School of Mechanical Engineering Purdue University. ME375 Frequency Response - 1

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1 Case Study ME375 Frequecy Respose -

2 Case Study SUPPORT POWER WIRE DROPPERS Electric trai derives power through a patograph, which cotacts the power wire, which is suspeded from a cateary. Durig high-speed rus betwee New Have, CT ad New York City, the trai experieces itermittet power loss at 4 km/hr ad km/hr. ME375 Frequecy Respose -

3 Case Study Patograph Model F (t) mz b b z k k z b z k z F c (t) m z mz bz kz bz kz F () c t k k m b b z For m = 3. kg, b = 5 N/(m/s), k = 96 N/m, m =.5 kg, b = 75 N/(m/s), ad k = 958 N/m: Z( s).87( s 9.78s834) 4 3 c () F s s 6.3s 79s 855s347,75 ME375 Frequecy Respose - 3

4 Case Study Frequecy Respose -6 ) Magitude (db) Phase (deg) Frequecy (rad/sec) ME375 Frequecy Respose - 4

5 Frequecy Respose Forced Respose to Siusoidal Iputs Frequecy Respose of LTI Systems Bode Plots ME375 Frequecy Respose - 5

6 Forced Respose to Siusoidal Iputs Ex: Let s fid the forced respose of a stable first order system: to a siusoidal iput: Forced respose: y 5yu ut () si() t Y() s G() s U() s where Gs ( ) ad Us ( ) Lsi( t) Y( s) PFE: A A Y() s A 3 Compare coefficiets to fid A, A ad A 3 : ME375 Frequecy Respose - 6

7 Forced Respose to Siusoidal Iputs Ex: (cot.) Use ILT to fid y(t) ) : yt ( ) L Y( s) L Useful lformula: Asi( i( t ) Bcos( t ) A B si( i( t ) Where ata( BA, ) = ( A jb) Usig this formula, the forced respose ca be represeted by 5t yt ( ) e si( t) ME375 Frequecy Respose - 7

8 Forced Respose of st Order System.5 Output Iput is si(t) Iput.5 Respose Time (sec) ME375 Frequecy Respose - 8

9 Forced Respose to Siusoidal Iputs Ex: Give the same system as i the previous example, fid the forced respose to u(t) ( ) = si( t). Y() s G() s U() s where Gs () ad Us () Lsi( t) Y( s) ME375 Frequecy Respose - 9

10 Forced Respose of st Order Systems Iput is si(t) Output Respose Iput Time (sec) ME375 Frequecy Respose -

11 Frequecy Respose Ex: Let s revisit the same example where y 5yu ad the iput is a geeral siusoidal iput: si(t). Ys () Gs () Us () s 5 s s 5 ( s j )( sj ) A A A3 Ys () s5 sj sj Istead of comparig coefficiets, use the residue formula to fid A i s: ( 5) ( ) 5 ( 5) A s Y s s s ( 5) s s s 5 A ( s j) Y( s) ( s j) Gs ( ) sj s sj A3 ( s j) Y( s) ( s j) Gs ( ) s j s s j ME375 Frequecy Respose -

12 Frequecy Respose Ex: (Cot.) A 5 A G( j ) j j 5 j j A 3 G ( j ) j j 5 j j The steady state respose Y SS (s) is: 3 A A YSS () s s j s j jt jt SS 3 y () t L Y ( s) A e A e SS yss ( t ) G ( j ) si( t ) where G( j) ME375 Frequecy Respose -

13 Frequecy Respose Frequecy Respose ME375 Frequecy Respose - 3

14 I Class Exercise For the curret example, y 5 yu Calculate the magitude ad phase shift of the steady state respose whe the system is excited by (i) si(t) ad (ii) si(t). Compare your result with the steady state respose calculated i the previous examples. Note: Gs () G( j ) s5 j 5 G( j ) ad G( j) ata(,5) 5 ME375 Frequecy Respose - 4

15 Frequecy Respose Frequecy respose is used to study the steady state output y SS (t) of a stable system due to siusoidal iputs at differet frequecies. I geeral, give a stable system: ( ) ( ) ( m) ( m) a y a y a ya yb u b u bub u m m m m bms bms bs b Ns () bm sz sz szm () as as as a Ds a s p s p s p ( )( ) ( ) Gs () ( )( ) ( ) If the iput is a siusoidal sigal with frequecy, i.e. ut () A si( t) u the the steady state output y SS (t) is also a siusoidal sigal with the same frequecy as the iput sigal but with differet magitude ad phase: y () t G( j ) A si( tg( j)) SS where G(j) ) is the complex umber obtaied by substitute for j s i G(s) ), i.e. G ( j ) G ( s ) s j u m m m b m ( j ) b ( j ) b ( j ) b a ( j) a ( j) a ( j) a ME375 Frequecy Respose - 5

