Chapter 9 Frequency-Domain Analysis of Dynamic Systems

Transcription

1 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes Chapter 9 Frequecy-Dai Aalysis f Dyaic Systes 9. INTRODUCTION A. Bazue The ter Frequecy Respse refers t the steady state respse f a syste t a siusidal iput. f t + τ f t fr all values f tie t, A iput f ( t is peridic with a perid τ if where τ is a cstat called the perid. Peridic iputs are cly fud i ay applicatis. The st c perhaps is ac vltage, which is siusidal. Fr the c ac frequecy f 60 Hz, the perid is τ 60 s. Rtatig ubalaced achiery prduces peridic frces the supprtig structures, iteral cbusti egies prduce a peridic trque, ad reciprcatig pups prduce hydraulic ad peuatic pressures that are peridic. Frequecy respse aalysis fcuses siusidal iputs. A sie fucti has the fr Asiω t, where A is the aplitude ad ω is its frequecy i radias/secds. Ntice π that a csie is siply a sie shifted by 90 r π rad, as csωt si ωt + 9. SINUSOIDAL TRANSFER FUNCTION (STF Whe a siusidal iput is applied t a LTI syste, the syste will ted t vibrate at its w atural frequecy, as well as fllw the frequecy f the iput. I the presece f dapig, that prti f ti sustaied by the siusidal iput will gradually die ut. As a result, the respse at steady-state is siusidal at the sae frequecy as the iput. The steady-state utput differs fr the iput ly i the aplitude ad the phase agle. See Figure 9- belw. X si ( ω t + φ A si ωt Iput Output Figure 9-. /6

2 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes Thus, the utput-iput aplitude rati ad the phase agle betwee the utput ad iput siusids are the ly paraeters eeded t predict the steady state utput f LTI systes whe the iput is a siusid. X : utput-iput aplitude rati A φ : phase agle betwee utput ad iput Frced Vibrati withut dapig Figure 9- illustrates a sprigass syste i which the ass is subjected t a siusidal iput frce p( t P siωt. Let us fid the respse f the syste whe it is iitially at rest. The equati f ti is x + x P siωt where x is the utput, P is the aplitude f the excitati ad ω is the frcig (excitati frequecy. The abve equati ca be writte i the fr Figure 9- p( t P siωt x Sprig-ass syste. P + x x siωt (9- where ω is w as the atural frequecy f the syste. The sluti f Equati ( csists f the vibrati at its atural frequecy (the cpleetary sluti ad that at the frcig frequecy (the particular sluti as shw i Figure 9-3. Thus, x( t cpleetary sluti + particular sluti Let us btai the sluti uder the cditi that the syste is at rest. Tae LT f bth sides f Equati (9- fr zer iitial cditis, i.e., where X ( s [ x( t ] s L. Substitutig P + X ( s s x 0 x 0 0. ω + ω ω ad Slvig fr X ( s yields P s ω + ω s + ω X s The abve equati ca be writte i partial fracti as: P ω A s + B A s + B s + ω s + ω s + ω s + ω + X s /6

3 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes where A, A, B ad B are left as a exercise fr the studet. The expressi fr X ( s is therefre Pω ω P ω ω ω s + ω ω s + ω + X s The iverse Laplace Trasfr f the abve equati is give by where A - [ ] L ( ω ω P P x ( t X s siω si t + ωt ω ω A B A siωt + B siωt ( ω ω P ( ω ω P ω De Cpleetary Sluti Particular Sluti P P ad B De ω (9- where De ω. P As ω 0, li A 0 ad li B static deflecti ω 0 ω 0 As ω icreases fr zer the deiatr De ω beces sall ad the aplitude icreases, therefre, bth A ad B icrease. The expressi f the deiatr De ca be writte as ω De ω ω ω ω ω ω ω ω It is clear that whe the deiatr beces zer ad the aplitude f vibratis icreases withut bud, therefre resace ccurs. Siusidal Trasfer Fucti (STF The siusidal Trasfer Fucti (STF is defied as the trasfer fucti G ( s i which the variable ( jω. G ( s s jω G ( j ω s is replaced by TF Figure 9-3 STF Siusidal Trasfer Fucti STF 3/6

