Analysis of Dynamic Systems

Transcription

1 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Chaper 8 Time-Domai Aalyi of Dyamic Syem 8. INTRODUCTION Pole a Zero of a Trafer Fucio A. Bazoue Pole: The pole of a rafer fucio are hoe value of for which he fucio i uefie (become ifiie). Zero: The zero of a rafer fucio are hoe value of for which he fucio i zero. Example of he Effec of Pole/Zero Locaio Figure 8-. From he evelopme ummarize above, he followig cocluio ca be raw: /33

2 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem A pole of he ipu fucio geerae he form of he force repoe. (i.e., he pole a he origi geerae a ep fucio a he oupu) A pole of he rafer fucio geerae he form of he aural repoe (i.e., he pole a σ geerae e σ ) A pole o he real axi geerae a expoeial repoe of he form pole locaio o he real axi. Thu, he pole farher he lef a pole i o he egaive e σ, where σ i he Pole-Zero Map exp(-) exp(-) exp(-) exp(-) exp(-3) exp(-4) I m a g A x i. -. R e p o e exp(-3) exp(-4) Real Axi Saar Form of he Fir Orer Syem Equaio B y + y u( ) The rafer fucio of he previou yem i efie by () G B / ( / ) Y U + where i kow a he ime coa. I ha imeio of ime for all phyical yem ecribe by he fir orer iffereial equaio above. The above equaio ca be repreee a /33

3 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem U ( ) G( ) Y ( ) Trafer Fucio Example i) Sprig-Damper Syem Equaio of moio or by + ky k x k y + y x b b k Comparig he above equaio wih he aar oe, o ha wih he ui of b / k ime coa [ ] [ b / k] [ N./m] [ N/m] [ ] k b x y ii) RC-Circui Equaio of moio wih he ui of RCe o + eo ei RC ime coa [ ] [ RC] [ V.] [ q] [ q] [ V] [ ] ei i R C eo iii) Liqui Level Syem Equaio of moio h + h q RC C RC ime coa wih he ui of [ ] [ RC] [ m] 3 m / m [ ] Q + q i Capaciace C H + h Reiace R Loa valve Q + q ο iv) Thermal Syem Equaio of moio 3/33

4 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem wih he ui of [ ] [ ] θ + θ qi RC C RC ime coa o C [ Kcal] RC o [ Kcal/ ] C Thu whe a fir orer yem i wrie i he form of Equaio () wih he coefficie of y / i equal o a he coefficie of he epea variable i ( / ) i which repree he ime coa of he yem a ha alway he imeio of [ ] ime regarle of he phyical yem uer coieraio. Iere i Aalyi of Dyamic Syem Afer obaiig a moel of a yamic yem, oe he ee o apply e igal o ee if he yem perform accorigly o cerai eig pecificaio pu forwar by he eig of he yem. I geeral, oe require he yem o be. able ( yem oe o grow ou uboue ). wih a fa repoe 3. ha a mall error a poible (eay ae error). Oher performace crieria may alo exi bu we will be maily cocere wih he above meioe hree. Uually he acual ipu o he yamic yem i ukow i avace; however, by ubjecig he yem o aar e igal, we ca ge a iicaio of he abiliy of he yem performace uer acual operaig coiio. For example, he iformaio we gai by aalyzig he yem abiliy a i pee of repoe a eay ae error ue o variou ype of aar e igal, will give a iicaio o he yem performace uer acual operaig coiio. Typical Te Sigal i) Sep Ipu θ Very commo ipu o acual yamic yem. A i repree a ue chage i he value of he referece ipu u( ) qi Ove Temperaure qo θ R ii) Ramp Ipu (Coa Velociy) Thi repree a iuaio where he ipu ha a coa rae of icreae wih ime (i.e. coa velociy) u( ) r α lope aα r 4/33

5 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem iii) Parabolic Ipu (Coa Acceleraio) Occur i iuaio where he ipu ha a coa acceleraio. u( ) u( ) a iv) Impulive Loa Ipu A large loa over a hor uraio of ime <<. yem ime coa ca be coiere a a impule. F ˆ δ, Fˆ F h of imp ul e reg u( ) u( ) F u( ) F impule u( ) Fˆ δ v) Siuoial Loa Ipu u Ai B co 4 3 A co() B i() Naural a Force Repoe The oluio y( ) o he homogeeou iffereial equaio () i compoe of wo par: Complemeary oluio: yc Paricular oluio : y p aural repoe ue o iiial coiio. force repoe uch ha: y( ) y ( ) + y ( ) p c 5/33

