Sound transmission of periodic composite structure lined with porous core: rib-stiffened double panel case
This paper develops a modeling procedure combining Biot theory and space harmonic series for the one-dimensional periodic composite structure with porous lining. The periodically rib-stiffened case is discussed to illustrate the proposed method. Detailed solution procedures are shown using the bonded-bonded boundary case. The modeling procedure is validated using theoretical and finite element method results. The influence of modeling parameters is studied. Qualitative results are obtained. Despite the cumbersome truncation, the method here can be promising in broadband sound modulation by combining the poroelastic modeling and periodic structures semi-analytically.
keywords:Sound transmission; Periodic composite structure; Porous media; rib-stiffened double panel; space harmonic series
Due to the high stiffness to weight ratio, multi-panel structures are widely used in engineering applications, such as aircraft, underwater and building structures. Their acoustic performance is studied for a long time.
Composite multi-panel structures without any attachments or fillings are always the simplest case. Concerning their sound transmission loss (STL), both theoretical, experimental and numerical methods are employed. Xin and Lu Xin et al. (2009) and Sakagami Sakagami et al. (2002) are representatives of theoretical models; experimental results are included in the latter. Semi-empirical models are also used, the representatives are the models by Sharp Sharp (1978) and Gu Gu and Wang (1983), improved models by Davy Davy (2009); the prediction models are reviewed and compared by Hongisto Hongisto (2006) and LegaultLegault and Atalla (2009) contemporarily. No theoretical or semi-empirical model appropriate for the case studied here though.
Much more attention is paid to the composite multi-panel structure with attachments or absorption fillings. The absorption fillings are complicated; nonetheless, they can be considered as porous materials in most cases, thus the models of porous media can be utilized. Two widely used theoretical models for porous media are the Biot theory Biot (1956) and the equivalent fluid model (EFM) Allard and Champoux (1992).
The EFM is widely used together with some numerical or semi-analytical treatment due to its simplicity. Double partition structures with porous cavity were studied by EFM Trochidis and Kalaroutis (1986), or together with semi-analytical method Xin and Lu (2010). Liu Liu and He (2016) used EFM together with the transfer matrix method (TMM), utilizing the simplification by Lee Lee et al. (2001); the formulation was different from Xin Xin and Lu (2010). STL of double panel structures filled with fiberglass was studied by Legault and Atalla, using EFM together with finite element method (FEM) Legault and Atalla (2010) or experiments Legault and Atalla (2009).
When it comes to elastic frame porous media, the EFM is invalid and the Biot theory Biot (1956) should be used. Two-dimensional (2D) poroelastic field variables within porous media are given by Bolton Bolton et al. (1996) using the Biot theory; the simplifications of Deresiewicz Deresiewicz and Rice (1962) and Allard Allard et al. (1989) are adopted. This 2D model was extended to the three-dimensional (3D) case by Zhou Zhou et al. (2013). Liu Liu and Sebastian (2015) investigated the effects of flow on the multi-panel structures using the 3D model. Numerical methods Alimonti et al. (2015) were also developed based on the Biot theory.
Meanwhile, multi-panel structures with attachments were prominent too. The current study is mainly about those with ribs, resilient mountings or elastic coatings; while the absorption fillings are absent. One of the most useful methods for these problems is the space harmonic series (SHS) introduced by Mead and Pujara Mead and Pujara (1971); it is still useful for periodic structures today. Utilizing the SHS, Xin studied the STL of rib-stiffened sandwich structures Xin and Lu (2010); Legault and Atalla investigated the effects of structure links Legault and Atalla (2009) and resilient mountings Legault and Atalla (2010) on the STL of double panel structures. The drawback of the SHS is illustrated by Legault Legault et al. (2011). The Fourier transform method (FTM) Mace (1980) is also useful for these periodic problems. Using the FTM, multi-panel structures were investigated by Lin Lin and Garrelick (1977) and Brunskog Brunskog (2005); Xin Xin and Lu (2010) studied these structures with absorption fillings together with EFM. In fact, the same nature is shared between FTM and SHS as pointed out by Mace Mace (1980). Another useful method for these structures is the modal approach. Using the orthonormal modal functions, the vibration Chung and Emms (2008), structural intensity Brunskog and Chung (2011), and modal characteristics Dickow et al. (2013) of simply supported ribbed structures were studied. For double panel structures, the work of Xin Xin and Lu (2009) and Liu Liu and Catalan (2017) can be representatives. However, for porous media, relevant orthonormal modal functions are not available for now.