4 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes G Whe ly the steady-state sluti (the particular sluti is wated, the STF jω ca siplify the sluti. I ur discussi, we are ccered with the behavir f stable, LTI syste uder steady state cditis, i.e., that is after the iitial trasiets died ut. We shall see that siusidal iputs prduce siusidal utputs i the steady state with the aplitude ad phase agle at each frequecy ω deteried by the agitude ad the G jω, respectively. agle f Derivig Steady State Output caused by Siusidal Iput Figure 9-4 shws a LTI syste fr which the iput X s. p( t P siωt P ( s G ( s TF P s ad the utput is X ( s Figure 9-4 Liear Tie Ivariat (LTI Syste The iput p( t is siusidal ad is give by p( t P siωt We shall shw that the utput x ( t at steady state is give by ( ω φ x ( t G j P si t + where G ( jω ad φ are the agitude ad phase agle f G i s ; that is jω, respectively. Suppse that the trasfer fucti G ( s ca be writte as a rati f tw plyials G s The Laplace trasfr X ( s is ( + ( + ( + ( s + s ( s + s ( s + s K s z s z s z where P( s [ p( t ] L. X ( s G ( s P ( s (9-3 Let us liit ur discussi t stable systes. Fr such systes, the real parts f the s i are egative. The steady state respse f a stable liear syste t a siusidal iput des t deped I. C s, s they ca be igred. 4/6

5 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes If G ( s has ly distict ples, the the partial fracti f Equati (9-3 yields Pω X ( s G ( s s + ω a a b b b s + jω s jω s + s s + s s + s where a ad bi ( i,,, respse (9-4 are cstats ad a is the cplex cjugate f a. The x t ca be btaied by taig the iverse Laplace trasfr f Equati (9-4 - j t j t st [ ] L ω ω s t s t x t X s ae + a e + b e + b e + + b e If G ( s ivlves ultiple ples s t h j t e (where h 0,,,, h j stable syste, the ters 0 Fr a stable syste these ters 0 as t sice they have egative real part s j, the x ( t will ivlve such ters as. Sice the real part f the s t t e whe t. s j is egative fr a Regardless f whether the syste ivlves ultiple ples, the steady state respse beces jωt jωt x ( t ae + a e (9-5 where the cstats a ad a ca be evaluated fr Equati (9-4: Pω P a G s s + j G j s + ω j s jω Pω P a G s s j G j s + ω j s jω Ntice that a is the cplex cjugate f a. Referrig t Figure 9-5, we ca write ( jω G y G ( jω φ φ G x σ G ( jω Figure 9-5 Cplex fucti ad its cplex cjugate. 5/6

6 ME 43 Systes Dyaics & Ctrl G j G + G x Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes y cs ( csφ siφ j G jω φ + j G jω siφ G j + j G j e φ G jω e φ φ. Siilarly, Ntice that j j ( ω ( ω φ G j G j e G j e Substitute the expressis f G ( jω ad G ( jω ca get The Equati (9-5 ca be writte as P a G ( jω e j P a G ( jω e j jφ it the expressis f a ad a, e jφ jφ x ( t G j P ( ω + φ ( ω + φ j t j t e e j si ( ωt + φ si( ω φ si( ωt φ G j P t + X + (9-6 where X G ( jω P ad φ G ( jω Sae frequecy p( t P siωt Iput G( jω Output aplitude X P φ Output G x( t Xsi ω t + φ ( ω ( jω G j Phase f the utput Figure 9-6 Iput utput relatiships fr siusidal iputs. 6/6

