Identification of mechanical properties of elastically restrained laminated composite plates using vibration data

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JOURNAL OF

SOUND AND

VIBRATION

Journal of Sound and Vibration 295 (2006) 999–1016

Identiﬁcation of mechanical properties of elastically restrained

laminated composite plates using vibration data

C.R. Lee, T.Y. Kam

Mechanical Engineering Department, National Chiao Tung University, Hsin Chu 300, Taiwan, Republic of China Received 18 March 2005; received in revised form 8 December 2005; accepted 31 January 2006

Available online 17 April 2006

Abstract

A nondestructive evaluation method using vibration data to determine mechanical properties (material and spring constants) of elastically restrained laminated composite plates is presented. The Rayleigh–Ritz method in which a set of Legendre’s polynomials is adopted to approximate the plate deflection is used to determine the theoretical natural frequencies of the elastically restrained laminated composite plates. A number of natural frequencies extracted from the impulse vibration test data of the laminated composite plates supported by elastic restraints at both the edges and centers of the plates are used in the present method to determine the mechanical properties of the plates. The sum of the squared differences function which measures the differences between the experimentally and theoretically predicted natural frequencies of the elastically restrained laminated composite plates is established. The identification of the plate mechanical properties is then formulated as a constrained minimization problem in which the mechanical properties are determined by making the sum of the squared differences function a global minimum. The feasibility and accuracy of the proposed method are studied by means of several numerical examples on the mechanical properties identification of elastically restrained laminated composite plates with different layups made of various composite materials. Experimental investigation of the mechanical properties identification of several elastically restrained laminated composite plates has been performed to illustrate the applications of the present method. It has been shown that the present method can produce good estimates of the mechanical properties of the elastically restrained laminated composite plates in an efficient and effective way.

1. Introduction

Owing to their many advantageous properties, the ﬁber-reinforced composite plates have been increasingly used in the aeronautical and aerospace industry as well as many other ﬁelds of modern technology. The attainment of the actual behavioral predictions of such structures usually depends on the correctness of the system parameters of the structures such as the elastic constants of the materials constituting the structures and the stiffnesses of the supports restraining the structures. As is well known, composite structures fabricated by different methods or curing processes may possess different mechanical properties and the material www.elsevier.com/locate/jsvi

Corresponding author. Tel.: +886 3 5712121x55124; fax: +886 3 5753735. E-mail addresses:[email protected], [email protected] (T.Y. Kam).

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constants of the structures in service will change due to structural and material degradations. Therefore, the material properties determined from standard specimens tested in laboratory in general may deviate from those of the laminated composite components manufactured in factory or the existing composite structures. In recent years, the determination of realistic material constants of structural components/structures has become an important topic of research and different techniques for elastic constants identification of beam and plate types of structures have been proposed. For instance, Castagne`de et al.[1]determined the elastic constants of thick composite plates via a quantitative ultrasonic approach. Fallstrom and Jonsson [2] determined the material constants of anisotropic plates using the frequencies and mode shapes measured by a real-time TV-holography system. Nielsenand and Toftegaard[3]used the ultrasonic measurement approach to obtain the elastic constants of fiber-reinforced polymer composites under the influence of absorbed moisture. Berman and Nagy [4] used measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure. Kam and his associates[5–10]developed methods to identify the element bending stiffnesses of beam structures using measured natural frequencies and mode shapes and determine elastic constants of shear deformable laminated composite plates using measured strains and/or displacements obtained from static testing of the plates. Recently, a number of researchers have used experimental natural frequencies to identify the elastic constants of laminated composite plates with free boundary conditions [11–18]. For instance, Moussu and Nivoit [14] used the method of superposition to determine the elastic constants of free rectangular plates from the measured experimental natural frequencies of the plates. Wilde and Sol[15]used the method of Bayesian estimation to study the identification of elastic constants from the experimental natural frequencies of free rectangular composite plates. Araujo et al. [16,17] used an optimization method to determine the elastic constants of free composite plates using the measured natural frequencies of the plates. In general, the previously proposed methods were only applicable for plates with simple boundary conditions and might require the use of 12–16 natural frequencies in the elastic constants identification process if obtaining results with satisfactory accuracy was desired. In view of the fact that the dynamical behaviors of plates with elastic restraints are very different from those with simple boundary conditions, when using vibration data to identify the mechanical properties of a flexibly supported plate, it is expected that the elastic restraints of the plate will play an important role in the identification. Therefore, if realistic mechanical properties of the plate are to be determined nondestructively, the effects of the elastic restraints on the identified properties must be taken into consideration. Although the system identification of plates with flexible supports is an important topic of research, so far not much work has been devoted to this area.

In this paper, a nondestructive evaluation method is presented for the identification of mechanical properties of laminated composite plates elastically restrained both at the edges and in the interior of the plates. The Rayleigh–Ritz method together with an appropriate set of characteristic functions is used to predict the natural frequencies of the flexibly supported laminated composite plates. Vibration tests of the flexibly supported laminated composite plates are performed to extract the natural frequencies of the plates from the measured vibration data. The sum of the squared differences function which measures the differences between the experimental and theoretical predictions of natural frequencies of the laminated composite plates is established. The identification of mechanical properties is then formulated as a constrained minimization problem in which the mechanical properties are determined by making the sum of the squared differences function a global minimum. A multi-start global minimization method is used to search for the global minimum and a normalization technique for normalizing the design variables is adopted to increase the convergence rate of the solution. A number of examples of the mechanical properties identification of elastically restrained laminated composite plates with different layups made of different composite materials are given to illustrate the accuracy and feasibility of the proposed method. Several flexibly supported laminated composite plates are subjected to impulse vibration testing. The measured natural frequencies of the composite plates are used in the present method to identify the mechanical properties of the plates.

