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Robust control of plate vibration via active constrained layer damping
Thin-Walled Structures 42 (2004) 427–448 www.elsevier.com/locate/tws
Robust control of plate vibration via active constrained layer damping TianXiong Liu a,b, HongXing Hua a,, Zhiyi Zhang a a
School of Mechanical Engineering, Vibration Shock and Noise Laboratory, Shanghai Jiao Tong University, Shanghai 200030, China b Chinese Academy of Space Technology, Beijing 100081, China Received 18 April 2002; received in revised form 6 March 2003; accepted 4 July 2003
Abstract In this paper, the theoretical modeling of a plate partially treated with active constrained layer damping (ACLD) treatments and its vibration control in an H1 approach is discussed. Vibration of the flat plate is controlled with patches of ACLD treatments, each consisting of a viscoelastic damping layer which is sandwiched between the piezo-electric constrained layer and the host plate. The piezo-electric constrained layer acts as an actuator to actively control the shear deformation of the viscoelastic damping layer according to the vibration response of the plate excited by external disturbances. In the first part of this paper, the Mindlin–Reissner plate theory is adopted to express the shear deformation characteristics of the viscoelastic damping layer, meanwhile GHM (Golla–Hughes–McTavish) model of viscoelastic damping material and FEM (finite element model) are incorporated to describe the dynamics of the plate partially treated with ACLD treatment. In the second part, particular emphasis is placed on the vibration control of the first four modes of the treated plate using H1 robust control method. For this purpose, an H1 robust controller is designed to accommodate uncertainties of the ACLD parameters, particularly those of the viscoelastic damping core which arise from the variation of the operation temperature and frequency. Disturbances and measurement noise are rejected in the closed loop by H1 robust controller. In the experimental validation, external disturbances of different types are employed to excite the treated plate. The results of the experimental clearly demonstrate that the proposed modeling method is correct and the ACLD treatments are very effective in fast damping out the structural vibration as compared to the conventional passive constrained layer damping (PCLD). # 2003 Elsevier Ltd. All rights reserved.
Corresponding author. Tel.: +86-21-653-70622. E-mail address: [email protected] (H. Hua).
0263-8231/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8231(03)00131-9
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Keywords: Active constrained layer damping; Finite element method; H1 control; Viscoelastic damping material
1. Introduction Viscoelastic shear layers integrated into structures have been widely used in engineering to suppress the vibration and noise of structures [1]. The oscillating energy, mainly concentrated at resonance peaks, is dissipated by the shearing deformation of the viscoelastic layer. Surface damping treatments are often effective when vibration frequency is high in thin-walled structures such as beams, plates and shells. The effective suppression of lower frequency modes usually requires the addition of an active vibration control scheme to augment the passive treatment [2]. One method that has been used successfully is to utilize piezoelectric patch to replace the conventional stiff constraining layer in the structure of constrained damping treatment. Shen and liao [3] has proposed the idea of ACLD to study the hybrid control of the vibration of a beam with a cover sheet of piezoelectric ceramics, and interesting results have been obtained. Also, Shen [4] investigated the vibration control of a plate attached completely with ACLD. The control used in his work is a proportional-plus-derivative control, but it is difficult to realize in real situations. If a lumped parameter model is derived, its order usually needs to be reduced prior to controller design. Balas [5] investigated the so-called spillover effects which result from ignoring higher order modes when implementing feedback control. It has been shown that spillover effects are liable to degrade or even destabilize the closed-loop system. It has long been recognized that PCLD is most effective in suppressing vibration of higher frequency modes. Using a hybrid approach, it appears feasible to use active control to suppress the vibration of lower modes, while the passive layer reduces spillover from the higher frequency modes. Baz [6] presented a variational formulation of the dynamics of beam which is fully treated with ACLD treatments, and H2 robust control law was utilized to control the vibration of the beam. In this paper, the governing equations of a multilayer plate with viscoelastic damping layer and piezoelectric constrained coversheet are formulated. At first, GHM and FEM are incorporated to describe the dynamics of plate partially treated with ACLD, then H1 robust control of the plate vibration is investigated. Experimental results demonstrate that the H1 robust control law cannot only effectively suppress the vibration of lower modes of plate, but also avoid spillover from the higher frequency modes. 2. Finite element modeling 2.1. Strategy The aim of this paper is to develop a model-based approach to implement the vibration control of a cantilever plate with ACLD treatments. Modeling of the
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plate is conveniently performed using finite element analysis in order to determine the equations of motion in terms of mass, damping, and stiffness matrices. Just as D.F. Golla [6] pointed out, ‘‘Arguably the most powerful and most popular technique for solving equations of motion is the finite element method. Any methodology for modeling material damping that does not merge naturally with the finite element method will never be incorporated into engineering practice’’. Before designing an optimal controller, model reduction based on FEM is necessary to obtain a lower order model. So an integrated system reduction scheme is proposed in this paper. First, the dynamic condensation method [7] is utilized to condense the model order in physical space, then Schur balanced model reduction [8] is utilized to reduce the model order in state space. Such integrated system reduction scheme combines the merit of the dynamic condensation and the balanced model reduction. The reduced model not only has lower order, but also is controllable and observable, which is necessary to design a perfect controller for feedback control. 2.2. Kinematic relationships The system under consideration involves the host plate to which a viscoelastic layer and a further piezo-electric constrained layer are added. Rectangular plate elements are used as shown in Fig. 1. The coordinate system and nodes of the element are illustrated in Fig. 2 and the kinematic relationships in Fig. 3. Basic hypothesis of modeling are as follows: (1) The analysis of the host plate and the piezo-electric constrained layer employs the assumptions of thin plate theory. Moreover, the Mindlin–Reissner plate theory is adopted to express the shear deformation characteristics of the viscoelastic layer. (2) The rotational inertia is assumed to be negligible. (3) The damping is considered only in shear deformation of the viscoelastic layer. (4) The transverse displacement in direction z is assumed to be
Fig. 1. A rectangular element of the ACLD plate.
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Fig. 2. Coordinate system and nodes of the element.
equal for all layers. (5) The deformations between layers are perfect continuity. (6) The applied voltage is uniform along the plate plane. 2.3. Degrees of freedom and shape functions At each node of an element, there are seven degrees of freedom, four longitudinal displacements of the constrained layer and the host plate, one transverse deflection w, two rotational displacements w;x and w;y of the deflection line. These displacements can be gathered together to form a vector: T ui ¼ uci vci upi vpi wi w;xi w;yi ; i ¼ 1; 2; 3; 4 ð1Þ where (,) is the partial derivative with respect to subscript variable, uci , vci and upi ,
Fig. 3. Kinematic relationships of the elements at x–z plane.