7 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes Therefre fr siusidal iputs, G ( jω ( jω ( jω X aplitude rati f the utput (9-7 P siusid t the iput siusid X j G j G ( jω ta P j real part f G j phase shift f the utput siusid with respect t the iput siusid iagiary part f (9-8 Exaple 9- (Textb Page 437 Csider the TF X s P s Fr the siusidal iput p( t P siωt G( s Ts + x t., what is the steady-state utput Sluti Substitutig jω fr s i G( s yields G j The utput-iput aplitude rati is ad the phase agle φ is Tjω + G j S, fr the iput p( t P siωt T ω + ta φ G jω Tω, the steady-state utput x t ca be fud as P ( ω ω x( t si t ta T T ω + (9-9 Exaple 9- Fid the steady state respse f the fllwig syste: 7/6

8 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes y + 5y 4p + p if the iput is p( t 0si 4t Sluti First btai the TF G s Y s 4 s s + P s s + 5 s + 5 Fr the iput p( t 0si 4t, it is clear that ω 4 rad/s. Therefre, the siusidal Trasfer fucti is The ad G G ( jω ( jω Y j j ω + 4 j j G j jω + 5 j j j j jω φ G jω + + jω + jω 5 + jω ω ω ta ta 0 + ta ta 0.53 rad The steady state respse is 4 4 ( 3 ( 5 ( ω φ ( t ( t y ( t G j P si t si si Exaple 9-3 (Exaple 9- i the Textb Page Suppse that a siusidal frce p( t P siωt is applied t the echaical syste shw i Figure 9-7. Assuig that the displaceet x is easured fr the equilibriu psiti, fid the steady-state utput. b p( t P siωt Figure 9-7 Mechaical syste x 8/6

9 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes Sluti The equati f ti fr the syste is x + bx + x p( t The Laplace Trasfr f this equati, assuig zer I.C s, is s + bs + X s P( s where X ( s L [ x( t ] ad P( s [ p( t ] steady state utput ad s ca be tae t be zer. The TF is L. (Ntice that the I.C s d t affect the X s G( s P( s s bs ( + + Sice the iput is a siusidal fucti p( t P siωt steady-state sluti. The STF is ( jω, we ca use the STF t btai the X G( jω P( jω ω + bjω + ω + jbω Fr Equati (9-6, the steady-state utput x( t ca be writte as where ad ( ω φ x( t G j P si t + G j ( ω ( ω + b ω φ G( jω ta + jbω bω ω therefre P x ( t si t ta ω + b ω ( ω bω ω Sice ω ad b ζ ω / /, the equati fr x t ca be writte as 9/6

10 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes r x t b ω P / si ta x t t ω b ω ω + ω si ω ta ( ω ω P / ζ ω ω t ( ω ω + ( ζ ( ω ω Let frequecy rati, the abve equati ca be writte as β ω ω x st si ωt ta x t β + ( ζβ ζβ β (9-0 x where st the aplitude rati P is the static deflecti. Writig the aplitude f x( t as X, we fid that X x ad the phase shift φ are / st X x st ( ζβ β + ad φ ta ζβ β The variatis f the aplitude rati ad 9-9 as a fucti f β fr differet values f ζ. X / x st ad the phase shift φ are shw i figures Frequecy Respse Magitude Rati ζ X/x st β ω /ω Figure 9-8 Variati f the aplitude rati X / x st with the frequecy rati β. 0/6

11 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes π ζ 0.0 Frequecy Respse Phase Agle φ(ω π/ ζ β ω / ω Figure 9-9 Variati f the phase φ with the frequecy rati β. 9.3 VIBRATIONS IN ROTATING MECHANICAL SYSTEMS Vibrati due t Rtatig Ubalace Frce iputs that excite vibratry ti fte arise fr rtatig ubalace, a cditi that arises whe the ass ceter f a rtatig rigid bdy ad the ceter f rtati d t cicide. Figure 9-0 shws a ubalaced achie restig shc uts. Assue that the rtr is rtatig at a cstat speed ω rad/s ad that the ubalaced ass is lcated at a distace r fr the ceter f rtati. The the ubalaced ass will prduce a cetrifugal frce ω r. The equati f ti fr the syste is Ttal Mass M r ω b x M x + bx + x p( t (9- where Figure 9-0 p( t ω r siωt Ubalace achie restig shc uts /6