6 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Traie a Seay Sae Repoe Traie repoe: repoe from iiial ae o fial ae. Seay ae Repoe: repoe a ime approache.. Sep Repoe Ampliue Traie Seay Sae. Noice ha if lim y ( ) a lim y ( ) c 5 5 o be eay ae; where he eay ae oluio i p Time (ec) a boue fucio of ime he he yem i ai lim y lim y y c p 8. TRANSIENT RESPONSE ANALYSIS OF FIRST-ORDER SYSTEM Roor moue i bearig i how i he figure below. Exeral orque T ( ) i applie o he yem. T ( ) J b Apply Newo eco law for a yem i roaio or Defie he ime coa ( J / b) 6/33 T ( ) M J θ J J J + b T + b / J T / J + T ( J / b) * ( ), he previou equaio ca be wrie i he form + T * ( ), ( ) b

7 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem which repree he equaio of moio a well a he mahemaical moel of he yem how. I repree a fir orer yem. Free Repoe T ( ) To fi he repoe ( ), ake LT of boh ie of he previou equaio. + Ω ( ) Ω( ) ( ) + Ω ( ) L[ ] L[ ] Ω + Takig ivere LT of he above equaio will give he expreio of ( ) ( b / J ) ( / ) e e I i clear ha he agular velociy ecreae expoeially a how i he figure below. Sice ( / ) lim e ; he for uch ecayig yem, i i coveie o epic he repoe i erm of a ime coa. A ime coa i ha value of ime ha make he expoe equal o -. For hi yem, ime coa J / b. Whe, he expoe facor i Thi mea ha whe ime coa We alo have ( ) ( ) / / e e e % ( ), he ime repoe i reuce o 36.8 % of i fial value. J / b ime coa /33

8 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Remember Force Repoe T ( ) + T * ( ) To fi he repoe ( ), ake LT of boh ie of he previou equaio for zero iiial coiio. * Ω( ) ( ) + Ω ( ) T L[ ] L [ ] + Ω T * The above equaio ca be wrie i he more geeral form a C R where C ( ) i he oupu or he repoe a equaio ca be repreee a i) Impule Repoe T ( ) * ( ) Ω + + R i he ipu or referece igal. The above R( ) G( ) C ( ) + ( / ) Trafer Fucio I hi cae, for a ui impule ipu of magiue δ R( ) a he above equaio ca be wrie i he form from which C r B, B B + ( / ) c( ) B e ( / ) The figure below how he repoe c( ) B e + mu chage iaaeouly from a ime ( ) o a ime ( ). Sice we aume zero I.C, he oupu. 8/33

9 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem B c( ) c B e ( / ) Figure. Impule repoe of a fir orer yem ii) Sep Repoe I hi cae, for a ui ep ipu of magiue B, R( ) a he above equaio ca be wrie i he form B where Therefore B a a C B a B + a a B + B + for which C B B B + + ( ) ( / ) ( / ) c B Be B e The repoe c( ) i repreee i he Figure below. 9/33

10 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem B.98 B.9 B c( ) lope of he age a origi c( ) B.63 B Rie Time, : The ime for he repoe o go from % o 9% of i fial value. T.* r T r Selig Time, T : The ime for he repoe o reach, a ay wihi, ± % of i fial value. T 4*. B T r T Time coa: I i he ime for Figure Sep repoe of a fir orer yem e o ecay o 37 % of i fial value, i.e., e e.37 Aleraively, he ime coa i he ime i ake for a ep repoe o rie o 63 % of i fial value, i.e., ( / ) c B e B e.63 B Rie Time: ime for he repoe o go from % o 9 % of i fial value. The rie ime fou by olvig he expreio for he ep repoe for he ifferece i ime.9 c( )., ha i or ( / ) ( e ).9 ( / ) ( / ) B e.9b ( ) e. / l(.) T i c a r imilarly or ( / ) B e.b /33

11 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem ( / ) ( e ). ( / ) Hece, he rie ime i ( ) e.9 / l(.9).5.5 Tr.9.. Selig Time: ime for he repoe o reach, a ay wihi % ime c. Thu, or or T i fou by olvig he expreio.98 ( / ) ( / ) ( ) ( / ) c B e.98b. ± of i fial value. The elig ( ) e.98 e. / l(.) Remark: T l(.) 4. The maller he ime coa, he faer i he repoe a he furhe i he pole of. Seay ae error B C ( ) ( + / ) e ue o ep ipu. lim c lim c p lie o he lef half of 3. The yem i able, i.e., provie ha he pole / he complex plae. Im plae Re / iii) Ramp Repoe I hi cae, for a ramp ipu of lope B, B r B R L r a he expreio of C ( ) above ca be wrie i he form where B a a b C /33

