# Research on Power Flow Transmission through Elastic Structure into a Fluid-Filled Enclosure

Authors: Rui Huo; Chuangye Li; Laizhao Jing; Weike Wang
Journal: Advances in Acoustics and Vibration
 (2018)

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## Abstract

The work of this paper is backgrounded by prediction or evaluation and control of mechanical self-noise in sonar array cavity. The vibratory power flow transmission analysis is applied to reveal the overall vibration level of the fluid-structural coupled system. Through modal coupling analysis on the fluid-structural vibration of the fluid-filled enclosure with elastic boundaries, an efficient computational method is deduced to determine the vibratory power flow generated by exterior excitations on the outside surface of the elastic structure, including the total power flow entering into the fluid-structural coupled system and the net power flow transmitted into the hydroacoustic field. Characteristics of the coupled natural frequencies and modals are investigated by a numerical example of a rectangular water-filled cavity with five acoustic rigid walls and one elastic panel. Influential factors of power flow transmission characteristics are further discussed with the purpose of overall evaluation and reduction of the cavity water sound energy.


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## Body

## 1. Introduction

### 1.1. Background

The work of this paper is backgrounded by prediction or evaluation and control of mechanical self-noise in sonar array cavity. The mechanical self-noise, which is caused by structural vibration of sonar cavity’s wall, might significantly weaken the detection performance of sonar at lower frequencies [1, 2]. The sources of mechanical self-noise might be multiple such as vibrating machines on the ship which diffuse vibration energy or second excitation of structure-borne sound. However, it is essential to comprehend the characteristics of interaction between the enclosed water sound field and its elastic boundary structures for the purpose of prediction, evaluation, and control of interior hydroacoustic noise [3].The subjects of cabin noise in various flight vehicles and automobiles are more familiar in the investigation of fluid-structural coupled vibration of acoustoelastic enclosure, which mainly focus on characteristics of sound transmission through the elastic wall into interior sound field resulting from exterior air-borne sound [4]. In these cases, weak coupling has been commonly assumed because of the low density of air and high stiffness of cabin wall, which means that the cavity’s interior sound pressure would have little influence on the vibration of cavity wall, and modals of interior sound field would also be affected very lightly [5]. In contrast, a much stronger coupling might be present when a water sound field takes place of the air [6].The sound pressure is most commonly used to represent the property of sound field in the study of acoustic-structural coupling of acoustoelastic enclosure. The ratio of sound pressure at the outside surface of the elastic cavity wallboard to that at internal surface, which is defined as “noise reduction,” is applied to evaluation of sound transmission characteristics [4, 7]. Since the sound pressure would change greatly at different points of the sound field, the value of noise reduction would also be very different, and a comprehensive measure, for example, power flow, would be expected for an overall evaluation of vibration level of the enclosed sound field. The power flow has been validated and widely utilized as a comprehensive measure for evaluation of overall level of vibration energy of vibration isolation systems mounted on flexible foundations [8], which could also be explained as average sound power when applied to sound field analysis.In this paper, through modal coupling analysis on the fluid-structural vibration of the water-filled enclosure with elastic boundaries, an efficient computational method is deduced to determine the vibratory power flow generated by exterior excitations on the outside surface of the elastic structure, including the total power flow entering into the fluid-structural coupled system and the net power flow transmitted into the hydroacoustic field. Characteristics of the coupled natural frequencies and modals are investigated by a numerical example of a rectangular water-filled cavity with five acoustic rigid walls and one elastic panel. Influential factors of power flow transmission characteristics are further discussed with the purpose of overall evaluation and reduction of the cavity water sound energy.


### 1.2. Theoretical Development

There has been a continuous effort for decades on investigation of fluid-structural mechanism of closed sound field with flexible boundaries. It has been recommended that the locally reactive acoustic normal impedance was the earlier theory to understand the sound absorption caused by the interaction between a reverberation room and its surrounding walls [9]. Later attention was paid to the modal coupling between the enclosed sound fields and the flexible walls to reveal the more complicated mechanism demonstrated by experimental results, which could not be interpreted by the locally reactive theory [10, 11].The modal responses of acoustoelastic enclosures were first developed by Dowell et al. [12, 13] by applying Green’s function to the inhomogeneous wave differential equation of the enclosed sound field and applying the classical modal and eigenvalue theorem to the simultaneous fluid-structural differential equations to result in a resolution of coupling modals. There are still other resolution methods for the same acoustoelasticity equations which could be referred to, such as Laplace transformation [14] and Ritz series [15]. In general, Dowell’s method is based on the familiar uncoupled acoustic enclosure modes and structural modes, could be more easily implemented, and has been successfully applied to the investigation of variety of fluid-structural interaction systems [16, 17]. Beginning with the “modal coupling method,” Pan and Bies gave an insight analysis of the weak-coupled and well-coupled modals and their decay characteristics of a rectangular panel-cavity coupled system [18, 19]; Davis put forward a method for approximate estimation of the coupled natural frequencies of acoustoelastic enclosures by “coupling coefficient” [20].Other important developments might lie in the field of discrete numerical techniques, such as FEM/BEM, for fluid-structural vibration analysis. However, these methods are usually preferred in the investigation of irregularly shaped cavities and targeting specific engineering problems. And that would be beyond the discussion of this paper, which would mainly focus on a general theoretical evaluation method for the overall vibration level of a fluid-filled enclosure through vibratory power flow calculation, especially based on Dowell’s modal coupling theory.


## 1.1. Background

The work of this paper is backgrounded by prediction or evaluation and control of mechanical self-noise in sonar array cavity. The mechanical self-noise, which is caused by structural vibration of sonar cavity’s wall, might significantly weaken the detection performance of sonar at lower frequencies [1, 2]. The sources of mechanical self-noise might be multiple such as vibrating machines on the ship which diffuse vibration energy or second excitation of structure-borne sound. However, it is essential to comprehend the characteristics of interaction between the enclosed water sound field and its elastic boundary structures for the purpose of prediction, evaluation, and control of interior hydroacoustic noise [3].The subjects of cabin noise in various flight vehicles and automobiles are more familiar in the investigation of fluid-structural coupled vibration of acoustoelastic enclosure, which mainly focus on characteristics of sound transmission through the elastic wall into interior sound field resulting from exterior air-borne sound [4]. In these cases, weak coupling has been commonly assumed because of the low density of air and high stiffness of cabin wall, which means that the cavity’s interior sound pressure would have little influence on the vibration of cavity wall, and modals of interior sound field would also be affected very lightly [5]. In contrast, a much stronger coupling might be present when a water sound field takes place of the air [6].The sound pressure is most commonly used to represent the property of sound field in the study of acoustic-structural coupling of acoustoelastic enclosure. The ratio of sound pressure at the outside surface of the elastic cavity wallboard to that at internal surface, which is defined as “noise reduction,” is applied to evaluation of sound transmission characteristics [4, 7]. Since the sound pressure would change greatly at different points of the sound field, the value of noise reduction would also be very different, and a comprehensive measure, for example, power flow, would be expected for an overall evaluation of vibration level of the enclosed sound field. The power flow has been validated and widely utilized as a comprehensive measure for evaluation of overall level of vibration energy of vibration isolation systems mounted on flexible foundations [8], which could also be explained as average sound power when applied to sound field analysis.In this paper, through modal coupling analysis on the fluid-structural vibration of the water-filled enclosure with elastic boundaries, an efficient computational method is deduced to determine the vibratory power flow generated by exterior excitations on the outside surface of the elastic structure, including the total power flow entering into the fluid-structural coupled system and the net power flow transmitted into the hydroacoustic field. Characteristics of the coupled natural frequencies and modals are investigated by a numerical example of a rectangular water-filled cavity with five acoustic rigid walls and one elastic panel. Influential factors of power flow transmission characteristics are further discussed with the purpose of overall evaluation and reduction of the cavity water sound energy.


