Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites

Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites Olaf Täger 1, Martin Dannemann n, Werner A. Hufenbach Technische Universität Dresden, Institute of Lightweight Engineering and Polymer Technology, Holbeinstr. 3, 01307 Dresden, Germany

a r t i c l e i n f o

abstract

Article history: Received 18 November 2013 Received in revised form 2 December 2014 Accepted 26 December 2014 Handling Editor: I. Lopez Arteaga

Lightweight structures for high-technology applications are designed to meet the increasing demands on low structural weight and maximum stiffness. These classical lightweight properties result in lower inertial forces that consequently lead to higher vibration amplitudes thereby increasing sound radiation. Here, special anisotropic multilayered composites offer a high vibro-acoustic lightweight potential. The authors developed analytical vibro-acoustic simulation models, which allow a material-adapted structural-dynamic and sound radiation analysis of anisotropic multilayered composite plates. Compared to numerical methods FEM/BEM these analytical models allow a quick and physically based analysis of the vibro-acoustic properties of anisotropic composite plates. This advantage can be seen by the presented extensive parameter studies, which have been performed in order to analyse the influence of composite-specific design variables on the resulting vibro-acoustic behaviour. Here, it was found that the vibroacoustic parameters like eigenfrequency and modal damping show direction-dependent properties. Furthermore, the investigations reveal that laminated composites show a so-called damping-dominated sound radiation behaviour. Based on these studies, a vibroacoustic design procedure is proposed and design guidelines are derived. & 2015 Elsevier Ltd. All rights reserved.

1. Introduction and literature survey The development of lightweight composite concepts, e.g. for vehicles, often presents special challenges of simultaneously realising lightweight structures and low sound radiation. This is due to the fact that the use of multilayered composite components with high stiffness and low structural weight as an increasingly used lightweight design concept can create a deterioration of the vibro-acoustic property profile. Here, the low coincidence frequencies result in a strong structure–fluid coupling already within the first eigenfrequencies of lightweight components and therefore produce high modal sound radiation. As a result, classical weight-intensive secondary measures, such as elastic bearings and additional sound insulating foils, are often used to improve the acoustic behaviour of the structural parts.

n

Corresponding author. Tel.: þ49 351 463 38134; fax: þ 49 351 463 38143. E-mail address: [email protected] (M. Dannemann). 1 Present address: Volkswagen AG, Group Research, Letter box 1499, 38436 Wolfsburg, Germany. http://dx.doi.org/10.1016/j.jsv.2014.12.040 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Idrisi, Johnson et al. [1,2] showed, e.g. that the transmission loss of double panel systems can be improved by the addition of mass inclusions to a poro-elastic layer, which significantly increases the structural mass of the construction. Lightweight solutions for acoustic optimised structures have been presented by Franco et al. [3]. The authors optimised the structural and acoustic behaviour of sandwich panels without considering anisotropic material damping. These solutions often face problems in industrial applications due to the limited draping and forming properties of the investigated sandwich panels resulting in major manufacturing problems. Anisotropic multilayered components allow to synergetically adapt the direction-dependent material damping as well as the modal spectrum to the external dynamic loads [4]. This paper makes a contribution to the use of such innovative lightweight-acoustic concepts by describing the complicated structural-dynamics and sound radiation of anisotropic multilayered composites with the aim to specifically include the vibro-acoustic behaviour into the design process. On the basis of extended solution methods, the authors developed advanced analytical models for the calculation of the structuraldynamic and damping behaviour as well as the sound radiation of anisotropic multilayered composite plates including shear effects. These models offer specific advantages for the performance of parameter studies containing a large number of composite-specific design variables. The vibro-acoustic properties can quickly be calculated and displayed, e.g. in threedimensional surface plots, which allows the calculation engineer to easily evaluate the physical dependencies and thus create design rules. In the past, the calculation of the structural-dynamic and acoustic behaviour of composite structures has been the subject to extensive scientific research. For example, composite beams have been investigated by Bachoo and Bridge [5], who minimised the modal density of a composite beam especially for Statistic Energy Analysis. The structural-dynamic analysis of plates was intensively investigated already by Leissa and Young [6,7], who presented an eigenfrequency analysis of plane load bearing structures by using variational principles in combination with the Ritz method. Whitney [8] applied these techniques in order to determine the vibrations of anisotropic laminate composite plates. The calculation of natural frequencies and vibration modes of plates with a fibre-orientation realised, e.g. by tailored fibre placement technique was performed by Honda and Narita [9] using special spline functions. Kayikci and Sonmez [10] published work on the design of composite laminates for optimal frequency response. The consideration of damping of vibrating structural components is of special importance for sound radiation problems. In the past, several authors took material damping effects into account (see, e.g. [11–14]). Saravanos and Lamancusa [15] showed an optimal structural design of robotic manipulators made of fibre-reinforced composite materials. A very detailed analysis of the modal damping properties of hybrid laminates using a global optimisation strategy was performed by Montemurro et al. [16]. The structural-dynamic and damping calculation for thick composites plates and sandwich structures was subject to intensive investigations by Ghinet and Atalla [17]. They assumed a discrete laminate model which describes each composite layer as thick laminate with orthotropic orientation, rotational inertia and transversal shearing, membrane and bending deformations using the First Order Shear Deformation Theory (FSDT). For the vibro-acoustic analysis the consideration of effects of the boundary conditions on modal damping and sound radiation is of practical relevance due to the large variation of boundary conditions in technically applications. Here, the papers of Hufenbach et al. [18] made a contribution to this aspect. Furthermore, the inclusion of transverse shear strain in modal and damping analysis, which is typical for anisotropic fibre-reinforced composite structures, was done by Alam and Asnani [19] and Täger [20]. More recent studies focus on an active vibration control of composite structures, e.g. by the use of piezoelectric fibres [21–23]. These approaches mainly focus on the use of Finite Element (FE) methods and control algorithms to simulate the influence of the active vibration control on the modal damping. The extension of the structural-dynamic analysis to the acoustic analysis of multilayered structures was done by Dym, Lang and Ventres [24–26] and others. The sound transmission behaviour of fibre-reinforced composite plates with infinite surfaces was calculated by Matsikoudi-Iliopoulou and Trochidis [27]. Shen et al. [28] and Mejdi et al. [29,30] investigated the sound radiation of laminated composite plates with especially orthogonal material characteristics and they focussed on the calculation of the transmission loss. A very detailed summary of possible structural–acoustic optimisation can be found in the publications of Denli and Sun [31]. The consideration of shear effects in laminated plates and the influence on the sound radiation was studied, e.g. by Cao et al. [32]. More detailed analyses of the influence of material anisotropy on the sound insulation were performed by Thamburaj and Sun [33] and others for the case of fibre-reinforced beams. Approaches for the calculation of the influences of elastic bearings on the sound radiation of laminated composites were presented by Jiang and Kam [34] as well as Park and Mongeau [35]. Among these papers, the consideration of the anisotropic material damping behaviour of composites remains still open. The minimisation of sound power through damping layer placement for isotropic material behaviour was investigated by Wodtke and Lamancusa [36]. The authors Wennhage et al. published extensive investigations on the structural and acoustic performance of sandwich panels as well as optimisation calculations for vehicle bodies [37–39]. These authors focussed on fibre-reinforced and sandwich materials in the low frequency range. Yamamoto et al. [40] and Tanneau et al. [41] performed extensive investigations on the optimisation of the acoustic transmission of multilayered structural components with isotropic material behaviour. Extensive numerical-based investigations for the structural–acoustic and shape optimisation of especially vehicle components have been presented by Marburg et al. [42,43]. The consideration of material-anisotropy and damping for large variety of multilayered composite materials was conducted within detailed theoretical and experimental studies by Täger and Dannemann [20,44]. They also included the influence of different boundary conditions especially on the sound radiation. The optimal vibro-acoustic design of flat panels was investigated by Jiang et al. [45], who considered different excitation locations for point-loads. In summary, it could be stated that the very important simultaneous consideration of anisotropic material damping and sound radiation within a holistic vibro-acoustic analysis approach still has to be developed in detail. This allows to realise a Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Fig. 1. Deformation according to the CLT at a randomly chosen time t ¼ τ.