2. Plate vibration analysis

Without loss of generality, consider the elastically restrained rectangular symmetrically laminated composite plate of area a0b0and constant thickness h composed of a ﬁnite number of orthotropic layers

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of same material properties and thickness inFig. 1. The x and y coordinates of the plate are taken in the mid-plane of the plate. The plate is supported continuously around the edges by ﬂexible strip-type pads of cross-sectional dimensions beheand at the center by an annulus-type ﬂexible restraint with inner radius ri. For the

ﬂexible supports considered in this study, it is further assumed that the dimensions, beand ri, of the elastic

supports are much smaller than the plate dimensions, a0and b0, so that the edge ﬂexible supports of the plate

can be modeled by longitudinal and torsional springs while the center support by a longitudinal spring as shown inFig. 2. It is noted that the plate size used in the vibration analysis is a b in which a ¼ a0beand

b ¼ b0be. For free vibration, the plate vertical displacement w(x, y, t) is assumed to be of the form

wðx; y; tÞ ¼ W ðx; yÞ sin ot, (1)

where W(x, y) is the deﬂection function and o is the angular frequency. According to the classical lamination theory with the neglect of the rotary inertia effect, the maximum strain energy UP and maximum kinetic

energy T of the plate are expressed as [19]

UP¼ 1 2 Z b 0 Z a 0 D11 q2W qx2 2 þ2D12 q2W qx2 q2W qy2 þD22 q2W qy2 2 þ4D66 q2W qxqy 2 " þ4D16 q2W qx2 _q2_W qxqy þ4D26 q2W qy2 _q2_W qxqy dx dy ð2Þ and T ¼1 2rho 2Z b 0 Z a 0 W2dx dy, (3) Center support a 0 b 0 Elastic pad h be he 2r i 2r i x y n

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where Dij are bending stiffness coefﬁcients and r is material density. The bending stiffness coefﬁcients are given by Dij¼ Z h=2 h=2 ¯ QðmÞ_ij z2dz ði; j ¼ 1; 2; 6Þ. (4)

The transformed lamina stiffness coefficients ¯QðmÞ_ij depend on the material properties and fiber orientation of the mth layer. For a layer with zero fiber angle, the lamina stiffness coefficients are expressed as

Q₁₁¼ E1 1 n12n21 ; Q₁₂ ¼ n12E2 1 n12n21 ¼Q₂₁; Q₂₂¼ E2 1 n12n21 , Q₆₆¼G12 with n12 E1 ¼n21 E2 , ð5Þ

where E1, E2 are Young’s moduli in the ﬁber and transverse directions, respectively; nij is the Poisson’s

ratio for transverse strain in the j-direction when stressed in the i-direction; G12 is shear modulus in the

1–2 plane. x h k L2 k L1 k L3 k L4 k c k R4 k R3 k R2 k R1 y a b k C n

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For the plate with spring-type elastic supports, additional strain energy stored in the supporting springs exists. The maximum strain energy of the ﬂexible restraints is

UB¼ kL1 2 Z b 0 W2dy x¼0 þkL2 2 Z b 0 W2dy x¼a þkL3 2 Z a 0 W2dy y¼0 þkL4 2 Z a 0 W2dy y¼b þkR1 2 Z b 0 qW qx 2 dy " # x¼0 þkR2 2 Z b 0 qW qx 2 dy " # x¼a þkR3 2 Z a 0 qW qy 2 dy " # y¼0 þkR4 2 Z a 0 qW qy 2 dy " # y¼b þkC 2 ½W 2 x¼a=2;y¼b=2, ð6Þ

where kLi and kRi (i ¼ 1;. . . ; 4) are spring constants per unit length of the edge longitudinal and torsional

springs, respectively; kC is spring constant of the center spring. The integrals in the brackets of the above

equation are evaluated at the four edges of the plate. Herein, the equivalent translational and rotational spring constants of the ﬂexible strip-type pad with cross-sectional area of beheand Young’s modulus Eeused as an

edge support as shown in Fig. 3 are to be approximated via the mechanics of materials approach. In the determination of the translational spring constant, it is assumed that the load is distributed uniformly on the top surface of the edge support where the top surface after deformation remains plane and horizontal as shown inFig. 3a. Hence, when treating the ﬂexible pad of unit length as an axial member, the translational spring constant per unit length is obtained as

kL¼

Eebe

he

. (7)

To determine the rotational spring constant, it is assumed that the moment-induced load is distributed linearly across the width of the support where the top surface of the pad remains plane after rotation as shown in Fig. 3b. Hence, when treating the top surface of the ﬂexible pad of unit length as a beam section, the rotational spring constant per unit length is obtained as

kR¼

Eeb3e

12he

. (8)

In view of Eqs. (2) and (6), the total strain energy, U, is then written as

U ¼ UPþUB. (9)

Based on the Rayleigh–Ritz method, the deﬂection function expressed in the nondimensional form is W ðx; ZÞ ¼X I i¼1 XJ j¼1 CijfiðxÞjjðZÞ, (10)

where Cijare undetermined displacement coefﬁcients, fi(x) and fj(Z) are the characteristic functions. In this

study, the Legendre’s orthogonal polynomials with x ¼ ð2x=aÞ 1 for 1pxp1 and Z ¼ ð2y=bÞ 1 for 1pZp1 are chosen to formulate the characteristic functions. In terms of the Legendre’s orthogonal polynomials, for instance, fi(x) can be written as

f₁ðxÞ ¼ 1, f₂ðxÞ ¼ x and if nX3,

fnðxÞ ¼ ½ð2n 3Þx fn1ðxÞ ðn 2Þ fn2ðxÞ=ðn 1Þ. (11)

It is noted that the above characteristic functions fi(x) satisfy the orthogonality condition

Z 1 1 f_nðxÞf_mðxÞ dx ¼ 0 if nam; 2 ð2n1Þ if n ¼ m: ( (12)

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Extremization of the functional P which is deﬁned as P ¼ U T with respect to the displacement coefﬁcients Cij leads to the following eigenvalue problem:

ð½K l2½MÞfCg ¼ 0 (13) with K ¼ KP+KB where l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rho2_a4_=D 0 p

, the nondimensionalized natural frequencies; {C} is the displacement coefﬁcient vector; D0¼E1h3=½12ð1 n12n21Þ; K is the structural stiffness matrix of the ﬂexibly

supported plate; KPand KBare portions of the structural stiffness matrix contributed by the stiffnesses of the

(a) Distributed Load he be (b) (c) be/2 he be

Plate Uniform Load

he

be

Elastic pad

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laminated plate and edge restraints, respectively. The elements of KP, KB, and M are obtained, respectively, as

½KPmnij ¼

16 D0

fD11E22miF00nj þa2D12ðE02miFnj20þE20miF02njÞ þa4D22E00miF22nj

þ2aD16ðE21miF01nj þE12miF10njÞ þ2a3D26ðEmi01F21nj þE10miF12njÞ þ4a2D66E11miF11njg, ð14Þ