## 1.2. Theoretical Development

There has been a continuous effort for decades on investigation of fluid-structural mechanism of closed sound field with flexible boundaries. It has been recommended that the locally reactive acoustic normal impedance was the earlier theory to understand the sound absorption caused by the interaction between a reverberation room and its surrounding walls [9]. Later attention was paid to the modal coupling between the enclosed sound fields and the flexible walls to reveal the more complicated mechanism demonstrated by experimental results, which could not be interpreted by the locally reactive theory [10, 11].The modal responses of acoustoelastic enclosures were first developed by Dowell et al. [12, 13] by applying Green’s function to the inhomogeneous wave differential equation of the enclosed sound field and applying the classical modal and eigenvalue theorem to the simultaneous fluid-structural differential equations to result in a resolution of coupling modals. There are still other resolution methods for the same acoustoelasticity equations which could be referred to, such as Laplace transformation [14] and Ritz series [15]. In general, Dowell’s method is based on the familiar uncoupled acoustic enclosure modes and structural modes, could be more easily implemented, and has been successfully applied to the investigation of variety of fluid-structural interaction systems [16, 17]. Beginning with the “modal coupling method,” Pan and Bies gave an insight analysis of the weak-coupled and well-coupled modals and their decay characteristics of a rectangular panel-cavity coupled system [18, 19]; Davis put forward a method for approximate estimation of the coupled natural frequencies of acoustoelastic enclosures by “coupling coefficient” [20].Other important developments might lie in the field of discrete numerical techniques, such as FEM/BEM, for fluid-structural vibration analysis. However, these methods are usually preferred in the investigation of irregularly shaped cavities and targeting specific engineering problems. And that would be beyond the discussion of this paper, which would mainly focus on a general theoretical evaluation method for the overall vibration level of a fluid-filled enclosure through vibratory power flow calculation, especially based on Dowell’s modal coupling theory.


## 2. Theory

### 2.1. Equations of Fluid-Structural Coupled Vibration

Consider that a fluid-filled enclosure occupies a volumeV. Its boundary D=DS+DF, where DF≠0 represents the flexible area of the surrounding wall and DS (might be zero) represents the acoustic rigid area.The fluid inside the enclosure satisfied the wave equation and associated boundary condition.(1)K0∇2pσ,t-ρ0∂2pσ,t∂t2=0σ∈V(2)∂pσ,t∂n=-ρ0anσ,tσ∈DF(3)∂pσ,t∂n=0σ∈DS,where pσ,t is the sound pressure at point σ(x,y,z)∈V; an(σ,t) is the acceleration of the flexible wall in the normal direction n (positive outward); ρ0 and K0 are the equilibrium fluid density and fluid volume stiffness, respectively.IfDF=0, (1) has modal solutions FArσ·exp⁡(jωArt),r=0,1,2,…, where ωAr is the rth acoustical natural frequency in the condition of rigid boundary and FArσ is the corresponding natural mode with orthogonality as follows:(4)∭VFArσFAsσρ0c02dv=0r≠sMArr=s,∭V∇FArσT·∇FAsσρ0dv=0r≠sωAr2MArr=s,where c0=K0/ρ0 is the acoustic velocity of the fluid, MAr is the rth acoustical modal mass in the condition of rigid boundary, and ∇FArσ=[∂FAr/∂x,∂FAr/∂y,∂FAr/∂z]T is the column gradient vector of modal function FArσ.Consider the solution of (1) with DF≠0 being in the form of modal superposition; that is,(5)pσ,t=∑r=0∞FArσPrt=FσTPtσ∈V,where F(σ) and P(t) are column vectors of modal function Fr(σ) and its corresponding modal coordinate Pr(t), respectively; that is, F(σ)=[FA0(σ),FA1(σ),FA2(σ),…]T and P(t)=[P0(t),P1(t),P2(t),…]T.After substituting (5) into (1), multiply both sides of the resultant equation with a left-multiplication matrix (vector) F(σ) and finally integrating the equation over volume V, one obtains(6)∭VFσ∇2pσ,tdv-1c02∭VFσFσTP¨tdv=0.By applying Green’s theorem to the first term of above equation, one has(7)1c02∭VFσFσTP¨tdv+∭V∇FσT∇FσPtdv=∯DFσ∂pσ,t∂ndA,where ∇F(σ) is the gradient matrix of modal function Fr(σ): ∇F(σ)=[∇FA0(σ),∇FA1(σ),∇FA2(σ),…].Now substitute the boundary condition equations (2)~(3) and orthogonality equation (5) into (7):(8)MAP¨t+ΩA2Pt=-∯DFσanσ,tdA,where MA and ΩA are diagonal matrices of acoustical modal masses and natural frequencies, respectively; that is, MA=diag[MA0,MA1,MA2,…] and ΩA=diag[ωA0,ωA1,ωA2,…].The flexible boundary of the cavity is assumed to be thin-wall structures, where linear partial differential equations would be adopted to fit the thin-wall structures’ vibration, such that(9)Swσ,t+mB∂2wσ,t∂t2=pAσ,t-pBσ,tσ∈DF,where S is a linear differential operator representing structural stiffness; mB is structural mass per unit area; pAσ,t and pBσ,t are excitations on the surface of the thin-wall structures due to the cavity acoustics and external dynamical forces (intensity of pressure), respectively; wσ,t is the displacement response of the thin-wall structures, which is defined in the normal direction of DF.The solution of (9) could be expressed as(10)wσ,t=∑j=1∞WBjσBjt=WσTBtσ∈DF,where WBjσ is the jth modal function that is defined on DF and concerned with the property of the thin-wall structures in vacuo and Bjt is the modal coordinate corresponding to WBjσ; W(σ) and B(t) are column vectors of WBjσ and Bjt, respectively; that is, W(σ)=WB1σ,WB2σ,WB3σ,…T and B(t)=B1t,B2t,B3t,…T.By substituting (10) into (9) and using the orthogonality of WBjσ, there would be a modal differential function as follows:(11)MBB¨t+ΩB2Bt=QAt+QBt,where MB and ΩB, expressed as MB=diag[MB1,MB2,MB3,…] and ΩB=diag[ωB1,ωB2,ωB3,…], are diagonal matrices of the modal masses and natural frequencies respectively, and ωBj and MBj=∬DFmBWBjσ2dA(j=1,2,3,…) represent the jth natural frequency and modal mass of the thin-wall structures in vacuo, respectively. QAt and QBt are column vectors of the general forces due to pAσ,t and pBσ,t loaded on the thin-wall structures in vacuo, respectively, and(12)QAt=∬DFWσpAσ,tdA,QBt=-∬DFWσpBσ,tdA.The right-hand term of (8) and the first term QAt on the right hand of (11) are of the fluid-structural interaction between the sound field inside the cavity and its flexible walls. Substituting (10) into the right-hand term of (8) and taking notice of an(σ,t)=∂2w(σ,t)/∂t2 at σ∈DF and an(σ,t)=0 at σ∈DS, one could define a coupling matrix L as follows:(13)L=∬DFFσWσTdA=Lrjr=0,1,2,…;j=1,2,3,…,where Lrj denotes the element of the coupling matrix L at the rth row and the jth column.And (8) turns into(14)MAP¨t+ΩA2Pt=-L·B¨t.Dealing withQAt, one could express pAσ,t in (12) with (5), and (11) would become(15)MBB¨t+ΩB2Bt=LT·Pt+QBt.


### 2.2. Modal Analysis

In order to carry out a modal analysis about the fluid-structural vibration system governed by (14) and (15), let QBt=0, and suppose that there exist vibration solutions as follows:(16)Pt=MA-1·χA·exp⁡jωt,Bt=ΩBMB-1·χB·exp⁡jωt,where MA=diagMA0,MA1,MA2,… and MB=diagMB1,MB2,MB3,… are square roots of the diagonal acoustical modal matrix MA and the diagonal structural modal matrix MB, respectively. χA=χA0,χA1,χA2,…T and χB=χB1,χB2,…T are column vectors of fluid-structural coupled modal shape coefficients related to the cavity sound field and the flexible boundary structures, respectively.Substituting (16) into (14) and (15), an eigenvalue problem could be obtained as(17)Aχ=A11A12A21A22χAχB=ω2χAχB=ω2χ,where χ could be named as vector of fluid-structural coupled modal shape coefficients and A is a symmetric characteristic matrix, and A’s partitioned matrices could be calculated by A11=ΩA2+MA-1LMB-1LTMA-1, A12=A21T=-MA-1LΩBMB-1, and A22=ΩB2.Equation (17) would give eigenvalues of matrix A, that is, ωCk2=ΩCk2(1+jηCk), and the accompanying eigenvectors χkT=χAkT,χBkT=[χA0k,χA1k,χA2k,…,χB1(k),χB2(k),χB3(k),…], k=0,1,2,…, where ΩCk corresponds to the kth fluid-structural natural frequency. ηCk is the loss factor associated with the kth damped normal mode, which might be resulting from the introduction of a complex stiffness of the flexible boundary or a complex volume stiffness K0 of the fluid in consideration of the damping properties of the fluid-structural system. It should also be noticed that [χk] might be complex vectors when ωCk2 are complex numbers.The fluid-structural coupled modal functions of the cavity’s sound field and the flexible boundaries would be expressed as(18)FCkσ=FσTMA-1χAkσ∈V,WCkσ=WσTΩBMB-1χBkσ∈DF.