material-adapted vibro-acoustic design of fibre-reinforced composite lightweight structures. Furthermore, the extensive analysis of different boundary conditions on the anisotropic vibro-acoustic behaviour has to be included within the design concept in order to fully exploit the high lightweight potential of this material group for practical use. This paper makes a contribution to this aspect by presenting an analytical based calculation approach, which allows to derive design rules. 2. Analysis of free damped vibration fields of hybrid anisotropic plates Multilayered anisotropic composites offer the possibility to specifically adjust the lightweight characteristics to the respective application demands. However, dynamically loaded lightweight structures of this kind often have high vibration amplitudes due to much smaller forces of inertia, which causes high levels of acoustic radiation. Thus, it is necessary to develop material-adapted design methods that realistically describe the complicated structural-dynamic behaviour of anisotropic composite structures. The determination of free standing vibration fields is of major importance for the dynamic analysis of anisotropic composite plates. These free standing plate waves show large resonances thus creating massive sound radiation. Therefore, the analysis of the eigenfrequency and mode shape characteristics is essential to the development of effective vibro-acoustic simulation models. 2.1. Elastomechanical fundamental equations 2.1.1. Classical lamination theory On the basis of the kinematic hypothesis of the classical (Kirchhoff) plate theory [46], the classical lamination theory (CLT) assumes that the normal to the mid-plane of the undeformed cross section remains normal to the mid-plane of the deformed cross section (Fig. 1). ^ v^ and w ^ within This results in the following description of the spatial and time-dependent displacement components u, the global x, y, z coordinate system2: ^ ∂wðx; y; tÞ ; u^ ðx; y; z; t Þ ¼ u^ 0 ðx; y; t Þ  z ∂x ^ ∂wðx; y; tÞ ; v^ ðx; y; z; t Þ ¼ v^ 0 ðx; y; t Þ z ∂y ^ ^ wðx; y; z; tÞ ¼ wðx; y; tÞ:

(1)

The functions u^ 0 ðx; y; tÞ and v^ 0 ðx; y; tÞ are the displacement components of the mid-plane z¼0 in x and y direction, respectively. Using this approach, the strain–displacement relation can be written as

ε^ x ¼ ϵ^ 0x þ zκ^ x with

ε^ 0x ¼ ∂∂xu^ 0 ;

κ^ x ¼  ∂∂xw2^ ;

ε^ y ¼ ϵ^ 0y þzκ^ y with

ε^ 0y ¼ ∂∂yv^ 0 ;

κ^ y ¼  ∂∂yw2^ ;

γ^ xy ¼ γ^ 0xy þzκ^ xy with ε^ z ¼ 0;

2

2

2

^ u^ 0 v^ 0 ∂ w γ^ 0xy ¼ ∂∂y þ ∂∂x ; κ^ xy ¼  2∂x∂y ; γ^ xz ¼ 0; γ^ yz ¼ 0: 2

(2)

^ is assumed to be small in the manner that sin ∂w=∂x ^ ^ The deflection w  ∂w=∂x (see, e.g. [8]).

Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Fig. 2. Cross-section deformation within the FSDT.

Neglecting the generally small displacements u^ 0 and v^ 0 of the mid-plane, the CLT describes the deformation of the ^ ¼ wðx; ^ composite plate only by using the displacement function w y; tÞ. The stresses of the kth single layer are calculated using the stress–strain relation 1 0 ðkÞ 1 0 ðkÞ e ðkÞ Q e ðkÞ 0 ε^ 1 e σ^ x Q Q 11 12 16 C x B ðkÞ C B C C ðkÞ B σ^ C B e ðkÞ ε^ : e ðkÞ Q e ðkÞ Q e ðkÞ CB (3) @ ε^ y A or σ^ ¼ Q B y C¼B Q 33 B C 12 22 26 @ A @ A γ^ ðkÞ ðkÞ ðkÞ xy τ^ ðkÞ e e e xy Q 16 Q 26 Q 36 Then the well-known structure law of multilayered composites can be written as ! !   ^ A B ε^ 0 N ¼ ^ B D κ^ M

(4)

