Damping optimisation of hybrid active–passive sandwich composite structures

Advances in Engineering Software 46 (2012) 69–74

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Damping optimisation of hybrid active–passive sandwich composite structures A.L. Araújo a,⇑, P. Martins a, C.M. Mota Soares a, C.A. Mota Soares a, J. Herskovits b a b

IDMEC/IST – Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal COPPE/UFRJ – Federal University of Rio de Janeiro, Caixa Postal 68503, 21945-970 Rio de Janeiro, Brazil

a r t i c l e

i n f o

Article history: Available online 19 November 2010 Keywords: Sandwich structures Passive damping Active damping Gradient optimisation Genetic algorithms Co-located control

a b s t r a c t Optimisation of active and passive damping is presented in this paper, using a new mixed layerwise finite element model developed for the analysis and optimisation of hybrid active–passive laminated sandwich plates. Optimisation is conducted through maximisation of modal loss factors, using as design variables the viscoelastic core thickness, the constraining elastic layers ply thicknesses and orientation angles, as well as the position of co-located sensor and actuator pairs. Optimal results for passive damping are compared with an alternative optimisation model, based on 3D finite elements included in commercial package ABAQUS. Optimal location for sensor–actuator pairs is also presented and results are discussed. Ó 2010 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

1. Introduction Optimal design of passive constrained layer damping treatments of vibrating structures has been a main subject of research, aiming at the maximisation of modal damping ratios and modal strain energies, by determining the optimal material and geometric parameters of the treatments, or minimising weight by selecting their optimal length and location. For example, Baz and Ro [1] optimised performance of constrained layer damping treatments by selecting the optimal thickness and shear modulus of the viscoelastic layer, and Marcelin et al. [2–5] used a genetic algorithm and beam finite elements to maximise the damping factor for fully and partially treated beams and skis, using as design variables the stacking sequences, thicknesses, dimensions and locations of the layers or patches. As verified by Nokes and Nelson [6], this layout optimisation can lead to significant saving in the amount of material used. For fully covered sandwich beams, Lifshitz and Leibowitz [7] determined the optimal passive constrained layer damping, with layer thickness as design variables. The vibration damping of fully covered passive constrained layer damping structures is determined by a large number of parameters which include material properties and thicknesses of both the constraining layers and the viscoelastic layer. Active constrained layer damping are hybrid treatments, that combine the high capacity of passive viscoelastic materials to dissipate vibrational energy at high frequencies, with the active

⇑ Corresponding author. Tel.: +351 218419462; fax: +351 218417915. E-mail addresses: [email protected] (A.L. Araújo), [email protected] (P. Martins), [email protected] (C.M. Mota Soares), [email protected] (C.A. Mota Soares), [email protected] (J. Herskovits).

capacity of piezoelectric materials at low frequencies. Therefore, in the same damping treatment, a broader control band is achieved. An extensive review on developments in active and passive constrained layer damping can be found in Trindade and Benjeddou [8]. In this paper we address a form of active constrained layer damping, where sensors and actuators are bonded to the exterior faces of a sandwich plate. This may be advantageous in increasing actuator’s authority, since one of the major problems with active constrained layer damping arises when actuators are bonded directly to the viscoelastic layer of much lower stiffness. An important aspect of hybrid active–passive damping is to be able to determine the optimal number of active control devices and their placement on the structure. Similarly, the number and placement of sensors can be critical to the robust functioning of active control systems. Also the size of actuators may be optimised in actuator placement problems, but the passive structure is always assumed to be of predetermined geometry and material. Suleman and Gonçalves [9] considered several objective functions simultaneously, such as maximising the average static vertical displacement of a beam and minimising the mass of the actuators and minimising the actuation voltage, where the design variables were the coordinates of actuator pairs and the size of rectangular actuator patches. Geometric constraints and upper and lower limits on design variables were considered. Adali et al. [10] considered a beam problem where the maximum vertical deflection of a laminated beam was to be minimised using one pair of actuators, through a robust design approach. A state of the art review in optimization of smart structures and actuators can be found in Frecker [11]. In this work, optimisation of modal loss factors of sandwich plates with elastic laminated constraining layers and a viscoelastic