Despite the fruitful research conducted on multi-panel structures, there are few papers on those with both periodic boundary and the porous core, or try to solve these problems using the Biot theory. For example, Xin Xin and Lu (2010) and Legault Legault and Atalla (2009, 2010) all studied these periodic multi-panel structures using EFM. Inspired by Bolton Bolton et al. (1996) and Zhou Zhou et al. (2013), the poroelastic field with periodic boundary conditions is deduced here utilizing the Biot theory and SHS; then the vibroacoustic model of the periodic composite structure with porous lining can be established. The problem can be solved by adopting the preconditioning method by Hull Hull and Welch (2010) then. The periodically rib-stiffened model is discussed as an example. The novelty here is the combination of the Biot theory and SHS for the poroelastic problem with periodic boundary conditions.
2 Modeling procedures for the periodic composite structure
The periodic composite structure is composed of a rib-stiffened double panel with porous lining, as shown in Fig.1; it is immersed in a stationary inviscid acoustic fluid. An incident wave transmits through the structure, is the incident wave vector, , . According to Fig.1, , here is the incident wavenumber, and are the incident elevation angle and azimuth angle respectively. The time dependent term is omitted henceforth as the incident wave is time harmonic Legault and Atalla (2009); Mace (1981). The space occupied by acoustic fluid on both sides is assumed to be semi-infinite and lossless; the density and sound velocity are designated as , and , for the incident and transmitted side respectively.
The ribs are periodically placed along x at a spacing , and extend infinitely along y; the thickness and height are , (as shown in Fig.1). The longitudinal deformation of ribs is ignored as the y dimension is infinite.
2.1 The velocity potentials in acoustic media
As periodic cavities between the ribs are formed, the SHS is used Xin and Lu (2010) to express the velocity potential of the incident side
where is the wave vector; is the unknown amplitude of reflected wave harmonics; , ; integer . According to wave equation , , .
The velocity potential of transmitted side can be expressed as
here is the wave vector; is the unknown amplitude of transmitted wave harmonics; integer . According to the law of refraction Morse and Ingard (1968), the wave vector component , ; substitute Eq.(2) in wave equation , then , .
Once the velocity potential is determined, the corresponding sound pressure and acoustic particle velocity can be obtained by , .
2.2 In-plane and transverse vibration of face panels
The two face panels are considered as isotropic thin plates. Their thickness, displacement along x, y and z are denoted as respectively, here correspond to the incident and transmitted side panel. When in-plane force and moment are present, the vibration equations are Axisa and Trompette (2005)
are the in-plane vibration and transverse vibration operator; are the external forces along z and x; the in-plane moment , and are unit vectors along x and y; is the density, is the thickness, is the in-plane stiffness, is the shear modulus, and is the bending stiffness. The panel displacement is expressed as
here , according to the law of refraction; is the unknown component amplitude vector; integer .
2.3 Flexural vibration and rotation of ribs
The flexural vibration of ribs is modeled by the Bernoulli-Euler model and Timoshenko model here; a comparison between them is made. The rotation of ribs is modeled by the torsional wave equation.
The Bernoulli-Euler beam (BE-B) equation is Junger and Feit (1986)
where is the displacement, is the density, is Young’s modulus, is the second moment of area, is the cross-section area and is the external force. The Timoshenko beam (TS-B) equation is Junger and Feit (1986)
where is the shear modulus, is the shear correction factor.
The rotation is determined by Rao (2007)
where is the polar moment of inertia, is the external moment; is the clockwise angle of rotation about the rib-panel interface, here is the panel displacement.
Then the forces exerted on the transmitted side plate can be obtained using the above equations and the displacement continuity condition. The results are
here , while
depending on the beam model.
The resultant force exerted by the periodic ribs can then be written as
here . It becomes
A double summation is present, which makes the problem complicated and cumbersome. An index separation identity is utilized to eliminate it.