^ and M ^ as vectors of the cutting forces and moments as well as with Aij, Bij and Dij as the tensile, coupling and bending with N stiffnesses (see, e.g. [47]) Z Aij ¼

 h=2

Z Bij ¼

h=2

¼

N X

  e ðkÞ z  z Q k1 ij k

ði; j ¼ 1; 2; 6Þ;

(5)

ði; j ¼ 1; 2; 6Þ;

(6)

ði; j ¼ 1; 2; 6Þ:

(7)

k¼1

N X   e ðkÞ  z dz ¼ 1 e ðkÞ z2 z2 Q Q ij k1 2 k ¼ 1 ij k  h=2

Z Dij ¼

e ðkÞ dz Q ij

h=2

N X   e ðkÞ z2 dz ¼ 1 e ðkÞ z3 z3 Q Q ij k1 3 k ¼ 1 ij k  h=2 h=2

Here, h is assumed to be the total thickness of the multilayered composite with N layers, zk  1 and zk (k ¼ 1; …; N) are supposed to be the lower and upper boundary of the kth layer (this results in the thickness of a kth layer being hk ¼ zk  zk  1 ). 2.1.2. First order shear deformation theory Within the FSDT, we study the influence of shear deformation on the basis of the models by Reissner and Mindlin analogously to the Timoshenko beam [48,49]. Each normal will remain a straight line in the deformed state. In contrast to the normals in the classical lamination theory (CLT), the normals to the mid-plane are not normal to the deformed midplane (see Fig. 2). Thus, it is necessary to consider the additional shear angles3 ψ^ x and ψ^ y which are not equal to the first ^ derivative of the deflection w. 3

^ The roof symbols denote the time dependence of the physical quantities ψ^ x , ψ^ y , w.

Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Fig. 3. Multilayered composite and coordinate systems.

Fig. 4. Geometry of the plate structure.

^ v^ and w ^ of the deformation vector u^ can be written in the form: With these assumptions, the components u, ^ y; z; tÞ ¼ u^ 0 ðx; y; tÞ þ zψ^ x ðx; y; tÞ; uðx; ^ y; z; tÞ ¼ v^ 0 ðx; y; tÞ þ zψ^ y ðx; y; tÞ; vðx; ^ ^ wðx; y; z; tÞ ¼ wðx; y; tÞ:

(8)

Here, the functions u^ 0 , v^ 0 are the deformations of the mid-plane z¼ 0 in x- and y-direction, respectively. Using Eq. (8), the strain field can be calculated by   ^ ∂u^ ∂ψ^ ∂w ; ε^ x ¼ 0 þ z x ; γ^ yz ¼ k1 ψ^ y þ ∂y ∂x ∂x   ∂ψ^ ^ ∂v^ ∂w ; ε^ y ¼ 0 þz y ; γ^ xz ¼ k2 ψ^ x þ ∂x ∂y ∂y

ε^ z ¼ 0;

γ^ xy ¼



   ∂u^ 0 ∂v^ 0 ∂ψ^ x ∂ψ^ y þ þz þ : ∂y ∂x ∂y ∂x

(9)

where k1 and k2 denote the necessary shear correction factors according to Reissner and Mindlin. The generalised Hookes law is used as the linear elastic material law for the kth layer 0

1

σ^ ðkÞ x B ðkÞ C B σ^ C B y C

0

e ðkÞ Q 11

B B e ðkÞ B Q 12 B C B B τ^ ðkÞ C B B yz C ¼ B 0 B C B B ^ ðkÞ C B B τ B xz C B 0 @ A @ τ^ ðkÞ e ðkÞ xy Q 16

e ðkÞ Q 12

0

0

e ðkÞ Q 16 e ðkÞ Q 26

e ðkÞ Q 22

0

0

0

e ðkÞ Q 44

e ðkÞ Q 45

0

e ðkÞ Q 45

e ðkÞ Q 55

e ðkÞ Q 26

0

0

1

0 1 C ε^ x CB CB ε^ y C C CB C CB C ^ C γ 0 CB yz C B CB γ^ C C xz C A 0 C@ A γ^ xy ðkÞ e Q 66

or

ðkÞ

σ^ ðkÞ ¼ Qe 55 ε^

(10)

e ðkÞ being the reduced stiffnesses for the kth layer. with the components Q ij For the description of the structural behaviour of a multilayered compound (see Fig. 3) the well-known structural law is used !  !  ^ A B ε^ 0 N ¼ (11) ^ B D κ^ M

Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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The extensional, coupling and flexural stiffnesses Aij, Bij and Dij are calculated on the basis of the CLT. Within the FSDT, additional shear stiffnesses are needed which are derived from Z h=2 N X e ðkÞ dz ¼ e ðkÞ ðz z ði; j ¼ 4; 5Þ: (12) Q Q Aij ¼ k k  1 Þ; ij ij  h=2

k¼1

N is the maximum number of layers and zk  1 denotes the lower border of the kth layer (see Fig. 3). 2.2. Modal and damping analysis The determination of free vibration fields in multilayered plate structures is done with respect to the special shear-elastic deformation characteristics of hybrid composite systems considering the large variety of different bearing conditions for anisotropic plates. The geometry and the global part coordinate systems of the analysed plate models are given in Fig. 4. The formulation of the main equations for the eigenfrequency and modal analysis is based on the Hamiltonian principle for conservative elastic systems. Here, the starting point is the Lagrange function b ðuðx; b uðx; ^ y; z; tÞÞ ¼ Tb ðuðx; ^ y; z; tÞÞ  Π ^ y; z; tÞÞ Lð

(13)

  T  ^ u; ^ v^ T u^ ¼ u^ z ; u^ x ; u^ y ¼ w;

(14)

with

b as the full elastic potential. as the time and spatial dependent deformation vector Tb as the kinetic energy and Π The kinetic energy of the analysed multilayered sandwich plate can be derived from  T Z   1 ∂u^ ∂u^ dV; ρðzÞ ¼ ρðkÞ for zk  1 rz r zk ; ρðzÞ Tb u^ ¼ 2 V ∂t ∂t 4

