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

Research on Power Flow Transmission through Elastic Structure into a Fluid-Filled Enclosure Hindawi Advances in Acoustics and Vibration Volume 2018, Article ID 5273280, 16 pages https://doi.org/10.1155/2018/5273280 Research Article Research on Power Flow Transmission through Elastic Structure into a Fluid-Filled Enclosure 1,2 1,2 1,2 1,2 Rui Huo , Chuangye Li , Laizhao Jing , and Weike Wang School of Mechanical Engineering, Shandong University, Jinan 250061, China Key Laboratory of High Ecffi iency and Clean Mechanical Manufacture (Shandong University), Ministry of Education, Jinan 250061, China Correspondence should be addressed to Rui Huo; huorui@sdu.edu.cn Received 30 October 2017; Revised 8 February 2018; Accepted 15 March 2018; Published 2 May 2018 Academic Editor: Kim M. Liew Copyright © 2018 Rui Huo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. eTh work of this paper is backgrounded by prediction or evaluation and control of mechanical self-noise in sonar array cavity. eTh vibratory power flow transmission analysis is applied to reveal th e overall vibration level of the uid-st fl ructural coupled system. Through modal coupling analysis on the uid-st fl ructural vibratio n of the uid-filled fl 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 uid-st fl ructural coupled system and the net power flow transmitted into the hydroacoustic field. Characteristics of t he coupled natural frequencies and modals are investigated by a numerical example of a rectangular water-filled cavity with vfi e acoustic rigid walls and one elastic panel. Inu fl ential factors of power flow transmission characteristics are further discussed with the purpose of overall evaluation and reduction of the cavity water sound energy. 1. Introduction from exterior air-borne sound [4]. In these cases, weak coupling has been commonly assumed because of the low 1.1. Background. The work of this paper is backgrounded by density of air and high stiffness of cabin wall, which means prediction or evaluation and control of mechanical self-noise that the cavity’s interior sound pressure would have little in sonar array cavity. eTh mechanical self-noise, which is influence on the vibration of cavity wall, and modals of caused by structural vibration of sonar cavity’s wall, might interior sound efi ld would also be aeff cted very lightly [5]. In signicfi antly weaken the detection performance of sonar at contrast, a much stronger coupling might be present when a lower frequencies [1, 2]. The sources of mechanical self- water sound field takes place of the air [6]. noise might be multiple such as vibrating machines on the eTh sound pressure is most commonly used to represent ship which diffuse vibration energy or second excitation of the property of sound field in the study of acoustic-structural structure-borne sound. However, it is essential to compre- coupling of acoustoelastic enclosure. The ratio of sound pres- hend the characteristics of interaction between the enclosed sure at the outside surface of the elastic cavity wallboard to water sound field and its elastic boundary structures for the that at internal surface, which is den fi ed as “noise reduction,” purpose of prediction, evaluation, and control of interior is applied to evaluation of sound transmission characteristics hydroacoustic noise [3]. [4, 7]. Since the sound pressure would change greatly at The subjects of cabin noise in various flight vehicles dieff rentpointsofthesoundfield,thevalueofnoisereduction and automobiles are more familiar in the investigation of would also be very different, and a comprehensive measure, uid-s fl tructural coupled vibration of acoustoelastic enclosure, for example, power flow, would be expected for an overall which mainly focus on characteristics of sound transmission evaluation of vibration level of the enclosed sound efi ld. through the elastic wall into interior sound efi ld resulting eTh power flow has been validated and widely utilized as 2 Advances in Acoustics and Vibration a comprehensive measure for evaluation of overall level of vibratory power flow calculation, especially based on Dowell’s vibration energy of vibration isolation systems mounted on modal coupling theory. flexible foundations [8], which could also be explained as averagesound powerwhenappliedtosoundfieldanaly- 2. Theory sis. 2.1. Equations of Fluid-Structural Coupled Vibration. Con- In this paper, through modal coupling analysis on the sider that a uid-fi fl lled enclosure occupies a volume 𝑉.Its uid-s fl tructural vibration of the water-filled enclosure with boundary 𝐷=𝐷 +𝐷 ,where 𝐷 =0 ̸ represents the flexi- elastic boundaries, an ecffi ient computational method is 𝑆 𝐹 𝐹 ble area of the surrounding wall and 𝐷 (might be zero) deduced to determine the vibratory power flow generated 𝑆 represents the acoustic rigid area. by exterior excitations on the outside surface of the elastic The u fl id inside the enclosure satisfied the wave equation structure, including the total power flow entering into the and associated boundary condition. uid-s fl tructural coupled system and the net power flow transmitted into the hydroacoustic efi ld. Characteristics of 𝜕 𝑝 ( 𝜎, 𝑡 ) (1) the coupled natural frequencies and modals are investigated 𝐾 ∇ 𝑝 ( 𝜎, 𝑡 )−𝜌 =0 (𝜎∈𝑉 ) 0 0 𝜕𝑡 by a numerical example of a rectangular water-filled cavity with vfi e acoustic rigid walls and one elastic panel. Inu fl ential ( 𝜎, 𝑡 ) (2) =−𝜌 𝑎 ( 𝜎, 𝑡 ) (𝜎 ∈ 𝐷 ) 0 𝑛 𝐹 factors of power flow transmission characteristics are further 𝜕𝑛 discussed with the purpose of overall evaluation and reduc- ( 𝜎, 𝑡 ) tion of the cavity water sound energy. (3) =0 (𝜎 ∈𝐷 ), 𝜕𝑛 1.2. eTh oretical Development. There has been a continuous where 𝑝(𝜎, 𝑡) is the sound pressure at point 𝜎(𝑥, 𝑦, 𝑧) ∈ 𝑉; effort for decades on investigation of uid-s fl tructural mech- 𝑎 (𝜎, 𝑡) is the acceleration of the flexible wall in the normal anismofclosedsound efi ldwithflexibleboundaries.Ithas direction 𝑛 (positive outward); 𝜌 and 𝐾 are the equilibrium 0 0 been recommended that the locally reactive acoustic normal uid fl density and uid fl volume stiffness, respectively. impedance was the earlier theory to understand the sound If 𝐷 =0,(1)hasmodal solutions 𝐹 (𝜎)⋅exp(𝑗𝜔 𝑡), 𝑟 = absorption caused by the interaction between a reverberation 0, 1, 2, . . .,where 𝜔 is the 𝑟th acoustical natural frequency in room and its surrounding walls [9]. Later attention was paid the condition of rigid boundary and 𝐹 (𝜎)is the correspond- to the modal coupling between the enclosed sound efi lds and ing natural mode with orthogonality as follows: theflexiblewalls torevealthemorecomplicatedmechanism 0𝑟 =𝑠̸ demonstrated by experimental results, which could not be 𝐹 ( 𝜎 )𝐹 ( 𝜎 ) ∭ dV = interpreted by the locally reactive theory [10, 11]. 𝜌 𝑐 0 𝑀 𝑟=𝑠, The modal responses of acoustoelastic enclosures were (4) first developed by Dowell et al. [12, 13] by applying Green’s 0𝑟 =𝑠̸ [∇𝐹 ( 𝜎 ) ] ⋅[∇𝐹 ( 𝜎 ) ] function to the inhomogeneous wave differential equation ∭ dV = of the enclosed sound field and applying the classical modal 𝜌 0 𝜔 𝑀 𝑟=𝑠, and eigenvalue theorem to the simultaneous uid-s fl tructural differential equations to result in a resolution of coupling where 𝑐 = √𝐾 /𝜌 is the acoustic velocity of the u fl id, 𝑀 0 0 0 modals. eTh re are still other resolution methods for the same is the 𝑟th acoustical modal mass in the condition of rigid acoustoelasticity equations which could be referred to, such boundary, and ∇𝐹 (𝜎) = [𝜕𝐹 /𝜕𝑥, 𝐹𝜕 /𝜕𝑦, 𝐹𝜕 /𝜕𝑧] is as Laplace transformation [14] and Ritz series [15]. In general, thecolumngradientvectorofmodalfunction 𝐹 (𝜎). Dowell’s method is based on the familiar uncoupled acoustic Consider the solution of (1) with 𝐷 =0 ̸ being in the form enclosure modes and structural modes, could be more of modal superposition; that is, easily implemented, and has been successfully applied to the investigation of variety of uid-s fl tructural interaction systems 𝑝 ( 𝜎, 𝑡 )= ∑𝐹 ( 𝜎 )𝑃 ( 𝑡 )= [F ( 𝜎 ) ] P ( 𝑡)( 𝜎∈𝑉 ), (5) [16, 17]. Beginning with the “modal coupling method,” Pan 𝑟=0 andBiesgaveaninsightanalysis oftheweak-coupledand where F(𝜎)and P(𝑡)are column vectors of modal function well-coupled modals and their decay characteristics of a 𝐹 (𝜎)and its corresponding modal coordinate 𝑃 (𝑡),respec- 𝑟 𝑟 rectangular panel-cavity coupled system [18, 19]; Davis put tively; that is, F(𝜎) = [𝐹 (𝜎), 𝐹 (𝜎), 𝐹 (𝜎), . . .] and P(𝑡) = 𝐴0 𝐴1 𝐴2 forward a method for approximate estimation of the coupled [𝑃 (𝑡), 𝑃 (𝑡), 𝑃 (𝑡), . . .]. 0 1 2 natural frequencies of acoustoelastic enclosures by “coupling Aeft r substituting (5) into (1), multiply both sides of the coefficient” [20]. resultant equation with a left-multiplication matrix (vector) Other important developments might lie in the field of F(𝜎)and finally integrating the equation over volume 𝑉,one discrete numerical techniques, such as FEM/BEM, for uid- fl obtains structural vibration analysis. However, these methods are usually preferred in the investigation of irregularly shaped 2 ∭ F ( 𝜎 )[∇ 𝑝 ( 𝜎, 𝑡 ) ]dV cavities and targeting specicfi engineering problems. And that (6) wouldbebeyondthe discussionofthis paper, whichwould mainly focus on a general theoretical evaluation method for − ∭ {F ( 𝜎 )[F ( 𝜎 ) ] P ( 𝑡 ) }dV =0. the overall vibration level of a uid-fil fl led enclosure through 0 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑠 𝐴𝑟 𝐴𝑟 𝐴𝑠 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝜕𝑝 𝜕𝑝 Advances in Acoustics and Vibration 3 By applying Green’s theorem to the rfi st term of above of the modal masses and natural frequencies respectively, equation, one has and 𝜔 and 𝑀 =∬ 𝑚 [𝑊 (𝜎)]d𝐴 (𝑗 = 1, 2, 3, . . .) represent the 𝑗th natural frequency and modal mass of the ∭ {F ( 𝜎 )[F ( 𝜎 ) ] P ( 𝑡 ) }dV thin-wall structures in vacuo, respectively. Q (𝑡)and Q (𝑡) 2 𝐴 𝐵 are column vectors of the general forces due to 𝑝 (𝜎, 𝑡) and 𝑝 (𝜎, 𝑡) loaded on the thin-wall structures in vacuo, (7) + ∭ [∇F ( 𝜎 ) ] ∇F ( 𝜎 )P ( 𝑡 )dV respectively, and ( 𝜎, 𝑡 ) Q ( 𝑡 )= ∬ W ( 𝜎 )𝑝 ( 𝜎, 𝑡 )d𝐴, 𝐴 𝐴 = ∯ F ( 𝜎 ) d𝐴, 𝜕𝑛 𝐹 (12) where ∇F(𝜎)is the gradient matrix of modal function 𝐹 (𝜎): Q ( 𝑡 )=− ∬ W ( 𝜎 )𝑝 ( 𝜎, 𝑡 )d𝐴. 