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Optimal design and parameter estimation of frequency dependent viscoelastic laminated sandwich composite plates
Composite Structures 92 (2010) 2321–2327
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Optimal design and parameter estimation of frequency dependent viscoelastic laminated sandwich composite plates A.L. Araújo a,*, C.M. Mota Soares b, C.A. Mota Soares b, J. Herskovits c a
ESTIG – Polytechnic Institute of Bragança, Campus de Sta. Apolónia, Apartado 1134, 5301-857 Bragança, Portugal IDMEC/IST – Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal c COPPE, Federal University of Rio de Janeiro, Caixa Postal 68503, 21945-970 Rio de Janeiro, Brazil b
a r t i c l e
i n f o
Article history: Available online 9 July 2009 Keywords: Gradient optimization Viscoelastic damping Sandwich structures Inverse problems
a b s t r a c t Recent developments in optimization and parameter estimation of frequency dependent passive damping of sandwich structures with viscoelastic core are presented in this paper. A finite element model for anisotropic laminated plate structures with viscoelastic frequency dependent core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach, by considering a higher order shear deformation theory (HSDT) to represent the displacement field of the viscoelastic core, and a first order shear deformation theory (FSDT) for the displacement fields of adjacent laminated face layers. The complex modulus approach is used for the viscoelastic material behaviour, and the dynamic problem is solved in the frequency domain, using viscoelastic material data for the core, assuming fractional derivative constitutive models. Constrained optimization of passive damping is conducted for the maximisation of modal loss factors, using the Feasible Arc Interior Point Algorithm (FAIPA). Identification of the frequency dependent material properties of the sandwich core is conducted by estimating the parameters that define the fractional derivative constitutive model. Optimal design and parameter estimation applications in sandwich structures are presented and discussed. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Structural damping can be defined as the process by which a structure or structural component dissipates mechanical energy or transfers it to connected structures or ambient media. These mechanisms have the effect of controlling the amplitude of resonant vibrations and modifying wave attenuation and sound transmission properties, increasing structural life through reduction in structural fatigue. Passive damping treatments are widely used in engineering applications in order to reduce vibration and noise radiation [1,2]. Passive layer damping, usually implemented as constrained layer damping, is the most common form of damping treatment, where the damping layer deforms in shear mode, thus dissipating energy in a more efficient way. Sandwich plates with viscoelastic core are very effective in reducing and controlling vibration response of lightweight and flexible structures, where the soft core is strongly deformed in shear, due to the adjacent stiff layers. Due to this high shear devel-
* Corresponding author. Tel.: +351 273 30 31 22; fax: +351 273 31 30 51. E-mail addresses: [email protected] (A.L. Araújo), [email protected] (C.M. Mota Soares), [email protected] (C.A. Mota Soares), [email protected] (J. Herskovits). 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.07.006
oped inside the core, equivalent single layer plate theories, even those based on higher order deformations, are not adequate to describe the behaviour of these sandwiches, mainly due to the high deformation discontinuities that arise at the interfaces between the viscoelastic core material and the surrounding elastic constraining layers. The usual approach to analyse the dynamic response of sandwich plates uses a layered scheme of plate and brick elements with nodal linkage, leading to a time consuming spatial modeling task. To overcome these difficulties, layerwise theories have been considered for constrained viscoelastic treatments, and most recently, Moreira et al. [3,4], among others, presented generalized layerwise formulations in this scope. Optimal design of 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 minimizing weight by selecting their optimal length and location. For example, Baz and Ro [5] optimized performance of constrained layer damping treatments by selecting the optimal thickness and shear modulus of the viscoelastic layer, and Marcelin et al. [6,7] used a genetic algorithm and beam finite elements to maximise the damping factor for partially treated beams, using as design variables the dimensions and locations of the patches. As verified by Nokes and Nelson [8], this
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layout optimization can lead to significant saving in the amount of material used. For fully covered sandwich beams, Lifshitz and Leibowitz [9] determined the optimal passive constrained layer damping, with layer thicknesses as design variables. The vibration damping of fully covered 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. Recently, Araújo et al. [10] presented optimal design formulations for damping maximisation in laminated composite material sandwiches, considering hysteretic damping, of which the present work is an extension for the case of frequency dependent viscoelastic material behaviour, considering also the corresponding identification of material parameters. In this work, optimization of modal loss factors of sandwich plates with elastic laminated constraining layers and a viscoelastic frequency dependent core is conducted with thicknesses and laminate layer ply orientation angles as design variables. The problem is solved through gradient based optimization associated to a mixed layerwise finite element model. Regarding identification of material parameters, eigenfrequency based methods for estimation of elastic [11], piezoelastic [12] and hysteretic damping parameters [13] have already been proposed by the authors, whilst very few research works are known in identification of frequency and temperature dependent viscoelastic material properties. Lekszycki et al. [14] conducted preliminary investigations concerning identification of constitutive parameters of viscoelastic materials in beam specimens, using a one-dimensional Voigt model associated with optimality conditions. Kostopoulos and Korontzis [15] conducted identification of frequency dependent viscoelastic properties of composite laminates, using laminated test coupons. The most usual non-destructive approach to the identification problem consists however, on the use of time and/or frequency domain response data [16,17]. In these approaches, the measurement error in experimental data can be an important issue that is reflected on the quality of the identified parameters. Thus, it would be convenient to develop an efficient alternative technique that relies only on experimental complex eigendata, which is less prone to measurement errors. In fact, very recently, Barkanov et al. [18] presented a technique that relies on experimental eigendata, associated to response surface analysis, which precludes the feasibility of such eigenvalue based approaches for viscoelastic material characterization. However, this approach approximates the frequency dependence of viscoelastic properties by mathematical expressions obtained through curve fitting techniques, and does not provide a mechanical model for the constitutive relations [19]. The eigenvalue based approach to the solution of the problem is explored in this paper using gradient optimization algorithms, along with analytic sensitivities, and parametric viscoelastic constitutive behaviour models, such as fractional derivative ones. The inverse problem is formulated as a constrained optimization problem, by fitting the response of the finite element numerical model to the corresponding experimental response of the viscoelastic sandwich structure. Application to the identification of viscoelastic frequency dependent core material parameters is presented and discussed in order to assess the feasibility of the proposed technique.
The basic assumptions in the development of the sandwich plate model are: 1. All points on a normal to the plate have the same transverse displacement wðx; y; tÞ, where t denotes time, and the origin of the z axis is the medium plane of the core layer. 2. No slip occurs at the interfaces between layers. 3. The displacement is C 0 along the interfaces. 4. Elastic layers are modeled with first order shear deformation theory (FSDT) and viscoelastic core with a higher order shear deformation theory (HSDT). 5. All materials are linear, homogeneous and orthotropic and the elastic layers ðe1 Þ and ðe2 Þ are made of laminated composite materials. 6. For the viscoelastic core, material properties are complex and frequency dependent. 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 i ðx; y; z; tÞ ¼ v i0 ðx; y; tÞ þ ðz zi Þhiy ðx; y; tÞ w 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 ðz ¼ 0Þ, and i ¼ e1 ; e2 is the layer index. 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: 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 w0 ðx; y; tÞ 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. The displacement continuity at the layer interfaces can be written as:
2. Mixed layerwise sandwich finite element model The development of a layerwise finite element model is presented here, to analyze sandwich laminated plates with a viscoelastic ðv Þ core and laminated anisotropic face layers ðe1 ; e2 Þ, as shown in Fig. 1.
ð1Þ
i
Fig. 1. Sandwich plate.
