VIBRATION CONTROL OF PIEZOELECTRIC FLEXIBLE STRUCTURE USING ROBUST CONTROL METHODOLOGY

Transcription

1 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: VIRTION CONTROL OF PIEZOELECTRIC FLEXILE STRUCTURE USING ROUST CONTROL METHODOLOGY JUNFENG HU, GUIYNG XU Shool of Mehaial & Eletrial Egieerig, Jiagxi Uiversity of Siee ad Tehology, Gazhou 4, Chia hjfsuper@6.om, @qq.om STRCT tive vibratio otrol of a smart flexible atilever beam was studied by usig mode theory ad μ sythesis method. The atuator is piezoerami path, the sesor is strai gauge. The fiite elemet method ad experimetal modal test was applied to obtai the dyami model of the smart struture. Takig aout ito uertaity of the exteral disturbae ad measuremet oise, ad uertaity of atural frequey, dampig ratio ad atuator parameters of the system, μ sythesis approah was applied to desig the vibratio otroller by seletig mode displaemet as evaluatio sigal ad seletig appropriate weightig futio aordig to amplitude ad frequey harateristis of the atual sigal. otroller with same weightigs was also desiged ad implemeted for ompariso aalysis. The performaes of the otroller are verified experimetally i this study. Experimetal results showed that µ -otroller a provide good disturbae rejetio ad is more robust to parameter variatios tha otroller. d the proposed otrollers a effetively suppress the vibratio respose of the flexible beam. Keywords: Smart Struture, tive Vibratio Cotrol, Fiite Elemet Method, µ Sythesis, Experimetal Modal Test. INTRODUCTION I preisio ad aerospae idustry, may researhes o lightweight ad miiaturized strutures have bee arried out to improve strutural performaes. mog the researhes, passive strutures usig omposite material are typially kow as oe of the effetive methods. However, the traditioal passive strutures are very sesitive to hage of iteral load oditio ad exteral eviromet oditio whih a eve ause sudde destrutio of strutures. Therefore, i order to satisfy striget requiremets for preisio otrol ad lightweight miiaturizatio, smart materials suh as shape memory alloys, piezoeramis, ad magetorheologial fluids are frequetly adopted for smart strutures. The performae requiremets of future spae strutures, jet fighters ad oept automobiles have brought muh iterest to the area of smart strutures. smart struture a be defied as a struture with boded or embedded sesors ad atuators as well as a assoiated otrol system, whih eable the struture to respod simultaeously to exteral stimuli exerted o it ad the suppresses udesired effets or ehae desired effets. mog various smart strutures, those with piezoeletri pathes have reeived muh attetio i reet years, due to the fat that piezoeletri materials have simple mehaial properties, small volume, light weight, large useful badwidth, effiiet oversio betwee eletrial ad mehaial eergy, good ability to perform vibratio otrol ad ease of itegratio ito metalli ad omposite strutures [-]. Differet otrol tehiques have bee ivestigated i the otrol of smart struture. breu oduted experimetal work for the vibratio otrol of flexible beam by usig piezoeletri sesors ad atuators with Liear Quadrati Gaussia (LQG) otroller [4]. There are may lassial strategies that a be used whe the mathematial model is available, for istae pole alloatio ad optimal otrol. However, if the model has uertaities these methods are ot idiated. There are may robust tehiques i strutural otrol literature. Li ivestigated two otrol strategies for robust vibratio otrol of 64

2 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: parameter uertai systems [5]. Mayha ombied itelliget otrol ad smart materials to produe a adaptive ad robust otroller to dampe the fudametal vibratio mode of the system i the presee of modelig uertaities [6]. Zhag et al. studied the ative vibratio otrol problem for the high-speed flexible mehaisms all of whose members were osidered as flexible by usig omplex mode method ad robust otrol sheme [7-8]. Kawabe utilized eural etworks (NN) theory for ative otrol i a logitudial atilevered-beam system by simulatio ad experimet. It is foud that fairly satisfatory ative dampig effet usig the NN otroller is obtaied [9]. ut the radom disturbae ad measuremet oise of the atual system were ot osidered