Finite Element Modelling of Vibro-Acoustic Systems for Active Noise Reduction

Transcription

1 ECHNISCHE MECHANIK, Band 25, Heft 3-4, (25), Manuskripteingang: 11. Dezember 25 Finite Element Modelling of Vibro-Aousti Systems for Ative Noise Redution J. Lefèvre, U. Gabbert Over the past years a lot of sientifi work has been done in the field of smart lightweight strutures to redue the strutural vibrations and the radiated sound. In the paper a virtual overall model of adaptive vibro-aousti systems, ompletely based on the finite element method (FEM), is presented. Beside the passive struture, this model ontains the ative piezoeletri elements, the aousti fluid, the vibro-aousti oupling, and the ontroller influene. Aording to the requirements of an effetive numerial analysis of thin walled strutures, the oupling between ative layered shell elements and aousti 3D hexahedron elements is implemented. It is possible to take into onsideration the interior as well as the exterior radiation problem. Beause of the large number of degrees of freedom of the FE model, a modal trunation tehnique based on a omplex eigenvalue analysis is performed. After transforming the model into the state spae form, the Matlab/Simulink software is used to design an appropriate ontroller. o show the auray and the performane of the developed software approah, a vibrating elasti plate and the resulting sound field are numerially investigated. 1 Introdution he investigation of smart strutural onepts strethes aross a wide range of appliations, suh as vibration suppression, noise attenuation, shape ontrol, damage detetion and others (see e.g. Gabbert, 22 and zou, 1998). In many ases strutures are atively influened by applying piezoeletri materials as distributed atuators and sensors onneted with an appropriate ontrol unit. he development and the design of suh smart strutures require powerful numerial analysis and simulation tools as well as suitable models inluding the main funtional parts of the system under investigation. In vibro-aousti systems the model should inlude the passive struture, the aousti fluid, the piezoeletri sensors and atuators and the ontrol algorithms. Suh an overall virtual model an be established on the basis of the finite element method, whih has to inlude the oupled eletro-mehanial fields of the piezoeletri materials as well as the ontrol (see Gabbert et al., 2, 22). Furthermore, the fluid-struture interations have to be taken into aount (see Everstine 1971). In order to solve suh multi-field vibro-aousti systems the finite element software COSAR has been extended by speial 1D, 2D, 3D and layered shell-type elements with oupled mehanial and eletrial degrees of freedom and brik-type finite aousti elements as well as semi-infinite aousti elements to study interior and exterior radiation problems, respetively. his results in large-sale finite element models of suh overall vibro-aousti smart systems, whih are in general infeasible for ontroller design purposes. herefore, model redution tehniques have to be applied to redue the number of degrees of freedom (see Gabbert et al., 22). he low frequeny range modal trunation seems to be an appropriate tehnique. But, in vibro-aousti systems a omplex eigenvalue problem has to be solved to perform an appropriate model redution. his requires a higher numerial effort as the solution of a positive definite eigenvalue problem. On the other hand the redued model an also be used to alulate the transient behavior of the system in a very effiient and fast way. For ontroller design purposes there are speial software pakages available, suh as Matlab/Simulink, whih an be oupled with the finite element pakage COSAR by a speial bi-diretional data interfae. he overall equations of motion in the time domain, generated by the FE software COSAR, are transformed into the state spae form and transferred to Matlab/Simulink through a data exhange interfae. In the paper the proedure is briefly explained and as a test example the ontrol of sound radiated from a vibrating plate is numerially studied. 2 Basi Equations and Finite Element Analysis he theoretial bakground of the simulation of piezoeletri smart strutures and aousti fluids is briefly presented here. All equations are developed in a Cartesian (x 1, x 2, x 3 )-oordinate system. 241

2 2.1 Finite Element Formulation of Piezoeletri Smart Strutures he derivation of a finite element formulation for the analysis of piezoeletri strutures is a well known proedure, published in a lot of papers during the last years (for an overview see Gabbert et al., 2). Based on the linear oupled eletromehanial onstitutive equations σ = Cε ee, (1) D = e ε + κe, (2) with the stress vetor σ, the vetor of eletri displaements D, the elastiity matrix C, the piezoeletri matrix e, the dieletri matrix κ, the strain vetor ε and the eletri field vetor E, the semi-disrete equations of motion of a system disretized by finite elements an be written as M w&& C + ϕ && w& K + ϕ& K wϕ K wϕ K ϕϕ w f = ϕ f w ϕ. (3) In equation (3) w ontains the nodal mehanial degrees of freedom, ϕ the nodal eletri potentials. M is the mass matrix, C the damping matrix, K the stiffness matrix, K ϕϕ the eletri matrix, K wϕ the piezoeletri oupling matrix, f w the mehanial load vetor and f ϕ the eletri load vetor. 2.2 Finite Element Formulation of the Aousti Fluid he homogeneous and invisid aousti fluid is modeled by using the linear aousti wave equation (see Kollmann, 2) 1 & Φ Φ =, (4) 2 onsidering the veloity potential Φ as a nodal degree of freedom. he veloity potential Φ is related to the fluid partile veloity v by v = Φ (5) and to the sound pressure p by p = ρ Φ &. (6) he Nabla operator is defined as = x x 1 2 x3. (7) Similar to the priniple of virtual displaements it is possible to formulate a priniple of virtual fluid potentials (Olson and Bathe, 1985). For these purposes equation (4) is multiplied by δφ and integrated over the entire volume. After applying the Gaussian integral theorem and onsidering imposed normal veloities v n and impedane funtions Z as boundary onditions, the following result is obtained: = 1 ρ χ = 2 δφ & Φ dv δφ Φ& do δφ Φ dv δφvn do (8) Z V f OZ V f Ov with Φ = Φ on O Φ, (9) 242

3 Furthermore, the whole fluid volume V f is divided into the inner and the outer region V i and V o. Both volumes are onneted at their interfae O k by the normal veloity boundary ondition. In order to take into aount the influene of the outer fluid region, whih extends to infinity, we follow the Doubly Asymptoti Approximation (DAA, Geers 1978). he behavior of the outer fluid is onsidered only in the low and high frequeny range. At low frequenies the fluid of the outer region is assumed to be inompressible. In the high frequeny range, the plane waves are onsidered. By superimposing these two effets the following equation is obtained χ = ρ δφ & Φ dv δφ Φ& 1 do Z 1 2 V f Τ OZ δφ Φ dv δφv Vo O v n Ok δφ Φ& dv Vi δφ Φ dv do =. (1) Following a standard finite element proedure with approximate funtion for the fluid potential Φ, the matrix equation of the aousti fluid is derived as a ( Ca + C I ) Φ& + ( K a + K I ) f a M Φ & + Φ =, (11) with the aousti mass matrix M a, the aousti damping matrix C a, the aousti stiffness matrix K a, the aousti load vetor f a and the matries C I and K I, whih haraterize the outer fluid. It is important to remark that for the alulation of K I a so-alled semi-infinite element has been developed (see Bettess, 1992), whih enables the modeling of domains having an infinite boundary at least in one diretion. 2.3 Vibro-aousti Coupling he vibro-aousti oupling effet results in additional loads that at on the fluid-struture interfae O S. here is load vetor due to the sound pressure f Φ & Φ& w = N wnpdo = ρ N wnn ado = Cw (12) OS OS and the load vetor due to the strutural vibrations f a 1 = Nan N wdo w& = C&. (13) ρ OS he matries N w and N a ontain shape funtions, n is the unity normal vetor to the surfae under onsideration. Introduing these oupling fores, finally, the semi-disrete system of oupled equations onsisting of the eletro-mehanial field (see equation (3)) and the aousti field equation (11) together with equation (12) and (13) an be written as M w& C C + ϕ&& Φ&& ρ M a Cw ρ K K wϕ + K wϕ Kϕϕ w w& ϕ& ( ) C + C Φ& a I w fw = ϕ fϕ ( ). (14) ρ K a + K I Φ ρ fa 2.4 Finite Element Implementation For the analysis and the simulation of piezoeletri smart strutures a number of finite elements with oupled eletro-mehanial degrees of freedom are available in the FE software COSAR. For smart thin-walled 243

