Finite Element Simulation of Smart Lightweight Structures for Active Vibration and Interior Acoustic Control

Transcription

1 ECHNISCHE MECHANIK, Bnd 3, Heft 1, (3), Mnuskripteingng: 3. Februr 3 Finite Element Simultion of Smrt Lightweight Strutures for Ative Vibrtion nd Interior Aousti Control J. Lefèvre, U. Gbbert Dedited to Dr.-Ing. Friedrih Whl on the osion of his 65 th birthdy. he pper presents numeril pproh to tive vibrtion nd noise ontrol of smrt lightweight strutures. he struture is provided with thin piezoeletri wfers s tutors nd sensors to ontrol vibrtions of the struture. Fully oupled eletromehnil field equtions were tken into ount where model bsed ontrollers re pplied for design purposes. he objetive of vibrtion ontrol of elsti strutures is to redue interior noise levels. Hene, the mehnil field is lso oupled with the ousti field, nd onsequently fully oupled eletro-mehnil-oustil problem needs to be solved. he numeril solution is bsed on the finite element method, introduing veloity potentil for the ousti fluid to reeive overll symmetri system mtries of the semi-disrete form of the eqution of motion. It is shown tht the vibro-ousti oupling n be negleted for ontroller design purposes, nd onsequently the modl truntion tehnique onsidering only the unoupled struturl modes, n be dpted to vibro-ousti systems. he behviour of smrt plte struture oupled with n ousti vity is studied s referene exmple. 1 Introdution Over the pst few yers smrt struturl onepts hve met with growing interest in mny engineering brnhes nd severl new pprohes nd tehnologies hve been developed (zou&gurn, 1998; Gbbert, ). Smrt strutures, or strutroni systems, re hrterized by synergisti integrtion of tive mterils into pssive struture onneted by ontrol system to filitte utomti djustment to hnging environmentl onditions. Piezoeletri mterils re widely used s distributed sensors nd tutors in smrt strutures. Commerilly vilble piezoeletri wfers re ommonly used s tive mterils for ontrolling struture vibrtions. Suh thin wfers my be glued on the surfe of the bse struture, or embedded in omposite mteril during the mnufturing proess. Intensified tivities in developing nd pplying piezoeletri smrt strutures require effetive nd relible design tools. he finite element method (FEM) is n exellent bsis to develop powerful softwre tools. Being widely spred, it hs beome theoretilly nd prtilly estblished method for solving oupled piezoeletri field problems. Bsed on the generl purpose finite element softwre COSAR (COSAR, 199), tool hs been designed nd grdully implemented over the pst few yers (Berger et l., ; Gbbert et l., ). he softwre pkge ontins n extended librry of multi-field finite elements for 1D, D nd 3D ontinu s well s for shell-type thin-wlled strutures (Seeger et l. ; Gbbert et l., b). he COSAR FEM softwre is linked with ontroller design tools, suh s MtLb/Simulink, through generl dt interfe, designing the ontroller on the bsis of finite element models (Gbbert et l. 1; Gbbert et l., ). Sound rdition is nother mjor problem in designing engineering strutures. Whenever n elsti struture gets into ontt with surrounding fluid, struturl vibrtions nd the ousti pressure field will influene eh other. his vibro-ousti oupling results in dditionl pressure lods on the struture-fluid interfe used by fluid pressure, s well s unwnted noise rdition used by struturl vibrtions. Strong vibro-ousti oupling effets our in thin-wlled lightweight strutures of lrge surfe. Ative noise ontrol by pplying piezoeletri pth tutors to the struture is n lterntive wy to redue noise rdition of elsti strutures (see Blhndrn et l., 1996; Ro&Bz, 1999; Kim et l., 1999; Gopinthn et l.,, nd others). Only reently, we extended our softwre tool COSAR by inluding new brik-type finite elements for disretizing ousti volumes (Lefèvre, ). he veloity potentil of the fluid ws onsidered s n dditionl nodl degree of freedom in order to reeive symmetri system mtries. Also, the vibro-ousti oupling effet ws tken into ount. Now, it is possible to solve the fully oupled eletro-mehnil-ousti field problem on the bsis of finite element disretiztion of the eletromehnil struture nd the fluid. hermopiezoeletri effets s disussed by Görnndt et l.,, re negleted. Further below, the theoretil bsis of this finite 59