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vpi are longitudinal displacements of the constrained layer and the host plate, respectively, in the x-z and y-z planes. uA and uB are the longitudinal displacement of interface between VEM layer. Given this displacement vector and the kinematic relationships of the elements, the shear strains bx and the shear strains by of viscoelastic layer rotating around the x and y axes can be given as bx ¼
uc u p d @w ; þ dv @x dv
by ¼
vc vp d @w þ dv @y dv
ð2Þ
where dc dp þ 2 2 the longitudinal displacements uvi ; vvi of the viscoelastic layer as 1 dc dP @w 1 dc dP @w uc þ up þ vc þ vp þ uv ¼ ; vv ¼ 2 @x 2 @y 2 2 d ¼ dv þ
ð3Þ
where dv , dP and dc represent the thickness of VEM, base plate and constrained layer, respectively. The spatial distribution of the longitudinal displacements uc , vc and up , vp w and the transverse displacement w over any element of the treated plate are assumed to be given by: uc ¼ a1 þ a2 x þ a3 y þ a4 xy
ð4Þ
vc ¼ a5 þ a6 x þ a7 y þ a8 xy
ð5Þ
up ¼ a9 þ a10 x þ a11 y þ a12 xy
ð6Þ
vp ¼ a13 þ a14 x þ a15 y þ a16 xy
ð7Þ
w ¼ a17 þ a18 x þ a19 y þ a20 x2 þ a21 xy þ a22 y2 þ a23 x3 þ a24 x2 y þ a25 xy2 þ a26 y3 þ a27 x3 y þ a28 xy3
ð8Þ
w;x ¼
@w ; @y
w;y ¼
@w @x
ð9Þ
where the constant fa1 ; a2 ; a3 ; . . . ; a28 g are determined in terms of the 28 components of the nodal displacements vector ui of the ith element which is bounded by the nodes 1, 2, 3, and 4 as shown in Fig. 2. The element displacements vector D of the ith element is given by D ¼ ½ u1
u2
u3
u4 T
ð10Þ
Therefore the displacement u ¼ uc vc up inside the ith element can be determined from u¼
uc
vc
up
vp
w
w;x
w;y
T
vp
¼ ND
w
w;x
w;y
at any location
ð11Þ
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where N ¼ ½N 1 N 2 N 3 N 4 N 5 N 6 N 7 T . N is the spatial interpolating matrix. N 1 ; N 2 ; N 3 ; N 4 ; N 5 ; N 6 ; N 7 are the spatial interpolating vectors corresponding to uc ; vc ; up ; vp ; w; w;x ; w;y , respectively. 2.4. Energy of plate with ACLD treatment 2.4.1. Potential energy The element potential energy associated with the extension, bending and shearing of the piezoelectric layer, the host plate, and the viscoelastic damping layer are as follows: ð 1
1 Pc ¼ rcx ecx þ rcy ecy þ scxy ccxy dv ¼ uT kc u ð12Þ 2 v 2 where kc is the element stiffness matrices of the piezoelectric layer. ðaðb
d3 dxdydz þ c 12 3
ðaðb
B T Dc B dxdydz 2 3 1 lc 1 N 1;x Ec 4 5 lc 1 0 5 B c ¼ 4 N 2;y Dc ¼ 1 l2c c N þ N 0 0 1l 2;x 2 ð 1;y 1
1 Pp ¼ rpx epx þ rpy epy þ spxy cpxy dv ¼ uT kp u 2 v 2
kc ¼ dc 2
0 0
BTc Dc Bc
0 0
2
3 N 5;xx B ¼ 4 N 5;yy 5 2N 5;xy
ð13Þ where kp is the element stiffness matrices of the host plate. ðaðb
d3p B TP Dp Bp dxdydz þ 12 0 3
ðaðb
B T Dp B dxdydz 2 3 1 lp 1 N 3;x E p 4l 5 0 5 B p ¼ 4 N 4;y Dp ¼ p 1 1 l2p 1lp N 3;y þ N 4;x 0 0 2 ð 1
1 T Pv ¼ rvx evx þ rvy evy þ svxy cvxy dv ¼ u kve u 2 v 2
kp ¼ dp 2
0
0 0
ð14Þ
where kve is one of the element stiffness matrices of the viscoelastic damping layer. ðaðb ð ð d3 a b T kve ¼ dv B Tv Dv Bv dxdydz þ v B Dv B dxdydz 12 0 0 2 0 0 3 2 3 1 lv 1 N 8;x E v 4 5 lv 1 0 5 B v ¼ 4 N 9;y Dv ¼ 2 1 l v v 0 0 1l 2 ðN 8;y þ N 9;x 1 1 Pb ¼ b Gb þ by Gby d v ¼ uT kvb u 2 v x x 2
ð15Þ
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where kvb is one of the element stiffness matrices of the viscoelastic damping layer.
kvb ¼
G dv
ðaðb 0 0
ðN T10 N 10 þ N T11 N 11 Þdxdy ¼ Gkv
Ec ; Ep and Ev are the Young’s modulus of the piezoelectric layer, the host plate and the viscoelastic damping layer respectively. G is shear modulus of viscoelastic damping layer. 2.4.2. Kinetic energy The element kinetic energy of the piezoelectric constrained layer, the host plate, and the viscoelastic damping layer (Tc ; Tp ; Tv , respectively) can be expressed as: 1 Tc ¼ 2
2 # ð " 2 @w @uc 2 @vc 1 qc þqc þqc dv ¼ u_ T mc u_ @t 2 @t @t v
ð16Þ
where mc is the element mass matrix of the piezoelectric constrained layer. ða ðb
T mc ¼ qc dc N 5 N 5 þ N T1 N 1 þ N T2 N 2 dxdy # ð " a b 2 @up 2 @vp 2 1 @w 1 Tp ¼ q þqp þqp dv ¼ u_ T mp u_ 2 v p @t 2 @t @t
ð17Þ
where mp is the element mass matrix of the host plate. ða ðb
T mp ¼ qp dp N 5 N 5 þ N T3 N 3 þ N T4 N 4 dxdy # ð " a b 2 1 @w @uv 2 @vv 2 1 Tv ¼ qv þqv þqv dv ¼ u_ T mv u_ 2 v @t 2 @t @t ða ðb
T mv ¼ qv dv N 5 N 5 þ N T8 N 8 þ N T9 N 9 dxdy
ð18Þ
a b
where mv is the element mass matrix of the viscoelastic damping layer, and qc ; qp and qv are the Young’s modulus of the piezoelectric layer, the host plate and the viscoelastic damping layer respectively. It is important to note that during the calculation of the element energy The shape function N i of corresponding displacements should be substituted into the corresponding equation.
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2.5. Mass matrices and stiffness matrices of an ACLD element According to the above equations, the mass matrices and stiffness matrices of an ACLD element can be given as follow m ¼ mp þ mc þ mv
ð19Þ
k ¼ kp1 þ kp2 þ kc1 þ kc2 þ kv1 þ kv2 þ kvb
ð20Þ
Eq. (20) can be rewritten as k ¼ ke þ kvb
ð21Þ
where ke ¼ kp1 þ kp2 þ kc1 þ kc2 þ kv1 þ kv2 . 2.6. Work done by applied voltage and external disturbance force Supposing deformation of the piezoelectric constrained layer meets the Kirchhoff hypothesis, the mechanics condition can be analyzed according to the plane stress condition. It is usually supposed that the polarization direction of the PZT is the thickness direction (Z axis). According to Refs [9] and [10], the relationship of the stress and strain can be given as
rcx rcy sxy ¼ Dc ecx ecy cxy Dc fd31 d32 d33 gE3 ; 2 3 1 lc 0 7 Ec 6 6 lc 1 0 7 Dc ¼ 4 5 1 lc 1 lc 0 0 2
ð22Þ
where rcx ; rcy ; sxy are plane stress. ecx ; ecy ; cxy are plane strain. Ec is the elastic modulus of PZT layer. E3 is the strength of electric field acting on the surface of PZT layer. The constants d31 ; d32 and d33 are the piezoelectric strain coefficients in the x, y and z directions. The formation of control forces generated by the control voltage is developed with the same approach described in Refs. [9] and [10]. The work done by the inplane piezoelectricity force f c is given by wc ¼ DT f c ¼
1 2
ð rce ec1 dv
ð23Þ
v
Substituting the shape function N i of corresponding displacements into Eq. (23),
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we get 1 wc ¼ E3 dc 2
3T 2 3 N 1 ;x d31 4 N 2 ;y 5 Dc 4 d32 5Ddxdy b N ; þN ; 0 1 y 2 x
ða ðb a
2
ð24Þ
So, the force vectors f c can be defined as 1 f c ¼ E3 dc 2
3T 2 3 N 1 ;x d31 4 N 2 ;y 5 Dc 4 d32 5dxdy: b N ; þN ; 0 1 y 2 x
ða ðb a
2
ð25Þ
The work done by the external disturbance force f is as follows wf ¼ DT f :
ð26Þ
2.7. Model of plate with partially covered ACLD Using Lagrange equation, the model of plate with partially covered ACLD treatments can be given as € þ ke D þ kvb D ¼ f c þ f : mD
ð27Þ
2.8. GHM method for constrained layer damping In this section, the GHM method proposed by Ref. [11] is introduced to model the viscoelastic damping material. The internal auxiliary coordinates are introduced by GHM method to describe the dynamic characteristic of viscoelastic damping material. The GHM method requires that the complex modulus of the viscoelastic core should be represented as a series of minioscillator terms. The complex modulus can be rewritten in Laplace domain as: " ~ ðsÞ ¼ G sG
1
N X 1þ ak k¼1
^ ks s2 þ 2^fk x _ ^2 s2 þ 2^fk xk þ x
# ð28Þ
k
where the factor G1 is the equilibrium value of the modulus, i.e. the final value of the relaxation function, G(t). The hatted terms (mini-oscillator terms) are obtained from the curving fitting of the complex modulus data for a particular viscoelastic damping material. Each mini-oscillator term is a second-order rational function ^ k ; ^fk g, which govern the shape of the involving three positive constants fak ; x modulus function in the complex s-plane. The number of the mini-oscillator terms depends on the high or low frequency dependence of the complex modulus.
