VIBRATION CONTROL OF A FLUID-LOADED CIRCULAR PLATE VIA POLE PLACEMENT

Transcription

1 MECHAICS vol. 7 o. 1 8 Lucya LEIOWSKA * VIBRATIO COTROL OF A FLUID-LOADED CIRCULAR PLATE VIA POLE PLACEMET SUMMARY This aer resets the alicatio of active cotrol syste to reduce the vibratio ad the soud radiatio of a circular late which is fluid-loaded ad claed at a fiite baffle. The late is drive by a uifor haroic riary force ad cotrolled by a distributed secodary force geerated by the iezodisks. The ai of the aer is to desig a cotroller to odify the resose of the lat i soe desired fashio. The atheatical odel of the syste icludes the ifluece the acoustic wave radiated by the late, Kelvi-Voigt daig i the late aterial ad viscous fluid daig. For the cosidered syste the trasfer fuctio of cotroller of 4th order, satisfyig the olyoial Diofatie equatio, is derived. The ifluece of the fluid-loadig ad daig effects o the syste resose are reseted grahically ad discussed. The results show that late odes have bee reduced very well by obtaied cotroller for a few assued values of daig coefficiets. Keywords: active ethods, vibratio cotrol, ole laceet, PZT actuators, iteral ad viscous daig STEROWAIE DRGAIAMI P YTY KO OWEJ Z UWZGLÊDIEIEM ODDZIA YWAIA OŒRODKA METOD LOKOWAIA BIEGUÓW W racy rzedstawioo zastosowaie aktywego sterowaia do redukcji drgañ i fal akustyczych roieiowaych rzez ³ytê ko³ow¹ utwierdzo¹ a obwodzie. P³yta obudzaa jest do drgañ si³¹ haroicz¹ o rozk³adzie rówoiery, a sterowaa za ooc¹ ary iezodysków PZT. Cele racy jest zarojektowaie uk³adu sterowaia, który odyfikuje odowiedÿ obiektu w o ¹day sosób. Model ateatyczy rozwa aego obiektu uwzglêdia oddzia³ywaie zwrote fali akustyczej, t³uieie wewêtrze ateria³u oraz t³uieie wiskotycze. Dla zastosowaego odelu wyzaczoo trasitacjê regulatora czwartego rzêdu, rozwi¹zuj¹c rówaie diofatycze. W³yw t³uieia wewêtrzego, wiskotyczego oraz srzê eia oiêdzy drgaj¹c¹ struktur¹ i oœrodkie a odowiedÿ uk³adu rzedstawioo w sosób graficzy i rzeaalizowao. Otrzyae wyiki okazuj¹, e zarojektoway regulator bardzo dobrze t³ui drgaia ³yty dla kilku za³o oych wartoœci wsó³czyików t³uieia. S³owa kluczowe: etody aktywe, sterowaie drgaiai, o³o eie bieguów, aktuatory PZT, tarcie wewêtrze i wiskotycze 1. ITRODUCTIO Plaar structures as ebraes or lates do ot usually vibrate i vacuo. They affect ad are affected by the surroudig edia which were i cotact with their surfaces. The vibratio ad soud radiatio of circular lates have bee studied by ay researchers sice it is a sigificat structural eleet i ay idustrial fields. Lord Rayleigh was the first erso who aalised the reactio of the air o a vibratig circular late (Rayleigh 199), showig that reactio to be equivalet to a virtual ass, ad radiatio daig to be added to the late ass ad the echaical daig. Modal coulig, radiatio efficiecy ad radiatio iedace were eloyed i the last several years, where the roble of atteuatig structure-bore oise was cosidered (Gu 1993; Juger 1986; Rosehouse 1). I this case the cost fuctio is derived fro the far-field radiated ressure (radiatio cotrol). Thus, for the closed-loo syste resose, the overall radiatio efficiecy is reduced while the overall late vibratio alitude is ot satisfactorily atteuated or soeties eve icreased. This aer ivestigates the alicatio of a active cotrol syste to reduce the vibratio of a circular late which is fluid-loaded ad claed at a fiite baffle. The foral solutio of the fluid-late couled equatio is reseted for a late drive by a uifor haroic riary force ad cotrolled by a distributed secodary force geerated by iezodisks. For