Application of model updating techniques in dynamics for the identification of elastic constants of composite materials

Composites: Part B 30 (1999) 79–85

Application of model updating techniques in dynamics for the identification of elastic constants of composite materials Jesiel Cunha a,*, Jean Piranda b,1 a

Department of Physical Sciences, Federal University of Uberlaˆndia, Campus Santa Moˆnica, Uberlaˆndia-MG, Brazil b Applied Mechanics Laboratory R. Chale´at, University of Franche-Comte´, Besanc¸on, France Received 30 July 1997, accepted 26 May 1998

Abstract This work consists of the identification of the stiffness properties of composite materials from dynamic tests. Unknown coefficients are identified by a technique of model updating. The formulation used (modal approach) is based on the minimization of the eigensolution residuals (sensitivity method). This technique allows the simultaneous identification of several properties from a single test. Stiffness properties of extension, bending, twisting and transverse shear have been identified. A discussion of the different identification approaches of elastic constants of composite materials from dynamic tests has been presented. Important points of the model updating in dynamics have been addressed: generalized mass errors, placement of sensors and approximate reanalysis of eigensolutions. Results obtained by numerical simulations show the efficiency of the proposed methodology. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: B. Vibration

1. Introduction The field of application of composite materials has strongly evolved in recent years. Used in the aerospace industry from early days, composite materials have reached extremely varied areas due to their versatility. Today, interesting features of composite materials such as low density associated with high strength characteristics have increased due to the development of new materials, together with new philosophies of design and fabrication. New generations of multi-functional structures such as hybrids or more recently, intelligent composites, have begun to emerge [1]. Thus, these new technologies require novel techniques of analysis of the mechanical behavior of the new materials. This work deals with the identification of the stiffness properties of composite materials from dynamic tests. The interest of the dynamic tests lies in the fact that in the vibratory behavior of a structure several energies of different natures intervene. This fact is perfectly coherent with the character of multiaxial identification inherent to composites. It is thought that this technique allows simultaneous identification of several properties from a single test. Unknown coefficients are identified by a model updating technique. The formulation used is based on the minimization of the * Corresponding author: e-mail: [email protected] 1 E-mail: [email protected]

eigensolution residuals, a technique commonly called the sensitivity method. The implementation of this procedure of parametric correction requires the analysis of different pertaining aspects. In this work, some important points of the updating procedure have been studied: the placement of sensors, the approximate reanalysis of eigensolutions and generalized mass errors. In this sense, the proposed procedure of identification from dynamic tests is more comprehensive than most works in the literature [2–6]. It can be applied to a vast range of structures of composite materials.

2. General formulation of the sensitivity method The sensitivity method consists of the minimization of a residual based on eigensolutions, which are considered as output quantities (Fig. 1) [7]. In elastodynamics, the advantages of the sensitivity method compared with other updating methods are, in a general sense: neither expansion nor condensation is required; it is exploitable when the number of sensors is reduced and is therefore well adapted to large systems; it is robust with respect to the measurement noise; it enables to ensure physical meaning to the updating. Its disadvantages are related to the difficult convergence or the possibility of reaching local minima, requiring an initial finite element model which is not too far from the real structure; the necessity of pairing modes; the utilization of

1359-8368/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S1359-836 8(98)00050-X

80

J. Cunha, J. Piranda / Composites: Part B 30 (1999) 79–85

paired eigensolutions of the model are used: …a† y…ex† n ˆ yn ⫹

l…ex† n

Fig. 1. General flowchart of the sensitivity method.

generalized masses and the numerical problems in the presence of multiples or quasi-multiples eigenvalues. Nevertheless, most of these disadvantages can be dealt using ad-hoc procedures [8]. In the updating procedure, corrections DK and DM for the stiffness and mass matrices of the model (a) related to the structure (ex) are determined: K …ex† ˆ K …a† ⫹ DK