12 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes Is the frce applied t the syste. Tae LT f bth sides f Equati. (9-, assuig zer I.C s, we have r The STF is ( jω M s + bs + X s P( s X s G( s P( s M s bs ( + + X G( jω P( jω M ω + jbω Fr the siusidal frcig fucti p( t, the steady-state utput is btaied fr Equati. (9-6 as ( ω φ x( t X si t + G( jω ω r si ωt ta ( ω bω M ω si t ta ω M + b ω ω r b ω M ω Divide the ueratr ad deiatr f the aplitude ad thse f the phase agle by ad substituteω / M ad b / M ζω it the result, the steady-state utput beces r si ωt ta x t ( ( ω ω ω r / ζ ω ω ( ω ω ( + ζ ( ω ω ω r / ζβ β ( ζβ β + si ωt ta x t where β ω ω. 9.4 VIBRATION ISOLATION Vibrati islati is a prcess by which vibratry effects are iiized r eliiated. The fucti f a vibrati islatr is: t reduce the agitude f frce trasitted fr a achie t its fudati. r t reduce the agitude f ti trasitted fr a vibratry fudati t a achie. /6

13 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes Figure 9-(a illustrates the case i which the surce f vibrati is a vibratig frce rigiatig withi the achie (frce excitati. The islatr reduces the frce trasitted t the fudati. I Figure 9-(b the surce f vibrati is a vibratig ti f the fudati (ti excitati. The islatr reduces the vibrati aplitude f the achie. Figure 9- Vibrati islati. (a Frce excitati; (b Mti excitati. Islati Systes Passive: It csists f a resiliet eber (stiffess ad eergy dissipater (dapig that have cstat prperties. Exaples are etal sprigs, cr, felt, peuatic sprigs, elaster (rubber sprigs. Active: Exteral pwer is required fr the islatr t perfr its fucti. A active islatr is cprised f a servechais with a sesr, sigal prcessr ad a actuatr. A typical vibrati islatr is shw i Figure 9-. (I a siple vibrati islatr, a sigle eleet lie sythetic rubber ca perfr the fuctis f bth the lad-supprtig eas ad the eergy-dissipatig eas. b Figure 9- Vibrati islatr. 3/6

14 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes Practical Exaples. Figure 9-3 (a- Udaped sprig ut;(b- Daped sprig ut;(c- Peuatic rubber Mut. Trasissibility. Trasissibilty is a easure f the reducti f a trasitted frce r ti affrded by a islatr Trasissibility fr Frce excitati. Fr the syste shw i Figure 9-8, the surce f vibrati is a vibratig frce resultig fr the ubalace f the achie. The trasissibility i this case is the frce aplitude rati ad is give by F Aplitude f the trasitted frce TrasissibilityTR t F Aplitude f the excitatry frce 0 Let us fid the trasissibility f this syste i ters f the dapig rati ζ ad the frequecy rati β ω ω. The excitati frce (i the vertical directi is caused by the ubalaced ass f the achie ad is p t ω r ωt F ωt si 0 si The equati f ti fr the syste is equati (9-, rewritte here fr cveiece: M x + bx + x p( t (9- where M is the ttal ass f the achie icludig the ubalace ass. The frce f t trasitted t the fudati is the su f the daper ad sprig frces, r 4/6

15 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes (9-3 f ( t bx + x F si( ωt + φ Taig the LT f Equatis. (9- ad (9-3, assuig zer I. C s, gives M s + bs + X s P( s t bs + X s F( s where X ( s L [ x( t ], P( s L [ p( t ] ad F ( s [ f ( t ] X s P( s M s + bs + F( s X s bs + Eliiatig X ( s fr the abve tw equatis yields L. Hece, The STF is thus F s F( s X s bs + P( s X s P s M s + bs + Substitutig btai bjω + ω + ω + ω + ω + ω + F j b M j M P j M bj b M j M M ω ad b M ζω Fr which it fllws that where F F it the last equati, ad siplifyig, we ( jω + j( ζω ω ( ω ω + j( ζω ω P j ( jω ( ζω ω P ( jω ( ζβ + + ( ω ω + ( ζω ω ( β + ( ζβ β ω ω. Ntig that the aplitude f the excitatry frce is F P( jω that the aplitude f the trasitted frce is F F ( jω t 0 ad, we btai the trasissibility: F F j t TR F P ( jω 0 + ( ζβ ( β + ( ζβ (9-4 5/6