12 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Therefore Hece B B a B + + B B a B + + B + B b B / + C / B B B B ( / ) ( / ) c L C B + B + B e B + e c( ) r ( ) B B ( / ) c B + e Figure Ramp repoe of a fir orer yem /33

13 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem 8.3 TRANSIENT RESPONSE ANALYSIS OF SECOND-ORDER SYSTEM Some Example of Seco Orer Syem Free Vibraio wihou ampig Coier he ma prig yem how i Figure 3-. The equaio of moio ca be give by 3/33

14 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem or where m x + k x k x + x x + x m i he aural frequecy of he yem a i expree i ra/. Takig LT of boh ie of he above equaio where x( ) a x ( ) x give X ( ) x x + X ( ) rearrage o ge a he repoe [ ] k m + L x x x + x X ( ), Remember pole are ± j I i clear ha he repoe x x + + X ( ) + complex cojugae m x( ) Figure 8- Ma Sprig Syem x i give by x x( ) i( ) + x co( ) x coi of a ie a coie erm a epe o he value of he iiial coiio x a x. Perioic moio uch ha ecribe by he above equaio i calle imple harmoic moio. k x( ) lope x Im plae Re if x Figure 8- x, π Perio T Free repoe of a imple harmoic moio a pole locaio o he -plae co( ) x x Free Vibraio wih Vicou ampig 4/33

15 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Dampig i alway pree i acual mechaical yem, alhough i ome cae i may be egligibly mall. Coier he ma prig amper yem how i he figure. The equaio of moio ca be give by m x + bx + k x () he characeriic equaio of he above equaio i + + () m b k a he wo roo of hi equaio are, ± 4 b b m k (3) m x( ) b m Figure 8-3 k We coier hree cae: b 4 m k < Roo are complex cojugae (uerampe cae) b 4 m k Roo are real a repeae b 4 m k > Roo are real a iic (overampe cae) (criically ampe cae) I olvig equaio () for he repoe x( ), i i coveie o efie k m uampe aural frequecy, [ ra/] a acual ampig value ζ ampi graio criical ampig value b km a rewrie equaio () i he form + ξ + (4) which i he aar form equaio of a eco orer yem. i) Uerampe Cae < ξ < Takig LT of boh ie of equaio () where x( ) x a x ( ) x, a rearrage o ge X ( ξ ) kowig ha equaio (4) ca be wrie a + x + x + ξ + + ξ + + ξ + ξ wich i a complee quare equaio. The aure of he roo a of equaio (4) wih varyig value of ampig raio ξ ca be how i he complex plae a how i he figure (5) 5/33

16 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem below. The emicircle repree he locu of he roo a i he rage < ξ < ξ for iffere value of ξ ξ ξ ξ for ξ > for ξ > ξ + ξ ξ ξ ξ Defie ζ ampe aural frequecy (ra/) i how The relaiohip bewee ζ a he o-imeioal frequecy ( / ) he figure below..9 Noimeioal frequecy / The X from which Dampig raio ζ b/b cr Figure No-imeioal frequecy veru he ampig raio. ξ x + x ( + ξ ) x ( ) + ( + ξ ) + ( ) + ξ + 6/33

17 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem or ξx + x ξ ξ x( ) L X ( ) e i + xe co ξ ξ x x e x i co + + x ξ x, he above equaio reuce o If he iiial velociy or where ξ ξ x x e i co + ξ (8) ξ i( φ ) x C e + (9) (6) (7) ξ φ a a ξ x( ) C x ξ Im () Ce ξ ξ + j plae j ξ Re j Dampe perio, π T ξ j Remark: Noice ha for hi cae (ueampe cae. he repoe i a ecayig iuoi. < ξ < ).. he frequecy of ocillaio i ( ξ ) 3. For poiive ampig ( ) ξ >, he pole a have egaive real a lie eirely o he le half of he complex plae. A a reul he raie repoe ecay wih ime a he yem i ai o be able. 4. The rae a which he raie repoe ecay epe o he coefficie i e ξ. Larger faer ecay of ξ of ξ (i.e., maller / ξ ) lea o faer raie repoe (i.e., x ). The erm / ξ i i hi cae he ime coa of he eco 7/33

