Multi-focus design of underwater noise control linings based on finite element analysis

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Applied Acoustics 69 (2008) 1141–1153 www.elsevier.com/locate/apacoust

Multi-focus design of underwater noise control linings based on finite element analysis S.N. Panigrahi, C.S. Jog, M.L. Munjal * Facility for Research in Technical Acoustics, Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India Received 9 May 2007; received in revised form 24 November 2007; accepted 30 November 2007 Available online 28 January 2008

Abstract Varied, counter-demanding objectives in designing the underwater noise control linings are addressed using a finite element model based methodology. Four different kinds of designs are proposed to attend to diverse and conflicting requirements concerning echo reduction (ER) and transmission loss (TL) performance of these linings. In this regard a slightly modified hybrid type finite element based on the Pian and Tong (PT) formulation has been used to make the computational efforts less demanding as compared to the original one. The adequacy of this formulation has been shown by comparing its results with the analytical, finite element analysis based, and experimental results. Different unit cell representations for different types of distributions of air cavities on the linings are discussed with respect to their limitations and applicability. Effect of static pressure is studied by using a simplified technique which can be used to simulate deep sea testing environment. Performance variation of different designs is investigated under different water depths to study their applicability in such situations. Ó 2007 Elsevier Ltd. All rights reserved. PACS: 43.30.Ky; 43.20.Fn Keywords: Underwater acoustics; Noise control; Anechoic lining; Insulation lining; Combination lining

1. Introduction Sound absorbing and insulating linings find numerous applications in the field of underwater acoustics. To create a free-field underwater environment for acoustic and vibration measurements on submerged bodies and structures, specially designed viscoelastic noise control linings can be fitted in an acoustic tank to reduce the acoustic reverberation. Two parameters are used to quantify two different phenomena of importance related to such linings. Echo reduction (ER) quantifies the effectiveness of the lining not to reflect any sound wave impinging on it, and transmission loss (TL) measures the efficiency of the lining not to transmit any sound wave through it.

*

Corresponding author. Tel.: +91 80 2293 2303; fax: +91 80 2360 0648. E-mail address: [email protected] (M.L. Munjal).

0003-682X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2007.11.012

Analysis and design of these linings containing circular air cavities started with the works of Oberst [1] and Mayer [2]. Oberst used a lumped system approximation in which cylindrical cells are analyzed on the assumption that they represent the entire lining due to symmetry of the distribution of these cavities, by evaluating approximate equivalent mass, stiffness and compliance of different portions of the circular cell. Gaunaurd [3,4] presented a one-dimensional analysis method for such layers and provided a physical explanation based on different modes of resonances with different physical dimensions of the cavities and those of the layer. Fiorito et al. [5] studied interaction of sound waves with these viscoelastic linings and gave analytical expressions for transmission and reflection of sound waves through such linings without any air cavities. More complex configurations with cavities in the coating have been analyzed using analytical methods [6–8]. Jackins and Gaunaurd [6] used their resonance scattering theory to

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study the effect of bi-laminar coatings where each layer was assumed to contain uniform clouds of air bubbles of known, different, and small concentrations. Specifically, they also analyzed the case of sound reflection by two pure rubber layers ideally bonded to each other. Gaunaurd et al. [7] then investigated the effect of hydrostatic pressure on the dynamic effective properties of such air filled layers. Strifors and Gaunaurd [8] have then studied the selective reflectivity of such bi-laminar plates and concluded that the coating should always be analyzed with the structure behind the coating. They also emphasized that it is unrealistic to expect very strong echo reduction over a very large frequency band, and the effective frequency band depends upon the right size of the air bubbles among other factors such as material properties, the backing structure and the surrounding media. Hennion et al. [9] then used a three dimensional finite element method to analyze these Alberich linings as a problem of scattering of plane wave by a doubly periodic elastic structure and corroborated their method with experimental results. Then, Easwaran and Munjal [10] presented a 3D FEM based analysis for these resonator linings. Recently Berry et al. [11] have used a 3D elasticity model to analyze the transmission of sound waves through such linings without any air channels under the influence of a heavy fluid. In all these works, however, only a single aspect of the noise control problem has been addressed. Though Easwaran and Munjal [10] have shown the linings to be very effective with respect to the echo reduction, the other aspect, that of transmission reduction, has not been taken care of. For use in such linings, certain class of materials (some special type of viscoelastic materials) have been developed [12–14] and are being used for the vibration as well as sound absorption. Their properties have also been studied and improved extensively to fit the exact requirements of these linings. These are generally rubbery materials, almost incompressible in nature. In order to minimize the reflected and the transmitted portions of sound energy, the absorption has to be maximized. This is achieved by using the material damping property [15] of the viscoelastic materials of the lining. A different perspective of designing of underwater noise control linings is taken up in this work using an inherently stable finite element formulation. Unlike the uni-focus design available in the literature, the four types of designs proposed here are for addressing two different types of