the cosidered fluid-late syste, the state sace realizatio is give. Three araeters which characterise fluid desity, late aterial iteral daig ad viscous fluid daig are icluded i the cosidered odel. It is well- kow that the dyaic behaviour of the liear systes deeds strogly o the locatio of the odels roots (zeros ad oles). To exaie the stability of the syste ad the ifluece of fluid-loadig as well as structural iteral daig ad viscous daig, the roots of the syste with * Istitute of Techology, Rzeszow Uiversity of Techology, [email protected] 18

2 MECHAICS Vol. 7 o. 1 8 assued araeters were lotted o a colex lae. This eables the use of a ethod for deteriig the trasfer fuctio of a cotroller which laces the oles of the closed-syste at soe desired locatios to suress a late vibratio. The effects of fluid-loadig ad viscous daig o the syste resose are observed ad reseted grahically. Fially, the feedback cotrol law is develoed for the iiizatio of the late vibratio.. FLEXURAL VIBRATIOS OF A CIRCULAR PLATE The structure uder study is a vibratig circular late of radius a, havig a costat thickess h (Fig. 1), surrouded by a lossless ediu with static desity. It is assued that the late, claed i a flat, rigid ad fiite baffle of radius b, (b>r>a, z=) is ade of a hoogeeous isotroic aterial with desity, ad has Kelvi-Voigt iteral daig. It is assued that the late is excited o oe side by a uifor eriodic force with costat alitude F geerated by a loudseaker f (, r t) F e for ra w it (.) ad it radiates ito free sace filled with fluid of desity. The syste odel is forulated whe takig ito accout the coulig effect betwee the structure ad the acoustic ediu, so the third cooet of the right had side of equatio (.), f (, r t), reresets the acoustic fluid-loadig actig o the late as a additioal force. The value of this force exerted by the fluid o the late surface ca be calculated as follows (Juger 1986): f (, r t) (, r z, t) (.3) where rz (,, t) is the acoustic ressure at the oit o the surface of the late. The acoustic waves roagate through the fluid ust satisfy the wave equatio (Malecki 1964): z 1 rzt (,, ) rzt (,, ) (.4) c t where is the two-diesioal Lalace oerator, ad c is soud velocity i the fluid. At the fluid-structure iterface, the ressure ust satisfy the boudary coditio (Juger 1986): Fig. 1. A circular late i a rigid baffle of radius b I the case beig cosidered, the alied loadig ad ed restraits of the circular late are ideedet of the agle (axially syetrical vibratios), thus we ca write the goverig differetial equatio of the forced otio of the late as follows (Leiowska 5): 4 4 B w(, r t) R [ wrt (, )] t t wrt (, ) (.1) h wrt (, ) f r t f r t f r t t w(, ) s(, ) (, ) where: 3 B Eh / 1( 1 v ) bedig stiffess of the late, E, ad R Youg s odulus, Poisso s ratio ad Kelvi-Voigt daig coefficiet for the late, desity for the cobied structure, viscous fluid daig coefficiet. The dislaceet w(r,t) ad its derivative wrt (, )/ r satisfy the boudary coditio for a claed late: they both equal zero at the edge of the late. rzt (,, ) z wrt (, ) wrt (, ) t (.5) with deotig the oral to the structure. I the case i which the acoustic ressure radiates fro the late vibratig haroically, the wave equatio reduces to the Helholtz equatio ( k ) ( r, z ) (.6) where: (r, z) ressure alitude, k = /c acoustic wave uber at frequecy. The goal i the cotrol roble is to deterie a cotrol force fs (, r t)which, whe alied to the late (realized via a voltage u(t) for the assued actuators), leads to a reduced level of vibratio. The secod cooet i the Eq. (.1) reresets such a wated cotrol force, fs (, r t), which will cacel the late vibratios. The locatio of actuators (a air of iezodisks) is assued to be i the iddle of the late. 3. STATE-SPACE SYSTEM MODEL To aroxiate the late dyaics, a Fourier-Bessel exasio of the late dislaceet is used to discretize the ifiite diesioal syste (.1). The late dislaceet ca be aroxiated by w (, r t) s () t w () r (3.1) 19