𝐵 𝐵 ∇F(𝜎) = [∇𝐹 (𝜎), ∇𝐹 (𝜎), ∇𝐹 (𝜎), . . .]. 𝐷 𝐴0 𝐴1 𝐴2 Now substitute the boundary condition equations (2)∼(3) The right-hand term of (8) and the rst fi term Q (𝑡)on and orthogonality equation (5) into (7): the right hand of (11) are of the u fl id-structural interaction between the sound efi ld inside the cavity and its flexible walls. M [P ( 𝑡 )+ Ω P ( 𝑡 )]=−∯ F ( 𝜎 )𝑎 ( 𝜎, 𝑡 )d𝐴, (8) 𝐴 𝑛 Substituting (10) into the right-hand term of (8) and taking 2 2 notice of 𝑎 (𝜎, 𝑡) = 𝜕 𝑤(𝜎, 𝑡)/𝜕𝑡 at 𝜎∈𝐷 and 𝑎 (𝜎, 𝑡) = 0 𝑛 𝐹 𝑛 where M and Ω are diagonal matrices of acoustical modal at 𝜎∈𝐷 , one could define a coupling matrix L as follows: 𝐴 𝐴 masses and natural frequencies, respectively; that is, M = diag[𝑀 ,𝑀 ,𝑀 ,...]and Ω = diag[𝜔 ,𝜔 ,𝜔 ,...]. 𝐴0 𝐴1 𝐴2 𝐴 𝐴0 𝐴1 𝐴2 L = ∬ F ( 𝜎 )[W ( 𝜎 ) ] d𝐴=[𝐿 ] The flexible boundary of the cavity is assumed to be thin- (13) wall structures, where linear partial differential equations (𝑟 = 0,1,2,...; 𝑗 = 1,2,3,...), wouldbeadoptedtofit thethin-wall structures’vibration, such that where 𝐿 denotes the element of the coupling matrix L at the 𝑟th row and the 𝑗th column. 𝜕 𝑤 ( 𝜎, 𝑡 ) 𝑆𝑤 ( 𝜎, 𝑡 )+𝑚 =𝑝 ( 𝜎, 𝑡 )−𝑝 ( 𝜎, 𝑡 ) 𝐵 𝐴 𝐵 And (8) turns into 𝜕𝑡 (9) ̈ ̈ (𝜎 ∈ 𝐷 ), M [P ( 𝑡 )+ Ω P ( 𝑡 )]=−L ⋅ B ( 𝑡 ). (14) 𝐹 𝐴 where 𝑆 is a linear differential operator representing struc- Dealing with Q (𝑡), one could express𝑝 (𝜎, 𝑡) in (12) 𝐴 𝐴 tural stiffness; 𝑚 is structural mass per unit area; 𝑝 (𝜎, 𝑡) with(5), and(11) wouldbecome 𝐵 𝐴 and 𝑝 (𝜎, 𝑡) areexcitations onthesurfaceofthethin-wall 2 𝑇 M [B 𝑡 + Ω B 𝑡 ]= L ⋅ P 𝑡 + Q 𝑡 . (15) () () () () structures due to the cavity acoustics and external dynamical 𝐵 𝐵 𝐵 forces (intensity of pressure), respectively; 𝑤(𝜎, 𝑡) is the displacement response of the thin-wall structures, which is 2.2. Modal Analysis. In order to carry out a modal analysis defined in the normal direction of 𝐷 . about the uid-s fl tructural vibration system governed by (14) eTh solution of (9) could be expressed as and (15), let Q (𝑡) = 0, and suppose that there exist vibration solutions as follows: 𝑤 ( 𝜎, 𝑡 )= ∑ 𝑊 ( 𝜎 )𝐵 ( 𝑡 )= [W ( 𝜎 ) ] B ( 𝑡 ) −1 P ( 𝑡 )=( M ) ⋅ 𝜒 ⋅ exp (𝑗𝜔𝑡) , 𝑗=1 (10) 𝐴 (16) (𝜎 ∈ 𝐷 ), −1 B 𝑡 =(Ω √M ) ⋅ 𝜒 ⋅ exp (𝑗𝜔𝑡) , () 𝐵 𝐵 𝐵 where 𝑊 (𝜎)is the 𝑗th modal function that is defined on 𝐷 and concerned with the property of the thin-wall structures where √M = diag⌊√𝑀 , √𝑀 , √𝑀 ,...⌋ and √M = 𝐴 𝐴0 𝐴1 𝐴2 𝐵 in vacuo and 𝐵 (𝑡)is the modal coordinate corresponding diag⌊√𝑀 , √𝑀 , √𝑀 ,...⌋ are square roots of the diago- 𝐵1 𝐵2 𝐵3 to 𝑊 (𝜎); W(𝜎) and B(𝑡)are column vectors of 𝑊 (𝜎) nalacousticalmodalmatrix M and the diagonal structural and 𝐵 (𝑡),respectively;that is, W(𝜎) = [𝑊 (𝜎), 𝑊 (𝜎), 𝑗 𝐵1 𝐵2 𝑇 modal matrix M ,respectively. 𝜒 =[𝜒 ,𝜒 ,𝜒 ,...] and 𝐵 𝐴 𝐴0 𝐴1 𝐴2 𝑇 𝑇 𝑊 (𝜎), . . .] and B(𝑡) = [𝐵 (𝑡), 𝐵 (𝑡), 𝐵 (𝑡), . . .]. 𝐵3 1 2 3 𝜒 =[𝜒 ,𝜒 ,...] arecolumnvectors offluid-structural 𝐵 𝐵1 𝐵2 By substituting (10) into (9) and using the orthogonality coupled modal shape coecffi ients related to the cavity sound of 𝑊 (𝜎),there wouldbeamodaldieff rential function as efi ld and the flexible boundary structures, respectively. follows: Substituting (16) into (14) and (15), an eigenvalue problem 2 could be obtained as M [B 𝑡 + Ω B 𝑡 ]= Q 𝑡 + Q 𝑡 , (11) () () () () 𝐵 𝐵 𝐴 𝐵 A A 𝜒 𝜒 11 12 𝐴 𝐴 2 2 where M and Ω , expressed as M = diag[𝑀 ,𝑀 ,𝑀 , A𝜒 =[ ][ ]=𝜔 [ ]=𝜔 𝜒, (17) 𝐵 𝐵 𝐵 𝐵1 𝐵2 𝐵3 A A 𝜒 𝜒 21 22 ...]and Ω = diag[𝜔 ,𝜔 ,𝜔 ,...], are diagonal matrices 𝐵 𝐵 𝐵 𝐵1 𝐵2 𝐵3 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝑟𝑗 𝑟𝑗 𝜕𝑝 𝐵𝑗 𝐵𝑗 𝐵𝑗 4 Advances in Acoustics and Vibration where 𝜒 could be named as vector of fluid-structural coupled transformation matrix to transform the general force P modal shape coefficients and A is a symmetric characteristic into its uid-s fl tructural expression (the derivation of the matrix, and A’s partitioned matrices could be calculated by matrices H and T has been explained via (A.5)∼(A.8) in 𝐻 𝐻 2 −1 −1 𝑇 −1 𝑇 Appendix A.2). M = X ⋅ Χ + X ⋅ Χ is a diagonal A = Ω +( √M ) LM L ( √M ) , A = A = 𝐶 𝐴 𝐴 𝐵 𝐵 11 𝐴 𝐴 12 𝐴 𝐵 21 −1 −1 2 matrix of the uid-s fl tructural coupled modal masses, and √ √ −( M ) LΩ ( M ) ,and A = Ω . 𝐴 𝐵 𝐵 22 𝐵 Ω = diag[𝜔 ,𝜔 ,𝜔 ,...]is a diagonal matrix of the u fl id- Equation (17) would give eigenvalues of matrix A,thatis, 𝐶 𝐶0 𝐶1 𝐶2 2 2 structural coupled natural frequencies. The superscript “𝐻” 𝜔 =Ω (1 + 𝑗𝜂 ), and the accompanying eigenvectors denotes Hermitian transposition of matrices. (𝑘) 𝑇 (𝑘) 𝑇 (𝑘) 𝑇 (𝑘) (𝑘) (𝑘) [𝜒 ] =[{𝜒 } ,{𝜒 } ]=[𝜒 ,𝜒 ,𝜒 ,..., 𝐴 𝐵 𝐴0 𝐴1 𝐴2 The power flow (density) inputted by exterior excitation (𝑘) (𝑘) (𝑘) 𝜒 ,𝜒 ,𝜒 ,...], 𝑘 = 0,1,2,... ,where Ω corre- 𝐵1 𝐵2 𝐵3 𝑝 (𝜎, 𝑡) into the uid-s fl tructural system is sponds to the 𝑘th uid-s fl tructural natural frequency. 𝜂 𝜔/2𝜋 is the loss factor associated with the 𝑘th damped normal 𝑝 ( 𝜎, 𝜔 )= ∫ Re {𝑝 ( 𝜎, 𝑡 ) } in 𝐵 mode, which might be resulting from the introduction of 2𝜋 a complex stiffness of the flexible boundary or a complex 𝜕𝑤 ( 𝜎, 𝑡 ) volume stiffness 𝐾 of the u fl id in consideration of the 0 (23) ⋅ Re {− } d𝑡 damping properties of the u fl id-structural system. It should 𝜕𝑡 (𝑘) also be noticed that [𝜒 ] might be complex vectors when =− Re {𝑗𝑊 𝜎, 𝜔 ⋅𝑃 𝜎 }(𝜎∈𝐷 ). ( ) ( ) 𝜔 are complex numbers. 𝐵 𝐹 The uid-s fl tructural coupled modal functions of the The total power flow input is cavity’s sound efi ld and the flexible boundaries would be expressed as 𝑇 𝑇 𝑃 ( 𝜔 )= ∬ 𝑝 ( 𝜎, 𝑡 )d𝐴=− Re {𝑗P T H ( 𝜔 ) in in 𝐵 −1 𝑇 (𝑘) 𝐹 ( 𝜎 )= [F ( 𝜎 ) ] ( M ) 𝜒 (𝜎∈𝑉 ), −1 𝑇 ∗ (18) √ (24) ⋅ X (Ω M ) ∬ W ( 𝜎 )⋅𝑃 ( 𝜎 )d𝐴} = −1 𝐵 𝐵 𝐵 𝐵 𝑇 (𝑘) 2 𝑊 ( 𝜎 )= [W ( 𝜎 ) ] (Ω √M ) 𝜒 (𝜎 ∈ 𝐷 ). 𝐵 𝐵 𝐵 𝐹 −1 𝑇 𝑇 𝑇 ∗ ⋅ Re {𝑗 P T H ( 𝜔 )⋅ X (Ω M ) P }, 𝐵 𝐵 𝐵 𝐵 𝐵 2.3. Vibratory Power Flow Transmission. In the condition that the flexible boundary structures of the cavity are subjected where the superscript “∗” denotes conjugation of complex to a harmonic exterior excitation, that is, 𝑝 (𝜎, 𝑡) = 𝑃 (𝜎) ⋅ 𝐵 𝐵 numbers. exp(𝑗𝜔𝑡),letP =−∬ W(𝜎)⋅𝑃 (𝜎)d𝐴; P indicates that the 𝐵 𝐵 𝐵 The power flow (density) transmitted through the uid- fl amplitudes of general forces belong to the uncoupled flexible structural interaction boundary of the cavity into the structures. eTh steady responses of the uid-s fl tructural cou- enclosed sound eld fi is plingcavitywouldbe 𝜔/2𝜋 𝑝 𝜎, 𝜔 = ∫ Re {𝑝 𝜎, 𝑡 } ( ) ( ) 󵄨 tr 󵄨𝜎∈𝐷 −1 𝐹 2𝜋 𝑝 𝜎, 𝑡 = F 𝜎 (√M ) Χ HTP exp (𝑗𝜔𝑡) ( ) [ ( ) ] 𝐴 𝐴 𝐵 (19) 𝜕𝑤 𝜎, 𝑡 ( ) (25) ⋅ Re {− 󵄨 } d𝑡 =𝑃 ( 𝜎, 𝜔 )⋅ exp (𝑗𝜔𝑡) (𝜎∈𝑉 ) 󵄨 𝜕𝑡 󵄨𝜎∈𝐷 −1 = Re {𝑗𝑊 ( 𝜎, 𝜔 )⋅𝑃 ( 𝜎, 𝜔 ) | }. 𝑤 𝜎, 𝑡 = W 𝜎 (Ω √M ) Χ HTP exp (𝑗𝜔𝑡) ( ) [ ( ) ] 𝜎∈𝐷 𝐵 𝐵 𝐵 𝐵 (20) The total transmission power flow is =𝑊 ( 𝜎, 𝜔 )⋅ exp (𝑗𝜔𝑡) (𝜎 ∈ 𝐷 ), where 𝑃(𝜎, 𝜔) and 𝑊(𝜎, 𝜔) denote the amplitudes of the 𝑃 ( 𝜔 )= ∬ 𝑝 ( 𝜎, 𝜔 )d𝐴 tr tr harmonic sound pressure in the cavity and harmonic dis- placement of the thin-wall structures. Χ andΧ are matrices 𝐴 𝐵 = Re { ∬ F 𝜎 ⋅ Λ 𝜔 ⋅ W 𝜎 d𝐴} [ ( ) ] ( ) ( ) composed of arrays of eigenvectors of the characteristic 2 (26) (0) (1) (2) matrix A;thatis, Χ =[𝜒 , 𝜒 , 𝜒 ,...]and Χ = 𝐴 𝐵 𝐴 𝐴 𝐴 (0) (1) (2) ∞ ∞ [𝜒 , 𝜒 , 𝜒 ,...].And 𝐵 𝐵 𝐵 { } = Re ∑ ∑ 𝐿 ⋅𝜆 ( 𝜔 ) , { } −1 2 2 𝑟=0 𝑗=1 (21) { } H ( 𝜔 )= (−𝜔 M + Ω ⋅ M ) 𝐶 𝐶 𝐶 −1 −1 −1 where Λ(𝜔) = 𝑗(√M ) X H(𝜔) ⋅ 𝐴 𝐴 𝐻 −1 𝐻 𝐻 𝐻 ∗ 𝐻 ∗ −1 (22) T =−X (√M ) LM + X Ω (√M ) , 𝐴 𝐴 𝐵 𝐵 𝐵 𝐵 TP P T [H(𝜔)]X (Ω√M ) =[𝜆 (𝜔)]might be 𝐵 𝐵 𝐵 𝐵 𝐵 named as a power transmission matrix and 𝜆 (𝜔) denotes where H couldbenamedasthecomplexfrequencyresponse the element of matrix Λ(𝜔) at the 𝑟th row and the 𝑗th matrix of the uid-s fl tructural coupled cavity and T is a column. 𝑟𝑗 𝑟𝑗 𝑟𝑗 𝑟𝑗 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 Advances in Acoustics and Vibration 5 For a simply supported plate, its natural frequencies and E Y modal functions are determined by 2 2 4 2 𝑚 𝑛 X O 𝜋 𝐸ℎ F 2 (29) 𝜔 = ( + ) 2 2 2 12𝜌 (1 − 𝜇 ) 𝐿 𝐿 𝑥 𝑦 𝑚 𝑛 𝑊 (𝑥,𝑦,0) = sin ( )sin ( ), 3 (30)  = 1000 kg/G 𝐿 𝐿 𝑥 𝑦 c =1500 m/s where 𝐸, 𝜌, ℎ,and 𝜇 are Young’s modulus, mass density, x thickness, and Poisson’s ratio of the plate, respectively; ∀(𝑚 ,𝑛 )∈𝑁 ,𝑗 ∈ 𝑁 ,let 𝜔 arrange in a sequence 2 2 2 Figure 1: A panel-cavity coupled system. |𝜔 |<|𝜔 |<|𝜔 |<⋅ ⋅ ⋅ . 