M …ex† ˆ M …a† ⫹ DM

…1†

To do so, we suppose that the finite element model is composed of sub-domains called macro-elements, which comprise elements depending on the same parameters: Kˆ

r X

Ke…E† 僆 RN;N

Mˆ

eˆ1

r X

Me…E† 僆 RN;N

…2†

eˆ1

where r is the number of elements in the macro-element, Ke…E† and Me…E† are respectively the stiffness and mass matrices associated with the element e. The corrections are made by acting on the p stiffness macro-elements and q mass macro-elements as follows: Ka ˆ

p X

ki Ki

Ma ˆ

iˆ1

q X

mj Mj

…3†

jˆ1

where ki and mj are unknown correction coefficients; K a and M a are the assembled stiffness and mass matrices. The residual is formed from the distances between the identified eigensolutions of the structure and those calculated by the finite element model evaluated on the c instrumented coordinates: …a† c;1 Dyn ˆ y…ex† n ⫺ yn 僆 R

n ˆ 1; :::; m

…a† Dln ˆ l…ex† n ⫺ ln 僆 R

…4†

In the sensitivity method, distances are expressed as a function of the macro-elements (correction parameters). To do so, first order Taylor series expansions in the vicinity of

p q X X 2y…a† 2y…a† n n dki ⫹ dmj 2 k 2 m i j iˆ1 jˆ1

…5†

p q X X 2l…a† 2l…a† n n ˆ l…a† dki ⫹ dmj n ⫹ 2k i 2mj iˆ1 jˆ1

In matrix form we have: 2 2y 2y 1 2y 1 2y 1 1 :: :: 6 2k 1 2 k 2 m 2 mq p 1 3 6 2 6 6 . Dy1 .. .. .. 6 7 6 .. 6 . : . : . 6 .. 7 6 6 . 7 6 2y m 2y m 2y m 2y m 7 6 6 :: :: 7 6 6 6 Dym 7 6 2k1 2k p 2m1 2mq 7 6 6 6 Dl 7 ˆ 6 6 6 1 7 2l1 2l1 7 6 2l1 :: 2l1 6 :: 6 . 7 6 2k 2k p 2m1 2mq 6 . 7 6 1 6 . 7 6 5 6 4 6 .. .. .. .. 6 . . : . : . Dlm 6 6 4 2lm 2lm 2lm 2lm :: :: 2k 1 2k p 2m1 2mq 3 2 dk1 7 6 6 .. 7 6 . 7 7 6 7 6 6 dkp 7 7 6  6 dm 7 6 1 7 7 6 6 . 7 6 . 7 6 . 7 5 4 dmq 2 3 h i h …k† i S…k† S…m† y y 5 Dp…m† ) DDyl ˆ4 Dp S…k† S…m† DZ Dp l l ‰m…c⫹1†;1Š

S ‰m…c⫹1†;p⫹qŠ

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

…6†

‰p⫹q;1Š

where m is the number of identified eigensolutions; c is the number of measured DOFs. In the sensitivity matrix S, expressions of the first derivatives of eigensolutions with respect to the stiffness and mass parameters are obtained by deriving the modal equilibrium equation of the model and by taking into account the orthonormality relations [7]. 2.1. Errors of generalized masses The definition of the eigenvector residual implies a correct normalization of the identified eigenvectors, which is a condition not easily satisfied in experimental modal identification. Thus, identification errors of generalized masses can lead to incorrect identification results. To solve this problem, generalized masses can be dealt with as additional unknown values. Let us call y^…ex† n the identified eigenvector without normalization errors and y…ex† the actun …ex† ˆ y …1 ⫺ an †, ally obtained eigenvector. We write: y^…ex† n n where an is an unknown real coefficient, accounting for