3 Lucya LEIOWSKA VIBRATIO COTROL OF A FLUID-LOADED CIRCULAR PLATE VIA POLE PLACEMET where is cosidered to be a fiite uber suitably large for the accurate odellig the syste dyaics ad w ()is r the (, ) late ode described as follows (Malecki, 1964) r J w () r u J a I ( ) I ( ) r (3.) a J( x), I( x)desigate the cylider fuctios, k ais the -th root of the frequecy equatio ad s ()is t the corresodig odal alitude i tie t. The eigefuctios satisfy the ortogoality coditio ad ca be oralized as follows: w () r ds a (3.3) S I a siilar way let us exad the right side of the late equatio of otio (.1) ito series: f (, r t) z () t w () r (3.4) w f (, r t) u () t w () r (3.5) s f (, r t) (, r z, t) () t w() r (3.6) Isertig above exasios ito the equatio (.1), ultilyig both sides by the orthogoal eigefuctio w (), r ad itegratig over the surface of the structure, the goverig equatio of otio ca be re-exressed as: 1 1 [ s () t ( ) s () t s () t z () t u () t ( t)] (3.7) where z u () t () t f j (, r t) w () r ds, jw,, s ; 1..,,, (3.8) () t S stad for the geeralised odal forces R/ B ad / h (3.9) 1 The odal odel reseted above ca ow be exressed i the state-sace forat. The state vector is defied as follows: s() t x() t s( t ) (3.1) Equatio (3.7) ca be exressed as (Leiowska 5) x () t Ax() t Bu() t Vz() t (3.11) where the dot deotes differetiatio with resect to tie, x is the (1) state vector, u is (1) cotrol vector, ad A is () state atrix, B is the () cotrol iut atrix, V is (1) disturbace atrix, described as follows: A 1 1 ( I+ E) ( )( I+ E) B 1 ( I+ E) K, V s ( ) 1 I E K w 1 (3.1) I above exressio I deotes idetity atrix, K S ad K W are the coefficiet vectors, E reresets fluid-late iteractio atrix, diag [ 1,,.., ]. It is assued, that the resose of the cosidered late to the alied force distributio is easured by a set of liearly ideedet oit sesors, situated at locatios r o the late. The outut equatio i atrix for is where y() t Cx() t (3.13) C C a C v w1( r1 ) w ( r1 ) : : : (3.14) w1 ( rc ) w ( rc ) w1( r1 ) w ( r1 ) : : : w1 ( rv ) w ( rv ) c ad v deote the uber of dislaceet ad velocity sesors resectively, wi( rj )is a value of i-th eigefuctio at j-th easureet oit. 4. SYSTEM DYAMICS AD DAMPIG EFFECTS To exaie the ossibility of istability of the syste ad the ifluece of fluid loadig as well as structural iteral daig ad viscous daig, it is coveiet to lot the roots of the syste for assued araeters o a colex lae. For the aalysis i this aer, water, roae, air ad aluiiu were chose to resectively rereset the acoustic fluid aterial ad aterial of the late, to esure strog, oderate ad light coulig betwee the structural ad acoustic resose. I the Figure the locatio of oles ad zeros of the eight-order syste is reseted i air, for two values of araeter 1 (ad costat ). It ca be see that the values of iteral daig coefficiet 1 have cosiderable ifluece o systes root locatios. The followig rules ay be forulated whe the araeter 1 icreases (Fig. ): the colex roots of the syste igrate to the circle with the radius of 1/( 1 ), the uber of colex roots diiishes they becoe real roots, the real roots ove alog the real axis: half of the igrate towards the iddle of the circle ad the secod half to the.