2 element pproh is presented, followed by disussion of the ontroller design for vibro-ousti problems. As lrge-sle finite element models nnot be used for ontroller design purposes, n pproprite model redution is required to redue the number of the degrees of freedom. Here, modl truntion seems to be the best suited method for estblishing the ontroller design of strutures bsed on the finite element disretiztion s flexible strutures exhibit low-pss hrteristis, whih llow to neglet high-frequeny dynmis (Strßberger, 1997). Bsed on seleted eigenmodes the finite element model is redued nd trnsformed into the stte spe form. All required dt of the modl stte spe model n be trnsferred to the ontroller design tool (MtLb/Simulink) vi generl dt exhnge interfe to design suitble model-bsed ontroller. o study the ontrolled behviour of the struture, the ontroller mtries n be retrnsferred to the originl finite element model. As shown in the pper, dequte results n be reeived for ontroller design purposes by tking into ount only the eigenmodes of the unoupled struture. his proedure hs been pplied to interior ousti problems with strong fluid-struture intertion. he simultion of smrt retngulr elsti plte struture oupled with n ousti vity is investigted s referene exmple. Governing Equtions nd Finite Element Anlysis his setion presents the theoretil bkground of our finite element softwre tool for simulting smrt strutures oupled with n ousti fluid. First, the finite element nlysis of oupled eletromehnil strutures nd ousti fluids re presented seprtely, followed by demonstrtion how both pprohes re oupled to reeive n overll vibro-ousti finite element model. he following equtions re bsed on Crtesin (x 1,x,x 3 )-oordinte system..1 Finite Element Model of Piezoeletri Smrt Strutures he oupled eletromehnil behviour of polrizble piezoeletri mteril in low voltge pplitions n be modelled with suffiient ury by mens of linerized onstitutive equtions. Furthermore, smll displements re onsidered. he omplete derivtion of the finite element model is disussed in detil in Gbbert et l.(). Hene, only brief summry of the equtions is presented here. In three-dimensionl ontinuum the finite element equtions re bsed on the mehnil equilibrium D p = ρ u, (1) u the eletri equilibrium (4 th Mxwell eqution) D ϕ D = () nd the liner oupled eletromehnil onstitutive equtions = C ee, (3) D = e (, (4) with the stress vetor = [ σ11 σ σ 33 σ1 σ 3 σ 31], the body fore vetor p = [ p1 p p3] displement vetor u = [ u1 u u3] = [ D1 D D3] [3 3], the strin vetor = [ ε11 ε ε33 ε1 ε 3 ε 31] E = [ E E ]. D u nd D ϕ desribe the following differentil mtrix opertors, the, the mss density ρ, the vetor of eletril displements D nd the elstiity mtrix C[6 6], the piezoeletri mtrix e [6 3], the dieletri mtrix 1 E3 nd the eletri field vetor nd x1 x x3 D u = (5) x x1 x3 x3 x x1 6