(15)

where ρðkÞ denotes the (constant) density of the kth layer. In the case of free vibrations and hyper-elasticity, the full elastic b is identical with the strain energy U b which is given by potential Π Z n o   b u^  ¼ U b u^ ¼ 1 (16) Π σ^ x ε^ x þ σ^ y ε^ y þ τ^ yz γ^ yz þ τ^ xz γ^ xz þ τ^ xy γ^ xy dV 2 V Within the Hamiltonian principle the deformation vector u^ is required, which gives a stationary value for the time integral of the Lagrange function Z t2 Z t2 h i b dt ¼ δ δTb  δUb dt ¼ 0: L (17) t1

t1

^ y; z; t 1 Þ ¼ δuðx; ^ y; z; t 2 Þ ¼ 0, (16) can be written with (14) by partial integration Considering the Lagrange variation δuðx; over time as

δUb þ

Z V

ρðzÞ

 2 T ∂ u^ δu^ dV ¼ 0: ∂t 2

(18)

^ y; z; tÞ are approximated for the stationary vibration state as timeHere, the free standing displacement fields uðx; harmonic displacements n u^ ðx; y; z; t Þ ¼ uðx; y; zÞeiωt ;

ω¼

2π : angular frequency TP

(19)

and satisfy the chosen boundary conditions. Considering the Lagrange variation and assuming harmonic time dependence ^ the following variational equation can be derived: for u, Z δU  ω2 ρðzÞuT δu dV ¼ 0 (20) V

b , ω the angular frequency, u the spatial part of u^ and V the plate volume. with U being the spatial part of U To solve this variational problem the Ritz approximation method is used, which finally yields the eigenvalue equation   (21) K  ΛM η ¼ 0: Here, K denotes the symmetric structural stiffness matrix, M the symmetric and positive definite mass matrix, η the eigenvector made of the Ritz coefficients and Λ ¼ ω2 the eigenvalues. The effects of material damping in anisotropic composites are included in the eigenfrequency and mode shape analysis by the use of the models of the theory of linear viscoelasticity. Here, the strain ε^ kl ðtÞ at the time t is related to the resulting 4

Here, the displacement vector u^ advantageously contains the z displacement u^ z as the first vector component.

Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Fig. 5. Dynamic material parameters for a UD single layer.

state of stress σ^ kl ðtÞ according to Boltzmann by the following material law: Z t ∂ε^ ðτÞ σ^ ij ðt Þ ¼ dτ ði; j; k; l ¼ 1; 2; 3Þ: C^ ijkl ðt  τÞ kl ∂τ 1

(22)

Using the Fourier transformation, this linear viscoelastic material law can be written in a complex valued form that is analogous to the generalised Hooke's law (10)

σ^ nij ðtÞ ¼ C nijkl ðωÞε^ nkl ðtÞ:

(23)

b ðtÞ in the The complex modulus functions C nijkl ðωÞ in the frequency domain are linked to the Hooke stiffness tensor C ijkl time domain by Z þ1 C nijkl ðωÞ ¼ iω (24) C^ ijkl ðξÞe  iωξ dξ; ξ ¼ t  τ: 0

The material law (23) can be transformed into the following vector–matrix form, which is applicable in this generalised form to different laminate theories (e.g., CLT, FSDT) 8 n e > ðCLTÞ e :Q ðFSDTÞ: 55

n

e of the complex stiffness matrix C e The elements C ij e 00 (loss moduli) by imaginary part C ij

n

e 0 (storage moduli) and the are separated into the real part C ij

e 0 þ iC e 00 : en ¼ C C ij ij ij

(26)

The storage and loss moduli for the global x–y–z coordinate system are calculated analogously to the corresponding elastic stiffnesses from the values for the fibre 1–2–3 coordinate system by polar transformation. The elements of the complex stiffness matrix are then built by the dynamic “engineering constants” and loss factors related to the fibre coordinate system (see Fig. 5). With these complex stiffnesses, the linear eigenvalue equation (21) becomes complex and can be written in the following form:  n  n K  Λ M ηn ¼ 0 (27)

Here, Λ denotes the complex eigenvalues and ηn the complex eigenvectors. The symmetric and complex structural e0 stiffness matrix Kn ¼ K0 þiK00 contains the complex stiffnesses. Thus, the real part K0 is connected with the storage moduli C ij 00 00 e . The symmetric and positive definite mass matrix M mainly and the imaginary part K is connected with the loss moduli C ij depends on the structural mass of the plate and consequently is real valued. e n are normally frequency dependent so that the In case of viscoelastic material behaviour, the complex stiffnesses C ij resulting complex eigenvalue equations become much more complicated to solve. Thus, the frequency range is usually split into limited frequency bands with frequency-independent material behaviour. In this way, the real material characteristics are incrementally approximated. A modal loss factor dg for the gth mode shape can be derived from the Z complex eigenvalues by using the following equation: n

dg ¼

Λ00g ðg ¼ 1; …; Z Þ Λ0g

(28)

Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Fig. 6. (Color online) Measuring setup for the resonance bending tests; left: resonance bending test stand, right: bearing and measuring device.

Fig. 7. Beam geometry used for resonance bending tests of composite materials.

Table 1 Material properties for different fibre-reinforced UD-layers. Property/Unit

CF/PEEK

CF/EP

GF/EP

GF/PP

GF/PA6

E0J /GPa E0? /GPa G0# /GPa ν J ? /– d J /% d ? /% d# /% ϱ/g/cm3

77.72 6.33 3.90 0.31 0.09 0.30 0.73 1.49

117.53 8.10 4.11 0.30 0.15 0.73 0.83 1.52

29.00 9.40 3.37 0.33 0.28 1.12 1.42 1.89

16.68 3.23 1.46 0.31 0.84 4.20 4.08 1.34

24.48 6.29 2.41 0.26 1.08 6.74 6.72 1.76

2.3. Experimental determination of dynamic composite material parameters The structural-dynamic design of hybrid anisotropic multilayered composites using the developed mechanical models requires the determination of validated dynamic composite material parameters. Here, the homogenised UD single layer is taken as the basis of the multilayered compound, the structural-dynamic behaviour of which is calculated and specifically designed using CLT or FSDT (see Section 2.1). For a material characterisation of each UD layer it is necessary to measure the dynamic elasticity parameters (engineering constants) and loss factors by adapted experiments. For that purpose, the authors used a resonance bending method that has been specifically adapted to fibre-reinforced composites (Figs. 6 and 7). The examinations of the directional-dependent dynamic material properties include tests on 01, 301, 451, 601 and 901 reinforced specimens. Testing of 01 and 901 specimens enables to determine the dynamic Young's modulus and material damping parallel and perpendicular to the fibre direction from the measured resonance frequencies and corresponding 3 dB-bandwidths. The shear modulus and shear damping is calculated using polar transformation relations (see, e.g. [50]). Based on this measurement setup, the material properties shown in Table 1 for several glass and carbon fibre-reinforced UD-laminates were determined.