2.4 The poroelastic field with periodic boundary conditions
In poroelastic problems, appropriate porous modeling procedure is quite absent when the periodic boundary conditions are present. An extended and generalized method for porous modeling is given, compared to the previous ones Bolton et al. (1996); Zhou et al. (2013). The wave components in the porous media are expressed as six groups of harmonics here, according to their propagation wavenumbers; then the poroelastic field with periodic boundary conditions is obtained. The porous material is assumed to be isotropic with homogeneous cylindrical pores to get a neat formulation.
where , , , , , , , , , ; is the density of ambient fluid; is the porosity; and are the bulk solid and fluid densities; is the geometrical structure factor; is the flow resistivity; auxiliary function , and are the Bessel functions of the first kind, first and zero order respectively. Parameter is the first Lame constant; is the shear modulus; the two coupling parameter , , here is the bulk modulus of fluid in the pores; is the sound velocity of fluid in pores; is the ratio of specific heats; auxiliary variable , is the Prandtl number in pores.
The poroelastic equations Eqs.(12)-(13) can be reduced to two wave equations Bolton et al. (1996); Zhou et al. (2013) (a fourth order equation and a second order equation) when two scalar potentials and two vector potentials are introduced; these potentials are the dilatational and rotational strains of corresponding phase Allard and Atalla (2009); Bolton et al. (1996). The wavenumbers correspond to the wave equations are
here the auxiliary term , , .
Utilizing SHS, the amplitude of solid phase strain , are written as
here are the unknown amplitude of harmonic components; , , are the z component of the corresponding wave vectors; the law of refraction is used here. According to Eqs.(12)-(13), the amplitude of fluid phase strain , are
It is assumed that the field variables are composed of six groups of harmonics (i.e. , , ) here, while only six wave components are included in previous papers Biot (1956); Bolton et al. (1996). According to the definition of potentials, the displacement can be deduced using a similar procedure as Bolton Bolton et al. (1996) with and Graff (1975)
where and are the forces in solid and fluid phase; is the normal strain or shear strain ; is the Kronecker delta
2.5 Boundary conditions
The boundary condition notation of Bolton Bolton et al. (1996) is used; the connection type can be bonded-bonded (BB), bonded-unbonded (BU) and unbonded-unbonded (UU). The related boundary conditions are not given here, as they can be found in Bolton Bolton et al. (1996), Zhou Zhou et al. (2013) and Liu Liu and Sebastian (2015) etc. Note that a right-handed coordinate system is used here, while Bolton and Zhou used the left-handed one, Liu used a right-handed one. The detailed expressions are identical for the 2D case, but in the 3D case, some modifications should be made between the two coordinate systems.
2.6 System equations and solution procedures
The bonded-bonded case is used as an example to demonstrate the solution procedure. Related boundary conditions are
To eliminate the summation index used, the orthogonal property below
and a double summation identity analogous to Hull Hull and Welch (2010)
then matrix can be rearranged to a block diagonal matrix
here the blank elements of are all zero; matrix can be rearranged to a full block matrix
where the blank elements of are in the same pattern as shown in Eq.(31); the vector can be rearranged to
here is a zero vector. Then Eq.(28) can be written as
The Eq.(33) needs to be truncated to obtain a solution, i.e. truncate the index number in , and to ; considerable accuracy can be ensured by appropriate convergence criteria. Then the matrices , are reduced to , and the vectors , are reduced to , ; thus Eq.(33) can be solved using .
The sound field here can be regarded as the sum of all harmonics, thus the transmission coefficient is Legault and Atalla (2009)
here is the real part operator of a complex variable.
The random STL can then be expressed as Zhou et al. (2013)
here is the minimum elevation angle.
3 Results and discussions
This part begins with overview of the convergence characteristics; then the validation is conducted; at last, several parameter analyses are performed. The ribs of rectangular cross-section and the porous parameters given in Bolton et al. (1996); Zhou et al. (2013); Liu and Sebastian (2015) are utilized. The detailed values are listed in Table 1; they are used if no other values are specified hereinafter. The random STL is calculated in 1/24 octave bands using the 2D Simpson’s rule; Bolton et al. (1996) as no convective flow is present; the integration domain of and are split up into 36 and 90 subdivisions respectively.