### 2.3. Vibratory Power Flow Transmission

In the condition that the flexible boundary structures of the cavity are subjected to a harmonic exterior excitation, that is,pBσ,t=PBσ·exp⁡(jωt), let PB=-∬DFWσ·PBσdA; PB indicates that the amplitudes of general forces belong to the uncoupled flexible structures. The steady responses of the fluid-structural coupling cavity would be(19)pσ,t=FσTMA-1ΧAHTPBexp⁡jωt=Pσ,ω·exp⁡jωtσ∈V(20)wσ,t=WσTΩBMB-1ΧBHTPBexp⁡jωt=Wσ,ω·exp⁡jωtσ∈DF,where Pσ,ω and Wσ,ω denote the amplitudes of the harmonic sound pressure in the cavity and harmonic displacement of the thin-wall structures. ΧA and ΧB are matrices composed of arrays of eigenvectors of the characteristic matrix A; that is, ΧA=[χA(0),χA(1),χA(2),…] and ΧB=[χB(0),χB(1),χB(2),…]. And(21)Hω=-ω2MC+ΩC2·MC-1(22)T=-XAHMA-1LMB-1+XBHΩBMB-1,where H could be named as the complex frequency response matrix of the fluid-structural coupled cavity and T is a transformation matrix to transform the general force PB into its fluid-structural expression (the derivation of the matrices H and T has been explained via (A.5)~(A.8) in Appendix A.2). MC=XAH·ΧA+XBH·ΧB is a diagonal matrix of the fluid-structural coupled modal masses, and ΩC=diag[ωC0,ωC1,ωC2,…] is a diagonal matrix of the fluid-structural coupled natural frequencies. The superscript “H” denotes Hermitian transposition of matrices.The power flow (density) inputted by exterior excitationpBσ,t into the fluid-structural system is(23)pinσ,ω=ω2π∫0ω/2πRepBσ,t·Re-∂wσ,t∂tdt=-ω2RejWσ,ω·PB∗σσ∈DF.The total power flow input is(24)Pinω=∬DFpinσ,tdA=-ω2RejPBTTTHω·XBTΩBMB-1∬DFWσ·PB∗σdA=ω2RejPBTTTHω·XBTΩBMB-1PB∗,where the superscript “∗” denotes conjugation of complex numbers.The power flow (density) transmitted through the fluid-structural interaction boundary of the cavity into the enclosed sound field is(25)ptrσ,ω=ω2π∫0ω/2πRepσ,tσ∈DF·Re-∂wσ,t∂tσ∈DFdt=ω2RejW∗σ,ω·Pσ,ωσ∈DF.The total transmission power flow is(26)Ptrω=∬DFptrσ,ωdA=ω2Re∬DFFσT·Λω·WσdA=ω2Re∑r=0∞∑j=1∞Lrj·λrjω,where Λω=jMA-1XAHω·TPBPBHTHHω∗XBHΩB∗MB-1=λrjω might be named as a power transmission matrix and λrjω denotes the element of matrix Λω at the rth row and the jth column.


## 2.1. Equations of Fluid-Structural Coupled Vibration

Consider that a fluid-filled enclosure occupies a volumeV. Its boundary D=DS+DF, where DF≠0 represents the flexible area of the surrounding wall and DS (might be zero) represents the acoustic rigid area.The fluid inside the enclosure satisfied the wave equation and associated boundary condition.(1)K0∇2pσ,t-ρ0∂2pσ,t∂t2=0σ∈V(2)∂pσ,t∂n=-ρ0anσ,tσ∈DF(3)∂pσ,t∂n=0σ∈DS,where pσ,t is the sound pressure at point σ(x,y,z)∈V; an(σ,t) is the acceleration of the flexible wall in the normal direction n (positive outward); ρ0 and K0 are the equilibrium fluid density and fluid volume stiffness, respectively.IfDF=0, (1) has modal solutions FArσ·exp⁡(jωArt),r=0,1,2,…, where ωAr is the rth acoustical natural frequency in the condition of rigid boundary and FArσ is the corresponding natural mode with orthogonality as follows:(4)∭VFArσFAsσρ0c02dv=0r≠sMArr=s,∭V∇FArσT·∇FAsσρ0dv=0r≠sωAr2MArr=s,where c0=K0/ρ0 is the acoustic velocity of the fluid, MAr is the rth acoustical modal mass in the condition of rigid boundary, and ∇FArσ=[∂FAr/∂x,∂FAr/∂y,∂FAr/∂z]T is the column gradient vector of modal function FArσ.Consider the solution of (1) with DF≠0 being in the form of modal superposition; that is,(5)pσ,t=∑r=0∞FArσPrt=FσTPtσ∈V,where F(σ) and P(t) are column vectors of modal function Fr(σ) and its corresponding modal coordinate Pr(t), respectively; that is, F(σ)=[FA0(σ),FA1(σ),FA2(σ),…]T and P(t)=[P0(t),P1(t),P2(t),…]T.After substituting (5) into (1), multiply both sides of the resultant equation with a left-multiplication matrix (vector) F(σ) and finally integrating the equation over volume V, one obtains(6)∭VFσ∇2pσ,tdv-1c02∭VFσFσTP¨tdv=0.By applying Green’s theorem to the first term of above equation, one has(7)1c02∭VFσFσTP¨tdv+∭V∇FσT∇FσPtdv=∯DFσ∂pσ,t∂ndA,where ∇F(σ) is the gradient matrix of modal function Fr(σ): ∇F(σ)=[∇FA0(σ),∇FA1(σ),∇FA2(σ),…].Now substitute the boundary condition equations (2)~(3) and orthogonality equation (5) into (7):(8)MAP¨t+ΩA2Pt=-∯DFσanσ,tdA,where MA and ΩA are diagonal matrices of acoustical modal masses and natural frequencies, respectively; that is, MA=diag[MA0,MA1,MA2,…] and ΩA=diag[ωA0,ωA1,ωA2,…].The flexible boundary of the cavity is assumed to be thin-wall structures, where linear partial differential equations would be adopted to fit the thin-wall structures’ vibration, such that(9)Swσ,t+mB∂2wσ,t∂t2=pAσ,t-pBσ,tσ∈DF,where S is a linear differential operator representing structural stiffness; mB is structural mass per unit area; pAσ,t and pBσ,t are excitations on the surface of the thin-wall structures due to the cavity acoustics and external dynamical forces (intensity of pressure), respectively; wσ,t is the displacement response of the thin-wall structures, which is defined in the normal direction of DF.The solution of (9) could be expressed as(10)wσ,t=∑j=1∞WBjσBjt=WσTBtσ∈DF,where WBjσ is the jth modal function that is defined on DF and concerned with the property of the thin-wall structures in vacuo and Bjt is the modal coordinate corresponding to WBjσ; W(σ) and B(t) are column vectors of WBjσ and Bjt, respectively; that is, W(σ)=WB1σ,WB2σ,WB3σ,…T and B(t)=B1t,B2t,B3t,…T.By substituting (10) into (9) and using the orthogonality of WBjσ, there would be a modal differential function as follows:(11)MBB¨t+ΩB2Bt=QAt+QBt,where MB and ΩB, expressed as MB=diag[MB1,MB2,MB3,…] and ΩB=diag[ωB1,ωB2,ωB3,…], are diagonal matrices of the modal masses and natural frequencies respectively, and ωBj and MBj=∬DFmBWBjσ2dA(j=1,2,3,…) represent the jth natural frequency and modal mass of the thin-wall structures in vacuo, respectively. QAt and QBt are column vectors of the general forces due to pAσ,t and pBσ,t loaded on the thin-wall structures in vacuo, respectively, and(12)QAt=∬DFWσpAσ,tdA,QBt=-∬DFWσpBσ,tdA.The right-hand term of (8) and the first term QAt on the right hand of (11) are of the fluid-structural interaction between the sound field inside the cavity and its flexible walls. Substituting (10) into the right-hand term of (8) and taking notice of an(σ,t)=∂2w(σ,t)/∂t2 at σ∈DF and an(σ,t)=0 at σ∈DS, one could define a coupling matrix L as follows:(13)L=∬DFFσWσTdA=Lrjr=0,1,2,…;j=1,2,3,…,where Lrj denotes the element of the coupling matrix L at the rth row and the jth column.And (8) turns into(14)MAP¨t+ΩA2Pt=-L·B¨t.Dealing withQAt, one could express pAσ,t in (12) with (5), and (11) would become(15)MBB¨t+ΩB2Bt=LT·Pt+QBt.