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The governing equation (27) can be expressed in Laplace domain as follows:
~ ðsÞkvv DðsÞ ¼ f c ðsÞ þ f ðsÞ: s 2 m þ ke þ s G
ð29Þ
^ is introduced such that: Now, an auxiliary coordinates Z ^ ðsÞ ¼ Z
^2 x DðsÞ: _ ^2 s 2 þ 2^ nx þ x
ð30Þ
Using this new auxiliary coordinates, a second order equation in time domain can be obtained by the inverse Laplace transform: q ¼ fc þ f: € m q þ c q_ þ k
ð31Þ
, damping matrix c and mass matrix m are given in time The stiffness matrix k domain by: 3 2 m 0 ::: 0 .. 7 60 a 1 K 0 . 1 2 7 6 ^ x 7 6 1 7 6 ¼ 6. m .. 7 . 7 6. 0 . 0 5 4 1 0 0 aN 2 K ^N 3 x 2 0 0 ::: 0 ^ .. 7 6 7 6 0 a1 2n1 K 0 . 7 6 ^1 x 7 6 c ¼ 6 . 7 . .. 0 7 6 .. 0 7 6 4 2^ nN 5 0 0 aN K ^N x 3 2 N P ~ ak a1 R ::: aN R 7 6 ke þ k 1 þ k¼1 7 6 7 T ¼6 a1 K 0 0 k 7 6 a1 R 7 6. . . 5 4 .. 0 . 0 T aN R 0 0 a K 0 1 N 0 1 f fc B0C B0 C D C B C q¼ fc ¼ B @ ... A f ¼ @ ... A Z 0
0
~ ¼ G 1 kvv , kvv ¼ Rv Kv RT , K ¼ G 1 Kv , R ¼ Rv K, Z ¼ RT Z ^ ; k¼ where k v v 1; 2; . . . ; N and Kv is a diagonal matrix of the nonzero eigenvalues of kvv and the corresponding orthonormalized eigenvectors from the columns of Rv . Eq. (31) describes the dynamic characteristic of a single plate with partial ACLD elements. Assembling the corresponding equations for different elements and
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applying proper boundary conditions yield the overall equation for the entire plate with partial ACLD elements. The overall equation for the entire plate with partial ACLD elements is as follows: € þ C x_ þ Kx ¼ F c w þ F d d Mx
ð32Þ
where, M, C and K denote the mass matrix, damping matrix and stiffness matrix respectively in the generalized coordinated system. F d is the distribution matrix of external disturbance force. F c is the distribution matrix of control force. w is control force vector caused by the applied voltage. d is the external disturbance force vectors. x is a new generalized coordinate.
3. Development of the robust controller 3.1. Overview We can now rewrite the overall equation of motion for the entire plate with partial ACLD elements in state space, calculate the frequency or time response and perform the design of controller. It is worth noting that the dimension of the overall motion in Eq. (32) must be reduced further before proceeding to the controller design. Model reduction is necessary to design a low order controller. So, it is necessary to perform the integrated system reduction scheme. In this section, particular emphasis is placed on the control of the first four modes of vibration of the treated plate using H1 robust control laws. For this purpose, an H1 robust controller is designed to accommodate uncertainties of the ACLD parameters, particularly those of the viscoelastic damping core which arise from the variation of the operating temperature and frequencies. Noise and external disturbances are taken into account in the design of the controller. In this study, the small gain theory is utilized to guide the selection of the controller gain in order to ensure system stability. The controller is designed to guarantee the stability of the closed loop in the presence of parametric uncertainties which may result from variation of the properties of the viscoelastic damping layer due to its operation over wide temperature and frequency ranges. At the same time, the controller should ensure that the disturbance rejection capability of the treatments is maximized over a desired frequency band. Eq. (32) can be rewritten in the standard state space form as: X_ ¼ AX þ Bw þ Dd y ¼ C1 X z ¼ C2 X
ð33Þ
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where 0 I 0 ; B ¼ ; M 1 K M 1 C M 1 F x X¼ ; C1 ¼ ½ cy 0 ; C2 ¼ ½ cz 0 : x_
A¼
D¼
0 ; M 1 Fd
3.2. Analysis of uncertainty ACLD treatments have been successfully utilized as effective means in damping out the vibration of various flexible structures. The effectiveness of the ACLD treatments is validated experimentally and theoretically using proportional and/or derivative feedback of the transverse deflection or the slope of the deflection line of the base structure [12]. Shen [13] used optimal control strategies devised by Baz and Ro [14] to avoid instability problems for structures with partial and full ACLD treatments. In all these attempts, no effort has been exerted to accommodate the uncertainties of the ACLD parameters. Also, in all these studies controllers are designed without any provisions for rejecting external disturbances. Moreover, no efforts were made to consider the system reduction scheme. Only recently, Baz [15] has theoretically developed a robust H2 controller to control the ACLD treatments in the presence of parameter uncertainty and external disturbances. Two kinds of uncertainties can be defined. The first one coming from the reduction scheme in frequency domain can be called frequency uncertainty, and the second one from characteristic variation of material can be referred to as parameter uncertainty. Fig. 4 shows a block diagram of a feedback system with uncertainties. Transfer functions of the controller, system and uncertainty are denoted by K, G and D, respectively. 3.3. Design of the robust controller Fig. 5 shows a block diagram of an H1 robust controller K that stabilizes the ACLD plate system G in the presence of frequency uncertainty and parameter uncertainty, when the system is subjected to external disturbances.
Fig. 4. Control structure of feedback with uncertainty.
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Fig. 5. H1 control mixed sensitivity design.
The controller K is designed to minimize the H1 norm of the transfer function Syd from the external disturbance d to the transverse displacement y at the free end of the plate, to guarantee the optimal disturbance rejection performance of the closed loop. Transfer function Syd denotes sensitivity function which represents the performance of the closed loop to suppress external disturbance. Generally, external disturbances mainly distribute in the band of low frequency, we can expect that the sensitivity function is small in this band. Mathematically, such a design index can be given as follows: kW1 ðsÞS ðsÞk1 < 1:
ð34Þ
From Fig. 5, we know the characteristic of the weighting function W1 ðsÞ determines the amount of damping and frequency response of the closed loop system. According to the small gain theory of robust control, the stability condition of the closed loop system for additive uncertainty is as follows kDðsÞT ðsÞk1 < 1:
ð35Þ
The transfer function from the external disturbance d to the input control signal u is denoted as Tud . The weighting function W2 ðsÞ can be selected to embed the additive uncertainty. Such design index can be denoted as kW2 ðsÞT ðsÞk1 < 1: So, the index of H1 robust control can be written as follows W1 ðsÞS ðsÞ W2 ðsÞT ðsÞ < 1: 1
ð36Þ
ð37Þ
The optimal H1 controller which meets Eq. (37) can be obtained by c iteration technique [16]. The choice of weighting functions ensures that the controller does not destabilize higher frequency modes, and also attenuates control signals at lower frequencies. However, the appropriate selection of weighting functions over the
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desired frequency range is not explicitly related to the performance objectives in a straightforward manner. Many trials of weighting functions are usually required in order to reach desired performance objectives. 4. Validation of the modeling and robust controller 4.1. Frequency response of the plate with the ACLD system The theoretical predictions of the ACLD model are compared with the experimental performance of an aluminum plate which is partially treated with a viscoelastic damping layer. The measurement of frequency response in this section also aims at demonstrating the accuracy of modeling and the merits of using PCLD to suppress the vibration of the flat plate. A cantilever sandwich plate with partial ACLD structure is depicted in Fig. 6. Elements 5 and 10 are ACLD elements. The other elements are ordinary thin plate elements. It is important to note that the first four modes of the plate are bending and torsion mode, So, the positions of ACLD treatments are elements 5 and 10 where strain is maximum. The physical and geometric properties of the plate with partially treated ACLD are as follows: The material parameters of the constrained layer are qc ¼ 7500 kg=m3 , Ec ¼ 3:6 1011 N=m2 , lc ¼ 0:3. The material parameters of the host plate layer are qp ¼ 2700 kg=m3 , Ep ¼ 9 1010 N=m2 , lp ¼ 0:3. The material parameters of the viscoelastic damping layer are qv ¼ 1250 kg=m3 , Ev ¼ 2 107 N=m2 , lv ¼ 0:3. The material of the viscoelastic damping layer is ZN-1. The relax function of VEM is derived from Ref. [17], GðtÞ ¼ 3:44 þ 7:089e193:39t þ 231:212e16345t þ 1744:4:2e485916:4t : The boundary condition: one short side of the base plate layer is fixed and the others are free. The dimensions of the sandwich plate are L1 ¼ 0:37 m, L2 ¼ 0:28 m,
Fig. 6. Cantilever sandwich plate with partial ACLD.