½KBmnij¼2 fK1F00njfmð1Þfið1Þ þ K2F00njfmð1Þfið1Þ þ a4ðK3E00mijnð1Þjjð1Þ þK4E00mijnð1Þjjð1ÞÞ þ 4 ½R1F00njf 0 mð1Þf 0 ið1Þ þ R2F00njf 0 mð1Þf 0 ið1Þ þa4ðR3E00mij 0 nð1Þj 0 jð1Þ þ R4E00mij 0 nð1Þj 0 jð1ÞÞg þ 4a3Kfmð0Þfið0Þjnð0Þjjð0Þ ð15Þ and ½M_mnij ¼E00_miF00_nj; m; i ¼ 1; 2; 3; . . . M; I ; n; j ¼ 1; 2; 3; . . . N; J; a ¼ a=b (16) with Ers_mi¼ Z 1 1 drf_mðxÞ dxr dsf_iðxÞ dxs dx; Frs_nj¼ Z 1 1 drj_nðZÞ dZr dsj_jðZÞ dZs dZ; r; s ¼ 0; 1; 2, (17) ðK1; K2; K3; K4Þ ¼ ðkL1a3=D0; kL2a3=D0; kL3b3=D0; kL4b3=D0Þ, (18) ðR1; R2; R3; R4Þ ¼ ðkR1a=D0; kL2a=D0; kL3b=D0; kL4b=D0Þ (19) and K ¼ kCb2=D0. (20)

The solution of Eq. (13) gives the theoretical natural frequencies of the ﬂexibly supported laminated composite plate. The theoretically predicted natural frequencies may deviate from the actual natural frequencies of the ﬂexibly supported laminated composite plate if incorrect mechanical properties such as E1, E2, G12, n12, Eeand

kCare used in the frequency analysis of the plate. In the following section, a method is presented to identify the

mechanical properties of ﬂexibly supported laminated composite plates by minimizing the differences between the theoretical and experimental predictions of natural frequencies of the plates.

3. The inverse problem

The problem of mechanical properties identiﬁcation of elastically restrained laminated composite plates is formulated as a minimization problem. In mathematical form it is stated as

Minimize eðxÞ ¼ ðx_Þt_ðx_Þ;

Subject to xL

ipxipxUi ; i ¼ 1 N;

(21) where x ¼ ½E1; E2; G12; n12; Ee; kC the vector containing the design variables used to denote the mechanical

properties of the elastically restrained laminated composite plates; x* is an N 1 vector containing the differences between the measured and predicted values of the natural frequencies; e(x) is the sum of the squared differences function measuring the differences between the predicted and measured data; xL

i, xUi are

the lower and upper bounds of the design variables. The elements in x* are expressed as o_i ¼opiomi

omi

; i ¼ 12N, (22)

where opi, omiare predicted and measured values of the natural frequencies, respectively. The above problem

of Eq. (21) is then converted into an unconstrained minimization problem by creating the following general augmented Lagrangian[20]: ¯ Wð ~x; l; g; rpÞ ¼eð ~xÞ þ X6 j¼1 ½m_jzjþrpz2j þZjfjþrpf2j (23)

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with zj¼max gjðx~jÞ; m_j 2rp ; g_jðx~jÞ ¼x~jx~Uj p0, f_j¼max Hjðx~jÞ; Z_j 2rp ; Hjðx~jÞ ¼x~Lj x~jp0; j ¼ 1 6, ð24Þ

where mj, Zj, gp are multipliers; max [*,*] takes on the maximum value of the numbers in the bracket. The

modiﬁed design variables ~x are deﬁned as ~x ¼ E1 a1 ;E2 a2 ;G12 a3 ; n12; Ee a4 ;kC a5 , (25)

where aiare normalization factors. It is noted that the values of aican affect the search direction and properly

selected values of aican help expedite the convergence of the solution. In general, the values of ~xiði ¼ 1; . . . ; 4Þ

are best chosen to be greater than 0 and less than 10. The modiﬁed design variables ~x are only used in the minimization algorithm while the original variables x are used in the Rayleigh–Ritz method to determine the natural frequencies of the plate. The updated formulas for the multipliers mj, Zj, and gpare

mnþ1_j ¼mn_j þ2rn_pzn_j; Znþ1_j ¼Zn_j þ2rn_pfn_j; j ¼ 126, rnþ1_p ¼ g₀rn p if rnþ1p ormaxp ; rmax_p if rnþ1_p Xrmax_p ; 8 < : ð26Þ

where the superscript n denotes iteration number; g0 is a constant; rmax_p is the maximum value of rp. The

parameters m0

j, Z0j, r0p, g0, rmaxp chosen based on experience are

m0_j ¼1:0; Z0_j ¼1:0; r_p0¼0:4; g0 ¼2:5; rmaxp ¼100: (27)

The constrained minimization problem of Eq. (23) has thus become the solution of the following unconstrained optimization problem:

Minimize ¯Wð ~x; l; g; rpÞ. (28)

The above unconstrained optimization problem is to be solved using a multi-start global optimization algorithm. In the adopted optimization algorithm, the objective function is treated as the potential energy of a traveling particle and the search trajectories for locating the global minimum are derived from the equation of motion of the particle in a conservative force ﬁeld[21,22]. The design variables, i.e., the plate elastic constants, Young’s modulus of the edge elastic restraints, and spring constant of the interior support, that make the potential energy of the particle, i.e., objective function, the global minimum constitute the solution of the problem. In the minimization process, a series of starting points for the design variables of Eq. (25) are selected at random from the region of interest. The lowest local minimum along the search trajectory initiated from each starting point is determined and recorded. A Bayesian argument is then used to establish the probability of the current overall minimum value of the objective function being the global minimum, given the number of starts and the number of times this value has been achieved. The multi-start optimization procedure is terminated when a target probability, typically 0.99, has been exceeded.