16 Frequecy Respose Iput u(t) U(s) LTI System G(s) Output y(t) Y(s) u / y SS / t t u () t A si( t) y () t G ( j ) A si( t G( j)) u A differet perspective of the role of the trasfer fuctio: Amplitude of the steady state siusoidal output G( j ) Amplitude of the siusoidal iput G( j ) Phase differece (shift) betwee yss ( t) ad the siusoidal iput SS u ME375 Frequecy Respose - 6

17 Frequecy Respose G Iput u(t) Output y(t) G ME375 Frequecy Respose - 7

18 I Class Exercise Ex: st Order System The motio of a pisto i a cylider ca be modeled by a st order system with force as iput ad pisto velocity as output: f(t) () Calculate the steady state output of the system whe the iput is Iput f(t) Steady State Output v(t) si( t) ) G(j) ) si( t + ) si(t) si(t + si(t) si(t + si(t) si(t + v The EOM is: si(3t) si(3t + Mv Bv f() t si(4t) si(4t + () Let M = kg. ad B = 5N/(m/s).5 N/(m/s), fid the trasfer fuctio of the system: si(5t) si(5t + si(6t) si(6t + ME375 Frequecy Respose - 8

19 I Class Exercise (3) Plot the frequecy respose plot Magitude ((m/s)/n) Phase (deg) Frequecy (rad/sec) Frequecy (rad/sec) ME375 Frequecy Respose - 9

20 Example - Vibratio Absorber (I) Without vibratio absorber: M z K B f(t) TF (from f(t) ) to z ): EOM: M z B z K z f () t () Let M = kg, K = N/m, B = 4 N/(m/s). Fid the steady state respose of the system for f(t) ) = (a) si(8.5t) (b) si(t) (c) si(.7t). Iput f(t) Steady State Output z (t) si( t) G(j) si( t + ) si(8.5t) si(8.5t + si(t) si(t + si(.7t) si(.7t + ME375 Frequecy Respose -

21 Example - Vibratio Absorber (I). f(t) = si(8.5 t) 5.5 z (m) f(t) = si( t) z (m) f(t) = si(.7 t) z (m) Time (sec) ME375 Frequecy Respose -

22 Example - Vibratio Absorber (II) With vibratio absorber: M K B M K B f(t) EOM: M z ( B B ) z K K z B z K z f ( t) z z TF (from f(t) )toz z ): Mz Bz Kz Bz Kz Let M = kg, K = N/m, B = 4 N/(m/s), M = kg, K = N/m, ad B =. N/(m/s). Fid the steady state respose of the system for f(t) ) = (a) si(8.5t) (b) si(t) (c) si(.7t). Iput f(t) Steady State Output z (t) si( t) G(j) si( t + ) si(8.5t) si(8.5t + si(t) si(t (t+ + si(.7t) si(.7t + ME375 Frequecy Respose -

23 Example - Vibratio Absorber (II).4 f(t) = si(8.5 t). z (m) f(t) = si( t) z (m) f(t) = si(.7 t) z (m) Time (sec) ME375 Frequecy Respose - 3

24 Example - Vibratio Absorber (II) Take a closer look at the poles of the trasfer fuctio: The characteristic ti equatio 4 3 s 5.s.4s 5s Poles: p.8.5 j, p3, j What part of the poles determies the rate of decay for the trasiet respose? (Hit: whe p = jthe respose is e t e j t ) ME375 Frequecy Respose - 4

25 Example - Vibratio Absorbers Frequecy Respose Plot No absorber added Frequecy Respose Plot Absorber tued at rad/sec added.5.5 Magitude (m/n) Magitude (m/n) Frequecy (rad/sec) Frequecy (rad/sec) Phase (d deg) Phase (d deg) Frequecy (rad/sec) Frequecy (rad/sec) ME375 Frequecy Respose - 5

26 Example - Vibratio Absorbers agitude (db) Phase (deg); M Bode Plot No absorber added Phase (deg); Ma agitude (db) Bode Plot Absorber tued at rad/sec added Frequecy (rad/sec) -8 Frequecy (rad/sec) ME375 Frequecy Respose - 6