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Fig. 7. Experimental set-up and instruments of frequency response function.
dc ¼ dv ¼ 0:001 m, dp ¼ 0:0025 m. The general arrangement of the frequency response test facility is shown in Fig. 7. A photograph of the set-up for frequency response measurement and control experiment is shown in Fig. 8. Firstly, frequency responses of the plate are presented by simulation. Fig. 9 shows a comparison between the FRF of the plate with PCLD elements and that without PCLD elements. In the case of the PCLD, the control loop that regulates the interaction between the piezoelectric actuator and the displacement sensor is kept open. The corresponding time responses are displayed in Fig. 10, where the solid line represents the response of the bare aluminum plate, while the dashed line
Fig. 8. Photograph of the experimental set-up used in FRF measurement and vibration control.
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Fig. 9. Frequency response functions of a cantilever plate and a cantilever sandwich plate with partial ACLD.
Fig. 10. The impulse responses of a cantilever plate and a cantilever sandwich plate with partial ACLD.
Table 1 Natural frequencies given by ANSYS5.6 and the experiment (E ¼ 910 Pa) Order
Calculation (Hz) A PCLD plate
ANSYS5.6 (Hz) B bare plate
FRF (Hz) PCLD plate
Error (%) A
B
1 2 3
17.97 55.56 111.49
17.58 53.93 109.67
17.0 47.0 111.5
5.71 18.21 0.01
3.41 14.74 1.64
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Fig. 11. Result of natural frequency of the FRF of PCLD plate.
represents the response of the PCLD plate. It is evident that the PCLD results in excellent attenuation compared to the pure aluminum plate. The natural frequencies of the plate partially treated PCLD elements are shown in Table 1. Secondly, frequency responses of the plate are presented by experimental testing. Figs. 11 and 12 show the frequency responses of the plate partially treated with PCLD elements. It is evident that the PCLD treatment has been very effective in attenuating the structural vibration of the plate over a wide frequency range. The results calculated with our modeling method are very close to the results of experimental test. So, it has been demonstrated that the model is accurate. We can utilize the dynamic model to design an H1 robust controller. 4.2. Investigation of performance in relation to ACLD The dynamic model of the plate with partial ACLD elements has been developed to enable ‘active damping’ as described in this paper. The vibration reduction
Fig. 12. The frequency response function of PCLD plate.
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Fig. 13. Experimental layout of control test.
achieved for a cantilever plate with partial ACLD elements subject to a base excitation is investigated using H1 robust control. The experimental layout is shown in Fig. 13. The photograph reflecting the relative position of the actuator and sensor is shown in Fig. 14. Suppose the node of intersection of elements 6, 7, 11 and 12 of the plate is the location of external exciting force. The direction of force is along the z-axis, the vertical direction. Elements 5 and 10 are ACLD elements. Figs. 15–18 show the comparison between the performance of the ACLD and PCLD in time domain when the plate with partial ACLD structure is excited with impulse, sinusoidal disturbances of the first resonance, square wave and the white noise. The period of square wave is 1 s. The thin solid line represents the response of the open loop system (without the action of voltage) and the thick dashed line represents the response of the closed system in Figs. 15–18.
Fig. 14. The photograph of the relative position of actuator and sensor.
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Fig. 15. Impulse response of the controlled structure.
It can be seen that the added damping obtained by applying an active constraining layer reduces the response of the plate considerably. It is evident that the ACLD results in excellent attenuation compared to conventional PCLD treatment. The consistent effectiveness of the ACLD is attributed to its built-in feedback that senses the adverse effects of the vibration of the plate. From the above discussion, it is certain that the new method presented for building the dynamic equation of sandwich plate with VEM core has good accuracy and reliability. The ACLD treatments combine the simplicity and reliability of PCLD with low weight and
Fig. 16. First resonance response of the controlled structure.
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Fig. 17. Response of the structure excited by square wave.
high efficiency of active control to attain high damping characteristics over a broad frequency band. The active component will provide adjustable damping, whereas the passive component will enhance gain and phase margins. Therefore, the spillover effects resulted from model reduction can be overcome.
Fig. 18. Response of the structure excited by white noise.
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5. Conclusions The theoretical analysis and experimental measurements of a plate partially treated with ACLD elements have been compared in this study. Based on experimentally validated finite element model, it is developed to describe the dynamics of the treated plate The results of FRF measurements have confirmed the accuracy of the new modeling method, and there is good agreement in the theoretical and experimental analysis of natural frequencies. It is evident that the proposed modeling method is accurate. The model is utilized to design an H1 robust controller which is stable in the presence of uncertainties of modeling and parameters, and ensures optimal disturbance rejection capability. Experimental results are presented to validate the effectiveness of the H1 robust controller in damping out structural vibrations when the ACLD treatments operate over a wide frequency band. The results show that the ACLD treatment has resulted in effective attenuation of plate vibrations. Hence, the theoretical and experimental techniques developed in this study constitute effective tools in designing and predicting the performance of smart laminated structures that can be used in many engineering applications. Acknowledgements The authors would like to thank the key scientific laboratory of dynamics of helicopter flexbeams of national defenses for partial funding under contract No. JS 52.4.3. Special thanks are due to Professor Mei Weisheng of the State Key Laboratory of Dynamics of Helicopter Flexbeams of Nanjing University of Aeronautics and Astronautics. References [1] Dai D. Application of viscoelastic material in engineering. Beijing: Tsinghua University Press; 1991. [2] Chantalakhana C, Stanway R. Active constrained layer damping of clamped-clamped plate vibrations. Journal of Sound and Vibration 2001;241(5):755–77. [3] Shen IY. Hybrid damping through intelligent constrained layer treatments. Journal of Vibration and Acoustics 1994;116(3):341–9. [4] Shen IY. Bending vibration control of composite plate structures through intelligent layer treatments. Proceedings SPIE 1994;2193:115–25. [5] Balas MJ. Feedback control of flexible systems. IEEE Transactions on Automatic Control 1978;AC-23:673–9. [6] Golla DF, Hughes PC. Dynamics of viscoelastic structures: a time-domain, finite element formulation. Journal of Applied Mechanics 1985;52:897–907. [7] ZuQing Q. A reduced order modeling technique for tall building with active tuned mass damper. Earthquake Engieering and Structural Dynamics 2001;30:349–62. [8] Safanov MG, Chiang RY. A Schur method for balanced model reduction. IEEE Trans Automatic Control 1989;34(7):729–33. [9] Tzou HS, Anderson GL. Intelligent structure system. Kluwer Academic Publishers; 1992. [10] Crawley EF, Lazarust KB. Induced strain actuation of isotropic and anisotropic plates. AIAA Journal 1991;29(6):944–51.