4. Experimental investigation

A number of elastically restrained square laminated composite plates with layups [01]8, [01/901]2S, and [451/

451/451]Swere fabricated for experimental investigation. The laminated composite plates were supported by

strip-type elastic pads with cross-sectional dimensions of be¼5:0 mm and he¼2:1 mm around the peripheries

of the plates with or without an annulus-type flexible support at the plate centers. The materials used to fabricate the laminated composite plates were T300/2500 graphite/epoxy prepreg tapes produced by Torayca Co., Japan. The elastic constants of the cured graphite/epoxy laminates were first determined experimentally using the standard testing procedure in accordance with the relevant ASTM specifications[23]. The means and

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coefﬁcients of variation (c.o.v.) of the elastic constants determined using three standard specimens for each test are as follows:

E1¼146:503 GPað0:72%Þ; E2¼9:223 GPað1:19%Þ; G12 ¼6:836 GPað3:16%Þ,

n12¼0:306ð0:19%Þ. ð29Þ

The values in the parentheses in the above equation denote the c.o.v.’s of the elastic constants of the composite material. The average layer thickness and mass density of the laminated composite plates were 0.125 mm and 1543 kg/m3, respectively. The elastic constant Eeof the edge supporting pads was also determined following

the standard ASTM tensile testing procedure. The mean and c.o.v. of Ee are 2.028 MPa and 2.3%,

respectively. The center annulus support, which was made of corrugate fabric as shown in Fig. 4, was connected to the plate via a hollow cylindrical tube with negligible mass. The inner and outer radii of the center annulus support were ri¼12:5 mm and r0¼16 mm, respectively. The translational spring constant of

the center annulus support determined from static testing was kC¼3:865 kN=m.

The elastically restrained laminated composite plates were subjected to impulse vibration testing using the experimental setup shown inFig. 5. In the vibration testing, a hand-held impulse hammer (Kistler 9722A500, Kistler Instrument, USA) was used to excite the composite plate at different points on the plate, a force transducer (Kistler 9904A, Kistler Instrument, USA) attached to the hammer’s head to measure the input forces, an accelerometer (AP19, APTechnology, Netherland) of mass 0.14 g, which is about 0.2% of the plate weight, located at different points on the plate to pick up the vibration response data, and a data acquisition and analysis system (B&K 3560C and B&K Pulse Labshop Version 6.1) to process the vibration data from which the natural frequencies of the composite plates were extracted. A series of tests had shown that the light accelerometer weight had negligible effects on the measured natural frequencies. Each flexibly supported composite plate was then tested for 15 times and each test produced a set of vibration data for constructing the frequency response spectrum of the plate. In general, the modal damping ratios of the plates were small, less than 2%, for the first seven modes of the plates. Therefore, without loss of generality, it is assumed that the effects of damping on the natural frequencies of the plate were negligible and not taken into consideration when extracting the natural frequencies from the frequency response spectrum of the plate. Herein, the first seven natural frequencies were extracted directly from the corresponding peaks in the frequency response spectra of the plates. For illustration purpose, Fig. 6 shows a typical frequency response spectrum of the

Plate

Hollow tube

Corrugate fabric annulus r_i

r_o

Movement direction Fig. 4. Schematic description of center support.

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Hammer (Kistler 9722A500)

Accelerometer Composite Laminated Plate

Elastic Restraint

(B&K 3560C) (AP19) Signal Analyzer B&K PULSE LabShop Version 6.1

Force transducer (Kistler 9904A)

Fig. 5. Experimental setup for impulse vibration testing.

40m

Autospectrum (signal 3) - Input Working : Input : Input : FFT Analyzer

36m 32m 28m 24m 20m 16m 12m 8m 4m 0 100 200 300 400 [Hz] [g] 500 600 700 800 158 270 309 458 534 575 677

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[451/451/451]Splate restrained peripherally and centrally. It is noted that the ﬁrst seven natural frequencies of

the [451/451/451]Splate can be easily identiﬁed from the peaks of the frequency response spectrum as shown

in the figure. The means and c.o.v.’s of the first seven measured natural frequencies of the peripherally restrained composite plates with or without center elastic supports determined from the impulse vibration testing of the plates are listed inTable 1. It is noted that the c.o.v.’s of the measured natural frequencies are less than or equal to 0.96%. In the elastic constants identification of the plates as will be described in the following section, the means of measured natural frequencies will be treated as the measured natural frequencies in Eq. (22) for identifying the mechanical properties of the plates.

5. Results and discussion

Before proceeding to the mechanical properties identification of the elastically restrained laminated composite plates which have been tested, the present method in predicting natural frequencies and identifying mechanical properties of elastically restrained composite plates made of different materials is worth studying. The present method is first used to predict the natural frequencies of several laminated composite plates with different boundary conditions. A convergence study has shown that the numbers of the characteristic functions in Eq. (10) being I ¼ J ¼ 10 are sufficient to make the solutions of the flexibly supported plates with or without center supports to converge. Therefore, the number of terms of I J ¼ 10 10 for the characteristic functions in the Rayleigh–Ritz method is chosen to evaluate the natural frequencies of the plates under consideration. The results obtained by the present method are listed inTable 2in comparison with those available in the literature[24,25]or obtained in the finite element analyses of the plates using the commercial code ANSYS[26]. For the cases with infinite kL, the value of kLis chosen as 108KN/m2in the analyses when

using the present method or ANSYS to solve the problems. It is noted that the present method can predict excellent natural frequencies for the laminated composite plates with or without center elastic supports. Next study the capability of the present method in mechanical properties identiﬁcation of various elastically restrained laminated composite plates made of graphite/epoxy (Gr/ep) or glass/epoxy (Gl/ep) composite materials. The sizes of the square and rectangular plates are 200 mm 200 mm and 200 mm 100 mm, respectively. The elastic constants of the Gr/ep and Gl/ep composite materials are as follows:

Gr=ep : E1¼131 GPa; E2¼11:2 GPa; G12¼6:55 GPa; n12¼0:28; r ¼ 1550 kg=m3,

Gl=ep : E1¼43:5 GPa; E2¼11:5 GPa; G12¼3:45 GPa; n12 ¼0:27; r ¼ 2000 kg=m3. ð30Þ Table 1