J. Cunha, J. Piranda / Composites: Part B 30 (1999) 79–85

normalization errors. Then, the complete system is written as: 3 3 2 2 Dy1 y…ex† 0 1 72 a 3 7 6 6 7 6 .. 7 6 …k† 1 76 S…m† :: 6 . 7 6 Sy y 76 . 7 7 6 6 7 7 6 6 .. 7 7 …ex† 76 6 Dym 7 6 7 6 0 y 7 6 6 m 76 7 6 am 7 6 Dl 7 ˆ 6 7 …7† 7 6 7 6 1 7 76 7 6 6 …k† 6 74 Dp 7 6 . 7 6 5 7 6 . 7 6 …k† …m† 7 6 . 7 6S 0 5 Dp…m† 5 4 l Sl 4 Dlm 2.2. Reanalysis of eigensolutions In the procedure of model updating by the sensitivity method, corrections of the model are made iteratively. Therefore, this requires the repeated solution of linear system of equations and eigenvalue problems. In practice, finite element models possess a large number of DOFs. This renders the exact solutions of the eigenvalue problem a very costly procedure. In this case, methods of approximate reanalysis, which undergo a model reduction, can be used. The problem of system reduction is inserted in the general context of model condensation. Basic techniques of reduction currently employed are: those which preserve a reduced set of physical DOFs (exact dynamic condensation, Guyan static condensation, etc.) and those which introduce generalized coordinates (Ritz methods). In the Ritz method, eigensolutions of the modified model are calculated from the initial solutions and the structural modifications of the model. This technique is accurate and relatively simple to implement. In this work, the principle of the modal superposition (Rayleigh–Ritz approach) has been used [8]. The equilibrium equation of the associated modified model is given by: ‰…K ⫹ DK† ⫺ l~ n …M ⫹ DM†Šy~n ˆ 0

…8†

where DK and DM 僆 Rn;n designate the introduced modifications. In the Rayleigh–Ritz procedure the eigenvector y~n of the modified problem is expressed on the sub-basis of the modal matrix of the initial model: y~n ˆ Y1 cn

…9†

where Y1 僆 Rn;m ; L1 ˆ diag {ln }, n ˆ 1; :::; m are the known modal and spectral matrices corresponding lowfrequency and cn 僆 Rm;1 is a vector of generalized coordinates. Using the orthonormality relations, the condensed problem of order m can be expressed as: ‰…L1 ⫹T Y1 DKY1 † ⫺ l~ n …I ⫹T Y1 DMY1 †Šcn ˆ 0

…10†

This system allows to determine m eigensolutions l~ n and y~n ˆ Y1 cn . This procedure of pure modal projection is sensitive to modal truncations. The method gives good results when the perturbations DK and DM are relatively small, which will be the case when updating composite materials.

81

Accuracy of the results can be improved by taking into account the dynamic and static residuals representing the contribution of the neglected modes which correspond to high-frequency [9]. 2.3. Optimal placement of sensors in view of model updating Choosing the DOF’s sensors is an important issue in model updating procedures. Indeed, although the finite element model possesses a large number of DOFs, in practice only a limited number of coordinates can be instrumented, hence the necessity of choosing adequately the location of the sensors in the structure. The optimal selection of sensors pursues one of the following objectives: observability, expansion, condensation, structural modification localization, conditioning of the sensitivity matrix, etc. Different techniques of selection exploiting kinetic or potential energies, measurements noise or modal matrix can be considered [10]. In this work, the objective of DOF sensor selection is to obtain the most orthogonal basis as possible, which will allow matching between calculated and identified eigenvectors. In view of the updating procedure, the goal can be considered as a preparation technique for the experimental test. The method consists of exploiting the modal matrix with a minimization of the condition number. We propose constructing a submatrix Y …a† 僆 Rc;n , c ⬎ n line by line. The proposed procedure is defined by the following steps: for a number n of given vectors of basis, the displacement vectors of each DOF i on n modes are formed as: h i …a† …a† …a† 1;n …11† y…a† i ˆ yi1 ; yi2 ; :::; yin 僆 R where y…a† ij is the displacement of the DOF i for the mode j. The first retained DOF (y…a† 1 ) has to maximize the norm of …a† 2;n with all j (j 苷 y…a† i . Then we construct all matrices Y2j 僆 R 1) retained DOFs: h i …a† T …12† Y2j…a† ˆ y…a† 1 yj The rank and the condition number of all matrices Y2j…a† are determined. The second selected DOF maximizes the rank and minimizes the condition number of Y2j…a† . Finally, all …a† 僆 Rp;n , with K 苷 1, K 苷 2,... are constructed: matrices YpK h i …a† …a† …a† T ˆ y…a† …13† YpK 1 y2 ::: yK …a† are evalThe conditioning and the rank of the matrices YpK uated. The p DOFs which maximize the rank and minimize …a† are selected. In the condition the conditioning of YpK number minimization method, the selected DOFs depend on the choice of the first DOF or the imposed DOFs that can generate several families of solutions. Thus, the selection technique remains sub-optimal. This aspect reveals the essentially numerical character of the method. There are no explicit physical considerations which allow, for example, a representation of kinetic or potential energies.