|density of incident side media||1.205 kg/|
|sound velocity of incident side media||343 m/|
|density of face panels||2700 kg/|
|Young’s modulus of face panels||70Pa|
|Poisson’s ratio of face panels||0.33|
|face panel thickness of incident side||1.27 mm|
|face panel thickness of transmitted side||0.762 mm|
|thickness of porous core||27 mm|
|rib spacing along x||50|
|rib thickness||1 mm|
|rib height||20 mm|
|density of solid phase||30 kg/|
|density of fluid phase||1.205 kg/|
|Young’s modulus of solid phase||8Pa|
|Poisson’s ratio of solid phase||0.4|
|loss factor of solid phase||0.265|
|geometrical structure factor||7.8|
|flow resistance||2.5 MKS Rayls/|
3.1 Overview of the convergence characteristics
A truncation procedure is needed to solve the infinite matrix equation Eq.(33). The convergence criteria is taken as 0.1dB under the maximum computation frequency (=10kHz); that is, when the change in STL by one additional item is less than , it is considered as converged. A typical convergence curve of the three boundary conditions is given in Fig.2. Different marks (circles, squares and hexagrams for BB, BU and UU cases respectively) here show the convergence points in the given figure.
The truncation item number needed for different parameter cases are shown in Fig.3. It can be seen that is dependent on the boundary conditions and parameter values. A related study concerning is in progress, however, no conclusion can be drawn for now.
As a result, one needs to determine in every numerical case when the method proposed here is used. This is the main drawback of the proposed method here.
3.2 Model Validation
The poroelastic field representation is validated using theoretical results of unribbed double panel structure with porous lining by Bolton Bolton et al. (1996) (2D), Zhou Zhou et al. (2013) (3D) and Liu Liu and Sebastian (2015) (3D). The unribbed case corresponds to is a zero matrix and tends to infinity here. The rib-stiffened configurations here are validated with FEM results, as no theoretical or experimental result can be found in the open literature so far, to the author’s knowledge.
3.2.1 Validation of poroelastic field representation
The model is first downgraded to 2D by setting =0. Result comparisons with Bolton are presented in Fig.4 and the consistency is good. Then validation with Zhou Zhou et al. (2013) and Liu Liu and Sebastian (2015) when external flow Mach number is performed. The results are shown in Fig.5. The correctness of the poroelastic field representation is confirmed by these two cases.
3.2.2 Oblique incident STL validation with FEM results
Several oblique incident cases are compared, as the random STL is clearly related to the oblique incident case according to Eq.(35). The oblique incident STL are used in Figs.7-9 for the vertical axis values.
The FEM models are formed by the mixed displacement-pressure form Biot-Allard equations Allard and Atalla (2009); the viscous and thermal characteristic length required are obtained using an equivalent relationship Allard and Champoux (1992); Allard and Atalla (2009) , , here is the dynamic viscosity of fluid in pores. Perfectly matched layer (PML) and periodic boundary conditions (periodic BC) are used. Symmetric boundary conditions (symmetric BC) for y coordinate are used, as the structure is infinite in the y direction. A schematic description of the FEM configuration is shown in Fig.6. The FEM calculations are performed using COMSOL.
The results are given in Figs.7-9 for the BB, BU and UU case respectively. They are in good overall coherence with FEM results, while critical local differences are satisfactory. For all oblique incidence cases, the STL differences are less than 2dB (absolute value) on average; in the meantime, the relative differences are all no more than 10% of the FEM results on average, despite few extrema occur asynchronously. This is another evidence of the usability of the proposed method here.
Time used in the 3D oblique incidence calculations are listed in Table 2. The efficiency is pronounced, while the FEM is expensive due to refined model required by large size change. However, the drawback should be noted, as the convergence check can take up to 69.76%, 73.34% and 75.45% of the total calculation time on average. For the 3D random incidence, more than 80% of the total calculation time can be used; thus an improvement is needed. A related study is in progress.
|Angle (,)||FEM (s)||Here (s)||FEM (s)||Here (s)||FEM (s)||Here (s)|
3.3 Influence of rib reinforcement model on the STL
The beam models mentioned above are used. The STL and its difference defined as the Timoshenko beam case minus the Bernoulli-Euler beam case for different boundary conditions are shown in Fig.10.
An overall consistency is found for the BB case, while 2dB or so differences (under most frequencies) are found for the BU and UU case. It is confirmed by FEM that combined flexural and torsional modes of ribs around 1372Hz appeared, one of these modes is shown in Fig.11 (UU case eigenfrequency 1375Hz). The FEM configuration of section 3.2.2 is used hereinafter. Due to the consistency of the two beam models, the Bernoulli-Euler beam model is more preferable. It is used in the following.