3 D ϕ =. (6) x1 x x3 ogether with stress boundry onditions nd the hrge boundry onditions, the mehnil nd the eletri blne equtions n be written in wek form s δu ( D ) dv ( u p ρu δϕ Dϕ )dv χ = D V δ ( q q) δϕ( Q Q) do = V O q u do (7) where δu is virtul displement, δϕ is virtul eletri potentil, q is given trtion vetor on the surfe O q nd Q is given hrge on the surfe O Q. In eqution (7) the strin-displement reltion = D u u (8) nd the reltion between the eletri field nd the eletri potentil E = D ϕ ϕ, (9) re introdued. he displements u 1, u, u 3 nd the eletri potentil ϕ re pproximted elementwise by shpe funtions, ontining the element nodl prmeters s unknown prmeters. Inluding the shpe funtions of finite element in the mtries G u nd G ϕ nd inorporting the unknown nodl prmeters in the vetors w nd, the pproximtion of the displement vetor u nd the eletri potentil ϕ n be written s ( x x, x ) G ( x, x x ) w u 1 3 u 1, 3, =, (1) ( x, x ) ( x, x x ) ϕ x1, 3 = Gϕ 1, 3. (11) Following the stndrd finite element proedure, the semi-disrete form of the equtions of motion of generl element (e) n be derived from eqution (7) s follows M uu w Kuu Kuϕ w fu () () () () =. (1) e e e e Kuϕ Kϕϕ fϕ he vetor w ontins the nodl displements nd the vetor ontins the nodl eletri potentils; M uu is the element mss mtrix, K uu is the element stiffness mtrix, K ϕϕ is the element eletri mtrix, K uϕ is the piezoeletri oupling mtrix, f u is the lod vetor resulting from mehnil lods nd f ϕ is the lod vetor resulting from eletri hrges.. Finite Element Model of the Aousti Fluid Aousti responses in fluid re usully regrded s smll perturbtions relted to n mbient referene stte. Hene, the finite element model of the fluid is derived from the liner ousti wve eqution (Fhy, 1994) 1 p = p, (13) onsidering the dibti wve propgtion in homogeneous, invisid fluid. In eqution (13) is the Lplin opertor, p is the ousti pressure nd is the sound speed. Although we re minly interested in sound pressure distribution, the use of pressure p s nodl vrible is rther disdvntgeous s in this se the finite element formultion of the oupled vibro-ousti problem results in un-symmetri system mtries (Desmet&Vndepitte, 1999). Everstine (1997) reommended the introdution of the veloity potentil Φ of the fluid s new degree of freedom in order to obtin symmetri mtries. he veloity potentil is slr field relted to the veloity v of the fluid prtiles by O Q 61

4 v = D Φ (14) nd to the sound pressure by p = ρ Φ, (15) with the differentil mtrix opertor D, hving the sme form s the opertor in eqution (6), nd the fluid density ρ (Kollmnn, ). Introduing equtions (14) nd (15) into eqution (13) results in the new form of the ousti wve eqution 1 Φ = Φ. (16) he next step onsiders n interior ousti problem, i.e. the whole fluid volume is enlosed by boundries in ll diretions with finite distne from ommon referene point. o obtin the wek form of the ousti wve eqution, eqution (16) is multiplied by ny test funtion τ nd integrted over the fluid volume V i. Using the Gussin integrl theorem the wve eqution is derived in wek form s 1 τ D D Φ dv τ Φ dv = ( τ D Φ) n do Vi Vi Oi, (17) where n is the outwrd direted norml vetor of the surfe O i of the volume V i. he right hnd side of eqution (17) orresponds to the boundry onditions. O i n be divided into the surfes O v with given norml veloity v n nd the surfes O z with n imposed impedne funtion Z n. he impedne funtion desribes the reltionship between the ousti pressure nd the norml omponent of the veloity p Z n =. (18) v n Following this proedure, it is possible to model boundry dmping effets (Kim et l., 1999) nd with the equtions (14) nd (15) eqution (17) n be written s 1 ρ D τ Φ do. (19) Z τ DΦ dv τ Φ dv = τ vn do Vi Vi O n v Oz In finite element the veloity potentil Φ nd the test funtion τ re pproximted by using the vetors of the nodl degrees of freedom, nd the mtrix G ontining the pproximte funtions s nd ( ) ( ), x, x = x, x x Φ x G 3 () 1 3 1, (, x, x ) = ( x, x x ) τ x G 3. (1) 1 3 1, With the bbrevition B = D G, () nd the equtions () nd (1) the eqution (19) n be written s 1 G G dv ρ Z () () () () G e e e e G G do B B dv = n Vi Oz Vi Ov v do. (3) n As this eqution must to be fulfilled for ny prmeter of the test funtion from eqution (3) we reeive M C K = f, (4) 6