2.4. Verification and comparison of the structural-dynamic simulation models The vibro-acoustic simulation models developed are verified using experimental and numerical techniques. For the experimental determination of the eigenfrequencies and mode shapes of the multilayered composite test plates, a laser scanning vibration interferometer (LSV) in combination with a multifunctional acoustic window test stand (AWS) was used (Fig. 8). In this AWS, the composite plate is mounted on special bearings inside a test window of a rigid wall. This wall separates the AWS into a rigid transmitter hall and a rigid receiver hall. Inside the transmitter hall, an acoustic sound source creates a diffuse sound field. The excited bending waves of the test plate are then analysed by the LSV inside the receiver hall. Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Fig. 8. Experimental setup of AWS and operating principle of LSV.

Fig. 9. Mode shapes of different multilayered composite plates measured by LSV (a) GF/EP ½ þ 30=  30= þ 60=  60s , mode (2,3), eigenfrequency 127.7 Hz; (b) CF/PEEK ½0=90=0=90= þ 45=  45=90=0=90=0s , mode (3,1), eigenfrequency 127.3 Hz; fibre orientation measured from the x-axis.

Fig. 10. (Color online) Comparison of calculated and measured eigenfrequencies; (a) GF/EP-MCP, (b) CF/PEEK-MCP.

Fig. 9 exemplarily shows measured mode shapes of a glass fibre epoxy (GF/EP) and a carbon fibre polyetheretherketon (CF/PEEK) multilayered composite plate (MCP). The geometry of the tested plates matches the window size of 900  600 mm. The numerical analysis of the experimentally investigated composite plates was performed by the finite element method (FEM) using the program system I-DEAS Master Series. The test plates have been modelled with the help of shell elements. Fig. 10 shows a comparison of the experimentally determined eigenfrequencies and mode shapes of these plates with the numerically and analytically calculated results. The analytical simulations were carried out using the described shear-elastic approach on the basis of the first order shear deformation theory (FSDT) as well as the classical laminate theory (CLT). Here, a good agreement between the experimental, numerical and analytical results has been achieved (for additional results see [20]). Thus, the performed verification clearly shows the accuracy of the developed structural-dynamic calculation model for multilayered anisotropic composites.

3. Sound radiation of anisotropic composite plates The full lightweight acoustic potential of multilayered anisotropic composites can only be exploited effectively if sound radiation is included in the design process. This vibro-acoustic design strategy should not only consider external excitation but also model the coupling between the structural wave field of the plate and the radiated sound pressure field inside the surrounding air volume. The structural-dynamic characteristics as well as the sound radiation behaviour of composite plates can only be specifically adjusted to the external dynamic loads if this coupling is part of the vibro-acoustic model. This paper presents a solution for the analytical modelling of the coupling conditions and the calculation of the radiated airwaves on Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Fig. 11. Composite plate with time-dependent surface load.

the basis of the developed analytical structural-dynamic simulation methods for multilayered composite plates. The radiated sound power and sound intensity are determined as practically oriented acoustic quantities. 3.1. Forced damped displacement and surface velocity fields The analysis of forced plate waves is performed on the basis of the continuum mechanics model of composite plates of length lx, width ly and thickness h on the surface z ¼ h=2 to which a time-dependent surface force P^ z ðx; y; z ¼  h=2; tÞ is applied (see Fig. 11). The external excitation of this composite plate creates a displacement field u^ S ðx; y; z; tÞ that requires the special displacement kinematics of anisotropic composite plates (see Section 2.1) to be taken into account. The Hamiltonian principle for conservative elastic systems is used analogously to the calculation of free standing plate waves. b which is composed of the kinetic energy Tb , the This variational principle is formulated by using the Lagrange function L b and the potential energy W c induced by the external surface load strain energy U b ðu^ S Þ  W c ðu^ S Þ: b u^ S Þ ¼ Tb ðu^ S Þ  U Lð b is given by The kinetic energy Tb and the strain energy U  T Z   1 ∂u^ S ∂u^ S dV; ρðzÞ Tb u^ S ¼ 2 V ∂t ∂t Z Z   e ε^ dV b u^ S ¼ 1 σ^ T ε^ dV ¼ 1 ε^ T C U 2 V 2 V Z T   1  e Gu^ S dV: Gu^ S C ¼ 2 V

(29)

(30)

(31)

Here, we use the real-valued deformation law analogously to (25) and the strain–displacement relations using the form

ε^ ¼ Gu^ S :

(32)

The differential operator matrix G is defined in the way that the strain–displacement relations for the CLT and FSDT can be written in a generalised form (see [20]). c of the external surface load is determined by The potential energy W Z lx Z ly T c ðu^ S Þ ¼  P^ u^ S dy dx W (33) 0

with the load vector

0

h i 8 ^ ^ > ðCLTÞ < P 11 ¼ P z h iT P^ ¼ > : P^ 31 ¼ P^ z 0 0 ðFSDTÞ:

By using the Hamiltonian principle, the Lagrange function (29) must satisfy the following equation: Z t2 Z t2 h i c dt ¼ 0: b dt ¼ δ δTb  δUb  δW L t1

(34)

(35)

t1

Assuming a harmonic time-dependent excitation frequency Ω of the load vector P^ and a complex-valued form of the standing displacement field   n h P^ x; y; z ¼  ; t ¼ Pðx; yÞeiΩt ; 2 n u^ S ðx; y; z; tÞ ¼ uS ðx; y; zÞeiΩt ;

(36)

Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Fig. 12. PSIB model for the calculation of the excited surface velocity field.