objectives and a combination thereof. The two main parameters of interest are the echo reduction and the transmission loss of the lining. It is a well known fact that the standard displacement based iso-parametric brick element performs poorly when it is highly distorted and/or when the material being analyzed is almost incompressible. Hence, a hybrid type brick element (proposed by Pian and Tong [16]) has been chosen and implemented with slight modification resulting in some computational savings without any loss in accuracy. 2. Proposed layer designs On analyzing a submerged steel plate in water [5], very low values of ER and TL are observed (in the frequency regions where they are crucial). Even the effect of padding the steel plate by thick viscoelastic material with good amount of inherent damping is not substantial. Relative values of the bulk modulus K and the shear modulus G of these materials ðK  500 GÞ [15] suggest that by allowing the volume deformation to transform into shear deformation by providing small air channels, Fig. 1a, a significant reduction of the effective elastic modulus of the layer can be achieved [2]. When it comes to the transmission of sound waves, the air pockets can be made larger (Fig. 1b) in order to bring the impedance mismatch and resonance into play, and good insulation of sound can be achieved in the lower frequency range. To have a compromise between these two situations, the natural generalization is to combine these two layers. The first configuration is called a coupled type combination lining, Fig. 2a, because the air channels in the two layers are connected to each other. The other type of lining has been obtained by putting these two layers on the opposite sides of the steel wall and this configuration is called the decoupled type combination lining shown in Fig. 2b. 3. Methodology To get the ER or TL characteristic of any of the above design under the water closure condition, an acoustic pressure wave of certain magnitude and frequency should impinge on the layers. Then the relative magnitude of the reflected and the transmitted waves can be evaluated to characterize the desired properties. To perform this analysis, however, the distribution of the air cavities has to be

Fig. 1. Two different types of designs for two dissimilar requirements; (a) anechoic layer and (b) insulation layer.

S.N. Panigrahi et al. / Applied Acoustics 69 (2008) 1141–1153

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Fig. 2. Two different types of designs for a balanced performance; (a) coupled type combination layer and (b) decoupled type combination layer.

taken into account while modelling the finite element domain. Generally, two different kinds of arrangements are made for placing the air cavities in the lining. As can be seen from Figs. 3a and b, the holes can be laid in a doubly periodic/ parallel fashion or in a staggered fashion or in a mixed hole pattern with different sizes of holes (see Fig. 3c). Therefore, depending on the distribution type, different shapes of domain which can represent the complete extent of the lining are selected for the FE analysis to exploit the symmetry in the pattern to reduce the computational effort. Modeling the vibro-acoustic phenomena of the resonator linings involves solution of a set of partial differential equations [10] satisfying a set of boundary conditions at some complex boundaries. Because of the symmetric nature of the unit cell the normal displacements on all the side surfaces are restricted. The analysis of these viscoelastic linings follows the steps enumerated below. 3.1. Steps involved (i) Considering symmetry of the distribution of perforations, a portion of the lining is selected which can represent the total domain when considered with appropriate boundary conditions. (ii) This unit cell domain is meshed with 8-noded elements for analysis. Formulation of the finite element follows this discussion. (iii) The mass, stiffness and the damping matrices are formed by discretizing the weak formulation of the system differential equations of motion while taking into account the proper material properties [12–14] and their frequency dependence.

(iv) Proper displacement boundary conditions are applied taking into account the symmetry of the domain [10]. Impedance boundary condition is applied on the water closure side. By applying this impedance boundary condition the semi-infinite water domain on the down-stream side is simulated. (v) A unit pressure loading is applied on the outer lining surface to simulate the impinging sound pressure. The actual amplitude of this pressure is not important as the ratio of the amplitude of the reflected or transmitted wave to that of the impinging wave is the actual measure of the effectiveness of the lining in echo reduction or transmission loss. (vi) The complex nodal displacement values are obtained by solving the equation of motion. (vii) The corresponding complex nodal pressure values and particle velocities are evaluated by using the above results. All these values obtained correspond to those of the standing waves existing in the domain. (viii) The particle velocities in the direction perpendicular to the surface of the wall (in the wave propagation direction) are averaged to obtain the average particle velocity in the wave propagation direction. (ix) Acoustic impedance in the wave propagation direction at the impinging surface is then obtained by: fin ¼ pst =vz

ð1Þ

where pst is the input pressure (unit standing wave pressure) at the lining surface and vz is the average particle velocity along the direction of wave propagation (considered to be the z-axis of the global co-ordinate system).

Fig. 3. Representative unit cells for a (a) parallel, (b) staggered and (c) mixed hole distribution of air cavities.

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(x) Using this acoustic impedance and the characteristic impedance of water, the reflection coefficient of the lining, defined as ratio of the pressure amplitudes of the reflected and the incident waves, is calculated by using the formula      p  f  Y w   R ¼  ref  ¼  in ð2Þ p f þY  inc

in

w

where Y w is the characteristic impedance of the water medium, and is given by qw cw , where qw and cw are the mass density and the sonic speed of the water medium. (xi) From the reflection coefficient of the lining, the two relevant parameters, i.e., the ER and the TL, are obtained by using the following formulas:     1 pinc    ð3Þ ER ¼ 20log10   ¼ 20log10   R p  ref    pst  TL ¼ 20log10  ð4Þ ð1 þ RÞptrans  where ptrans is the transmitted standing wave pressure, pinc is the incident pressure wave component, and pst is the standing wave pressure at the lining surface.