0965-9978/$ - see front matter Ó 2010 Civil-Comp Ltd and Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2010.09.007

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core is conducted with thicknesses and laminate layer ply orientation angles as design variables. The problem is solved through gradient based optimisation using a new mixed layerwise finite element model, developed by the authors for the analysis and optimisation of hybrid active–passive laminated sandwich plates as, to the author’s knowledge, few active laminated plate/shell sandwich models with viscoelastic core exist, and are mostly limited to isotropic materials. The results are compared to an alternative approach, using 3D finite elements from the commercial package ABAQUS [12] and an implementation of a genetic algorithm (GA). Using the new layerwise model, we also study the best locations for sensor–actuator pairs in a sandwich structure, in order to maximise modal loss factors of one, or a set of selected modes of interest. It is assumed that sensors and actuators are co-located and simple control laws will be used (direct proportional and velocity feedback). A genetic algorithm is used to solve the discrete unconstrained optimization problem. 2. Mixed layerwise sandwich finite element model Fig. 1 shows schematically the hybrid active–passive sandwich laminated plate with a viscoelastic (v) core, laminated anisotropic face layers (e1, e2) and piezoelectric sensor (s) and actuator (a) layers. The basic assumptions in the development of the sandwich plate model are: all points on a normal to the plate have the same transverse displacement w(x, y, t), where t denotes time, and the origin of the z-axis is the medium plane of the core layer; No slip occurs at the interfaces between layers; the displacement is C0 along the interfaces; elastic and piezoelectric layers are modelled with first order shear deformation theory (FSDT) and viscoelastic core with a higher order shear deformation theory (HSDT); all materials are linear, homogeneous and orthotropic and the elastic layers (e1) and (e2) are made of laminated composite materials; for the viscoelastic core, material properties are complex and frequency dependent; upper and lower layers play the roles of sensor and actuator, respectively, and are connected via feedback control laws, considering co-located control. The FSDT displacement field of the face layers may be written in the general form:

ui ðx; y; z; tÞ ¼ ui0 ðx; y; tÞ þ ðz  zi Þhix ðx; y; tÞ

v ðx; y; z; tÞ ¼ v i

i 0 ðx; y; tÞ

þ ðz 

zi Þhiy ðx; y; tÞ

v

uv ðx; y; z; tÞ ¼ uv0 ðx; y; tÞ þ zhvx ðx; y; tÞ þ z2 u0 v ðx; y; tÞ þ z3 hx ðx; y; tÞ

v v ðx; y; z; tÞ ¼ v v0 ðx; y; tÞ þ zhvy ðx; y; tÞ þ z2 v 0 v ðx; y; tÞ þ z3 hy v ðx; y; tÞ wv ðx; y; z; tÞ ¼ w ðx; y; tÞ 0

ð2Þ where uv0 and v v0 are the in-plane displacements of the mid-plane of the core, hvx and hvy are rotations of normals to the mid-plane of the core about the y-axis (anticlockwise) and x-axis (clockwise), respectively, w0 is the transverse displacement of the core (same for all v v layers in the sandwich). The functions u0 v ; v 0 v , hx and hy are higher order terms in the series expansion, defined also in the mid-plane of the core layer. Applying displacement continuity conditions at the layer interfaces allows us to retain the rotational degrees of freedom of the face layers, while eliminating the corresponding in-plane displacement ones. Hence, the generalized displacement field has 17 mechanical unknowns [13]. We consider that fibre-reinforced laminae in elastic multi-layers (e1) and (e2), viscoelastic core (v), and piezoelectric sensor (s) and actuator (a) layers are characterised as orthotropic. Furthermore, piezoelectric material is assumed to be polarised in the thickness direction. Constitutive equations for each lamina in the sandwich may then be expressed in the principal material directions, and for the zero transverse normal stress situation, as [14]:

8 9 r11 > > > > > > > > > > r > > 22 < =

2

0 0 Q 11 Q 12 6 0 0 6 Q 12 Q 22 6 0 0 Q r23 ¼ 6 0 44 > > 6 > 6 > > 0 0 Q 55 r13 > > > 4 0 > > > > : ; 0 0 0 0 r12 3 2 0 0 e31 78 9 6 E1 > 0 e32 7> 6 0 7< = 6 7 6 6 0 e24 0 7> E2 > 7: ; 6 0 5 E3 4 e15 0 0

ð1Þ

wi ðx; y; z; tÞ ¼ w0 ðx; y; tÞ where ui0 and v i0 are the in-plane displacements of the mid-plane of the layer, hix and hiy are rotations of normals to the mid-plane about the y-axis (anticlockwise) and x-axis (clockwise), respectively, w0 is the transverse displacement of the layer (same for all layers in the sandwich), zi is the z coordinate of the mid-plane of each layer, with reference to the core layer mid-plane (z = 0), and i = s, e1, e2, a is the layer index.

Fig. 1. Hybrid active–passive sandwich plate.

For the viscoelastic core layer, the HSDT displacement field is written as a second order Taylor series expansion of the in-plane displacements in the thickness coordinate, with constant transverse displacement:

0

8 9 2 0 > < D1 > = 6 D2 ¼ 4 0 > : > ; e31 D3 2

11

6 þ4 0 0

0

22 0

0 0

0

0

e24

e32

0

0

3e

7 0 5

33

9 3E 8 e11 > 0 > > > > > 7 > > > e 0 7 > > > 22 = 7 < 7 0 7 c23 > > 7 > >c > > 0 5 > 13 > > > > > : c12 ; Q 66

8 9 e11 > > > > > > 3> > > e15 0 > > < e22 > = 7 0 0 5 c23 > > > > 0 0 > > c13 > > > > > > : c12 ; 8 9 > < E1 > = E2 > : > ; E3

ð3Þ

where rij are stress components, eij and cij are strain components, Ei and Di are the electric field and electric displacement components, respectively, Q Eij are reduced stiffness coefficients at constant electric field, eij and eij are piezoelectric and reduced piezoelectric constants, respectively, and eij and 33e are dielectric and reduced dielectric constants, measured at constant strain. Expressions for the reduced quantities mentioned above can be found in [14,15]. The complete set of equations in (3) is only used for the piezoelectric sensor and actuator layers, while for the remaining elastic and viscoelastic layers, only the first equation is considered, without the piezoelectric part. For the viscoelastic core layer, the reduced stiffness coefficients Q Eij are complex quantities, since the complex modulus approach was used in this work, using the elastic-viscoelastic

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principle. In this case, the usual engineering moduli may be represented by complex quantities:

E1 ðixÞ ¼ E01 ðxÞð1 þ igE1 ðxÞÞ

G23 ðixÞ ¼ G023 ðxÞð1 þ igG23 ðxÞÞ

ð10Þ

where, the complex eigenvalue

kn

is written as:

kn ¼ kn ð1 þ ign Þ

E2 ðixÞ ¼ E02 ðxÞð1 þ igE2 ðxÞÞ G12 ðixÞ ¼ G012 ðxÞð1 þ igG12 ðxÞÞ

   K ðxÞ  kn Muu un ¼ 0

ð4Þ

G13 ðixÞ ¼ G013 ðxÞð1 þ igG13 ðxÞÞ

ð11Þ

and kn ¼ x2n is the real part of the complex eigenvalue and gn is the corresponding modal loss factor. The non-linear eigenvalue problem is solved iteratively with a shift-invert transformation [13]. Eigenvalues and modal loss factors can also be determined from the frequency domain response [17].

m12 ðixÞ ¼ m012 ðxÞð1 þ igm12 ðxÞÞ where the prime quantities denote storage moduli, associated material loss factors are represented pffiffiffiffiffiffiffi by the letter g, x represents frequency of vibration and i ¼ 1 is the imaginary unit. Additionally, in Eq. (4), E, G and m denote Young’s moduli, shear moduli and Poisson’s ratio, respectively. The definition of constitutive relations of a laminate is usually made in terms of stress resultants. These forces and moments are defined separately for the viscoelastic core (v), the elastic multilayered laminates (e1) and (e2) and the piezoelectric sensor (s) and actuator (a) layers [16]. The equations of motion for the plate are obtained by applying Hamilton’s principle, using an eight node serendipity plate element with 17 mechanical degrees of freedom per node, and one electric potential degree of freedom per piezoelectric layer, assuming that the potential varies linearly in the thickness direction:



Muu 0

#      " Kuu ðxÞ Ku/ € u Fu u ¼ þ T € Ku/ K// / 0 / 0 0

ð5Þ

€ are mechanical degrees of freedom and corre€ ; / and / where u; u sponding accelerations, electric potential and corresponding second time derivatives, respectively. Muu and Kuu(x) are the mass and complex stiffness matrices, respectively, corresponding to purely mechanical behaviour, while K// is the dielectric stiffness matrix, Ku/ is the stiffness matrix that corresponds to the coupling between the mechanical and the piezoelectric effects, and Fu is the externally applied mechanical load vector. The feedback control law is based on direct proportional or velocity feedback, and can be written in the following form:

/a ¼ Gd /s þ Gv /_ s

ð6Þ

where Gd and Gv are the constant displacement and the constant velocity feedback gains, respectively. The vectors of actuator (a) and sensor (s) potentials are /a and /s, while /_ s is the vector of sensor potential time derivatives. Assuming harmonic vibrations, the final equilibrium equations are given by:

½K ðxÞ  x2 Muu u ¼ Fu

ð7Þ

where the condensed stiffness matrix is written as:

h i 1 T K ðxÞ ¼ Kuu ðxÞ  ðGd þ ixGv ÞKau/ þ Ksu/ Ks// Ksu/

ð8Þ

and Kuu(x) is a complex matrix. It is worthwhile noting that when electroded surfaces exist in a given patch or layer, equipotential conditions should be imposed before condensing the electric degrees of freedom. The forced vibration problem is solved in the frequency domain, which implies the solution of the following linear system of equations for each frequency point:

½K ðxÞ  x2 Muu uðxÞ ¼ Fu ðxÞ

ð9Þ

where Fu ðxÞ ¼ F½Fu ðtÞ is the Fourier transform of the time domain force history Fu(t). For the free vibration problem, Eq. (9) reduces to the following non-linear eigenvalue problem:

3. Optimal sandwich design formulations 3.1. Passive design The objective of this study is to maximise damping in passive sandwich plate structures. If the structure is subjected to a given load or load set, this maximisation of damping must be conducted with design constraints, such as maximum displacement, total mass, failure criteria, as well as physical constraints on design variables and objective function. Two different approaches are used, namely: the layerwise/FAIPA model, associating the finite element model described in the previous section to a gradient based optimisation algorithm, and the ABAQUS/GA model which uses 3D finite elements from commercial program ABAQUS [12] associated to a specifically developed implementation of a genetic algorithm. Both approaches are described next. 3.1.1. Layerwise/FAIPA approach The formulated problem is solved using the Feasible Arc Interior Point Algorithm (FAIPA) [18], along with the developed finite element sandwich model. For damping maximisation with passive treatments in sandwich type structures, the overall goal will be to maximise the modal loss factor of a particular mode of interest, or of a particular set of modes of interest within some frequency range. Thus, a weighted sum of reciprocal loss factors was chosen as the objective function to be minimised in this framework, subjected to design constraints:

min

f ¼

N X

wi

i¼1

1

gi

s:t: g j : gj 6 0; j ¼ 1; . . . ; N m g Nþ1 : 160 mmax w g Nþ2 : 160 wmax g Nþ3 : F TH  1 6 0 xli 6 xi 6 xui ;

ð12Þ

i ¼ 1; . . . ; n

where wi are weighting factors associated with each modal loss factor gi, N is the total number of modes of interest, m and mmax are the overall mass and maximum allowable mass of the structure, respectively, w and wmax are the maximum displacement of the structure and the maximum allowable value of the displacement, respectively, and FTH is the Tsai–Hill failure criteria parameter for the elastic composite material layers, defined as:

F TH ¼



1 1 1  þ  r11 r22 X Y Z X2 Y 2 Z2

r 2 1 1 1 1 1 1 23  þ 2  2 r22 r33  þ 2  2 r11 r33 þ 2 2 R Y Z X Z X Y r 2 r 2 13 12 þ þ <1 ð13Þ S T r 2 11

þ

r 2 22

þ

r 2 33

where the stress components are calculated for each elastic layer ply and refer to the principal material directions of the ply, X, Y

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and Z are lamina failure stresses in the associated principal directions, which must respect the sign of the stresses, and one must consider different values in traction and compression. R, S and T are failure stresses in shear for the associated planes in Eq. (13). Assuming a uniform sandwich plate structure made of a given set of materials, with fixed in-plane dimensions, the natural choice for the design variables xi in Eq. (12) are the thicknesses of the constituent layers and the orientation angles of the laminated elastic composite material plies. In Eq. (12), xli and xui are the lower and upper bounds on the design variables. Calculation of the objective function is done by solving the eigenvalue problem of Eq. (10) iteratively, for a frequency dependent complex stiffness matrix and real mass matrix. The gradient of the objective function with respect to the design variables is calculated as:



@ 1 wn @ gn ¼ 2 wn @xi gn gn @xi

ð14Þ

where



  @R kn @ gn 1 @I kn ¼  gn @xi @xi kn @xi

ð15Þ

  with R kn and I kn being the real and imaginary parts of the complex eigenvalue kn , respectively. The derivative of the complex eigenvalue with respect to design variables is obtained using the following expression [19]:

h i uu ðxi Þ uTn @Kuu@xðxi ;xi Þ  kn @M@x un i i ¼ h @xi uT Muu  xn @Kuu ðx;xi Þ j x¼xn un n 2k @x

@kn

ð16Þ

n

Calculation of response quantities such as displacements and stresses are done after the eigenvalue problem has been solved. This problem is solved in the frequency domain, by first making a forward Fourier transform of the applied load time history, and then solving Eq. (9) in order to the displacement vector, for the resonant frequency of interest. Afterwards, stresses in each elastic material layer ply are calculated and the Tsai–Hill factor FTH in Eq. (13) is evaluated. Sensitivities of complex eigenvalues, displacement and stress quantities can be calculated analytically, semi-analytically, or using a global finite difference approach [19]. 3.1.2. ABAQUS/GA approach A genetic algorithm has been developed and linked with commercial code ABAQUS [12] for passive damping design in sandwich structures [20]. For the analysis, quadratic complete integration solid elements were used (C3D20H for the viscoelastic core, and C3D20 for the other laminae). Maximisation of modal loss factors can be achieved through minimisation of the following objective function:

f ¼

N X i¼1

wi

1

gi

P1 P2 P3

ð17Þ

where Pi are penalty functions that allow for design constraints satisfaction, expressed as:

ðP1 ; P2 ; P3 Þ ¼ 1:1



w m ; F TH ; wmax mmax

ð18Þ

A genetic algorithm [21] with binary encoding is used to solve the problem [22]. The algorithm initialises a random sample of individuals with different parameters to be optimised using evolution via survival of the fittest. The selection scheme used is tournament selection with a shuffling technique for choosing random pairs for mating.

