Geometrical and environmental properties [2]

## 1. Introduction

Although the researchers have done many efforts to perform the numerical model such as FEM (Finite Elements Method) to investigate the wave prorogation through the shells, the analytical vibro-acoustic modeling of the composite shells is unavoidable because of the accuracy of the model in a broadband frequency. Bolton *et. al.* [1] investigated sound transmission through sandwich structures lined with porous materials and following Lee *et. al.* [2] proposed a simplified method to analyze curved sandwich structures. Daneshjou *et. al.* [3-5] studied an exact solution to estimate the transmission loss of orthotropic and laminated composite cylindrical shells with considering all three displacements of the shell. Recently the authors [6] have presented an exact solution of free harmonic wave propagation in a double-walled laminated composite cylindrical shell whose walls sandwich a layer of porous material using an approximate method. This investigation is focused on sound transmission through the sandwich structure, which includes the porous material core between the two laminated composite cylindrical shells to predict the reliable results for all structures used foam as an acoustic treatment.

Wave propagation through a composite cylindrical shell lined with porous materials is investigated, based on classical laminated theory. The porous material is completely modeled using elastic frame. The vibro-acoustic equations of the shell are derived considering both the shell vibration equations and boundary conditions on interfaces. These coupled equations are solved simultaneously to calculate the Transmission Loss (TL). Moreover, the results are verified with a special case where the porosity approaches zero. Finally, the numerical results are illustrated to properly study the geometrical and physical properties of composite and porous material. In addition, the effects of the stacking sequence of composite shells and fiber directions are properly studied.

## 2. Propagation of sound in porous media

If the porous material is assumed a homogeneous aggregate of the elastic frame and the fluid trapped in pores, its acoustic behavior can be considered by the following two wave equations (See Eq. (22) and Eq. (25) of [1]):

Eqs. (1) and (2) determine 2 elastic longitudinal waves and 1 rotational wave, respectively. In Eqs. (1) and (2)

where

where

The complex wave numbers of the two compression (longitudinal) waves,

where

and the wave number of the shear (rotational) wave is:

## 3. Simplified method

As the full method is too complicated to model the porous layer in the curved sandwich structures, thus a simplified method is expanded for this category of structures [2]. The foundation of this approximate method considers the strongest wave between those ones. It includes two steps. At the first step a flat double laminated composite with infinite extents with the same cross sectional construction is considered using the full method. Then, only the strongest wave number is chosen from the results and the material is modeled using the wave number and its corresponding equivalent density. Thus, the material is modeled as an equivalent fluid.

The strain energy which is related to the displacement in the solid and fluid phases can be defined for each wave component. The energy terms can be represented as follows;

## 4. Model specification

Figure 1 shows a schematic of the cylindrical double shell of infinite length subjected to a plane wave with an incidence angle

## 5. Applying full method to two-dimensional problem

For a two-dimensional problem as shown in the

where

Three kinds of the waves propagate in porous material, therefore six traveling waves, which have the same trace wave numbers, are induced by an oblique incident wave in a finite depth layer of porous material, as shown in Fig. 2. The

The displacements in the fluid phase are:

The stresses in the solid phase are:

The stresses in the fluid phase are:

where

These complex relations and six constants

where

where

## 6. Prediction of ratios of the energy

The energy related to the waves in the fluid phase and solid phase are descript as follows [2].

The airborne wave:

And the frame wave:

where the subscripts

## 7. Formulation of the problem

In the external space, the wave equation becomes [3]:

where

The shell motions are described by classic theory, fully considering the displacements in all three directions. Let the axial coordinate be

In which the subscripts

The forces and moments are:

where the extensional, coupling and bending stiffness,

In Eqs. (44 - 46)

The boundary conditions at the two interfaces between the shells and fluid are [2]:

where

Considering the circular cylindrical geometry, the pressures are expanded as:

where

Substitution of the expressions in the displacements of the shell (Eq. (44)-Eq.(49)) and the acoustic pressures (Eq. (54)-Eq.(58)) equations into six shell equations (Eq. (34)-Eq.(36)) and four boundary conditions (Eq. (50)-Eq.(53)) yields ten equations, which can be decoupled for each mode if the orthogonality between the trigonometric functions is utilized. These ten equations can be sorted into a form of a matrix equation as follow:

where

The ten unknown coefficients

## 8. Calculation of transmission losses (TLs)

The transmission coefficient,

where

To consider the random incidences,

where

In the following, the averaged TLs of the structure is calculated in terms of the 1/3 octave band for random incidences.

## 9. Convergence algorithm

Eqs. (44) to (49) and (54) to (58) are obtained in series form. Therefore, enough numbers of modes should be included in the analysis to make the solution converge. Therefore, an iterative procedure is constructed in each frequency, considering the maximum iteration number. Unless the convergence condition is met, it iterates again. When the TLs calculated at two successive calculations are within a pre-set error bound, the solution is considered to have converged. An algorithm for the calculation of TLs at each frequency is followed as:

REPEAT

## 10. Numerical results

Parametric numerical studies of transmission loss (TLs) are conducted for a double-walled composite laminated cylindrical shell lined with porous material specified as follows, considering 1/3 octave band frequency. Table 1 presents the geometrical and environmental properties of a sandwich cylindrical structure. Each layer of the laminated composite shells are made of graphite/epoxy, see Table 2. The plies were arranged in a

The results are verified by those investigated by the authors’ previous work for an especial case in which the porous material properties go into fluid phase (In other world, the porosity is close to 1). The comparison of these results shown in Fig. 3, indicates a good agreement.