where c and sc represent the engineering strains and stresses, b are the body forces, t are the prescribed tractions, C is the material constitutive matrix, and v and rc represent the variations of the displacement and stress fields, respectively. Let the interpolations be given by u ¼ N u ^u; v ¼ N u^v;

sc ¼ P^s;

ð7Þ

r: rc ¼ P^

ð8Þ

Substituting these interpolations into Eqs. (5) and (6), and using the arbitrariness of the variations, one gets the matrix equations      H G ^s g^ ¼ ^ ; ð9Þ t ^u G 0 f where Z Pt C 1 P dX; H¼ X Z G¼ Pt B u dX; X

g^ ¼ 0; Z Z N tu b dX þ N tut dC: f^ ¼ X

In the aforementioned methodology, the actual physical problem has been approximated with the assumption that the plane sound waves impinge on the lining-coated infinite flat steel plate with normal incidence.

Ct

In the form given above, the N u shape functions have to be C 0 continuous, though P can be discontinuous. Hence, ^s ¼ H 1 G^u can be eliminated at the element level to yield K^u ¼ f^;

4. Finite element modelling of the domain The unit-cell described above, has to be discretized with finite elements. Moreover, it is a well-known fact that the standard iso-parametric Q8 brick element performs poorly when it is distorted and when the material being analyzed is almost incompressible. Since both these situations arise in the analysis of the above mentioned linings, the hybrid brick element that is relatively immune to the above-mentioned shortcomings has been employed for the analysis. It is a slightly modified version of the one proposed by Pian and Tong [16]; therefore, this element henceforth has been referred to as Modified PT element [17]. The PT element has been modified in a manner that requires inversion of a 12  12 matrix instead of an 18  18 matrix as in the PT element (without any loss in accuracy), thus resulting in some computational savings. Details of the formulations are discussed below.

ð10Þ t

1

where K ¼ G H G. It can be noted that unlike a standard iso-parametric formulation, a matrix inversion is required to form the element stiffness matrix. The accuracy of the element hinges crucially on the choice of shape functions P, a detailed discussion of which can be found in Ref. [17]. Let the master element be as shown in Fig. 4. The

4.1. Formulation of the modified PT element The PT hybrid element is based on a two-field HellingerReisner variational formulation. The variational equations for the static problem are Z Z Z tc ðvÞsc dX ¼ v  b dX þ v  t dC 8v ð5Þ X X Ct Z ½rtc c ðuÞ  rtc C 1 sc  dX ¼ 0 8rc ; ð6Þ X

Fig. 4. Node and face numbering of the master element for the 8-noded finite element formulation.

S.N. Panigrahi et al. / Applied Acoustics 69 (2008) 1141–1153

Jacobian evaluated at the origin ðn; g; fÞ ¼ ð0; 0; 0Þ is given by 2 ox oy oz 3 2 3 a1 b1 c 1 on on on 6 ox oy oz 7 6 7 7 ¼ 4 a2 b2 c 2 5 ; J0 ¼ 6 ð11Þ 4 og og og 5 oy ox oz a3 b3 c 3 of

of

of

ðn;g;fÞ¼ð0;0;0Þ

2

a 1 b1

a2 b2

1145

a3 b3

3

6 7 N21 ¼ 4 b1 c1 b2 c2 b3 c3 5; a1 c 1 a2 c 2 a3 c 3 2 3 ða1 b2 þ b1 a2 Þ ða2 b3 þ a3 b2 Þ ða1 b3 þ a3 b1 Þ 6 7 N22 ¼ 4 ðb2 c1 þ b1 c2 Þ ðb2 c3 þ b3 c2 Þ ðb1 c3 þ b3 c1 Þ 5: ða1 c2 þ a2 c1 Þ ða3 c2 þ a2 c3 Þ ða3 c1 þ a1 c3 Þ

where In what follows, Xe denotes the domain of the element, R and V e ¼ Xe dX denotes the volume of the element. Let

1 a1 ¼ ðx1 þ x2 þ x3  x4  x5 þ x6 þ x7  x8 Þ; 8 1 b1 ¼ ðy 1 þ y 2 þ y 3  y 4  y 5 þ y 6 þ y 7  y 8 Þ; 8 1 c1 ¼ ðz1 þ z2 þ z3  z4  z5 þ z6 þ z7  z8 Þ; 8 1 a2 ¼ ðx1  x2 þ x3 þ x4  x5  x6 þ x7 þ x8 Þ; 8 1 b2 ¼ ðy 1  y 2 þ y 3 þ y 4  y 5  y 6 þ y 7 þ y 8 Þ; 8 1 c2 ¼ ðz1  z2 þ z3 þ z4  z5  z6 þ z7 þ z8 Þ; 8 1 a3 ¼ ðx1  x2  x3  x4 þ x5 þ x6 þ x7 þ x8 Þ; 8 1 b3 ¼ ðy 1  y 2  y 3  y 4 þ y 5 þ y 6 þ y 7 þ y 8 Þ; 8 1 c3 ¼ ðz1  z2  z3  z4 þ z5 þ z6 þ z7 þ z8 Þ: 8 2

1 60 6 6 60 ~ P¼6 60 6 6 40 0

f1 ¼ n  n; f 2 ¼ g  g; f 3 ¼ f  f;    f4 ¼ ng  ng; f 5 ¼ gf  gf; f 6 ¼ nf  nf; where ð12Þ

