Modeling of a fluid-loaded smart shell structure for active noise and vibration control using a coupled finite element–boundary element approach This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2010 Smart Mater. Struct. 19 105009 (http://iopscience.iop.org/0964-1726/19/10/105009) Download details: IP Address: 141.44.130.82 The article was downloaded on 30/09/2010 at 14:37 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
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Modeling of a fluid-loaded smart shell structure for active noise and vibration control using a
coupled finite element–boundary element approach
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
Modeling of a fluid-loaded smart shellstructure for active noise and vibrationcontrol using a coupled finiteelement–boundary element approachS Ringwelski and U Gabbert
Institute of Mechanics, Otto-von-Guericke University of Magdeburg, Universitatsplatz 2,39106 Magdeburg, Germany
Received 9 April 2010, in final form 9 June 2010Published 6 August 2010Online at stacks.iop.org/SMS/19/105009
AbstractA recently developed approach for the simulation and design of a fluid-loaded lightweightstructure with surface-mounted piezoelectric actuators and sensors capable of actively reducingthe sound radiation and the vibration is presented. The objective of this paper is to describe thetheoretical background of the approach in which the FEM is applied to model the activelycontrolled shell structure. The FEM is also employed to model finite fluid domains around theshell structure as well as fluid domains that are partially or totally bounded by the structure.Boundary elements are used to characterize the unbounded acoustic pressure fields. Theapproach presented is based on the coupling of piezoelectric and acoustic finite elements withboundary elements. A coupled finite element–boundary element model is derived byintroducing coupling conditions at the fluid–fluid and fluid–structure interfaces. Because of thepossibility of using piezoelectric patches as actuators and sensors, feedback control algorithmscan be implemented directly into the multi-coupled structural–acoustic approach to provide aclosed-loop model for the design of active noise and vibration control. In order to demonstratethe applicability of the approach developed, a number of test simulations are carried out and theresults are compared with experimental data. As a test case, a box-shaped shell structure withsurface-mounted piezoelectric actuators and four sensors and an open rearward end isconsidered. A comparison between the measured values and those predicted by the coupledfinite element–boundary element model shows a good agreement.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
In recent years, increasing attention has been paid to vibrationand acoustic noise control in many industrial applications.The control of noise and vibration is essential in the designprocess of a product, since it contributes to comfort, efficiencyand safety. There are two different approaches to achievenoise and vibration attenuation. On the one hand, there isthe widely used passive approach. Passive control techniquesmostly reduce vibration and sound emission of structures byapplying additional damping materials. These methods arebest suited for a frequency range above 1 kHz. A drawback
of this traditional way can be an increase in weight due to theapplication of additional damping materials. It is also possibleto improve the structural–acoustic properties by modifyingthe structural geometry. An overview of the developments instructural design optimization for passive noise control can befound in Marburg [1].
Active control techniques provide an alternative way tominimize unwanted structural vibration and noise that movesmore and more into the field of vision for designers, sincethey avoid an increase in structural weight. A very usefulcontrol technique that was first introduced by Fuller [2] isactive structural–acoustic control (ASAC). In this concept
Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
actuators are directly attached to the shell structure in orderto reduce sound radiation by changing the dynamic behaviorof the structure. ASAC is similar to active vibration control(AVC) [3] since control forces are applied to the structure,but the goals are different. The concept of AVC is developedwith the goal of reducing structural vibration instead of theacoustic response. ASAC also differs from active noisecontrol (ANC) [4]. The purpose of ANC is to reduce theunwanted sound of a fluid–structure system by producingan anti-noise through loudspeakers. In ANC and ASACsystems piezoelectric ceramics are widely used as sensorsand actuators, because they can easily be mounted onto thevibrating structure. Piezoelectric ceramics are available inseveral configurations such as stacks or thin patches. Activecontrol techniques are usually employed in applications wherethe frequency range of interest is between 50 Hz and 1 kHz.Over the past few years, several researchers have studieddifferent control strategies for ASAC. For example, Li andZhao [5] investigated ASAC on a rectangular fluid-loaded platewith piezoelectric layers using a velocity feedback controlalgorithm. In this study, velocity feedback has been provento be a robust and effective control strategy. Ruckman andFuller [6] applied a feedforward control approach to reduceactively the acoustic radiation of a fluid-loaded spherical shellstructure. This approach uses linear quadratic optimal controlto minimize the total radiated power.
The development and industrial application of smartstructural–acoustic systems for active noise and vibrationcontrol require efficient and reliable simulation tools. Virtualmodels are of particular interest in the design process ofa product, since they enable the testing of several controlstrategies and they are required to determine optimal sensorand actuator locations. A simulation of an active structural–acoustic system includes the modeling of the mechanicalstructure, the piezoelectric actuators and sensors, the interiorand exterior fluids as well as the control algorithm. Whendealing with a fluid-loaded shell structure these subsystemscannot be modeled separately, because it is well known thatthe presence of a surrounding fluid strongly influences thevibration behavior of thin-walled structures. The subsystemsare connected to an overall model by introducing couplingconditions that take into account the fluid–structure interactionand the fluid–fluid interaction.
The accurate modeling of active noise and vibrationcontrol has become a topic of wide interest. Wang et al[7] modeled analytically the controlled behavior of plate-likestructures. The effect of piezoelectric actuation is introducedin this model through equivalent bending moments. Gu andFuller [8] used an analytic approach to describe the control offluid-loaded plates. Active control of fluid-loaded plates hasalso been investigated analytically by Lee and Park [9].
More complex problems cannot be solved analytically,and it is necessary to use numerical methods. Severalnumerical techniques such as the finite element method (FEM)and the boundary element method (BEM) are available topredict the behavior of active structural–acoustic systems. Avirtual model established entirely on the basis of the FEM issuggested by Khan et al [10]. In this work, infinite acoustic
elements are applied to construct a two-dimensional modelfor harmonic analysis. An extension of this work is given byLefevre and Gabbert [11]. They developed a three-dimensionaltime-domain model which is suitable to design an optimallinear quadratic regulator (LQR).
Only a few studies deal with the coupling of piezoelectricfinite elements with acoustic boundary elements. In Kaljevicand Saravanos [12] a coupled FE–BE approach for computingthe steady-state response of acoustic cavities bounded bypiezoelectric composite shell structures is presented. TheFE–BE approach is performed by describing the piezoelectricstructure with finite elements and the fluid medium withboundary elements. Lerch et al [13] proposed a combinedFE–FE–BE scheme for the modeling of acoustic antennas.Here, finite elements are applied to predict the behavior of thepiezoelectric structure as well as the acoustic near-field, andboundary elements are employed to describe the far-field.
In both approaches the FE–BE model and the FE–FE–BE model are applied only to analyze the interaction of thepiezoelectric structure with the surrounding fluid, whereasthe influence of control is not considered. Zhang et al[14] developed an approach for modeling active vibrationisolation and sound radiation control of underwater structuresby integrating different control algorithms into the FE–BEmodel. This approach utilizes the added mass concept toprovide a closed-loop model for time-domain simulations. Adrawback of this approach is the restriction to pure mechanicalstructures without piezoelectric actuators and sensors.
In order to successfully employ piezoelectric patches inactive noise and vibration control, the coupled response offluid-loaded smart shell structures needs to be accurately andreliably predicted. Therefore the present paper proposes a FE–FE–BE formulation in the frequency domain, which allowsthe modeling of arbitrary-shaped structural–acoustic systemsincluding the employed control strategy. The constructionof the coupled formulation consists of three parts. First,the piezoelectric shell structure as well as the enclosed andsurrounding fluids have to be modeled. In the next step,the subsystems are coupled to obtain a multi-coupled system.Lastly, the employed control is implemented in the overallmodel.
In the present approach, the FEM is applied to modelthe shell structure as well as the piezoelectric actuators.The finite element selected for structural modeling is astandard eight-node layered Mindlin-type shell element withelectromechanical properties [15]. The FEM is capable ofanalyzing shell structures with irregularly shaped geometriesand complex boundary conditions in statics and dynamicsincluding piezoelectric materials by applying linear coupledelectromechanical constitutive equations. Finite elementsare also employed to model finite fluid domains around theshell structure as well as fluid domains that are partially ortotally bounded by the structure, such as cavities. Quadratichexahedron elements with twenty nodes are used to discretizethe finite fluid domains. The FE analysis of acoustic fieldsoffers the possibility to take into account inhomogeneitiesinside the fluid. A disadvantage of FE models is thelarge number of degrees of freedom. Quadratic conforming
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
boundary elements with eight nodes are utilized to characterizethe unbounded acoustic pressure field around the shell structureand the finite fluid domains. In contrast to FEM, the BEMautomatically fulfills the Sommerfeld radiation condition anddoes not produce, as the FEM does, reflections at the far-field boundaries. For this reason the BEM allows numericalprediction of sound pressure fields in unbounded domains. TheBEM is able to reduce the computational effort when dealingwith unbounded problems, since only the boundary of theradiating domains has to be discretized. One drawback of theclassical BEM is the resulting type of matrices, which are fullypopulated, non-symmetric and frequency-dependent.
A coupled FE–FE–BE approach is derived by introducingcoupling conditions which describe the fluid–structure andfluid–fluid interaction. The coupling conditions are obtainedby assuming kinematic and dynamic continuity of thestructural and acoustic variables at the interfaces. Aftercoupling the subsystems, either the unknown structuraldisplacements or the acoustic pressures can be eliminatedto establish a system of equations in terms of acoustic orstructural variables only. For the present study a differentprocedure is employed, where both the structural and theacoustic variables are retained in the final system of equations.
The modeling of fluid-loaded smart shell structuresrequires not only the construction of an appropriate structural–acoustic model, but also consideration of the involved control.In simulations of active noise and vibration suppression,the influence of control can be taken into account simplyby implementing the corresponding control algorithm in thestructural–acoustic model. The present study makes use ofvelocity feedback control to demonstrate how a closed-loopmodel can be simulated [4, 14]. The velocity feedbackalgorithm generates forces which are aimed to minimize thevibration of the structure. The reduction of the sound radiationin the far-field is not directly influenced by the control.Consequently, in the far-field the sound pressure may not besuppressed as much as the vibration of the structure itself,which can be seen as a drawback of the velocity feedbackcontrol.
In order to check the accuracy and the quality of theproposed modeling approach, numerous test examples werestudied in detail, and partially the numerical solutions werealso compared with measurements. It could be observed that ingeneral the numerical predictions are in very good agreementwith the experimental results.
This paper is structured as follows. First, the theoreticalbackground of the numerical approach is presented in detail.The applied finite elements and boundary elements formodeling the mechanical, electromechanical and fluid fieldsas well as their coupling is given. Then it is shownhow the control can be included in the overall multi-fieldmodel in the frequency domain, which results in an overallclosed-loop model. Finally, as a test example, a box-shaped shell structure is presented to evaluate the numericalresults. The dynamic behavior of the cover plate of the boxcan be actively influenced by a group of surface-mountedpiezoelectric patches. The experiments are performed inreality and virtually based on the proposed coupled FE–FE–BE modeling approach. The comparison of the measured
results and the numerical predictions shows good agreement.Additionally, the experimental and the numerical results revealthat the radiated sound field and structural vibrations of thebox-shaped shell structure are significantly reduced by meansof velocity feedback control. The paper concludes with asummary and an outlook to ongoing activities.
2. Finite element and boundary element modeling
In this section, the FEM is applied to develop a planepiezoelectric composite shell element for the analysis of smartstructures and a hexahedron element for the modeling offinite acoustic fluid domains. The BEM is used to derivea quadrilateral boundary element for simulating infinite fluidregions. All equations are described using a Cartesianx1, x2, x3-coordinate system.
2.1. Finite element model of piezoelectric shell structures
In the following, the theoretical background of a FEformulation for analyzing laminated plates with integratedpiezoelectric actuators and sensors is briefly discussed. Thefinite element selected for structural modeling is a simplepiezoelectric composite Mindlin-type shell element with eightnodes (figure 1). It is assumed that the thickness of the layersis the same at each node. Additionally it is presumed that themodeled composite laminate plate consists of perfectly bondedlayers and the bonds are infinitesimally thin as well as nonsheardeformable. To describe the electromechanical properties ofthe element in terms of the natural coordinates ξ1 and ξ2, thefollowing quadratic serendipity shape functions are used forthe corner nodes
and for typical mid-side nodes located at ξ1i = 0 and ξ2i ± 1
Ni = 12 (1 − ξ 2
1 )(1 + ξ2iξ2). (2)
The shell element has five degrees of freedom u1, u2, u3,θx2, θx1 at each node for elastic behavior and there isone potential degree of freedom ϕ per layer to model thepiezoelectric effect. The normal rotation θx3 is considered to bezero. The strain–displacement relations for the used plane shellelement are based on the Mindlin first order shear deformationtheory.
Assuming small deformations, the strain–displacementrelationship reads
ε = Buu, (3)
where ε is the linear strain vector and u is the vector with thenodal displacements and rotations. The strain–displacementmatrix Bu is given by
Bu = [ B1u · · · Bi
u · · · B8u ] , (4)
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
where the submatrix Biu , which is associated with the node i of
the shell element, has the form
Biu =
⎡⎢⎢⎢⎢⎢⎣
∂ Ni∂x1
0 0 x3∂ Ni∂x1
0
0 ∂ Ni∂x2
0 0 −x3∂ Ni∂x2
∂ Ni∂x1
∂ Ni∂x2
0 x3∂ Ni∂x2
−x3∂ Ni∂x1
0 0 ∂ Ni∂x2
0 −Ni
0 0 ∂ Ni∂x1
Ni 0
⎤⎥⎥⎥⎥⎥⎦
. (5)
The poling direction of the piezoelectric layers is assumedto be coincident with the thickness direction x3, which meansthat the electric field acts only perpendicular to the layers.Moreover, the difference in the electric potential ϕ is supposedto be constant in each layer of the shell element. Theelectric field, which varies linearly through the thicknessof a piezoelectric layer, causes an in-plane expansion orcontraction. This behavior is called the transverse piezoelectriceffect. In Marinkovic et al [16] it is shown that theseassumptions are accurate enough in thin structure applications.For modeling the electric field only one electric degree offreedom per layer has to be specified within the element. Thusthe electric field-potential relation can be written in the simplescalar form
Ek = Bϕkϕk, (6)
with
Bϕk = − 1
tk. (7)
Here Ek is the electric field, ϕk is the difference in the electricpotential and tk is the thickness of the kth piezoelectric layer ofthe shell element.
The coupled electromechanical behavior of piezoelectricmaterials in low voltage, strain and stress applications canbe modeled with sufficient accuracy by means of linearizedconstitutive equations. In matrix form, the constitutiverelations for a piezoelectric layer k are defined as [17]
σ k = Qkε − ek Ek, (8)
Dk = eTk ε − κk Ek, (9)
where σ k denotes the stress vector and Dk is the dielectricdisplacement in thickness direction. Qk and ek are the plane-stress reduced elastic stiffness and the piezoelectric matricesrespectively. The coefficient κk represents the plane-stressreduced dielectric permittivity of the kth piezoelectric layer.
The piezoelectric constitutive relations given above haveto be used within the weak form of the mechanical equilibriumequations [18] to derive the electromechanical FE equationsof a piezoelectric layer by applying a standard Galerkinprocedure. After adding the local equations of all layers andelements to a global model, the resulting system of coupledalgebraic equations can be expressed as[
Mu 00 0
] [uϕ
]+
[Cu 00 0
] [uϕ
]+
[Ku Kuϕ
KTuϕ −Kϕ
] [uϕ
]=
[fu
fϕ
],
(10)
where Mu is the mass matrix, Ku is the stiffness matrix andKϕ is the dielectric matrix of the discretized piezoelectric
composite shell structure. The piezoelectric coupling arisesin the piezoelectric coupling matrix Kuϕ . For convenience,a Rayleigh damping is introduced into the system ofequation (10) assuming that the matrix Cu is a linearcombination of the matrix Mu and Ku . The external loadsare stored in the mechanical load vector fu and in the electricload vector fϕ . Applying the Fourier transform to the obtainedsystem of equation (10) leads to an equivalent system ofequations in the frequency domain
[−�2Mu + i�Cu + Ku Kuϕ
KTuϕ −Kϕ
] [uϕ
]=
[fu
fϕ
], (11)
where the vectors u and ϕ represent the complex amplitudes ofthe structural displacements as well as rotations and electricpotentials. Here, � is the excitation frequency and i is theimaginary unit.
2.2. Finite element model of the acoustic fluid
Computational prediction of noise can be achieved by theFEM as well as BEM. When dealing with far-field problemsthe BEM is very efficient. On the other hand, the FEM ismore suitable for bounded fluid regions. For this reason aclassical FE formulation is employed to model the finite fluiddomains around the smart shell structure as well as the fluiddomains that are partially or totally bounded by the structure.The development of acoustic FE elements to calculate timeharmonic acoustic pressure waves in homogeneous fluids isbased on the Helmholtz equation
� p + k2 p = 0, (12)
where p is the complex amplitude of the time harmonicpressure and k is the wavenumber. In equation (12) � is theLaplacian operator. The wavenumber k is given by
k = �
c, (13)
where c is the speed of sound in the fluid. In the present study,a 20-node hexahedron element is chosen to discretize the fluiddomain. The pressure amplitude p is considered as the nodaldegree of freedom in the finite element. To approximate theacoustic pressure within the hexahedron element, the followingquadratic shape functions are used for the corner nodes
Ni = 18 (1+ξ1ξ1i )(1+ξ2ξ2i )(1+ξ3ξ3i)(ξ1ξ1i +ξ2ξ2i +ξ3ξ3i −2)
(14)and for typical mid-side nodes located at ξ1i = 0, ξ2i ± 1 andξ3i = ±1
Ni = 14 (1 − ξ 2
1 )(1 + ξ2ξ2i )(1 + ξ33ξ3i). (15)
As in structural mechanics, the Helmholtz equation canbe cast in a weak form to derive mass-like and stiffness-like element matrices. Following a standard FE assemblyprocedure, the matrix equation of the discretized fluid domainbecomes
(−�2Mp + Kp)p = −iρ0�fp, (16)
with the acoustic mass matrix Mp, the acoustic stiffness matrixKp and the acoustic load vector fp due to the prescribed normalvelocities vn. The variable ρ0 stands for the density of theacoustic medium.
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
Boundary elements are utilized instead of infinite acousticelements to characterize the unbounded acoustic pressure fieldaround the shell structure and the finite fluid domains. Infiniteelements do not fulfill exactly the Sommerfeld radiationcondition, and do not perform sufficiently well when locatedin the near-field. Boundary elements, on the other hand, canbe placed at any arbitrary position. Furthermore, boundaryelements lead to a significantly smaller system of equations,because the unknown variables are introduced only on theboundary of the radiating domains. In the present paper, aquadrilateral boundary element with eight nodes is applied todiscretize the boundary of the radiating domains. The acousticpressure p and the normal velocity vn are the nodal degrees offreedom. They are linked by the relationship [19]
∂ p
∂n= −iρ0�vn, (17)
where n is the unit normal vector directed outwards fromthe fluid domain. The development of a BE formulation isbased on the Helmholtz equation (12), which by means ofthe weighted residual method and Green’s identity can betransformed into the following integral equation
∫O
p∂g
∂ndO + cp p = −iρ0�
∫O
gvn dO . (18)
Here g is the fundamental solution of the Helmholtzequation (12) and cp is a factor describing the surface angleof the source point located on the boundary of the radiatingdomain O. To derive the BE formulation for the acoustic fluid,the standard boundary element discretization together with thecollocation method are applied to the integral equation (18).In the same way as in the FEM, the unknown variables areapproximated by shape functions. To interpolate the acousticpressure and the normal velocity within the boundary element,quadratic serendipity shape functions given in equations (1)and (2) are applied. The resulting direct BE matrix equationreads
Hp = −iρ0�Gvn, (19)
where H and G are the influence matrices and the vectors p andvn are the nodal values of the acoustic pressure and the normalvelocity. As mentioned before, the BEM only has to dealwith a two-dimensional surface model. However, the influencematrices H and G are fully populated and have to be computedfor each frequency �, since they are frequency-dependent.
Figure 2. A multi-coupled structural–acoustic system.
3. Multi-structural-acoustic coupling
The dynamic behavior of lightweight structures with interiorand exterior fluid loading is strongly affected by the interactionbetween the subsystems. Especially in the case of thin-walled structures with openings, fluid–structure and fluid–fluid interactions take place. The purpose of the followingsection is to present a FE–FE–BE approach to model the multi-coupling that occurs in systems with fluid–structure and fluid–fluid interfaces.
The approach is illustrated using the multi-coupledstructural–acoustic system shown in figure 2. As illustrated,S is a flexible thin-walled piezoelectric structure that radiatessound into the neighboring fluid domains VH and VB . Thefluid domain VH consists of two parts: an inner region thatis substantially enclosed by the structure S, and an outerregion that includes sound fields around the openings andacoustic near-fields relevant for controller design purposes. VB
is an unbounded exterior fluid domain surrounding the shellstructure S and the finite fluid region VH .
In the approach, the FEM is applied to model the flexiblepiezoelectric shell structure as well as the finite fluid domainVH . The FE modeling of sound fields allows the designof direct acoustic control, since the acoustic pressure can beprovided as nodal degree of freedom in order to develop aclosed-loop model. To predict the harmonic behavior of theunbounded exterior fluid domain VB the BEM is applied.A coupled FE–FE–BE formulation is derived by introducingcoupling conditions at the fluid–fluid interface OH B andthe fluid–structure interfaces OSH and OSB . The couplingconditions are obtained by assuming kinematic and dynamiccontinuity of structural and acoustic variables at the interfaces.The coupling conditions are given by
pH = pB, vnH = vnB on OH B,
pH = σun, vnH = i�un on OSH ,
pB = σun, vnB = i�un on OSB .
(20)
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
At the fluid–fluid interface OH B the acoustic pressure andthe normal particle velocity of the finite and the infinite fluidregions are equal. At the fluid–structure interfaces OSH andOSB , the particles of the fluid and the boundary of the structuremove together. As a result, the normal components of theparticle velocity and of the first time derivative of the structuraldisplacement are equal, and the acoustic pressure and thenormal stress at the surface of the structure are in equilibrium.
Using equation (11) and the interface relations (20), theFE formulation of the fluid-loaded piezoelectric shell structureS can be written as
[Ku Kuϕ
KTuϕ −Kϕ
] [uϕ
]=
[fu + LSH pS
H + LSB pSB
fϕ
], (21)
with
Ku = −�2Mu + i�Cu + Ku . (22)
In equation (21) there are two additional load vectorsdescribing the sound pressure that acts on the vibratingstructure. The matrices LSH and LSB are coupling matricesincluding the shape functions of the shell element and thecorresponding hexahedron and boundary element respectively.
The model of the finite fluid region VH is based on theequations (16) and (20). If the vector of the acoustic pressurep in equation (16) is split into the inner degrees of freedom pV
Hand degrees of freedom along the boundaries OSH and OH B ,the FE formulation of the finite fluid region VH can be writtenas
⎡⎣
K11p K12
p K13p
K21p K22
p K23p
K31p K32
p K33p
⎤⎦
[ pSH
pVH
pBH
]=
[ρ0�
2LTSH u
0iρ0�LH B vH
nB
], (23)
withK11
p = −�2M11p + K11
p
K12p = −�2M12
p + K12p
...
K33p = −�2M33
p + K33p .
(24)
Due to the load of the mechanical structure and theexterior fluid, there are two additional load vectors. The matrixLH B is a further coupling matrix, which contains the shapefunctions of the hexahedron and the corresponding boundaryelement.
In an analogous manner, the BE formulation (19) can besplit into degrees of freedom along the boundary OSB anddegrees of freedom along the boundary OH B to predict thebehavior of the unbounded exterior fluid domain VB . Theresulting equation reads[
H11 H12
H21 H22
] [pS
BTH BpB
H
]= − iρ0�
[G11 G12
G21 G22
] [i�TSB u
vHnB
],
(25)
where TH B is a transformation matrix between the nodalpressures of the hexahedron and the boundary element. Thematrix TSB transforms the nodal displacement of the shellelement into the normal velocity of the boundary element.
Combining equations (21), (23) and (25), and moving allunknowns to the left-hand side, the following coupled FE–FE–BE matrix equation is obtained⎡⎢⎢⎢⎢⎢⎣
Ku Kuϕ −LS H 0 0 −LS B 0KT
uϕ −Kϕ 0 0 0 0 0
−ρ0�2LT
S H 0 K11p K12
p K13p 0 0
0 0 K21p K22
p K23p 0 0
0 0 K31p K32
p K33p 0 −i�ρ0LH B
−ρ0�2G11TS B 0 0 0 H12TH B H11 iρ0�G12
−ρ0�2G21TS B 0 0 0 H22TH B H21 iρ0�G22
⎤⎥⎥⎥⎥⎥⎦
× [ u ϕ pSH pV
H pBH pS
B vHnB ]T = [ fu fϕ 0 0 0 0 0 ]T.
(26)
It should be noted that throughout the whole approachconforming discretizations are assumed. In general, structuralFE and acoustic BE meshes do not match at their interfaces.This may be due to different shape functions or differentelement sizes. To overcome the limitations of the classicalnode-to-node contact algorithms, Bernardi et al [20] developeda segment-to-segment discretization technique called mortarmethod. In contrast to the node-to-node discretization, in themortar method the continuity constraints are not enforced atdiscrete finite or boundary element nodes, but are formulatedalong the entire coupling boundary in a weak integral sense.The mortar method offers a universal coupling strategy thatcan be easily applied to the present system in order to obtainthe required coupling matrices between the three subsystemsindependently of a given discretization.
4. Velocity feedback control
The analysis of fluid-loaded smart shell structures requiresnot only the design of an appropriate structural–acousticmodel but has also to take into consideration the involvedcontrol. In numerical analyses, the influence of control can betaken into account simply by implementing the correspondingcontrol algorithm in the coupled structural–acoustic model.The present study makes use of velocity feedback controlto demonstrate how a closed-loop model can be obtained.Collocated piezoelectric actuators and sensors are utilized toform the closed-loop control system. The collocated designof an actuator/sensor pair guarantees control stability. In theconsidered feedback control system, the sensor output voltageis differentiated, amplified by a constant gain g and directlyfed back to the collocated actuator. Due to the feedback, thecollocated actuator generates counteracting moments whichsuppress the vibrations and the resulting sound radiation ofthe shell structure. The velocity feedback control law of acollocated actuator/sensor pair reads in the frequency domain
ϕa = i�gϕs, (27)
where ϕs is the voltage taken from the piezoelectric sensor andϕa is voltage applied to the piezoelectric actuator.
Considering a fluid-loaded shell structure with multiplecollocated actuator/sensor pairs, the vector of the electricpotentials ϕ in equation (26) can be split into degrees of
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
freedom of the sensor layers ϕs and the actuator layers ϕa. Thesystem of equation (26) then becomes⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
Ku Kauϕ Ks
uϕ −LS H 0 0 −LS B 0
KaTuϕ −Ka
ϕ 0 0 0 0 0 0
KsTuϕ 0 −Ks
ϕ 0 0 0 0 0
−ρ0�2LT
S H 0 0 K11p K12
p K13p 0 0
0 0 0 K21p K22
p K23p 0 0
0 0 0 K31p K32
p K33p 0 −i�ρ0LH B
−ρ0�2G11TS B 0 0 0 0 H12TH B H11 iρ0�G12
−ρ0�2G21TS B 0 0 0 0 H22TH B H21 iρ0�G22
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
× [ u ϕa ϕs pSH pV
H pBH pS
B vHn B ]T = [ fu fa
ϕ 0 0 0 0 0 0 ]T.
(28)
Here it is assumed that each actuator/sensor pair ismodeled by a separate independent piezoelectric compositeshell element with one potential degree of freedom for eachactuator and the sensor layer respectively. As a result, thedielectric matrix Kϕ becomes diagonal [5]. In addition, it ispresumed that for the sensor layer the applied charge fs
ϕ is zero.In equation (28), the subscript a refers to the actuator layers andthe subscript s refers to the sensor layers.
The electric potential on the sensors can be obtained interms of the structural displacements from equation (28) asfollows
ϕs = Ks−1
ϕ KsT
uϕu. (29)
In feedback control systems, the electric potential on theactuators is known from the applied control law. In such cases,the global system of equations can be expressed in terms of thestructural and acoustic variables only. Thus equation (28) canbe rewritten as⎡⎢⎢⎢⎢⎢⎢⎢⎣
Ku + KsuϕKs−1
ϕ KsT
uϕ −LS H 0 0 −LS B 0
−ρ0�2LT
S H K11p K12
p K13p 0 0
0 K21p K22
p K23p 0 0
0 K31p K32
p K33p 0 −i�ρ0LH B
−ρ0�2G11TS B 0 0 H12TH B H11 iρ0�G12
−ρ0�2G21TS B 0 0 H22TH B H21 iρ0�G22
⎤⎥⎥⎥⎥⎥⎥⎥⎦
× [ u pSH pV
H pBH pS
B vHnB ]T = [ fu − Ka
uϕ ϕa 0 0 0 0 0 ]T .
(30)
In equation (30), the known electric potentials on theactuators ϕa appear as external force. Using velocity feedbackcontrol, the control algorithm of the multiple independent localfeedback loops can be written in the following vector-matrixnotation
ϕa = i�Gasϕs. (31)
The control matrix Gas in equation (31) accomplishes twothings. It determines the gains within the local feedback loopsand relates the potential degrees of freedom of the sensor layersto those of the corresponding actuator layers. With the controlalgorithm (31) the prescribed electric potentials ϕa can alsobe expressed in terms of the structural displacements u, andthus all the electric degrees of freedom in equation (28) can be
condensed. The condensed system takes the following form⎡⎢⎢⎢⎢⎢⎢⎣
Ku + KsuϕKs−1
ϕ KsT
uϕ + i�Cas −LSH 0 0 −LSB 0
−ρ0�2LT
SH K11p K12
p K13p 0 0
0 K21p K22
p K23p 0 0
0 K31p K32
p K33p 0 −i�ρ0LH B
−ρ0�2G11TSB 0 0 H12TH B H11 iρ0�G12
−ρ0�2G21TSB 0 0 H22TH B H21 iρ0�G22
⎤⎥⎥⎥⎥⎥⎥⎦
× [ u pSH pV
H pBH pS
B vHnB ]T = [ fu 0 0 0 0 0 ]T ,
(32)
where Cas is the active damping matrix [17]
Cas = KauϕGasKs−1
ϕ KsT
uϕ. (33)
The obtained system of equation (32) describes thecontrolled behavior of fluid-loaded smart shell structures.Due to the implementation of velocity feedback control, anadditional damping term occurs on the left-hand side ofequation (32). The feedback control forces generated by thefeedback voltage increase actively the damping of the system.From equation (31) it is known that the intensity of the activedamping depends only on the chosen feedback gains. Thus,by adjusting the feedback gains, the goal of controlling thevibration and the resulting sound radiation of the shell structurecan be achieved.
A drawback of velocity feedback control is that the soundradiation of the shell structure is not controlled directly, sincethe control forces are generated to minimize the vibration of thestructure. Due to the indirect control, sound pressure in the far-field may not be suppressed as much as the structural vibration.This is due to the fact that the applied control suppresses notonly the vibration but also causes changes in the shape of thevibration modes. Consequently it is possible that the modifiedmode shapes produce a higher sound pressure at some points, ifthe radiated sound waves enhance each other. To overcome theproblem, several ASAC strategies have been proposed, suchas optimal positioning of the actuator/sensor pairs to minimizeradiated sound power of the structure.
It should be noted that modeling a fluid-loaded smart shellstructure with another feedback control strategy can be realizedin an analogous way as it has been demonstrated for velocityfeedback control.
5. Numerical studies
The purpose of this section is to demonstrate the validity ofthe proposed modeling approach. For this reason a number oftest simulations are carried out and the results are comparedwith experimental data. Although the present formulation isapplicable to any geometry such as curved shell structures,this paper utilizes a box-shaped shell structure with a partiallyenclosed acoustic cavity and an unbounded surrounding fluiddomain. The box-shaped shell structure consists of a flexibleplate, four rigid walls and an open side, where the inner andouter fluids interact. The flexible plate is clamped at all sidesto the rigid walls, which are assumed to be acoustically hardsurfaces. Eight piezoceramic patches are bonded to the top
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
Table 1. Geometrical parameters of the cavity and materialproperties of the inner and outer fluid.
Dimensionslx1 × lx2 × lx3 (mm3)
Speed of soundc (mm s−1)
Density ρ0
(kg · 103 mm−3)
600 × 900 × 1000 340 000 1.29 × 10−12
Table 2. Geometrical and material parameters of the aluminumplate.
Dimensionslx1 × lx2 × lx3
(mm3)Density ρs
(kg · 103 mm−3)
Young’smodulus Es
(N mm−2)Poisson’s ratioνs (−)
600 × 900 × 4 2.63 × 10−9 70 000 0.34
and bottom surfaces of the flexible plate to form a set of fourcollocated actuator/sensor pairs for active vibration and noisesuppression. A velocity feedback control algorithm is appliedto relate the sensor voltage to the actuator voltage in fourindependent closed feedback loops. Figure 3 shows the layoutof the considered fluid-loaded smart shell structure.
The results presented in this section are obtained with thegeometrical and material parameters given in tables 1, 2 and 3.
The center of the first collocated actuator/sensor pair islocated at x1 = 300 mm, x2 = 300 mm, the second atx1 = 300 mm, x2 = 450 mm, the third at x1 = 450 mm,x2 = 300 mm and the fourth at x1 = 450 mm, x2 = 450 mm.
In the approach, the FEM is applied to model the acousticcavity VH and the shell structure S including the rigid walls,the flexible plate as well as the surface attached piezoelectricactuators and sensors. The BEM is employed to model theexterior sound field VB . A number of 404 standard finite shellelements and four piezoelectric composite shell elements areused to model the box-shaped structure (figure 4). The acousticcavity is approximated by 832 finite hexahedron elements.For the unbounded exterior fluid, a conforming discretizationwith 508 boundary elements is applied. In all these elements,quadratic serendipity shape functions are employed. It shouldbe mentioned that the experimental setup does not exactlyfulfill the clamped boundary conditions of the plate. For
Figure 4. Conforming FE–FE–BE discretization of the shellstructure as well as enclosed and surrounding fluid.
this reason model updating has been carried out by applyingpartially clamped boundary conditions.
The box-shaped structure is excited by a harmonic force,and consequently, the structural and the acoustic responses arealso harmonic. In the present study, a harmonic point forceis applied perpendicular to the outer surface of the plate atx1 = 500 mm and x2 = 500 mm. The matrix equation (32)models the overall behavior of the box-shaped shell structurein the frequency domain. It can be used to determine thefrequency response functions of the uncontrolled as well as thecontrolled system. For the comparison of the numerical resultswith the experiments (figure 5), frequency response functionsbetween the excitation force and the plate displacement atx1 = 100 mm, x2 = 200 mm are computed. Additionally,frequency response functions for the acoustic pressure in thecavity at x1 = 300 mm, x2 = 650 mm and x3 = 525 mm arecalculated.
Figure 6 compares computed and measured structuraland acoustic responses of the uncontrolled system. In thesimulations a frequency range from 40 to 200 Hz is chosento guarantee that several eigenfrequencies of the multi-coupledstructural–acoustic system are within this band. To achieve adamping effect through velocity feedback control, a feedbackgain of g = 0.05 s is selected.
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
Figure 5. Photographs of the experimental setup.
(a) (b)
Figure 6. Comparison of measured and calculated frequency response functions. (a) Displacement. (b) Sound pressure.
Table 3. Geometrical and material parameters of the piezoelectric patches PIC 151.
A comparison of the measured values and those predictedby the coupled FE–FE–BE model shows good agreement. Asshown in figure 6, the frequency range of interest includesthe first five eigenfrequencies of the coupled system. Anexamination of the computed mode shapes reveals that thefirst, third, fourth and fifth mode can be interpreted as modesmainly caused by the shell structure. Due to the presenceof the acoustic cavity and the exterior fluid, the in vacuoeigenfrequencies of the plate modes shift. The second peak infigure 6 denotes the first resonance frequency of the acousticcavity. The computed deformed shape of the box-shapedshell structure at the first five eigenfrequencies is provided infigure 7.
From the resonance peaks it can be observed that thesound pressure in the acoustic cavity is related to the vibration
of the plate. Only the fourth eigenmode of the coupled systemdoes not influence the sound pressure at the measurementposition. This is due to the position of the measurement point,which is placed close to the center of the acoustic cavity. Atthis position the radiated sound waves of the fourth plate modeshape cancel each other out.
In figures 8 and 9 the results of the controlled systemare presented. Figure 8 compares calculated and measureddisplacement responses of the uncontrolled and the controlledplate and figure 9 presents sound pressure responses for theuncontrolled and controlled case. In both figures it can beobserved that measured data and numerical predictions agreevery well. Additionally, the results show that a significantdamping effect is achieved at resonance regions due to theimplementation of the velocity feedback control. According to
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
Figure 7. Computed deformed shapes of the box-shaped shell structure.
(a) (b)
Figure 8. Uncontrolled and controlled frequency response of the plate displacement. (a) Simulation. (b) Experiment.
figure 8, attenuation of the plate displacement at the resonanceregions is about 53%. The same behavior can be seen infigure 9, where the response of the pressure amplitudes isreduced by about 41%.
As mentioned before, when the shell structure is excitedby a harmonic force, the harmonically vibrating plate radiatessound into the enclosed and surrounding fluid. Thus theacoustic pressure in radiated sound fields oscillates at a single
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
(a) (b)
Figure 9. Uncontrolled and controlled frequency response of the acoustic sound pressure. (a) Simulation. (b) Experiment.
(a) (b)
Figure 10. Sound pressure distribution of the uncontrolled system over the cross-section. (a) Simulation. (b) Measurement.
frequency. For further comparison with the experiments,steady-state acoustic pressure fields of the uncontrolled andcontrolled system are computed by solving the system ofequation (32). The following figures illustrate the computedsound pressure distribution over the cross-section of the shellstructure. Simulated and measured contour plots are obtainedfor the case that the plate is excited by a harmonic force withan amplitude of 1 N and a frequency of 60 Hz.
In both figures it can be noted that the experimentallymeasured pressure fields show high consistency with thecomputed ones. Moreover, the figures reveal that for theuncontrolled and controlled case, the sound pressure levelinside the acoustic cavity is significantly higher than outside.From the contour plots in figures 10 and 11 it can also be seenthat due to the controller influence the sound pressure level isreduced by approximately 5 dB. The attenuation demonstratesthat velocity feedback control reduces in the considered systemthe responses of acoustic pressure locally as well as globally.