A.L. Araújo et al. / Composite Structures 92 (2010) 2321–2327
he hv ue1 x; y; ze1 1 ; t ¼ uv x; y; ; t 2 2 h e1 hv e1 v v x; y; ze1 ; t ¼ v x; y; ; t 2 2 h hv e ue2 x; y; ze2 þ 2 ; t ¼ uv x; y; ; t 2 2 h h e v e2 x; y; ze2 þ 2 ; t ¼ v v x; y; v ; t 2 2
ð3Þ
where the coordinates of layer mid-planes are:
hv he1 ze1 ¼ þ 2 2 zv ¼ 0 ze2 ¼
hv he2 2 2
2
3
he1 e1 hv h h v h þ uv0 þ hvx þ v u0 v þ v hx 2 x 2 4 8
v e0
¼
h e1 e1 hv h h h þ v v0 þ hvy þ v v 0 v þ v hy v 2 y 2 4 8
ue02
he hv h h v ¼ 2 hex2 þ uv0 hvx þ v u0 v v hx 2 2 4 8
2
3
2
v
ð5Þ
3
2
e2 0
2.1. Constitutive relations We consider that fibre-reinforced laminae in elastic multi-layers ðe1 Þ and ðe2 Þ, and viscoelastic core ðv Þ are characterized as orthotropic. Constitutive equations for each lamina in the sandwich may then be expressed in the principal material directions ðx1 ; x2 ; x3 ¼ zÞ, and for the zero transverse normal stress situation, as [12]:
Q 11
6 6 Q 12 6 r23 ¼ 6 60 > > > > > 6 > r > 40 13 > > > > > ; : r12 0
Q 12
0
0
0
Q 22
0
0
0
0
Q 44
0
0
0
0
Q 55
0
0
0
0
Q 66
9 38 e11 > > > > > > 7> > e22 > > > 7> = < 7 7 c23 7> > > > 7> c13 > > 5> > > > > ; :
ð6Þ
E1 ðjxÞ ¼ E01 ðxÞð1 þ jgE1 ðxÞÞ E2 ðjxÞ ¼ E02 ðxÞð1 þ jgE2 ðxÞÞ G23 ðjxÞ ¼ G023 ðxÞð1 þ jgG23 ðxÞÞ G13 ðjxÞ ¼ G013 ðxÞð1 þ jgG13 ðxÞÞ
m12 ðjxÞ ¼ m012 ðxÞð1 þ jgm12 ðxÞÞ
ð9Þ
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:
½KðxÞ x2 MuðxÞ ¼ FðxÞ
ð7Þ
ð10Þ
where FðxÞ ¼ FðFðtÞÞ is the Fourier transform of the time domain force history FðtÞ. For the free vibration problem, Eq. (10) reduces to the following non-linear eigenvalue problem:
½KðxÞ kn Mun ¼ 0
ð11Þ
where the complex eigenvalue kn is written as:
kn ¼ kn ð1 þ ign Þ
ð12Þ
and kn is the real part of the complex eigenvalue and gn is the corresponding modal loss factor. The non-linear eigenvalue problem is solved iteratively using ARPACK [21] with a shift-invert transformation. The iterative process is considered to have converged when:
kxi xi1 k
c12
where rij are stress components, eij and cij are strain components, and Q ij are reduced stiffness coefficients. Expressions for the reduced quantities mentioned above can be found in [12]. For the viscoelastic core layer, the reduced stiffness coefficients Q ij are complex quantities, since the complex modulus approach was used in this work, using the elastic–viscoelastic principle. In this case, the usual engineering moduli may be represented by complex quantities:
G12 ðjxÞ ¼ G012 ðxÞð1 þ jgG12 ðxÞÞ
€ are mechanical degrees of freedom and correspondwhere u and u ing accelerations, respectively, M and K are the mass and complex stiffness matrices, respectively, and F is the externally applied mechanical load vector. One should note that the viscoelastic behaviour of the core translates into a complex frequency dependent stiffness matrix KðxÞ. Assuming harmonic vibrations, the final equilibrium equations are given by:
½KðxÞ x2 Mu ¼ F
These relations allow us to retain the rotational degrees of freedom of the face layers, while eliminating the corresponding inplane displacement ones. Hence, the generalized displacement field has 13 mechanical unknowns.
2
ð8Þ
3
he hv h h v ¼ 2 hey2 þ v v0 hvy þ v v 0 v v hy 2 2 4 8
9 8 r11 > > > > > > > > > > > = < r22 >
The equations of motion for the plate are obtained by applying the extended Hamilton’s principle, using an eight node serendipity plate element with 13 mechanical degrees of freedom per node:
€ þ Ku ¼ F Mu
Applying displacement continuity conditions at the layer interfaces, one obtains:
1
where the prime quantities denote storage moduli, associated material loss factors are represented bypthe ffiffiffiffiffiffiffi letter g; x represents angular frequency of vibration and j ¼ 1 is the imaginary unit. Furthermore, in Eq. (7), 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 ðv Þ and the elastic multilayered laminates ðe1 Þ and ðe2 Þ [20]. 2.2. Finite element formulation
ð4Þ
ue01 ¼
2323
xi1
6
ð13Þ
where xi and xi1 are current and previous iteration values for the real part of the particular eigenfrequency of interest, respectively, and is the convergence tolerance.
3. Optimal design formulation The objective of this study is to maximise damping in 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. The formulated problem is solved using the Feasible Arc Interior Point Algorithm (FAIPA) [22], along with the developed finite element sandwich model, using semi-analytic sensitivities.
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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 minimized in this framework, subjected to design constraints:
min f ¼ xi
N P i¼1
s:t:
gj :
wi g1
i
gj 6 0;
j ¼ 1; . . . ; N
g Nþ2 :
m mmax w wmax
g Nþ3 :
F TH 1 6 0
g Nþ1 :
xli
6 xi 6
1 6 0
ð14Þ
1 6 0
xui ;
4. Estimation of core viscoelastic parameters
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 F TH 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 X 2 Y 2 Z2 1 1 1 1 1 1 þ 2 2 r22 r33 þ 2 2 r11 r33 2 2 Y Z X Z X Y r 2 r 2 r 2 23 13 12 þ þ þ <1 R S T r 2 11
þ
r 2 22
ð15Þ
where
@ gn 1 @Iðkn Þ @Rðkn Þ ¼ gn kn @xi @xi @xi
ð17Þ
with Rðkn Þ and Iðkn Þ being the real and imaginary parts of the complex eigenvalue kn , respectively. The derivative of the complex eigenvalue with respect to design variables can be obtained using the following expression [23]:
@kn h ¼ x @xi uTn M 2kn n
@Kðx;xi Þ @x
xi
s:t:
33
ð16Þ
iÞ kn @Mðx @xi
min U P 0
r 2
þ
@ 1 wn @ gn ¼ 2 wn @xi gn gn @xi
h
The inverse eigenvalue problem of estimation of the material properties can be solved in a number of different ways, and the approach that is used in this work consists on minimizing an error function, with respect to the elastic and frequency dependent damping material parameters:
i
i
un
un
g j 6 0; xli
where the stress components are calculated for each elastic layer ply and refer to the principal material directions of the ply, X; Y 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. (15). 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. (14) are the thicknesses of the constituent layers and the orientation angles of the laminated elastic composite material plies. In Eq. (14), 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. (11) iteratively, for a frequency dependent complex stiffness matrix and real mass matrix. Semi-analytic sensitivities are used to evaluate the gradient of the objective function with respect to the design variables:
uTn @Kð@xx;xi Þ i
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. (10) 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 F TH in Eq. (15) is evaluated. Sensitivities of displacement and stress quantities can be calculated analytically, semi-analytically, or using a global finite difference approach [23].
ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi where xn ¼ Rðkn Þ is the resonant frequency, and all derivatives are evaluated at x ¼ xn .
6 xi 6
j ¼ 1; . . . ; m xui ;
ð19Þ
i ¼ 1; . . . ; n
where U is the error function, xi are the material parameters design variables, and g j are design constraints based on thermodynamic requirements on isothermal linear viscoelasticity. The error function is a weighted least squares error estimator, which expresses the deviation between experimental and numerical complex eigenvalues:
U¼
2 X I kn g 2 w kn 1 þ wgn 1 n ~kn g~ n n¼1 n¼1
I X
ð20Þ
where ~ kn are the real parts of the experimental eigenvalues, kn are the corresponding real parts of the eigenvalues predicted by the ~ n and gn are, respectively, experimental and numerical model, g numerical modal loss factors, wkn and wgn are weights used to express the confidence level in each experimental eigenvalue and corresponding loss factor, respectively, and I is the total number of experimental eigenpairs under consideration. The constrained minimization problem is solved using FAIPA, the Feasible Arc Interior Point Algorithm [22], and analytic sensitivities. 4.1. Fractional derivative models As a first approach to the identification of frequency dependent viscoelastic material parameters, an isotropic material was assumed for the core of a passive sandwich and fractional derivative parametric models were chosen for the complex shear modulus. A four parameter fractional derivative model is considered first [24]:
GðjxÞ ¼ G0
1 þ aðjxÞa 1 þ bðjxÞa
ð21Þ
where a; b; a, and G0 are parameters to be identified, with G0 the static shear modulus. It can be shown that this model is causal and satisfies the thermodynamic constraints on linear viscoelasticity [24], as long as the model parameters are non-negative. It is furthermore assumed that 0 6 a 6 1. Next, a five parameter fractional derivative model was considered [25]:
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" GðjxÞ ¼ G0 1 þ
#
aðjxÞa 1 þ bðjxÞ
b
ð22Þ
where a; b; a; b, and G0 are the parameters to be identified. For this model to be causal, it follows from thermodynamic requirements that all parameters should be non-negative, and additionally a > b, where we have considered a b < 0:06, from experimental evidence on polymeric damping materials [25]. Again, it is assumed that 0 6 a 6 1 and 0 6 b 6 1. 4.2. Design sensitivities Analytic sensitivities are used in this work due to computational efficiency reasons. Hence, the derivatives of the objective function have been obtained analytically as follows:
I X @U kn ¼ 2 w kn 1 ~kn @xi n¼1
I X 1 @kn g 1 @ gn 2 wgn 1 n ~kn @xi g~ n g~ n @xi n¼1
ð23Þ
where @@xgn is obtained by Eq. (17) and the derivative of the complex i eigenvalue with respect to design variables is obtained using Eq. (18). The derivatives of the system matrices are obtained analytically at the element level, where the usual finite element assembly procedure is used. 5. Applications 5.1. Passive design of a simply supported sandwich plate A simply supported sandwich plate of in-plane dimensions 300 mm 200 mm with symmetric layout of layers is considered. The composite elastic layers ðe1 Þ and ðe2 Þ are made of three plies each with equal thickness, which are design variables: ð3Þ ð1Þ ð2Þ ð2Þ ð1Þ ð3Þ he1 ¼ he2 ; he1 ¼ he2 , and he1 ¼ he2 (from outer to inner plies). As for the viscoelastic core, its thickness hv will also be a design variable. The thickness design variables can take values from 0.5 mm to 10 mm. The orientation angles of the composite elastic layer plies are considered also to be design variables: ð1Þ ð2Þ ð2Þ ð1Þ ð3Þ (from outer to inner plies), hð3Þ e1 ¼ he2 ; he1 ¼ he2 , and he1 ¼ he2 assuming values between 0° and 175°. The material properties for the elastic material layers are: E1 ¼ 98:0 GPa, E2 ¼ 7:9 GPa, G12 ¼ G13 ¼ G23 ¼ 5:6 GPa, m12 ¼ 0:28, q ¼ 1520 kg=m3 . For the isotropic viscoelastic core, the five parameter fractional derivative constitutive model of Eq. (22) is used to describe the complex shear modulus of the material, where G0 ¼ 20 MPa; a ¼ 0:011; b ¼ 7:932 106 ; a ¼ 0:566, and b ¼ 0:558. Additionally m ¼ 0:49 and q ¼ 1300 kg=m3 are also considered. The fundamental flexural modal loss factor of the plate will be maximised, with a maximum allowable mass mmax ¼ 0:5 kg and a maximum allowable displacement wmax ¼ h=5, where h ¼ he1 þ hv þ he2 is the total thickness of the plate. The failure stresses in Eq. (15) are, for the elastic layers, X ¼ 820 MPa; Y ¼ Z ¼ 45 MPa, both in tension and compression, and R ¼ S ¼ T ¼ 45 MPa. The excitation consisted of a 10 N force applied at the mid-point of the plate at t ¼ 0, and a 6 4 finite element mesh was used, with a total of 1505 degrees of freedom. Results are presented in Table 1, where the initial and final designs are shown. For the final design, the only active constraints are the mass constraint, and the lower bounds on elastic layer thicknesses. Figs. 2 and 3 show the magnitude of the frequency response and time response, respectively, for the initial and final sandwich plate designs. A comparison of the optimization results produced by the present technique and an alternative one based on Genetic Algorithms and a commercial finite element code has been established [10],
Table 1 Optimal design results for the simply supported sandwich plate (DV: design variable, OBJ: objective function). Initial
Optimal
ð3Þ
ð1Þ
0.60
0.50
ð2Þ
ð2Þ
0.60
0.50
ð3Þ
0.60
0.50
0.60 90
2.90 100
¼ heð2Þ (deg.) DV: heð2Þ 1 2
45
97
¼ heð3Þ (deg.) DV: heð1Þ 1 2
45
72
OBJ: f x1 (Hz) m (g)
12.9 349.7 375 46.1
4.5 352.5 500 71.1
0.28 7.7
0.08 22.4
DV: he1 ¼ he2 (mm) DV: he1 ¼ he2 (mm) DV:
ð1Þ he1
¼ he2 (mm)
DV: hv (mm) DV: heð3Þ ¼ heð1Þ (deg.) 1 2
F TH ð103 Þ w (mm) g1 (%)
with very good agreement between the two approaches, with less computational effort for the present approach. 5.2. Parameter estimation A 300 mm 200 mm laminated sandwich plate with all edges clamped and made of carbon fibre plies and a central ISD-112 viscoelastic damping polymer core is considered. The stacking sequence is ½0 c =90 c = þ 45 c =0 v =þ45 c =90 c =0 c , where subscripts c and v stand for carbon fibres and viscoelastic damping material, respectively. The thickness of each carbon fibre ply is 0.5 mm, and the viscoelastic core is 2.5 mm thick. Material properties for ISD-112 viscoelastic damping polymer are taken from the literature [18], for the frequency range f ¼ 5; . . . ; 1600 Hz:
G ¼ 4:759 0:9266=z þ 2:405z2 ðMPaÞ with z ¼ 0:1918 þ 0:0005148f
gG ¼ gE ¼ 1:385 0:03673z 0:01342=z
ð24Þ
with z ¼ 0:01 þ 0:0006306f
m ¼ 0:49; q ¼ 1000 kg=m3
Fig. 2. Frequency response (magnitude) at the point of maximum displacement for the initial and final designs of the simply supported sandwich plate.