16 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes which depeds β ad ζ ly. Figure 9-4 shws plts f TR fr differet values f ζ as a fucti f β. It iediately fllws fr Figure 9-4 that the cditi β > ust be et i rder that TR <, which eas the trasitted frce aplitude is less tha the excitati frce aplitude. 6 Curves f Trasissibility T 5 0. ζ TR Aplificati Regi Islati Regi TR > TR < β ω / ω Figure 9-4 Curves f trasissibility TR versus β ω ω Therefre, i rder t achieve trasitted frce reducti, fte called suppressi, it is iprtat t desig the sprig cstat such that ω satisfies the cditi that β ω ω > r frequecy ω. Whe β the value f ζ. ω ω < fr a give ass M ad a specified frcig Figure 9-4 shws se curves f the trasissibility versus, hwever the Trasissibility is equal t uity regardless f all the curves pass thrugh a critical pit where β < β ω ω. It is clear that: TR ad β. Fr, as the dapig rati ζ icreases, the trasissibility at resace decreases. Fr β > Fr, as ζ icreases, the trasissibility icreases. β <, r ω ω β >, r ω ω Fr islati <, icreasig dapig iprves the vibrati islati. >, icreasig dapig adversely affects the vibrati 6/6

17 ME 43 Systes Dyaics & Ctrl Ntice that sice fudati is P ( jω F0 ω r Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes, the aplitude f the frce trasitted t the t F F j ω r + ( ζβ ( β + ( ζβ (9-5 Exaple 9-4 Suppse a achie is uted a elastic bearig, which i turs sits a rigid fudati. The bearig dapig is egligible. I perati the achie geerates a haric frce havig a frequecy f 000 rp. If the ass f the achie is 50 g, fid the cditi the equivalet sprig cstat f the elastic bearig fr suppressi f the trasitted frce. Als fid the percetage f the dyaic frce, geerated by the achie, that is trasitted it the fudati if the stiffess f the bearig is 00 N/. Sluti The cditi fr suppressi f the trasitted frce is give by β ω ω > r ω ω < The frcig frequecy is ω 000 rp (000( π rad s. Substitutig the values f ω ad it the abve equati gives r 04.7 ω < rad s < N/ which is the desired cditi the stiffess f the bearig. Whe the stiffess f the fudati is 00 N/, the atural frequecy f the achie-bearig syste is give by ω 63. rad s 50 Therefre, the frequecy rati is β ω ω Substitute this value f β ad ζ 0 it Equati (9-4 yields ( ζβ + TR ( β + ( ζβ (.657 Therefre 57.3% f the achie-geerated dyaic frce is trasitted it the fudati. A assesset f whether this is a adequate reducti ust be based up the ifrati t prvided i the prble stateet. 7/6

18 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes Exaple 9-5 (Textb Page 445 I the syste shw i Figure 9-8 ad shw belw fr cveiece, the ass. M 5 g, b 450 N-s/, 6000 N/, g, r 0. ad ω 6 rad/s, what is the frce trasitted t the fudati? Sluti The equati f ti fr the syste is (0.005(6 (0.si6 Csequetly, x x x t 6000 ω 0rad s ζω ζ We ca fid that β ω ω Fr Equati. (9-5, we have ( ζβ ( β ( ζβ ω r Ft 0.39N Autbile Suspesi Syste. Figure 9-5(a shws a autbile syste. Figure 9-5(b is a scheatic diagra f a autbile suspesi syste. As the car ves alg the rad, the vertical displaceets at the tire act as ti excitati t the autbile suspesi syste. The ti f this syste csists f a traslatial ti f the ceter f ass ad a rtatial ti abut the ceter f ass. A cplete aalysis f the suspesi syste wuld be very ivlved. b b Figure 9-5 (a Autbile syste; (b Scheatic diagra f a autbile suspesi syste. 8/6