3.2. Active design The objective of this study is to determine the location of a given number of sensor and actuator pairs, in order to maximise modal loss factors of hybrid sandwich plate structures. It is assumed here that passive damping optimization has already been dealt with, and we formulate a discrete unconstrained minimisation problem, using as design variables the element numbers in the finite element mesh where sensor–actuator pairs are to be applied. The number of design variables is chosen to be equal to the number of available sensor–actuator pairs for the sandwich structure. If contiguous elements have patches, we consider that they are independent and individually co-located. Modal loss factors are obtained from the solution of the associated non-linear eigenvalue system. The same genetic algorithm described previously was used to solve this problem with the micro-GA [23] option, uniform crossover and elitism. The micro-GA option is used to reduce population size, and consequently the number of function evaluations, as these algorithms evolve very small populations that are very efficient in locating promising areas of the search space. However, since small populations are unable to maintain diversity for many generations, the population is restarted whenever diversity is lost, keeping only the very best fit (elitism of one individual). 4. Application A simply supported sandwich plate of in-plane dimensions 300  200 mm2 is considered. For passive design, the plate has a viscoelastic soft core and two symmetric face layers each with three orthotropic elastic plies. These elastic plies have material properties: E1 = 98.0 GPa, E2 = 7.9 GPa, m12 = 0.28, G12 = G13 = G23 = 5.6 GPa, q = 1520 kg/m3. For the isotropic viscoelastic core, the material properties are: G = 20(1 + 0.3i) MPa, m = 0.49, q = 1140 kg/m3. Additionally, for active design, piezoelectric patches are considered for which the material properties are: E1 = E2 = 47 GPa, m12 = 0.33, G12 = 16.3 GPa, G13 = G23 = 17.4 GPa, q = 8036 kg/m3, e31 ¼ e32 ¼ 14:7 N=Vm, and 33 e ¼ 2:12  108 F=m. 4.1. Passive design We consider the plate with no piezoelectric patches, where the design variables are the three elastic layer ply thicknesses in the symmetric layout, the viscoelastic core thickness, and the orientation angles of the elastic layer plies. Thickness design variables can take values from 0.5 mm to 10 mm, and for the ABAQUS/GA approach, in increments of 0.5 mm. The orientation angles can take values between 0° and 175°, with increments of 15° for the ABAQUS/GA approach. The fundamental flexural modal loss factor of the plate will be maximised, with a maximum allowable mass mmax = 0.5 kg and a maximum allowable displacement wmax = h/5, where h ¼ he1 þ hv þhe2 is the total thickness of the plate. The failure stresses in Eq. (13) are, for the elastic layers, X = 820 MPa, Y = Z = 45 MPa, both in tension and compression, and R = S = T = 45 MPa. The excitation consists of a 10 N force applied at the mid-point of the plate at t = 0. For the layerwise model, a 6  4 finite element mesh was used, with a total of 1133 degrees of freedom. As for the ABAQUS model, each lamina was discretised with 24 elements (six along the length, four along the width, and one along the thickness), with a total of 168 elements and 2931 degrees of freedom. For the GA, the population size was 10, the mutation probability 0.01, and a total of 228 evaluations were made. Results are presented in Table 1, where the initial/intermediate and final designs are shown for both approaches used. A good

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corresponding to those of an element of the regular 6  6 finite element mesh used. The objective is the determination of the best locations for the four pairs of co-located sensors and actuators, in order to maximise the weighted sum of the first seven flexural modal loss factors, where the weights wi in Eq. (12), and associated to each modal loss factor are assumed to be linearly decreasing with modal index i, and the last weight was fixed (w7 = 0.01), allowing for the determination of the other six, by assuming a unit sum of weights. A negative velocity feedback control law with gain Gv = 0.01 was used for each pair of co-located sensor and actuator. Modal loss factors are obtained by iteratively solving the associated non-linear eigenvalue problem. A population size of five individuals was used for the genetic algorithm, with a maximum of 100 generations, and mutation probability of 0.02. The finite element mesh is presented in Fig. 2, along with the optimal (shaded) position for the patches. The first seven flexural natural frequencies and modal loss factors for the sandwich plate are presented in Table 2 for the initial design, optimal passive design and final active design. Fig. 3 shows the obtained frequency response functions for the centre node of the plate (corresponding to a 10 N impulse excitation at the same node, and at t = 0) for the initial design, optimal passive design and final active design. Only the response of the first and third flexural modes can be observed at this particular point.

agreement between the two models and approaches is observed. A perfect match is not obtained for the optimal results due to the continuous versus discrete nature of the two optimization approaches. Also, for the Tsai–Hill factors, since the finite elements used in the two approaches are quite different in nature, the slight discrepancies observed were expected. It is also interesting to note that the computational effort associated to the ABAQUS/GA approach is much greater than the corresponding effort for the Layerwise/FAIPA approach (at least 15 times). 4.2. Active design The plate considered for active design is the one corresponding to the optimal passive design configuration in the previous section. For the active design, we consider four pairs of piezoelectric patches of thickness ha = hs = 0.5 mm with in-plane dimensions Table 1 Passive design results for the simply supported sandwich plate. Layerwise/FAIPA

hei (mm) hv (mm) hi (°) f (objective) x1 (Hz) m (g) (mass) FTH (Tsai–Hill) wmax (mm) g1 (%)