3.4 Influence of torsional motion on the STL
The influence of torsion is discussed, though the transverse motion is more concerned in acoustics; detailed results are shown in Fig.12. The STL decrease emerges and can be larger than 5dB when the frequency exceeds 1454Hz (indicated in Fig.12). It is due to the emergence of the torsional mode of ribs (confirmed by FEM as shown in Fig.13). The STL increases temporarily around the eigenfrequency, due to energy consumed by the torsional mode; then the torsion deteriorates the STL when apart from the eigenfrequency. Thus the torsion should be considered for its correlativity with the STL.
3.5 Influence of rib spacing on the STL
The results when varies among versus the unribbed case are discussed. As the results of BB case are analogous to BU and UU case, the details are given in the supplementary materials to be brief.
In all the three cases, the STL is positively related to in general; while a degradation occurs compared to the unribbed one. The overall trends are analogous to previous unribbed results Bolton et al. (1996); Zhou et al. (2013); Liu and Sebastian (2015). As decreases, more fluctuations emerge and the trough frequencies shift to lower ones (indicated in Figs.14-15). In fact, to avoid wave interference between ribs, should be far greater than the coincidence wavelength of the face panels, which is Junger and Feit (1986)
here is the sound velocity of adjacent medium; for the incident and transmitted side panel and (mm) respectively. When is not large enough, the interference can be complex and fluctuations emerge.
In general, the STL is positively related to , while the presence of ribs lower it, though direct transfer path is absent. This is due to the increase in flexural wave speed of the composite structure, i.e. the influence of stiffness increase is stronger than that of the mass increase Hambric et al. (2016) when ribs are present. To obtain better sound insulation performance, the ribs should be removed; at least, should be increased, as is related to the fluctuations and trough frequencies. Further investigations on the dispersion relation are needed to draw a quantitative conclusion.
3.6 Influence of area moment of inertia on the STL
To exclude the influence of mass change, the cross-section area of ribs is kept constant in this parameter case; for rectangular ribs, the area moment of inertia is inversely proportional to then. Thus the investigation is performed at .
For the BB case, the influence of is weak and the STL characteristics of unribbed structure dominate, as the structural connections are strong. The results are given in the supplementary materials. As shown in Figs.16-17, the STL decreases with the increase of (as decreases) under relatively low frequency range (i.e. between 100Hz and 1kHz); it’s due to the increase of the phase velocity of ribs Junger and Feit (1986)
which is proportional to the group velocity here; thus better energy transmission and lower insulation as anticipated. In higher frequency range, the STL curve is analogous to the unribbed case Bolton et al. (1996); Zhou et al. (2013); it is the superposition of unribbed STL and the fluctuations caused by ribs.
In general, the influence of occurs in the low-frequency range; within this range, is negatively related to the STL. To obtain better sound insulation in the low-frequency range, smaller area moment of inertia is preferred for similar structures.
This paper develops a modeling procedure for the one-dimensional periodic composite structure with porous lining. The periodically rib-stiffened case is discussed utilizing the proposed method. The influence of the modeling parameters is studied, such as beam models, torsional motion, rib spacing and area moment of inertia.
It is found that the Bernoulli-Euler beam is enough to model the overall frequency response of ribs here. The torsion should not be ignored for its correlativity with the STL. In general, the STL is positively related to the rib spacing, while the use of ribs lowers it; smaller rib spacing means more STL fluctuations and worse sound insulation performance. The influence of area moment of inertia is mainly in the low-frequency range; it is negatively related to the STL there. The periodic ribs make the STL performance complicated, with fluctuations and decrease.
Despite the cumbersome truncation, the modeling procedure here for periodic poroelastic problems is efficient with considerable accuracy. As a feasible tool for combined periodic structure (metamaterial) and poroelastic problems, it can be promising in broadband sound modulation.
This work is supported by the National Natural Science Foundation of China (NSFC) No.11572137. The authors wish to thank some anonymous researchers and reviewers for their warm-hearted suggestions and comments.
Appendix A Elements of porous field variable coefficient matrix
To be brief, here , , , and are replaced by , , , and respectively, the elements of coefficient matrix are