19 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes A highly siplified versi appears i Figure 9-6. Let us aalyze this siple del whe the ti iput is siusidal. We shall derive the trasissibility ti excitati syste. b Figure 9-6 Siplified versi f the autbile suspesi syste f Figure 9-3. Trasissibility fr Mti Excitati. I the echaical syste shw i Figure 9-7, the ti f the bdy is i the vertical directi ly. The ti p( t f pit A is the iput t the syste; the vertical ti x ( t f the bdy is the x t is easured fr the equilibriu psiti i the absece f the iput p( t. We assue that p( t is siusidal, r p( t P siωt utput. The displaceet The equati f ti fr the syste is r ( 0 x + b x p + x p x + bx + x bp + p Tae LT f bth sides f the abve equati, assuig zer I.C s Hece, ( s + bs + X ( s ( bs + P ( s The STF is the ( + X s bs P s s bs ( + + ( ω + X j bj P jω ω jbω ( + Figure 9-7 b p( t P siωt Mechaical syste The steady-state utput x ( t has the aplitude X P give by jω. The iput aplitude is jω. The trasissibility TR i this case is the displaceet aplitude rati ad is Aplitude f the utput displaceet TR Aplitude f the iput displaceet 9/6

20 ME 43 Systes Dyaics & Ctrl Thus Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes X j b ω + P ( jω + b ω ( ω Substitutig btai ω ad b ζω TR it the last equati, ad siplifyig, we + ( ζβ ( β + ( ζβ (9-6 where β ω ω. This equati is idetical t Equati 9-4. Exaple 9-6 (Exaple 9-4 i the Textb Page 447 A rigid bdy is uted a islatr t reduce vibratry effects. Assue that the ass f the rigid bdy is 500 g, the dapig cefficiet f the islatr is very sall ( ζ 0.0, ad the sprig cstat f the islatr is,500 N/. Fid the percetage f ti trasitted t the bdy if the frequecy f the ti excitati f the base f the islatr is 0 rad/s. Sluti The udaped atural frequecy s ω f the syste is.,500 ω 5 rad s 500 β ω ω Substitutig ζ 0.0 ad β 4 it Equati 9-6, we get ( ζβ ( β ( ζβ TR The islatr thus reduces the vibratry ti f the rigid bdy t 6.69% f the vibratry ti f the base f the islatr. 9.5 DYNAMIC VIBRATION ABSORBERS If a echaical syste perates ear a critical frequecy, the aplitude f vibrati icreases t a degree that cat be tlerated, because the achie ight brea dw r ight trasit t uch vibrati t the surrudig achies. This secti discusses a way t reduce vibratis ear a specified peratig frequecy that is clse t the atural frequecy (i.e., the critical frequecy f the syste by the use f a dyaic vibrati absrber. This secti is cvered with re details i the Lab. Refer t Labratry tes. 0/6

21 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes 9.6 FREE VIBRATION IN MULTI-DEGREES-OF-FREEDOM SYSTEMS Tw-degrees-f-freed syste. A tw-degrees-f-freed syste requires tw idepedet crdiates t specify the syste s cfigurati. Csider the echaical syste shw i Figure 9-8, which illustrates the tw-degrees-f-freed-case. 3 Figure 9-8 Mechaical syste with tw-degrees-f-freed. Let us derive the atheatical del f this syste. Apply Newt s secd law t ass ad ass, we have Mass F x x x x x Mass F x x x x x Rearragig ters yields 3 (9-9 x x x (9-0 x x x The abve equatis represet a atheatical del f the syste. Free Vibratis i tw-degrees-f-freed syste. Csider the echaical syste shw i Figure 9-9, which is a special case f the syste shw i Figure 9-8. Figure 9-9 Mechaical syste with tw-degrees-f-freed. The equatis f ti fr the syste f Figure 9-7 ca be btaied by substitutig ad 3 it Equatis (9-9 ad (9-0, yieldig (9- (9- x + x x 0 x + x x 0 /6