 ¼ ng

n dX

R ; g ¼

Ve ng dX Xe Ve

0 0 0 0

0 0

f2 0

f3 0

0 f1

0 f3

0 0

0 0

0 0

0 0

0 0

f5 0

0 f6

0

1

0 0

0

0

0

0

0

f1

f2

0

0

0

0

0

0 0

0 0

1 0 0 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

f3 0

0 f1

0 0

0 0

0 0

3 0 07 7 7 f4 7 7; 07 7 7 05

0

0

0 0

1

0

0

0

0

0

0

0

0

f2

0

0

0

3 a23 7 b23 5;

c21

c22

c23

g dX

Ve R Xe

R ; f ¼

gf dX Ve

Xe

f dX

Ve

 ¼ ; nf

;

R

nf dX

Xe

Ve

ð15Þ ;

~ b, ~ The stress distribution ~sc is assumed to be given by P where

0 0

a22 b22

Xe

 ¼ ; gf

0 1

a21 6 2 N11 ¼ 4 b1 2

Xe

R

ð13Þ 2

R n ¼

The stresses are assumed to be given by s ¼ J t0~sJ 0 , where ~s are the stress components based on the natural coordi~ b, ~ then the internates n, g and f. If ~sc is interpolated as P polation for the engineering stress components can be ~b ^ where written as sc ¼ T P   N11 N12 1 T ¼ ½ T1 T2 T3 T4 T5 T6  ¼ ; 2=3 N21 N22 jJ 0 j and

ð14Þ

ð16Þ

which in turn leads to the following assumed stress distribution for the physical components of stresses: 2 3 3 2 sxx b1 6s 7 7 6 yy 7 6 6 7 6b 7 6 szz 7  6 2 7 6 7 ¼ Pb ¼ I 66 P612 6 . . . 7; ð17Þ 7 6s 7 6 7 6 xy 7 6 6 7 4... 5 4 syz 5 b18 sxz where  P612 ¼

2a1 a2 6 N12 ¼ 4 2b1 b2

2a2 a3 2b2 b3

3 2a1 a3 7 2b1 b3 5;

2c1 c2

2c2 c3

2c1 c3

f2 T 1

f3 T 1

f1 T 2 f1 T 5

f3 T 2 f2 T 6

f1 T 3 f5 T 1

f2 T 3 f6 T 2

 f3 T 4 : f4 T 3 ð18Þ

The difference in the above assumed distribution as compared with PT is that in PT, the coefficients f1 ; . . . ; f6 given by Eq. (14) are defined without the ‘overbar’ terms

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(i.e., f1 ¼ n, f2 ¼ g, and so on). This makes it necessary to invert an 18  18 matrix in PT, whereas in the current formulation, one needs to invert a 12  12 matrix while computing the element stiffness matrix as shown below. Let S ¼ C 1 be the material compliance matrix, and B be the strain-displacement matrix, i.e.,  ¼ B^ u. Assuming that C is constant over each element, and as a result of the assumed stress distribution, the element stiffness matrix and stress distribution are given by K e ¼ G te H 1 e G e;

ð19Þ

Pe H 1 u; e G e^

ð20Þ

se ¼

where (omitting the subscript ‘e’ for " Z ½V e S66 t H 1818 ¼ Pe SPe dX ¼ 0126 Xe

G 1224

convenience) 0612 R ½ Xe Pt SP dX1212

" R # ½ Xe B dX624 ¼ R ; ½ Xe Pt B dX1224

#

ð21Þ ð22Þ

Simplifying the above expressions for K and s, one gets K 2424 ¼ s61 ¼

1 t G CG 1 þ G t2 H 1 G 2 ; Ve 1

1  1 G 2 ^ CG 1 ^ u þ PH u; Ve

ð23Þ ð24Þ

Fig. 5. Viscoelastic layer with water closure condition.

4.2.1. Solid viscoelastic plate immersed in water For this test problem, a 20 mm thick, infinite viscoelastic plate immersed in water (fluid medium) is considered. Acoustic waves are impinging on the plate from a semi-infinite domain of water medium and the other side of the plate is exposed to another semi-infinite water domain so that the waves emerging from the plate never get reflected back. This condition, precisely, is termed as the water closure condition. The following material constants have been used for the viscoelastic material of the immersed plate: Storage modulus Er ¼ 1:4  108 N=m2 Poisson’s ratio m ¼ 0:49 Loss factor g ¼ 0:23 Mass density q ¼ 1100 kg=m3

where ½G 1 624 ¼

Z B dX;

ð25Þ

X

½G 2 1224 ¼ ½H1212 ¼

Ze Z

Pt B dX;

ð26Þ

Pt SP dX:

ð27Þ

Xe

Xe

Thus, as mentioned earlier, in the original PT element, the 18  18 H matrix needs to be inverted, while in the current formulation, one needs to invert the 12  12 H matrix. Further reduction of the computational cost has been achieved by evaluating H 1 G 2 not via an explicit inversion of H, but by solving the system Hq ¼ G 2 for q. The mass matrix is given simply by Z M¼ N t N dX: ð28Þ