by these urretly proposed vibratio otrol strategies. The issue of robustess agaist exteral disturbaes was ot addressed, ad therefore the proposed vibratio otrollers aot be effetively applied to the smart struture uder the radom uertai disturbaes. Very few attempts have bee made toward the appliatio of robust otrol methodology to otrol vibratios i lightweight flexible strutures. Robust vibratio otrol methodology has reeived muh attetio due to their wide appliatios, ad led to a rapid developmet of various otrol strategies suh as the LQG otrol ad the fuzzy otrol, et. I reet years, a great deal of attetio has bee paid to the otrol beause it ot oly provides a uified ad geeral otrol framework for all otrol strutures, but also yields a otroller with guarateed margis. However, otrol models all uertaities as a sigle omplex full blok, whih results i a rather oservative desig. Uder suh irumstaes, the µ sythesis tehique, whih ivolves the use of optimizatio for sythesis ad strutured sigular value for aalysis, has bee developed. Durig the otroller desig, a issue to be osidered is the proess of the mathematial model betwee the disturbae fore ad the maipulator, whih is always assumed to be kow. I egieerig pratie, however, it is diffiult to obtai this model eve though system idetifiatio or theoretial approah. Hee, how to desig a µ - otroller without requirig a mathematial model betwee the disturbae fore ad struture is of great iterest i the appliatio of µ -sythesis tehique. We aim here to deal with the ative vibratio redutio problem i flexible struture with uertaities through desigig reasoable µ - otroller. I this paper, the vibratio otrol of a flexible beam is ivestigated by usig µ sythesis ad experimetal modal test method, ad takig ito aout the radom disturbae uertaity, modal parameter uertaity ad measuremet oise. The paper is orgaized as follows. I setio, a dyami model of a flexible beam boded with piezoeletri atuators ad strai gauge sesors is ostruted by usig fiite elemet method. I setio, the µ otroller is proposed. I setio 4, experimetal idetifiatio of the flexible atilever beam is performed to obtai its modal parameters. d the experimetal validatio test is performed based o the dspce DS platform. The olusios are give i setio 5.. DYNMIC MODELING OF SMRT STRUCTUE The modelig of smart struture with piezoeletri atuators ad sesors has bee a subjet of itese researh for a log time ad is oly briefly desribed here. y u u u 4 u Figure : Plaar beam elemet showig odal degrees of freedom ad oordiate systems The flexible struture is modeled by usig a two-ode beam elemet. The beam elemet is show i Fig., whih has two odes with four degrees of freedom at eah ode; amely u, u 5, the logitudial displaemet, ad u, u 6,the trasverse displaemet, ad u, u 7, the slope, ad u 4, u 8, the urvature. L e is the legth of elemet. The odal displaemet vetor u with respet to referee frame xyis expressed as u = [ u ] T u u u4 u5 u6 u7 u8 () The trasverse ad logitudial displaemet fields of two-ode beam elemet are ostruted usig the quiti hermite ad liear iterpolatio polyomials, respetively. V, L e u 6 u 7 u 5 u 8 x 65

3 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: W deotes logitudial ad trasverse elasti displaemet of arbitrary poit, respetively. Subsript T deotes matrix traspose. They a be writte by the followig form V( xt, ) N ( x) = u () W ( xt, ) N ( x) where ( x) ( x) ( x) T N = N N is shape futio, u is the odal displaemet vetor. The system dyami equatios a be obtaied by usig fiite elemet method MU + CU + KU = P () where M, C, K are the systemati mass, dampig, stiffess, respetively. U,U, U are the geeralized displaemet, veloity, ad aeleratio vetors of the system, respetively. P is the systemati geeralized disturbae fore vetor orrespodig to vetor U. The piezoeletri pathes work as atuators are perfetly boded o the upper ad lower surfaes of the beam at the same loatio. For the modelig of PZT atuators, literature [, ] provides a detailed derivatio of ouplig of PZT atuators ad a host beam. These bedig momets idued by the atuators is give by ( ) T = T = d Eb t + t V (4) p p p a i where d, E p, b p, t p is piezoeletri ostat, elasti module, width, ad thikess of PZT path, respetively. t a is the