Measured natural frequencies of peripherally and elastically restrained square composite plates with or without center support Layup Plate dimensions and

weight Center support kC(kN/m) Natural frequency 1st 2nd 3rd 4th 5th 6th 7th [01]8 Length (cm) 20.5 0 120 187 311 417 467 490 552 Thickness (mm) 1 (0.45%)a _(0.24%) _(0.27%) _(0.55%) _(0.70%) _(0.18%) _(0.28%) 3.865 143 183 310 418 459 474 549 Weight (g) 64.63 (0.49%) (0.77%) (0.61%) (0.66%) (0.39%) (0.37%) (0.51%) [01/901]2S Length (cm) 20.5 0 122 281 364 469 574 710 773 Thickness (mm) 1 (0.67%) (0.43%) (0.75%) (0.11%) (0.47%) (0.57%) (0.59%) 3.865 145 284 367 467 582 707 781 Weight (g) 64.72 (0.24%) (0.37%) (0.35%) (0.28%) (0.53%) (0.62%) (0.06%) [451/451/451]S Length (cm) 19.5 0 126 261 301 444 523 554 660 Thickness (mm) 0.75 (0%) (0%) (0.18%) (0.12%) (0.22%) (0.15%) (0.25%) 3.865 158 270 309 458 534 575 677 Weight (g) 44.23 (0.74%) (0.33%) (0.67%) (0.96%) (0.32%) (0.27%) (0.62%) a_{The values in parentheses denotes the coefﬁcient of variation of the measured natural frequency.}

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Each laminated composite plate is peripherally supported by same strip-type elastic pads and centrally supported by an elastic spring. Different values for the Young’s modulus Ee of the elastic pads and spring

constant kCof the center elastic spring are adopted in the study. The ﬁrst seven actual natural frequencies of

the elastically restrained composite plates under consideration are listed in Table 3. The actual natural frequencies inTable 3will be treated as ‘‘measured’’ natural frequencies and used in the numerical study to Table 2

Natural frequencies of square plates predicted by different methods

Layup Edge support Center

support

Method Natural frequency l

kL kR kC 1st 2nd 3rd 4th 5th 6th 7th [01/901/01] 2 (MN/m2) 800 (N) 5 (kN/m) Presenta 28.95 29.68 48.89 57.18 61.02 63.11 69.93 ANSYS[26] 28.90 29.46 48.53 57.00 60.84 63.05 69.76 [451/451/451] 100 (kN/m2₎ _{100 (N) 1 (kN/m)} _Present _17.60 _22.93 _28.95 _31.33 _37.59 _39.79 _44.58 ANSYS[26] 17.49 22.83 28.89 31.24 37.57 39.72 44.55 [01/901/01] N 0 0 Present 13.95 21.76 38.64 51.21 55.79 63.99 66.98 ANSYS[26] 13.94 21.74 38.61 51.17 55.74 63.88 66.90 Masoud[24] 13.95 — — — — — — Reddy[25] 13.948 — — — — — — 20 0 Present 25.90 33.33 49.59 69.39 74.04 74.79 84.85 ANSYS[26] 25.71 33.16 49.41 68.92 73.58 74.46 84.40 Masoud[24] 25.91 — — — — — — 20 0 Present 31.24 38.64 55.73 82.94 84.51 88.81 99.34 ANSYS[26] 30.96 38.39 55.48 82.53 83.80 88.12 98.67 Masoud[24] 31.24 — — — — — —

a_{Material property and deﬁnition of normalized natural frequency for the analysis: E}

1¼200 GPa, E2¼10 GPa, G12¼6 GPa, n12¼0:25, l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rho2_a4_=D 0 p . Table 3

Actual natural frequencies of the Gr/ep and Gl/ep plates supported by elastic restraints with different rigidities

Material Layup Shape Edge

support Ee (MPa) Center support kC (kN/m) Natural frequency 1st 2nd 3rd 4th 5th 6th 7th Gr/ep [01/901/01]S Square 1 1 106.159 210.290 287.920 360.594 424.408 531.693 612.360 10 163.290 210.290 287.920 360.594 444.188 531.693 627.521 Rectangular 15 1 310.451 446.202 755.975 852.199 941.920 1158.707 1213.974 10 368.062 446.202 783.529 852.199 941.920 1158.707 1213.974 [451/451]2S Square 1 1 152.074 317.208 348.261 526.847 611.322 630.898 777.177 10 189.412 317.208 348.261 530.342 611.322 647.641 777.177 Rectangular 15 1 411.270 638.073 969.090 1145.644 1394.426 1432.504 1840.852 10 445.426 638.073 984.205 1145.644 1394.426 1432.566 1840.852 Gl/ep [01/901/01]S Square 1 1 71.817 132.677 161.183 215.186 266.810 329.155 338.956 10 120.301 132.677 161.183 215.186 284.263 329.155 363.741 Rectangular 15 1 218.688 288.550 455.579 562.911 618.972 696.449 739.844 10 274.087 288.550 494.273 562.911 618.972 696.449 739.844 [451/451]2S Square 1 1 92.877 190.224 202.845 319.178 367.807 374.863 482.650 10 132.318 190.224 202.845 321.233 367.807 400.575 482.650 Rectangular 15 1 265.755 387.479 580.410 705.413 831.212 860.262 1089.689 10 304.749 387.479 601.264 705.413 831.212 860.303 1089.689

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identify the mechanical properties of the elastically restrained plates. The square [451/451]2Splate made of

Gr/ep material supported by edge elastic pads with Ee¼1:0 MPa and center support with spring constant

kC¼1 kN=m is used as an example to demonstrate the identiﬁcation process of the present method. The

upper and lower bounds of the mechanical properties adopted in solving the identiﬁcation problem are chosen based on experience

0pE1p400 GPa; 0pE2p40 GPa; 0pG12p20 GPa;

0pn12p0:5; 0pEep20 MPa; 0pkCp20 kN=m: ð31Þ

The modiﬁed design variables of Eq. (25) are obtained via the use of the following normalization factors

a1 ¼100 (32a)

and

ai¼10 ði ¼ 2 5Þ. (32b)

It is noted that the use of the above normalization factors can adjust the search direction in such a way that the convergence of the solution can be expedited. The randomly generated starting points, the lowest local minima for the starting points, numbers of iterations required to obtain the lowest local minima, and the global minimum for the Gr/ep [451/451]2S plate using 5 and 6 ‘‘measured’’ natural frequencies in

identifying the mechanical properties are listed in Tables 4 and 5, respectively. For the cases under consideration, six starting points are sufficient to find the global minima and around seven iterations to obtain the lowest local minima for the starting points during the minimization process. A further study has shown that the actual mechanical properties can definitely be identified when more than six ‘‘measured’’ natural frequencies are used in the present method. Similarly, the mechanical properties of the other elastically restrained composite plates inTable 3can be identified using the same identification procedure. Herein, six ‘‘measured’’ natural frequencies have been used in the identification process to identify the mechanical properties of the elastically restrained Gr/ep and Gl/ep plates. The results have shown that the actual mechanical properties of the plates can be obtained for all the cases under consideration irrespective of the rigidities of the flexible supports. It is worth noting that if the number of spring constants and elastic constants

Table 4

Mechanical properties identiﬁcation of the square Gr/ep [451/451]2Splate using ﬁve ‘‘measured’’ natural frequencies Starting

point no.