4 MECHAICS Vol. 7 o. 1 8 Fig.. Distributio of oles ad zeros of the cosidered syste i air for 5. ;a) 1. ;b) 1. 4; zeros; oles O the basis of the ole locatios we ca ake a coclusio about the syste dyaics which ay be observed o the Bode diagra (Fig. 3). Figure 3 resets the results obtaied i air for three values of iteral daig coefficiet 1 ad costat value of fluid daig araeter, whe the acoustic ad structural wave ubers are equal: kv = k. It ca be see that for 1 =.11, the resoace frequecy of the first ad secod ode of the cosidered late are doiated. However, for lower values of 1 the third resoace frequecy close to 5 Hz aears ad the hase characteristic becoes ore colicated. Fig. 3. Bode diagra of the fluid-late syste i air: 1) 1.11; ) 1.;3) 1.44 It is worth to ote that the odificatio of araeter, assuig the liear badwidth.1 1 s/ 3, does ot chage root locatios oticeably ad ca be observed o the Bode diagra i the viciity of the resoace eaks. The third araeter icluded i the odel derived, the fluid-loadig ter (Crighto 1989), is helful for hk exaiig the ifluece of the fluid surroudig the cosidered late. Takig soe values of the fluid desity ( 1 ad are costat ow) it ay be see (Fig. 4) how the localizatio of syste roots is chaged for differet values of. For the icreasig desity of the surroudig fluid ediu, the roots of the cosidered syste ove to oit (, ) o the colex lae. The Bode diagra (Fig. 4) reveals a additioal feature which is iortat for the correct desig of cotroller trasfer fuctio, aely a hase shift, esecially for low frequecy radiated acoustic waves (Fig. 4a). It ca be see that whe the fluid desity alters (icreases), the resoace frequecies ad the aroriate hase characteristics diiish. As a result of the fluid coulig the resose of the late i fluid ca be sigificatly differet fro resoses i vacuo. It ca also be observed, that the effect of the fluid-loadig o the cosidered late is deedet o the frequecy of the vibratio. I the case of lower frequecies, the shift of the resoace eak is greater (Fig. 5a) ad whe the oeratig frequecy icreases it diiishes. To best illustrate this effect, Figure 6 shows the Bode diagras of the syste i water (where the strog coulig ca be theoretically assued) for the lower (Fig. 6a) ad higher frequecies of acoustic waves (Fig. 6b). A exaiatio of the syste dyaics shows that the waveuber sectru of the soud ressure ca be divided ito two doais. Below the ull frequecy defied as c f, h the surroudig fluid ass-loads the late ad the effective ass of the late icreases. I this case the fluid-loadig acts aily as a ass-loadig ad the resoaces eaks are oved towards lower frequecies. 1

5 Lucya LEIOWSKA VIBRATIO COTROL OF A FLUID-LOADED CIRCULAR PLATE VIA POLE PLACEMET a) Fig. 4. Distributio of oles of the cosidered syste for 1., 5. ad three kids of the fluid desity: (*) =1; (o) =5; (+) o =1. b) a) b) Fig. 6. Bode diagra of the fluid-late syste i water, for 1., 1. ad three value of the frequecy of the acoustic ressure: a) ( )k =.4 (1 Hz); ( )k = 1.6 (3 Hz); ( )k =.9 (5 Hz); b) ( )k = 5.13 (6 Hz); ( )k = (1 Hz); ( )k = 6.83 (15 Hz) O the other had, for frequecies of acoustic resose f > f, the fluid acts as a daeer the effect of the fluid-loadig is sall ad it has dissiatig character. 5. FEEDBACK COTROL The ai of the roject is to desig a cotrol syste to odify the resose of the lat i soe desired fashio. The closed-loo setu is sketched i the Figure 7. Fig. 5. Bode diagra of the fluid-late syste for 1., 1. ad three kids of the fluid desity: ( ) =1; ( ) =5; ( ) = 1.; a) f =1 Hz; b) f = 875 Hz Fig. 7. Closed-loo setu