5 with the ousti element mss mtrix 1 M = V i the ousti element dmping mtrix ρ = Z C G n O z the ousti element stiffness mtrix G G dv, (5) G do, (6) K = B B dv (7) V i nd the ousti element lod vetor f = G v do. (8) O v.3 Vibro-Aousti Coupling n First, it is ssumed tht the ousti pressure represents n dditionl lod with respet to the struture. Developing our finite element model, we must lso onsider n dditionl lod vetor f () u e resulting from the ousti pressure. Consequently, eqution (1) n be ugmented to M uu w K uu Kuϕ w fu fu () () () () =. (9) e e e e Kuϕ Kϕϕ fϕ With eqution (15) the vetor f n be lulted s u fu = Gu npdo = Gu ng do = C ρ u, (3) O O S S where O s is the fluid struture interfe nd C u is the oupling mtrix with regrd to the ousti pressure. Furthermore the veloity of the vibrting struture ts on O s s new ousti lod of the fluid M C K = f f. (31) Considering tht the norml veloity of the struture n be expressed by u = n G w, (3) n u we reeive n expression similr to eqution (8) f = G n G do w = C w, (33) O S u with C s the oupling mtrix regrding struturl vibrtions. Compring eqution (3) nd eqution (33), symmetri system mtries re reeived by multiplying ll lines relted to fluid degrees of freedom by (-ρ ). herefore nd when the element mtries re inluded in globl mtrix nd the oupling mtries re inserted 63

6 in the left hnd side, the semi-disrete system of equtions of the eletro-mehnil-ousti field problem n be written s M uu w Cuu Cu w M ρ C C u ρ Kuu Kuϕ w fu = Kuϕ Kϕϕ fϕ ρ K ρ f. (34) In eqution (34) the Ryleigh dmping hs been introdued s the struturl dmping C uu. Mention should be mde of the ft tht the vetor w lso represents the nodl displements of the pssive prt of the struture. Bsed on eqution (34) two ousti brik-type elements with 8 nd nodes of the Serendipity element fmily hve been developed nd implemented in our COSAR softwre tool. he isoprmetri element onept ws used to properly pproximte the element geometry. Furthermore, lso the vibro-ousti oupling proedures s presented in equtions (9)-(3) were inorported in the COSAR softwre tool. ogether with initil onditions eqution (34) n be integrted numerilly, e.g. by using the Newmrk formuls. 3 Control of Smrt Strutures Numeril simultions of smrt strutures under the finite element onept require n overll finite element model omprising the pssive struture, the tive sensor nd the tutor elements s well s suitble model of the ontroller. ody, omprehensive design tools suh s MtLb/Simulink re vilble to support the designing proess. A generl dt exhnge interfe is required to exhnge dt nd informtion between the finite element model nd the ontroller design tool. Only reently, suh dt interfe ws developed to ouple our finite element COSAR softwre with MtLb/Simulink (see Gbbert et l., 1, ). 3.1 Controller Design For ontroller design purposes the vibro-ousti oupling effet is negleted in the tive struture. Combining the displement nd the eletri potentil degrees of freedom in one vetor w = [ w ] eqution (1) n be written s, (35) () t Bu () t M w Cw Kw = Ef, (36) r where C is n dditionl dmping mtrix nd u r (t) is the vetor of the ontroller influene of the struture. he mtries E nd B desribe the positions of the externl fores nd the ontroller prmeters in the finite element model, respetively. o use the modl truntion tehnique the liner eigenvlue problem ( K M) wˆ = ω, (37) k k =, where m is the number of eigen- needs to be solved. he result is the modl mtrix Q [ wˆ wˆ... wˆ 1 m ] modes onsidered. Ortho-normlizing Q with Q MQ = I = dig() 1 nd Q KQ = = dig( ω ) Q CQ = = dig ( δ ω ) k k, onsidering proportionl dmping. Introduing modl oordintes qˆ s k, we lso obtin w = Qqˆ (38) into eqution (36), we obtin the modl trunted system of (m m) differentil equtions () t Q Bu () t q ˆ qˆ qˆ = Q Ef r. (39) 64