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Fig. 3. Time response at the point of maximum displacement for the initial and final designs of the simply supported sandwich plate.
For the carbon fibre plies, material properties are E1 ¼ 130:8 GPa, E2 ¼ 10:6 GPa, G12 ¼ 5:6 GPa, G13 ¼ 4:2 GPa, G23 ¼ 3:0 GPa, m12 ¼ 0:36, and q ¼ 1543 kg=m3 . The first twelve flexural natural frequencies of free vibration of the sandwich plate and corresponding modal loss factors, using the given properties, are considered for identification, as reference values (the numerical equivalent to experimental data in the simulation). As for the fractional derivative constitutive models, we consider both four and five parameter ones. In order to assess if the static
Four parameter model
Five parameter model
G0 not DV
G0 DV
G0 not DV
G0 DV
2.681
0.542
2.702
1.268
b G0 (MPa)
7:517 107 0.593 – 0.016a
1:302 105 0.597 – 0.078
2:923 1012 0.592 0.589 0.016a
1:448 1011 0.594 0.705 0.034
U
1:421 103
1:106 103
1:389 103
1:356 103
a
a
shear modulus G0 can be identified with dynamic data, two identification simulations are conducted for each fractional derivative model: one considering G0 as design variable, and the other not. Results are presented in Table 2. In Table 3, reference natural frequencies ~f n and modal loss fac~ n , calculated with the data from Eq. (24) and used in the identors g tifications, are presented along with the corresponding residuals after identification, which are defined as:
r fn ¼
Table 2 Identified parameters (DV: design variable).
a b
Fig. 4. Frequency response functions (magnitude) for ISD-112 and identified fractional derivative models. The curves are obtained at the center point of the plate, corresponding to a 10 N impulsive load applied at t ¼ 0 at the same point.
Calculated from Eq. (24).
fn ~f n 100; ~f n
r gn ¼
gn g~ n 100 g~ n
ð25Þ
where fn and gn are the natural frequencies and modal loss factors of the identified model. The original frequency response curves and the ones based on the identified parameters (considering G0 as design variable) are presented in Fig. 4. Except for the low frequency region, the curves exhibit a perfect match, indicating the feasibility to conduct parametric identification of viscoelastic material properties just with eigenfrequency based experimental data. The low frequency behaviour reflects the fact that we identified the static shear
Table 3 Natural frequencies, modal loss factors and residuals after identification. n
~f n (Hz)
g~ n (%)
Four parameter model G0 not DV
1 2 3 4 5 6 7 8 9 10 11 12
211.72 382.63 473.22 630.85 674.83 876.43 963.66 995.54 1080.30 1393.95 1440.67 1461.00
46.58 41.87 42.52 39.41 31.93 39.59 32.92 31.43 33.41 30.16 33.90 27.51
Five parameter model G0 DV
G0 not DV
G0 DV
r fn (%)
r gn (%)
rfn (%)
r gn (%)
r fn (%)
r gn (%)
r fn (%)
r gn (%)
0.11 0.53 0.43 0.16 0.07 0.11 0.15 0.11 0.13 0.05 0.10 0.10
2.80 0.86 1.13 0.64 0.50 0.25 0.33 0.62 0.57 0.13 0.39 0.52
0.19 0.39 0.30 0.07 0.01 0.18 0.21 0.15 0.18 0.00 0.05 0.06
1.89 1.26 1.38 0.74 0.53 0.31 0.38 0.79 0.73 0.08 0.13 0.25
0.03 0.47 0.37 0.11 0.04 0.16 0.18 0.14 0.17 0.02 0.07 0.07
2.81 0.86 1.13 0.65 0.49 0.23 0.28 0.64 0.55 0.17 0.42 0.53
0.13 0.58 0.48 0.21 0.12 0.06 0.10 0.07 0.09 0.09 0.15 0.13
2.47 1.02 1.25 0.69 0.55 0.26 0.39 0.61 0.60 0.03 0.30 0.44
A.L. Araújo et al. / Composite Structures 92 (2010) 2321–2327
modulus G0 using dynamic data. However, when no feasible information exists regarding G0 , the current parameter estimation procedure might be used to estimate this quantity, although the estimated parameters should be used with caution outside the frequency range of the experimental eigendata used for identification. 6. Conclusions Damping maximisation in passive sandwich laminated plates with frequency dependent viscoelastic core and laminated face layers has been addressed in this paper. The optimization problem was formulated with thickness design variables, as well as elastic ply angles of laminated face layers, and the maximisation of fundamental modal loss factors was sought. A gradient based approach was used, employing the Feasible Arc Interior Point Algorithm (FAIPA) along with a new layerwise sandwich model that was developed for this purpose. Results for the optimization of a simply supported sandwich plate show that the present technique can improve substantially modal loss factors for these structures. Identification of frequency dependent properties of frequency dependent viscoelastic core materials in a composite sandwich configuration has also been presented. The obtained results indicate that the inverse problem can be solved using only information from eigenfrequencies and modal loss factors, associated to fractional derivative constitutive models. Acknowledgements The authors thank the financial support of Fundaçao para a Ciência e a Tecnologia, Portugal, through: POCTI/FEDER, and POCI(2010)/FEDER, CNPq (Brazil), CAPES (Brazil), and FAPERJ (Brazil). References [1] Nashif AD, Jones DIG, Henderson JP. Vibration damping. New York: John Wiley & Sons; 1985. [2] Sun CT, Lu YP. Vibration damping of structural elements. New Jersey: Prentice Hall PTR; 1995. [3] Moreira RAS, Rodrigues JD, Ferreira AJM. A generalized layerwise finite element for multi-layer damping treatments. Comput Mech 2006;37:426–44. [4] Moreira RAS, Rodrigues JD. A layerwise model for thin soft core sandwich plates. Comput Struct 2006;84:1256–63.
