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Natural Frequencies and Acoustic Radiation ModeAmplitudes of Laminated Composite Plates Basedon the Layerwise FEMJinWu Wu and LingZhi HuangSchool of Aircraft Engineering, Nanchang Hangkong University, Nanchang, People’s Republic of China

(Received 21 April 2012; revised 27 December 2012; accepted 7 February 2013)

In this paper, the natural frequencies and acoustic radiation mode amplitudes of laminated composite plates arestudied. The layerwise finite element model is imposed to determine the natural frequencies and velocity distri-butions of laminated composite plates. The amplitude of the laminated composite plates are then discussed basedon the acoustic radiation mode, the effects of the panel orientation angle, the elastic modulus ratio, the width-depth ratio, and the damping ratio on the first acoustic radiation mode. A sixteen-layer laminated plate was usedas an example, and the numerical simulations and experimental results show that the natural frequencies of thelaminated composite plate can be analysed accurately using the proposed model. Furthermore, it is found thatthe effects of the panel orientation angle and width-depth ratio on the acoustic radiation mode amplitude of thelaminated composite plates are significant.

1. INTRODUCTION

Laminated composite plates have been widely used inaerospace vehicles, maritime carriers, and wind turbine blades,where high strength, high stiffness, and low weight are im-portant properties.1 When these composite structures are usedin dynamic environments, vibration control and noise reduc-tion become of great technical significance. The vibration oflaminated composite structures is generally in low-frequencyranges. Thus, noise reduction of laminated composite struc-tures in low-frequency ranges is very important. Althoughactive vibration control could be used to decrease structuralnoise,1 low noise design is the most reliable means of reducingradiated noise. The most suitable function of a low noise de-sign is in sound radiation power.2, 3 A low noise design optimi-sation is considered with the goal of minimising the total soundpower of the structure. The sound radiation power is related tothe characteristics of the structure.2–4 Therefore, it is importantto study the relation of the laminated composite structure pa-rameters to the sound radiation power in low-frequency ranges.

For planar radiators, it is well known that the low-frequencysound radiation is directly related to the velocity distributionof the structure’s surface. Different velocity distributions havedifferent contributions to the radiation sound power. Velocitydistributions corresponding to certain mode shapes are moreefficient radiators. It has been widely accepted that structuralacoustic problems can be analysed based on so-called radiationmodes.1 Radiation modes are sets of independent radiatingvelocity distributions. The dominant radiation modes are thefirst few order modes at low-frequency ranges. Controlling thefirst few order radiation modes’ amplitudes has been shownto reduce the total sound power efficiently in low-frequencyranges.1, 5, 6

For laminated composite plates, the dynamic response ofthe structures must first be analysed in order to study the firstfew order radiation modes’ amplitudes of the plates in low-frequency ranges. Laminated plate theories are essential to

providing an accurate analysis of laminated composite plates,and a variety of laminated plate theories have been developedand reported in the literature. A review of the various equiva-lent single layer and layerwise laminated plate theories can befound in Reddy’s work.7 Because of their complex behaviourin the analysis of laminated composite plates, some technicalaspects must be taken into consideration. For example, theclassical laminate plate theory (CLPT) is based on the Kirch-hoff plate theory. It is the simplest theory, but the shear de-formation effects are neglected.8 Furthermore, it results in anunderestimate of the deflection and an overestimate of the nat-ural frequencies. The first and higher order shear deformationtheories are improvements over classical theories. For the firstand higher order shear deformation theories, transverse sheardeformation through the thickness of the structure is not ig-nored.9

Another aspect in the analysis of composite structures is theexistence of couplings among stretching, shearing, bending,and twisting. These couplings can significantly change the re-sponse of composite structures and need to be considered. Thelayerwise lamination theory assumes a displacement represen-tation formula in each layer.10 The layerwise finite elementtheory can be seen as a three-dimensional theory. The assumedlayerwise displacement field uses a linear Lagrange polyno-mial approximation for the thickness of each lamina and a con-stant transverse displacement for the entire thickness. The in-terlaminar stresses can be predicted accurately, and the layer-wise finite element theory can better adapt to the combinationof boundary conditions. In this paper, the dynamic responsesof the laminated composite plates are examined, based on alayerwise finite element theory.