7 Eqution (39) is trnsformed into the stte spe form, whih is more onvenient for the ontrol theory. Introduing the stte spe vetor qˆ z = q ˆ (4) in eqution (39) results in I z = z ur() t f() t = Aqz() t Bqur() t Eqf() t. (41) Q B Q E In smrt strutures vriety of signls n be mesured, first of ll eletri potentils by using piezoeletri sensors; but lso other signls, suh s exiting fores or displements, n be mesured in order to hrterize the stte spe of the struture. Hene, the mesurement mtrix is used in generl form s () t C z D u () t F f() t y =, (4) q q r q where C q, D q nd F q re mpping mtries desribing the reltions between the mesured quntities of the vetors z, u r nd f with respet to the mesuring vetor y. he mtries A q, B q, E q, C q, D q, F q re trnsferred to MtLb/Simulink where the ontroller is designed. Designing time independent LQ ontroller, the tutor signl is generted by the ontroller mtrix R s u r () t Rz() t 3. Solution Conept =. (43) he semi-disrete form of the oupled vibro-ousti eqution of motion inluding ontrol re s follows M w C Cu M ρ Cu ρc K w w Ef = ρ K ρ () t Bur () t f () t. (44) C u is the oupling mtrix with respet to the vetor w. o solve eqution (44), the tutor signl (43) must be expressed in terms of w nd w, respetively. If only few seleted eigenmodes re onsidered, Q is not squre mtrix. o get the modl oordintes from eqution (38), the pseudo-inverse mtrix of Q, 1 ( Q Q) Q Q =, (45) is required. With eqution (45) the stte spe vetor hs the following form Q z = Q w. (46) w Mention should be mde tht the use of Q in eqution (46) does not provide n ext solution with respet to the modl oordintes. However, if the motion of the struture is dominted by the seleted eigenmodes, there is very good greement between the pproh in eqution (46) nd the ext solution s we ould estblish. Following the integrtion proedure of the initil finite element model, the tutor signl u r n be inluded s n dditionl fore vetor in the right hnd side of eqution (44). 4 Exmples his setion ontins numeril demonstrtion of the ft tht ontroller designed with modl redued system s bove n be pplied with suffiient ury to fully oupled vibro-ousti system. he finite element 65

8 model of lmped retngulr plte struture oupled with n ousti vity is used s test exmple (Fig. 1). Four piezoeletri pthes tthed to the plte s two olloted tutor/sensor pirs (tutors on the top nd sensors on the bottom of the plte) re employed. he struture is exited by hrmoni fore t given point. All dimensions nd mteril properties of the plte, the ousti vity nd the tutors/sensors re given in bles 1-4. he plte is pproximted by 96 pssive hexhedron elements ontining only mehnil degrees of freedom; the vity is disretized with 384 ousti hexhedron elements. Eh tutor or sensor is pproximted by oupled piezoeletri hexhedron element ontining mehnil s well s eletril degrees of freedom. In ll these finite elements qudrti shpe funtions re used. he Newmrk formuls re employed for numeril time integrtion of the equtions of motion. sensor/tutor pirs F(t) x h l 3 x 1 x 3 l 1 l Figure 1. Smrt Plte Struture Coupled with n Aousti Cvity. Plte/Cvity System Atutors/Sensors l 1 6mm Length in x 1 -diretion 1mm l 4mm Length in x -diretion 5mm l 3 4mm Pth thikness.mm h (Plte thikness) mm ble 1. Geometry of the Vibro-ousti System Inluding Atutors nd Sensors. Elsti Constnts Piezoeletri Constnts Dieletri Constnts N/mm² 33 14N/mm² e N/(mV)mm κ N/(mV)² 1 631N/mm² N/mm² e N/(mV)mm κ N/(mV)² N/mm² 66 N/mm² e N/(mV)mm Density ρ Ns²/mm 4 ble. Mteril Properties of the Piezoeletri Atutors nd Sensors. Young s modulus E 1N/mm² Poisson s rtio ν.3 Density ρ P Ns²/mm 4 ble 3. Mteril Properties of the Elsti Plte. Speed of sound 34mm/s Fluid density ρ Ns²/mm 4 ble 4. Mteril Properties of the Aousti Fluid. Number Frequeny Hz 9.9Hz Hz Hz Hz ble 5. Eigenfrequenies of the Unoupled Elsti Plte. 66