𝐵1 𝐵2 𝐵3 The geometrical and material properties adopted for numerical computation are as follows: 𝐿 = 0.4 m, 𝐿 = 𝑥 𝑦 0.6 m, 𝐿 = 0.7 m, and ℎ = 0.005 m; 𝜌 = 1000 kg/m and 3. Numerical Simulation and Analysis 𝑧 0 9 −4 11 −3 𝐾 = 2.25×10 Pa×(1+10 j);𝐸 = 2.0×10 Pa×(1+10 j), 3 3 3.1. Simulation Model. A panel-cavity coupled system shown 𝜌 = 7.8 × 10 kg/m ,and 𝜇 = 0.28 (steel). in Figure 1 consists of a rectangular water-filled room with five rigidwalls andone simplysupportedplate subjecttoexterior 3.2. Modal Analysis. There is a convergence investigation harmonic distributed force (pressure) 𝑝 (𝜎, 𝑡). about the uid-s fl tructural coupled natural frequencies result- In discussion of the distribution shape of exterior exci- ing from (17) at rfi st. In Table 1, with a fixed number of plate tation, the plane harmonic wave incident is a common modals involved in calculation, the convergence of coupled assumption. Suppose that 𝑝 (𝜎, 𝑡) = 𝑃 ⋅exp[𝑗(𝜔𝑡−𝑘 𝑥 sin 𝜃)], natural frequencies could be observed by increasing the 𝐸 𝐸 where 𝑃 is the amplitude, 𝑘 is the wave number, and 𝜃 number of water sound modals involved. The convergence is the incident angle (𝑝 is uniform along the 𝑦 direction); couldalsobeobservedbyincreasingthenumberofplate 𝑝 has a perpendicular component 𝑝 (𝜎, 𝑡) = 𝑃 cos𝜃⋅ modals involved in Table 2. It could be suggested that the 𝐸 𝐵 𝐸 exp(−𝑗𝑘 𝑥 sin 𝜃) ⋅ exp(𝑗𝜔𝑡) = 𝑃 (𝜎) ⋅ exp(𝑗𝜔𝑡). Generally, modal coupling method could achieve good convergence in 𝑃 (𝜎)wouldbeacomplexfunctionandhaveinnfi itevariety solving the uid-s fl tructural coupled problem described here. of distribution shapes when the wave frequency, velocity, And it could also be observed that the convergence at lower and incident angle changed. Figure 2 shows one example of frequencies is more rapid than that at higher frequencies, distribution shape of 𝑃 (𝜎)with wave frequency 𝑓 = 500 Hz, and the solution precision would be more dependent on velocity 𝑐 = 344 m/s, and incident angle 𝜃=𝜋/3 .Itis accounting for more water sound modals. However, there is true that the modal coupling method is valid in dealing with noneedtocarryoutahighaccuracycalculation herefora those variant distribution shapes of 𝑃 (𝜎).However,some theoretical qualitative analysis, and in the later part of this specicfi analysis on the special case with 𝜃=0 ,thatis,a paper, 50 plate modals and 500 water sound efi ld modals uniform 𝑃 over theplate surface, wouldalsogiveindications are taken into consideration, by which totally 550 coupled of general significance. eTh uniform excitation had been modals could be revealed. And also, because there is no need adopted by other authors previously [7, 15]. And, moreover, to list all550 modes here,onlypartial data(thefirst several taking sonar array cavities as examples, they are regularly modes) are listed in Tables 1 and 2 mounted on related ship structures through rubber blankets; Figure 3 gives a comparison of sound pressure solutions the uniform structural excitation assumption would be a between the modal coupling approach and FEM (by the basic consideration. software of LMS Virtual.Lab Acoustics) as a theoretical val- For the water-filled rectangular room, the natural fre- idation verification. eTh differences between the two results quencies and modal functions of the sound eld fi with rigid in Figure 3(a) are due to modal truncation; more modes are boundaries are determined by involved in the FEM/BEM software. The uid-s fl tructural coupled natural frequencies resulting 2 2 from (17) are compared with those of sound efi ld with rigid 𝐾 𝑚 𝜋 𝑛 𝜋 𝑙 𝜋 𝜔 = [( ) +( ) +( ) ] (27) boundaries obtained by (27) and simply supported steel plate 𝜌 𝐿 𝐿 𝐿 0 𝑥 𝑦 𝑧 obtained by(29),whicharelistedinTable 3.Asmentioned above, 550 coupled modals have been obtained by involving 𝐹 (𝑥,𝑦,𝑧) 50 plate modals and 500 water sound efi ld modals in modal coupling calculation, but only the first 11 coupled modal (28) 𝑚 𝑛 𝑙 = cos ( )cos ( )cos ( ), frequencies are presented. 𝐿 𝐿 𝐿 𝑥 𝑦 𝑧 As a whole, it could be concluded that the u fl id-structural frequencies are very different from those of the water sound where ∀(𝑚 ,𝑛 ,𝑙 )∈𝑁 , 𝑟∈𝑁 ,and let 𝜔 arrange in a field (with rigid boundaries) and the flexible boundary plate 2 2 2 sequence |𝜔 |<|𝜔 |<|𝜔 | < ⋅⋅⋅ . (in vacuo), which means that there would be a strong 𝐴0 𝐴1 𝐴2 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝜋𝑧 𝜋𝑦 𝜋𝑥 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝜋𝑦 𝜋𝑥 𝐵𝑗 𝐵𝑗 𝐵𝑗 6 Advances in Acoustics and Vibration Table 1: Coupled natural frequencies with different number of water sound modals involved. Fluid-structural coupled natural frequency 𝑓 (Hz) Number of water sound modals involved 50 200 500 1000 2000 Number of plate modals involved 50 50 50 50 50 0 0 0000 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 10 682.5 617.7 599.4 592.1 584.7 Table 2: Coupled natural frequencies with different number of plate modals involved. Fluid-structural coupled natural frequencies 𝑓 (Hz) Number of water sound modals involved 500 500 500 500 500 Number of plate modals involved 25 50 100 200 500 0 0 0000 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 10 599.5 617.7 599.3 599.3 599.3 Table 3: Natural frequencies of the water sound field, plate, and uid-st fl ructural coupled cavity. Fluid-structural coupled Natural frequencies of water sound field Natural frequencies of plate natural frequencies (𝑚 ,𝑛 ) 𝑓 (Hz) 𝑟 (𝑚 ,𝑛 ,𝑙 ) 𝑓 (Hz) 𝑗 𝑘𝑓 (Hz) 0 (0, 0, 0)00 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 10 599.4 𝐶𝑘 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐶𝑘 𝐶𝑘 Advances in Acoustics and Vibration 7 0.5 0.5 −0.5 −1 −0.5 80 80 60 60 50 50 40 40 40 40 30 30 20 20 20 20 10 10 0 0 0 0 (a) Real part (b) Imaginary part Figure 2: One example of nonuniform distribution shape of 𝑃 (𝜎)(wave frequency 𝑓 = 500 Hz, velocity 𝑐 = 344 m/s, and incident angle 𝜃=𝜋/3 ). Pressure (nodal values).1 Occurrence 540 (./G ) 64.9 36.5 8.05 −20.4 −48.8 −77.2 −50 −106 −134 −162 −191 −100 −219 On Boundary −150 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz) matlab VL (b) Cloud picture of boundary sound pressure by VL (a) Comparison of sound pressure solutions (540 Hz) Figure 3: Comparison of sound pressure solutions between the modal coupling approach and FEM. coupling between the enclosed water sound field and its unneglectable departure of coupled natural frequencies from elastic surrounding structures. If the u fl id in the cavity was every natural frequency of the uncoupled flexible structures and cavity. air (𝜌 ≈ 1.29 kg/m ; 𝑐 ≈ 344 m/s), it could be found that 0 0 Except 𝑓 =0, the natural frequencies of the water the coupled natural frequencies 𝑓 would approximately be 𝐴0 sound efi ld are much higher than those of the simply equal to either some uncoupled structural natural frequencies supported plate, and the uid-s fl tructural coupled natural 𝑓 or some uncoupled cavity acoustical natural frequencies frequencies are inclined to come to be rather lower. It 𝑓 , and that is a situation of weak coupling. eTh motion seemed that one might carry out a comparison between of each subsystem in a weakly coupled system will not be the coupled natural frequencies and those of plate, and the essentially different from that of the uncoupled systems. change regulation of differences of adjacent fluid-structural However, if the density of the medium in the cavity is much natural frequencies 𝑓 −𝑓 is similar to those 𝑓 −𝑓 denser than air, such as water, the coupling may turn out to (𝑘+1) 𝐶 𝐵(−𝑗 1) of the plate. However, the frequency distribution of coupled be strong, and big deformation of the resulting modes from naturalfrequencieswouldturntobelowerandmorecrowded the uncoupled panel and the cavity modes may be expected as the order 𝑘 increased. [18]. In this sense, the strong coupling could be judged by any Sound pressure (dB) 𝐵𝑗 𝐶𝑘 𝐴𝑟 𝐵𝑗 𝐶𝑘 8 Advances in Acoustics and Vibration 0.8 0.5 0.6 0.4 −0.5 0.2 0 −1 80 80 60 60 50 50 40 40 40 40 30 30 y y 20 20 20 20 x x 10 10 0 0 0 0 (a) 𝑊 = sin(𝜋𝑥/𝐿 ) sin(𝜋𝑦/𝐿 ),𝑓 ≈108.0 Hz (b) 𝑊 = sin(𝜋𝑥/𝐿 ) sin(3𝜋𝑦/𝐿 ),𝑓 ≈373.4 Hz 𝐵1 𝑥 𝑦 𝐵1 𝐵4 𝑥 𝑦 𝐵4 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 80 80 60 60 50 50 40 40 40 40 30 30 y y 20 20 20 20 x x 10 10 0 0 0 0 (c) 𝑊 = sin(3𝜋𝑥/𝐿 ) sin(𝜋𝑦/𝐿 ),𝑓 ≈705.9 Hz (d) 𝑊 = sin(𝜋𝑥/𝐿 ) sin(5𝜋𝑦/𝐿 ),𝑓 ≈905.2 Hz 𝐵8 𝑥 𝑦 𝐵4 𝐵11 𝑥 𝑦 𝐵11 Figure 4: Some uncoupled plate modals. In order to reveal modal coupling characteristics, Figures the plane 𝑧=𝐿 /2, and Figure 6(b) illustrates the appur- 4 and 5 show several uncoupled plate modals 𝑊 (𝜎)which tenant participant coefficients of the uncoupled acoustical are expressed by (30) and uid-s fl tructural coupled plate cavity modals in the constitution of coupled acoustical cavity modals 𝑊 (𝜎)which are determined by (29) through modal modal. coupling calculation. The figure shows that 𝑊 is almost the 𝐶0 same as 𝑊 , while it is true according to 𝑊 ’s expression 3.3. Power Flow Transmission. In the simulation model of 𝐵1 𝐶0 and that is just an example of weak-coupled modal. It seemed Figure 1, the vibratory power flow inputted by exterior that the coupled modal shape 𝑊 is similar to the uncoupled excitation into the whole uid-s fl tructural coupled system and 𝐶3 modal shape 𝑊 ; however, they are quite different in fact the enclosed water sound efi ld, that is, 𝑃 calculated by (24) 𝐵4 in according to the expression of 𝑊 ,and that is astrong and 𝑃 calculated by (26), is dissipated by system damping. 