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[5] Baz A, Ro J. Optimum design and control of active constrained layer damping. J Mech Eng Des 1995;117:135–44. [6] Marcelin JL, Trompette P, Smati A. Optimal constrained layer damping with partial coverage. Finite Elem Anal Des 1992;12:273–80. [7] Marcelin JL, Shakhesi S, Pourroy F. Optimal constrained layer damping of beams: experimental numerical studies. Shock Vib 1995;2:445–50. [8] Nokes DS, Nelson FC. Constrained layer damping with partial coverage. Shock Vib Bull 1968;38:5–10. [9] Lifshitz JM, Leibowitz M. Optimal sandwich beam design for maximum viscoelastic damping. Int J Solids Struct 1987;23:1027–34. [10] Araújo AL, Martins P, Mota Soares CM, Mota Soares CA, Herskovits J. Damping optimization of viscoelastic laminated sandwich composite structures. Struct Multidiscip Optim 2009. doi:10.1007/s00158-009-0390-4. [11] Mota Soares CM, Freitas MJM, Araújo AL, Pedersen P. Identification of material properties of composite plate specimens. Compos Struct 1993;25:277–85. [12] Araújo AL, Lopes HMR, Vaz MAP, Mota Soares CM, Herskovits J, Pedersen P. Parameter estimation in active plate structures. Comput Struct 2006;84:1471–9. [13] Araújo AL, Mota Soares CM, Herskovits J, Pedersen P. Estimation of piezoelectric and viscoelastic properties in laminated structures. Compos Struct 2009;87:168–74. [14] Lekszycki T, Olhoff N, Pedersen JJ. Modelling and identification of viscoelastic properties of vibrating sandwich beams. Compos Struct 1992;22:15–31. [15] Kostopoulos V, Korontzis DTh. A new method for the determination of the viscoelastic properties of composite laminates: a mixed analyticalexperimental approach. Compos Sci Technol 2003;63:1441–52. [16] Castello DA, Rochinha FA, Roitman N, Magluta C. Constitutive parameter estimation of a viscoelastic model with internal variables. Mech Syst Signal Process 2008;22:1840–57. [17] Kim S-Y, Lee D-H. Identification of fractional-derivative-model parameters of viscoelastic materials from measured FRFs. J Sound Vib 2009. doi:10.1016/ j.jsv.2009.02.040. [18] Barkanov E, Skukis E, Wesolowski M, Chate A. Characterization of adhesive layers in sandwich composites by nondestructive technique. Proc World Acad Sci Eng Technol 2009:38. [19] Dietrich L, Lekszycki T, Turski K. Problems of identification of mechanical characteristics of viscoelastic composites. Acta Mech 1998;126:153–67. [20] Araújo AL, Mota Soares CM, Herskovits J, Pedersen P. Development of a finite element model for the identification of mechanical and piezoelectric properties through gradient optimisation and experimental vibration data. Compos Struct 2002;58:307–18. [21] Sorensen DC. Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations. Department of Computational and Applied Mathematics, Rice University, Technical Report TR95-13, Houston, Texas, 1995. [22] Herskovits J, Mappa P, Goulart E, Mota Soares CM. Mathematical programming models and algorithms for engineering design optimization. Comput Methods Appl Mech Eng 2005;194:3244–68. [23] Haftka RT, Gurdal Z. Elements of structural optimization. Dordrecht: Kluwer Academic Publishers.; 1992. [24] Pritz T. Analysis of four-parameter fractional derivative model of real solid materials. J Sound Vib 1996;195:103–15. [25] Pritz T. Five-parameter fractional derivative model for polymeric damping materials. J Sound Vib 2003;265:935–52.
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Optimal design and parameter estimation of frequency dependent viscoelastic laminated sandwich composite plates A.L. Araújo a,*, C.M. Mota Soares b, C.A. Mota Soares b, J. Herskovits c a
ESTIG – Polytechnic Institute of Bragança, Campus de Sta. Apolónia, Apartado 1134, 5301-857 Bragança, Portugal IDMEC/IST – Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal c COPPE, Federal University of Rio de Janeiro, Caixa Postal 68503, 21945-970 Rio de Janeiro, Brazil b
a r t i c l e
i n f o
Article history: Available online 9 July 2009 Keywords: Gradient optimization Viscoelastic damping Sandwich structures Inverse problems
a b s t r a c t Recent developments in optimization and parameter estimation of frequency dependent passive damping of sandwich structures with viscoelastic core are presented in this paper. A finite element model for anisotropic laminated plate structures with viscoelastic frequency dependent core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach, by considering a higher order shear deformation theory (HSDT) to represent the displacement field of the viscoelastic core, and a first order shear deformation theory (FSDT) for the displacement fields of adjacent laminated face layers. The complex modulus approach is used for the viscoelastic material behaviour, and the dynamic problem is solved in the frequency domain, using viscoelastic material data for the core, assuming fractional derivative constitutive models. Constrained optimization of passive damping is conducted for the maximisation of modal loss factors, using the Feasible Arc Interior Point Algorithm (FAIPA). Identification of the frequency dependent material properties of the sandwich core is conducted by estimating the parameters that define the fractional derivative constitutive model. Optimal design and parameter estimation applications in sandwich structures are presented and discussed. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Structural damping can be defined as the process by which a structure or structural component dissipates mechanical energy or transfers it to connected structures or ambient media. These mechanisms have the effect of controlling the amplitude of resonant vibrations and modifying wave attenuation and sound transmission properties, increasing structural life through reduction in structural fatigue. Passive damping treatments are widely used in engineering applications in order to reduce vibration and noise radiation [1,2]. Passive layer damping, usually implemented as constrained layer damping, is the most common form of damping treatment, where the damping layer deforms in shear mode, thus dissipating energy in a more efficient way. Sandwich plates with viscoelastic core are very effective in reducing and controlling vibration response of lightweight and flexible structures, where the soft core is strongly deformed in shear, due to the adjacent stiff layers. Due to this high shear devel-
* Corresponding author. Tel.: +351 273 30 31 22; fax: +351 273 31 30 51. E-mail addresses: [email protected] (A.L. Araújo), [email protected] (C.M. Mota Soares), [email protected] (C.A. Mota Soares), [email protected] (J. Herskovits). 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.07.006
oped inside the core, equivalent single layer plate theories, even those based on higher order deformations, are not adequate to describe the behaviour of these sandwiches, mainly due to the high deformation discontinuities that arise at the interfaces between the viscoelastic core material and the surrounding elastic constraining layers. The usual approach to analyse the dynamic response of sandwich plates uses a layered scheme of plate and brick elements with nodal linkage, leading to a time consuming spatial modeling task. To overcome these difficulties, layerwise theories have been considered for constrained viscoelastic treatments, and most recently, Moreira et al. [3,4], among others, presented generalized layerwise formulations in this scope. Optimal design of 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 minimizing weight by selecting their optimal length and location. For example, Baz and Ro [5] optimized performance of constrained layer damping treatments by selecting the optimal thickness and shear modulus of the viscoelastic layer, and Marcelin et al. [6,7] used a genetic algorithm and beam finite elements to maximise the damping factor for partially treated beams, using as design variables the dimensions and locations of the patches. As verified by Nokes and Nelson [8], this
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layout optimization can lead to significant saving in the amount of material used. For fully covered sandwich beams, Lifshitz and Leibowitz [9] determined the optimal passive constrained layer damping, with layer thicknesses as design variables. The vibration damping of fully covered 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. Recently, Araújo et al. [10] presented optimal design formulations for damping maximisation in laminated composite material sandwiches, considering hysteretic damping, of which the present work is an extension for the case of frequency dependent viscoelastic material behaviour, considering also the corresponding identification of material parameters. In this work, optimization of modal loss factors of sandwich plates with elastic laminated constraining layers and a viscoelastic frequency dependent core is conducted with thicknesses and laminate layer ply orientation angles as design variables. The problem is solved through gradient based optimization associated to a mixed layerwise finite element model. Regarding identification of material parameters, eigenfrequency based methods for estimation of elastic [11], piezoelastic [12] and hysteretic damping parameters [13] have already been proposed by the authors, whilst very few research works are known in identification of frequency and temperature dependent viscoelastic material properties. Lekszycki et al. [14] conducted preliminary investigations concerning identification of constitutive parameters of viscoelastic materials in beam specimens, using a one-dimensional Voigt model associated with optimality conditions. Kostopoulos and Korontzis [15] conducted identification of frequency dependent viscoelastic properties of composite laminates, using laminated test coupons. The most usual non-destructive approach to the identification problem consists however, on the use of time and/or frequency domain response data [16,17]. In these approaches, the measurement error in experimental data can be an important issue that is reflected on the quality of the identified parameters. Thus, it would be convenient to develop an efficient alternative technique that relies only on experimental complex eigendata, which is less prone to measurement errors. In fact, very recently, Barkanov et al. [18] presented a technique that relies on experimental eigendata, associated to response surface analysis, which precludes the feasibility of such eigenvalue based approaches for viscoelastic material characterization. However, this approach approximates the frequency dependence of viscoelastic properties by mathematical expressions obtained through curve fitting techniques, and does not provide a mechanical model for the constitutive relations [19]. The eigenvalue based approach to the solution of the problem is explored in this paper using gradient optimization algorithms, along with analytic sensitivities, and parametric viscoelastic constitutive behaviour models, such as fractional derivative ones. The inverse problem is formulated as a constrained optimization problem, by fitting the response of the finite element numerical model to the corresponding experimental response of the viscoelastic sandwich structure. Application to the identification of viscoelastic frequency dependent core material parameters is presented and discussed in order to assess the feasibility of the proposed technique.