## 2.2. Modal Analysis

In order to carry out a modal analysis about the fluid-structural vibration system governed by (14) and (15), let QBt=0, and suppose that there exist vibration solutions as follows:(16)Pt=MA-1·χA·exp⁡jωt,Bt=ΩBMB-1·χB·exp⁡jωt,where MA=diagMA0,MA1,MA2,… and MB=diagMB1,MB2,MB3,… are square roots of the diagonal acoustical modal matrix MA and the diagonal structural modal matrix MB, respectively. χA=χA0,χA1,χA2,…T and χB=χB1,χB2,…T are column vectors of fluid-structural coupled modal shape coefficients related to the cavity sound field and the flexible boundary structures, respectively.Substituting (16) into (14) and (15), an eigenvalue problem could be obtained as(17)Aχ=A11A12A21A22χAχB=ω2χAχB=ω2χ,where χ could be named as vector of fluid-structural coupled modal shape coefficients and A is a symmetric characteristic matrix, and A’s partitioned matrices could be calculated by A11=ΩA2+MA-1LMB-1LTMA-1, A12=A21T=-MA-1LΩBMB-1, and A22=ΩB2.Equation (17) would give eigenvalues of matrix A, that is, ωCk2=ΩCk2(1+jηCk), and the accompanying eigenvectors χkT=χAkT,χBkT=[χA0k,χA1k,χA2k,…,χB1(k),χB2(k),χB3(k),…], k=0,1,2,…, where ΩCk corresponds to the kth fluid-structural natural frequency. ηCk is the loss factor associated with the kth damped normal mode, which might be resulting from the introduction of a complex stiffness of the flexible boundary or a complex volume stiffness K0 of the fluid in consideration of the damping properties of the fluid-structural system. It should also be noticed that [χk] might be complex vectors when ωCk2 are complex numbers.The fluid-structural coupled modal functions of the cavity’s sound field and the flexible boundaries would be expressed as(18)FCkσ=FσTMA-1χAkσ∈V,WCkσ=WσTΩBMB-1χBkσ∈DF.


## 2.3. Vibratory Power Flow Transmission

In the condition that the flexible boundary structures of the cavity are subjected to a harmonic exterior excitation, that is,pBσ,t=PBσ·exp⁡(jωt), let PB=-∬DFWσ·PBσdA; PB indicates that the amplitudes of general forces belong to the uncoupled flexible structures. The steady responses of the fluid-structural coupling cavity would be(19)pσ,t=FσTMA-1ΧAHTPBexp⁡jωt=Pσ,ω·exp⁡jωtσ∈V(20)wσ,t=WσTΩBMB-1ΧBHTPBexp⁡jωt=Wσ,ω·exp⁡jωtσ∈DF,where Pσ,ω and Wσ,ω denote the amplitudes of the harmonic sound pressure in the cavity and harmonic displacement of the thin-wall structures. ΧA and ΧB are matrices composed of arrays of eigenvectors of the characteristic matrix A; that is, ΧA=[χA(0),χA(1),χA(2),…] and ΧB=[χB(0),χB(1),χB(2),…]. And(21)Hω=-ω2MC+ΩC2·MC-1(22)T=-XAHMA-1LMB-1+XBHΩBMB-1,where H could be named as the complex frequency response matrix of the fluid-structural coupled cavity and T is a transformation matrix to transform the general force PB into its fluid-structural expression (the derivation of the matrices H and T has been explained via (A.5)~(A.8) in Appendix A.2). MC=XAH·ΧA+XBH·ΧB is a diagonal matrix of the fluid-structural coupled modal masses, and ΩC=diag[ωC0,ωC1,ωC2,…] is a diagonal matrix of the fluid-structural coupled natural frequencies. The superscript “H” denotes Hermitian transposition of matrices.The power flow (density) inputted by exterior excitationpBσ,t into the fluid-structural system is(23)pinσ,ω=ω2π∫0ω/2πRepBσ,t·Re-∂wσ,t∂tdt=-ω2RejWσ,ω·PB∗σσ∈DF.The total power flow input is(24)Pinω=∬DFpinσ,tdA=-ω2RejPBTTTHω·XBTΩBMB-1∬DFWσ·PB∗σdA=ω2RejPBTTTHω·XBTΩBMB-1PB∗,where the superscript “∗” denotes conjugation of complex numbers.The power flow (density) transmitted through the fluid-structural interaction boundary of the cavity into the enclosed sound field is(25)ptrσ,ω=ω2π∫0ω/2πRepσ,tσ∈DF·Re-∂wσ,t∂tσ∈DFdt=ω2RejW∗σ,ω·Pσ,ωσ∈DF.The total transmission power flow is(26)Ptrω=∬DFptrσ,ωdA=ω2Re∬DFFσT·Λω·WσdA=ω2Re∑r=0∞∑j=1∞Lrj·λrjω,where Λω=jMA-1XAHω·TPBPBHTHHω∗XBHΩB∗MB-1=λrjω might be named as a power transmission matrix and λrjω denotes the element of matrix Λω at the rth row and the jth column.


## 3. Numerical Simulation and Analysis

### 3.1. Simulation Model

A panel-cavity coupled system shown in Figure1 consists of a rectangular water-filled room with five rigid walls and one simply supported plate subject to exterior harmonic distributed force (pressure) pE(σ,t).Figure 1
A panel-cavity coupled system.In discussion of the distribution shape of exterior excitation, the plane harmonic wave incident is a common assumption. Suppose thatpE(σ,t)=PE·exp⁡[j(ωt-k′xsin⁡θ)], where PE is the amplitude, k′ is the wave number, and θ is the incident angle (pE is uniform along the y direction); pE has a perpendicular component pB(σ,t)=PEcos⁡θ·exp⁡-jk′xsin⁡θ·exp⁡jωt=PBσ·exp⁡(jωt). Generally, PB(σ) would be a complex function and have infinite variety of distribution shapes when the wave frequency, velocity, and incident angle changed. Figure 2 shows one example of distribution shape of PB(σ) with wave frequency f=500 Hz, velocity c=344 m/s, and incident angle θ=π/3. It is true that the modal coupling method is valid in dealing with those variant distribution shapes of PB(σ). However, some specific analysis on the special case with θ=0, that is, a uniform PB over the plate surface, would also give indications of general significance. The uniform excitation had been adopted by other authors previously [7, 15]. And, moreover, taking sonar array cavities as examples, they are regularly mounted on related ship structures through rubber blankets; the uniform structural excitation assumption would be a basic consideration.Figure 2
One example of nonuniform distribution shape ofPB(σ) (wave frequency f=500 Hz, velocity c=344 m/s, and incident angle θ=π/3).
(a)
Real part (b)
Imaginary partFor the water-filled rectangular room, the natural frequencies and modal functions of the sound field with rigid boundaries are determined by(27)ωAr2=K0ρ0mArπLx2+nArπLy2+lArπLz2(28)FArx,y,z=cos⁡mArπxLxcos⁡nArπyLycos⁡lArπzLz,where ∀mAr,nAr,lAr∈N3, r∈N, and let ωAr arrange in a sequence ωA02<ωA12<ωA22<⋯.For a simply supported plate, its natural frequencies and modal functions are determined by(29)ωBj2=π4Eh212ρ1-μ2mBj2Lx2+nBj2Ly22(30)WBjx,y,0=sin⁡mBjπxLxsin⁡nBjπyLy,where E, ρ, h, and μ are Young’s modulus, mass density, thickness, and Poisson’s ratio of the plate, respectively; ∀mBj,nBj∈N2,j∈N+, let ωBj arrange in a sequence ωB12<ωB22<ωB32<⋯.The geometrical and material properties adopted for numerical computation are as follows:Lx=0.4 m, Ly=0.6 m, Lz=0.7 m, and h=0.005 m; ρ0=1000 kg/m3 and K0=2.25×109Pa×(1+10-4j); E=2.0×1011Pa×(1+10-3j), ρ=7.8×103 kg/m3, and μ=0.28 (steel).