82

J. Cunha, J. Piranda / Composites: Part B 30 (1999) 79–85

2.4. Optimization method The optimization method choosen to calculate the solution Dp in the sensitivity Eq. (6) is of the gradient type with inequality constraints. The principle of the method, called Modified Method of Feasible Directions [11], consists of determining the solution that minimizes the cost function J(p), with the advantage of specifying the search direction and the amount of change in the parameter space. The cost function is formed by the distances between the eigensolutions and the distances of the correction parameters: J…p† ˆT Dy…p†Wy Dy…p† ⫹T Dl…p†Wl Dl…p† ⫹T Dp…p†Wp Dp…p†

equation of the laminate in contracted form is: 2 3 2 32 3 N A B 0 em 6 7 6 76 7 6 M 7 ˆ 6 B D 0 76 k 7 4 5 4 54 5 Q

0

H

gc

where N represents normal forces; M, bending and twisting moments; Q, transverse shear forces; em , normal strains; k , curvatures; g c, shear strains; ‰AŠ 僆 R3;3 , extensional stiffness matrix; ‰BŠ 僆 R3;3 , extension–bending/twisting coupling stiffness matrix; ‰DŠ 僆 R3;3 , bending/twisting stiffness matrix; ‰HŠ 僆 R2;2 , transverse shear stiffness matrix. These matrices can be expressed as:

…14† sup and subject to inequality constraints: Dpinf i ⱕ Dpi ⱕ Dpi inf sup …ex† …a† mc;1 is the pi ⱕ pi ⱕ pi , where Dy ˆ y ⫺ y 僆 R eigenvector distances vector; Dl ˆ l…ex† ⫺ l…a† 僆 Rm;1 is the eigenvalue distances vector; Dp ˆ p ⫺ p…0† 僆 Rp⫹q;1 is the parametric correction vector; and Wy 僆 Rmc;mc , Wl 僆 Rm;m and Wp 僆 Rp⫹q;p⫹q are the weighting matrices chosen according to the specificity of the problem.

0

…15†

Aij ˆ

n X

…Q ij †k …hk ⫺ hk⫺1 † Bij ˆ

kˆ1

Dij ˆ

1 3

n X

1 2

n X

…Q ij †k …h2k ⫺ h2k⫺1 †

kˆ1

…Q ij †k …h3k ⫺ h3k⫺1 †

kˆ1

Hij ˆ kij

n X

…Q ij †k …hk ⫺ hk⫺1 †

kˆ1

…16† 2.5. Dynamic and mathematical aspects of the problem The implementation of a parametric correction procedure requires the analysis of different related physical and mathematical aspects. These points of theoretical and practical order are very important to identify the elastic parameters of composite materials. Some aspects have been approached by Cunha [8]: initial parameter estimation by genetic algorithms approach; regularization techniques of the optimization problem; model updating procedure with second order approximation. The solutions proposed rest on a methodology allowing applications in a general context, even if the appropriate adaptations to each structure have to be considered.