Stage Mechanical property

E1(GPa) E2(GPa) G12(GPa) n12 Ee(MPa) kC(kN/m) Sum of squared differences Number of iterations 1 Initial 310.387 27.119 13.090 0.1464 14.643 0.978 2.2251E+0 5 Final 132.894 11.008 7.637 0.0322 1.016 0.980 1.40E16 2 Initial 188.674 23.736 7.596 0.2094 17.884 0.274 7.2722E1 10 Final 132.895 11.007 7.638 0.0320 1.016 0.980 1.072E16 3 Initial 130.031 23.142 16.456 0.0698 9.797 16.291 4.5275E1 9 Final 132.894 11.007 7.638 0.0321 1.016 0.980 1.139E16 4 Initial 193.400 6.969 5.425 0.3481 14.904 9.830 6.9981E1 8 Final 102.814 9.063 7.957 0.2908 7.471 0.291 5.1718E4 5 Initial 130.976 11.236 18.951 0.0447 10.671 0.077 1.1503E1 9 Final 132.894 11.007 7.638 0.0321 1.016 0.980 1.45E16 6 Initial 149.957 32.578 12.079 0.3394 5.183 2.197 3.4467E1 9 Final 132.895 11.007 7.638 0.0320 1.016 0.980 1.073E16 Global minimum 132.895 (1.45%)a 11.007 (1.72%) 7.638 (16.61%) 0.0320 (88.57%) 1.016 (1.56%) 0.980 (1.97%) a_{The values in parentheses denote percentage difference between predicted and measured data.}

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of the elastic supports to be identiﬁed is larger than two, six ‘‘measured’’ natural frequencies will be insufﬁcient and more natural frequencies will be required in the present method to identify the mechanical properties of the plate.

Now the present method is applied to the mechanical properties identiﬁcation of the elastically estrained laminated composite plates which have been tested. The measured frequencies of the [01]8 plate

with or without a center support in Table 1 are first used to illustrate the identification process. Tables 6 and 7 list the randomly generated starting points, the lowest local minima obtained for the starting points, the numbers of iterations required for getting the lowest local minima, and the global minimum for the plate with or without a center support, respectively, using different numbers of measured natural frequencies in the identification processes. In view of the results inTable 6, due to the existence of noise in the measurements, the use of the first six measured natural frequencies in the identification process is unable to produce acceptable result for the [01]8plate with a center support while the use of seven measured

natural frequencies can produce satisfactory estimates of the mechanical properties with percentage differences less than or equal to 6.48%. It is worth noting that for the case with the use of seven measured natural frequencies, only four starting points are needed to obtain the global minimum with probability exceeding 0.99 and around eight iterations to find the lowest local minima for the starting points during the minimization process. As for the case without a center support, the results inTable 7show that the use of five rather than four measured natural frequencies in the identification process can produce better estimates of the mechanical properties with percentage differences less than or equal to 5.99% for the [01]8plate. It is also noted that the use of five measured natural frequencies only requires four starting points

to find the global minimum and around seven iterations to obtain the lowest local minima for the starting points in the minimization process. The mechanical properties of the other elastically restrained composite plates which have been tested and listed in Table 1 are then identified using the present method with the use of seven and five measured natural frequencies for the cases with or without a center support, respectively. The identified mechanical properties and their associated percentage differences of the plates are listed in Table 8. Again, it is noted that very good estimates of the mechanical properties with percentage differences less than or equal to 7.63% have been obtained for the plates. In view of the small percentage differences between the actual and identified mechanical properties obtained for the plates, the neglect of the damping effects on the measured natural frequencies is found to be acceptable. It is also worth pointing out that if the number of the unknown spring constants and elastic constants of the elastic supports are larger than two, it is required to use more than seven measured natural frequencies to identify the mechanical properties of the plates.

Table 5

Mechanical properties identiﬁcation of the square Gr/ep [451/451]2Splate using six ‘‘measured’’ natural frequencies Starting

point no.

Stage Mechanical property

E1(GPa) E2(GPa) G12(GPa) n12 Ee(MPa) kC(kN/m) Sum of squared differences Number of iterations 1 Initial 174.214 1.738 8.137 0.3526 10.378 16.747 6.2286E1 6 Final 131.000 11.200 6.550 0.2800 1.000 1.000 1.499E16 2 Initial 69.822 29.244 9.650 0.1984 7.450 6.899 3.6506E2 7 Final 131.000 11.200 6.550 0.2800 1.000 1.000 1.1476E16 3 Initial 285.288 3.545 6.267 0.0411 0.362 0.045 1.1060E1 5 Final 131.000 11.200 6.550 0.2800 1.000 1.000 1.0456E16 4 Initial 331.548 33.018 7.028 0.3314 3.440 17.652 2.7603E+0 10 Final 131.000 11.200 6.550 0.2800 1.000 1.000 1.2525E16 Global minimum 131.000 (0%)a 11.200 (0%) 6.550 (0%) 0.2800 (0%) 1.000 (0%) 1.000 (0%) a_{The values in parentheses denote percentage difference between predicted and measured data.}