22 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes T fid the atural frequecies f the vibrati, we assue that the ti is haric. That is, we assue that The x A si ωt, x B siωt x Aω si ωt, x Bω siωt If the precedig expressis are substituted it Equatis (9- ad (9-, the resultig equatis are ( ω ( ω A + A B siωt 0 B + B A siωt 0 Sie these equatis ust be satisfied at all ties ad sice siω t cat be zer at all ties, the quatities i paretheses ust be equal t zer. Thus, Aω + A B 0 Bω + B A 0 Rearragig ters we have ( ω ( ω A B 0 (9-3 A + B 0 (9-4 Fr cstats A ad B t be zer, the deteriat f the cefficiets f Equatis (9-3 ad (9-4 ust vaish, r ( ω ( ω 0 r ( ω 0 r 4 ω 4 ω (9-5 Equati (9-5 ca be factred as fr which ω ω 3 0 /6

23 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes ω, ω 3 Csequetly, ω has tw values, the first represetig the first atural frequecy ω (first de ad the secd represetig the secd atural frequecy ω (secd de: ω, ω 3 Ntice that i the e-degree-f-freed syste ly e atural frequecy exists, whereas the tw degrees-f-freed syste has tw atural frequecies. Ntice that, fr Equati (9-3, we have B A ω (9-6 als, fr Equati (9-4, we btai A B ( ω (9-7 If we substitute bth cases, ω (first de it either Equati (9-6 r (9-7, we btai, i A B Figure 9-0 Mde shape crrespdig the first de If we substitute ω 3 (secd de it either Equati (9-6 r (9-7, we have A B Figure 9- Mde shape crrespdig the secd de. 3/6

24 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes If the syste vibrates at either f its tw atural frequecies, the tw asses ust vibrate at the sae frequecy. Exaple 9-7 I the syste shw i Figure, the displaceets x, x ad x 3 are easured fr the staticequilibriu psiti f the syste. x x x3 3 Figure 9- Syste with three degrees f freed. Sluti The Free Bdy Diagra (FBD f the abve syste is shw i Figure. x x x 3 ( ( x x x x 3 ( ( x x x x 3 3 Figure 9-3 FBD f the syste abve. Neglect the gravitatial frce the three asses ad apply Newt s secd law f ti fr a syste i traslati, e ca get Mass Mass Mass 3, x ( x x F x ( ( + 0 (3, x ( x x ( x x , x ( x x ( I atrix fr, the previus syste ca be writte as 0 0 x 0 x x ( + + x x 0 x (5 r i re cpact fr 4/6

25 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes [ M ]{ x} + [ K ]{ x} { 0} (6 where [ M ] ass atrix f the syste K + stiffess atrix f the syste 0 [ ] (7 { x} x x vectr f displaceet crdiates x 3 Assue a sluti f the fr { x} { v} e iωt (8 Substituti f eq. (8 it (6 yields the fllwig assciated eigevalue prble 0 v 0 0 v ( v λ v 0 v 0 0 v (9 where N/, ad g. The script that λ ω. Assue that 3 deteries the eigevalues ad assciated eigevectrs is MATLAB PROGRAM: >> ; ;; >> ; ;; >> K[ - 0;- * -;0 - ]; >> M[ 0 0;0 0;0 0 ]; >>[Mdes, Eigevalues]eig(K,M Executi f the script gives Mdes Eigevalues 5/6

26 ME 43 Systes Dyaics & Ctrl Chapter 9: Frequecy Dai Aalyis f Dyaic Systes Systes sice λ ω, the the frequecies are 3.73 rad/s ω rad/s. ω, ω rad/s ad Whe the syste abve is exaied, it is fud that, sice the asses at each ed are t restraied, a rigid-bdy de i which all asses ve i the sae directi by the sae aut is pssible. This is reflected i the crrespdig vibrati de, which is depicted i the third clu f the atrix f vibrati des. The sprigs are either stretched r cpressed i this case /6