18 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem orer yem. Therefore, he ime coa of he eco orer yem ca be mae maller (i.e., i pee faer) by movig he real par ξ farher away from he origi of he complex plae. ii) Criically ampe Cae ζ I hi cae, he pole he pole a become ξ a he repoe x( ) ca be obaie from equaio (5). Thu ( ) X ( ) + + ( + ) from which + x + x + x + x + x x + x x + ( + ) ( + ) ξ + ( + ) x x e x x e ξ which i ecayig expoeially a how i he figure below x( ) Im x plae Re ξ iii) Overampe Cae boh real a he repoe ξ > I hi cae, he pole he pole ξ + ξ x become ξ ξ ξ + ξ x x( ) x e ξ ξ ξ ξ + + where he repoe i how i he figure below + x x ξ ξ e a are ( ξ ξ ) + ( ξ ξ ) 8/33

19 ME 43 Syem Dyamic & Corol x( ) x Icreaig ξ Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Im plae Re Remark: The repoe i hi cae (overampe cae ξ > ) i imilar o ha of he fir orer yem a he oher. a i he um of wo expoeial. The fir ha a ime coa The ifferece bewee hee wo ime coa icreae a he ξ icreae o ha he expoeial erm correpoig o he maller oe (i.e., ) ecay much faer ha ha correpoig o. Uer uch cae he eco orer yem may be approximae by a fir orer oe wih ime coa equal o. From uy of he fir orer yem we fou ha he repoe remai wihi % of i fial value i > 4 ime coa 4 ( ime coa). For a eco orer uerampe yem ζ a he ime require for he oluio o remai wihi % of i fial value i calle he elig ime T which from above i give by T 4 4 ζ Free Repoe of a Seco Orer Syem by MATLAB MATLAB PROGRAM: >> w; >> zea[ ]; >> for k:6 um[ w^]; e[ *zea(k) w^]; yf(um,e) impule(y); hol o >> e 9/33

20 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem 5 Free Repoe of a eco orer yem a iffere value of he ampig raio ζ ζ.5 (Uerampe ) ζ.5 ( Uerampe) ζ.7 (Uerampe) ζ (Criically ampe) Free Repoe x() 5-5 ζ 5 (Overampe) ζ (Overampe) Time () /33

21 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Experimeal Deermiaio of ampig raio (Logarihmic Decreme) I i omeime eceary o eermie he ampig raio a ampe aural frequecie of recorer a oher irume. To eermie he ampig raio a ampe aural frequecy of a yem experimeally, a recor of ecayig or ampe ocillaio, uch a ha how i he Figure below i eee. /33

22 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem x( ) x π T + T x x 3 3 x π T The raio x x i equal o ξ co( φ ) ξ co( φ ) x x C e x x C e x x co co( φ ) ξ φ e ice a are elece co Hece T eco apar, oe ca wrie ( φ ) co ( + T ) φ co( + T φ ) co( + π φ ) co( φ ) x x e ξ T The Logarihmic Decreme δ i efie a he aural logarihm of he raio of ay wo ucceive iplaceme ampliue, o ha by akig he aural logarihm of boh ie of he above equaio x π πζ j δ l ζ T ζ x j+ ζ ζ (*) olvig he above equaio for ζ, /33

23 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem ζ δ π + δ Noice from Eq. (*) ha if ζ << (ha i, very low ampig, which quaiaively mea b << b cr ), ζ a hu δ πζ (**) The figure below how a compario bewee Eq. (*) a (**) veru he ampig raio ζ. Logarimic ecreme δ δ π ζ / (-ζ ).5 δ π ζ Dampig raio ζ For o-ucceive ampliue, ay for ampliue x a oberve ha x x x x3 x4 x x x x x x x Takig he aural logarihm of boh ie of he above equaio give x +, where i a ieger, we So x x x x l l l l x+ x x3 x+ δ + δ + + δ δ x x δ l or δ l x x + Table- Logarihmic ecreme for Variou Type of Srucure Type of Srucure Approximae Rage of 3/33

24 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Example Logarihmic Decreme, δ Muliory Seel Builig.. Seel Brige.5.5 Muliory Cocree Builig.. Cocree Brige..3 Machiery Fouaio.4.6 Sep Repoe of a Seco Orer yem: Coier he mechaical yem how i he Figure below. Aume ha he yem i a re for <. A, he force u a ( ) [where a i a coa a ( ) i a ep force of magiue N] i applie o he ma m. The iplaceme i meaure from he equilibrium poiio before he ipu force u i applie. Aume ha he yem i uerampe ( ζ < ) k x m u b The equaio of moio for he yem i 4/33