4 strutures a speial layered shell type element, onsisting of any number of passive and ative (piezoeletri) layers (see Fig. 1a), is already available in the COSAR pakage (for details see Gabbert et al., 22; Seeger, 24). he development of this element is based on the lassial SemiLoof-type shell element originally proposed by B. Irons (198). Reently, new finite aousti (see Fig. 1b) and semi-infinite aousti finite elements (see Fig. 1) have been developed and implemented into the COSAR software. Additionally, the oupling terms between the strutural SemiLoof elements and the 3D aousti finite elements have been developed and implemented, and, onsequently, fully oupled vibro-aousti simulations taking into aount the interior as well as the exterior sound radiation an be performed numerially. ative SemiLoof shell element n ξ 3 ξ 2 u1 u2 u3 ϕ ξ 1 finite aousti hexahedron element Φ Semi-infinite aousti hexahedron element (a) (b) () 13 [ Φ] Figure 1. Coupling of shell elements and aousti hexahedron elements: (a) piezoeletri SemiLoof-type shell element, (b) aousti hexahedron element, () semi-infinite aousti element 3 Model Redution and Controller Design As mentioned before due to the large number of degrees of freedom of a finite element model a model redution tehnique is needed to design a ontroller. As it is well-known from the analysis of strutural vibrations this redution an be performed using a few preseleted eigenmodes of the system. In the paper a modal trunation is applied, whih is briefly presented in the following. 3.1 Modal runation o apply the modal trunation tehnique to smart vibro-aousti systems, equation (14) is rewritten in the following ompat form Mr &&+ Cr& + Kr = f. (15) Introduing the state spae vetor [ r r& ] [ w ϕ Φ w& ϕ& Φ& ] z = = (16) from equation (15) it follows C M M K f z& + z = Bz& + Az = (17) M From equation (16) the linear eigenvalue problem an be derived ( A B) qˆ = λ. (18) i i he solution of equation (18) results in the modal matrix Q with 2k pairs of onjugate omplex eigenvetors [ qˆ qˆ... q ] Q =. (19) 1 2 ˆ 2k 244

5 If the modal matrix Q is ortho-normalized with Q BQ = I = diag( 1) and Q AQ = Λ = diag( λ ) oordinates z = Qq are introdued in equation (18), the redued state spae form is obtained as i, and new f q& + Λq = Q. (2) 3.2 Controller Design If the state spae equation (2) is extended by the measurement equation the set of equations, whih an be used to design an appropriate ontroller is obtained. B E q & +, (21a) = Λq + Q u() t + Q f () t = Aq + Bu() t Ef () t () t Cq + Du() t Ff () t y = + (21b) For this reason the state matries A, B, E, C, D and F are transferred to Matlab/Simulink via a speial data exhange interfae. Based on these model matries in Matlab/Simulink a time independent LQ-ontroller was designed, whih results in the following ontroller matrix R (for details see Nestorović et al., 25) u () t Rq() t =. (22) 4 Example he finite element model of a simply supported smart retangular plate struture is used to demonstrate the apability of the presented software approah. Four piezoeletri pathes as two olloated sensor/atuator pairs are attahed to an elasti plate (see Fig. 2). he time-dependent behavior of the sound radiated into the upper Figure 2. Smart plate struture oupled with aousti half spae half spae was analyzed for the ontrolled and the unontrolled ase. he plate has been meshed with 96 layered SemiLoof-type finite shell elements. For the aousti half spae a disretization with 516 finite aousti hexahedron elements and 236 semi-infinite aousti hexahedron elements was performed to desribe the behavior in the far field. For ontrolling the system a time-independent LQ-ontroller was designed taking into aount the first five pairs of onjugate omplex eigenvetors of the oupled system. In order to demonstrate the ontrolled and the unontrolled behavior the plate is exited by a harmoni fore ontaining the first three eigenfrequenies of the system. he designed ontroller is swithed on after 1.5 s. In Figure 3 the measured sound pressure at a distane 245