27 Bode Diagrams (Plots) Bode Diagrams (Plots) A uique way of plottig the frequecy respose fuctio, G(j), w.r.t. frequecy of systems. Cosists of two plots: Magitude Plot : plots the magitude of G(j) ) i decibels w.r.t. logarithmic frequecy, i.e. Gj ( ) log Gj ( ) vs log db Phase Plot : plots the liear phase agle of G(j) ) w.r.t. logarithmic frequecy, i.e. G( j) vs log To plot Bode diagrams, oe eeds to calculate the magitude ad phase of the correspodig trasfer fuctio. Ex: Gs () s s s ME375 Frequecy Respose - 7

28 Bode Diagrams Revisit the previous example: s ( j) Gs () G( j) s s j ( j ) 3 5 G( j) G ( j ) G( j) log Gj ( ) G( j ) Phase (deg g); Magitude (db B) Frequecy (rad/sec) ME375 Frequecy Respose - 8

29 Bode Diagrams Recall that if The m m bm s bm s b s b bm ( s z )( s z ) ( s zm ) G ( s ) as a s asa a ( s p )( s p ) ( s p ) Gj ( ) bm( j z )( j z ) ( j zm) a ( j p )( j p ) ( j p ) bm a ( j p ) ( j p ) ( j p ) F I F I H G KJ HG c h c h ( j z ) ( j z ) ( j z ) bm log ( Gj ( ) ) log log log a ( j p ) KJ ( j p ) ad log ( j z ) log ( j z ) bm( j z)( j z) ( j zm) Gj ( ) a ( j p )( j p ) ( j p ) ( j z ) ( j z ) ( j z ) ( j p ) ( j p ) ( j p ) m F HG I KJ m ME375 Frequecy Respose - 9

30 Example Ex: Fid the magitude ad the phase of the followig trasfer fuctio: G ( s ) 3 3s s 9s s 3 s 76s8 ( )( ) ME375 Frequecy Respose - 3

31 Bode Diagram Buildig Blocks st Order Real Poles Trasfer Fuctio: Gp() s, s Frequecy Respose: Gp( j), j R Gp ( j ) S T Gp ( j) ata(, ) ta a f Phase (deg g); Magitude (db B) Q: By just lookig at the Bode diagram, ca you determie the time costat ad the steady state gai of the system? -9././ / / / Frequecy (rad/sec) ME375 Frequecy Respose - 3

32 Example st Order Real Poles Trasfer Fuctio: 5 Gs () s 5 Plot the straight lie approximatio of G(s) s Bode diagram: ) ); Magitude (db Phase (deg) Frequecy (rad/sec) ME375 Frequecy Respose - 3

33 Bode Diagram Buildig Blocks st Order Real Zeros Trasfer Fuctio: Gz( s) s, Frequecy Respose: G ( j ) j, R S T z G G z z ( j) ( j) ata(, ) ta a f B) Phase (deg g); Magitude (d /./ / / / Frequecy (rad/sec) ME375 Frequecy Respose - 33

34 Example st Order Real Zeros Trasfer Fuctio: G() s 7. s 7. Plot the straight lie approximatio of G(s) s ( ) Bode diagram: Phase (de eg); Magitude (d db) Frequecy (rad/sec) ME375 Frequecy Respose - 34

35 Example Lead Compesator Trasfer Fuctio: 35s 35 Gs () s 5 Plot the straight lie approximatio of G(s) s Bode diagram: Phase (de eg); Magitude (d db) Frequecy (rad/sec) ME375 Frequecy Respose - 35

36 st Order Bode Diagram Summary st Order Poles G p( s), s Break Frequecy b rad/s Mag. Plot ta Approximatio db from DC to b ad a straight lie with db/decade slope after b. Phase Plot Approximatio deg from DC to. Betwee ad b b, a straight lie from deg to 9 deg (passig 45 deg at b ). For frequecy higher tha b, straight lie o 9 deg. st Order Zeros G z( s ) s, Break Frequecy b rad/s Mag. Plot Approximatio db from DC to b ad a straight lie with db/decade slope after b. Phase Plot Approximatio deg from DC to. Betwee b b ad b, a straight lie from deg to 9 deg (passig 45 deg at b ). For frequecy higher tha b, straight lie o 9 deg. b Note: By lookig at a Bode diagram you should be able to determie the relative order of the system, its break frequecy, ad DC (steady-state) state) gai. This process should also be reversible, i.e. give a trasfer fuctio, be able to plot a straight lie approximated Bode diagram. ME375 Frequecy Respose - 36

37 Bode Diagram Buildig Blocks Itegrator (Pole at origi) Trasfer Fuctio: Gp() s s Frequecy Respose: Gp( j) j R Gp ( j ) S Gp( j) 9 T Phase (deg); Magitud de (db) Frequecy (rad/sec) ME375 Frequecy Respose - 37