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[11] McTavish DJ, Hughes PC. Modeling of linear viscoelastic space structures. J of Vibration and Acoustics 1993;115:103–10. [12] Lesieutre UL. A finite element for beams having segmented active constrained layers with frequency-dependent viscoelastics. Journal of Smart Material and Structures 1996;5:615–27. [13] Shen IY. Hybrid damping through intelligent constrained layer damping treatments. Journal of Vibration and Acoustics 1994;116(3):341–9. [14] Baz A, Ro J. Optimum design and control of active constrained layers damping. Journal of Vibration and Acoustics 1995;117(3):135–44. [15] Baz A, Ro J. Robust control of active constrained layers damping. Journal of Sound and Vibration 1998;211(3):467–80. [16] Doley JC, Glover K, Khargonekar PP, et al. State space solutions to standard H2 and H1 control problems. IEEE Trans Auto Contr 1989;34:831–47. [17] Shen S, Leng W. Dynamic analysis of viscoelastic damping structures. Ship Building of China 1996;133(2):44–52.
Robust control of plate vibration via active constrained layer damping TianXiong Liu a,b, HongXing Hua a,, Zhiyi Zhang a a
School of Mechanical Engineering, Vibration Shock and Noise Laboratory, Shanghai Jiao Tong University, Shanghai 200030, China b Chinese Academy of Space Technology, Beijing 100081, China Received 18 April 2002; received in revised form 6 March 2003; accepted 4 July 2003
Abstract In this paper, the theoretical modeling of a plate partially treated with active constrained layer damping (ACLD) treatments and its vibration control in an H1 approach is discussed. Vibration of the flat plate is controlled with patches of ACLD treatments, each consisting of a viscoelastic damping layer which is sandwiched between the piezo-electric constrained layer and the host plate. The piezo-electric constrained layer acts as an actuator to actively control the shear deformation of the viscoelastic damping layer according to the vibration response of the plate excited by external disturbances. In the first part of this paper, the Mindlin–Reissner plate theory is adopted to express the shear deformation characteristics of the viscoelastic damping layer, meanwhile GHM (Golla–Hughes–McTavish) model of viscoelastic damping material and FEM (finite element model) are incorporated to describe the dynamics of the plate partially treated with ACLD treatment. In the second part, particular emphasis is placed on the vibration control of the first four modes of the treated plate using H1 robust control method. For this purpose, an H1 robust controller is designed to accommodate uncertainties of the ACLD parameters, particularly those of the viscoelastic damping core which arise from the variation of the operation temperature and frequency. Disturbances and measurement noise are rejected in the closed loop by H1 robust controller. In the experimental validation, external disturbances of different types are employed to excite the treated plate. The results of the experimental clearly demonstrate that the proposed modeling method is correct and the ACLD treatments are very effective in fast damping out the structural vibration as compared to the conventional passive constrained layer damping (PCLD). # 2003 Elsevier Ltd. All rights reserved.
Corresponding author. Tel.: +86-21-653-70622. E-mail address: [email protected] (H. Hua).
0263-8231/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8231(03)00131-9
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Keywords: Active constrained layer damping; Finite element method; H1 control; Viscoelastic damping material
1. Introduction Viscoelastic shear layers integrated into structures have been widely used in engineering to suppress the vibration and noise of structures [1]. The oscillating energy, mainly concentrated at resonance peaks, is dissipated by the shearing deformation of the viscoelastic layer. Surface damping treatments are often effective when vibration frequency is high in thin-walled structures such as beams, plates and shells. The effective suppression of lower frequency modes usually requires the addition of an active vibration control scheme to augment the passive treatment [2]. One method that has been used successfully is to utilize piezoelectric patch to replace the conventional stiff constraining layer in the structure of constrained damping treatment. Shen and liao [3] has proposed the idea of ACLD to study the hybrid control of the vibration of a beam with a cover sheet of piezoelectric ceramics, and interesting results have been obtained. Also, Shen [4] investigated the vibration control of a plate attached completely with ACLD. The control used in his work is a proportional-plus-derivative control, but it is difficult to realize in real situations. If a lumped parameter model is derived, its order usually needs to be reduced prior to controller design. Balas [5] investigated the so-called spillover effects which result from ignoring higher order modes when implementing feedback control. It has been shown that spillover effects are liable to degrade or even destabilize the closed-loop system. It has long been recognized that PCLD is most effective in suppressing vibration of higher frequency modes. Using a hybrid approach, it appears feasible to use active control to suppress the vibration of lower modes, while the passive layer reduces spillover from the higher frequency modes. Baz [6] presented a variational formulation of the dynamics of beam which is fully treated with ACLD treatments, and H2 robust control law was utilized to control the vibration of the beam. In this paper, the governing equations of a multilayer plate with viscoelastic damping layer and piezoelectric constrained coversheet are formulated. At first, GHM and FEM are incorporated to describe the dynamics of plate partially treated with ACLD, then H1 robust control of the plate vibration is investigated. Experimental results demonstrate that the H1 robust control law cannot only effectively suppress the vibration of lower modes of plate, but also avoid spillover from the higher frequency modes. 2. Finite element modeling 2.1. Strategy The aim of this paper is to develop a model-based approach to implement the vibration control of a cantilever plate with ACLD treatments. Modeling of the
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plate is conveniently performed using finite element analysis in order to determine the equations of motion in terms of mass, damping, and stiffness matrices. Just as D.F. Golla [6] pointed out, ‘‘Arguably the most powerful and most popular technique for solving equations of motion is the finite element method. Any methodology for modeling material damping that does not merge naturally with the finite element method will never be incorporated into engineering practice’’. Before designing an optimal controller, model reduction based on FEM is necessary to obtain a lower order model. So an integrated system reduction scheme is proposed in this paper. First, the dynamic condensation method [7] is utilized to condense the model order in physical space, then Schur balanced model reduction [8] is utilized to reduce the model order in state space. Such integrated system reduction scheme combines the merit of the dynamic condensation and the balanced model reduction. The reduced model not only has lower order, but also is controllable and observable, which is necessary to design a perfect controller for feedback control. 2.2. Kinematic relationships The system under consideration involves the host plate to which a viscoelastic layer and a further piezo-electric constrained layer are added. Rectangular plate elements are used as shown in Fig. 1. The coordinate system and nodes of the element are illustrated in Fig. 2 and the kinematic relationships in Fig. 3. Basic hypothesis of modeling are as follows: (1) The analysis of the host plate and the piezo-electric constrained layer employs the assumptions of thin plate theory. Moreover, the Mindlin–Reissner plate theory is adopted to express the shear deformation characteristics of the viscoelastic layer. (2) The rotational inertia is assumed to be negligible. (3) The damping is considered only in shear deformation of the viscoelastic layer. (4) The transverse displacement in direction z is assumed to be
Fig. 1. A rectangular element of the ACLD plate.
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Fig. 2. Coordinate system and nodes of the element.
equal for all layers. (5) The deformations between layers are perfect continuity. (6) The applied voltage is uniform along the plate plane. 2.3. Degrees of freedom and shape functions At each node of an element, there are seven degrees of freedom, four longitudinal displacements of the constrained layer and the host plate, one transverse deflection w, two rotational displacements w;x and w;y of the deflection line. These displacements can be gathered together to form a vector: T ui ¼ uci vci upi vpi wi w;xi w;yi ; i ¼ 1; 2; 3; 4 ð1Þ where (,) is the partial derivative with respect to subscript variable, uci , vci and upi ,
Fig. 3. Kinematic relationships of the elements at x–z plane.