The analysis has been performed in the 0–20 kHz frequency range as this is the general frequency range of interest. In this case, the mesh size used for the structure conforms to the k=4 criterion. In such a situation, the pressure reflection coefficient is given by [5,10]   c s c a  s2 1 1 R¼ þ ð30Þ cs þ ca cs þ js ca  js and the transmission coefficient is given by   1 1 þ T ¼ js cs þ js ca  js

ð31Þ

4.2. Validation of the finite element formulation

and where s ¼ q0 c0 =qc, cs ¼ cotðdÞ, ca ¼ tanðdÞ d ¼ xd=2cd . cd is the dilatational wave speed in the viscoelastic material. From the above equations, echo reduction and the transmission loss, the two parameters that are generally used in this work, can be evaluated using Eqs. (3) and (4). As can be seen from Figs. 6 and 7, the results obtained through the finite element analysis match exactly with the analytical results over the complete range of frequency. This provides the first validation of the developed model.

In order to validate the Finite Element model, first a simple test problem has been considered. Wave propagation through an infinite, homogeneous and isotropic viscoelastic plate (see Fig. 5) is analyzed. Analytical solution is available [5] for this case.

4.2.2. Viscoelastic plate with air cavities immersed in water The next level of complexity is added by introducing air channels inside the submerged viscoelastic plate. Cylindrical air inclusions of 15 mm height and 20 mm diameter, arranged in a doubly periodic configuration with a grating

X

The final equation of motion is given by   ½K  x2 ½M  x½S fqg ¼ ½f 

ð29Þ

S.N. Panigrahi et al. / Applied Acoustics 69 (2008) 1141–1153 45

1147

30 Analytical FEM

40

Present Model FEM (Ref. [5]) 25

Transmission Loss (dB)

Echo Reduction (dB)

35 30 25 20 15

20

15

10

10 5 5 0

2

3

4

5

6

7 8 9 10 Frequency (kHz)

20

30

0 0.2

40

Fig. 6. Echo reduction for an infinite viscoelastic plate immersed in water termed as water closure condition.

0.6

0.8 1.0 2 Frequency (kHz)

3

4

5

6 7 8

Fig. 8. Transmission loss for a viscoelastic plate containing large sized air cavities with water closure condition.

5

45

Analytical FEM

4.5

0.4

Present Model Experiment

40

4

Transmission Loss (dB)

Transmission Loss (dB)

35

3.5 3 2.5 2 1.5 1

25 20 15 10

0.5 0

30

5

2

3

4

5

6

7 8 9 10 Frequency (kHz)

20

30

40

Fig. 7. Transmission loss for an infinite viscoelastic plate water closure condition.

spacing of 30 mm, are introduced into a 20 mm thick viscoelastic plate. Physical properties for the plate material are same as those for the previous case. The results obtained from the present model have been compared with those presented by Hennion and Decarpigny [9] in Fig. 8 and can be seen to be matching reasonably well. The discrepancy can be attributed to the different formulations of finite elements used. In their work they have shown a frequency shift with mesh refinement but no convergence can be noticed. Results from the present model match with their coarse mesh model better than those of the refined mesh models. This can be explained by the established fact that the normal brick elements are in general inadequate to handle viscoelastic materials because of their high incompressibility, and the inadequacy, in fact, may result in higher errors with mesh refinements. The elements used in the present work are known to be inherently immune to such difficulties. Another level of validation is carried out to

0 0.2

0.4

0.6

0.8 1.0 2 Frequency (kHz)

3

4

5

6 7 8

Fig. 9. Transmission loss for a viscoelastic plate containing relatively smaller sized air cavities with water closure condition.

establish the adequacy of the model. In this case the material incompressibility is of very high order ðm ¼ 0:49976Þ. Results obtained from the present model have been compared with those obtained experimentally [9] and shown in Fig. 9. The trends can be seen to be matching. Discrepancy in the results (mainly the frequency shift) can definitely be attributed to the inaccuracies in the knowledge of the properties of the material used. In their work [9], Hennion et al. have shown that these kinds of inaccuracies in the measured material properties and the negligence of the frequency dependence of the these properties can cause such frequency shifts. 5. Design of different layers The four kinds of design introduced earlier in Figs. 1 and 2 shall now be discussed here keeping their respective

S.N. Panigrahi et al. / Applied Acoustics 69 (2008) 1141–1153 40 Holes are present in the lining No holes in the lining

35

Echo Reduction (dB)

30 25 20 15 10 5 0

2

3

4

5

6

7

8

9 10

20

Frequency (kHz)

Fig. 10. Echo reduction (ER) for a viscoelastic anechoic layer, backed by a steel wall of 6 mm thickness, with water closure condition.