The basic assumptions in the development of the sandwich plate model are: 1. All points on a normal to the plate have the same transverse displacement wðx; y; tÞ, where t denotes time, and the origin of the z axis is the medium plane of the core layer. 2. No slip occurs at the interfaces between layers. 3. The displacement is C 0 along the interfaces. 4. Elastic layers are modeled with first order shear deformation theory (FSDT) and viscoelastic core with a higher order shear deformation theory (HSDT). 5. All materials are linear, homogeneous and orthotropic and the elastic layers ðe1 Þ and ðe2 Þ are made of laminated composite materials. 6. For the viscoelastic core, material properties are complex and frequency dependent. 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 i ðx; y; z; tÞ ¼ v i0 ðx; y; tÞ þ ðz zi Þhiy ðx; y; tÞ w 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 ðz ¼ 0Þ, and i ¼ e1 ; e2 is the layer index. 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: 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 w0 ðx; y; tÞ 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. The displacement continuity at the layer interfaces can be written as:
2. Mixed layerwise sandwich finite element model The development of a layerwise finite element model is presented here, to analyze sandwich laminated plates with a viscoelastic ðv Þ core and laminated anisotropic face layers ðe1 ; e2 Þ, as shown in Fig. 1.
ð1Þ
i
Fig. 1. Sandwich plate.
A.L. Araújo et al. / Composite Structures 92 (2010) 2321–2327
he hv ue1 x; y; ze1 1 ; t ¼ uv x; y; ; t 2 2 h e1 hv e1 v v x; y; ze1 ; t ¼ v x; y; ; t 2 2 h hv e ue2 x; y; ze2 þ 2 ; t ¼ uv x; y; ; t 2 2 h h e v e2 x; y; ze2 þ 2 ; t ¼ v v x; y; v ; t 2 2
ð3Þ
where the coordinates of layer mid-planes are:
hv he1 ze1 ¼ þ 2 2 zv ¼ 0 ze2 ¼
hv he2 2 2
2
3
he1 e1 hv h h v h þ uv0 þ hvx þ v u0 v þ v hx 2 x 2 4 8
v e0
¼
h e1 e1 hv h h h þ v v0 þ hvy þ v v 0 v þ v hy v 2 y 2 4 8
ue02
he hv h h v ¼ 2 hex2 þ uv0 hvx þ v u0 v v hx 2 2 4 8
2
3
2
v
ð5Þ
3
2
e2 0
2.1. Constitutive relations We consider that fibre-reinforced laminae in elastic multi-layers ðe1 Þ and ðe2 Þ, and viscoelastic core ðv Þ are characterized as orthotropic. Constitutive equations for each lamina in the sandwich may then be expressed in the principal material directions ðx1 ; x2 ; x3 ¼ zÞ, and for the zero transverse normal stress situation, as [12]:
Q 11
6 6 Q 12 6 r23 ¼ 6 60 > > > > > 6 > r > 40 13 > > > > > ; : r12 0
Q 12
0
0
0
Q 22
0
0
0
0
Q 44
0
0
0
0
Q 55
0
0
0
0
Q 66
9 38 e11 > > > > > > 7> > e22 > > > 7> = < 7 7 c23 7> > > > 7> c13 > > 5> > > > > ; :
ð6Þ
E1 ðjxÞ ¼ E01 ðxÞð1 þ jgE1 ðxÞÞ E2 ðjxÞ ¼ E02 ðxÞð1 þ jgE2 ðxÞÞ G23 ðjxÞ ¼ G023 ðxÞð1 þ jgG23 ðxÞÞ G13 ðjxÞ ¼ G013 ðxÞð1 þ jgG13 ðxÞÞ
m12 ðjxÞ ¼ m012 ðxÞð1 þ jgm12 ðxÞÞ
ð9Þ
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:
½KðxÞ x2 MuðxÞ ¼ FðxÞ
ð7Þ
ð10Þ
where FðxÞ ¼ FðFðtÞÞ is the Fourier transform of the time domain force history FðtÞ. For the free vibration problem, Eq. (10) reduces to the following non-linear eigenvalue problem:
½KðxÞ kn Mun ¼ 0
ð11Þ
where the complex eigenvalue kn is written as:
kn ¼ kn ð1 þ ign Þ
ð12Þ
and kn is the real part of the complex eigenvalue and gn is the corresponding modal loss factor. The non-linear eigenvalue problem is solved iteratively using ARPACK [21] with a shift-invert transformation. The iterative process is considered to have converged when:
kxi xi1 k
c12
where rij are stress components, eij and cij are strain components, and Q ij are reduced stiffness coefficients. Expressions for the reduced quantities mentioned above can be found in [12]. For the viscoelastic core layer, the reduced stiffness coefficients Q ij are complex quantities, since the complex modulus approach was used in this work, using the elastic–viscoelastic principle. In this case, the usual engineering moduli may be represented by complex quantities:
G12 ðjxÞ ¼ G012 ðxÞð1 þ jgG12 ðxÞÞ
€ are mechanical degrees of freedom and correspondwhere u and u ing accelerations, respectively, M and K are the mass and complex stiffness matrices, respectively, and F is the externally applied mechanical load vector. One should note that the viscoelastic behaviour of the core translates into a complex frequency dependent stiffness matrix KðxÞ. Assuming harmonic vibrations, the final equilibrium equations are given by:
½KðxÞ x2 Mu ¼ F
These relations allow us to retain the rotational degrees of freedom of the face layers, while eliminating the corresponding inplane displacement ones. Hence, the generalized displacement field has 13 mechanical unknowns.
2
ð8Þ
3
he hv h h v ¼ 2 hey2 þ v v0 hvy þ v v 0 v v hy 2 2 4 8
9 8 r11 > > > > > > > > > > > = < r22 >
The equations of motion for the plate are obtained by applying the extended Hamilton’s principle, using an eight node serendipity plate element with 13 mechanical degrees of freedom per node:
€ þ Ku ¼ F Mu
Applying displacement continuity conditions at the layer interfaces, one obtains:
1
where the prime quantities denote storage moduli, associated material loss factors are represented bypthe ffiffiffiffiffiffiffi letter g; x represents angular frequency of vibration and j ¼ 1 is the imaginary unit. Furthermore, in Eq. (7), 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 ðv Þ and the elastic multilayered laminates ðe1 Þ and ðe2 Þ [20]. 2.2. Finite element formulation
ð4Þ
ue01 ¼
2323
xi1
6
ð13Þ
where xi and xi1 are current and previous iteration values for the real part of the particular eigenfrequency of interest, respectively, and is the convergence tolerance.
3. Optimal design formulation The objective of this study is to maximise damping in 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. The formulated problem is solved using the Feasible Arc Interior Point Algorithm (FAIPA) [22], along with the developed finite element sandwich model, using semi-analytic sensitivities.