From the good agreement of the results, it can beconcluded that the proposed FE–FE–BE formulation allowsthe modeling of structural–acoustic systems in combination
with control. The developed approach differs from previousapproaches, which focused either on the electromechanicalor the structural–acoustic modeling, in that it involves allsubsystems. In addition, the new approach enables not onlythe modeling of velocity feedback control, but also of otherfeedback control strategies. Moreover, there are no particularrestrictions on the shape of the lightweight structures as wellas the geometry of the surrounding acoustic fluids.
The experimental reference solutions were obtained withhardware-in-the-loop experiments using a dSPACE controllerboard which determines the necessary control outputs forthe actuators. A shaker was employed to excite the platewith a harmonic force and an accelerometer was appliedto detect the plate vibration. To measure the acousticpressure, a microphone was positioned inside the cavity. Thesound pressure distribution was measured with the help ofa uniformly distributed microphone array consisting of 3 ×7 microphones with a grid spacing of 100 mm. Duringthe measurements, the microphone array was moved severaltimes to cover the whole cross-section of the box-shaped shellstructure. Experimental and numerical determination of the
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
(a) (b)
Figure 11. Sound pressure distribution of the controlled system over the cross-section. (a) Simulation. (b) Measurement.
pressure fields was carried out with the same feedback gainas for the frequency response functions.
During the tests the box-shaped shell structure was placedin an anechoic chamber with a lower cut-off frequency of100 Hz, which ensures a free-field environment above thisfrequency. Since measurement of the acoustic fields wascarried out with a harmonic excitation of 60 Hz, the measuredpressure distribution outside the cavity was non-uniform. Moreexact measurements could be achieved if the radiating structurewas placed in an anechoic chamber for low frequencies.
Experimental testing revealed that velocity feedbackcontrol is a more robust strategy than acceleration ordisplacement feedback as far as phase shifts are concerned.Phase shifts are the main reason why the performance offeedback controllers is limited in real systems. Phase shiftsare primarily caused by the dynamic response of the installedsensors, actuators and filters as well as the dSPACE controllerboard. The attenuation achieved with the testing rig could beimproved by compensating the phase lag using the dSPACEcontroller board to modify the response of the plant within thefrequency range of interest.
6. Conclusions
In this paper, a coupled FE–FE–BE formulation has beenpresented to simulate a fluid-loaded smart lightweight structurewith surface-mounted piezoelectric actuators and sensors. Inthis approach, the FEM is applied to model the piezoelectricshell structure. For structural modeling a plane piezoelectriccomposite Mindlin-type shell element with eight nodes isused. The FEM is also adopted to predict the behavior of thefinite fluid domains around the shell structure as well as fluiddomains that are partially or totally bounded by the structure.To discretize the finite fluid domains acoustic hexahedronelements are chosen. Conforming boundary elements are usedinstead of infinite acoustic elements to model the unboundedacoustic pressure field around the shell structure and the finitefluid domains. A procedure has been presented to obtain aFE–FE–BE formulation for modeling the interaction at the
fluid–fluid and fluid–structure interfaces. The resulting systemof equations is obtained in terms of displacements of theshell structure, electric potentials of the piezoelectric layersand acoustic pressures of the fluid domains. This modelhas been constructed particularly to simulate the active noiseand vibration suppression in the frequency domain. Forthis reason, it has been shown how a control algorithm canbe implemented into the multi-coupled structural–acousticformulation to provide a closed-loop model. In the presentstudy a velocity feedback control is used to demonstrate theimplementation. In addition, collocated piezoelectric actuatorsand sensors are assumed to form a simple closed-loop controlsystem. The approach developed is very useful in order todetermine the optimal number of piezoelectric actuator/sensorpairs, the dimension of each pair and their locations on thestructure. In addition, the approach developed is easy toimplement in existing FE–BE codes, and there is also norestriction regarding the shape of the structure when usingthe proposed approach. The coupled FE–FE–BE formulationleads to a significant smaller fluid mesh compared to the pureFE model, since boundary elements can be used in the near-field. With infinite elements, which are the counterparts toboundary elements, the whole near-field has to be discretized,because infinite elements perform sufficiently well only whenthey are positioned in the far-field. In contrast to the indirectboundary element method (IBEM), which is also well suitedto model fluid-loaded shell structures, the approach developedoffers the possibility to model inhomogeneities inside finitefluid domains.
To demonstrate the applicability and validity of thedevelopments, test simulations are carried out and the resultsare compared with measurements. As a test case, a box-shapedshell structure with surface-mounted piezoelectric actuatorsand four sensors and an open rearward end is considered.It is shown that the measured values and those predictedby the coupled FE–FE–BE model are in a good agreement.It can be concluded that the proposed approach is able tomodel the uncontrolled as well as controlled behavior of fluid-loaded smart shell structures with a good quality. The present
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Smart Mater. Struct. 19 (2010) 105009 S Ringwelski and U Gabbert
developments provide a solid basis to model and to designfluid-loaded piezoelectric devices for active noise and vibrationsuppression.
Furthermore, experimental and numerical results revealedthat the radiated sound field and structural vibrations ofthe box-shaped shell structure were significantly reduced bymeans of the designed velocity feedback control. Efficiencyof the control could be further improved by introducingphase-lag compensators and by changing the position of thesensor/actuator pairs. In relation to ongoing research projects,an experimental setup consisting of a stripped engine will bebuilt in order to test the performance of the methods developedon complex industrial-like applications.
Acknowledgments
The work is supported financially by the German Federal Stateof Saxony-Anhalt and by the European Commission, in theframe of the research project ‘COmpetence in MObility’. Thissupport is gratefully acknowledged.
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