𝐶3 tr coupled modal. Phenomena of strong modal coupling are And thus the higher or lower power flow level would be a obvious when inspecting 𝑊 and 𝑊 shown in Figure 5. comprehensive indicator to measure the vibration level or 𝐶7 𝐶8 The similar couplings have also happened to the acous- energy level of the panel-cavity coupled system and water tical cavity modals. And, moreover, except that the coupled sound efi ld. acoustical modal 𝐹 is practically equal to 𝐹 ,thatis,the Figure 7showsthespectrumofinput powerflow 𝑃 and 𝐶0 𝐴0 in rigid body modal of the uncoupled cavity sound efi ld, all transmitted power flow 𝑃 ,inwhichthedropbetween 𝑃 and tr in other uid-s fl tructural coupled acoustical cavity modals are 𝑃 is the dissipation power of the plate’s damping. Because the tr composed in a strong coupling manner; that is, they are linear exterior excitation is symmetric (uniform 𝑃 as mentioned combinations of several 𝐹 ,theuncoupledacousticalmodals in Section 3.1), only symmetric modals are present, and the of the rigid wall cavity. As 𝐹 (𝜎)is defined in three-dimen- spectrum peaks at 0 Hz, 217.6 Hz, 451.8 Hz, and 537.2 Hz could sional space and it is inconvenient to plot it by a planar figure, be associated with the modals shown in Figures 5 and 6. Figure 6(a) illustrates one coupled acoustical modal shape in There is some similarity between 𝑃 or 𝑃 and the water in tr B8 W B1 W B4 C11 𝐶𝑟 𝐴𝑟 𝐶𝑘 𝐵𝑗 Advances in Acoustics and Vibration 9 f ≈ 217.6 Hz C3 f =0 Hz C0 0.025 0.02 0.02 0.01 0.015 0.01 −0.01 0.005 −0.02 80 80 50 50 40 40 40 40 30 30 y 20 20 20 10 10 0 0 0 0 (a) 𝑊 ≈0.0248(𝑊 +0.05𝑊 +0.02𝑊 +0.01𝑊 ) (b) 𝑊 ≈0.0132(−0.21𝑊 +𝑊 +0.04𝑊 +0.05𝑊 +0.03𝑊 ) 𝐶0 𝐵1 𝐵4 𝐵8 𝐵11 𝐶3 𝐵1 𝐵4 𝐵8 𝐵11 𝐵12 f ≈ 451.8 Hz f ≈ 537.2 Hz C7 C8 0.01 0.01 0.005 0.005 0 0 −0.005 −0.005 −0.01 −0.01 50 50 40 40 40 40 30 30 y 20 y 20 x x 10 10 0 0 0 (c) 𝑊 ≈0.0074(−0.28𝑊 −0.26𝑊 +𝑊 +0.14𝑊 +0.07𝑊 ) (d) 𝑊 ≈0.0062(−0.11𝑊 −0.16𝑊 −𝑊 +0.18𝑊 +0.08𝑊 ) 𝐶7 𝐵1 𝐵4 𝐵8 𝐵11 𝐵12 𝐶8 𝐵1 𝐵4 𝐵8 𝐵11 𝐵12 Figure 5: Some coupled plate modals. ×10 ×10 8.8 4 8.75 8.7 8.65 −2 8.6 80 −4 −6 y 20 0 10 20 30 40 50 60 70 80 90 100 0 Modal serial number (a) 𝐹 (𝑥,𝑦,𝐿 ),𝑓 ≈537.2 Hz (b) Participant coefficients of the uncoupled cavity modals 𝐶8 𝑧/2 𝐶8 Figure 6: Simple illustration of one coupled cavity modal shape. C7 C0 F (x,y,L /2) C8 z C3 C8 Modal participant coefficients 10 Advances in Acoustics and Vibration −150 has been predicted by Table 3. u Th s the alteration of elasticity modulus of plate would lead to a phenomenon of “frequency −200 shifting.” eTh 𝑃 spectrum with smaller plate’s elasticity mod- tr ulus could be regarded as a contraction of that with greater −250 elasticity modulus toward lower frequencies, and the peaks of 𝑃 would occur at relatively lower frequencies and become −300 tr more crowded. However, since the variation range of Young’s −350 modulus would be limited in practice, its influence on power flow transmission might not be very serious, and it could also −400 be observed that reduction of Young’s modulus might bring about a benefit of slight reduction of 𝑃 ’s peak valleys. −450 tr In order to make an inspection of the alteration of differ- −500 ences between 𝑃 and 𝑃 with different plate elasticity mod- in tr ulus, Figure 9(b) shows the spectra of power flow ratio PR = −550 10 log(𝑃 /𝑃 ).ThepeaksofPR wouldalwaysappeararound 0 200 400 600 800 1000 1200 1400 1600 1800 2000 in tr the resonance frequencies except at 𝑓 ,which couldbeeasily 𝐶0 Frequency (Hz) explained by the fact that the plate’s damping consumes more CH energy when system resonances take place. In the lowest NL frequency range around 𝑓 =0,the plateconsumeslittle 𝐶0 energy, and therefore 𝑃 ≈𝑃 and PR ≈0.Itshouldbenoted tr in Figure 7: Spectrum of 𝑃 and 𝑃 . in tr that greaterPRs mightnot implylowerlevelsoftransmitted power flow 𝑃 ;thefactisprobablyjust theoppositebecause tr the power flow input 𝑃 might be in much higher levels at in thesametime.In this sense, minority of PR peaknumbers 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 𝑃 attenuation, which could tr be discovered through a synthesized analysis of the figures shown. In Figure 10(a), different mass density values were given −50 to the elastic plate, in which the value of 2770 kg/m is by reference to aluminium. A signicfi ant feature of the spectra in thefigureisthat changesinplate’s massdensity wouldappar- −100 ently change the average level of transmitted power flow 𝑃 . tr According to (29), the increase of plate’s mass density would −150 cause decreasing of plate’s natural frequencies and “frequency 0 200 400 600 800 1000 1200 1400 1600 1800 2000 shifting” of uid-s fl tructural coupled modals, as shown in Fig- Frequency (Hz) ure 10(a), similar to the situation of decreasing Young’s mod- P(L /2, L ,0) x y ulus in Figure 9. However, the increase of plate’s mass density P(L /2, L ,L /2) x y z would increase its modal masses at the same time. And that is the reason why 𝑃 ’s average level is cut down even tr Figure 8: Spectrum of water sound pressure. though more resonance modals would come into being in the relative lower frequency band. It could also be explained by thefactthataheaviervibrating masswouldgenerate sound pressure at the center of the plate’s interior surface, a greater reduction in dynamic force (or pressure) trans- that is, 𝑃((𝐿 /2, 𝐿 /2, 0), 𝜔)(refer to (21)), which is shown in mission. Figure 10(b) is about the power flow ratio PR. PR 𝑥 𝑦 Figure 8. However, the power flow is of evaluation of sound could not effectively reveal the apparent 𝑃 reduction at tr power. nonresonance frequencies, such as 800 Hz∼1200 Hz, because As a theoretical investigation, hypothesize that the mate- the transmitted power flow 𝑃 is very close to the power flow tr rial property parameters of the plate, that is, Young’s modulus input 𝑃 at those frequencies. in 𝐸, mass density 𝜌, Poisson’s ratio 𝜇,and dampinglossfactor, Another important influential factor that should be paid could be altered independently. In Figure 9(a), transmitted attention is the damping loss factor of the plate. Figure 11(a) power flows are compared under different Young’s modulus demonstrates a conflictive situation where smaller damping loss factorwouldincreasethepeaksoftransmitted power oftheelasticplate, wherethevalueof 7.24 × 10 Pa is by reference to aluminium. When the plate’s elasticity modulus flow at resonance frequencies, while at the broadband non- decreases, the plate’s natural frequencies would decrease resonance frequencies smaller damping would be beneficial simultaneously,andthiswouldmovethefluid-structuralcou- to reduction of transmitted power flow. To explain this result, pled natural frequencies into lower frequency ranges, which one might make an analogy with the vibration isolation P and P (dB) Sound pressure (dB) CH NL Advances in Acoustics and Vibration 11 −200 80 −250 −300 −350 −400 −450 −500 −550 −10 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Frequency (Hz) 11 −3 11 −3 E = 2.0 × 10 0; × (1 + 10 D) E = 2.0 × 10 0; × (1 + 10 D) 10 −3 10 −3 E = 7.24 × 10 0; × (1 + 10 D) E = 7.24 × 10 0; × (1 + 10 D) 10 −3 E = 2.0 × 10 0; × (1 + 10 D) (a) Transmitted power flow (b) Power flow ratio Figure 9: Effect of plate’s elasticity modulus on power flow transmission. −200 90 −250 −300 −350 −400 −450 −500 −550 −10 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Frequency (Hz)  = 7800 EA/G  = 7800 EA/G = 2770EA/G = 2770EA/G 4 3 = 1 × 10 EA/G (a) Transmitted power flow (b) Power flow ratio Figure 10: Effect of plate’s mass density on power flow transmission. theory. If the elastic plate was considered as some kind of be favorable in the nonresonance frequency ranges, which elastic isolator which was specially designed to attenuate the would not be counted in power flow ratio. transmission of exterior excitation energy into the water- Poisson’s ratio 𝜇 would aeff ct the power flow transmission filledcavity, thedampingwouldincreasethepowertrans- the same way Young’s modulus does as shown in Figure 9. mission and would not be expected except for attenuation Referring to (29), increasing Young’s modulus could be equiv- of resonance peak. Through illustration of power flow ratio alent to increasing Poisson’s ratio. However, the variation PR as that in Figure 11(b), it could be confirmed that greater scope of Poisson’s ratio is much smaller than that of Young’s damping could be used to obstruct energy transmission modulus, and the power flow transmission would be affected at resonance frequencies, whereas smaller damping would more by Young’s modulus than by Poisson’s ratio. And, for this P (dB) P (dB) NL NL PR (dB) PR (dB) 12 Advances in Acoustics and Vibration −200 160 −250 −300 −350 −400 −450 −500 −550 −600 −20 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Frequency (Hz) 11 −3 11 −3 E = 2.0 × 10 0; × (1 + 10 D) E = 2.0 × 10 0; × (1 + 10 D) 11 −4 11 −4 E= 2.0 × 10 0; × (1 + 3.310 D) E= 2.0 × 10 0; × (1 + 3.310 D) (a) Transmitted power flow (b) Power flow ratio Figure 11: Eeff ct of damping loss factor of plate on power flow transmission. reason, no additional repetitive gfi ures would be put forward reduction of peak valleys of transmitted power flow could also here. be expected on the other hand. (4) Denser material could be beneficial to an apparent attenuation of average level of power flow transmission 4. Conclusions into the water-filled enclosure, even though it would be Backgrounded by evaluation or control of mechanical self- accompanied with a decrease of system natural frequencies. noise in sonar array cavity, transmitted power flow or sound (5) Smaller inner damping of enclosure’s thin-wall struc- power input calculation is carried out by modal coupling tures, which were distinguished from that specially designed analysis on the uid-s fl tructural vibration of the uid-fil fl led for sound absorption destination and only for the purpose enclosure with elastic boundaries. Power flow transmission of suppression of resonance peaks, might be proposed to analysis is presented through a numerical simulation exam- attenuate the average level of power flow