3. Adaptation of the sensitivity method to composite materials Stiffness properties to be identified constitute the behavior law of the structure which expresses the resultants and moments as functions of extension and shear deformations and curvatures. In the context of the Laminated Plate and Shell Theories, given the variety and the complexity of the parameters influencing the mechanical behavior of the laminate, choice of the displacement field, constraints and associated boundary conditions lead to the use of various approaches [12]. In the present work, the First-Order Shear Deformation Theory has been used [13,14]. This theory ensures good results for most laminated composites including the transverse shear effects. The constitutive

where Q ij are the stiffness constants of the lamina in the laminate reference; hk is the normal coordinate of the layer k; n is the number of layers and kij are the correction factors of the transverse shear. The identification of stiffness properties of composite materials by typical model updating methods in dynamics is relatively new. The anisotropic character, the diversity of the materials and the variety of the architectures render the mechanical behavior of the composite materials very particular. For example, the coupling and delamination effects are typical phenomena of the laminates. Thus, the general formulations which would allow approaching all cases concerning the identification do not exist in the literature. The sensitivity method with Bayesian estimation is the most used. Some authors model the structure as global homogeneous identifying the laminate engineering constants (Ex, Ey, Gxy, n xy, etc.) [2,3,5]. This is, in fact, a sort of simplification of the problem and does not allow access to the lamina engineering constants. Another possibility, particularly in the case of identical layers, is to identify directly the lamina engineering constants. Nevertheless, these constants would be strongly nonlinear with respect to the eigenvectors and eigenvalues. A very interesting approach to the problem consists of the utilization of stiffness constants of the constitutive equation which expresses the resulting forces and moments as a function of extension and shear deformations and curvatures [4,6]. Indeed, the utilization of matrices [A], [B], [D] and [H] presents some advantages: the ease to understand the mechanical behavior of the structure (these ‘global properties’ express the extension, bending, twisting, shear and coupling effects); the identification of [A], [B], [D] and [H] that allows the further calculation of the engineering

J. Cunha, J. Piranda / Composites: Part B 30 (1999) 79–85

Fig. 2. Simulated plate.

83

constants in case of identical layers; the possibility of interfacing of the sensitivity method with the finite element codes, since stiffness constants are linear with respect to the elementary matrices; finally, the current finite element codes allow to use the constants [A], [B], [D] and [H] directly as input data avoiding the identification of lamina engineering constants. In the sensitivity method, corrections are made on the stiffness macro-elements, corresponding to the components of matrices [A], [B], [D] and [H]. Thus, each macro-element possesses a well defined role on the mechanical behavior of the structure. To illustrate this idea, for a symmetric laminated plate, corrections on the p macro-elements are

Fig. 3. Evolution of the updating parameters versus iterations.

84

J. Cunha, J. Piranda / Composites: Part B 30 (1999) 79–85

Table 1 Final stiffness constants values Stiffness constants

Initial model

Perturbed model

Updated model

D11 D12 D22 D16 D26 D66 H44 H55

449 324 449 59 59 355 2.06 × 10 7 2.06 × 10 7

539 (20%) 389 (20%) 374 (17%) 49 (17%) 71 (20%) 426 (20%) 1.72 × 10 7 (17%) 2.47 × 10 7 (20%)

449 (0%) 324 (0%) 449 (0%) 59 (0%) 59 (0%) 355 (0%) 2.06 × 10 7 (0%) 2.06 × 10 7 (0%)