Eq. (35) can be transformed after integration over time into Z Z δU  Ω2 ρðzÞuTS δuS dV  V

0

lx

Z 0

ly

PT δuS dy dx ¼ 0:

(37)

This equation has to be solved by special functions for the displacement vector uS which fulfil the forced standing displacement field according to the given boundary conditions of the composite plate [20]. 3.2. Modelling of sound radiation The starting point for the calculation of the sound radiation of anisotropic and hybrid composite plates is the acoustic model of a plane sound source in an infinite rigid baffle (PSIB) (see Fig. 12). The plane sound source is the vibrating composite plate that is assumed to be acoustically excited by an idealised diffuse (ID) sound field. The literature shows a large number of theoretical and experimental investigations concerning diffuse sound fields. Here, the authors used a modelling based on the publications of Witting [51] and Schultz [52]. This method uses a mathematical description of the statistic properties of a diffuse sound field. This sound field creates a exciting n pressure P^ ex on a plane plate surface in the form of a constant surface load with harmonic time-dependence (see Fig. 12). The resulting surface velocity field of the plate creates a sound radiation into the two half-infinite air volumes by compressing the adjacent air. Here, the radiated sound field inside the positive half-space is calculated. This calculation is done for the low frequency range using the well-qualified assumption that the exciting pressure of the ID sound field on the surface of the plate (length lx; width ly; thickness h) can be approximated as a spatially constant harmonic surface load n P^ ex ðx; y; z ¼ h; tÞ  P A eiΩt

for 0 r x rlx and 0 ry rly :

(38)

n

The resulting surface velocity field v^ S of the composite plate can be derived from z 9 8 ðgÞ n M Rz X N Rz Z =
g

g

(39)

g

with R

I ng ¼ lx ly P A

R

Mz X Nz X m¼1n¼1

nðgÞ ηmnz

Z 0

1

Z

1 0

Φxzm ðx○ ÞΦyzn ðy○ Þ dx○ dy○ ;

J ng ¼ ηngT Mηng

(40)

(41)

The functions Φxzm and Φyzn are the corresponding beam solution and are used to build the Ritz approximation of nðgÞ the mode shapes (solution level: MRz in x-direction, NRz in y-direction). The Ritz coefficients ηmnz are determined as the n z-component of the gth eigenvector ηg (see Section 2.2). The surface velocity field of the rigid wall is zero due to the total reflection of the incident ID sound waves. Thus, the total surface velocity distribution in the infinite plane z ¼ 0 is given by (see Fig. 12) ( n ðplateÞ; v^ S ðx; y; tÞ: 0 r x rlx ; 0 r y r ly n z v^ z ðx; y; z ¼ 0; tÞ ¼ (42) 0 : x o 0; x 4 lx ; y o0; y 4 ly ðwallÞ: This structural surface velocity field of the rigid wall and the vibrating composite plate is transferred into the adjacent air volume of the positive half-space by vibro-acoustic coupling. This coupling is formulated as a boundary condition for the n n plane z ¼ 0 by setting the zcomponent v^ z;air of the air velocity equal to the structural surface velocity v^ z n n v^ z;air ðx; y; z ¼ 0; tÞ  v^ z ðx; y; z ¼ 0; tÞ:

(43)

Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Table 2 Investigated multilayered composites. Name

Lay-up

No. of layers

Thickness in mm

GF/EP-MCP CF/EP-MCP CF/PEEK-MCP

½ þ 30=  30= þ 60=  60s ½ þ 45=  45=90=0=0=90=  45= þ 45s ½0=90=0=90= þ 45=  45=90=0=90=0s

8 16 20

2.11 4.1 2.78

Fig. 13. (Color online) Qualitative comparison of the calculated and measured surface velocity fields for different plates. n Here, the surface velocity v^ z of the wall and the composite plate, respectively, is given by (42). Assuming that the n radiated airwave propagates as an undisturbed plane wave into the positive half-space, the air velocity v^ z;air inside the total positive half-space z Z 0 can be derived from n n v^ z;air ðx; y; z; tÞ ¼ v^ z;air ðx; y; z ¼ 0; tÞe  ikz z

(44)

by using (39), (42) and (43). n The corresponding sound pressure P^ is then calculated from the linear Euler equation of motion n v^ z;air ðx; y; z; t Þ ¼ 

n

∂P^ ðx; y; z; tÞ : iΩρ0 ∂z 1

(45)

This pressure field must satisfy the wave equation for all points in the positive half-space n

ΔP^ 

n

1 ∂2 P^ ¼ 0: c20 ∂t 2

(46)

With these basic equations, the time averaged sound power, which is radiated through an infinite plane z ¼ z 0 in the positive half-space, is given by

Z þ 1 Z þ 1  1 P¼ R P n ðx; y; z ¼ z 0 Þvz;n ðx; y; z ¼ z 0 Þ dx dy (47) air 2 1 1

3.3. Experimental and numerical verification The semi-analytical models developed for the calculation of the modal sound radiation of acoustically excited multilayered composite plate structures have been verified by experimental and numerical investigations using practically oriented laminate lay-ups (see Table 2). This analytical vibro-acoustic simulation model has been implemented into the Cþ þ-program DEVAM that allows effective parameter studies for the sensitivity analysis of the influence of different textile-specific material properties on the sound radiation. 3.3.1. Experimental verification Starting point of the experimental verification is the determination of the structure-borne surface velocity fields of the investigated plates. Fig. 13 displays for chosen eigenfrequencies the analytically calculated surface velocity fields (left) and the via laser scanning vibration interferometry determined surface velocity fields (right). The measured surface velocity fields show a very good agreement of the areal distribution compared to the calculated surface velocity fields (see also [20]). The calculated and the measured eigenfrequencies show differences below 10 Hz. This Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Fig. 14. (Color online) Measured and calculated sound power levels (left) and eigenfrequencies (right) of different multilayered composites.