transmission. ple of water-filled rectangular panel-cavity coupled system. Detailed discussion is carried out about power flow transmis- Appendix sion characteristics affected by variation of material property parameters of cavity’s elastic boundary structure, aiming at A. Derivation Procedures of Related reduction of water sound level inside the cavity. From the Matrices Equations results, one could draw the following conclusions. (1) Power flow or sound power transmission analysis A.1. Eigenvalue Problem. Equations (14) and (15) could be could be a valuable method for evaluation or prediction of easilyveriefi dtobeequivalenttothe followingsimultaneous water sound level in dealing with strong coupled vibration modal differential equations, which are fundamental in Dow- problems of water-filled acoustoelastic enclosure systems. ell’s modal coupling method: (2) eTh uid-s fl tructural coupled natural frequencies of a water-filledacoustoelasticenclosurewouldbegreatlyaeff cted by interaction between the hydroacoustic field and its sur- ̈ ̈ 𝑀 𝑃 +𝑀 𝜔 𝑃 =−∑𝐿 𝐵 𝑟 = 0, 1, 2, . . . ( ) 𝑟 𝑟 𝑗 rounding elastic boundaries. There would be a tendency that 𝑗=1 the uid-s fl tructural coupled natural frequencies turn to be smaller in value and more crowded in frequency distribution (A.1) 𝑀 𝐵 +𝑀 𝜔 𝐵 = ∑𝐿 𝑃 +𝑄 𝑗 𝑗 𝑟 than those of elastic boundary structures and inner water 𝑟=0 sound field in rigid boundary condition. (3) Decreasing elasticity modulus or Poison’s ratio of (𝑗 = 1,2,...), water sound cavity’s thin-wall structures would cause the frequency distribution of system modals to contract toward where 𝐿 =∬ 𝐹 𝑊 d𝐴, 𝑄 =−∬ 𝑊 𝑝 d𝐴,and all lower frequency ranges and result in more power flow 𝐷 𝐷 𝐹 𝐹 transmission peaks in the lower frequency band, while slight other symbols have been mentioned previously. P (dB) NL PR (dB) 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐴𝑟 𝑟𝑗 𝐵𝑗 𝐵𝑗 𝑟𝑗 𝐵𝑗 𝐵𝑗 𝑟𝑗 𝐴𝑟 𝐴𝑟 𝐴𝑟 Advances in Acoustics and Vibration 13 󸀠 2 −1 −1 𝑇 −1 2 󸀠 Through some simple algebraic algorithm, (14) and (15) 𝜒 Ω + M LM L −M LΩ 𝜒 𝐴 𝐴 𝐵 𝐴 𝐵 2 𝐴 𝐴 𝜔 [ ]=[ ][ ] couldbegrouped togetherandwrittenas 󸀠 󸀠 −1 𝑇 2 𝜒 −M L Ω 𝜒 𝐵 𝐵 𝐵 𝐵 (A.3) 2 −1 −1 𝑇 −1 2 P Ω + M LM L −M LΩ P 𝐴 𝐴 𝐵 𝐴 𝐵 󸀠 [ ]+[ ][ ] = A [ ]. −1 𝑇 2 B 𝜒 B −M L Ω 𝐵 𝐵 (A.2) −1 −1 The above equation raises an eigenvalue problem, but it −M LM Q 𝐴 𝐵 𝐵 =[ ]. is nonstandard because the matrix A is asymmetric. eTh −1 M Q transformation expressed by (16) would result in a standard eigenvalue problem, which has been expressed by (17), where While carrying out a modal analysis, one could simply let the matrix A is symmetric. And, moreover, one could prove 󸀠 󸀠 P = 𝜒 ⋅ exp(𝑗𝜔𝑡), B = 𝜒 ⋅ exp(𝑗𝜔𝑡),and Q =0;thus that 𝐴 𝐵 −1 −1 −1 −1 (√M ) L (√M ) Ω (√M ) L (√M ) −Ω 𝐴 𝐵 𝐴 𝐵 𝐴 𝐵 A =[ ][ ] −Ω O Ω O 𝐵 𝐴 (A.4) −1 −1 −1 −1 (√M ) L (√M ) Ω (√M ) L (√M ) Ω 𝐴 𝐵 𝐴 𝐴 𝐵 𝐴 =[ ][ ] , −Ω O −Ω O 𝐵 𝐵 where O denotes a zero matrix with proper dimension rank. = T ⋅ X ⋅ 𝛼 ( 𝑡 ), Before any damping factors are introduced, (A.4) proves (A.5) that the matrix A is positive semidefinite; that is, all its eigenvalues would be nonnegative. That would provide a where 𝛼(𝑡) isacolumnvectorvariabletotakeplace of P(𝑡) mathematical guarantee that the eigenvalue calculation of and B(𝑡). matrix A would not fail at any occurrences of negative square Aeft r substituting the above equation into (A.2), multiply natural frequencies. both sides of the resultant equation with a le-ft multiplication 𝐻 −1 By substituting any 𝑘th eigenvalue of matrix A and its matrix X T ;one has accompanying eigenvector, which have been symbolled with 2 (𝑘) 𝑇 (𝑘) 𝑇 (𝑘) 𝑇 𝜔 and [𝜒 ] =[{𝜒 } ,{𝜒 } ],respectively,into(16) 𝐻 −1 𝐻 −1 󸀠 𝐴 𝐵 X T T X ⋅ 𝛼 ̈ + X T A T X ⋅ 𝛼 0 0 0 0 (𝑘) (𝑘) (i.e., let 𝜔=𝜔 and 𝜒 = 𝜒 and 𝜒 = 𝜒 ), the 𝑘th 𝐴 𝐴 𝐵 𝐵 −1 −1 uid-s fl tructural coupled modal of the cavity sound field and (A.6) −M LM Q 𝐴 𝐵 𝐻 −1 = X T [ ]. its flexible boundary structures, which have been symbolled −1 M Q 𝐵 𝐵 with 𝐹 and 𝑊 , respectively, are obtained by taking out the time-independent part of P(𝑡)and B(𝑡), that is, (18). −1 󸀠 𝐻 𝐻 Take notice that T A T = A,and X X and X AX are all diagonal matrices according to the theory of orthogonality A.2. Harmonic Solution. In the situation of forced vibration, of eigenvectors. In fact, the diagonal elements of matrix that is, Q =0 ̸ , there is a standard decoupling procedure 𝐻 𝐻 𝐻 M = X X are just the modal masses of the u fl id-structural for (A.2) by applying an eigenmatrix X =[X ,X ] 𝐴 𝐵 coupled modals, and the diagonal elements of matrix Ω = which is composed of the series of A’s eigenvectors, where −1 𝐻 the detailed definitions of X and X have been mentioned M (X AX)arethecomplexfluid-structuralcouplednatural 𝐴 𝐵 previously in Section 2.2 aer ft (19) and (20), and note that frequencies; these two matrices have already been defined in Hermitian transposition of matrices has been employed here Section 2.2 aer ft (20). With this knowledge, (A.2) or (A.6) is in consideration of the introduction of damping loss factors. turned into its decoupled expression: Replacing the real elasticity modulus of the flexible bound- −1 −1 ary structures or real volume stiffness of the cavity sound −M LM Q 𝐴 𝐵 2 −1 𝐻 −1 field with a complex elasticity modulus or complex volume 𝛼 ̈ + Ω 𝛼 = M X T [ ]. (A.7) 𝐶 𝐶 0 −1 M Q stiffness is a regular method when damping factors are to be 𝐵 involved in analysis, which has been declared in Sections 2.2, Obviously, the transient or steady solution of the above 3.1, and 3.3; the method has also been adopted by many other equation under arbitrary deterministic Q (𝑡) could be authors. obtained by the method of convolution integral. Only har- Now let monic solution is concerned here. Referring to (12), when −1 𝑝 (𝜎, 𝑡) is harmonic, Q (𝑡)is harmonic, too. And suppose 𝐵 𝐵 P ( 𝑡 ) (√M ) O X [ ] [ ]= [ ]𝛼 ( 𝑡 ) that Q (𝑡) =P ⋅ exp(𝑗𝜔𝑡) and 𝛼(𝑡) = 𝛼 exp(𝑗𝜔𝑡);bysub- 𝐵 𝐵 −1 B ( 𝑡 ) X √ 𝐵 O (Ω M ) stituting them into (A.7), the time-dependent part exp(𝑗𝜔𝑡) 𝐵 𝐵 [ ] 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 14 Advances in Acoustics and Vibration −1 would be eliminated, and the time-independent part of 𝛼(𝑡) 𝑇 ∗ ⋅ X [(Ω M ) ] W ( 𝜎 )⋅𝑃 ( 𝜎 )d𝐴} . 𝐵 𝐵 𝐵 𝐵 is obtained; that is, (A.10) −1 −1 −M LM P −1 2 2 −1 𝐻 −1 𝐴 𝐵 𝛼 =(−𝜔 I + Ω ) M X T [ ] 𝐶 𝐶 0 −1 Note that in the last expression of the right hand of M P the above equation, the matrices P , T, H(𝜔), X , Ω ,and 𝐵 𝐵 𝐵 −1 2 2 M are all 𝜎-independent; only W(𝜎) ⋅ 𝑃 (𝜎)would join in 𝐵 𝐵 =(−𝜔 M + Ω M ) 𝐶 𝐶 𝐶 ∗ the integral in (A.10). The integral would result in (P),in −1 −1 which P has been den fi ed in the beginning paragraph of √M O −M LM P 𝐴 𝐴 𝐵 𝐵 𝐻 𝐻 Section 2.3, and the superscript “∗” denotes conjugation of ⋅[X X ][ ][ ] 𝐴 𝐵 −1 (A.8) O Ω √M M P 𝐵 𝐵 𝐵 𝐵 complex numbers. Other matters need attention; H(𝜔),Ω , andM are all diagonal matrices; that is, [H(𝜔)]= H(𝜔)and = H ( 𝜔 ) −1 𝑇 −1 [(Ω√M ) ] =(Ω √M ) . Aeft r taking into account all 𝐵 𝐵 𝐵 𝐵 −1 −1 above factors, (24) is realized. 𝐻 −1 𝐻 √ √ ⋅[−X ( M ) LM + X Ω ( M ) ]P 𝐴 𝐵 𝐵 𝐵 𝐴 𝐵 𝐵 The obtainment of (25) is similar to that of (23), in which the cavity sound pressure at the interior surface of the thin- = H 𝜔 ⋅ T ⋅ P , ( ) wall structures takes place of the exterior excitation. And also the integral of 𝑝 (𝜎, 𝜔) is processed similar to that of 𝑝 (𝜎, 𝜔) tr in at the beginning by quotation of (22) and (25). where I isaunitmatrixofproperdimension rank. A.3. Power Flow Formulation. eTh basic formulation for 𝑃 ( 𝜔 )= ∬ Re {𝑗𝑊 ( 𝜎, 𝜔 )⋅𝑃 ( 𝜎, 𝜔 ) | }d𝐴 tr 𝜎∈𝐷 power flow computation is = ∬ Re {𝑗𝑃 ( 𝜎, 𝜔 ) 2𝜋/𝜔 𝑃 = ∫ 𝐹 cos ⋅ 𝑉 cos (𝜔𝑡 + 𝜑) d𝑡 2𝜋 󵄨 𝜔 ⋅ 𝑊 𝜎, 𝜔 󵄨 } d𝐴= [ ( ) ] 2𝜋/𝜔 󵄨𝜎∈𝐷 = ∫ Re {𝐹 exp (𝑗𝜔𝑡)} 2𝜋 0 −1 ⋅ Re { ∬ 𝑗 [F ( 𝜎 ) ] ( M ) (A.9) 𝐷 ⋅ Re {𝑉 exp (𝑗𝜔𝑡 + 𝑗𝜑)} d𝑡= 𝐹𝑉 cos 𝜑 −1 𝐻 𝐻 𝐻 𝐻 ∗ ⋅ Χ HTP P T H X [(Ω √M ) ] 𝐴 𝐵 𝐵 𝐵 𝐵 𝐵 (A.11) = Re {[𝐹 exp (𝑗𝜔𝑡)] [𝑉 exp (𝑗𝜔𝑡 + 𝑗𝜑)]} 1 𝜔 = Re {[𝐹 exp (𝑗𝜔𝑡)] [𝑉 exp (𝑗𝜔𝑡 + 𝑗𝜑)] }, ⋅ W ( 𝜎 )d𝐴} = Re { ∬ [F ( 𝜎 ) ] 2 2 −1 where 𝐹 cos or 𝐹 exp(𝑗𝜔𝑡) is a harmonic excitation force, ⋅[𝑗 (√M ) 𝑉 cos or 𝑉 exp(𝑗𝜔𝑡 + 𝑗𝜑) is the harmonic velocity response at theactionpointof theexcitationforce,and 𝜑 is the phase −1 𝐻 𝐻 ∗ 𝐻 ∗ difference between the excitation force and velocity response. ⋅ Χ HTP P T H X (Ω √M ) ] 𝐴 𝐵 𝐵 𝐵 𝐵 𝐵 By applying the above formulation, (23) would be appar- ent. And because 𝑃 (𝜎)is of 𝜎-dependent distribution over 𝐷 , 𝑝 (𝜎, 𝜔) in (23) is of distribution of power flow density. ⋅ W ( 𝜎 )d𝐴} . 𝐹 in Equation (25) is an integral of 𝑝 over 𝐷 to count up the in 𝐹 total power flow. The integral is initiated by quotation of (22) The basis of the above transformation includes the fol- and (23) as follows: lowing: W(𝜎)is real, H and Ω are diagonal, and M is real 𝐵 𝐵 anddiagonal.Itcouldbeexaminedthatthe productofthe matrices between [F(𝜎)]and W(𝜎)in the last expression of 𝑃 ( 𝜔 )=− ∬ Re {𝑗𝑊 ( 𝜎, 𝜔 )⋅𝑃 ( 𝜎 ) }d𝐴 in 𝐵 the right hand of the above equation, which has been den fi ed as Λ(𝜔)in (26), is 𝜎-independent. And the expansion of the 𝑇 ∗ =− ∬ Re {𝑗 𝑊 𝜎, 𝜔 ⋅𝑃 𝜎 }d𝐴 [ ( ) ] ( ) 𝐵 product [F(𝜎)]Λ(𝜔)W(𝜎)is 𝑇 𝑇 𝑇 𝜆 ( 𝜔 ) 𝜆 ( 𝜔 ) ⋅⋅⋅ 𝑊 ( 𝜎 ) =− Re { ∬ 𝑗 T [H ( 𝜔 ) ] 11 12 𝐵1 ∞ ∞ 𝑇 [ ] [ ] 𝜆 ( 𝜔 ) 𝜆 ( 𝜔 ) ⋅⋅⋅ 𝑊 ( 𝜎 ) 21 22 𝐵2 F ΛW =[𝐹 ( 𝜎 ) 𝐹 ( 𝜎 ) 𝐹 ( 𝜎 ) ⋅⋅⋅][ ] [ ] = ∑ ∑𝜆 ( 𝜔 )⋅𝐹 ( 𝜎 )⋅𝑊 ( 𝜎 ). (A.12) 𝐴0 𝐴1 𝐴2 . . . 