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vpi are longitudinal displacements of the constrained layer and the host plate, respectively, in the x-z and y-z planes. uA and uB are the longitudinal displacement of interface between VEM layer. Given this displacement vector and the kinematic relationships of the elements, the shear strains bx and the shear strains by of viscoelastic layer rotating around the x and y axes can be given as bx ¼
uc u p d @w ; þ dv @x dv
by ¼
vc vp d @w þ dv @y dv
ð2Þ
where dc dp þ 2 2 the longitudinal displacements uvi ; vvi of the viscoelastic layer as 1 dc dP @w 1 dc dP @w uc þ up þ vc þ vp þ uv ¼ ; vv ¼ 2 @x 2 @y 2 2 d ¼ dv þ
ð3Þ
where dv , dP and dc represent the thickness of VEM, base plate and constrained layer, respectively. The spatial distribution of the longitudinal displacements uc , vc and up , vp w and the transverse displacement w over any element of the treated plate are assumed to be given by: uc ¼ a1 þ a2 x þ a3 y þ a4 xy
ð4Þ
vc ¼ a5 þ a6 x þ a7 y þ a8 xy
ð5Þ
up ¼ a9 þ a10 x þ a11 y þ a12 xy
ð6Þ
vp ¼ a13 þ a14 x þ a15 y þ a16 xy
ð7Þ
w ¼ a17 þ a18 x þ a19 y þ a20 x2 þ a21 xy þ a22 y2 þ a23 x3 þ a24 x2 y þ a25 xy2 þ a26 y3 þ a27 x3 y þ a28 xy3
ð8Þ
w;x ¼
@w ; @y
w;y ¼
@w @x
ð9Þ
where the constant fa1 ; a2 ; a3 ; . . . ; a28 g are determined in terms of the 28 components of the nodal displacements vector ui of the ith element which is bounded by the nodes 1, 2, 3, and 4 as shown in Fig. 2. The element displacements vector D of the ith element is given by D ¼ ½ u1
u2
u3
u4 T
ð10Þ
Therefore the displacement u ¼ uc vc up inside the ith element can be determined from u¼
uc
vc
up
vp
w
w;x
w;y
T
vp
¼ ND
w
w;x
w;y
at any location
ð11Þ
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where N ¼ ½N 1 N 2 N 3 N 4 N 5 N 6 N 7 T . N is the spatial interpolating matrix. N 1 ; N 2 ; N 3 ; N 4 ; N 5 ; N 6 ; N 7 are the spatial interpolating vectors corresponding to uc ; vc ; up ; vp ; w; w;x ; w;y , respectively. 2.4. Energy of plate with ACLD treatment 2.4.1. Potential energy The element potential energy associated with the extension, bending and shearing of the piezoelectric layer, the host plate, and the viscoelastic damping layer are as follows: ð 1
1 Pc ¼ rcx ecx þ rcy ecy þ scxy ccxy dv ¼ uT kc u ð12Þ 2 v 2 where kc is the element stiffness matrices of the piezoelectric layer. ðaðb
d3 dxdydz þ c 12 3
ðaðb
B T Dc B dxdydz 2 3 1 lc 1 N 1;x Ec 4 5 lc 1 0 5 B c ¼ 4 N 2;y Dc ¼ 1 l2c c N þ N 0 0 1l 2;x 2 ð 1;y 1
1 Pp ¼ rpx epx þ rpy epy þ spxy cpxy dv ¼ uT kp u 2 v 2
kc ¼ dc 2
0 0
BTc Dc Bc
0 0
2
3 N 5;xx B ¼ 4 N 5;yy 5 2N 5;xy
ð13Þ where kp is the element stiffness matrices of the host plate. ðaðb
d3p B TP Dp Bp dxdydz þ 12 0 3
ðaðb
B T Dp B dxdydz 2 3 1 lp 1 N 3;x E p 4l 5 0 5 B p ¼ 4 N 4;y Dp ¼ p 1 1 l2p 1lp N 3;y þ N 4;x 0 0 2 ð 1
1 T Pv ¼ rvx evx þ rvy evy þ svxy cvxy dv ¼ u kve u 2 v 2
kp ¼ dp 2
0
0 0
ð14Þ
where kve is one of the element stiffness matrices of the viscoelastic damping layer. ðaðb ð ð d3 a b T kve ¼ dv B Tv Dv Bv dxdydz þ v B Dv B dxdydz 12 0 0 2 0 0 3 2 3 1 lv 1 N 8;x E v 4 5 lv 1 0 5 B v ¼ 4 N 9;y Dv ¼ 2 1 l v v 0 0 1l 2 ðN 8;y þ N 9;x 1 1 Pb ¼ b Gb þ by Gby d v ¼ uT kvb u 2 v x x 2
ð15Þ
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where kvb is one of the element stiffness matrices of the viscoelastic damping layer.
kvb ¼
G dv
ðaðb 0 0
ðN T10 N 10 þ N T11 N 11 Þdxdy ¼ Gkv
Ec ; Ep and Ev are the Young’s modulus of the piezoelectric layer, the host plate and the viscoelastic damping layer respectively. G is shear modulus of viscoelastic damping layer. 2.4.2. Kinetic energy The element kinetic energy of the piezoelectric constrained layer, the host plate, and the viscoelastic damping layer (Tc ; Tp ; Tv , respectively) can be expressed as: 1 Tc ¼ 2
2 # ð " 2 @w @uc 2 @vc 1 qc þqc þqc dv ¼ u_ T mc u_ @t 2 @t @t v
ð16Þ
where mc is the element mass matrix of the piezoelectric constrained layer. ða ðb
T mc ¼ qc dc N 5 N 5 þ N T1 N 1 þ N T2 N 2 dxdy # ð " a b 2 @up 2 @vp 2 1 @w 1 Tp ¼ q þqp þqp dv ¼ u_ T mp u_ 2 v p @t 2 @t @t
ð17Þ
where mp is the element mass matrix of the host plate. ða ðb
T mp ¼ qp dp N 5 N 5 þ N T3 N 3 þ N T4 N 4 dxdy # ð " a b 2 1 @w @uv 2 @vv 2 1 Tv ¼ qv þqv þqv dv ¼ u_ T mv u_ 2 v @t 2 @t @t ða ðb
T mv ¼ qv dv N 5 N 5 þ N T8 N 8 þ N T9 N 9 dxdy
ð18Þ
a b
where mv is the element mass matrix of the viscoelastic damping layer, and qc ; qp and qv are the Young’s modulus of the piezoelectric layer, the host plate and the viscoelastic damping layer respectively. It is important to note that during the calculation of the element energy The shape function N i of corresponding displacements should be substituted into the corresponding equation.
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2.5. Mass matrices and stiffness matrices of an ACLD element According to the above equations, the mass matrices and stiffness matrices of an ACLD element can be given as follow m ¼ mp þ mc þ mv
ð19Þ
k ¼ kp1 þ kp2 þ kc1 þ kc2 þ kv1 þ kv2 þ kvb
ð20Þ
Eq. (20) can be rewritten as k ¼ ke þ kvb
ð21Þ
where ke ¼ kp1 þ kp2 þ kc1 þ kc2 þ kv1 þ kv2 . 2.6. Work done by applied voltage and external disturbance force Supposing deformation of the piezoelectric constrained layer meets the Kirchhoff hypothesis, the mechanics condition can be analyzed according to the plane stress condition. It is usually supposed that the polarization direction of the PZT is the thickness direction (Z axis). According to Refs [9] and [10], the relationship of the stress and strain can be given as
rcx rcy sxy ¼ Dc ecx ecy cxy Dc fd31 d32 d33 gE3 ; 2 3 1 lc 0 7 Ec 6 6 lc 1 0 7 Dc ¼ 4 5 1 lc 1 lc 0 0 2
ð22Þ
where rcx ; rcy ; sxy are plane stress. ecx ; ecy ; cxy are plane strain. Ec is the elastic modulus of PZT layer. E3 is the strength of electric field acting on the surface of PZT layer. The constants d31 ; d32 and d33 are the piezoelectric strain coefficients in the x, y and z directions. The formation of control forces generated by the control voltage is developed with the same approach described in Refs. [9] and [10]. The work done by the inplane piezoelectricity force f c is given by wc ¼ DT f c ¼
1 2
ð rce ec1 dv
ð23Þ
v
Substituting the shape function N i of corresponding displacements into Eq. (23),
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we get 1 wc ¼ E3 dc 2
3T 2 3 N 1 ;x d31 4 N 2 ;y 5 Dc 4 d32 5Ddxdy b N ; þN ; 0 1 y 2 x
ða ðb a
2
ð24Þ
So, the force vectors f c can be defined as 1 f c ¼ E3 dc 2
3T 2 3 N 1 ;x d31 4 N 2 ;y 5 Dc 4 d32 5dxdy: b N ; þN ; 0 1 y 2 x
ða ðb a
2
ð25Þ
The work done by the external disturbance force f is as follows wf ¼ DT f :
ð26Þ
2.7. Model of plate with partially covered ACLD Using Lagrange equation, the model of plate with partially covered ACLD treatments can be given as € þ ke D þ kvb D ¼ f c þ f : mD
ð27Þ
2.8. GHM method for constrained layer damping In this section, the GHM method proposed by Ref. [11] is introduced to model the viscoelastic damping material. The internal auxiliary coordinates are introduced by GHM method to describe the dynamic characteristic of viscoelastic damping material. The GHM method requires that the complex modulus of the viscoelastic core should be represented as a series of minioscillator terms. The complex modulus can be rewritten in Laplace domain as: " ~ ðsÞ ¼ G sG
1
N X 1þ ak k¼1
^ ks s2 þ 2^fk x _ ^2 s2 þ 2^fk xk þ x
# ð28Þ
k
where the factor G1 is the equilibrium value of the modulus, i.e. the final value of the relaxation function, G(t). The hatted terms (mini-oscillator terms) are obtained from the curving fitting of the complex modulus data for a particular viscoelastic damping material. Each mini-oscillator term is a second-order rational function ^ k ; ^fk g, which govern the shape of the involving three positive constants fak ; x modulus function in the complex s-plane. The number of the mini-oscillator terms depends on the high or low frequency dependence of the complex modulus.
