Chap8 - Freq 1. Frequency Response

Transcription

1 Chp8 - Freq Frequecy Repoe

2 Chp8 - Freq Aged Prelimirie Firt order ytem Frequecy repoe Low-p filter Secod order ytem Clicl olutio Frequecy repoe Higher order ytem

3 Chp8 - Freq 3 Frequecy repoe Stedy-tte behvior of ytem to hrmoic excittio over rge of iput frequecie Determitio of importt behviorl chrcteritic of dymic ytem by ubjectig them to hrmoic iput d obervig the repoe Experimetlly, Alyticlly, or Numericlly.

4 Prelimirie - ODE equtio cott- coefficiet differetil lier, Ordiry, CE Thi expreio i clled thechrcteritic equtio Ce e C e C e C Ce t x form The olutio cbe writte ithe x dt dx dt x d dt x d Homogeeou equtio t x t x t x Solutio cbeexpreed t Gu x dt dx dt x d dt x d t t t t t h p h Chp8 - Freq 4

5 Prelimirie - Lplce trform k where U G U X U G t f L i iput preet derivtive re If G U X TF GU X k k k k k k k k , Chp8 - Freq 5

6 Chp8 - Freq 6 Firt Order Sytem - Clicl Solutio Firt order equtio x x Gu t Homogeeou equtioi x x olutio x t h Ae t SubtitutioitoCE, / For hrmoic excittio, Gu t Theclicl olutio : x t homogeeou prticulr triet tedy tte where B Gu Ae t Gu t cot Bco t d Ai foud from iitil coditio

7 Chp8 - Freq 7 Frequecy Repoe The mplitude d the phe of Let RC d E B d t Gu E, the tedy - tte repoe Normlized output repoemplitude E E Repreet repoe mplitude o be db log B / Gu logrithm,

8 Frequecy Repoe, co E U E EU E E RC Let t E e e e RC Chp8 - Freq 8

9 Chp8 - Freq 9 Frequecy Repoe The reultig TF corer frequecy TF E j U j E j TF MAG TF t MAG E MAG j E TF

10 Chp8 - Freq Low-P Filter Typicl ytem repoe

11 Chp8 - Freq Secod-order Sytem Sprig-m-dmper ytem mx bx kx m b F x x x k k k x x x where x b b c Theudmped turl k / m F dmpig rtio F / k cot cot x cot frequecy i

12 Secod-Order Sytem Chrcteritic equtio 4 4,, Chp8 - Freq

13 Secod-Order Sytem / t / co co, x A t A t x t x j i le th If Chp8 - Freq 3

14 Secod-Order Sytem The trfer fuctio of d -order ytem: The frequecy repoe of thi ytem c be modeled : Whe : log db j H H 4 db/decde Chge by 8 t 4log log db j H j H

15 Trfer fuctio: H 5 The criticl frequecie re = zero, pole, d 5 pole. MATLAB exct rep.: w = logpce-,3,3; = j*w; H = *+./+./+5; mgdb = *logbh; phe = gleh*8/pi; MATLAB Bode: um = [ ]; de = cov[ o], [ 5]; bodeum, de; Bode plot re ueful lytic tool. Pole d Zeroe j H j j j 5 4 j j j 5

16 Chp8 - Freq 6 Exmple Fid the time cott for the followig RC circuit tht h prmeter R=8 Ω d C=.μf. Sketch the mplitude frequecy repoe d determie the corer frequecy. Approximtely how log doe it tke the triet prt of the olutio to thi ytem to die out?

17 Chp8 - Freq 7 Solutio

18 Chp8 - Freq 8 Exmple A hrmoic igl of mplitude d frequecy 7 Hz i the iput to lier firt-order ytem whoe time cott i.5 ecod. Wht i the mplitude of the output? Wht i the phe of the output with repect to the iput? Doe the output lg behid the iput or led it?

19 Chp8 - Freq 9 Solutio

20 Chp8 - Freq Pir-Shre: Exmple 3 Awer the me quetio i the previou problem, except tht ow the ytem i ecod-order lier ytem with turl frequecy of rd/ec d dmpig rtio of.5

21 Chp8 - Freq Solutio

22 Chp8 - Freq Pir-Shre - Exmple 4 It i propoed to ttch dditiol prig-m-dmper ytem to primry prig-m-dmper ytem how i figure below. Fid the tedy-tte mplitude of the diplcemet repoe of the primry m, d plot it with repect to the iput frequecy for the ce with without the ttched ytem. Such ttched ytem c be ued to borb uwted vibrtio. Commet o the effectivee of reducig vibrtio for thi ytem. The prmeter of the problem re: w lbf k b 8,5lbf / i lbf ec/ i w k b 5lbf 46lbf / i 5lbf ec/ i

23 Chp8 - Freq 3 Solutio

24 Chp8 - Freq 4 Solutio cot.

25 Chp8 - Freq 5 Solutio cot.

26 Chp8 - Freq 6 Exmple 5 Determie the trfer fuctio reltig the output voltge eto the iput voltge e for the followig RLC circuit: Develop the correpodig Bode plot uig dt from Exmple 4.3. Wht i the turl frequecy of the ytem? Redeig the circuit o to icree it turl frequecy by 3 percet, but keep the dmpig rtio t.77.

27 Chp8 - Freq 7 Solutio cot.

28 Chp8 - Freq 8 Solutio

29 Chp8 - Freq 9 Solutio cot.

30 Chp8 - Freq 3 Exmple 6 Fid the root of the chrcteritic equtio of the followig ytem. Wht re the time cott d/or turl frequecie? Wht re the dmpig rtio? Commet o the tbility of ech. Solve by d check by digitl computtio.. x 8x 3x 5u t b. x x 5 34u t c. 7 z 5 z z 3z.5 f t Pir Shre d. z z 4z 5z 4 f t Pir Shre

31 Chp8 - Freq 3. x 8x 3x 5u t Solutio b. x x 5 34u t

32 Chp8 - Freq 3 Solutio cot. c. 7 z 5 z z 3z.5 f t d. z z 4z 5z 4 f t

33 Chp8 - Freq 33 Exmple 7 Problem 8.,c,e,g Fid the trfer fuctio for ech of the ytem below, reltig the output x or z to the iput u or f.. b. c. d. x 8x x x x 5 z z v 3v 3x 4x 4z v 5u t 5 34u t 3v 3z u t 4v 8 f t Pir Pir Shre Shre

34 Chp8 - Freq 34 Solutio Tke Lplce trform of ech equtio d olve for the repoe vrible divided by the iput vrible. For d, firt put ito clicl form.. x 8x 3x 5u t c. x 4x u t b. x x 5 34u t d. 5 z z 4z 3v 4v 8 f t v 3v v 3z

35 Exmple 8 Fid the et of tte-pce equtio for ech of the below ytem: z v v v t f v v z z z d t u x x c t u x x b t u x x x Chp8 - Freq 35

36 Chp8 - Freq 36 Solutio

37 Chp8 - Freq 37 Solutio cot.

38 Chp8 - Freq 38 Solutio Cot b. x x 5 34u t c. x 4x u t

39 Chp8 - Freq 39 Solutio Cot d. 5 z z 4z 3v 4v 8 f t v 3v v 3z

40 Chp8 - Freq 4 Exmple 9 Determie the trfer fuctio coefficiet for the circuit : If: R C f L mh R C f Fid the eige vlue, d check the tbility of the ytem. If the ytem i tble, determie it Bode plot.

41 Chp8 - Freq 4 Solutio

42 Chp8 - Freq 4 Solutio cot.

43 Chp8 - Freq 43 Solutio cot. The Bode plot i