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Ta ble 6 Ide ntiﬁed mechan ical pro perties of the [0 1]8 plate with a ce nter supp ort using differ ent numbe rs of measu red natu ral freque ncies No . o f me asured natu ral fr equencies Startin g point no. Stag e Identiﬁ ed me chanical prop erty E1 (GPa) E2 (GPa) G12 (GPa) n12 Ee (MPa ) kC (N/m ) Sum of square d differ ences Numbe r o f iteratio ns 6 1 Initia l 72.73 8 10.611 10.343 0.231 14 6.272 15.03 3 1.247 3E 16 Final 136.3 4 8.563 5.59 0.3 3.061 3.643 7.762 0E 7 2 Initia l 394.8 32 34.915 7.625 0.371 45 15.16 8 6.887 3.883 8E+0 6 Final 136.3 43 8.563 5.593 0.3 3.058 3.644 7.760 5E 7 3 Initia l 278.2 23 8.977 18.075 0.319 06 7.187 16.21 3 1.020 0E+0 1 0 Final 165.2 2 6.164 1.925 0.297 64 1.951 18.00 7 2.473 5E 2 4 Initia l 272.7 9 38.101 4.232 0.394 51 8.252 10.86 8 2.580 4E+0 9 Final 136.3 43 8.563 5.593 0.3 3.058 3.644 7.760 3E 7 5 Initia l 23.31 3 21.975 8.017 0.437 36 4.375 12.57 4 1.879 9E 15 Final 136.3 42 8.563 5.592 0.3 3.059 3.644 7.760 4E 7 6 Initia l 315.0 4 7.485 17.158 0.141 78 9.933 2.138 8.061 4E 18 Final 136.3 43 8.563 5.593 0.3 3.058 3.644 7.760 4E 7 Globa l minimum 136.3 43 ( 6.94%) a 8.563 ₍7.16%) 5.593 ₍18.18 %) 0.3 ( 1.96% ) 3.058 ₍₅0.79%) 3.644 ₍5.72% ) 7 1 Initia l 175.0 47 2.213 13.979 0.316 02 14.82 9 11.28 4 1.850 1E 16 Final 138.4 92 8.625 6.625 0.3 2.118 3.811 2.586 4E 5 2 Initia l 294.1 96 15.885 7.743 0.297 97 1.583 0.695 5.900 2E 18 Final 138.4 95 8.626 6.626 0.3 2.118 3.811 2.586 4E 5 3 Initia l 146.6 93 26.308 18.35 0.166 04 15.90 1 15.63 1.922 2E+0 6 Final 138.4 92 8.625 6.625 0.300 01 2.118 3.811 2.586 4E 5 4 Initia l 245.8 51 1.123 13.69 0.245 54 6.003 17.79 6 2.492 4E 18 Final 138.4 91 8.625 6.625 0.3 2.119 3.811 2.586 4E 5 Globa l minimum 138.4 92 ( 5.47%) 8.625 ₍6.48%) 6.625 ₍3.09% ) 0.300 01 ( 1.96% ) 2.118 ₍₄.44%) 3.811 ₍1.40% ) a The values in paren theses denot e p ercentag e differen ce betw een predic ted and measured data.

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Ta ble 7 Ide ntiﬁed mechan ical pro perties of the [0 1]8 plate witho ut a center support usin g differ ent nu mbers of measured natural freq uencies No . o f measu red natu ral fr equencies Starting poi nt no. Stage Identiﬁ ed mech anical prop erty E1 (GPa) E2 (GPa) G12 (GP a) n12 Ee (MPa ) Sum of square d differ ences Numb er of iteratio ns 4 1 Initia l 187.593 8.076 10.86 1 0.168 56 19.587 2.863 5E 18 Final 137.128 9.943 6.195 0.286 1 1.999 1.568 1E 4 2 Initia l 239.461 22.07 3 10.15 5 0.027 48 13.113 8.057 8E 19 Final 137.047 9.936 6.135 0.298 3 1.999 1.567 1E 4 3 Initia l 57.795 16.60 1 9.77 0.297 26 1.813 3.950 7E 27 Final 137.036 9.934 6.128 0.299 98 1.999 1.567 0E 4 4 Initia l 321.878 19.59 6 14.71 1 0.208 62 8.341 1.134 4E+0 7 Final 137.074 9.938 6.152 0.294 76 1.999 1.567 4E 4 Global min imum 137.036 ( 6.46% ) a 9.934 _(7.71% ) 6.128 ₍10.36 %) 0.299 98 ( 1.97% ) 1.999 ₍1.43%) 5 1 Initia l 224.312 1.477 8.309 0.301 08 7.357 1.712 9E 17 Final 137.728 9.645 6.727 0.300 01 1.997 3.883 0E 4 2 Initia l 76.698 26.26 5 18.62 6 0.469 79 12.917 6.011 3E 17 Final 137.728 9.645 6.726 0.300 01 1.997 3.883 0E 4 3 Initia l 275.026 36.96 5 17.58 4 0.432 26 0.409 8.583 5E 19 Final 137.726 9.645 6.727 0.300 01 1.997 3.883 0E 4 4 Initia l 347.444 29.50 1 11.22 9 0.160 32 1.476 1.220 1E+0 6 Final 137.724 9.645 6.728 0.300 01 1.997 3.883 0E 4 Global min imum 137.724 ( 5.99% ) 9.645 _(4.58% ) 6.728 ₍1.58% ) 0.300 01 ( 1.96% ) 1.997 ₍1.53%) a The values in paren theses denot e p ercentag e differen ce betw een predic ted and measured data.

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6. Conclusions

The nondestructive evaluation of mechanical properties of a number of laminated composite plates elastically restrained at the centers and peripheries of the plates using measured natural frequencies extracted from the vibration data of the plates have been studied via both theoretical and experimental approaches. The nondestructive evaluation method used for the mechanical properties identification of the plates has been established on the basis of the Rayleigh–Ritz method together with a multi-start global minimization method. The theoretical natural frequencies which are obtained in the Rayleigh–Ritz method using trial mechanical properties and the measured natural frequencies of the plates have been used to construct the sum of the squared differences function for measuring the differences between the theoretical and experimental natural frequencies of the plates. The multi-start global minimization method together with several measured natural frequencies has been used to identify the mechanical properties of each of the plates by making the sum of the squared differences function of the plate a global minimum. A normalization technique has also been used in the identification process to expedite the convergence of the solution. In the theoretical study, the mechanical properties identifications of several peripherally and centrally restrained plates made of Gr/ep or Gl/ep composite materials with different layups and dimensions have been performed to demonstrate the capability and accuracy of the present method. It has been shown that the use of six actual natural frequencies, which are treated as measured ones, can identify the actual mechanical properties of the plates with peripheral and central elastic supports in an efficient and effective way. In the experimental study, several flexibly supported laminated composite plates have been fabricated and subjected to impulse vibration testing. For the plates with peripheral and central elastic supports, seven measured natural frequencies have been used to identify the plate mechanical properties of which the percentage differences are less than or equal to 7.63%. For the plates with only peripheral elastic supports, five measured natural frequencies have been used to identify the plate mechanical properties of which the percentage differences are less than or equal to 5.99%. The experimental investigation has demonstrated the applications and validated the capability of the present method.