𝑟=0 𝑗=1 . . . . . d . [ ] [ ] 𝐵𝑗 𝐴𝑟 𝑟𝑗 𝜔𝑡 𝜔𝑡 𝐹𝑉 𝜔𝑡 Advances in Acoustics and Vibration 15 Table 4: Dimension ranks of related matrices with R cavity acoustical modals and N thin-wall structural modals. Matrices Dimension rank Matrices Dimension rank Matrices Dimension rank ∇𝐹 3×1 F 𝑅×1 P 𝑅×1 ∇F 3×𝑅 M 𝑅×𝑅 Ω 𝑅×𝑅 𝐴 𝐴 W 𝑁×1 B 𝑁×1 M 𝑁×𝑁 Ω 𝑁×𝑁 Q 𝑁×1 Q 𝑁×1 𝐵 𝐴 𝐵 (𝑘) 󸀠 (𝑘) 󸀠 L 𝑅×𝑁 𝜒 ,𝜒 , 𝜒 𝑅×1 𝜒 ,𝜒 ,𝜒 𝑁×1 𝐴 𝐴 𝐴 𝐵 𝐵 𝐵 (𝑘) 󸀠 𝜒, 𝜒 (𝑅 + 𝑁) × 1 A,A (𝑅 + 𝑁) × (𝑅 + 𝑁) A 𝑅×𝑅 A 𝑅×𝑁 A 𝑁×𝑅 A 𝑁×𝑁 12 21 22 X 𝑅×(𝑅 +𝑁) X 𝑁×(𝑅 +𝑁) X (𝑅 + 𝑁) × (𝑅 + 𝑁) 𝐴 𝐵 S (𝑅 + 𝑁) × 1 H (𝑅 + 𝑁) × (𝑅 + 𝑁) T (𝑅 + 𝑁) × 𝑁 S 𝑁×1 M (𝑅 + 𝑁) × (𝑅 + 𝑁) Ω (𝑅 + 𝑁) × (𝑅 + 𝑁) 𝑄 𝐶 𝐶 P 𝑁×1 Λ 𝑅×𝑁 T (𝑅 + 𝑁) × (𝑅 + 𝑁) 𝐵 0 𝛼, 𝛼 (𝑅 + 𝑁) × 1 Since 𝜆 (𝜔) is 𝜎-independent, when the integral of [5] G. B. Warburton, “Vibration of a cylindrical shell in an acoustic 𝑇 medium,” Journal of Mechanical Engineering Science,vol.3,no. [F(𝜎)]Λ(𝜔)W(𝜎)over 𝐷 is proceeding, only 𝐹 (𝜎)𝑊 (𝜎) 1, pp.69–79,1961. is involved, and that would result in 𝐿 ,whichhasbeen [6] J.-M. David and M. Menelle, “Validation of a medium-frequen- defined in (13) or (A.1). And, at last, 𝑃 (𝜔) is formulated by tr cy computational method for the coupling between a plate and (26). a water-filled cavity,” Journal of Sound and Vibration,vol.265, no. 4, pp. 841–861, 2003. B. Dimension Ranks of Related Matrices under [7] J.Yang, X. Li,and J.Tian,“eTh oreticalanalysisofsoundtrans- Modal Truncation mission through viscoelastic sandwich panel backed by a cavity,” ACTA Acustica,vol.24, no.6,pp.617–626, 1999. Modal truncation has to be implemented in the application [8] J. Yang, Y. P. Xiong, and J. T. Xing, “Vibration power flow and of modal coupling method, because there are no numerical force transmission behaviour of a nonlinear isolator mounted techniques that could provide a computation in infinity on a nonlinear base,” International Journal of Mechanical Sci- mannerbynow.Andbecauseofthemodalconvergenceprop- ences, vol. 115-116, pp. 238–252, 2016. erty, modal truncation errors could be controlled properly. [9] P.M.Morse,“Some aspectsofthe theory of room acoustics,” Suppose that there are totally 𝑅 cavity acoustical modals and eTh JournaloftheAcousticalSocietyofAmerica ,vol.11, no.1,pp. 𝑁 thin-wall structural modals to be accounted for the uid- fl 56–66, 1939. structural modal coupling; dimension ranks of associated matrices mentioned in this paper are listed in Table 4. [10] J. Pan, “eTh forced response of an acoustic-structural coupled system,” The Journal of the Acoustical Society of America ,vol.91, no. 2, pp. 949–956, 1992. Conflicts of Interest [11] S. R. Bistafa and J. W. Morrissey, “Numerical solutions of the eTh authors declare that there are no conflicts of interest acousticeigenvalueequationinthe rectangularroomwitharbi- trary (uniform) wall impedances,” JournalofSoundandVibra- regarding the publication of this paper. tion,vol.263,no. 1, pp.205–218,2003. [12] E. H. Dowell, G. F. Gorman III, and D. A. Smith, “Acoustoe- References lasticity: General theory, acoustic natural modes and forced [1] H. Guo, D. Luo, M. Chen, and F. Zhou, “Prediction method response to sinusoidal excitation, including comparisons with for low frequency of self-noise in submarine’s fore body sonar experiment,” JournalofSoundandVibration,vol.52,no.4,pp. platform area,” Ship Science and Technology,vol.27,no.4,pp. 519–542, 1977. 74–77, 2005. [13] E. H. Dowell, “Comment on ‘On Dowell’s simplification for [2] M.Yu,J. Ye,Y.Wu, andS.Lu,“Prediction andcontrolmethod acoustic cavity-structure interaction and consistent alterna- of self-noise in ship’s sonar domes,” Journal of Ship Mechanics, tives’,” eJ Th ournaloftheAcoustical SocietyofAmerica ,vol.133, vol. 6, no. 5, pp. 80–94, 2002. no. 3, pp. 1222–1224, 2013. [3] T. Musha and T. Kikuchi, “Numerical calculation for determin- [14] R. W. Guy, “eTh response of a cavity backed panel to external ing sonar self noise sources due to structural vibration,” Applied airborne excitation: A general analysis,” The Journal of the Acoustics,vol.58, no.1,pp. 19–32, 1999. Acoustical Society of America,vol.65,no.3,pp. 719–731, 1979. [15] J. H. Ginsberg, “Derivation of a Ritz series modeling technique [4] S. Narayanan and R. L. 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Research on Power Flow Transmission through Elastic Structure into a Fluid-Filled Enclosure

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Hindawi Publishing Corporation
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Copyright © 2018 Rui Huo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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10.1155/2018/5273280
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Hindawi Advances in Acoustics and Vibration Volume 2018, Article ID 5273280, 16 pages https://doi.org/10.1155/2018/5273280 Research Article Research on Power Flow Transmission through Elastic Structure into a Fluid-Filled Enclosure 1,2 1,2 1,2 1,2 Rui Huo , Chuangye Li , Laizhao Jing , and Weike Wang School of Mechanical Engineering, Shandong University, Jinan 250061, China Key Laboratory of High Ecffi iency and Clean Mechanical Manufacture (Shandong University), Ministry of Education, Jinan 250061, China Correspondence should be addressed to Rui Huo; huorui@sdu.edu.cn Received 30 October 2017; Revised 8 February 2018; Accepted 15 March 2018; Published 2 May 2018 Academic Editor: Kim M. Liew Copyright © 2018 Rui Huo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. eTh work of this paper is backgrounded by prediction or evaluation and control of mechanical self-noise in sonar array cavity. eTh vibratory power flow transmission analysis is applied to reveal th e overall vibration level of the uid-st fl ructural coupled system. Through modal coupling analysis on the uid-st fl ructural vibratio n of the uid-filled fl 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 uid-st fl ructural coupled system and the net power flow transmitted into the hydroacoustic field. Characteristics of t he coupled natural frequencies and modals are investigated by a numerical example of a rectangular water-filled cavity with vfi e acoustic rigid walls and one elastic panel. Inu fl ential factors of power flow transmission characteristics are further discussed with the purpose of overall evaluation and reduction of the cavity water sound energy. 1. Introduction from exterior air-borne sound [4]. In these cases, weak coupling has been commonly assumed because of the low 1.1. Background. The work of this paper is backgrounded by density of air and high stiffness of cabin wall, which means prediction or evaluation and control of mechanical self-noise that the cavity’s interior sound pressure would have little in sonar array cavity. eTh mechanical self-noise, which is influence on the vibration of cavity wall, and modals of caused by structural vibration of sonar cavity’s wall, might interior sound efi ld would also be aeff cted very lightly [5]. In signicfi antly weaken the detection performance of sonar at contrast, a much stronger coupling might be present when a lower frequencies [1, 2]. The sources of mechanical self- water sound field takes place of the air [6]. noise might be multiple such as vibrating machines on the eTh sound pressure is most commonly used to represent ship which diffuse vibration energy or second excitation of the property of sound field in the study of acoustic-structural structure-borne sound. However, it is essential to compre- coupling of acoustoelastic enclosure. The ratio of sound pres- hend the characteristics of interaction between the enclosed sure at the outside surface of the elastic cavity wallboard to water sound field and its elastic boundary structures for the that at internal surface, which is den fi ed as “noise reduction,” purpose of prediction, evaluation, and control of interior is applied to evaluation of sound transmission characteristics hydroacoustic noise [3]. [4, 7]. Since the sound pressure would change greatly at The subjects of cabin noise in various flight vehicles dieff rentpointsofthesoundfield,thevalueofnoisereduction and automobiles are more familiar in the investigation of would also be very different, and a comprehensive measure, uid-s fl tructural coupled vibration of acoustoelastic enclosure, for example, power flow, would be expected for an overall which mainly focus on characteristics of sound transmission evaluation of vibration level of the enclosed sound efi ld. through the elastic wall into interior sound efi ld resulting eTh power flow has been validated and widely utilized as 2 Advances in Acoustics and Vibration a comprehensive measure for evaluation of overall level of vibratory power flow calculation, especially based on Dowell’s vibration energy of vibration isolation systems mounted on modal coupling theory. flexible foundations [8], which could also be explained as averagesound powerwhenappliedtosoundfieldanaly- 2. Theory sis. 2.1. Equations of Fluid-Structural Coupled Vibration. Con- In this paper, through modal coupling analysis on the sider that a uid-fi fl lled enclosure occupies a volume 𝑉.Its uid-s fl tructural vibration of the water-filled enclosure with boundary 𝐷=𝐷 +𝐷 ,where 𝐷 =0 ̸ represents the flexi- elastic boundaries, an ecffi ient computational method is 𝑆 𝐹 𝐹 ble area of the surrounding wall and 𝐷 (might be zero) deduced to determine the vibratory power flow generated 𝑆 represents the acoustic rigid area. by exterior excitations on the outside surface of the elastic The u fl id inside the enclosure satisfied the wave equation structure, including the total power flow entering into the and associated boundary condition. uid-s fl tructural coupled system and the net power