38 Bode Diagram Buildig Blocks Differetiator (Zero at origi) Trasfer Fuctio: Gz( s) s Frequecy Respose: G ( j) j R S T z S Gz ( j) G ( j) z 9 Phase (deg); Magitud de (db) Frequecy (rad/sec) ME375 Frequecy Respose - 38

39 Example Combiatio of Systems Trasfer Fuctio: 3 35s 35 Gs () ss ( 5) Plot the straight lie approximatio of G(s) s Bode diagram: ); Magitude (db B) Phase (deg) Frequecy (rad/sec) ME375 Frequecy Respose - 39

40 Example Combiatio of Systems Trasfer Fuctio: 5 Gs () ss ( 55s5) Plot the straight lie approximatio of G(s) s Bode diagram: Phase (de eg); Magitude (d db) Frequecy (rad/sec) ME375 Frequecy Respose - 4

41 Bode Diagram Buildig Blocks d Order Complex Poles Trasfer Fuctio: Gp() s s s, - Frequecy Respose: G G p p j j j 4 (db) Phase (d deg); Magitude G j ta p -8 Frequecy (rad/sec) ME375 Frequecy Respose - 4

42 Example Secod-Order System Trasfer Fuctio: 5 Gs () s s 5 Plot the straight lie approximatio of G(s) s ( ) Bode diagram: (db) eg); Magitude ( Phase (d Frequecy (rad/sec) ME375 Frequecy Respose - 4

43 Bode Diagram Buildig Blocks d Order Complex Zeros Trasfer Fuctio: 6 s s 4 Gz( s), Frequecy Respose: Gz j j - p G j G j z 4 p G j G j z ta Phase (deg g); Magitude (db B) Frequecy (rad/sec) ME375 Frequecy Respose - 43

44 Bode Diagrams of Poles ad Zeros Bode Diagrams of stable complex zeros are the mirror images of the Bode diagrams of the idetical stable complex poles w.r.t. the db lie ad the deg lie, respectively. Let G ( s ) R S T R S G p p ( j) G () s z G z ( j) G ( j) G ( j) p z Magitude (db B) e p j c z h log G ( j) log G ( j) T Gp( j) Gz( j) Ph hase (deg) -8 Frequecy (rad/sec) ME375 Frequecy Respose - 44

45 d Order Bode Diagram Summary d Order Complex Poles G p s s s (), Break Frequecy b rad/s Mag. Plot Approximatio db from DC to ad a straight lie with 4 db/decade slope after. Peak value occurs at: r Gp( j r) MAX Phase Plot Approximatio deg from DC to 5. Betwee ad 5, a straight lie from deg to 8 deg (passig 9 deg at ). For frequecy higher h tha, straight lie o 8 deg. d Order Complex Zeros G s s ( s), z Break Frequecy b rad/s Mag. Plot Approximatio db from DC to ad a straight lie with 4 db/decade slope after. Phase Plot Approximatio deg from DC to 5. Betwee 5 ad a straight lie from deg to 8, deg (passig 9 deg at ). For frequecy higher tha, straight lie o 8 deg. ME375 Frequecy Respose - 45

46 d Order System Frequecy Respose A Closer Look: Gp( s), s s Frequecy Respose Fuctio: Magitude: G p j ( j) ( j) j Phase: Gp j Gp j j ata, The maximum value of G(j) occurs at the Peak (Resoat) Frequecy r : r ad Gp ( j r) ME375 Frequecy Respose - 46

47 d Order System Frequecy Respose 4 (db) Phase (de eg); Magitude Frequecy (rad/sec) ME375 Frequecy Respose - 47

48 d Order System Frequecy Respose A Few Observatios: Three differet characteristic frequecies: Natural Frequecy ( ) Damped Natural Frequecy ( d ): d Resoat t(p (Peak) k)f Frequecy ( r ): r d Whe the dampig ratio,, there is o peak i the Bode magitude plot. DO NOT cofuse this with the coditio for over-damped ad uder-damped damped systems: whe the system is uder-damped damped (has overshoot) ad whe the system is over-damped d( (o overshoot). As, r ad G(j) icreases; also the phase trasitio from degto 8 deg becomes sharper. r ME375 Frequecy Respose - 48

49 Example Combiatio of Systems Trasfer Fuctio: G ( s ) ( s s 5) ss ( )( s s 5) Plot the straight lie approximatio of G(s) s Bode diagram: ME375 Frequecy Respose - 49

50 Example 4 Magitude (db) Phas se (deg) Frequecy (rad/sec) ME375 Frequecy Respose - 5