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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 minimized in this framework, subjected to design constraints:
min f ¼ xi
N P i¼1
s:t:
gj :
wi g1
i
gj 6 0;
j ¼ 1; . . . ; N
g Nþ2 :
m mmax w wmax
g Nþ3 :
F TH 1 6 0
g Nþ1 :
xli
6 xi 6
1 6 0
ð14Þ
1 6 0
xui ;
4. Estimation of core viscoelastic parameters
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 F TH 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 X 2 Y 2 Z2 1 1 1 1 1 1 þ 2 2 r22 r33 þ 2 2 r11 r33 2 2 Y Z X Z X Y r 2 r 2 r 2 23 13 12 þ þ þ <1 R S T r 2 11
þ
r 2 22
ð15Þ
where
@ gn 1 @Iðkn Þ @Rðkn Þ ¼ gn kn @xi @xi @xi
ð17Þ
with Rðkn Þ and Iðkn Þ being the real and imaginary parts of the complex eigenvalue kn , respectively. The derivative of the complex eigenvalue with respect to design variables can be obtained using the following expression [23]:
@kn h ¼ x @xi uTn M 2kn n
@Kðx;xi Þ @x
xi
s:t:
33
ð16Þ
iÞ kn @Mðx @xi
min U P 0
r 2
þ
@ 1 wn @ gn ¼ 2 wn @xi gn gn @xi
h
The inverse eigenvalue problem of estimation of the material properties can be solved in a number of different ways, and the approach that is used in this work consists on minimizing an error function, with respect to the elastic and frequency dependent damping material parameters:
i
i
un
un
g j 6 0; xli
where the stress components are calculated for each elastic layer ply and refer to the principal material directions of the ply, X; Y 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. (15). 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. (14) are the thicknesses of the constituent layers and the orientation angles of the laminated elastic composite material plies. In Eq. (14), 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. (11) iteratively, for a frequency dependent complex stiffness matrix and real mass matrix. Semi-analytic sensitivities are used to evaluate the gradient of the objective function with respect to the design variables:
uTn @Kð@xx;xi Þ i
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. (10) 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 F TH in Eq. (15) is evaluated. Sensitivities of displacement and stress quantities can be calculated analytically, semi-analytically, or using a global finite difference approach [23].
ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi where xn ¼ Rðkn Þ is the resonant frequency, and all derivatives are evaluated at x ¼ xn .
6 xi 6
j ¼ 1; . . . ; m xui ;
ð19Þ
i ¼ 1; . . . ; n
where U is the error function, xi are the material parameters design variables, and g j are design constraints based on thermodynamic requirements on isothermal linear viscoelasticity. The error function is a weighted least squares error estimator, which expresses the deviation between experimental and numerical complex eigenvalues:
U¼
2 X I kn g 2 w kn 1 þ wgn 1 n ~kn g~ n n¼1 n¼1
I X
ð20Þ
where ~ kn are the real parts of the experimental eigenvalues, kn are the corresponding real parts of the eigenvalues predicted by the ~ n and gn are, respectively, experimental and numerical model, g numerical modal loss factors, wkn and wgn are weights used to express the confidence level in each experimental eigenvalue and corresponding loss factor, respectively, and I is the total number of experimental eigenpairs under consideration. The constrained minimization problem is solved using FAIPA, the Feasible Arc Interior Point Algorithm [22], and analytic sensitivities. 4.1. Fractional derivative models As a first approach to the identification of frequency dependent viscoelastic material parameters, an isotropic material was assumed for the core of a passive sandwich and fractional derivative parametric models were chosen for the complex shear modulus. A four parameter fractional derivative model is considered first [24]:
GðjxÞ ¼ G0
1 þ aðjxÞa 1 þ bðjxÞa
ð21Þ
where a; b; a, and G0 are parameters to be identified, with G0 the static shear modulus. It can be shown that this model is causal and satisfies the thermodynamic constraints on linear viscoelasticity [24], as long as the model parameters are non-negative. It is furthermore assumed that 0 6 a 6 1. Next, a five parameter fractional derivative model was considered [25]:
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" GðjxÞ ¼ G0 1 þ
#
aðjxÞa 1 þ bðjxÞ
b
ð22Þ
where a; b; a; b, and G0 are the parameters to be identified. For this model to be causal, it follows from thermodynamic requirements that all parameters should be non-negative, and additionally a > b, where we have considered a b < 0:06, from experimental evidence on polymeric damping materials [25]. Again, it is assumed that 0 6 a 6 1 and 0 6 b 6 1. 4.2. Design sensitivities Analytic sensitivities are used in this work due to computational efficiency reasons. Hence, the derivatives of the objective function have been obtained analytically as follows:
I X @U kn ¼ 2 w kn 1 ~kn @xi n¼1
I X 1 @kn g 1 @ gn 2 wgn 1 n ~kn @xi g~ n g~ n @xi n¼1
ð23Þ
where @@xgn is obtained by Eq. (17) and the derivative of the complex i eigenvalue with respect to design variables is obtained using Eq. (18). The derivatives of the system matrices are obtained analytically at the element level, where the usual finite element assembly procedure is used. 5. Applications 5.1. Passive design of a simply supported sandwich plate A simply supported sandwich plate of in-plane dimensions 300 mm 200 mm with symmetric layout of layers is considered. The composite elastic layers ðe1 Þ and ðe2 Þ are made of three plies each with equal thickness, which are design variables: ð3Þ ð1Þ ð2Þ ð2Þ ð1Þ ð3Þ he1 ¼ he2 ; he1 ¼ he2 , and he1 ¼ he2 (from outer to inner plies). As for the viscoelastic core, its thickness hv will also be a design variable. The thickness design variables can take values from 0.5 mm to 10 mm. The orientation angles of the composite elastic layer plies are considered also to be design variables: ð1Þ ð2Þ ð2Þ ð1Þ ð3Þ (from outer to inner plies), hð3Þ e1 ¼ he2 ; he1 ¼ he2 , and he1 ¼ he2 assuming values between 0° and 175°. The material properties for the elastic material layers are: E1 ¼ 98:0 GPa, E2 ¼ 7:9 GPa, G12 ¼ G13 ¼ G23 ¼ 5:6 GPa, m12 ¼ 0:28, q ¼ 1520 kg=m3 . For the isotropic viscoelastic core, the five parameter fractional derivative constitutive model of Eq. (22) is used to describe the complex shear modulus of the material, where G0 ¼ 20 MPa; a ¼ 0:011; b ¼ 7:932 106 ; a ¼ 0:566, and b ¼ 0:558. Additionally m ¼ 0:49 and q ¼ 1300 kg=m3 are also considered. The fundamental flexural modal loss factor of the plate will be maximised, with a maximum allowable mass mmax ¼ 0:5 kg and a maximum allowable displacement wmax ¼ h=5, where h ¼ he1 þ hv þ he2 is the total thickness of the plate. The failure stresses in Eq. (15) are, for the elastic layers, X ¼ 820 MPa; Y ¼ Z ¼ 45 MPa, both in tension and compression, and R ¼ S ¼ T ¼ 45 MPa. The excitation consisted of a 10 N force applied at the mid-point of the plate at t ¼ 0, and a 6 4 finite element mesh was used, with a total of 1505 degrees of freedom. Results are presented in Table 1, where the initial and final designs are shown. For the final design, the only active constraints are the mass constraint, and the lower bounds on elastic layer thicknesses. Figs. 2 and 3 show the magnitude of the frequency response and time response, respectively, for the initial and final sandwich plate designs. A comparison of the optimization results produced by the present technique and an alternative one based on Genetic Algorithms and a commercial finite element code has been established [10],
Table 1 Optimal design results for the simply supported sandwich plate (DV: design variable, OBJ: objective function). Initial
Optimal
ð3Þ
ð1Þ
0.60
0.50
ð2Þ
ð2Þ
0.60
0.50
ð3Þ
0.60
0.50
0.60 90
2.90 100
¼ heð2Þ (deg.) DV: heð2Þ 1 2
45
97
¼ heð3Þ (deg.) DV: heð1Þ 1 2
45
72
OBJ: f x1 (Hz) m (g)
12.9 349.7 375 46.1
4.5 352.5 500 71.1
0.28 7.7
0.08 22.4
DV: he1 ¼ he2 (mm) DV: he1 ¼ he2 (mm) DV:
ð1Þ he1
¼ he2 (mm)
DV: hv (mm) DV: heð3Þ ¼ heð1Þ (deg.) 1 2
F TH ð103 Þ w (mm) g1 (%)
with very good agreement between the two approaches, with less computational effort for the present approach. 5.2. Parameter estimation A 300 mm 200 mm laminated sandwich plate with all edges clamped and made of carbon fibre plies and a central ISD-112 viscoelastic damping polymer core is considered. The stacking sequence is ½0 c =90 c = þ 45 c =0 v =þ45 c =90 c =0 c , where subscripts c and v stand for carbon fibres and viscoelastic damping material, respectively. The thickness of each carbon fibre ply is 0.5 mm, and the viscoelastic core is 2.5 mm thick. Material properties for ISD-112 viscoelastic damping polymer are taken from the literature [18], for the frequency range f ¼ 5; . . . ; 1600 Hz:
G ¼ 4:759 0:9266=z þ 2:405z2 ðMPaÞ with z ¼ 0:1918 þ 0:0005148f
gG ¼ gE ¼ 1:385 0:03673z 0:01342=z
ð24Þ
with z ¼ 0:01 þ 0:0006306f
m ¼ 0:49; q ¼ 1000 kg=m3
Fig. 2. Frequency response (magnitude) at the point of maximum displacement for the initial and final designs of the simply supported sandwich plate.