flow transmitted into the hydroacoustic efi ld. Characteristics of 𝜕 𝑝 ( 𝜎, 𝑡 ) (1) the coupled natural frequencies and modals are investigated 𝐾 ∇ 𝑝 ( 𝜎, 𝑡 )−𝜌 =0 (𝜎∈𝑉 ) 0 0 𝜕𝑡 by a numerical example of a rectangular water-filled cavity with vfi e acoustic rigid walls and one elastic panel. Inu fl ential ( 𝜎, 𝑡 ) (2) =−𝜌 𝑎 ( 𝜎, 𝑡 ) (𝜎 ∈ 𝐷 ) 0 𝑛 𝐹 factors of power flow transmission characteristics are further 𝜕𝑛 discussed with the purpose of overall evaluation and reduc- ( 𝜎, 𝑡 ) tion of the cavity water sound energy. (3) =0 (𝜎 ∈𝐷 ), 𝜕𝑛 1.2. eTh oretical Development. There has been a continuous where 𝑝(𝜎, 𝑡) is the sound pressure at point 𝜎(𝑥, 𝑦, 𝑧) ∈ 𝑉; effort for decades on investigation of uid-s fl tructural mech- 𝑎 (𝜎, 𝑡) is the acceleration of the flexible wall in the normal anismofclosedsound efi ldwithflexibleboundaries.Ithas direction 𝑛 (positive outward); 𝜌 and 𝐾 are the equilibrium 0 0 been recommended that the locally reactive acoustic normal uid fl density and uid fl volume stiffness, respectively. impedance was the earlier theory to understand the sound If 𝐷 =0,(1)hasmodal solutions 𝐹 (𝜎)⋅exp(𝑗𝜔 𝑡), 𝑟 = absorption caused by the interaction between a reverberation 0, 1, 2, . . .,where 𝜔 is the 𝑟th acoustical natural frequency in room and its surrounding walls [9]. Later attention was paid the condition of rigid boundary and 𝐹 (𝜎)is the correspond- to the modal coupling between the enclosed sound efi lds and ing natural mode with orthogonality as follows: theflexiblewalls torevealthemorecomplicatedmechanism 0𝑟 =𝑠̸ demonstrated by experimental results, which could not be 𝐹 ( 𝜎 )𝐹 ( 𝜎 ) ∭ dV = interpreted by the locally reactive theory [10, 11]. 𝜌 𝑐 0 𝑀 𝑟=𝑠, The modal responses of acoustoelastic enclosures were (4) first developed by Dowell et al. [12, 13] by applying Green’s 0𝑟 =𝑠̸ [∇𝐹 ( 𝜎 ) ] ⋅[∇𝐹 ( 𝜎 ) ] function to the inhomogeneous wave differential equation ∭ dV = of the enclosed sound field and applying the classical modal 𝜌 0 𝜔 𝑀 𝑟=𝑠, and eigenvalue theorem to the simultaneous uid-s fl tructural differential equations to result in a resolution of coupling where 𝑐 = √𝐾 /𝜌 is the acoustic velocity of the u fl id, 𝑀 0 0 0 modals. eTh re are still other resolution methods for the same is the 𝑟th acoustical modal mass in the condition of rigid acoustoelasticity equations which could be referred to, such boundary, and ∇𝐹 (𝜎) = [𝜕𝐹 /𝜕𝑥, 𝐹𝜕 /𝜕𝑦, 𝐹𝜕 /𝜕𝑧] is as Laplace transformation [14] and Ritz series [15]. In general, thecolumngradientvectorofmodalfunction 𝐹 (𝜎). Dowell’s method is based on the familiar uncoupled acoustic Consider the solution of (1) with 𝐷 =0 ̸ being in the form enclosure modes and structural modes, could be more of modal superposition; that is, easily implemented, and has been successfully applied to the investigation of variety of uid-s fl tructural interaction systems 𝑝 ( 𝜎, 𝑡 )= ∑𝐹 ( 𝜎 )𝑃 ( 𝑡 )= [F ( 𝜎 ) ] P ( 𝑡)( 𝜎∈𝑉 ), (5) [16, 17]. Beginning with the “modal coupling method,” Pan 𝑟=0 andBiesgaveaninsightanalysis oftheweak-coupledand where F(𝜎)and P(𝑡)are column vectors of modal function well-coupled modals and their decay characteristics of a 𝐹 (𝜎)and its corresponding modal coordinate 𝑃 (𝑡),respec- 𝑟 𝑟 rectangular panel-cavity coupled system [18, 19]; Davis put tively; that is, F(𝜎) = [𝐹 (𝜎), 𝐹 (𝜎), 𝐹 (𝜎), . . .] and P(𝑡) = 𝐴0 𝐴1 𝐴2 forward a method for approximate estimation of the coupled [𝑃 (𝑡), 𝑃 (𝑡), 𝑃 (𝑡), . . .]. 0 1 2 natural frequencies of acoustoelastic enclosures by “coupling Aeft r substituting (5) into (1), multiply both sides of the coefficient” [20]. resultant equation with a left-multiplication matrix (vector) Other important developments might lie in the field of F(𝜎)and finally integrating the equation over volume 𝑉,one discrete numerical techniques, such as FEM/BEM, for uid- fl obtains structural vibration analysis. However, these methods are usually preferred in the investigation of irregularly shaped 2 ∭ F ( 𝜎 )[∇ 𝑝 ( 𝜎, 𝑡 ) ]dV cavities and targeting specicfi engineering problems. And that (6) wouldbebeyondthe discussionofthis paper, whichwould mainly focus on a general theoretical evaluation method for − ∭ {F ( 𝜎 )[F ( 𝜎 ) ] P ( 𝑡 ) }dV =0. the overall vibration level of a uid-fil fl led enclosure through 0 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑠 𝐴𝑟 𝐴𝑟 𝐴𝑠 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝜕𝑝 𝜕𝑝 Advances in Acoustics and Vibration 3 By applying Green’s theorem to the rfi st term of above of the modal masses and natural frequencies respectively, equation, one has and 𝜔 and 𝑀 =∬ 𝑚 [𝑊 (𝜎)]d𝐴 (𝑗 = 1, 2, 3, . . .) represent the 𝑗th natural frequency and modal mass of the ∭ {F ( 𝜎 )[F ( 𝜎 ) ] P ( 𝑡 ) }dV thin-wall structures in vacuo, respectively. Q (𝑡)and Q (𝑡) 2 𝐴 𝐵 are column vectors of the general forces due to 𝑝 (𝜎, 𝑡) and 𝑝 (𝜎, 𝑡) loaded on the thin-wall structures in vacuo, (7) + ∭ [∇F ( 𝜎 ) ] ∇F ( 𝜎 )P ( 𝑡 )dV respectively, and ( 𝜎, 𝑡 ) Q ( 𝑡 )= ∬ W ( 𝜎 )𝑝 ( 𝜎, 𝑡 )d𝐴, 𝐴 𝐴 = ∯ F ( 𝜎 ) d𝐴, 𝜕𝑛 𝐹 (12) where ∇F(𝜎)is the gradient matrix of modal function 𝐹 (𝜎): Q ( 𝑡 )=− ∬ W ( 𝜎 )𝑝 ( 𝜎, 𝑡 )d𝐴. 𝐵 𝐵 ∇F(𝜎) = [∇𝐹 (𝜎), ∇𝐹 (𝜎), ∇𝐹 (𝜎), . . .]. 𝐷 𝐴0 𝐴1 𝐴2 Now substitute the boundary condition equations (2)∼(3) The right-hand term of (8) and the rst fi term Q (𝑡)on and orthogonality equation (5) into (7): the right hand of (11) are of the u fl id-structural interaction between the sound efi ld inside the cavity and its flexible walls. M [P ( 𝑡 )+ Ω P ( 𝑡 )]=−∯ F ( 𝜎 )𝑎 ( 𝜎, 𝑡 )d𝐴, (8) 𝐴 𝑛 Substituting (10) into the right-hand term of (8) and taking 2 2 notice of 𝑎 (𝜎, 𝑡) = 𝜕 𝑤(𝜎, 𝑡)/𝜕𝑡 at 𝜎∈𝐷 and 𝑎 (𝜎, 𝑡) = 0 𝑛 𝐹 𝑛 where M and Ω are diagonal matrices of acoustical modal at 𝜎∈𝐷 , one could define a coupling matrix L as follows: 𝐴 𝐴 masses and natural frequencies, respectively; that is, M = diag[𝑀 ,𝑀 ,𝑀 ,...]and Ω = diag[𝜔 ,𝜔 ,𝜔 ,...]. 𝐴0 𝐴1 𝐴2 𝐴 𝐴0 𝐴1 𝐴2 L = ∬ F ( 𝜎 )[W ( 𝜎 ) ] d𝐴=[𝐿 ] The flexible boundary of the cavity is assumed to be thin- (13) wall structures, where linear partial differential equations (𝑟 = 0,1,2,...; 𝑗 = 1,2,3,...), wouldbeadoptedtofit thethin-wall structures’vibration, such that where 𝐿 denotes the element of the coupling matrix L at the 𝑟th row and the 𝑗th column. 𝜕 𝑤 ( 𝜎, 𝑡 ) 𝑆𝑤 ( 𝜎, 𝑡 )+𝑚 =𝑝 ( 𝜎, 𝑡 )−𝑝 ( 𝜎, 𝑡 ) 𝐵 𝐴 𝐵 And (8) turns into 𝜕𝑡 (9) ̈ ̈ (𝜎 ∈ 𝐷 ), M [P ( 𝑡 )+ Ω P ( 𝑡 )]=−L ⋅ B ( 𝑡 ). (14) 𝐹 𝐴 where 𝑆 is a linear differential operator representing struc- Dealing with Q (𝑡), one could express𝑝 (𝜎, 𝑡) in (12) 𝐴 𝐴 tural stiffness; 𝑚 is structural mass per unit area; 𝑝 (𝜎, 𝑡) with(5), and(11) wouldbecome 𝐵 𝐴 and 𝑝 (𝜎, 𝑡) areexcitations onthesurfaceofthethin-wall 2 𝑇 M [B 𝑡 + Ω B 𝑡 ]= L ⋅ P 𝑡 + Q 𝑡 . (15) () () () () structures due to the cavity acoustics and external dynamical 𝐵 𝐵 𝐵 forces (intensity of pressure), respectively; 𝑤(𝜎, 𝑡) is the displacement response of the thin-wall structures, which is 2.2. Modal Analysis. In order to carry out a modal analysis defined in the normal direction of 𝐷 . about the uid-s fl tructural vibration system governed by (14) eTh solution of (9) could be expressed as and (15), let Q (𝑡) = 0, and suppose that there exist vibration solutions as follows: 𝑤 ( 𝜎, 𝑡 )= ∑ 𝑊 ( 𝜎 )𝐵 ( 𝑡 )= [W ( 𝜎 ) ] B ( 𝑡 ) −1 P ( 𝑡 )=( M ) ⋅ 𝜒 ⋅ exp (𝑗𝜔𝑡) , 𝑗=1 (10) 𝐴 (16) (𝜎 ∈ 𝐷 ), −1 B 𝑡 =(Ω √M ) ⋅ 𝜒 ⋅ exp (𝑗𝜔𝑡) , () 𝐵 𝐵 𝐵 where 𝑊 (𝜎)is the 𝑗th modal function that is defined on 𝐷 and concerned with the property of the thin-wall structures where √M = diag⌊√𝑀 , √𝑀 , √𝑀 ,...⌋ and √M = 𝐴 𝐴0 𝐴1 𝐴2 𝐵 in vacuo and 𝐵 (𝑡)is the modal coordinate corresponding diag⌊√𝑀 , √𝑀 , √𝑀 ,...⌋ are square roots of the diago- 𝐵1 𝐵2 𝐵3 to 𝑊 (𝜎); W(𝜎) and B(𝑡)are column vectors of 𝑊 (𝜎) nalacousticalmodalmatrix M and the diagonal structural and 𝐵 (𝑡),respectively;that is, W(𝜎) = [𝑊 (𝜎), 𝑊 (𝜎), 𝑗 𝐵1 𝐵2 𝑇 modal matrix M ,respectively. 𝜒 =[𝜒 ,𝜒 ,𝜒 ,...] and 𝐵 𝐴 𝐴0 𝐴1 𝐴2 𝑇 𝑇 𝑊 (𝜎), . . .] and B(𝑡) = [𝐵 (𝑡), 𝐵 (𝑡), 𝐵 (𝑡), . . .]. 𝐵3 1 2 3 𝜒 =[𝜒 ,𝜒 ,...] arecolumnvectors offluid-structural 𝐵 𝐵1 𝐵2 By substituting (10) into (9) and using the orthogonality coupled modal shape coecffi ients related to the cavity sound of 𝑊 (𝜎),there wouldbeamodaldieff rential function as efi ld and the flexible boundary structures, respectively. follows: Substituting (16) into (14) and (15), an eigenvalue problem 2 could be obtained as M [B 𝑡 + Ω B 𝑡 ]= Q 𝑡 + Q 𝑡 , (11) () () () () 𝐵 𝐵 𝐴 𝐵 A A 𝜒 𝜒 11 12 𝐴 𝐴 2 2 where M and Ω , expressed as M = diag[𝑀 ,𝑀 ,𝑀 , A𝜒 =[ ][ ]=𝜔 [ ]=𝜔 𝜒, (17) 𝐵 𝐵 𝐵 𝐵1 𝐵2 𝐵3 A A 𝜒 𝜒 21 22 ...]and Ω = diag[𝜔 ,𝜔 ,𝜔 ,...], are diagonal matrices 𝐵 𝐵 𝐵 𝐵1 𝐵2 𝐵3 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝑟𝑗 𝑟𝑗 𝜕𝑝 𝐵𝑗 𝐵𝑗 𝐵𝑗 4 Advances in Acoustics and Vibration where 𝜒 could be named as vector of fluid-structural coupled transformation matrix to transform the general force P modal shape coefficients and A is a symmetric characteristic into its uid-s fl tructural expression (the derivation of the matrix, and A’s partitioned matrices could be calculated by matrices H and T has been explained via (A.5)∼(A.8) in 𝐻 𝐻 2 −1 −1 𝑇 −1 𝑇 Appendix A.2). M = X ⋅ Χ + X ⋅ Χ is a diagonal A = Ω +( √M ) LM L ( √M ) , A = A = 𝐶 𝐴 𝐴 𝐵 𝐵 11 𝐴 𝐴 12 𝐴 𝐵 21 −1 −1 2 matrix of the uid-s fl tructural coupled modal masses, and √ √ −( M ) LΩ ( M ) ,and A = Ω . 