(N.m) (N.m) (N.m) (N.m) (N.m) (N.m) (N/m) (N/m)

such as: Ka ˆ

p X

ki Ki ˆ k1 D11 K1 ⫹ k2 D12 K2 ⫹ k3 D22 K3 ⫹ k4 D16 K4

iˆ1

⫹ k5 D26 K5 ⫹ k6 D66 K6 ⫹ k7 A11 K7

Stiffness constants of the initial model have been disturbed by ^ 20%. Excellent results of the parametric correction have been noticed. Fig. 3 shows the evolution of the updating parameters versus iterations: evolution of the correction coefficients (ki) of the stiffness constants; evolution of the objective function (J(p)), formed by the eigensolution residuals; evolution of the stiffness constants sensitivities (Sj); evolution of the frequential errors (Dfn =fn ) and vectorial errors (MAC, jjDyn jj=jjyn jj). The graphs show a very good parameter convergence, with a significant reduction of the distances between eigensolutions (frequential and vectorial errors). The sensitivity of the bending/twisting properties is greater than that of transverse shear properties. As expected, a best convergence of these most sensitive constants is observed. Table 1 shows that the initial stiffness constants have been found (the number between parentheses represents the distances with respect to the initial model).

⫹ k8 A12 K8 ⫹ k9 A22 K9 ⫹ k10 A16 K10 ⫹ k11 A26 K11 ⫹ k12 A66 K12 ⫹ k13 H44 K13 ⫹ k14 H55 K14 …17† where D11 K1 ; D12 K2 ; :::; H55 K14 represents the stiffness matrices of the macro-elements; ki is the correction coefficient of the macro-element i; K a is the assembled stiffness matrix; D11 is bending about x; D12 is bending about x/bending about y coupling; D22 is bending about y; D16 is bending about x/twisting coupling; D26 is bending about y/twisting coupling; D66 is twisting; A11 is extension x; A12 is extension x/extension y coupling; A22 is extension y; A16 is extension x/ shear xy coupling; A26 is extension y/shear xy coupling; A66 is shear xy; H44 is transverse shear yz; H55 is transverse shear xz.

4. Numerical application The principle of the numerical simulation consists of artificially perturbing the model and then trying to find these perturbations by means of the model updating procedure. Optimal placement of sensors and the approximate reanalysis of eigensolutions have been used (Sections 2.2 and 2.3). Concerning the placement of the sensors on the structure, several solutions have been obtained. Fig. 2 shows a possible configuration. The studied test is a free-free angle-ply symmetric laminate. The material is a graphite/ epoxy-(Hercules AS1/3501-5) prepreg, which is composed of 16 orthotropic layers at ^ 45⬚. Characteristics of the structure and ‘test conditions’ are the following (Fig. 2): free-free boundary conditions; number of sensors (accelerometers) ˆ 17 (in z direction); number of measured modes ˆ 10 (obtained by ‘experimental’ finite element model).

5. Remarks concerning the sensitivity of elastic constants Results obtained by numerical simulations in other composite structures (anti-symmetric, angle-ply and crossply laminates, sandwiches, etc.), and application to the real structures have demonstrated the accuracy and the robustness of the sensitivity method [8]. Favorable conditions for the sensitivity and consequently for the identification of elastic constants are as follows. • H44 and H55 transverse shear: influence of the transverse shear deformations increases as the length/thickness ratio decreases; influence of the transverse shear deformations increases as G13 and G23 coefficients decrease, that is, when the Young’s modulus/transverse shear modulus ratio becomes great; clamped boundary conditions emphasize the transverse shear effects; influence of the transverse shear deformations increases for high modes. • D12 bending about x/bending about y coupling: free-free condition favors D12 sensitivity. • D16 and D26 bending/twisting coupling: the most important aspect concerning the influence of D16 and D26 on the mechanical behavior of the laminate is the number of layers and their orientation. The greater the number of layers, the smaller the effects of D16 and D26. Indeed, when the number of layers is too large, the laminate becomes globally an orthotropic homogeneous material and coupling effects disappear. The constants mentioned above are generally less sensitive and therefore their identification is more difficult. For the other elastic constants (D11, D22 and D66), particular conditions which guarantee their sensitivity are not necessary. In most cases, these constants have a significant influence on the dynamic behavior and can therefore be correctly identified.