In recent years, there has been much research on the vi-bration and acoustical characteristics of laminated compositeplates. For example, Li et al.11, 12 deal with the structural vi-bration and acoustic radiation of a fluid-loaded laminated platebased on the first order shear deformation theory (FSDT) andthe classical Kirchhoff-Love thin plate theory. Cao et al.13

134 (pp. 134–140) International Journal of Acoustics and Vibration, Vol. 18, No. 3, 2013

JinWu Wu, et al.: NATURAL FREQUENCIES AND ACOUSTIC RADIATION MODE AMPLITUDES OF LAMINATED COMPOSITE PLATES. . .

present the structural and acoustical characteristics of stiffenedplates in the form of the transverse displacement spectra andsound pressure. The equations of motion for the compositelaminated plate are derived on the basis of the first-order sheardeformation plate theory. Acoustic radiation is analysed usingthe Fourier wave number transform and the stationary phasemethod. The numerical study of the vibration and acoustic re-sponse characteristics of a fibre-reinforced composite plate in athermal environment is present by Jeyaraj et al.,14 and the crit-ical buckling temperature and vibration response are obtainedusing the finite element method based on the classical lami-nate plate theory (CLPT), while the sound radiation character-istics are obtained using a coupled FEM/BEM technique. Niuet al.2–4 propose optimizing the vibrating laminated compositeplates for minimum sound radiation. However, little has beenreported on the relation of the laminated composite structureparameters to the sound radiation in the low-frequency range.This study aims to disclose the vibrating features and acous-tic radiation mode amplitudes of laminated composite platesso that effective composite structure parameter approaches canbe implemented to address the low noise design problem ofsuch structures in low-frequency ranges.

In this paper, the structure vibration and acoustic radia-tion mode amplitudes of laminated composite plates based onlayerwise finite element models are presented. Furthermore,based on the acoustic radiation mode, the effects of the orien-tation angle, the elastic modulus ratio, the width-depth ratio,and the damping ratio on the amplitude of the first acousticradiation mode for laminated composite plates are discussed.

2. RADIATION MODES THEORY

Consider a planar structure vibrating with an angular fre-quency ω and radiating sound into the upper half space V ex-terior to the panel surface S for z > 0. The density of themedium is ρ, and the sound speed is c. The surface area of thevibrating plate is S. The plate is divided into J elements withequal areas. Assume that the elemental source is small com-pared to the acoustic wavelength. The vector of the normalvelocities of each element is denoted as U(ω). The acousticpower can be expressed as1

W(ω) =ρcS

2U(ω)HR ·U(ω); (1)

where the superscript H denotes the complex conjugate trans-pose. The matrix R is a J × J matrix of the real partof the acoustic transfer impedance between each pair of el-ements. Because the matrix R is real, symmetric and pos-itive definite,1 the following eigenvalue decomposition is al-ways possible: R = QΛQT, where the superscript T de-notes the transpose. The matrix Λ is a diagonal matrix witheigenvalues λi, which are related to the radiation efficiency.Q = [Q1, Q2, . . . , Qi, . . . , QJ ] is a group of orthogonal sur-face vibration patterns, and Qi is a real vector representingthe ith radiation mode shape. Because the matrix R is relatedto the frequency, the radiation mode shapes Qi are also fre-quency dependent. It has been demonstrated that the radiationmode shapes can be chosen to be independent of the frequencyat low-frequency ranges.15, 16

Any vibration velocity distribution U(ω) has a representa-

Figure 1. Laminated plate geometry and coordinate system.

tion in terms of the radiation modes Qi:

U(ω) =J∑i=1

vi(ω)Qi; (2)

wherevi(ω) = QT

iU(ω) (3)

and vi(ω) is called the ith radiation mode’s amplitude.1

Because R = QΛQT, the total radiation acoustic power canbe rewritten as

W (ω) =ρcS

2

J∑i=1

λi|vi(ω)|2. (4)

From Eq. (4), it can be demonstrated that each radiationmode contributes to the sound power independently. Equa-tion (4) shows that the total radiation acoustic power W (ω) isthe sum of the individual mode amplitudes squared, multipliedby their corresponding eigenvalues. The relative contributionof the first radiation mode to the radiated sound power is onlydominant for low frequencies.1, 17 At low-frequency ranges,the sound power of the first radiation mode is the main part ofthe total sound power. Moreover, after cancelling the soundpower of the first radiation mode, the total acoustic power canbe reduced significantly.1 Therefore, acquiring informationon the amplitudes of the first radiation modes is important tocontrol the acoustic radiation power of the vibrating structure.However, for higher frequencies, all radiation modes have sim-ilar radiation efficiencies, so more radiation modes have to betaken into account to reduce the radiated noise. Moreover, inorder to obtain reduction at higher frequencies, i.e., ka > 1,where k is the wave number and a is the characteristic radiusof the structure, more radiation modes are needed.

In this paper, the vibration and sound radiations of a lami-nated composite plate at low-frequency ranges are presented.This will be discussed in the next section using the layerwiseFEM.

3. THE NORMAL VELOCITY DISTRIBUTION

Based on the layerwise theory, each laminate in the direc-tion of the thickness needs to be interpolated twice, and thelaminated plate geometry and coordinate system are shown inFig. (1). The displacement of the laminated plate can be writ-ten as:18

U(x, y, z, t) =2n+1∑i=1

µi(x, y, t)Ψi(z); (5)

International Journal of Acoustics and Vibration, Vol. 18, No. 3, 2013 135


V (x, y, z, t) =2n+1∑i=1

vi(x, y, t)Ψi(z); (6)

W (x, y, z, t) =2n+1∑i=1

wi(x, y, t)Ψi(z). (7)

U(x, y, z, t), V (x, y, z, t), and W (x, y, z, t) are the displace-ments in the x, y, z directions, respectively; n denotes thelayer number; 2n + 1 denotes the interpolation surface num-ber; ui(x, y, t), vi(x, y, t). and wi(x, y, t) denote the displace-ments of the ith interpolation surface in the x, y, z directions,respectively. Ψi(z) is the interpolation coefficient, where:18

Ψ1(z) = φ11(z) (z1 ≤ z ≤ z3); (8)Ψ2i(z) = φ2i(z) (z2j−1 ≤ z ≤ z2j+1); (9)

Ψ2i+1(z) =

ßφ3i(z) (z2j−1 ≤ z ≤ z2j+1)φ1(i+1)(z) (z2j+1 ≤ z ≤ z2j+3)

; (10)

Ψn(z) = φ3n(z) (z2n−1 ≤ z ≤ z2n+1); (11)

where j denotes the layer number of the laminated plate, zkdenotes the ordinate of the kth interpolation layer, and φ1i, φ2i,and φ3i denote:

φ1j(z) = (1− zz/hj)(1− 2zz/hj); (12)φ2j(z) = 4zz/hj(1− zz/hj); (13)φ3j(z) = −zz/hj(1− 2zz/hj); (14)

where j = (1, 2.....n), hj is the jth layer thickness, and zz isthe local coordinate of the plate thickness.

According to finite element theory, ui(x, y, t), vi(x, y, t)and wi(x, y, t) in Eqs. (5–7) can be rewritten as:

ui(x, y, t) =m∑k=1

Nk(x, y)ukT (t); (15)

vi(x, y, t) =m∑k=1

Nk(x, y)vkT (t); (16)

wi(x, y, t) =m∑k=1

Nk(x, y)wkT (t). (17)

wherem denotes the finite element node. Nk(x, y) is the shapefunction expression, uk, vk, wk are the x, y, z coordinates, re-spectively, of the finite element kth node, and T (t) is the timefunction. In this paper, a finite element unit is eight noderectangular elements, and the length by width of each unit is2a× 2b. The shape function expression of the eight-node rect-angular elements can be written as:

N1 = (1− x/a)(1− y/b)(−x/a− y/b− 1)/4; (18)

N2 = (1− x/a)2(1− y/b)/2; (19)N3 = (1 + x/a)(1− y/b)(x/a− y/b− 1)/4; (20)

N4 = (1− (y/b)2)(1 + x/a)/2; (21)N5 = (1 + x/a)(1 + y/b)(x/a+ y/b− 1)/4; (22)

N6 = (1− (x/a)2)(1 + y/b)/2; (23)N7 = (1− x/a)(1 + y/b)(−x/a+ y/b− 1)/4; (24)

N8 = (1− (y/b)2)(1− x/a)/2. (25)

Substituting Eqs. (15–17) and (18–25) into Eqs. (5–7) yieldsthe unit shape matrix N:

N = [N11,N12, . . . ,Nlk, . . . ,N38]

(l = 1, 2, 3; k = 1, 2, . . . , 8); (26)

where

Nlk =

NkφikNkφik

Nkφik

. (27)

According to the elastic displacement strain, the unit strain ma-trix can be written as

B = [B11,B12, · · · ,Blk, · · · ,B38]; (28)

where

Blk =

∂Nk

∂x φik∂Nk

∂y φik

Nk∂φik

∂z

Nk∂φik

∂z∂Nk

∂y φik

Nk∂φik

∂z∂Nk

∂x φik∂Nk

∂y φik∂Nk

∂x φik

. (29)

The stress-strain relationship of orthotropic materials is

[ε]i = [S]i × [σ]i; (30)

where the matrix S can be written as

S =

1

E1

−v12E2

−v13E3

−v12E2

1

E2

−v23E3

−v31E1

−v23E3

1

E3

1

G231

G311

G12

j

.

(31)If there is a viscoelastic laminate plate, the elastic modulus

ratio of each viscoelastic layer can be denoted as E = E · (1 +ηl), where l =

√−1 and η is the loss factor. According to the

shaft formula, when the x, y, z coordinate axis is inconsistentwith the layer material coordinate axis, the displacement-strainrelationship can be expressed as:

[σ]i = [D]i × [ε]i; (32)

where D denotes the stiffness matrix.From Eqs. (28) and (32), we obtain the B matrix and D

matrix, and the unit stiffness matrix K can be expressed as

K =

∫∫∫V

BTDBdxdydz. (33)

The unit mass matrix M can be expressed as

M =

∫∫∫V

ρNTNdxdydz. (34)

136 International Journal of Acoustics and Vibration, Vol. 18, No. 3, 2013


Table 1. Material properties of the viscoelastic laminate plate.

Property ValueLength (x direction) 0.3048 mLength (y direction) 0.3480 m

Thickness h1 = h3 = 0.007 62mYoung’s modulus E1 = E3 = 68.9GPa

Poisson ratio υ1 = υ3 = 0.3, υ2 = 0.49Shear modulus G2 = 0.869MPa

Loss factor η = 0.5

Density ρ1 = 2 740 kg/m3, ρ2 = 999 kg/m3

Table 2. Natural frequencies and the loss factor of the viscoelastic laminatedplate.

Mode Natural frequency (Hz) Loss factorRef.17 Ref.18 Present Ref.15 Ref.16 Present

1 60.3 60.24 58.3 0.190 0.1901 0.17182 115.4 115.22 114.5 0.203 0.2034 0.19243 130.6 130.43 130.0 0.199 0.1991 0.17204 178.7 178.46 178.3 0.181 0.1806 0.17005 195.7 195.42 196.6 0.174 0.1737 0.1562

Generally, the damping value of the composite laminateplate in the low-modal range is related to the degree ofanisotropy and each mode of the composite laminate plate. TheRayleigh damping model is used in this paper, and the dampingmatrix C can be written as

C = αM + βK; (35)

where α, β are the mass and stiffness factors, respectively. Theelement equivalent nodal force can be expressed as

F =

∫∫∫V

NTpvdxdydz. (36)

From the unit stiffness matrix K, mass matrix M, dampingmatrix C, and element nodal force F, the total power equationcan be obtained:

MX + CX + KX = F. (37)

From Eq. (37), the surface normal displacement vector Xcan be determined, and the normal velocity vector UX(x, y, ω)and natural frequency ω can be obtained from

∣∣K− ω2M∣∣ =

0.

4. NUMERICAL EXAMPLES

In order to check the quality of the layerwise FEM method,the natural frequencies of a viscoelastic laminate plate and aT300 laminated plate are studied. The first acoustic radiationmode amplitude of a 16-layer laminate plate with a simply-supported boundary condition is presented.

4.1. Natural Frequencies of a Laminate PlateThe physical properties of a viscoelastic laminate plate un-

der a simply-supported boundary condition are summarized inTable 1. The natural frequencies and loss factor of the vis-coelastic laminate plate are determined: ω′ =

√re(ω)2 and

η′ = im(ω)2/re(ω)2. We used regular grids of 60×60 ele-ments in the x- and y-directions.

The result is obtained from Eq. (37) and is compared withthe result in references19 and.20 A comparison of the results isshown in Table 2. The results show good agreement for boththe frequencies and loss factors in this paper. Reference19 takesa first-order shear deformation theory to describe the defor-mation of the faces. The first-order shear deformation theory

Table 3. Material properties of the T300 laminated plate.

Property ValueComposite material T300Length (x direction) 0.27 mLength (y direction) 0.27 m

Thickness h = 0.002mYoung’s modulus E1 = 1.24GPa,E2 = 7.5GPa,E3 = E2

Shear modulus G12 = 4.6GPa,G13 = G12 = G23Poisson ratio υ12 = υ13 = 0.31

(a)

(b)

Figure 2. Experimental set-up of the T300 laminated plate.

(FSDT) provides a balance between computational efficiencyand accuracy for the global structural behaviour of thin andmoderately thick laminated composite plates. Reference20 alsotakes an analytic solution based on the layerwise theory anduses eigenfunctions as the plate boundary conditions.

Consider sixteen anti-symmetric 0-degree angle-ply T300laminated plates with four sides clamped; the physical prop-erties of the plate are summarized in Table 3. In order to checkthe accuracy of the method in this paper, the numerical simu-lations and experimental results of the natural frequency of theT300 laminated plate are analysed.

The experimental set-up is shown in Fig. (2). In the sim-ulations, clamped boundary conditions are assumed. In theexperiments, the boundary conditions approximate clampedboundary conditions. A T300 laminated plate is mounted ona cast-iron box with a B&K4808 hammer attached at the topof the plate to provide the primary driving force, and a pointforce located at x = 0.25 · Lx, y = 0, 75 · Ly is applied toexcite the plate. The accelerometer output signal of the plateis measured using a B&K4370 accelerometer. Sixteen points,which are uniformly distributed on the vibrating plate, are se-lected for measurement in the experiments. When the relatedparameters of the modal analysis software in the host PC havebeen well adjusted, the output signal and the input force ham-



Figure 3. Transfer function of the T300 laminated plate.

Table 4. Natural frequencies of the T300 laminated plate.

Result/mode 1 2 3 4 5 6Simulation result (Hz) 169 226 364 513 552 648

Experimental result (Hz) 153 220 360 502 531 661Error (%) 10.05 2.65 1.09 2.14 3.80 2.01

mer signal are simultaneously measured by the modal analysissoftware. The transfer functions of the sixteen points are anal-ysed in the modal analysis software. Curve fitting the sixteentransfer functions give the natural frequencies of the structure.A typical measured transfer function is shown in Fig. (3). Allexperiment apparatuses are demarcated, and all measurementsignals are transformed by the engineering unit and limited toa frequency range below 800 Hz. The numerical simulationsand experimental results of the natural frequencies are listed inTable 4. Table 4 shows that there are errors between the ex-perimental and numerical simulation results. The first reasonis because the boundary condition is slightly different in theanalytical model and in the experimental set-up. The secondreason is that the density and Poisson ratio of the laminatedcomposite plate are not constant.

4.2. Factors of the Acoustic Radiation ModeAmplitude

The physical properties of the 16-layer laminate simply-supported plate are as follows: Lx = 0.40 m, Ly = 0.25 m, thesingle thickness is 0.125 mm, the total thickness is h = 2 mm,Young’s modulus is E1 = 181 GPa, and v12 = v13 = v23 =0.28. One point force (force amplitude of 10 N), located atx = 0.5 · Lx, y = 0.5 · Ly , is applied to excite the plate. As-sume that the bonding layer materials damping ratio is ignored.

4.2.1. The Effect of the Laying Angle

Assume that the laminated plate damping ratio is 0.2, theshear modulus is G23 = G21 = G13 = 7.17 GPa, andE2 = E3 = 10.3 GPa. The four orientation angles areα = (−15/15)8, (−30/30)8, (−45/45)8 and (−0/90)8. Thenatural frequencies and the first acoustic radiation mode ampli-tude of the laminated plates with different orientation anglesare shown in Table 5 and Fig. (4). It is found that the firstacoustic radiation mode amplitude of the laminated simply-supported plate is only related to the (odd, odd) order struc-ture mode. Frequencies corresponding to the maximum am-plitude are equal to frequencies of the (odd, odd) order struc-ture mode. These characteristics are the same as those of theacoustic radiation of the general single-layer structure. In ad-dition, Table 5 and Fig. (4) show that the acoustic radiationmode is only related to the geometrical shape of the radiating

Table 5. Natural frequencies of the laminated plate with different orientationangles.

Mode Natural frequencies (Hz)α = 15◦ α = 30◦ α = 45◦ α = (0/90)◦

1 96.3 127 152 1302 187 291 288 2303 278 301 427 4424 327 507 480 4705 383 544 604 5226 520 571 735 6617 538 790 844 7568 557 806 876 7939 577 852 885 917

Figure 4. Effect of the orientation angle on radiation mode amplitude.

structure and is independent of the material properties. Fur-thermore, besides (−0/90)8, the smaller the orientation angleof a laminated plate is, the smaller is the natural frequencyof the corresponding structure mode values. This means thatthe more natural the frequency number is, the greater the first-order acoustic radiation mode amplitude is and the greater thecorresponding radiated power value is at the same external ex-citation frequency. Therefore, if a 45 degree orientation anglefor the laminated plate is used, the laminated plate noise canbe reduced significantly.

4.2.2. The Effect of the Young’s Modulus Ratio

Assuming that the orientation angle of the laminated plateis α = (−45/45)8 and the damping is 0.2, the shear modu-lus is G23 = G21 = G13 = 0.6E2 and Young’s modulus isE2 = E3. In addition, the elastic modulus ratio E1/E2 is, re-spectively, 5, 10, 15, and 20 for the four orientation angles. Thenatural frequency and the first order acoustic radiation modeamplitude of the laminated plates with different Young’s mod-ulus ratios are shown in Table 6 and Fig. (5).

Table 6 and Fig. (5) show that the natural frequency of thesame structural mode is almost equal at different Young’s mod-ulus ratios, which means that the corresponding frequency ofthe maximum amplitude of the first order acoustic radiationmode is almost identical. This illustrates that the Young’s mod-ulus ratio has little effect on the acoustic radiation mode am-plitude.

Table 6. Natural frequencies of the laminated plate with different E1/E2.

Mode Natural frequencies (Hz)E1/E2 = 5 E1/E2 = 10 E1/E2 = 15 E1/E2 = 20

1 159 154 152 1512 298 291 288 2873 476 442 429 4424 510 489 481 4775 635 614 605 6016 803 755 737 7287 871 853 845 8248 1001 912 880 8419 1152 945 887 862



Figure 5. Effect of E1/E2 on the radiation mode amplitude.

Table 7. Natural frequencies of the laminated plate with different b/h.

Mode Natural frequencies (Hz)b/h = 75 b/h = 100 b/h = 125 b/h = 150

1 251 190 152 1272 476 359 288 2403 476 534 428 3574 793 598 480 4015 884 753 605 5056 997 885 735 6057 1186 916 844 7068 1214 1050 886 7329 1389 1092 885 885

4.2.3. The Effect of the Thickness Ratio

Assuming that the orientation angle of the laminated plateis α = (−45/45)8 and the Young’s modulus is E2 = E3 =10.3 GPa, the shear modulus is G23 = G21 = G13 = 0.6E2.In addition, b/h is, respectively, 150, 125, 100 and 75 for thefour orientation angles, where b is the defined side length ofthe plate. The natural frequency and the first order acousticradiation mode amplitude of the laminated plates with differentb/h ratios are shown in Table 7 and Fig. (6).

Table 7 and Fig. (6) show that the b/h ratio has a great effecton the acoustic radiation mode amplitude. The correspondingnatural frequencies of the same order structure mode vary withthe b/h ratio, and the larger the b/h ratio is, the smaller thenatural frequencies of the same order structure mode are. Theacoustic radiation mode amplitude decreases as the peak cor-responding to the frequency increases, and the total radiatedsound power decreases accordingly when the thickness of thelaminated plate increases. Therefore, a smaller b/h ratio isbetter in order to reduce the laminated plate noise. However,increasing the thickness is limited by the material and process.

Figure 6. Effect of b/h on the acoustic radiation mode amplitude.

(a)

(b)

Figure 7. Effects of the damping ratio on acoustic radiation mode amplitudeand the sound radiation power.

4.2.4. The Effect of Damping

Assuming that the Young’s modulus of the laminated plateis E2 = E3 = 10.3 GPa, the shear modulus is G23 = G21

= G13 = 0.6E2 and the orientation angle of the laminatedplate is α = (−45/45)8. The first-order acoustic radiationmode amplitude and the acoustic power of the laminated plateswith different dampings are shown in Fig. (7).

It is well known that the damping does not affect the struc-ture of the natural frequency, which is also reflected in Fig. (7).In addition, the figure shows that the damping ratio is largerand that the first order acoustic radiation mode amplitude andthe acoustic power of the laminated plate are smaller at thesame external excitation frequency.

5. CONCLUSIONS

In this paper, the natural frequency and acoustic radiationmode amplitude of laminated composite plates have been in-vestigated. Based on the layerwise finite element theory, thenatural frequency and dynamic response of laminated com-posite plates are obtained. The effects of the laminated com-posite plate’s parameters on the first-order acoustic radiationmode amplitude of the laminated composite plates are dis-cussed when the excitation point is located at the centre ofthe plate. The numerical calculation and experimental resultsshow that the natural frequencies of the laminated compositeplate can be analysed accurately using the layerwise finite el-ement model. In addition, the layerwise finite element modelhas been successfully used to analyse the acoustic radiation ofclamped laminated composite plates. The proposed methodcan provide the structural parameters of the optimal design oflow-noise laminated plates.

There is still a need to further investigate the influence of thedifferent excitation points on the results. Furthermore, the case



in which the multi-excitation points simultaneously impact thelaminated composite plate may be considered in future work.

ACKNOWLEDGMENTS

This work is supported by the National Natural ScienceFoundation of China (No: 51265038) and the AeronauticalScience Foundation of China (No: 2011ZA56002).

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