9 In ll simultions fore exittion with n mplitude of 1N nd frequeny ontining the first eigenfrequeny (see b. 5) ws pplied, whih is very importnt exittion regrding the influene to the ousti pressure in the enlosure. In the simultions the ontroller ws swithed on fter period of 1s. A LQ ontroller for the elsti plte ws designed on the bsis of redued model ontining the first five struturl modes (see b. 5). o verify the reeived ontroller mtrix, the response of the plte ws simulted without ny vibro-ousti oupling. Fig. displys the resulting signl of the sensor. 8.E3 6.E3 4.E3.E3.E M-.E3-4.E3-6.E3-8.E sensor signl t [s] Figure. Sensor Signl of the Unoupled Plte Resulting from Hrmoni Exittion. [mv] As n be seen in Fig., the designed ontroller redues struturl vibrtions to pproximtely qurter of its mximum mplitude. he seond simultion dditionlly onsidered the vibro-ousti oupling. For this simultion the pressure relese ondition (p=) ws pplied to ll boundries of the fluid in ddition to oupling with n elsti plte, s inluded in the oupling equtions (3) nd (33). Fig. 3 displys the sensor signl. [mv] 8.E3 6.E3 4.E3.E3.E M-.E3-4.E3-6.E3-8.E sensor signl t [s] Figure 3. Resulting Sensor Signl with Respet to the Coupled Vibro-ousti System under the Assumption of Boundry Pressure Relese Conditions. A omprison between Fig. nd Fig. 3 revels tht sound pressure does not exert strong influene on the elsti plte when pressure relese onditions re ssumed t the boundries. For the lst simultion we ssumed tht the pressure relese ondition only ts on the boundry x 3 =l 3. Aprt from the plte ll other four boundries of the vity were onsidered s oustilly hrd surfes (v n =). Fig. 4 displys the signl reeived by the sensors. Fig. 5 shows the orresponding sound pressure in the entre of the vity. Fig. 4 nd Fig. 5 disply strong oupling effet. he oupled fluid serves s n dditionl dmping with respet to the vibrting plte. Furthermore, the motion of the plte is no longer only dominted by the first unoupled eigenmode. he reson is the ousti fluid ltering the eigenfrequenies nd eigenmodes of the whole system. [mv] 8.E3 6.E3 4.E3.E3.E -.E3-4.E3-6.E3-8.E sensor signl t [s] M Figure 4. Resulting Sensor Signl of the Coupled Vibro-ousti System Considering Aoustilly Hrd Boundries. 67

10 p [N/mm²] 5.E-6 4.E-6 3.E-6.E-6 1.E-6.E -1.E-6 -.E-6-3.E-6-4.E-6-5.E t [s] sound pressure Figure 5. Resulting Sound Pressure in the Cvity Centre when Considering Aoustilly Hrd Boundries. As n be seen in ll simultions, the ontroller redues the sensor signls to lmost the sme level. In our opinion the fluid-struture intertion n be negleted for ontroller design purposes if there is no strong oupling between the struture nd the fluid. However, we often enounter problems where strong vibro-ousti oupling ours. Bsilly, the intertion of fluid nd struture needs to be onsidered for ontroller design whenever lrge surfe re of thin-wlled lightweight struture is overed by fluid. his is lso of gret importne when suh models re used for lulting optiml tutor nd sensor positions t the struture. In suh ses the modl truntion should be bsed on the fully oupled eletro-mehnil-oustil eigenvlue problem. 5 Conlusions he pper presents the theoretil bkground of new finite element-bsed softwre tool for solving threedimensionl eletro-mehnil-oustil field problems, inluding ontrol lgorithms. his tool ws inorported in our finite element nlysis softwre COSAR, filitting simultions of the ontrolled behviour of oupled vibro-ousti systems with distributed piezoeletri tutors nd sensors with regrd to the interior ousti rdition problem. A generl dt exhnge interfe is used to design ontrollers by MtLb/Simulink. Furthermore, numeril investigtions were performed whether the modl truntion tehnique using the unoupled struturl