might be caused by a variation of the material parameters (e.g., dynamic Young's modulus) between the tested composite plates and the specimens used for the determination of the material data. The developed vibro-acoustic model uses the radiated sound power level of plate structures made of different fibrereinforced composite materials. The scalar energy quantity sound power as a spatially integral value is well suited for a quantitative comparison of the sound radiation of different materials. This sound power level can experimentally be determined in a very good manner using the sound intensity measurement technique in combination with a vibro-acoustic window test stand [20,53]. This measurement technique allows to determine the modal sound power levels of specific mode shapes. Fig. 14 shows a comparison of the measured and calculated sound power levels of the investigated composites for the (1,3) and (3,1) mode shapes.5 The differences between the measured and calculated sound power levels are between 0.4 and 1.7 dB, which indicates a very good agreement between the calculation model and the experiment. The eigenfrequencies show a difference up to 9 percent between calculation and measurement due to possible influences of plate manufacturing. 3.3.2. Numerical verification The finite element and boundary element method was used to verify the analytical vibro-acoustic model. Here, the finite element structural-dynamic model provides the modal parameters, such as eigenfrequency and modal damping, and the boundary element model calculates the radiated sound power on the basis of the exciting surface loads. Table 3 shows the comparison of the numerically and analytically determined sound power levels for the first mode shape for the chosen multilayered composite plates as well as the corresponding eigenfrequencies. Furthermore, a comparison of the calculated sound power levels of higher modes was done (see Table 4). The analytical model shows very small differences for the first three modes. For the fourth mode (3,3) a larger difference between the numerical and analytical calculation of the sound power level occurs. Here, the chosen number of the used ansatz functions for the Ritz solution (3  3) is a limiting factor for the accuracy of the analytical model for higher frequencies. Nevertheless, using high numbers results in high computational time even for the analytical solution. Thus, the authors have chosen a number of ansatz functions limited for the lower modes, which are of practical relevance due to the higher amplitudes and sound radiation. This opens the possibility to use the analytical model with its – compared to the numerical solution – short calculation times as a pre-design tool to evaluate a first set of composite-specific material parameters for a latter detailed numerically based design. 4. Vibro-acoustic parameter studies The sensitivity of the composite-specific design variables (e.g., fibre orientation, fibre matrix combination and lay-up) has been analysed by extensive analytical vibro-acoustic parameter studies [20,44]. Due to the many possibilities of combining the different design variables, this paper focuses on examples of these parameter studies with practical relevance for the calculation engineer. The vibro-acoustic analyses were based on symmetrical and balanced multilayered fibre-reinforced composite plates with glass and carbon fibres as well as different thermoset and thermoplastic matrix systems. The used material properties of the single UD-layer are taken from Table 1. 4.1. Influence of the composite material on the vibro-acoustic behaviour Fig. 15 shows the dependence of the vibro-acoustic properties eigenfrequency f1, modal loss factor d1 and radiated modal sound power LW jf ¼ f 1 of the first mode shape on the laminate angle φ of a rectangular plate. Additionally, the composite material was varied. Here, the authors investigated epoxy (EP) as matrix material in combination with glass (GF) and carbon fibres (CF) as well as polypropylene (PP) and polyamide 6 (PA6) as matrix material in combination with glass fibres. 5

The necessary load cases for the analytical calculation have been determined on the basis of scanning laser vibration measurements [20].

Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Table 3 Comparison of numerical and analytically determined sound power levels and eigenfrequencies (mode (1,1)). Name

Sound power levels (dB)

GF/EP-MCP CF/EP-MCP CF/PEEK-MCP

Eigenfrequency (Hz)

FEM/BEM

Analytical

FEM

Analytical

31.36 49.54 59.73

31.47 49.29 59.70

23.5 80.4 52.3

23.5 79.9 53.5

Table 4 Comparison of numerical and analytically determined sound power levels and eigenfrequencies of different mode shapes of a GF/EP-plate. Mode

(1,1) (3,1) (1,3) (3,3)

Sound power levels (dB)

Eigenfrequency (Hz)

FEM/BEM

Analytical

Difference

FEM

Analytical

31.36 23.91 24.11 41.46

31.47 23.78 24.07 37.99

0.11 0.13 0.04 3.47

23.5 65.4 101.4 149.1

23.5 65.5 100.9 149.2

Fig. 15. (Color online) Polar diagrams showing the influence of fibre and matrix material of unidirectional composite plates on the vibro-acoustic properties f1, d1 and LW jf ¼ f 1 .

Fig. 16. (Color online) Influence of the fibre angle difference αbetween two layers in the case of asymmetric cross-ply lay-ups.

Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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Fig. 17. (Color online) Influence of the thickness on the eigenfrequency, modal damping and sound radiation of the first mode shape of a plate clamped on all edges.

The chosen fibre material (GF or CF) has significantly more influence on the eigenfrequencies than the matrix material. This is caused by the high impact of the chosen fibre material on the dynamic Young's modulus or composite stiffness and thus the eigenfrequencies. In contrast, damping is dominated by the matrix material. Matrices with high material damping values – especially thermoplastic matrices like PP or PA6 – cause high modal loss factors over the whole angle range. The modal radiated sound power levels show a characteristic curve that is dominated by the damping properties. Here, fibre angles, which cause high modal loss factors, lead to low sound power levels even if these orientations are connected with low eigenfrequencies. All examined materials show this ‘damping dominated behaviour’. Consequently, the use of composite materials with high damping values results in low modal sound power levels. Fig. 16 shows the fibre-dependent characteristic of the eigenfrequency, modal damping and sound radiation of a UD and an asymmetric cross-ply laminate. The characteristics of the vibro-acoustic parameters of the UD composites show a typical strong anisotropy, whereas the characteristics of the asymmetric cross-ply composites exhibit a typical bi-directional characteristic. Generally, the direction-dependent dynamic material parameters of the composites create an anisotropy of the vibroacoustic properties eigenfrequency, modal damping and radiated sound power. Furthermore, the plate geometry complexly interacts with this material anisotropy. As a result, the detailed direction-dependence of the radiated sound power has to be calculated for each individual case.