### 3.2. Modal Analysis

There is a convergence investigation about the fluid-structural coupled natural frequencies resulting from (17) at first. In Table 1, with a fixed number of plate modals involved in calculation, the convergence of coupled natural frequencies could be observed by increasing the number of water sound modals involved. The convergence could also be observed by increasing the number of plate modals involved in Table 2. It could be suggested that the modal coupling method could achieve good convergence in solving the fluid-structural coupled problem described here. And it could also be observed that the convergence at lower frequencies is more rapid than that at higher frequencies, and the solution precision would be more dependent on accounting for more water sound modals. However, there is no need to carry out a high accuracy calculation here for a theoretical qualitative analysis, and in the later part of this paper, 50 plate modals and 500 water sound field modals are taken into consideration, by which totally 550 coupled modals could be revealed. And also, because there is no need to list all 550 modes here, only partial data (the first several modes) are listed in Tables 1 and 2Table 1
Coupled natural frequencies with different number of water sound modals involved.
Fluid-structural coupled natural frequencyfCk (Hz) Number of water sound modals involved 50 200 500 1000 2000  Number of plate modals involved 50 50 50 50 50  k 0 0 0 0 0 0 1 106.8 103.1 101.5 100.9 100.3 2 218.5 208.4 206.0 203.9 202.5 3 234.4 222.7 217.6 215.5 213.4 4 291.1 270.9 264.7 262.2 259.7 5 447.6 415.1 404.7 400.2 395.9 6 455.4 420.7 409.1 404.5 399.8 7 476.1 457.3 451.8 447.1 443.9 8 561.9 543.1 537.2 533.4 530.5 9 626.7 581.5 566.2 559.0 552.3 10 682.5 617.7 599.4 592.1 584.7Table 2
Coupled natural frequencies with different number of plate modals involved.
Fluid-structural coupled natural frequenciesfCk (Hz) Number of water sound modals involved 500 500 500 500 500  Number of plate modals involved 25 50 100 200 500  k 0 0 0 0 0 0 1 101.5 103.1 101.5 101.5 101.5 2 206.1 208.4 206.0 206.0 206.0 3 217.6 222.7 217.5 217.5 217.5 4 264.7 270.9 264.7 264.7 264.7 5 404.7 415.1 404.6 404.6 404.6 6 409.2 420.7 409.1 409.1 409.1 7 451.8 457.3 451.6 451.6 451.6 8 537.2 543.1 537.1 537.1 537.1 9 566.2 581.5 566.0 566.0 566.0 10 599.5 617.7 599.3 599.3 599.3Figure3 gives a comparison of sound pressure solutions between the modal coupling approach and FEM (by the software of LMS Virtual.Lab Acoustics) as a theoretical validation verification. The differences between the two results in Figure 3(a) are due to modal truncation; more modes are involved in the FEM/BEM software.Figure 3
Comparison of sound pressure solutions between the modal coupling approach and FEM.
(a)
Comparison of sound pressure solutions (b)
Cloud picture of boundary sound pressure by VL (540 Hz)The fluid-structural coupled natural frequencies resulting from (17) are compared with those of sound field with rigid boundaries obtained by (27) and simply supported steel plate obtained by (29), which are listed in Table 3. As mentioned above, 550 coupled modals have been obtained by involving 50 plate modals and 500 water sound field modals in modal coupling calculation, but only the first 11 coupled modal frequencies are presented.Table 3
Natural frequencies of the water sound field, plate, and fluid-structural coupled cavity.
Natural frequencies of water sound field Natural frequencies of plate Fluid-structural couplednatural frequencies r (mAr,nAr,lAr) f A r (Hz) j (mBj,nBj) f B j (Hz) k f C k (Hz) 0 ( 0,0 , 0 ) 0    0 0 1 ( 0,0 , 1 ) 1071.4 1 ( 1,1 ) 108.0 1 101.5 2 ( 0,1 , 0 ) 1250.0 2 ( 1,2 ) 207.6 2 206.0 3 ( 0,1 , 1 ) 1646.3 3 ( 2,1 ) 332.2 3 217.6 4 ( 1,0 , 0 ) 1875.0 4 ( 1,3 ) 373.7 4 264.7 5 ( 0,0 , 2 ) 2142.9 5 ( 2,2 ) 431.9 5 404.7 6 ( 1,0 , 1 ) 2159.5 6 ( 2,3 ) 598.0 6 409.1 7 ( 1,1 , 0 ) 2253.5 7 ( 1,4 ) 606.3 7 451.8 8 ( 0,1 , 2 ) 2480.8 8 ( 3,1 ) 705.9 8 537.2 9 ( 1,1 , 1 ) 2495.2 9 ( 3,2 ) 805.6 9 566.2 10 ( 0,2 , 0 ) 2500.0 10 ( 2,4 ) 830.5 10 599.4As a whole, it could be concluded that the fluid-structural frequencies are very different from those of the water sound field (with rigid boundaries) and the flexible boundary plate (in vacuo), which means that there would be a strong coupling between the enclosed water sound field and its elastic surrounding structures. If the fluid in the cavity was air (ρ0≈1.29 kg/m3; c0≈344 m/s), it could be found that the coupled natural frequencies fCk would approximately be equal to either some uncoupled structural natural frequencies fBj or some uncoupled cavity acoustical natural frequencies fAr, and that is a situation of weak coupling. The motion of each subsystem in a weakly coupled system will not be essentially different from that of the uncoupled systems. However, if the density of the medium in the cavity is much denser than air, such as water, the coupling may turn out to be strong, and big deformation of the resulting modes from the uncoupled panel and the cavity modes may be expected [18]. In this sense, the strong coupling could be judged by any unneglectable departure of coupled natural frequencies from every natural frequency of the uncoupled flexible structures and cavity.ExceptfA0=0, the natural frequencies of the water sound field are much higher than those of the simply supported plate, and the fluid-structural coupled natural frequencies are inclined to come to be rather lower. It seemed that one might carry out a comparison between the coupled natural frequencies and those of plate, and the change regulation of differences of adjacent fluid-structural natural frequencies fC(k+1)-fCk is similar to those fBj-fB(j-1) of the plate. However, the frequency distribution of coupled natural frequencies would turn to be lower and more crowded as the order k increased.In order to reveal modal coupling characteristics, Figures4 and 5 show several uncoupled plate modals WBj(σ) which are expressed by (30) and fluid-structural coupled plate modals WCk(σ) which are determined by (29) through modal coupling calculation. The figure shows that WC0 is almost the same as WB1, while it is true according to WC0’s expression and that is just an example of weak-coupled modal. It seemed that the coupled modal shape WC3 is similar to the uncoupled modal shape WB4; however, they are quite different in fact according to the expression of WC3, and that is a strong coupled modal. Phenomena of strong modal coupling are obvious when inspecting WC7 and WC8 shown in Figure 5.Figure 4
Some uncoupled plate modals.
(a)
W B 1 = sin ⁡ ( π x / L x ) sin ⁡ ( π y / L y ), fB1≈108.0 Hz (b)
W B 4 = sin ⁡ ( π x / L x ) sin ⁡ ( 3 π y / L y ), fB4≈373.4 Hz (c)
W B 8 = sin ⁡ ( 3 π x / L x ) sin ⁡ ( π y / L y ), fB4≈705.9 Hz (d)
W B 11 = sin ⁡ ( π x / L x ) sin ⁡ ( 5 π y / L y ), fB11≈905.2 HzFigure 5
Some coupled plate modals.
(a)
W C 0 ≈ 0.0248 ( W B 1 + 0.05 W B 4 + 0.02 W B 8 + 0.01 W B 11 ) (b)
W C 3 ≈ 0.0132 ( - 0.21 W B 1 + W B 4 + 0.04 W B 8 + 0.05 W B 11 + 0.03 W B 12 ) (c)
W C 7 ≈ 0.0074 ( - 0.28 W B 1 - 0.26 W B 4 + W B 8 + 0.14 W B 11 + 0.07 W B 12 ) (d)
W C 8 ≈ 0.0062 ( - 0.11 W B 1 - 0.16 W B 4 - W B 8 + 0.18 W B 11 + 0.08 W B 12 )The similar couplings have also happened to the acoustical cavity modals. And, moreover, except that the coupled acoustical modalFC0 is practically equal to FA0, that is, the rigid body modal of the uncoupled cavity sound field, all other fluid-structural coupled acoustical cavity modals are composed in a strong coupling manner; that is, they are linear combinations of several FAr, the uncoupled acoustical modals of the rigid wall cavity. As FCr(σ) is defined in three-dimensional space and it is inconvenient to plot it by a planar figure, Figure 6(a) illustrates one coupled acoustical modal shape in the plane z=Lz/2, and Figure 6(b) illustrates the appurtenant participant coefficients of the uncoupled acoustical cavity modals in the constitution of coupled acoustical cavity modal.Figure 6
Simple illustration of one coupled cavity modal shape.
(a)
F C 8 ( x , y , L z / 2 ), fC8≈537.2 Hz (b)
Participant coefficients of the uncoupled cavity modals