modes for ontroller design n be dpted to vibro-ousti systems. If there is no strong oupling between the fluid nd the struture this method yields suffiient results. Otherwise, the eigenmodes of the ompletely oupled system need to be onsidered. Aknowledgement his work hs been supported by the postgrdute progrm of the federl stte of Shsen-Anhlt. his support is grtefully knowledged. Referenes Blhndrn, B., Smprth, A., Prk, J.: Ative ontrol of interior noise in three-dimensionl enlosure, J. of Smrt Mterils nd Strutures, Vol. 5, (1996), pp Berger, H., Gbbert, U., Köppe, H., Seeger, F.: Finite Element Anlysis nd Design of Piezoeletri Controlled Smrt Strutures, J. of heoretil nd Applied Mehnis, 3, 38, (), pp COSAR Generl Purpose Finite Element Pkge: Mnul, FEMCOS mbh Mgdeburg (see lso (199). Desmet, W., Vndepitte, D.: Finite Element Method in Aoustis, Pro. of the 1 th Interntionl Seminr on Advned ehniques in Applied nd Numeril Aoustis, Leuven, Belgium, (1999). Everstine, G.C.: Finite Element Formultions of Struturl Aousti Problems, J. Computers & Strutures, Vol. 65, No. 3, (1997), pp

11 Fhy, F.: Sound nd Struturl Vibrtion. Rdition, rnsmission nd Response, Ademi Press, London, (1994). Gbbert, U., Berger, H., Köppe, H., Co, X.: On Modelling nd Anlysis of Piezoeletri Smrt Strutures by the Finite Element Method, J. of Applied Mehnis nd Engineering, Vol. 5, No. 1, (), pp Gbbert, U., Köppe, H., Seeger, F.: Overll design of tively ontrolled smrt strutures by the finite element method, SPIE Proeedings Series, Vol. 436, (1), pp *DEEHUW8 SSH1HVWRURYLü7UDMNRY7(QWZXUILQWHOOLJHQWHU6WUukturen unter Einbeziehung der Regelung. t Automtisierungstehnik 5, 9, (), pp Gbbert, U., Köppe, H., Seeger, F., Berger, H.: Modeling of smrt omposite shell strutures, J. of heoretil nd Applied Mehnis, 3, 4, (b), pp Gbbert, U.: Reserh tivities in smrt mterils nd strutures nd expettions to future developments, J. of heoretil nd Applied Mehnis, 3, 4, (), pp Görnndt, A., Gbbert, U. (): Finite Element Anlysis of hermopiezoeletri Smrt Strutures. At Mehni, 154, (), pp Gopinthn, S. V., Vrdn, V. V., Vrdn, V. K.: Finite Element/Boundry Element Simultion of Interior Noise Control Using Ative-Pssive Control ehnique, SPIE Proeedings Series, Vol. 3984, (), pp. -3. Kim, J., Ko, B., Lee, J.-K., Cheong, C.-C.: Finite element modeling of piezoeletri smrt struture for the bin noise problem, J. of Smrt Mterils nd Strutures, Vol. 8, (1999), pp Kim, J., Lee, J.-K., Im, B.-S., Chung, C.-C.: Modeling of Piezoeletri Smrt Strutures Inluding Absorbing Mterils for Cbin Noise Problems, SPIE Proeedings Series, Vol. 3667, (1999), pp Kollmnn, F. G.: Mshinenkustik. Grundlgen, Meßtehnik, Berehnung, Beeinflussung, Springer-Verlg, Berlin-Heidelberg, (). Lefèvre, J.: Modellierung und Simultion geregelter Leihtbustrukturen zur ktiven Reduktion der Shllbstrhlung, Diplomrbeit, Otto-von-Guerike-Universität Mgdeburg, Institut für Mehnik, (). Ro, J., Bz, A.: Control of sound rdition from plte into n ousti vity using tive onstrined lyer dmping, J. of Smrt Mterils nd Strutures, Vol. 8, (1999), pp Seeger, F., Gbbert, U., Köppe, H., Fuhs, K.: Anlysis nd Design of hin-wlled Smrt Strutures in Industril Applitions, SPIE Proeedings Series, Vol. 4698, (), pp Strßberger, M.: Aktive Shllreduktion durh digitle Zustndsregelung der Strukturshwingungen mit Hilfe piezo-kermisher Aktoren, Mitteilungen us dem Institut für Mehnik, Ruhr-Universität Bohum, (1997). zou, H.-S., Gurn, A. (Eds.): Strutroni Systems: Smrt Strutures, Devies nd Systems, Prt 1: Mterils nd Strutures, Prt : Systems nd Control, World Sientifi, (1998). Address: Dipl.-Ing. Jen Lefèvre, Prof. Dr.-Ing. hbil. Dr. h.. Ulrih Gbbert, Otto-von-Guerike-Universität Mgdeburg, Universitätspltz, D-3916 Mgdeburg, e-mil: jen.lefevre@mb.uni-mgdeburg.de, ulrih.gbbert@mb.uni-mgdeburg.de 69