thikess of the beam elemet, V i is the vetor of iput voltage to the piezoeletri atuators. The momets are assembled as a part of the exteral momets exerted o ode. Strai is the amout of deformatio of a struture due to a applied fore. Strai gauge is the most ommo sesor for measurig strai. strai gauge s eletrial resistae varies i proportio to the amout of strai plaed o it. The deformatios maily ilude ompressio or tesio ad bedig deformatio for the flexible beam. The straiε i x diretio is give by ε = εl + ε (5) where ε L is ompressio or tesio strai ad ε are logitudial strai due to bedig deformatio, respetively. They a be give by ε L = N u = N U, (6) ε = hn u = hn U where h is the distae betwee the eutral axis of the beam ad the outer surfae of the beam, N, N are the first-order ad the seod-order differetial of the shape futio N ad N, respetively, deotes the trasformatio matrix. Thus, the exogeous perturbatio ad the otrol iputs have o diret effet o the measured outputs. There are a piezoeletri atuators ad s strai gage sesors o the flexible struture. Combiig Eq. ()~ Eq.(6), the dyami equatios of the struture equipped with piezoeletri atuators ad strai gauge trasduer a be expressed as MU + CU + KU = P + DaVi (7) y= DU s where D a is the systemati otrol matrix related to ofiguratio of atuators, Ds is the systemati output matries determied by ofiguratio of sesors, a a -by- vetor, ad y is the strai from the sesors, a s -by- vetor. It was show that the dyami respose of the flexible struture is omposed maily of the lower modes. I order to otrol the lower modes, the physial-oordiate equatios must be first trasformed ito modal oordiates. Here, we hoose the first order modes as otrol modes. pplyig the modal theory, the ormalized modal trasformatio is itrodued by U = ψ η (8) where ψ is the otrolled ormalized modal matrix, η is otrolled modal oordiate vetors. Substitutig Eq. (8)ito Eq.(7), the system dyami equatios are rewritte as η + Cη + Kη = N + DV a i (9) y = D η s T T T where N = ψ P, Da = ψ Da, Ds = ψ D C s, K is diagram matrix, whih is determied by system atural frequey ad dampig ratio. For otrol sythesis, the system must be writte as a system of first-order ordiary differetial equatios (ODEs). We a defie the otrolled state variables by the followig form X = η η η η () [ ] T Due to the otrolled mode umber, the umber of otrolled state variables is. Takig ito aout measuremet oise ν, the state-spae model for the system a be writte as 66

4 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: X = X + V i + N () y= CX + ν where I a = K C, = Da, C = [ Ds s ], I is uit matrix, a, s are a, s zero matrix, respetively.. ROUST µ -CONTROLLER DESIGN Robust otrol theory is exploited to develop a more simple ad reliable otroller for the flexible struture vibratio otrol appliatios. The robust otroller requires both omial model ad uertaity model. So i order to make atual otrol beome robust, it is eessary to provide a aurate ad detail modelig of the uertaities that is relevat to the struture model. eause the plat to be otrolled is a flexible struture modal model is best oe desribed its dyami behavior. I geeral, modal parameters of a flexible struture model are: resoae frequeies, dampig ratios, modal vibratio shape. Eah of these modal parameters otais uertaity. I additio, the truated modes i the model redutio ad sigal oise i atuators ad sesors also irease the uertaity i the assumed model. The mathematial models for uertaities established i robust theory are additive uertaity. Sie the atural frequeies ad mode shapes of the system are futios of the time, Equatios () are otiuous time-varyig state spae represetatio of the mehaism system. Desigig a otroller with time-varyig statespae matries is out of the sope of this study. The prevailig pratie i the aalytial studies available is to assume ostat state spae matries, i.e., ostat atural frequeies ad mode shapes for the etire motio yle of the system. lthough these results are ot olusive, similar assumptio will be made i the otroller desig of urret work. ased o the model obtaied i setio, a otroller is desiged for the smart flexible beam. The goal of the otroller is to atteuate the vibratios of the smart beam at its first two flexural frequeies