### 3.3. Power Flow Transmission

In the simulation model of Figure1, the vibratory power flow inputted by exterior excitation into the whole fluid-structural coupled system and the enclosed water sound field, that is, Pin calculated by (24) and Ptr calculated by (26), is dissipated by system damping. And thus the higher or lower power flow level would be a comprehensive indicator to measure the vibration level or energy level of the panel-cavity coupled system and water sound field.Figure7 shows the spectrum of input power flow Pin and transmitted power flow Ptr, in which the drop between Pin and Ptr is the dissipation power of the plate’s damping. Because the exterior excitation is symmetric (uniform PB as mentioned in Section 3.1), only symmetric modals are present, and the spectrum peaks at 0 Hz, 217.6 Hz, 451.8 Hz, and 537.2 Hz could be associated with the modals shown in Figures 5 and 6. There is some similarity between Pin or Ptr and the water sound pressure at the center of the plate’s interior surface, that is, P((Lx/2,Ly/2,0),ω) (refer to (21)), which is shown in Figure 8. However, the power flow is of evaluation of sound power.Figure 7
Spectrum ofPin and Ptr.Figure 8
Spectrum of water sound pressure.As a theoretical investigation, hypothesize that the material property parameters of the plate, that is, Young’s modulusE, mass density ρ, Poisson’s ratio μ, and damping loss factor, could be altered independently. In Figure 9(a), transmitted power flows are compared under different Young’s modulus of the elastic plate, where the value of 7.24×1010 Pa is by reference to aluminium. When the plate’s elasticity modulus decreases, the plate’s natural frequencies would decrease simultaneously, and this would move the fluid-structural coupled natural frequencies into lower frequency ranges, which has been predicted by Table 3. Thus the alteration of elasticity modulus of plate would lead to a phenomenon of “frequency shifting.” The Ptr spectrum with smaller plate’s elasticity modulus could be regarded as a contraction of that with greater elasticity modulus toward lower frequencies, and the peaks of Ptr would occur at relatively lower frequencies and become more crowded. However, since the variation range of Young’s modulus would be limited in practice, its influence on power flow transmission might not be very serious, and it could also be observed that reduction of Young’s modulus might bring about a benefit of slight reduction of Ptr’s peak valleys.Figure 9
Effect of plate’s elasticity modulus on power flow transmission.
(a)
Transmitted power flow (b)
Power flow ratioIn order to make an inspection of the alteration of differences betweenPin and Ptr with different plate elasticity modulus, Figure 9(b) shows the spectra of power flow ratio PR=10log⁡(Pin/Ptr). The peaks of PR would always appear around the resonance frequencies except at fC0, which could be easily explained by the fact that the plate’s damping consumes more energy when system resonances take place. In the lowest frequency range around fC0=0, the plate consumes little energy, and therefore Ptr≈Pin and PR≈0. It should be noted that greater PRs might not imply lower levels of transmitted power flow Ptr; the fact is probably just the opposite because the power flow input Pin might be in much higher levels at the same time. In this sense, minority of PR peak numbers would be a good design for noise isolation, which requires a greater plate elasticity modulus. And, instead, high PR values between adjacent resonance peaks of the PR spectra would be of real benefit for the purpose of Ptr attenuation, which could be discovered through a synthesized analysis of the figures shown.In Figure10(a), different mass density values were given to the elastic plate, in which the value of 2770 kg/m3 is by reference to aluminium. A significant feature of the spectra in the figure is that changes in plate’s mass density would apparently change the average level of transmitted power flow Ptr. According to (29), the increase of plate’s mass density would cause decreasing of plate’s natural frequencies and “frequency shifting” of fluid-structural coupled modals, as shown in Figure 10(a), similar to the situation of decreasing Young’s modulus in Figure 9. However, the increase of plate’s mass density would increase its modal masses at the same time. And that is the reason why Ptr’s average level is cut down even though more resonance modals would come into being in the relative lower frequency band. It could also be explained by the fact that a heavier vibrating mass would generate a greater reduction in dynamic force (or pressure) transmission. Figure 10(b) is about the power flow ratio PR. PR could not effectively reveal the apparent Ptr reduction at nonresonance frequencies, such as 800 Hz~1200 Hz, because the transmitted power flow Ptr is very close to the power flow input Pin at those frequencies.Figure 10
Effect of plate’s mass density on power flow transmission.
(a)
Transmitted power flow (b)
Power flow ratioAnother important influential factor that should be paid attention is the damping loss factor of the plate. Figure11(a) demonstrates a conflictive situation where smaller damping loss factor would increase the peaks of transmitted power flow at resonance frequencies, while at the broadband nonresonance frequencies smaller damping would be beneficial to reduction of transmitted power flow. To explain this result, one might make an analogy with the vibration isolation theory. If the elastic plate was considered as some kind of elastic isolator which was specially designed to attenuate the transmission of exterior excitation energy into the water-filled cavity, the damping would increase the power transmission and would not be expected except for attenuation of resonance peak. Through illustration of power flow ratio PR as that in Figure 11(b), it could be confirmed that greater damping could be used to obstruct energy transmission at resonance frequencies, whereas smaller damping would be favorable in the nonresonance frequency ranges, which would not be counted in power flow ratio.Figure 11
Effect of damping loss factor of plate on power flow transmission.
(a)
Transmitted power flow (b)
Power flow ratioPoisson’s ratioμ would affect the power flow transmission the same way Young’s modulus does as shown in Figure 9. Referring to (29), increasing Young’s modulus could be equivalent to increasing Poisson’s ratio. However, the variation scope of Poisson’s ratio is much smaller than that of Young’s modulus, and the power flow transmission would be affected more by Young’s modulus than by Poisson’s ratio. And, for this reason, no additional repetitive figures would be put forward here.


## 3.1. Simulation Model

A panel-cavity coupled system shown in Figure1 consists of a rectangular water-filled room with five rigid walls and one simply supported plate subject to exterior harmonic distributed force (pressure) pE(σ,t).Figure 1
A panel-cavity coupled system.In discussion of the distribution shape of exterior excitation, the plane harmonic wave incident is a common assumption. Suppose thatpE(σ,t)=PE·exp⁡[j(ωt-k′xsin⁡θ)], where PE is the amplitude, k′ is the wave number, and θ is the incident angle (pE is uniform along the y direction); pE has a perpendicular component pB(σ,t)=PEcos⁡θ·exp⁡-jk′xsin⁡θ·exp⁡jωt=PBσ·exp⁡(jωt). Generally, PB(σ) would be a complex function and have infinite variety of distribution shapes when the wave frequency, velocity, and incident angle changed. Figure 2 shows one example of distribution shape of PB(σ) with wave frequency f=500 Hz, velocity c=344 m/s, and incident angle θ=π/3. It is true that the modal coupling method is valid in dealing with those variant distribution shapes of PB(σ). However, some specific analysis on the special case with θ=0, that is, a uniform PB over the plate surface, would also give indications of general significance. The uniform excitation had been adopted by other authors previously [7, 15]. And, moreover, taking sonar array cavities as examples, they are regularly mounted on related ship structures through rubber blankets; the uniform structural excitation assumption would be a basic consideration.Figure 2
One example of nonuniform distribution shape ofPB(σ) (wave frequency f=500 Hz, velocity c=344 m/s, and incident angle θ=π/3).
(a)
Real part (b)
Imaginary partFor the water-filled rectangular room, the natural frequencies and modal functions of the sound field with rigid boundaries are determined by(27)ωAr2=K0ρ0mArπLx2+nArπLy2+lArπLz2(28)FArx,y,z=cos⁡mArπxLxcos⁡nArπyLycos⁡lArπzLz,where ∀mAr,nAr,lAr∈N3, r∈N, and let ωAr arrange in a sequence ωA02<ωA12<ωA22<⋯.For a simply supported plate, its natural frequencies and modal functions are determined by(29)ωBj2=π4Eh212ρ1-μ2mBj2Lx2+nBj2Ly22(30)WBjx,y,0=sin⁡mBjπxLxsin⁡nBjπyLy,where E, ρ, h, and μ are Young’s modulus, mass density, thickness, and Poisson’s ratio of the plate, respectively; ∀mBj,nBj∈N2,j∈N+, let ωBj arrange in a sequence ωB12<ωB22<ωB32<⋯.The geometrical and material properties adopted for numerical computation are as follows:Lx=0.4 m, Ly=0.6 m, Lz=0.7 m, and h=0.005 m; ρ0=1000 kg/m3 and K0=2.25×109Pa×(1+10-4j); E=2.0×1011Pa×(1+10-3j), ρ=7.8×103 kg/m3, and μ=0.28 (steel).


## 3.2. Modal Analysis

There is a convergence investigation about the fluid-structural coupled natural frequencies resulting from (17) at first. In Table 1, with a fixed number of plate modals involved in calculation, the convergence of coupled natural frequencies could be observed by increasing the number of water sound modals involved. The convergence could also be observed by increasing the number of plate modals involved in Table 2. It could be suggested that the modal coupling method could achieve good convergence in solving the fluid-structural coupled problem described here. And it could also be observed that the convergence at lower frequencies is more rapid than that at higher frequencies, and the solution precision would be more dependent on accounting for more water sound modals. However, there is no need to carry out a high accuracy calculation here for a theoretical qualitative analysis, and in the later part of this paper, 50 plate modals and 500 water sound field modals are taken into consideration, by which totally 550 coupled modals could be revealed. And also, because there is no need to list all 550 modes here, only partial data (the first several modes) are listed in Tables 1 and 2Table 1
Coupled natural frequencies with different number of water sound modals involved.
Fluid-structural coupled natural frequencyfCk (Hz) Number of water sound modals involved 50 200 500 1000 2000  Number of plate modals involved 50 50 50 50 50  k 0 0 0 0 0 0 1 106.8 103.1 101.5 100.9 100.3 2 218.5 208.4 206.0 203.9 202.5 3 234.4 222.7 217.6 215.5 213.4 4 291.1 270.9 264.7 262.2 259.7 5 447.6 415.1 404.7 400.2 395.9 6 455.4 420.7 409.1 404.5 399.8 7 476.1 457.3 451.8 447.1 443.9 8 561.9 543.1 537.2 533.4 530.5 9 626.7 581.5 566.2 559.0 552.3 10 682.5 617.7 599.4 592.1 584.7Table 2
Coupled natural frequencies with different number of plate modals involved.
Fluid-structural coupled natural frequenciesfCk (Hz) Number of water sound modals involved 500 500 500 500 500  Number of plate modals involved 25 50 100 200 500  k 0 0 0 0 0 0 1 101.5 103.1 101.5 101.5 101.5 2 206.1 208.4 206.0 206.0 206.0 3 217.6 222.7 217.5 217.5 217.5 4 264.7 270.9 264.7 264.7 264.7 5 404.7 415.1 404.6 404.6 404.6 6 409.2 420.7 409.1 409.1 409.1 7 451.8 457.3 451.6 451.6 451.6 8 537.2 543.1 537.1 537.1 537.1 9 566.2 581.5 566.0 566.0 566.0 10 599.5 617.7 599.3 599.3 599.3Figure3 gives a comparison of sound pressure solutions between the modal coupling approach and FEM (by the software of LMS Virtual.Lab Acoustics) as a theoretical validation verification. The differences between the two results in Figure 3(a) are due to modal truncation; more modes are involved in the FEM/BEM software.Figure 3
Comparison of sound pressure solutions between the modal coupling approach and FEM.
(a)
Comparison of sound pressure solutions (b)
Cloud picture of boundary sound pressure by VL (540 Hz)The fluid-structural coupled natural frequencies resulting from (17) are compared with those of sound field with rigid boundaries obtained by (27) and simply supported steel plate obtained by (29), which are listed in Table 3. As mentioned above, 550 coupled modals have been obtained by involving 50 plate modals and 500 water sound field modals in modal coupling calculation, but only the first 11 coupled modal frequencies are presented.Table 3
Natural frequencies of the water sound field, plate, and fluid-structural coupled cavity.
Natural frequencies of water sound field Natural frequencies of plate Fluid-structural couplednatural frequencies r (mAr,nAr,lAr) f A r (Hz) j (mBj,nBj) f B j (Hz) k f C k (Hz) 0 ( 0,0 , 0 ) 0    0 0 1 ( 0,0 , 1 ) 1071.4 1 ( 1,1 ) 108.0 1 101.5 2 ( 0,1 , 0 ) 1250.0 2 ( 1,2 ) 207.6 2 206.0 3 ( 0,1 , 1 ) 1646.3 3 ( 2,1 ) 332.2 3 217.6 4 ( 1,0 , 0 ) 1875.0 4 ( 1,3 ) 373.7 4 264.7 5 ( 0,0 , 2 ) 2142.9 5 ( 2,2 ) 431.9 5 404.7 6 ( 1,0 , 1 ) 2159.5 6 ( 2,3 ) 598.0 6 409.1 7 ( 1,1 , 0 ) 2253.5 7 ( 1,4 ) 606.3 7 451.8 8 ( 0,1 , 2 ) 2480.8 8 ( 3,1 ) 705.9 8 537.2 9 ( 1,1 , 1 ) 2495.2 9 ( 3,2 ) 805.6 9 566.2 10 ( 0,2 , 0 ) 2500.0 10 ( 2,4 ) 830.5 10 599.4As a whole, it could be concluded that the fluid-structural frequencies are very different from those of the water sound field (with rigid boundaries) and the flexible boundary plate (in vacuo), which means that there would be a strong coupling between the enclosed water sound field and its elastic surrounding structures. If the fluid in the cavity was air (ρ0≈1.29 kg/m3; c0≈344 m/s), it could be found that the coupled natural frequencies fCk would approximately be equal to either some uncoupled structural natural frequencies fBj or some uncoupled cavity acoustical natural frequencies fAr, and that is a situation of weak coupling. The motion of each subsystem in a weakly coupled system will not be essentially different from that of the uncoupled systems. However, if the density of the medium in the cavity is much denser than air, such as water, the coupling may turn out to be strong, and big deformation of the resulting modes from the uncoupled panel and the cavity modes may be expected [18]. In this sense, the strong coupling could be judged by any unneglectable departure of coupled natural frequencies from every natural frequency of the uncoupled flexible structures and cavity.ExceptfA0=0, the natural frequencies of the water sound field are much higher than those of the simply supported plate, and the fluid-structural coupled natural frequencies are inclined to come to be rather lower. It seemed that one might carry out a comparison between the coupled natural frequencies and those of plate, and the change regulation of differences of adjacent fluid-structural natural frequencies fC(k+1)-fCk is similar to those fBj-fB(j-1) of the plate. However, the frequency distribution of coupled natural frequencies would turn to be lower and more crowded as the order k increased.In order to reveal modal coupling characteristics, Figures4 and 5 show several uncoupled plate modals WBj(σ) which are expressed by (30) and fluid-structural coupled plate modals WCk(σ) which are determined by (29) through modal coupling calculation. The figure shows that WC0 is almost the same as WB1, while it is true according to WC0’s expression and that is just an example of weak-coupled modal. It seemed that the coupled modal shape WC3 is similar to the uncoupled modal shape WB4; however, they are quite different in fact according to the expression of WC3, and that is a strong coupled modal. Phenomena of strong modal coupling are obvious when inspecting WC7 and WC8 shown in Figure 5.Figure 4
Some uncoupled plate modals.
(a)
W B 1 = sin ⁡ ( π x / L x ) sin ⁡ ( π y / L y ), fB1≈108.0 Hz (b)
W B 4 = sin ⁡ ( π x / L x ) sin ⁡ ( 3 π y / L y ), fB4≈373.4 Hz (c)
W B 8 = sin ⁡ ( 3 π x / L x ) sin ⁡ ( π y / L y ), fB4≈705.9 Hz (d)
W B 11 = sin ⁡ ( π x / L x ) sin ⁡ ( 5 π y / L y ), fB11≈905.2 HzFigure 5
Some coupled plate modals.
(a)
W C 0 ≈ 0.0248 ( W B 1 + 0.05 W B 4 + 0.02 W B 8 + 0.01 W B 11 ) (b)
W C 3 ≈ 0.0132 ( - 0.21 W B 1 + W B 4 + 0.04 W B 8 + 0.05 W B 11 + 0.03 W B 12 ) (c)
W C 7 ≈ 0.0074 ( - 0.28 W B 1 - 0.26 W B 4 + W B 8 + 0.14 W B 11 + 0.07 W B 12 ) (d)
W C 8 ≈ 0.0062 ( - 0.11 W B 1 - 0.16 W B 4 - W B 8 + 0.18 W B 11 + 0.08 W B 12 )The similar couplings have also happened to the acoustical cavity modals. And, moreover, except that the coupled acoustical modalFC0 is practically equal to FA0, that is, the rigid body modal of the uncoupled cavity sound field, all other fluid-structural coupled acoustical cavity modals are composed in a strong coupling manner; that is, they are linear combinations of several FAr, the uncoupled acoustical modals of the rigid wall cavity. As FCr(σ) is defined in three-dimensional space and it is inconvenient to plot it by a planar figure, Figure 6(a) illustrates one coupled acoustical modal shape in the plane z=Lz/2, and Figure 6(b) illustrates the appurtenant participant coefficients of the uncoupled acoustical cavity modals in the constitution of coupled acoustical cavity modal.Figure 6
Simple illustration of one coupled cavity modal shape.
(a)
F C 8 ( x , y , L z / 2 ), fC8≈537.2 Hz (b)
Participant coefficients of the uncoupled cavity modals