J. Cunha, J. Piranda / Composites: Part B 30 (1999) 79–85

6. Conclusion The evolution of the composite material applications is largely dependent, among other factors, on the accurate knowledge of the phenomena governing their mechanical behavior. In this sense, the diversity of materials and the complexity of the structural configurations make the elaboration of the general behavior models not always easy. Similar reasoning for the identification methods of the mechanical properties can be made. In this context, the procedure of identification from dynamic tests is very interesting, because it can be applied to a vast range of structures. This procedure of a non-destructive type has a relatively simple implementation. The approach used is well adapted to the anisotropic character of the composite materials. Compared with other identification techniques, the method has the advantage of considerably simplifying the tests since several different properties are identified simultaneously. Another advantage lies in the fact that the procedure of identification can be applied directly to the structure (plates, shells, tubes, or more complex geometrical shapes). In experimental tests, adapting specimens according to the parameter to be identified is not necessary. Thus, stiffness constants which are generally accurately obtained can be further directly introduced as input data in finite element codes. Important aspects in the model updating procedure are: sensitivity and linear independence of the constants to be identified, sufficient modal basis and number of sensors. The technique presents several sources of possible errors, which play a fundamental role in the results. The modeling errors in finite element mesh, simplified calculation hypotheses, etc. The model updating procedure (sensitivity method) errors include poor representation in the objective function (linearization of the problem, reduced measured modal basis, parameter weighting, etc.), non-uniqueness and instability of the solutions, and approximate reanalysis of eigensolutions. The specimen topology errors are heterogeneity of the physical properties, geometrical imperfections (thickness, fiber orientations, etc.), and defects in general.

85

The measurement errors include random and systematic errors in the dynamic test (placement of sensors, noise, etc.). The quality of the results depends directly on the quality of the finite element model, the model updating strategy and the precision of the experimental data. It is the association of these factors that can generate a reliable solution in the identification procedure.

References [1] Chou T-W. Microstructural design of fiber composites. Cambridge University Press, 1992. [2] Deobald LR, Gibson RF. Determination of elastic constants of orthotropic plates by a modal analysis/Rayleigh–Ritz technique. Proceedings of the 4th IMAC, 1986:682–690. [3] Frederiksen PS. Estimation of elastic moduli in thick composite plates by inversion of vibrational data. Proceedings of the 2nd International Symposium on Inverse Problems, Paris, 1994:111–118. [4] Link M, Zhiqing Z. Vibration test of CFRP sandwich plate and material parameter updating. Rapport Universite´ de Kassel, 1993. [5] Pedersen P. Laminates: analysis, sensitivity, optimal design, identification of material parameters. Notes for Lectures, 1988. [6] Sol H. Identification of anisotropic plate rigidities using free vibration data. PhD Thesis, Vrije Universiteit Brussel (V.U.B.), Belgium, 1986. [7] Piranda J. Analyse modale et recalage de mode`le. HDR Project, Universite´ de Franche-Comte´, Besanc¸on, 1994. [8] Cunha J. Application des techniques de recalage en dynamique a` l’identification des constantes e´lastiques des mate´riaux composites. PhD Thesis, Universite´ de Franche-Comte´, Besanc¸on, 1997. [9] Ennaime S-E. Contribution a` l’identification de structures me´caniques: localisation des de´fauts dominants et re´analyse, estimation des forces exte´rieures. PhD Thesis, Universite´ de Franche-Comte´, Besanc¸on, 1996. [10] Ramanitranja AN. Identification parame´trique: adaptation entre la structure et son mode`le par expansion. PhD Thesis, Universite´ de Franche-Comte´, Besanc¸on, 1996. [11] VMA Engineering. DOT users manual. Version 2.04, 1990. [12] Bert CW. A critical evaluation of new plate theories applied to laminated composites. Composite Structures 1984;2:329–347. [13] Pagano NJ. Exact solutions for composite laminates in cylindrical bending. Jounal of Composite Materials 1969;3:398–411. [14] Reddy JN. Free vibration of antisymmetric, angle-ply laminated plates including transverse shear deformation by the finite element method. Journal of Sound and Vibration 1979;66:565–576.