ad obtai gai stabilize the umodeled high frequey modes. I otrol desig framework, the objetive is to miimize the orm of the weighted trasfer futios from the iput disturbae sigals to the output error sigals. The uertaities i the plat model a be put i suh a form that some of the disturbaes ad error sigals orrespod to the haels through whih the omial model iterats with a orm bouded uertaity blok. This geerates the set of plats i whih the true plat is assumed to exist. This framework is represeted i Fig.. N V i W at W d Geeralized model of the system P Cotroller K z W per W Figure : The augmeted dyami losed-loop system From the perspetive of robust otrol, the suppressio of the vibratio respose of elasti mehaism is to ihibit the effet of the disturbae sigal, suh as modal fore ad measuremet oise, o the output of the system. ordig to otrol theory, that is, the better the otrol performae is, the smaller sesitivity futio is. This is a miimum sesitivity otrol problem. ordig to the model of flexible maipulator, we a obtai the augmeted dyami losed-loop system as show i Fig.. I the Figure, N is the modal fore, is treated as exteral disturbae term, ad Wd is the weighted futio of the sigal. N is the measuremet oise, ad W is the weighted futio orrespodig to the oise sigal. I geeral, i order to fid a otroller to suppress N i desired frequey badwidth y should be weighted by a performae weightig futio matrix. z represets the evaluatio sigal ad W per is the weighted futio of the sigal. V i is the otrol voltage. I additio, the otrol voltage should be weighted by the weightig futio matrix so as to prevet the saturatio of otroller output. I order to limit the voltage rage, the weighted futio Wat is used to pealize the sigal. y is the measuremet output of the system, the output strais from strai gages. Measuremet oise represets eletrial oise, samplig errors, driftig alibratio, ad other effets that impair measuremet preisio ad auray. I the desig of the µ otroller, we a hoose the two differet evaluatio sigals i aordae with the atual physial meaig of the y N 67

5 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: otrolled plat, the first type sigal is the otrolled mode displaemet, ad the other is the output strai sigal from the sesor. d takig ito aout the rage of iput voltage of atuator, the two differet evaluatio sigals z ad z are represeted by the followig form η C z = = X + Vi () Vi Na I Na I where C =, I, uit matrix, respetively. I N a is N a ad Na Na is Na Na zero matrix. The exteral disturbae sigal w iludes the modal fore sigal ad measuremet oise sigal, ad it a deotes [, ] w = N N. Combied equatio() ad equatio(), the state-spae equatio of the geeralized otrolled plat a be writte as follow X = X + V i + N y = CX () η C z = = + i i N X a N V V I a The determiatio of weightig futio W s, W s, W s, W s is a matries ( ) ( ) ( ) ( ) d ruial step i the otroller desig. ordig to the amplitude ad frequey harateristis of modal fore ad measuremet oise sigal, the s W s are set to weightig futio W ( ) ad ( ) respetively where N max fore sigal, ( s) d at Wd = Nmax f W = M d ( s) per (4) is the maximum amplitude of the mode f deotes its haraters i the d frequey domai, M is the amplitude of the measuremet oise sigal. W s is The weightig futio matrix ( ) seleted as to prevet the saturatio of otroller output due to the limits of DS D/ overter output. Suppose the maximum allowable value of the atuatorv max, the frequey harater of the sigal f at, the weightig futio W at ( s) is Wat = fat ( s) (5) Vmax The itrodutio of Wper ( s) is to redue the ifluee of disturbae o sesor outputs. I at geeral, it is hose with large amplitude so as to suppress the low-frequey vibratio. There are two type evaluatio sigals. Oe is modal displaemet ad W per is its weightig futio; the other is output strai ad W per deotes the weightig futio orrespodig to the sigal. They are give by Wper = Pfper ( s) (6) Wper = P fper ( s) where P ad P is the weightig oeffiiet, f per ( s) ad f per ( s) is hose as a oe-order diagoal matrix, ad they a be give by f per ( s) = diag,, s / ω + s / ω + (7) f per ( s) = diag,, s / ωo + s / ωo + where ω ~ ω is hose as the larger value tha the first otrolled mode frequey, ω o is a slightly