## 3.3. Power Flow Transmission

In the simulation model of Figure1, the vibratory power flow inputted by exterior excitation into the whole fluid-structural coupled system and the enclosed water sound field, that is, Pin calculated by (24) and Ptr calculated by (26), is dissipated by system damping. And thus the higher or lower power flow level would be a comprehensive indicator to measure the vibration level or energy level of the panel-cavity coupled system and water sound field.Figure7 shows the spectrum of input power flow Pin and transmitted power flow Ptr, in which the drop between Pin and Ptr is the dissipation power of the plate’s damping. Because the exterior excitation is symmetric (uniform PB as mentioned in Section 3.1), only symmetric modals are present, and the spectrum peaks at 0 Hz, 217.6 Hz, 451.8 Hz, and 537.2 Hz could be associated with the modals shown in Figures 5 and 6. There is some similarity between Pin or Ptr and the water sound pressure at the center of the plate’s interior surface, that is, P((Lx/2,Ly/2,0),ω) (refer to (21)), which is shown in Figure 8. However, the power flow is of evaluation of sound power.Figure 7
Spectrum ofPin and Ptr.Figure 8
Spectrum of water sound pressure.As a theoretical investigation, hypothesize that the material property parameters of the plate, that is, Young’s modulusE, mass density ρ, Poisson’s ratio μ, and damping loss factor, could be altered independently. In Figure 9(a), transmitted power flows are compared under different Young’s modulus of the elastic plate, where the value of 7.24×1010 Pa is by reference to aluminium. When the plate’s elasticity modulus decreases, the plate’s natural frequencies would decrease simultaneously, and this would move the fluid-structural coupled natural frequencies into lower frequency ranges, which has been predicted by Table 3. Thus the alteration of elasticity modulus of plate would lead to a phenomenon of “frequency shifting.” The Ptr spectrum with smaller plate’s elasticity modulus could be regarded as a contraction of that with greater elasticity modulus toward lower frequencies, and the peaks of Ptr would occur at relatively lower frequencies and become more crowded. However, since the variation range of Young’s modulus would be limited in practice, its influence on power flow transmission might not be very serious, and it could also be observed that reduction of Young’s modulus might bring about a benefit of slight reduction of Ptr’s peak valleys.Figure 9
Effect of plate’s elasticity modulus on power flow transmission.
(a)
Transmitted power flow (b)
Power flow ratioIn order to make an inspection of the alteration of differences betweenPin and Ptr with different plate elasticity modulus, Figure 9(b) shows the spectra of power flow ratio PR=10log⁡(Pin/Ptr). The peaks of PR would always appear around the resonance frequencies except at fC0, which could be easily explained by the fact that the plate’s damping consumes more energy when system resonances take place. In the lowest frequency range around fC0=0, the plate consumes little energy, and therefore Ptr≈Pin and PR≈0. It should be noted that greater PRs might not imply lower levels of transmitted power flow Ptr; the fact is probably just the opposite because the power flow input Pin might be in much higher levels at the same time. In this sense, minority of PR peak numbers would be a good design for noise isolation, which requires a greater plate elasticity modulus. And, instead, high PR values between adjacent resonance peaks of the PR spectra would be of real benefit for the purpose of Ptr attenuation, which could be discovered through a synthesized analysis of the figures shown.In Figure10(a), different mass density values were given to the elastic plate, in which the value of 2770 kg/m3 is by reference to aluminium. A significant feature of the spectra in the figure is that changes in plate’s mass density would apparently change the average level of transmitted power flow Ptr. According to (29), the increase of plate’s mass density would cause decreasing of plate’s natural frequencies and “frequency shifting” of fluid-structural coupled modals, as shown in Figure 10(a), similar to the situation of decreasing Young’s modulus in Figure 9. However, the increase of plate’s mass density would increase its modal masses at the same time. And that is the reason why Ptr’s average level is cut down even though more resonance modals would come into being in the relative lower frequency band. It could also be explained by the fact that a heavier vibrating mass would generate a greater reduction in dynamic force (or pressure) transmission. Figure 10(b) is about the power flow ratio PR. PR could not effectively reveal the apparent Ptr reduction at nonresonance frequencies, such as 800 Hz~1200 Hz, because the transmitted power flow Ptr is very close to the power flow input Pin at those frequencies.Figure 10
Effect of plate’s mass density on power flow transmission.
(a)
Transmitted power flow (b)
Power flow ratioAnother important influential factor that should be paid attention is the damping loss factor of the plate. Figure11(a) demonstrates a conflictive situation where smaller damping loss factor would increase the peaks of transmitted power flow at resonance frequencies, while at the broadband nonresonance frequencies smaller damping would be beneficial to reduction of transmitted power flow. To explain this result, one might make an analogy with the vibration isolation theory. If the elastic plate was considered as some kind of elastic isolator which was specially designed to attenuate the transmission of exterior excitation energy into the water-filled cavity, the damping would increase the power transmission and would not be expected except for attenuation of resonance peak. Through illustration of power flow ratio PR as that in Figure 11(b), it could be confirmed that greater damping could be used to obstruct energy transmission at resonance frequencies, whereas smaller damping would be favorable in the nonresonance frequency ranges, which would not be counted in power flow ratio.Figure 11
Effect of damping loss factor of plate on power flow transmission.
(a)
Transmitted power flow (b)
Power flow ratioPoisson’s ratioμ would affect the power flow transmission the same way Young’s modulus does as shown in Figure 9. Referring to (29), increasing Young’s modulus could be equivalent to increasing Poisson’s ratio. However, the variation scope of Poisson’s ratio is much smaller than that of Young’s modulus, and the power flow transmission would be affected more by Young’s modulus than by Poisson’s ratio. And, for this reason, no additional repetitive figures would be put forward here.


## 4. Conclusions

Backgrounded by evaluation or control of mechanical self-noise in sonar array cavity, transmitted power flow or sound power input calculation is carried out by modal coupling analysis on the fluid-structural vibration of the fluid-filled enclosure with elastic boundaries. Power flow transmission analysis is presented through a numerical simulation example of water-filled rectangular panel-cavity coupled system. Detailed discussion is carried out about power flow transmission characteristics affected by variation of material property parameters of cavity’s elastic boundary structure, aiming at reduction of water sound level inside the cavity. From the results, one could draw the following conclusions.(1) Power flow or sound power transmission analysis could be a valuable method for evaluation or prediction of water sound level in dealing with strong coupled vibration problems of water-filled acoustoelastic enclosure systems.(2) The fluid-structural coupled natural frequencies of a water-filled acoustoelastic enclosure would be greatly affected by interaction between the hydroacoustic field and its surrounding elastic boundaries. There would be a tendency that the fluid-structural coupled natural frequencies turn to be smaller in value and more crowded in frequency distribution than those of elastic boundary structures and inner water sound field in rigid boundary condition.(3) Decreasing elasticity modulus or Poison’s ratio of water sound cavity’s thin-wall structures would cause the frequency distribution of system modals to contract toward lower frequency ranges and result in more power flow transmission peaks in the lower frequency band, while slight reduction of peak valleys of transmitted power flow could also be expected on the other hand.(4) Denser material could be beneficial to an apparent attenuation of average level of power flow transmission into the water-filled enclosure, even though it would be accompanied with a decrease of system natural frequencies.(5) Smaller inner damping of enclosure’s thin-wall structures, which were distinguished from that specially designed for sound absorption destination and only for the purpose of suppression of resonance peaks, might be proposed to attenuate the average level of power flow transmission.



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*Source: 5273280-2018-05-02.xml*