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The governing equation (27) can be expressed in Laplace domain as follows:
~ ðsÞkvv DðsÞ ¼ f c ðsÞ þ f ðsÞ: s 2 m þ ke þ s G
ð29Þ
^ is introduced such that: Now, an auxiliary coordinates Z ^ ðsÞ ¼ Z
^2 x DðsÞ: _ ^2 s 2 þ 2^ nx þ x
ð30Þ
Using this new auxiliary coordinates, a second order equation in time domain can be obtained by the inverse Laplace transform: q ¼ fc þ f: € m q þ c q_ þ k
ð31Þ
, damping matrix c and mass matrix m are given in time The stiffness matrix k domain by: 3 2 m 0 ::: 0 .. 7 60 a 1 K 0 . 1 2 7 6 ^ x 7 6 1 7 6 ¼ 6. m .. 7 . 7 6. 0 . 0 5 4 1 0 0 aN 2 K ^N 3 x 2 0 0 ::: 0 ^ .. 7 6 7 6 0 a1 2n1 K 0 . 7 6 ^1 x 7 6 c ¼ 6 . 7 . .. 0 7 6 .. 0 7 6 4 2^ nN 5 0 0 aN K ^N x 3 2 N P ~ ak a1 R ::: aN R 7 6 ke þ k 1 þ k¼1 7 6 7 T ¼6 a1 K 0 0 k 7 6 a1 R 7 6. . . 5 4 .. 0 . 0 T aN R 0 0 a K 0 1 N 0 1 f fc B0C B0 C D C B C q¼ fc ¼ B @ ... A f ¼ @ ... A Z 0
0
~ ¼ G 1 kvv , kvv ¼ Rv Kv RT , K ¼ G 1 Kv , R ¼ Rv K, Z ¼ RT Z ^ ; k¼ where k v v 1; 2; . . . ; N and Kv is a diagonal matrix of the nonzero eigenvalues of kvv and the corresponding orthonormalized eigenvectors from the columns of Rv . Eq. (31) describes the dynamic characteristic of a single plate with partial ACLD elements. Assembling the corresponding equations for different elements and
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applying proper boundary conditions yield the overall equation for the entire plate with partial ACLD elements. The overall equation for the entire plate with partial ACLD elements is as follows: € þ C x_ þ Kx ¼ F c w þ F d d Mx
ð32Þ
where, M, C and K denote the mass matrix, damping matrix and stiffness matrix respectively in the generalized coordinated system. F d is the distribution matrix of external disturbance force. F c is the distribution matrix of control force. w is control force vector caused by the applied voltage. d is the external disturbance force vectors. x is a new generalized coordinate.
3. Development of the robust controller 3.1. Overview We can now rewrite the overall equation of motion for the entire plate with partial ACLD elements in state space, calculate the frequency or time response and perform the design of controller. It is worth noting that the dimension of the overall motion in Eq. (32) must be reduced further before proceeding to the controller design. Model reduction is necessary to design a low order controller. So, it is necessary to perform the integrated system reduction scheme. In this section, particular emphasis is placed on the control of the first four modes of vibration of the treated plate using H1 robust control laws. For this purpose, an H1 robust controller is designed to accommodate uncertainties of the ACLD parameters, particularly those of the viscoelastic damping core which arise from the variation of the operating temperature and frequencies. Noise and external disturbances are taken into account in the design of the controller. In this study, the small gain theory is utilized to guide the selection of the controller gain in order to ensure system stability. The controller is designed to guarantee the stability of the closed loop in the presence of parametric uncertainties which may result from variation of the properties of the viscoelastic damping layer due to its operation over wide temperature and frequency ranges. At the same time, the controller should ensure that the disturbance rejection capability of the treatments is maximized over a desired frequency band. Eq. (32) can be rewritten in the standard state space form as: X_ ¼ AX þ Bw þ Dd y ¼ C1 X z ¼ C2 X
ð33Þ
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where 0 I 0 ; B ¼ ; M 1 K M 1 C M 1 F x X¼ ; C1 ¼ ½ cy 0 ; C2 ¼ ½ cz 0 : x_
A¼
D¼
0 ; M 1 Fd
3.2. Analysis of uncertainty ACLD treatments have been successfully utilized as effective means in damping out the vibration of various flexible structures. The effectiveness of the ACLD treatments is validated experimentally and theoretically using proportional and/or derivative feedback of the transverse deflection or the slope of the deflection line of the base structure [12]. Shen [13] used optimal control strategies devised by Baz and Ro [14] to avoid instability problems for structures with partial and full ACLD treatments. In all these attempts, no effort has been exerted to accommodate the uncertainties of the ACLD parameters. Also, in all these studies controllers are designed without any provisions for rejecting external disturbances. Moreover, no efforts were made to consider the system reduction scheme. Only recently, Baz [15] has theoretically developed a robust H2 controller to control the ACLD treatments in the presence of parameter uncertainty and external disturbances. Two kinds of uncertainties can be defined. The first one coming from the reduction scheme in frequency domain can be called frequency uncertainty, and the second one from characteristic variation of material can be referred to as parameter uncertainty. Fig. 4 shows a block diagram of a feedback system with uncertainties. Transfer functions of the controller, system and uncertainty are denoted by K, G and D, respectively. 3.3. Design of the robust controller Fig. 5 shows a block diagram of an H1 robust controller K that stabilizes the ACLD plate system G in the presence of frequency uncertainty and parameter uncertainty, when the system is subjected to external disturbances.
Fig. 4. Control structure of feedback with uncertainty.
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Fig. 5. H1 control mixed sensitivity design.
The controller K is designed to minimize the H1 norm of the transfer function Syd from the external disturbance d to the transverse displacement y at the free end of the plate, to guarantee the optimal disturbance rejection performance of the closed loop. Transfer function Syd denotes sensitivity function which represents the performance of the closed loop to suppress external disturbance. Generally, external disturbances mainly distribute in the band of low frequency, we can expect that the sensitivity function is small in this band. Mathematically, such a design index can be given as follows: kW1 ðsÞS ðsÞk1 < 1:
ð34Þ
From Fig. 5, we know the characteristic of the weighting function W1 ðsÞ determines the amount of damping and frequency response of the closed loop system. According to the small gain theory of robust control, the stability condition of the closed loop system for additive uncertainty is as follows kDðsÞT ðsÞk1 < 1:
ð35Þ
The transfer function from the external disturbance d to the input control signal u is denoted as Tud . The weighting function W2 ðsÞ can be selected to embed the additive uncertainty. Such design index can be denoted as kW2 ðsÞT ðsÞk1 < 1: So, the index of H1 robust control can be written as follows W1 ðsÞS ðsÞ W2 ðsÞT ðsÞ < 1: 1
ð36Þ
ð37Þ
The optimal H1 controller which meets Eq. (37) can be obtained by c iteration technique [16]. The choice of weighting functions ensures that the controller does not destabilize higher frequency modes, and also attenuates control signals at lower frequencies. However, the appropriate selection of weighting functions over the
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desired frequency range is not explicitly related to the performance objectives in a straightforward manner. Many trials of weighting functions are usually required in order to reach desired performance objectives. 4. Validation of the modeling and robust controller 4.1. Frequency response of the plate with the ACLD system The theoretical predictions of the ACLD model are compared with the experimental performance of an aluminum plate which is partially treated with a viscoelastic damping layer. The measurement of frequency response in this section also aims at demonstrating the accuracy of modeling and the merits of using PCLD to suppress the vibration of the flat plate. A cantilever sandwich plate with partial ACLD structure is depicted in Fig. 6. Elements 5 and 10 are ACLD elements. The other elements are ordinary thin plate elements. It is important to note that the first four modes of the plate are bending and torsion mode, So, the positions of ACLD treatments are elements 5 and 10 where strain is maximum. The physical and geometric properties of the plate with partially treated ACLD are as follows: The material parameters of the constrained layer are qc ¼ 7500 kg=m3 , Ec ¼ 3:6 1011 N=m2 , lc ¼ 0:3. The material parameters of the host plate layer are qp ¼ 2700 kg=m3 , Ep ¼ 9 1010 N=m2 , lp ¼ 0:3. The material parameters of the viscoelastic damping layer are qv ¼ 1250 kg=m3 , Ev ¼ 2 107 N=m2 , lv ¼ 0:3. The material of the viscoelastic damping layer is ZN-1. The relax function of VEM is derived from Ref. [17], GðtÞ ¼ 3:44 þ 7:089e193:39t þ 231:212e16345t þ 1744:4:2e485916:4t : The boundary condition: one short side of the base plate layer is fixed and the others are free. The dimensions of the sandwich plate are L1 ¼ 0:37 m, L2 ¼ 0:28 m,
Fig. 6. Cantilever sandwich plate with partial ACLD.