larger value tha the highest the otrolled atural frequey value. That is, the output sigal whose frequey is less tha ω o a be atteuated uder the atio of the otroller. I geeral, the desig of a µ -otrol system iludes the sythesis of the otroller ad the seletio of weightig futios. The osideratio of the physial system is ruial. The so-alled physial system iludes both the system to be otrolled ad the atuator/sesor ofiguratio. Differet seletios of atuator/sesor loatios lead to very differet sesor/atuator trasfer futios ad aordigly affet the desig of µ otrollers. s is kow, the atuator/sesor ofiguratio determies the otrollability ad observability of the system, whih has diret impat o the modes to be otrolled ad the otrol eergy required. Whe piezoeletri atuators are used, they should be usually loated at regios where the strai deformatios of the most domiat modes are large eough. Ideally, sesors should be olloated (or at least very lose to the atuators) to esure the plat model desribed by the trasfer futio as a miimum phase system i the otrolled frequey bad. The umber of sesors ad atuators to be used depeds o the umber of modes to be otrolled. Combiig equatio()~(7), the trasfer futio of the geeralized otrolled plat P a be writte by the followig form 68

6 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: z w P P w = P = (8) y Vi P P Vi Suppose the otroller is give by Vi = Ky (9) Substitutig equatio (9) ito equatio (8), the losed-loop trasfer futio from w to z a be obtaied l (, ) = + ( ) F PK P PK I PK P () The otroller K a obtaied by solvig two algebrai Riati equatios whih the losedloop system is stable ad the orm of the losedloop trasfer futio a reah to the miimum value. Vi [ ] [ ] F F P K z or z y Figure : lok diagram of µ sythesis augmeted model Sie it is diffiult to establish the exat mathematial model of the dyami system by usig fiite elemet method, the strutural parameters suh as atural frequey, dampig ratio ad the parameter relative to the atuator are ot aurate. ω, ζ ad g are used to represet the uertaity of these parameters, the uertaity of oeffiiet matrix of state equatios are desribed as ( ω, ζ ) = ( ω, ζ ) () g = g ( ) ( ) where (, ) ad ( g ) ω ζ is the atual oeffiiet matrix, ad is the omial value, ( ω, ζ ) ad ( g ) represet the bouded uertaity of these parameters. I order to desig the otroller by usig µ -sythesis method, ad a be desribed as ( ω ζ ) ( ω ζ ) ( ω ζ ) ( g) ( g) ( g), = D, E, () = D E where D, E, D ad E are uertaity strutured ostat matrix, respetively. ( ω, ζ) ad ( g ) is ukow uertaity parameters whih is a bouded perturbatio. ordig to the performae idex itrodued by above part, the µ -sythesis augmeted model is show as Fig.. The dotted lies idiate the hypothetial perturbatio [ ]. [ ] F F is the atual perturbatios. Thus, the diagoal blok p of uertaity matrix is obtaied to ostrut a stadard µ -sythesis framework. F F Δ p = () Obviously, this is the stadard µ -otrol problem, ad the desig a be based o the Matlab µ -toolbox, i whih the D-K iteratio is adopted to perform the sythesis proedure. D-K iteratio is a two-step miimizatio proess: the first step is a miimizatio of the orm over all stabilizig otrollers K while the salig matrix D is held fixed ad seod step is a miimizatio over a set of salig D while the otroller K is held fixed. 4. EXPERIMENTL INVESTIGTION I this setio, we shall experimetally evaluate the effetiveess of the proposed otrol method i the vibratio otrol of a flexible beam. 4. Experimetal setup The legth of the atilever beam is mm, its width ad thikess are mm,.5 mm, respetively. Its material is steel with elasti module, desity ad Poisso s ratio GPa, 78 kg/m,., respetively. The beam is divided ito 7 beam elemets ad 8 geeralized oordiates show as i the Fig. 4. The legths of the elemets are 5 mm, 4 mm, 4 mm, 4 mm, 4 mm, 4 mm, 5 mm, respetively. The symbols N, E,, S deote ode, atuator, sesor, respetively, i the Fig. 4. The atuators are made of PZT-5H piezoeletri erami, whih its thikess is.5 mm, legth is 4 mm ad width is mm, piezoeletri ostat d, elasti 69

7 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: module ad desity is m/v, GPa, 765 kg/m, respetively. The sesors are resistae strai gage. The ofiguratio of the atuators ad sesors are show i Fig.4. The three pairs atuator boded o the lik are loated at elemets E, E 4 ad E 6. Two sesors S, S are loated at the midpoit of elemets E ad E 5. Three atuators,, are loated at elemets E, E 4 ad E 6, respetively. atuator sesor N N S S N N 4 N5 N N N E E 5 E E 4 E E7 E6 Figure 4: Cofiguratio of the elemets ad odes, atuator ad sesor, N, E,, S deote ode, atuator, sesor, respetively (h) (i) (j) (l) (k) (e) (f) Figure 5: Experimetal setup (a) sigal geerator (b) atilever beam () aelerometer (d) dyami sigal aquisitio (e) hammer (f) dyami strai gauge (g) strai gauge (h) piezoeletri path (i) DS oetor pael (j) eletri bridge box (k) voltage amplifier (l) idustrial omputer Fig. 5 illustrates the experimetal setup that osists of a flexible beam boded o PZT ad strai gauge. s the bolt is tighte, the rotatioal pair beomes a fixed pair, that is, the flexible lik a be osidered as a atilever beam. The highspeed aalog iput ad output ports are provided with dspce DS. The eletri amplifier made i Harbi ore tomorrow siee ad tehology Co., Ltd is used to drive piezoeletri path. Resistae strai gauges are made i ZEMIC Co., Ltd, its type is E- (), the resistae is (d) (a) () (b) (g) Ω, ad the sesitivity oeffiiet is.7. The strai gauge is oeted to dyami strai gauge through /4 eletri bridge whih is used to trasform the strai sigal to voltage. 4.. Experimetal modal test Experimetal modal aalysis is a ase of system idetifiatio where a priori model form osistig of modal parameters is assumed. eause it is hard to obtai dampig ratio of struture by fiite elemet method (FEM), the experimetal modal test is a good method to get aurate atural frequey ad dampig ratio, whih provide a basis for adjustmet of the otrol model of the flexible struture. The setup of the experimetal modal test is show as Fig.5. Kistler 869C5-type piezoeletri aelerometer is used as aeleratio sesor. The Zoiook/68E is the dyami sigal aquisitio system. The ez-alyst software is real-time vibratio aalysis software equipped with a dyami sigal aquisitio system whih provides a real-time aalysis apability i the frequey domai ad time domai. ME sopeves software is used to be a post-proessig test whih is apable of aalyzig mehaial ad strutural stati ad dyami harateristis to obtai modal parameters. Impulse hammer method is applied to perform experimetal modal test. To exite the bedig vibratio, the atilever beam was hit with a hammer at the speified exitatio poits. Usig a hammer to produe a wide bad of exitatio, it a exite eah mode i a wider frequey rage. The loatios of hammer ad aelerometers are loated at midpoits of elemet E ad elemets E ~ E 7, respetively. miiaturized aelerometer PC is sequetially plaed at differet loatios. The tap positio of the hammer is fixed, ad the measuremet poits are 7 differet positios. I order to elimiate measuremet oise, the multiple measuremets value of measuremet poit is averaged, the times of measuremet is set to 5. The ez-alyst software is used to ollet data of exitatio poit ad all measuremet poits, to obtai the frequey respose futio of the measurig poit, ad the export the data format whih ME sopeves a support. The ME sopeves software is used to idetify the modal parameters of the flexible beam. The sythesis of the otroller is based o a omial model ostruted by low-frequey modes. I the preset ase, the first four-order mode of the lik is idetified. The first four-order ature frequeies by usig experimetal idetifiatio ad FEM are show i Table, respetively, ad the orrespodig dampig ratios are idetified ad 7

8 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: tabulated i Table. s a be see from the table, relative error of alulated ad experimetal values of atural frequey is lose to -6 %, whih idiates that the fiite elemet model is ot fully osistet i the atual system. That is to say, the model used to desig otroller is uertai. Mode order Figure 6: Fittig urve of frequey respose data Table : The first four-order atural frequey ad dampig ratio 4 Natural frequey (Hz) Calulatio value Experimetal value Relative error Dampig ratio (%) Experimetal results Shemati diagram of vibratio otrol experimet is show as Fig. 7. The priipal of the vibratio otrol is desribed as follows. Whe the exogeous disturbaes are exerted o the beam, the vibratio respose will be geerated. The output of the strai gage is give as iput to the dyami strai gauge whih filters out the oise otets. The oditioed sesor sigal is give as aalog iput ard through the eletri bridge oetor box. The vibratio sigal measured by a sesor is trasformed ito a voltage sigal by a dyami strai trasduer, ad through a low-pass filter ad a /D overter, the aalog voltage sigal is overted ito a digital sigal to the dspce otroller board. d the otrol voltage applied to atuator a be obtaied through the desiged otroller. s the otrol voltage from D/ port is relatively low, the voltage exerted o the piezoeletri path must be amplified by the voltage amplifier to implemet the vibratio otrol. The otrol sigal alulated by the dspce is overted ito a aalog sigal by a D/ overter, ad the is magified 5 times by a voltage amplifier. Sigals are the amplified ad fed to a digital otrol system. The otrol algorithms are implemeted usig dspce DS system with eessary Matlab/Simulik software istalled i a idustrial omputer. The otrol algorithm is implemeted usig Simulik software ad Real Time Workshop (RTW) is used to geerate C ode from the developed Simulik model. The C ode is the overted to target speifi ode by real time iterfae (RTI) ad target laguage ompiler (TLC) supported by DS otroller board. The we a desig a vibratio otrol experimet i real time by usig CotrolDestk software provided by dspce. The otrol objetive is to miimize the output strai of two sesor outputs withi the otrol badwidth uder the exitatio of the disturbae fore idued by exteral fore. Voltage amplifier DS platform Dyami strai gauge S S Exitatio Figure7: Shemati diagram of experimet Exitatio The objetive i the experimetal studies is to otrol the first three vibratio modes. I order to geerate the disturbae fore exerted o the beam, we use the atuator as exitatio soure. The, are applied to vibratio otrol. atuators The gai of the dyami strai gauge is set to, the bridge voltage 6 V, utoff frequey of filter is set to Hz. The samplig frequey of aalog iput ad output port is set to Hz. The ovariae of measuremet oise is estimated as 4. The samplig period of the otroller is ms. The real-time otrol time is set to s. The shemati diagram of the feedbak otrol system is depited i Fig. 7. I order to experimetally ivestigate the otrol performae, the two exitatio soures are exerted to the atilever beam, respetively. The first exitatio soure alled exitatio is the free-vibratio oe by 7

9 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: tapig istatly the free ed of the beam. The seod exitatio soure amed as exitatio is the fored oe by applyig a voltage to the piezoeletri erami path. The exitatio voltage is geerated by the sigal geerator whih geerates a white oise sigal, amplified by voltage amplifier. d the ovariae of the exitatio sigal is set to V. I the followig setio, the otrol effet of the otroller is aalyzed. I order to meet better the situatio of the atual system, the measuremet oise is osidered i the experimetal validatio, ad the sigal is treated as white oise whose amplitude is set to 4. Sie the frequey bad to otrol is betwee ad Hz iludig the first two modes of the struture, the ut-off frequey of low-pass filter of dyami strai gauge is set to 5 Hz. The weightig matrix of performae idex futio show as equatio(4), (5) ad (6) are set as follows where diag ( ) ( ) 4 5,, Wd = W = diag s + Wat = diag (,) 5 s (4) Wper = 5 diag, s + 75 s Wper = 5 diag, s + 9 s + 9 deotes diagoal matrix. The miimum ad maximum value of iput voltage is k = k = set to V ( ) V ad V ( ) V. mi 5 max 5 The effetiveess ad otrol performae of the four otrollers are aalyzed from the time domai poit of view. The otroller sythesis ad aalysis tehiques desribed above are applied to the smart beam. Two otrollers ad two µ otrollers are desiged aordig to above two performae idex. I subsequet setio, the otroller with modal displaemet for evaluatio sigal is alled otroller. The otroller with output strai for evaluatio sigal is alled otroller. The µ otroller with modal displaemet for evaluatio sigal is alled otroller, ad the µ otroller with output strai for evaluatio sigal is amed as otroller4. Figure 8 (a), (b) shows the output strai from two sesors without otroller. Figure 9 ad (a), (b), (), (d) represets the output strai from two sesors with otroller, otroller, otroller ad otroller4, respetively. y omparig Fig.9 ad with Fig.8, we a see that the two output strais from sesors are redued rapidly uder the atio of the four otrollers. The results show that the four kids of otrollers a suppress vibratio respose of the flexible maipulator. y omparig the aalysis of Figure 9, Figure (a), (b), the aalysis results showed that two otrollers have the same otrol effet. Similarly, the two µ otrollers also have the same otrol performae. However, the results of omparative aalysis of Fig.9- (a), (b), (), (d) show that the suppressio of the output strai with µ otroller is faster tha oe with, whih it also idiates the otrol performae of µ otroller is better tha otroller. Figure ad show the two atuator iput voltages orrespod to the otroller,,, 4, respetively. From these diagrams, we a observe that these iput voltages are with the available rage. Through omparative aalysis of the above two iput voltages, the iput voltage of the otroller ad otroller4 are higher tha the otroller ad otroller. This idiates that the µ otroller a osume more eergy to obtai better otrol performae. strai( - ) -.5 time(s) strai( - ) -.5 time(s) (a) strai from sesor S (b) strai from sesor S Figure 8: Strai of from two sesors i ase of without otroller strai( - ) strai( - ) (a) otroller () otroller strai( - ) strai( - ) (b) otroller (d) otroller Figure 9: Strais from sesor S with otroller,, ad 4 7

10 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: strai( - ) strai( - ) (a) otroller () otroller strai( - ) strai( - ) (b) otroller (d) otroller Figure : Strais from sesor S with otroller,, ad 4 otrol voltage (V) otrol voltage (V) (a) otroller () otroller otrol voltage (V) otrol voltage (V) (b) otroller (d) otroller Figure : Cotrol iput voltage for atuator otrol voltage (V) otrol voltage (V) (a) otroller () otroller otrol voltage (V) otrol voltage (V) (b) otroller (d) otroller Figure : Cotrol iput voltage for atuator CKNOWLEDGMENT It is a projet supported by the Natioal Natural Siee Foudatio of Chia (No. 5656), the Natural Siee Foudatio of Jiagxi Provie (No. 69), the Eduatio Fud of Jiagxi Proviial Departmet (No. GJJ58) ad the Researh Fud of Jiagxi Uiversity of Siee ad Tehology(No. jxxj5). REFRENCES: [] S.. Choi, tive strutural aousti otrol of a smart plate featurig piezoeletri atuators, Joural of Soud ad Vibratio, Vol. 94, 6, pp [] K. Ma, daptive oliear otrol of a lamped retagular plate with PZT pathes, Joural of Soud ad Vibratio, Vol. 64, No. 4,,pp [] G. L. C. M. breu, J. F. Ribeiro, ad V. Steffe, Experimets o optimal vibratio otrol of a flexible beam otaiig piezoeletri sesors ad atuators, Shok ad vibratio, Vol.,,pp. 8. [4] J. F. Hu, D. C. Zhu, Vibratio otrol of smart struture usig slidig mode otrol with observer, Joural of Computer, Vol. 7, No.,, pp [5] J. F. Hu, Vibratio Cotrol of a High-Speed Parallel Robot Usig Disrete Variable Struture Cotrol Methodology, Iteratioal Joural of dvaemets i Computig Tehology, Vol. 4, No. 5,, pp [6] X. M. Zhag, C. J. Shao, ad. G. Erdma, tive vibratio otroller desig ad ompariso study of flexible likage mehaism systems, Mehaism & Mahie Theory, Vol. 7,, pp [7] X. M. Zhag, C. J. Shao, ad S. Li, et al, Robust vibratio otrol for flexible likage mehaism systems with piezoeletri sesors ad atuators, Joural of Soud ad Vibratio, Vol. 4, No.,, pp [8] H. Kawabe, N. Tsukiyama, ad K. Yoshida, tive vibratio dampig based o eural etwork theory, Materials Siee ad Egieerig, Vol. 44, 6, pp [9] L. Iorga, H. aruh, ad I. Ursu, Review of robust otrol of piezoeletri smart strutures, ppl. Meh. Rev., Vol. 6, No. 4, 8, pp. -6. [] X. Y. Wag, K. M. James, S. X. Guo, Experimetal idetifiatio ad ative otrol of ofiguratio depedet likage vibratio i a plaar parallel robot, IEEE Trasatios o Cotrol Systems Tehology, Vol. 7, No. 4, 9, pp

11 Joural of Theoretial ad pplied Iformatio Tehology th May. Vol. 5 No. 5 - JTIT & LLS. ll rights reserved. ISSN: E-ISSN: [] N. el Hadj li, I.F.C. Smith, Dyami behavior ad vibratio otrol of a tesegrity struture, Iteratioal Joural of Solids ad Strutures, Vol. 47, No. 9,, pp