6 MECHAICS Vol. 7 o. 1 8 For the assued easureet oit (SISO syste), the cosidered structure has a followig trasfer fuctio 1 bs () b s b 1s... b Gs () (5.1) as () 1 a s a s... a 1 The closed-loo trasfer fuctio is give by GsRs bss Gz () () () () () s 1 RsGs () () ass () () bsqs ()() We seek a cotroller 1 (5.) qs () qs q 1s... q Rs () (5.3) s () 1 s s... 1 a b d a1 a b1 b d 1 1 a1 b1 a a b b a a1 b b1 q a b q d (5.5) or grahically by usig MATLAB Root Locus Tool (i this case the Diofatie equatio is solved as well). Sice we assued two colex oles ad two colex zeros corresodig to the desired cotroller dyaics (Fig. 9). of order satisfyig the olyoial Diohatie equatio (Astro 199): ass () () bsqs ()() ds () (5.4) 1 d s d s... d 1 where d(s) deotes a desiged olyoial of + order which oved the syste roots to soe redefied locatio. The order of the cotroller should be equal to the uber of cosidered odes i the syste odel. Sice we are basically iterested i daig the doiatig vibratig odes ad a loworder cotroller is geerally referred for hysical ileetatio reasos, we take ito accout first four odes oly. The oe-loo zeros ad oles locatios of the lat are deicted i the Figure 8. Fig. 9. Poles ad zeros of closed-loo fluid-late syste i air, zeros; x oles; active oit of root locus After a series of attets we obtaied fourth-order cotroller with the followig values (Tab. 1). Table 1 Poles ad zeros of desiged cotroller Gai Poles Zeros 9 ±39 i 96 ±5 i 79 ±15 i 5 8 i Fig. 8. Poles ad zeros of oe-loo fluid-late syste i air: 1. 4; zeros; oles I theory, if the syste is cotrollable, the oles ad zeros ca be laced aywhere to irove close syste erforace. It ca be doe i two ways: aalytically by solvig the liear syste of equatio with (+)(+) o-sigular Sylvester atrix: O the Bode agitude lot of the Figure 1, we ca otice the sigificat reductio i the resoace eaks of the doiatig vibratig odes of the closed-loo syste i air. It ca be see that the ucotrolled late resose vibrates sigificatly while the cotroller causes that late doiat odes have bee reduced very well for two values of iteral daig coefficiet. However, there are soe disadvatages of ole laceet cotrol. First of the is steady-state error which aears usually i closed-loo syste (Fig. 11). 3

7 Lucya LEIOWSKA VIBRATIO COTROL OF A FLUID-LOADED CIRCULAR PLATE VIA POLE PLACEMET a) This roble ca be iroved by a aroriate referece sigal correctio or by alyig cascade structure of cotrol syste. The ext disadvatage of olyoial ole laceet cotrol is uerical troubles for higher order of Diofatie equatio. It is well-kow that liear algebra robles ivolvig olyoials are geerally ill-coditioed. A frequecy scalig or alterative olyoial bases (Chebyshev, Berstei, orthogoal olyoials) ca robably hel to irove uerical coditioig. 6. FIAL REMARKS b) Fig. 1. Coared Bode agitude lots of the oe-loo ad closed-loo syste i air for ad two values of iteral daig: a) 1 =.11; b) =.; oe-loo; closed-loo The roble of active suressio of the late vibratio of the fluid-loaded late has bee solved by eas of usig ole laceet ethod. The atheatical odel of the cosidered syste icludes the ifluece the acoustic wave radiated by the late iteractig o its surface due to the coulig echais. I additio, Kelvi-Voigt daig i the late aterial ad viscous fluid daig has bee take ito accout. It has bee observed that the fluid-loadig has two ai effects o the vibratig late i deedece o the oeratig frequecy it causes either daig or a ass-loadig. The obtaied cotroller was verified by testig the late dislaceet resose obtaied for chir disturbace. Coarig the structural ad acoustic resoses for the oe ad close-loo systes, it ca be see that the obtaied cotroller was daed the outut sigal very well. The ai disadvatage of this ethod: a steady-state error ca be iroved by a aroriate correctio of referece sigal or by alyig a cascade structure of cotrol syste. REFERECES Fig. 11. The ste resose of the closed-loo syste for 1. ad Astro K.L., Witteark B. 199: Couter Cotrolled Systes. Theory ad Desig, d ed., Pretice Hall, Ic., ISB: Crighto D.G., 1989: Fluid loadig the iteractio betwee soud ad vibratio. Joural of Soud ad Vibratio, 133, Issue 1, 1 7. Gu Y., Fuller C.R. 1993: Active cotrol of soud radiatio fro a fluid- -loaded rectagular uifor late. The Joural of the Acoustical Society of Aerica, vol. 93, Issue 1, Juger M.C., Feit D. 1986: Soud, Structures ad Their Iteractio. Massachusetts Istitute of Techology Press, Bosto, ISB: Leiowska L. 5: Modellig of iezoceraic actuators of circular late to reduce oise ad vibratio, Molecular ad Quatu Acoustics, vol. 6, Malecki I. 1964: Theory of waves ad acoustic systes (i Polish). PW, Warszawa. Rayleigh J.W. 199: Theory of Soud. MacMilla, Lodo. Rosehouse G. 1: Active oise Cotrol. WIT Press, Lodo.