The calculated transmission loss for the laminated composite shell is compared with those of other authors for a special case of isotropic materials. In other word, in this model the mechanical properties of the lamina in all directions are chosen the same as an isotropic material such as Aluminum, and then the fiber angles are approached into zero. Fig. 4 compares the TL values of the special case of laminated composite walls obtained from present model and those of aluminum walls from Lee’s study [2]. The results show an excellent agreement.

We are going to verify the model in behavior comparing the results of cylindrical shell in a case where the radius of the cylindrical shell becomes large or the curvature becomes negligible with the results of the flat plate done by Bolton [1] (See Fig. 5). It should be also noted that both structures sandwich a porous layer and have the same thickness. Although it is not expected to achieve the same results as the derivation of the shell equations is quite different comparing with derivation of plate equations, however, the comparison between the two curves indicate that they behave in a same trend in the broadband frequencies.

Ambient | Outer Shell | Porous Core | Inner Shell | Cavity | |

Material | Air | Composite | Porous Material | Composite | Air |

Density (kg/m^{3}) | 1.21 | - | - | - | 0.94 |

Speed of Sound (m/s) | 343 | - | - | - | 389 |

Radius (mm) | - | 172.5 | - | 150 | - |

Thickness (mm) | - | 2 | 20 | 3 | - |

Bulk Density of Solid Phase^{*} (kg/m^{3}) | - | - | 30 | - | - |

Bulk Young’s Modulus^{*} (kPa) | - | - | 800 | - | - |

Bulk Poisson’s Ratio (-) | - | - | 0.4 | - | - |

Flow Resistivity (MKs) | - | - | 25000 | - | - |

Tortuosity (-) | - | - | 7.8 | - | - |

Porosity (-) | - | - | 0.9 | - | - |

Loss Factor (-) | - | - | 0.265 | - | - |

Graphite/epoxy | Glass/epoxy | |

Axial Modulus (GPa) | 137.9 | 38.6 |

Circumferential Modulus (GPa) | 8.96 | 8.2 |

Shear Modulus (GPa) | 7.1 | 4.2 |

Density (kg/m^{3}) | 1600 | 1900 |

Major Poisson’s Ratio (-) | 0.3 | 0.26 |

Figure 6 indicates that whenever the radius of the shell descends, the TL of the shell is ascending in low frequency region. It is due to the fact that the flexural rigidity of the cylinder will be increased with reduction of shells radii. In addition, decreasing the radius of the shell leads to weight reduction and then in high frequency especially in Mass-controlled region the power transmission into the structure increases.

Figure 7 shows the effect of the composite material on TL. Materials chosen for the comparison are graphite/epoxy and glass/epoxy (Table 2). The figure shows that material must be chosen properly to enhance TL at Stiffness-control zone. The results represent a desirable level of TL at Stiffness-control zone (Lower frequencies) for graphite/epoxy. It is readily seen that, in higher frequency, as a result of density of materials, the TL curves are ascending. Therefore, the TL of glass/epoxy is of the highest condition in the Mass-controlled region.

It is well anticipated that increase of porous layer thickness leads to increase of TL. As illustrated in Fig. 8, a considerable increase due to thickening the porous layer is obtained. As it is well obvious from this figure, the weight increase of about 12% (

Figure 9 shows a comparison between the transmission loss for a ten-layered composite shell and an aluminum shell with the same radius and thickness. Since, the composite shell is stiffer than the aluminum one, its TL is upper than that of aluminum shell in the Stiffness-controlled region. However, as a result of lower density of composite shell, it does not appear to be effective as an aluminum shell in Mass-controlled region.

The effect of stacking sequence is shown in Fig. 10. Two patterns

## 11. Conclusions

Transmission losses (TLs) of double-walled composite laminated shells sandwiching a layer of porous material were calculated. It is also considered the acoustic-structural coupling effect as well as the effect of the multi-waves in the porous layer. In order to make the problem solvable, one dominant wave was used to model the porous layer. In general the comparisons indicated the benefits of porous materials. Also, a considerable increase due to thickening the porous layer was obtained. For example, the weight increase of about 12% and 25% may respectively lead to an increase of 35% or 60% in amount of averaged TL values in broadband frequency. In addition, it was shown that increasing the axial modulus of plies made the TL be increased in low frequency range. Moreover, the comparison of double-walled shell with a gap and the one sandwiched with porous materials (where the porosity is close to 1) indicated a good agreement. Eventually the arrangement of layers in laminated composite can be so effective in Stiffness-controlled region. Therefore, optimizing the arrangement of layers can be useful in future study.

## Nomenclature

## Appendix

The non-zero components of the matrix

Here: