ol oly i h lf half la Thu ym of qual aural frquy ar ho wih ol o h mi-irl r a h origi a of magiu M Sym wih aural frqui highr ha hi valu aar oui of h miirl Damig raio Th agl of h ol i giv by R( φ o M ( Th agl of h ohr ol a b fou o b φ φ i h ral ar ar rily h am a h imagiary ar ar gaiv ivr O ubiuio, φ o o ( Th aramr i of all h amig raio of a ym Thu ym wih qual amig raio hav ol ha li alog wo mirror (i h ral axi vor maaig from h origi a agl giv by ± φ ± o ( Sym wih highr amig raio will aar ii h o ha rgio bou by h limiig vor

6 Figur of ral a imagiary ar of ol, li of oa aural frquy a amig raio W oiu h im omai rforma aalyi of o orr ym by xamiig h aliaio of a ui iu o h o orr ym i aar form Tha i, l U ( i U G X ( ( ( Uig ivr Lala raform, w a fi ha i( o( ( x whr whih i of all h am aural frquy, a, whih i qual o h magiu of h ral ar of h ym ol Th ra of hag of hi fuio i giv by o( i( i( o( ( x D Collig rm, ( ( i( i( o( ( x D I i o ha h iiial valu of h ym i ( i( o( ( x Th iiial lo of h ouu i

7 xd ( Alo, h fial valu of h ouu i lim x ( x f S Ro ro of a o orr ym 8 l iu m A 6 4 o: iiial zro lo Tim ( Byo h bai alula arifa of h ouu, h ha of h ro of h ym o a ui iu i ially fi by four aramr: Th ri im, r, whih i fi hr a h fir im h ym ro i vual fial valu afr h aliaio of h iu Thi i a maur of h of ro of h ym Th ak im,, whih i h im a whih h ym rah i high or ak valu afr a iu bfor rurig bak ow a vually lig a a fial valu Thi i alo a maur of h of ro of a ym, a i uually oly u i ojuio wih fiig h ovrhoo i a ym Th maximum rag ovrhoo, M, i fi a how far h ym go a i vual fial valu a h ak afr a iu Thi i a maur of how oillaory h ym i Th lig im,, whih i fi a h im byo whih h ym ay wihi a rifi rag bou of h fial valu afr a iu Thi i a ombi maur of h of ro of h ym a i oillaory aur I i a goo raial maur of how fa h ym a rah a w ay oraig oiio afr a iurbig iu Th ri im, r Thi aramr a b fi i vral way, bu for our uro will b fi a h im afr a iu i ali uil h ym fir ro i vual fial valu Thrfor i orr o omu h ri im, w o fi h fir im a whih x ( x f By obrvig h ym ouu oluio, hi will our wh h rigoomri ar of h ouu ar zro, lavig oly h oa ou fro, whih i rily h fial valu Tha i a h ri im,

8 o( i( By rarragig hi w g a( a( a Thu h ri im a b xr r a Thi fuio a b vry roughly aroxima by 8 r whr h aroximaio ak io aou h oibiliy of xra ol a zro affig h ro of h ym Thu h aroximaio i oly a orvaiv maur Howvr, h aroximaio i uful i ha i mak lar ha h ri im i ially o a fuio of amig raio a all, bu rahr oly a fuio of h ym aural frquy Ri im variaio wih amig r a o f m i r i amig raio Th figur iia h valu β i h fuio β r whr β a

9 I how ha β o o vary igifialy ovr a wi rag of amig raio Thu oirig hi faor a a oa i a raoabl aroximaio I i alo o ha β aar o vary from 4 o Thi i i ora o h ho valu of 8 i h aroximaio, a iffr whih a b aribu o ohr faor uh a xra ol a zro i ral ym Th ak im, Th ak im i aohr maur of h of ro of a ym, muh lik o h ri im I i mo of u i ojuio wih a maur of h maximum ovrhoo of a ym Th ak im i fi a h mom wh h ym rah i maximum valu Thi i quival o ayig wh h lo of h ym rah zro for h fir im afr h iiial i ali Tha i h fir im afr wh ( ( x i( Thi oiio will our wh i( for h o im Th fir im our wh Thu h ak im,, aifi π or π π I a b how ha hi fuio ha rogr variaio wih amig raio ha h ri im, bu alo i i ha h variaio wih aural frquy i rily h am a for ri im Th maximum rag ovrhoo, M Th maximum rag ovrhoo i h larg x, xr a a rag, by whih h ouu of h ym x i fial valu Thi will our a rily h ak im Tha i, h maximum ovrhoo will our wh h ouu i x( o( i( a giv ha π π, π π x( o( i( ( Alo, oig ha h fial valu of h ouu i x f, h x abov h fial valu, ivi by h fial valu, a xr a a rag i

10 M π π % I i o ha h maximum rag ovrhoo i a fuio of amig raio alo, a o o of aural frquy Th ivr fuio for amig raio from ovrhoo a b fou a M l M π l Th lig im, Th lig im i a maur of h im ha i ak bfor h ouu of a ym r a rgio ar o i fial valu a o logr lav ha rgio From xamiaio of h ouu quaiio x( o( i( i i ha h rigoomri rm i h brak will oiuouly oilla wih a oa amliu Thrfor if h ym i o l o om ay a oiio, h xoial rm mu ay o zro To fi h lig im i i uffii o fi h im a whih h xoial rm ha ay o mallr ha h bou rquir for h lig rgio L u fi h lig rgio bou a b abou h fial valu, x f Thu h lig im our a uh ha b A h limi of h oiio, b l(b l( b l( b Tyial valu for h bou ar %, %, a 5 % ig o igr rfr Uig h oiio, h lig im formula a b xliily a a Bou, b % % 5% Slig im formula,

11 Examl A ym of gral form i oru wih X ( G( U (, G( a a ui iu or U ( Th im ro of four vrio of hi ym ar how i h figur Th abl ummariz h im omai rforma rori of h ym a ri by h ig quaio Sym Naural frquy, Damig raio, Ri im, r Pak im, Maximum rag ovrhoo, M / / / /3 / Slig im, 6 S Ro So orr ym ro 4 u li m A Tim (