6 of 639 mm above the middle of the plate is shown. After swithing the ontroller on, the two olloated pairs of piezoeletri path atuators and sensors result in a good noise redution above the plate. able 1. Material properties and dimensions Piezoeletri Material Properties: E 11 = E 22 = 6935 N/mm², G 12 = N/mm², ν=.37, e 31 = N/(mV)mm, κ 33 = N/(mV)², ρ= Ns²/mm 4 Material Properties of the Elasti Plate: E=7 N/mm², ν=.3, ρ P = Ns²/mm 4 Material Properties of the Aousti Fluid: =34 mm/s, ρ = Ns²/mm 4 Dimensions of the Plate: l 1 =6 mm, l 2 =4 mm, h= 2 mm Dimensions of the Pathes: 1 mm 5 mm.2 mm Figure 3. Resulting sound pressure 5 Conlusions In the present paper the theoretial bakground of a general finite element simulation tool for the design of atively ontrolled thin lightweight strutures to redue the noise radiating is presented. Besides the passive struture, the finite element model inludes ative piezoeletri elements, the aousti fluid, the vibro-aousti oupling, and the ontroller influene. Piezoeletri layered shell type finite elements developed on the basis of the SemiLoof element have been extended to inlude a vibro-aousti oupling with 3D aousti finite elements and infinite elements for the far field. Beause of the large number of degrees of freedom of the FE model, a modal trunation tehnique based on a omplex eigenvalue analysis is performed in COSAR and the redued model is transformed into the state spae form. Based on a data interfae the state matries are transferred to Matlab/Simulink, where an appropriate ontroller an be designed and tested. he developed proedure is applied to a smart plate struture and the noise redution after swithing the ontroller on is presented. he effiieny of the smart vibro-aousti system an be improved by alulating optimal positions of the atuators and sensors at the struture. Aknowledgement: his work has been supported by the postgraduate program of the federal state of Sahsen- Anhalt, whih is gratefully aknowledged. 246

7 Referenes Bettess, P.: Infinite Elements. Penshaw Press, (1992). COSAR General Purpose Finite Element Pakage, Manual. (FEMCOS mbh Magdeburg, 1992). (see also Everstine, G.C.: Finite Element Formulations of Strutural Aousti Problems. J. Computers & Strutures, 65 (3), (1971), Gabbert, U.: Researh ativities in smart materials and strutures and expetations to future developments, J. of heoretial and Applied Mehanis. 4 (3), (22), Gabbert, U., Berger, H., Köppe, H., Cao, X.: On Modelling and Analysis of Piezoeletri Smart Strutures by the Finite Element Method. J. of Applied Mehanis and Engineering, 5 (1), (2), Gabbert, U., Köppe, H., Nestorović-rajkov,.: Controller Design for Engineering Smart Strutures Based on Finite Element models. Rao, V.S. (Eds.): Modeling, Signal Proessing and Control, SPIE, Vol. 4693, (22), Gabbert, U., Köppe, H., Seeger, F., Berger, H.: Modeling of Smart Composite Shell Strutures. Journal of heoretial and Applied Mehanis, 4 (3), (22), Geers,.L.: Doubly Asymptoti Approximations for ransient Motions of Submerged Strutures. he Journal of the Aoustial Soiety of Ameria, 64 (5), (1978), Irons, B.M., Ahmad, S.: ehniques of Finite Element. Ellis Horwood Ltd., Chihester, (198). Kollmann, F. G.: Mashinenakustik. Grundlagen, Meßtehnik, Berehnung, Beeinflussung. Springer-Verlag, Berlin-Heidelberg, (2). Nestorović-rajkov., Köppe H., Gabbert U.: Ative vibration ontrol using optimal LQ traking system with additional dynamis. International Journal of Control, 78 (15), (25), Olson, L.G., Bathe, K.-J.: An Infinite Element for Analysis of ransient Fluid-Struture Interations. Engineering Computations, 2 (4), (1985), Seeger, F.: Simulation und Optimierung adaptiver Shalenstrukturen. Fortshritt-Berihte VDI-Reihe 2, Nr. 383, VDI-Verlag, Düsseldorf (24). zou, H.-S., Guran, A. (Eds.), Strutroni Systems: Smart Strutures, Devies and Systems, Part 1: Materials and Strutures, Part 2: Systems and Control, World Sientifi, (1998). Address: Dipl.-Ing. Jean Lefèvre and Prof. Dr.-Ing. habil. Ulrih Gabbert, Institut für Mehanik, Otto-von- Guerike-Universität Magdeburg, Universitätsplatz 2, D-3916 Magdeburg. jean.lefevre@mb.uni-magdeburg.de; ulrih.gabbert@mb.uni-magdeburg.de. 247