25 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem m x + bx + k x ai( ) The TF for he yem i X U m + b + k Hece Defie L X a m + + m b k b k + + m m k m uampe aural frequecy, [ ra/] a a The acual ampig value ζ ampi graio criical ampig value b km Hece, X L ( ) X a m + ξ + a m + ζ + a m + ζ + ζ + ( ) ( ) ( ) a ζ * + ζ m ( + ζ ) + ( + ζ ) + (8-6) + ζ + ζ + ζ + a ζ ζ + m ( + ζ ) + ( + ζ ) + 5/33

26 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem where ζ. The ivere Laplace raform of he la equaio give a ζ ζ ζ x( ) e i co e m ζ a ζ ζ e i + co m ζ ζ a e ζ i + a m ζ ζ The repoe ar from ( ) x a reache of he repoe curve i how i he figure below. x a m. The geeral hape.4 Sep Repoe. a/m.8 Ampliue.6.4 Figure 8- Sep repoe of a eco orer yem. The repoe curve how correpo o he cae where ζ.7 a ra/ Time (ec) Aume ha he yem i uerampe ( ζ < ) MATLAB PROGRAM: >> w; >> zea[ ]; >> for k:6 um[ w^]; e[ *zea(k) w^]; yf(um,e) ep(y); hol o u( ) x( ) U ( ) Ipu X + ζ + ( ) Oupu? 6/33

27 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem >> e.6.4 ζ. Sep Repoe..5.7 Ampliue Time (ec) Impule Repoe of a Seco Orer yem: MATLAB PROGRAM: >> w; >> zea[ ]; >> for k:6 um[ w^]; e[ *zea(k) w^]; yf(um,e) impule(y); hol o >> e u( ) x( ) U ( ) Ipu X + ζ + ( ) Oupu? 7/33

28 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Impule Repoe.8 ζ. Ampliue.6.4. ζ.5 ζ.7 ζ. ζ. ζ Time (ec) 8/33

29 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Ipu Table 8. u ( ) y( ) u ( ) B (Ramp of Slope B ) u B, (Sep of magiue B ) * δ u B (Impule of magiue B ) The Repoe of he Fir Orer Liear Syem y + y u ( ), y( ) y where u ( ) y oup u a ipu Repoe y( ) if y( ) y Repoe y( ) if ( / ) y e - ( / ) ( / ) ( ) + + y B e y e ( ) ( / ) ( / ) y y e + B e y B + y e ( ) / y ( / ) y B + e ( / ) ( ) y B e y B e ( ) / 9/33

30 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Table 8. The Free Repoe of he Seco Orer Liear Syem mx + bx + kx, x x a x x k The aural frequecy ra/ m b b The ampig raio ζ b mk The ampe frequecy cr for ζ < ζ Dampig raio ζ < ζ ζ > Repoe y( ) ζ ζ x x e x i co + + x ζ x x, hi implifie o If ζ ζ x( ) xe i co + ζ x + ζ ζ e i a ζ ζ x ζ ζ e co a ζ ζ + ( + ) If x( ) x x( ) x ( + ) e x x e x x e, hi implifie o ζ + ζ x x( ) x e ζ ζ ξ + ζ x + x e + ζ ζ x x, hi implifie o If x ( ζ ζ ) + ( ζ ζ ) ζ + ζ ζ ζ x ζ + ζ e + ξ + ζ e ζ 3/33

31 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Table 8.3 The Force Repoe of he Seco Orer Liear Syem, a mx + bx + kx f x x k The aural frequecy ra/ m b b The ampig raio ζ b mk The ampe frequecy cr ζ for ζ < ζ The phae agle ψ a for ζ < ζ For overampe yem ζ >, he ime coa are / ζ ζ / ζ ζ +, ur ( ) ramp wih lope Ipu u ( ) ui ep δ ( ) ui impule Dampig raio ζ < ζ Ipu Repoe y( ) f u r y ζ a ζ e ζ r + ζ co i + y ( ) ( ψ ) f u ζ e co ζ δ ( ) i δ f r r f u f u δ ( ) y ( ) e f ζ > f ( ) u ( ) ζ e y ζ y + e + e y e e δ ζ + ζ r ( yr e e ) ζ 3/33

32 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem f u y ( ) ( e e ) δ ( ) f yδ ( ) ( e e ) ζ ζ 3/33

33 ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem 33/33