strengths and weaknesses in view. As the name suggests, the first layer is designed to be a non-reflecting surface as opposed to the second layer which acts as a sound insulation surface. The third layer is created by coupling these two layers on the same side of the steel wall and the fourth layer is named as a decoupled layer as these two types of layers are provided on opposite sides of the steel plate. 5.1. Anechoic layer A typical anechoic layer containing very small diameter holes with considerable spacing in between has been analyzed with the developed model, and the ER results are presented in Fig. 10. The same figure also shows the performance of a lining with the same thickness and same material but without holes. A comparison of the ER curves for the two conditions explains the importance of this layer and in particular the usefulness of the presence of the air channels in the lining. It can be clearly seen that the presence of the holes tends to increase the echo reduction of the lining. The TL performance has also been evaluated but it has not been observed to improve substantially. The improvement in ER is a result of the presence of the air cavities which allow the bulk deformation of the layer to transform into shear deformation near the cavity walls and in this process some of the acoustic energy gets dissipated due to the damping property of the layer material. 5.1.1. Effect of the distribution of holes The holes that are introduced in the layer of the viscoelastic material are generally distributed in two different ways. One is the doubly periodic or the parallel configuration and the second one is the symmetric staggered distribution. This latter distribution has the advantage of being mechanically stronger than the doubly periodic one which is weak in the lines of the holes. The unit cells that can be used for analysis, however, are different in these

two cases as has already been discussed (see Fig. 3). If one compares the results from the square and the hexagonal unit cells (with the same overall porosity, i.e., the ratio of hole area to the total area of the unit cell) of Fig. 11, it can be observed that the staggered distribution has no extraordinary performance as compared to the parallel ones. Therefore, one can go with the parallel distribution of the holes for analysis to exploit the simplicity of the unit cell geometry. Later on the equivalent staggered configuration can be designed from the parallel distribution data. Furthermore, the circular unit cell approximation also can be noticed to be giving very accurate results. So, one can still simplify the unit cell by choosing the circular unit cell to approximate the parallel configuration. Therefore, in summary, the staggered distribution can be approximated to the parallel one by keeping the overall porosity same and then further reduction can be made in the mesh size by going for a circular unit cell approximation. Another point to be noted is that thickness of the cover layer (which is provided to restrict water from entering into these air cavities and to achieve a high surface smoothness) has been taken as 5 mm. 5.1.2. Effect of static pressure on anechoic layer performance The previous analysis was performed with a zero static gauge pressure on the interacting surface of the layer. But in reality, when deeply immersed in the fluid medium, the layer may have to perform at non-zero static gauge pressures. Such a case has been analyzed in this work employing an indirect finite element method with some inherent assumptions. A static analysis is first carried out on the unit cell in a commercial finite element software (ANSYS) by applying the desired static load on the free surface of the lining and by treating the steel–wall outer surface as a fixed surface, other boundary conditions remaining the same. Deformations are evaluated under the applied pressure. These deformations are then used 40 Square Cell Circular Cell Hexagonal cell

35 30

Echo Reduction (dB)

1148

25 20 15 10 5 0

2

3

4

5

6

7

8

9 10

20

Frequency (kHz)

Fig. 11. Comparison of the echo reduction curves for different unit cell approximation for various domains.

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along with the original nodal information to get the deformed nodal positions. Then a new datafile is prepared using these new nodal information and further analysis is done in the usual way. In this process, two assumptions have been made; namely, the residual stresses which are generated during the static deformation are not taken into consideration in the next stage of analysis and the material properties are not considered to change with these deformations. In fact, properties of these viscoelastic materials are known to be insensitive to the pressure field; therefore, it is assumed that only the changes in the shape and sizes of the cavities under the static pressure dictate the overall layer characteristics. Such an analysis when performed on a sample of anechoic lining revealed that for different static pressures either of the performance curves does not vary much. This effect is very much expected as the size of the holes is very small and the relative changes in their dimensions are also very small. Hence, these kinds of layers are very stable under high static pressures and their performance does not get affected when used in deep water conditions. 5.2. Insulation layer It is clear from the preceding discussion that the anechoic lining has got very effective ER characteristics. However, the TL performance is very inadequate in the low frequency region, i.e., from a few hundred hertz to a few thousand hertz. It is well known that the TL performance of steel wall itself is very good when used in the air medium. But, when used for underwater application the performance is not very good. This happens because of the relatively lower impedance mismatch between the steel– water medium pair ðY steel =Y water ¼ 31Þ as compared to that between the steel–air medium pair ðY steel =Y air ¼ 113400Þ. Hence a proper impedance mismatch is created by introducing big sized air cavities (as compared to the cavities in the anechoic layer) in the viscoelastic layer. Fig. 12

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shows the TL performance for a typical insulation layer. It has also been compared with the performance curve for the pure steel wall. It can be seen that the TL performance has been improved by carpeting the steel wall with the insulation layer. The improvement in the TL characteristic is very important as it has been in the low frequency regions where it is more crucial. The ER has also been evaluated and is seen to be very poor. It is worth noting that the ER performance of the pure steel layer is also not good in the higher frequency range. So, the worsening of the ER performance in the low frequency region is of not much consequence. 5.2.1. Analysis of different hole distributions Fig. 13 shows the TL performance (this is the prime criterion for this lining) predicted for different configurations using appropriate unit cells. The overall porosity of all the three configurations has been kept the same. It is observed form the figure that the results of the hexagonal unit cell are very similar to those of the circular cell. The difference is actually of the order of 2–3 dB. Given the uncertainty in the measurements of the material properties and estimation of the performance parameters, these discrepancies are actually insignificant. Yet, the discrepancy in Fig. 13 is more as compared to the anechoic case. As opposed to the anechoic lining, where the hole sizes are very small, holes are bigger in size in this case. So, the boundary conditions play a very important role here and for this reason a hexagonal domain is more closely approximated than a square domain by the circular unit cell. If one needs to analyze a symmetric staggered configuration, one can always go with the circular cell. 5.2.2. Effect of static pressure on insulation lining performance The insulation lining is expected to be more sensitive towards the static pressure head because of the presence 35 Circular unit cell Square unit cell Hexagonal unit cell

35 With the Insulation Lining Only Steel Plate of 6 mm with no Lining

30

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Fig. 13. Transmission loss curves for different unit cell approximation for various distributions.