Acknowledgment

This research work was supported by the National Science Council of the Republic of China under Grant No. NSC 93-2218-E-009-014. Their support is gratefully appreciated.

References

[1] B. Castagne`de, J.T. Jenkins, W. Sachse, S. Baste, Optimal determination of the elastics constants of composite materials from ultrasonic wave-speed measurements, Journal of Applied Physics 67 (1990) 2753–2761.

[2] K.E. Fallstrom, M. Jonsson, A nondestructive method to determine material properties in anistropic plate, Polymer Composites 12 (5) (1991) 293–305.

[3] S.A. Nielsen, H. Toftegaard, Ultrasonic measurement of elastic constants in ﬁber-reinforced polymer composites under inﬂuence of absorbed moisture, International Journal of Ultrasonic 38 (1998) 242–246.

[4] A. Berman, E.J. Nagy, Improvement of a large analytical model using test data, AIAA Journal 21 (1983) 1168–1173. Table 8

Identified mechanical properties of flexibly supported laminated composite plates using measured natural frequencies Layup Support condition Identified mechanical property

E1(GPa) E2(GPa) G12(GPa) n12 Ee(MPa) kC(N/m)

[01/901]2S Edge and center 142.181 (2.95%) a

8.76 (5.02%) 6.439 (5.81%) 0.3 (1.96%) 2.126 (4.83%) 3.834 (0.80%) Edge 139.72 (4.63%) 8.818 (4.39%) 6.818 (0.26%) 0.30002 (1.95%) 1.998 (1.48%) —

[451/451/451]S Edge and center 147.149 (0.44%) 9.927 (7.63%) 6.833 (0.04%) 0.3 (1.96%) 2 (1.38%) 3.777 (2.28%)

Edge 138.925 (5.17%) 9.158 (0.70%) 7.158 (4.71%) 0.30001 (1.96%) 1.999 (1.43%) —

a

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[5] T.Y. Kam, T.Y. Lee, Crack size identiﬁcation using and expanded mode method, International Journal of Solids and Structures 31 (1994) 925–940.

[6] T.Y. Kam, T.Y. Lee, Detection of cracks from modal test data, International Journal of Engineering Fracture Mechanics 42 (1992) 381–387.

[7] T.Y. Kam, T.Y. Lee, Identiﬁcation of crack size via and energy approach, Journal of Nondestructive Evaluation 13 (1994) 1–11. [8] T.Y. Kam, C.K. Liu, Stiffness identiﬁcation of laminated composite shafts, International Journal of Mechanical Sciences 40 (9) (1998)

927–936.

[9] W.T. Wang, T.Y. Kam, Material characterization of laminated composite plates via static testing, Composite Structures 50 (4) (2000) 347–352.

[10] W.T. Wang, T.Y. Kam, Elastic constants identiﬁcation of shear deformable laminated composite plates, ASCE, Journal of Engineering Mechanics Division 127 (11) (2001) 1117–1123.

[11] L.R. Deobald, R.F. Gibson, Determination of elastic constants of orthotropic plates by a modal analysis/Rayleigh–Ritz technique, Journal of Sound and Vibration 124 (2) (1988) 269–283.

[12] P.S. Frederiksen, Estimation of elastic moduli thick composite plates by inversion of vibrational data, in: H.D. Bui, M. Tanaka (Eds.), Inverse Problem in Engineering Mechanic, A.A. Balkema, Rotterdam, 1994, pp. 111–118.

[13] E.O. Ayorinde, R.F. Gibson, Improved method for in-situ elastic constants of isotropic and orthotropic composite materials using plate modal data with trimodal and hexamodal Rayleigh formulations, Journal of Vibration and Accoustics 117 (1995) 180–186. [14] F. Moussu, M. Nivoit, Determination of elastic constants of orthotropic plates by a modal analysis/Rayleigh–Ritz technique, Journal

of Sound and Vibration 124 (2) (1988) 269–283.

[15] W.P. Wilde, H. Sol, Anisotropic material identiﬁcation using measured resonant frequencies of rectangular composite plates, Proceedings of the Fourth International Conference on Composite Structures, Paisley College of Technology, Scotland, 1987, pp. 2.317–2.324.

[16] A.L. Araujo, C.M. Mota Soares, M.J. Moreira, Characterization of material parameters of composite plate specimens using optimization and experimental vibrational data, Composites: Part B 27 (B) (1996) 185–191.

[17] A.L. Araujo, C.M. Mota Soares, J. Herskovits, P. Pedersen, Development of a ﬁnite element model for the identiﬁcation of mechanical and piezoelectric properties through gradient optimization and experimental vibration data, Composite Structures 58 (2002) 307–318.

[18] S.F. Hwang, C.S. Chang, Determination of elastic constants of materials by vibration testing, Composite Structures 49 (2000) 183–190.

[19] K.C. Hung, M.K. Lim, K.M. Liew, Boundary beam characteristics orthonormal polynomials in energy approach for vibration of symmetric laminates—II: elastically restrained boundaries, Composite Structures 26 (1993) 185–209.

[20] G.N. Vanderplaats, Numerical Optimization Techniques for Engineering Design with Applications, McGraw-Hill, New York, 1984. [21] T.Y. Kam, J.A. Snyman, Optimal design of laminated composite structures using a global optimization technique, Journal of

Composite Structure 19 (1991) 351–370.

[22] J.A. Snyman, L.P. Fatti, A multi-start global minimization algorithm with dynamic search trajectories, Journal of Optimization Theory and Applications 54 (1) (1987) 121–141.

[23] ASTM Standards and literature references for composite materials, second ed., West Conshohocken, PA, 1990.

[24] R.R. Masoud, M. Pierre, Buckling and vibration analysis of composite sandwich plates with elastic rotational edge restraints, AIAA Journal 37 (35) (1999) 579–587.

[25] J.N. Reddy, Mechanics of Laminated Composite Plates Theory and Analysis, CRC Press, Boca Raton, FL, 1997 (pp. 332–333). [26] The ANSYS user’s manual, Swanson Analysis System Inc, 1995.