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Fig. 7. Experimental set-up and instruments of frequency response function.
dc ¼ dv ¼ 0:001 m, dp ¼ 0:0025 m. The general arrangement of the frequency response test facility is shown in Fig. 7. A photograph of the set-up for frequency response measurement and control experiment is shown in Fig. 8. Firstly, frequency responses of the plate are presented by simulation. Fig. 9 shows a comparison between the FRF of the plate with PCLD elements and that without PCLD elements. In the case of the PCLD, the control loop that regulates the interaction between the piezoelectric actuator and the displacement sensor is kept open. The corresponding time responses are displayed in Fig. 10, where the solid line represents the response of the bare aluminum plate, while the dashed line
Fig. 8. Photograph of the experimental set-up used in FRF measurement and vibration control.
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Fig. 9. Frequency response functions of a cantilever plate and a cantilever sandwich plate with partial ACLD.
Fig. 10. The impulse responses of a cantilever plate and a cantilever sandwich plate with partial ACLD.
Table 1 Natural frequencies given by ANSYS5.6 and the experiment (E ¼ 910 Pa) Order
Calculation (Hz) A PCLD plate
ANSYS5.6 (Hz) B bare plate
FRF (Hz) PCLD plate
Error (%) A
B
1 2 3
17.97 55.56 111.49
17.58 53.93 109.67
17.0 47.0 111.5
5.71 18.21 0.01
3.41 14.74 1.64
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Fig. 11. Result of natural frequency of the FRF of PCLD plate.
represents the response of the PCLD plate. It is evident that the PCLD results in excellent attenuation compared to the pure aluminum plate. The natural frequencies of the plate partially treated PCLD elements are shown in Table 1. Secondly, frequency responses of the plate are presented by experimental testing. Figs. 11 and 12 show the frequency responses of the plate partially treated with PCLD elements. It is evident that the PCLD treatment has been very effective in attenuating the structural vibration of the plate over a wide frequency range. The results calculated with our modeling method are very close to the results of experimental test. So, it has been demonstrated that the model is accurate. We can utilize the dynamic model to design an H1 robust controller. 4.2. Investigation of performance in relation to ACLD The dynamic model of the plate with partial ACLD elements has been developed to enable ‘active damping’ as described in this paper. The vibration reduction
Fig. 12. The frequency response function of PCLD plate.
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Fig. 13. Experimental layout of control test.
achieved for a cantilever plate with partial ACLD elements subject to a base excitation is investigated using H1 robust control. The experimental layout is shown in Fig. 13. The photograph reflecting the relative position of the actuator and sensor is shown in Fig. 14. Suppose the node of intersection of elements 6, 7, 11 and 12 of the plate is the location of external exciting force. The direction of force is along the z-axis, the vertical direction. Elements 5 and 10 are ACLD elements. Figs. 15–18 show the comparison between the performance of the ACLD and PCLD in time domain when the plate with partial ACLD structure is excited with impulse, sinusoidal disturbances of the first resonance, square wave and the white noise. The period of square wave is 1 s. The thin solid line represents the response of the open loop system (without the action of voltage) and the thick dashed line represents the response of the closed system in Figs. 15–18.
Fig. 14. The photograph of the relative position of actuator and sensor.
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Fig. 15. Impulse response of the controlled structure.
It can be seen that the added damping obtained by applying an active constraining layer reduces the response of the plate considerably. It is evident that the ACLD results in excellent attenuation compared to conventional PCLD treatment. The consistent effectiveness of the ACLD is attributed to its built-in feedback that senses the adverse effects of the vibration of the plate. From the above discussion, it is certain that the new method presented for building the dynamic equation of sandwich plate with VEM core has good accuracy and reliability. The ACLD treatments combine the simplicity and reliability of PCLD with low weight and
Fig. 16. First resonance response of the controlled structure.
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Fig. 17. Response of the structure excited by square wave.
high efficiency of active control to attain high damping characteristics over a broad frequency band. The active component will provide adjustable damping, whereas the passive component will enhance gain and phase margins. Therefore, the spillover effects resulted from model reduction can be overcome.
Fig. 18. Response of the structure excited by white noise.
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5. Conclusions The theoretical analysis and experimental measurements of a plate partially treated with ACLD elements have been compared in this study. Based on experimentally validated finite element model, it is developed to describe the dynamics of the treated plate The results of FRF measurements have confirmed the accuracy of the new modeling method, and there is good agreement in the theoretical and experimental analysis of natural frequencies. It is evident that the proposed modeling method is accurate. The model is utilized to design an H1 robust controller which is stable in the presence of uncertainties of modeling and parameters, and ensures optimal disturbance rejection capability. Experimental results are presented to validate the effectiveness of the H1 robust controller in damping out structural vibrations when the ACLD treatments operate over a wide frequency band. The results show that the ACLD treatment has resulted in effective attenuation of plate vibrations. Hence, the theoretical and experimental techniques developed in this study constitute effective tools in designing and predicting the performance of smart laminated structures that can be used in many engineering applications. Acknowledgements The authors would like to thank the key scientific laboratory of dynamics of helicopter flexbeams of national defenses for partial funding under contract No. JS 52.4.3. Special thanks are due to Professor Mei Weisheng of the State Key Laboratory of Dynamics of Helicopter Flexbeams of Nanjing University of Aeronautics and Astronautics. References [1] Dai D. Application of viscoelastic material in engineering. Beijing: Tsinghua University Press; 1991. [2] Chantalakhana C, Stanway R. Active constrained layer damping of clamped-clamped plate vibrations. Journal of Sound and Vibration 2001;241(5):755–77. [3] Shen IY. Hybrid damping through intelligent constrained layer treatments. Journal of Vibration and Acoustics 1994;116(3):341–9. [4] Shen IY. Bending vibration control of composite plate structures through intelligent layer treatments. Proceedings SPIE 1994;2193:115–25. [5] Balas MJ. Feedback control of flexible systems. IEEE Transactions on Automatic Control 1978;AC-23:673–9. [6] Golla DF, Hughes PC. Dynamics of viscoelastic structures: a time-domain, finite element formulation. Journal of Applied Mechanics 1985;52:897–907. [7] ZuQing Q. A reduced order modeling technique for tall building with active tuned mass damper. Earthquake Engieering and Structural Dynamics 2001;30:349–62. [8] Safanov MG, Chiang RY. A Schur method for balanced model reduction. IEEE Trans Automatic Control 1989;34(7):729–33. [9] Tzou HS, Anderson GL. Intelligent structure system. Kluwer Academic Publishers; 1992. [10] Crawley EF, Lazarust KB. Induced strain actuation of isotropic and anisotropic plates. AIAA Journal 1991;29(6):944–51.
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