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of the large sized holes under the cover layer. The same methodology, as discussed earlier, has been adopted for this lining too. The deformed shape of the unit cell under certain static loading, is shown in Fig. 14. This deformed

unit cell is then used as the unit cell for the dynamic analysis (with the assumptions discussed earlier). The performance curves for different pressure loading conditions have been compared in Fig. 15. As was expected, the static pressure affects the performance curves to a reasonable extent, and hence cannot be neglected. The variations also follow the trends of a parametric investigation carried out for such linings. As the pressure increases, the average thickness of the layer gets reduced and the average diameter of the hole also gets smaller keeping the unit cell side length constant. So, the combined effect is also expected to have the same trend. It can be seen from the figure that the same trend is followed over the complete frequency range. 5.3. Combination layer: coupled type From the foregoing discussions it is clear that each lining has its own characteristics which are very different in

Fig. 14. Deformed shape of the unit cell under static pressure.

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4 Zero gauge pressure 0.5 MPa gauge pressure 1.0 MPa gauge pressure

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Fig. 15. Echo reduction and transmission loss curves for different static head on a typical insulation layer lining.

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Fig. 16. (a) Echo reduction and (b) transmission loss curves for a coupled layer lining.

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nature; in fact, they are opposite in character. So, to have a compromise between these two situations, the natural generalization is to combine these two layers. In the first design, the anechoic layer is made to face the incident wave and the insulation layer is sandwiched between the steel wall and the anechoic layer as shown in Fig. 2a. The performance curves for this type of lining have been depicted in Fig. 16. It can be observed that the TL performance is very good but the ER performance is very poor for this kind of a design. The behavior is more close to the behavior of the insulation lining. This behavior can be attributed to the fact the two air channels are coupled together. So, for the anechoic layer also, the compliance of the cavity of the insulation layer contributes and the system acts more like the latter. The dimensions used are as outlined below. Cover layer thickness; tca ¼ 5 mm Anechoic layer thickness; ta ¼ 10 mm

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acteristic has improved reasonably over a broad frequency band. However, the thickness has been increased by 20 mm from the previous case. That amounts to an increase of about 22 kg/m2 in the surface density of the lining. This increase in weight may some time become very undesirable. 5.4. Combination layer: decoupled type To overcome the aforementioned shortcomings, the combination has been modified by disconnecting the two air channels and locating them on the two sides of the steel wall to form the fourth type of combination layer. The schematic of this kind of layer has been depicted in Fig. 2b. Using the same dimensions for both the layers (and the respective holes) as in the case of Fig. 16, the decoupled configuration has been formed. The only change that has been introduced is the cover layer for the insulation layer of thickness tcr , which is set to 3 mm. The results are shown in Fig. 18 where a comparison has been made to emphasize

Insulation layer thickness; tr ¼ 7 mm Steel layer thickness; ts ¼ 6 mm Hole radius of anechoic layer; ra ¼ 0:5 mm

a 10

Hole radius of insulation layer; rr ¼ 6 mm Cell radius; rc ¼ 7:5 mm

9

However, the ER performance can be improved by increasing the anechoic layer thickness, ðta Þ. By increasing this thickness, the deformation of the anechoic layer as a cover layer for the insulation layer is restricted and the layer acts more on the absorption than the resonance principle. Fig. 17 shows the performance curves for the coupled-type combination layer with a larger thickness of the anechoic layer ðta ¼ 30 mmÞ. All the other dimensions and the material properties have been kept the same as in the previous case. It is observed that the TL performance has improved slightly. The important point, however, is that the ER char-

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Fig. 17. Echo reduction curve for a coupled layer lining with very thick anechoic layer.

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Fig. 18. Comparison of (a) echo reduction and (b) transmission loss curves for a coupled and decoupled type combination layer lining.

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Fig. 19. Different shapes of the hole in the resonator layer of the decoupled layer lining.

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6. Conclusions

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the effectiveness of the decoupled type combination lining as compared to the coupled type. It can be seen from the ER and the TL characteristic curves that the ER performance has improved whereas the TL curve remains almost unchanged over the entire frequency range. In fact, the ER curves for the present configuration have almost become comparable to that of Fig. 17 with the thicker anechoic layer. So it is worth noting that on decoupling the two layers by putting them on opposite sides of the steel wall, both the performance parameters have become reasonably good. If this configuration is compared with that corresponding to Fig. 17, there is a weight reduction of about 18.7 kg/ m2 of the lining with approximately the same overall performance in both aspects. Further, to improve the balance between these layer effects, a conical hole is made in place of the cylindrical one in the insulation layer of the lining (Fig. 19). The performances of these two kinds of linings have been presented in Fig. 20. It is clear from the curves that the echo reduction performance has improved at the low frequencies too. Though, the transmission loss curve has come down, it can be considered acceptable as there is a 6 dB of transmission loss even at 500 Hz. There is also a 3 dB of echo reduction even at 2 kHz, and exceeds 6 dB for frequencies beyond 4 kHz.