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Fig. 3. Time response at the point of maximum displacement for the initial and final designs of the simply supported sandwich plate.
For the carbon fibre plies, material properties are E1 ¼ 130:8 GPa, E2 ¼ 10:6 GPa, G12 ¼ 5:6 GPa, G13 ¼ 4:2 GPa, G23 ¼ 3:0 GPa, m12 ¼ 0:36, and q ¼ 1543 kg=m3 . The first twelve flexural natural frequencies of free vibration of the sandwich plate and corresponding modal loss factors, using the given properties, are considered for identification, as reference values (the numerical equivalent to experimental data in the simulation). As for the fractional derivative constitutive models, we consider both four and five parameter ones. In order to assess if the static
Four parameter model
Five parameter model
G0 not DV
G0 DV
G0 not DV
G0 DV
2.681
0.542
2.702
1.268
b G0 (MPa)
7:517 107 0.593 – 0.016a
1:302 105 0.597 – 0.078
2:923 1012 0.592 0.589 0.016a
1:448 1011 0.594 0.705 0.034
U
1:421 103
1:106 103
1:389 103
1:356 103
a
a
shear modulus G0 can be identified with dynamic data, two identification simulations are conducted for each fractional derivative model: one considering G0 as design variable, and the other not. Results are presented in Table 2. In Table 3, reference natural frequencies ~f n and modal loss fac~ n , calculated with the data from Eq. (24) and used in the identors g tifications, are presented along with the corresponding residuals after identification, which are defined as:
r fn ¼
Table 2 Identified parameters (DV: design variable).
a b
Fig. 4. Frequency response functions (magnitude) for ISD-112 and identified fractional derivative models. The curves are obtained at the center point of the plate, corresponding to a 10 N impulsive load applied at t ¼ 0 at the same point.
Calculated from Eq. (24).
fn ~f n 100; ~f n
r gn ¼
gn g~ n 100 g~ n
ð25Þ
where fn and gn are the natural frequencies and modal loss factors of the identified model. The original frequency response curves and the ones based on the identified parameters (considering G0 as design variable) are presented in Fig. 4. Except for the low frequency region, the curves exhibit a perfect match, indicating the feasibility to conduct parametric identification of viscoelastic material properties just with eigenfrequency based experimental data. The low frequency behaviour reflects the fact that we identified the static shear
Table 3 Natural frequencies, modal loss factors and residuals after identification. n
~f n (Hz)
g~ n (%)
Four parameter model G0 not DV
1 2 3 4 5 6 7 8 9 10 11 12
211.72 382.63 473.22 630.85 674.83 876.43 963.66 995.54 1080.30 1393.95 1440.67 1461.00
46.58 41.87 42.52 39.41 31.93 39.59 32.92 31.43 33.41 30.16 33.90 27.51
Five parameter model G0 DV
G0 not DV
G0 DV
r fn (%)
r gn (%)
rfn (%)
r gn (%)
r fn (%)
r gn (%)
r fn (%)
r gn (%)
0.11 0.53 0.43 0.16 0.07 0.11 0.15 0.11 0.13 0.05 0.10 0.10
2.80 0.86 1.13 0.64 0.50 0.25 0.33 0.62 0.57 0.13 0.39 0.52
0.19 0.39 0.30 0.07 0.01 0.18 0.21 0.15 0.18 0.00 0.05 0.06
1.89 1.26 1.38 0.74 0.53 0.31 0.38 0.79 0.73 0.08 0.13 0.25
0.03 0.47 0.37 0.11 0.04 0.16 0.18 0.14 0.17 0.02 0.07 0.07
2.81 0.86 1.13 0.65 0.49 0.23 0.28 0.64 0.55 0.17 0.42 0.53
0.13 0.58 0.48 0.21 0.12 0.06 0.10 0.07 0.09 0.09 0.15 0.13
2.47 1.02 1.25 0.69 0.55 0.26 0.39 0.61 0.60 0.03 0.30 0.44
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modulus G0 using dynamic data. However, when no feasible information exists regarding G0 , the current parameter estimation procedure might be used to estimate this quantity, although the estimated parameters should be used with caution outside the frequency range of the experimental eigendata used for identification. 6. Conclusions Damping maximisation in passive sandwich laminated plates with frequency dependent viscoelastic core and laminated face layers has been addressed in this paper. The optimization problem was formulated with thickness design variables, as well as elastic ply angles of laminated face layers, and the maximisation of fundamental modal loss factors was sought. A gradient based approach was used, employing the Feasible Arc Interior Point Algorithm (FAIPA) along with a new layerwise sandwich model that was developed for this purpose. Results for the optimization of a simply supported sandwich plate show that the present technique can improve substantially modal loss factors for these structures. Identification of frequency dependent properties of frequency dependent viscoelastic core materials in a composite sandwich configuration has also been presented. The obtained results indicate that the inverse problem can be solved using only information from eigenfrequencies and modal loss factors, associated to fractional derivative constitutive models. Acknowledgements The authors thank the financial support of Fundaçao para a Ciência e a Tecnologia, Portugal, through: POCTI/FEDER, and POCI(2010)/FEDER, CNPq (Brazil), CAPES (Brazil), and FAPERJ (Brazil). References [1] Nashif AD, Jones DIG, Henderson JP. Vibration damping. New York: John Wiley & Sons; 1985. [2] Sun CT, Lu YP. Vibration damping of structural elements. New Jersey: Prentice Hall PTR; 1995. [3] Moreira RAS, Rodrigues JD, Ferreira AJM. A generalized layerwise finite element for multi-layer damping treatments. Comput Mech 2006;37:426–44. [4] Moreira RAS, Rodrigues JD. A layerwise model for thin soft core sandwich plates. Comput Struct 2006;84:1256–63.
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