4.2. Influence of the plate geometry (plate thickness) In order to investigate the vibro-acoustic influence of the plate geometry on the modal parameters eigenfrequency, modal damping, and modal sound radiation this paper focusses especially on the plate thicknesses, which gives the possibility to derive basic design vibro-acoustic advices for this major geometry design variable (for further geometric influences see [20,54]). The influence of the plate thickness on the vibro-acoustic parameters eigenfrequency f1, modal damping d1 and radiated sound power level LW jf ¼ f 1 is shown in Fig. 17. Here, plates with a constant length-to-width ratio lx ly ¼ 860 =560 made from several unidirectional reinforced GF/EP-composites or aluminium are considered. Furthermore, either the mass m (blue curves in Fig. 17) or the surface area A (yellow and red curves in Fig. 17) of the plate are kept constant. The calculations show an increasing eigenfrequency with increasing plate thickness for all materials investigated. In contrast to this, there is no influence of the plate thickness on the resulting modal damping as the mode shape is not influenced by the thickness. Caused by the increasing plate stiffness, the modal sound radiation decreases with increasing plate thickness assuming a constant acoustic excitation. In case of the composite plates made from GF/EP (blue curves), a change of the fibre orientation leads to a shift of the resulting curves. The stiffness increases with increasing fibre angles, while the modal damping values decrease. The disproportional decrease of the modal damping compared to the increase of stiffness leads to an increasing sound power level with increasing fibre angle. Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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O. Täger et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

.

.

.

.

.

.

Fig. 18. (Color online) Influence of the fibre orientation and the number of layers on the vibro-acoustic properties of the first mode shape of a rectangular CF/EP-plate.

The plate with constant surface area made of GF/EP (yellow curve) shows a lower sensitivity to changes of the plate thickness (i.e., a lower slope) compared to the plate with constant weight (blue curves). Plate thicknesses above the intersection of the yellow and the blue solid curves (both with a fibre orientation of 01) result in a smaller sound radiating surface for the plate with a constant weight than for the plate with a constant surface area. Hence, the assumed constant excitation pressure causes an apparent increase in the modal sound power level of the plate with constant surface area. The curves of eigenfrequency, modal damping and radiated sound power level of the aluminium plate (red curve) are parallel to the curves of the corresponding GF/EP plate (yellow curve). In fact, the aluminium plate has a higher mass per unit area as well as higher eigenfrequencies; however, caused by the low modal damping values, the plate shows higher sound power levels.

4.3. Influence of a set of multiple parameters – number of layers and fibre angle The analysis of the influence of different composite-specific design variables on the vibro-acoustic properties of composite structures can easily be performed by using the presented analytical approach. Furthermore, the physical dependencies of the design variables can be displayed in problem-oriented surface plots of the eigenfrequency, modal damping and sound radiation. This allows to generate a better understanding of the inherent physical effects. As an example, Fig. 18 shows the influence of the number of composite layers and their corresponding fibre angle on the resulting vibro-acoustic properties eigenfrequency, modal damping and sound radiation. Numerous calculations with different sets of these design variables using the developed analytical model are the basis of the presented results. As expected, an increasing number of layers leads to an increase of the eigenfrequency and thus to an increase of the composite stiffness. Furthermore, the influence of the fibre orientation on the eigenfrequency is higher for composites with a higher number of layers (higher total thickness). In contrast, the composite modal damping is found to be only slightly influenced by the number of layers, if the number of layers is above four (n 4 2). The sound radiation is calculated to be at minimum for composites with a high number of layers. Thus, the sound radiation is mainly determined by stiffness and mass effects. The fibre orientation is of minor influence for the sound radiation of thick-walled composite plates. In summary, this paper demonstrates that the radiated sound power level of the investigated GF/EP and CF/EP composite plates decreases with increasing plate thickness. Higher fibre angles result in higher sound power levels for plates of the same thickness due to the significant lower modal damping values. In case of a constant plate surface area, the sensitivity of the radiated sound power level as a function of the plate thickness decreases compared to the plates with a constant weight. In contrast to the aluminium plates, the composite plates show significantly lower sound radiation caused by the higher modal damping values.

5. Conclusions The increasing need to develop lightweight solutions with integrated functions using cost- and weight-effective design approaches results in an increased use of high-performance fibre-reinforced composites. Here, the increasing customer demands for comfort and safety require not only highly specific lightweight characteristics but also reduced vibrations and sound radiation. Thus, the main precondition for the development of load-adapted and acoustically optimised lightweight structures is a holistic design which includes direction-dependent stiffnesses and material damping values as well as an adjusted sound radiation. This vibro-acoustic design strategy is provided by the material-adapted simulation methods developed for multilayered anisotropic composites. These analytical vibro-acoustic simulation methods allow a quick determination of the structural-dynamic and acoustic property profile in dependence of the main composite specific design parameters. Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i

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The parameter studies performed clearly reveal the complicated physically based coupling of the manifold dynamic and composite-specific design variables. Therefore, a detailed structural-dynamic and acoustic analysis is necessary for a material-adapted vibro-acoustic design of fibre-reinforced composites. Here, the analytical vibro-acoustic simulation model developed is a useful tool for the calculation engineer in order to exploit the great lightweight acoustic potential of composite structures. The paper presents several new results of which the most important are:

 An analytical model for the calculation of the vibro-acoustic modal parameters eigenfrequency f, modal damping d and modal sound radiation level LW of composite plate structures was developed and verified using FEM and measurements.

 Composite-specific dynamic material properties were determined. For this purpose, a bending resonance test stand and the appropriate measuring procedure were adapted to fibre-reinforced materials.

 The direction-dependent material properties lead to direction-dependent vibro-acoustic properties of the composite structures as well.

 Achieving low sound power levels requires the adjustment of high damping values. This can be called ‘dampingdominated sound radiation’ behaviour.

 The acoustic properties of fibre-reinforced plate structures are a result of the complex interaction between material and geometry parameters, the design of which requires the use of powerful material-adapted tools. The developed analytical vibro-acoustic model can be the basis for a hybrid analytical and numerical vibro-acoustic design concept. Here, the fast computing analytical approach can be used to determine a first well suited set of compositespecific material parameters, which can serve as an input for a detailed numerical analysis. Consequently, the number of necessary numerical calculations can be significantly reduced by this hybrid design approach. References [1] Kamal Idrisi, Marty E. Johnson, Daniel Theurich, James P. Carneal, A study on the characteristic behavior of mass inclusions added to a poro-elastic layer, Journal of Sound and Vibration 329 (20) (2010) 4136–4148. 10.1016/j.jsv.2010.04.001 ISSN 0022–460X. [2] Kamal Idrisi, Marty E. 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Please cite this article as: O. Täger, et al., Analytical study of the structural-dynamics and sound radiation of anisotropic multilayered fibre-reinforced composites, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2014.12.040i