35 30 25 20 15 10 5 0 0.5

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Fig. 20. (a) Echo reduction and (b) transmission loss curves for different shapes of the hole in the resonator layer of the decoupled layer lining.

A multi-focus approach for designing the underwater noise control linings has been carried out to tackle some conflicting requirements. The coupled-type combination layers behave more like the insulation layers. With very high thicknesses, however, the benefits of the anechoic layer can be observed. For the same dimensions, the decoupled type lining is preferable to the coupled one as it gives better ER performance without affecting the TL curve over the entire frequency range of interest. This type of lining also results in reduction of the thickness of the outer layer (if the anechoic layer of the combination lining is used as the outer layer), thereby cutting down the weight which will reduce the drag force in the case of underwater moving objects. This also simplifies the process of manufacturing, thereby reducing the overall production cost. Further it is observed that providing conical air cavities in the insulation layer improves the ER performance at the cost of TL performance. A finite element model based methodology has been used to analyze the proposed designs of the noise control linings. A hybrid type finite element based on the Pian and Tong formulation has been modified and used. The modified formulation requires a 12  12 matrix inversion against an 18  18 matrix inversion in the original one. As expected, the hybrid finite element has been observed to be immune to the near incompressible characteristics of the viscoelastic materials used and the high distortion of the finite elements of the mesh. The correctness of this

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formulation has been shown by comparing its results with the analytical, the FE based and the experimental results. Unit cell representation for different types of distributions of air cavities on the linings have been discussed and the symmetry of the distribution has been exploited to reduce the mesh size and thereby the computational effort. A circular unit cell is adequate to model the anechoic lining for any kind of distribution. This can also be used in place of the exact hexagonal unit cell for representing a staggered distribution of the holes in the case of the insulation layer lining. A square unit cell, obtained by keeping the overall porosity constant, cannot be used to represent a staggered distribution. Effect of static pressure has been studied by using a simplified finite element technique to simulate deep sea testing environment. It has been shown that the effect of this static head is more prominent in the case of the insulation layer than the anechoic layer. As the performance of anechoic layer does not get affected under static pressure head, the decoupled type combination lining is preferable under conditions where the static pressure may be very high, without affecting the performance. Acknowledgement Financial support of the Facility for Research in Technical Acoustics (FRITA) by the Department of Science and Technology of the Government of India is gratefully acknowledged. References [1] Oberst H. Resonant sound absorbers. In: Richardson EG, editor. Technical aspects of sound, vol. II. Amsterdam: Elsevier; 1957.

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[2] Mayer E, Brendel K, Tamm K. Pulsation oscillations of cavities in rubber. J Acoust Soc Am 1958;30(12):1115–24. [3] Gaunaurd G. One dimensional model for acoustic absorption in a viscoelastic medium containing short cylindrical cavities. J Acous Soc Am 1977;62:298–307. [4] Gaunaurd G. Coments on absorption mechanism for water borne sound in Alberich anechoic layers. Ultrasonics 1985;23:90–1. [5] Fiorito R, Madigosky W, Uberall H. Resonance theory of acoustic waves interacting with an elastic plate. J Acous Soc Am 1989;66: 1857–66. [6] Jackins PD, Gaunaurd GC. Resonance reflection of acoustic waves by a perforated bilaminar rubber coating model. J Acoust Soc Am 1983;73(5):1456–63. [7] Gaunaurd G, Callen E, Barlow J. Pressure effects on the dynamic effective properties of resonating perforated elastomers. J Acoust Soc Am 1984;76(1):173–7. [8] Strifors HC, Gaunaurd GC. Selective reflectivity of viscoelastically coated plates in water. J Acoust Soc Am 1990;88(2):901–10. [9] Hennion AC, Decarpigny JN. Analysis of the scattering of a plane wave by a doubly periodic elastic structure using the finite element method: application to Alberich anechoic coatings. J Acous Soc Am 1991;90:3356–67. [10] Easwaran V, Munjal ML. Analysis of reflection characteristics of a normal incident plane wave on resonant sound absorbers: a finite element approach. J Acous Soc Am 1993;93(3):1308–18. [11] Berry A, Foin O, Szabo JP. Three-dimensional elasticity model for a decoupling coating on a rectangular plate immersed in a heavy fluid. J Acous Soc Am 2001;109(6):2704–13. [12] Nolle AW. Dynamic mechanical properties of rubberlike materials. J Polym Sci 1950;5:1–54. [13] Rosen SL. Fundamental principles of polymeric materials. Wiley; 1993. [14] Aklonis JJ. Introduction to polymer viscoelasticity. Wiley; 1995. [15] Lakes RS. Viscoelastic solids. CRC Press; 1999. [16] Pian TH, Tong P. Relations between incompatible displacement model and hybrid stress model. Int J Num Meth Eng 1986;22:173–81. [17] Sze KY, Fan H. An economical assumed stress brick element and its implementation. Finite Elem Anal Des 1996;21:179–200.