ABAQUS/GA

Initial design

Final design

Intermediate

Final design

0.5, 0.5, 0.5 1.0 90, 45, 45 10.4 298.8 342.0 4.7  102 0.34 9.6

0.5, 0.5, 0.5 3.3 102, 100, 61 5.2 320.7 500.0 3.9  103 0.11 19.3

0.5, 0.5, 0.5 1.0 90, 45, 45 10.0 299.0 342.0 6.8  102 0.33 10.0

0.5, 0.5, 0.5 3.0 105, 105, 60 5.2 316.7 478.8 4.9  103 0.11 19.2

Fig. 2. Optimal sensor/actuator placement for the hybrid active–passive simply supported sandwich plate, where shaded elements represent co-located sensor and actuator pairs.

Fig. 3. Frequency responses (magnitude) for the initial design, optimal passive design and final active–passive design of the hybrid simply supported sandwich plate.

Table 2 First seven flexural natural frequencies and modal loss factors for the initial design, optimal passive design and final active design of the simply supported laminated sandwich plate. Initial design

Optimal passive design

Final active design

Mode

fn (Hz)

gn (%)

Mode

fn (Hz)

gn (%)

Mode

fn (Hz)

gn (%)

(0,0) (1,0) (2,0) (0,1) (3,0) (1,1) (2,1)

298.69 414.17 612.48 815.67 880.36 891.69 1055.63

9.60 10.92 11.98 15.45 12.75 15.25 14.09

(0, 0) (1, 0) (2, 0) (0, 1) (1, 1) (3, 0) (2, 1)

320.68 426.54 610.76 728.65 796.95 845.78 941.26

19.30 17.78 17.92 21.84 20.77 18.79 19.14

(0, 0) (1, 0) (2, 0) (0, 1) (1, 1) (3, 0) (2, 1)

304.04 432.40 671.95 685.57 803.72 867.25 960.05

32.82 30.50 56.80 27.68 27.20 35.28 29.50

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From the results one can conclude that all modes become significantly damped after passive and active designs. 5. Conclusions Damping maximisation in hybrid active–passive sandwich laminated plates with viscoelastic core and laminated face layers with co-located piezoelectric patches has been addressed in this paper. The optimisation problem was formulated with thickness design variables, as well as elastic ply angles of laminated faces and piezoelectric sensor and actuator placement. Since few plate/shell hybrid active–passive sandwich models exist, and are mostly limited to isotropic materials, a new mixed layerwise finite element model was developed for optimization purposes. For passive damping optimisation, two approaches were used: one gradient based with the new layerwise sandwich model, and another one based on genetic algorithms and ABAQUS 3D finite elements. Results for both approaches present a good agreement. Active damping maximisation has been conducted using a genetic algorithm to find the optimal position of a predefined number of sensor–actuator co-located piezoelectric patches. Optimal position of four sensor–actuator pairs has been determined for a simply supported hybrid active–passive sandwich plate, for the first seven flexural modes of vibration. Results show an appreciable increase in modal loss factors for the frequency band under consideration. Acknowledgements The authors thank the financial support of FCT (Portugal) through: POCTI/FEDER, and POCI(2010)/FEDER, Projects POCI/ EME/56316/2004 and PPCDT/ EME/56316/2004, EU through FP6STREP Project CASSEM, Contract No. FP6-NMP3-CT-2005-013517, CNPq (Brazil), CAPES (Brazil) through Project CAPES/FCT No. 268/ 2010 and FAPERJ (Brazil). References [1] Baz A, Ro J. Optimum design and control of active constrained layer damping. J Mech Eng Des 1995;117:135–44. [2] Marcelin JL, Trompette Ph, Smati A. Optimal constrained layer damping with partial coverage. Finite Elem Anal Des 1992;12:273–80.

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