𝐴 𝐵 𝐵 22 𝐵 Ω = diag[𝜔 ,𝜔 ,𝜔 ,...]is a diagonal matrix of the u fl id- Equation (17) would give eigenvalues of matrix A,thatis, 𝐶 𝐶0 𝐶1 𝐶2 2 2 structural coupled natural frequencies. The superscript “𝐻” 𝜔 =Ω (1 + 𝑗𝜂 ), and the accompanying eigenvectors denotes Hermitian transposition of matrices. (𝑘) 𝑇 (𝑘) 𝑇 (𝑘) 𝑇 (𝑘) (𝑘) (𝑘) [𝜒 ] =[{𝜒 } ,{𝜒 } ]=[𝜒 ,𝜒 ,𝜒 ,..., 𝐴 𝐵 𝐴0 𝐴1 𝐴2 The power flow (density) inputted by exterior excitation (𝑘) (𝑘) (𝑘) 𝜒 ,𝜒 ,𝜒 ,...], 𝑘 = 0,1,2,... ,where Ω corre- 𝐵1 𝐵2 𝐵3 𝑝 (𝜎, 𝑡) into the uid-s fl tructural system is sponds to the 𝑘th uid-s fl tructural natural frequency. 𝜂 𝜔/2𝜋 is the loss factor associated with the 𝑘th damped normal 𝑝 ( 𝜎, 𝜔 )= ∫ Re {𝑝 ( 𝜎, 𝑡 ) } in 𝐵 mode, which might be resulting from the introduction of 2𝜋 a complex stiffness of the flexible boundary or a complex 𝜕𝑤 ( 𝜎, 𝑡 ) volume stiffness 𝐾 of the u fl id in consideration of the 0 (23) ⋅ Re {− } d𝑡 damping properties of the u fl id-structural system. It should 𝜕𝑡 (𝑘) also be noticed that [𝜒 ] might be complex vectors when =− Re {𝑗𝑊 𝜎, 𝜔 ⋅𝑃 𝜎 }(𝜎∈𝐷 ). ( ) ( ) 𝜔 are complex numbers. 𝐵 𝐹 The uid-s fl tructural coupled modal functions of the The total power flow input is cavity’s sound efi ld and the flexible boundaries would be expressed as 𝑇 𝑇 𝑃 ( 𝜔 )= ∬ 𝑝 ( 𝜎, 𝑡 )d𝐴=− Re {𝑗P T H ( 𝜔 ) in in 𝐵 −1 𝑇 (𝑘) 𝐹 ( 𝜎 )= [F ( 𝜎 ) ] ( M ) 𝜒 (𝜎∈𝑉 ), −1 𝑇 ∗ (18) √ (24) ⋅ X (Ω M ) ∬ W ( 𝜎 )⋅𝑃 ( 𝜎 )d𝐴} = −1 𝐵 𝐵 𝐵 𝐵 𝑇 (𝑘) 2 𝑊 ( 𝜎 )= [W ( 𝜎 ) ] (Ω √M ) 𝜒 (𝜎 ∈ 𝐷 ). 𝐵 𝐵 𝐵 𝐹 −1 𝑇 𝑇 𝑇 ∗ ⋅ Re {𝑗 P T H ( 𝜔 )⋅ X (Ω M ) P }, 𝐵 𝐵 𝐵 𝐵 𝐵 2.3. Vibratory Power Flow Transmission. In the condition that the flexible boundary structures of the cavity are subjected where the superscript “∗” denotes conjugation of complex to a harmonic exterior excitation, that is, 𝑝 (𝜎, 𝑡) = 𝑃 (𝜎) ⋅ 𝐵 𝐵 numbers. exp(𝑗𝜔𝑡),letP =−∬ W(𝜎)⋅𝑃 (𝜎)d𝐴; P indicates that the 𝐵 𝐵 𝐵 The power flow (density) transmitted through the uid- fl amplitudes of general forces belong to the uncoupled flexible structural interaction boundary of the cavity into the structures. eTh steady responses of the uid-s fl tructural cou- enclosed sound eld fi is plingcavitywouldbe 𝜔/2𝜋 𝑝 𝜎, 𝜔 = ∫ Re {𝑝 𝜎, 𝑡 } ( ) ( ) 󵄨 tr 󵄨𝜎∈𝐷 −1 𝐹 2𝜋 𝑝 𝜎, 𝑡 = F 𝜎 (√M ) Χ HTP exp (𝑗𝜔𝑡) ( ) [ ( ) ] 𝐴 𝐴 𝐵 (19) 𝜕𝑤 𝜎, 𝑡 ( ) (25) ⋅ Re {− 󵄨 } d𝑡 =𝑃 ( 𝜎, 𝜔 )⋅ exp (𝑗𝜔𝑡) (𝜎∈𝑉 ) 󵄨 𝜕𝑡 󵄨𝜎∈𝐷 −1 = Re {𝑗𝑊 ( 𝜎, 𝜔 )⋅𝑃 ( 𝜎, 𝜔 ) | }. 𝑤 𝜎, 𝑡 = W 𝜎 (Ω √M ) Χ HTP exp (𝑗𝜔𝑡) ( ) [ ( ) ] 𝜎∈𝐷 𝐵 𝐵 𝐵 𝐵 (20) The total transmission power flow is =𝑊 ( 𝜎, 𝜔 )⋅ exp (𝑗𝜔𝑡) (𝜎 ∈ 𝐷 ), where 𝑃(𝜎, 𝜔) and 𝑊(𝜎, 𝜔) denote the amplitudes of the 𝑃 ( 𝜔 )= ∬ 𝑝 ( 𝜎, 𝜔 )d𝐴 tr tr harmonic sound pressure in the cavity and harmonic dis- placement of the thin-wall structures. Χ andΧ are matrices 𝐴 𝐵 = Re { ∬ F 𝜎 ⋅ Λ 𝜔 ⋅ W 𝜎 d𝐴} [ ( ) ] ( ) ( ) composed of arrays of eigenvectors of the characteristic 2 (26) (0) (1) (2) matrix A;thatis, Χ =[𝜒 , 𝜒 , 𝜒 ,...]and Χ = 𝐴 𝐵 𝐴 𝐴 𝐴 (0) (1) (2) ∞ ∞ [𝜒 , 𝜒 , 𝜒 ,...].And 𝐵 𝐵 𝐵 { } = Re ∑ ∑ 𝐿 ⋅𝜆 ( 𝜔 ) , { } −1 2 2 𝑟=0 𝑗=1 (21) { } H ( 𝜔 )= (−𝜔 M + Ω ⋅ M ) 𝐶 𝐶 𝐶 −1 −1 −1 where Λ(𝜔) = 𝑗(√M ) X H(𝜔) ⋅ 𝐴 𝐴 𝐻 −1 𝐻 𝐻 𝐻 ∗ 𝐻 ∗ −1 (22) T =−X (√M ) LM + X Ω (√M ) , 𝐴 𝐴 𝐵 𝐵 𝐵 𝐵 TP P T [H(𝜔)]X (Ω√M ) =[𝜆 (𝜔)]might be 𝐵 𝐵 𝐵 𝐵 𝐵 named as a power transmission matrix and 𝜆 (𝜔) denotes where H couldbenamedasthecomplexfrequencyresponse the element of matrix Λ(𝜔) at the 𝑟th row and the 𝑗th matrix of the uid-s fl tructural coupled cavity and T is a column. 𝑟𝑗 𝑟𝑗 𝑟𝑗 𝑟𝑗 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 Advances in Acoustics and Vibration 5 For a simply supported plate, its natural frequencies and E Y modal functions are determined by 2 2 4 2 𝑚 𝑛 X O 𝜋 𝐸ℎ F 2 (29) 𝜔 = ( + ) 2 2 2 12𝜌 (1 − 𝜇 ) 𝐿 𝐿 𝑥 𝑦 𝑚 𝑛 𝑊 (𝑥,𝑦,0) = sin ( )sin ( ), 3 (30)  = 1000 kg/G 𝐿 𝐿 𝑥 𝑦 c =1500 m/s where 𝐸, 𝜌, ℎ,and 𝜇 are Young’s modulus, mass density, x thickness, and Poisson’s ratio of the plate, respectively; ∀(𝑚 ,𝑛 )∈𝑁 ,𝑗 ∈ 𝑁 ,let 𝜔 arrange in a sequence 2 2 2 Figure 1: A panel-cavity coupled system. |𝜔 |<|𝜔 |<|𝜔 |<⋅ ⋅ ⋅ . 𝐵1 𝐵2 𝐵3 The geometrical and material properties adopted for numerical computation are as follows: 𝐿 = 0.4 m, 𝐿 = 𝑥 𝑦 0.6 m, 𝐿 = 0.7 m, and ℎ = 0.005 m; 𝜌 = 1000 kg/m and 3. Numerical Simulation and Analysis 𝑧 0 9 −4 11 −3 𝐾 = 2.25×10 Pa×(1+10 j);𝐸 = 2.0×10 Pa×(1+10 j), 3 3 3.1. Simulation Model. A panel-cavity coupled system shown 𝜌 = 7.8 × 10 kg/m ,and 𝜇 = 0.28 (steel). in Figure 1 consists of a rectangular water-filled room with five rigidwalls andone simplysupportedplate subjecttoexterior 3.2. Modal Analysis. There is a convergence investigation harmonic distributed force (pressure) 𝑝 (𝜎, 𝑡). about the uid-s fl tructural coupled natural frequencies result- In discussion of the distribution shape of exterior exci- ing from (17) at rfi st. In Table 1, with a fixed number of plate tation, the plane harmonic wave incident is a common modals involved in calculation, the convergence of coupled assumption. Suppose that 𝑝 (𝜎, 𝑡) = 𝑃 ⋅exp[𝑗(𝜔𝑡−𝑘 𝑥 sin 𝜃)], natural frequencies could be observed by increasing the 𝐸 𝐸 where 𝑃 is the amplitude, 𝑘 is the wave number, and 𝜃 number of water sound modals involved. The convergence is the incident angle (𝑝 is uniform along the 𝑦 direction); couldalsobeobservedbyincreasingthenumberofplate 𝑝 has a perpendicular component 𝑝 (𝜎, 𝑡) = 𝑃 cos𝜃⋅ modals involved in Table 2. It could be suggested that the 𝐸 𝐵 𝐸 exp(−𝑗𝑘 𝑥 sin 𝜃) ⋅ exp(𝑗𝜔𝑡) = 𝑃 (𝜎) ⋅ exp(𝑗𝜔𝑡). Generally, modal coupling method could achieve good convergence in 𝑃 (𝜎)wouldbeacomplexfunctionandhaveinnfi itevariety solving the uid-s fl tructural coupled problem described here. of distribution shapes when the wave frequency, velocity, And it could also be observed that the convergence at lower and incident angle changed. Figure 2 shows one example of frequencies is more rapid than that at higher frequencies, distribution shape of 𝑃 (𝜎)with wave frequency 𝑓 = 500 Hz, and the solution precision would be more dependent on velocity 𝑐 = 344 m/s, and incident angle 𝜃=𝜋/3 .Itis accounting for more water sound modals. However, there is true that the modal coupling method is valid in dealing with noneedtocarryoutahighaccuracycalculation herefora those variant distribution shapes of 𝑃 (𝜎).However,some theoretical qualitative analysis, and in the later part of this specicfi analysis on the special case with 𝜃=0 ,thatis,a paper, 50 plate modals and 500 water sound efi ld modals uniform 𝑃 over theplate surface, wouldalsogiveindications are taken into consideration, by which totally 550 coupled of general significance. eTh uniform excitation had been modals could be revealed. And also, because there is no need adopted by other authors previously [7, 15]. And, moreover, to list all550 modes here,onlypartial data(thefirst several taking sonar array cavities as examples, they are regularly modes) are listed in Tables 1 and 2 mounted on related ship structures through rubber blankets; Figure 3 gives a comparison of sound pressure solutions the uniform structural excitation assumption would be a between the modal coupling approach and FEM (by the basic consideration. software of LMS Virtual.Lab Acoustics) as a theoretical val- For the water-filled rectangular room, the natural fre- idation verification. eTh differences between the two results quencies and modal functions of the sound eld fi with rigid in Figure 3(a) are due to modal truncation; more modes are boundaries are determined by involved in the FEM/BEM software. The uid-s fl tructural coupled natural frequencies resulting 2 2 from (17) are compared with those of sound efi ld with rigid 𝐾 𝑚 𝜋 𝑛 𝜋 𝑙 𝜋 𝜔 = [( ) +( ) +( ) ] (27) boundaries obtained by (27) and simply supported steel plate 𝜌 𝐿 𝐿 𝐿 0 𝑥 𝑦 𝑧 obtained by(29),whicharelistedinTable 3.Asmentioned above, 550 coupled modals have been obtained by involving 𝐹 (𝑥,𝑦,𝑧) 50 plate modals and 500 water sound efi ld modals in modal coupling calculation, but only the first 11 coupled modal (28) 𝑚 𝑛 𝑙 = cos ( )cos ( )cos ( ), frequencies are presented. 𝐿 𝐿 𝐿 𝑥 𝑦 𝑧 As a whole, it could be concluded that the u fl id-structural frequencies are very different from those of the water sound where ∀(𝑚 ,𝑛 ,𝑙 )∈𝑁 , 𝑟∈𝑁 ,and let 𝜔 arrange in a field (with rigid boundaries) and the flexible boundary plate 2 2 2 sequence |𝜔 |<|𝜔 |<|𝜔 | < ⋅⋅⋅ . (in vacuo), which means that there would be a strong 𝐴0 𝐴1 𝐴2 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝜋𝑧 𝜋𝑦 𝜋𝑥 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝜋𝑦 𝜋𝑥 𝐵𝑗 𝐵𝑗 𝐵𝑗 6 Advances in Acoustics and Vibration Table 1: Coupled natural frequencies with different number of water sound modals involved. Fluid-structural coupled natural frequency 𝑓 (Hz) Number of water sound modals involved 50 200 500 1000 2000 Number of plate modals involved 50 50 50 50 50 0 0 0000 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 10 682.5 617.7 599.4 592.1 584.7 Table 2: Coupled natural frequencies with different number of plate modals involved. Fluid-structural coupled natural frequencies 𝑓 (Hz) Number of water sound modals involved 500 500 500 500 500 Number of plate modals involved 25 50 100 200 500 0 0 0000 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 10 599.5 617.7 599.3 599.3 599.3 Table 3: Natural frequencies of the water sound field, plate, and uid-st fl ructural coupled cavity. Fluid-structural coupled Natural frequencies of water sound field Natural frequencies of plate natural frequencies (𝑚 ,𝑛 ) 𝑓 (Hz) 𝑟 (𝑚 ,𝑛 ,𝑙 ) 𝑓 (Hz) 𝑗 𝑘𝑓 (Hz) 0 (0, 0, 0)00 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 10 599.4 𝐶𝑘 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐴𝑟 𝐶𝑘 𝐶𝑘 Advances in Acoustics and Vibration 7 0.5 0.5 −0.5 −1 −0.5 80 80 60 60 50 50 40 40 40 40 30 30 20 20 20 20 10 10 0 0 0 0 (a) Real part (b) Imaginary part Figure 2: One example of nonuniform distribution shape of 𝑃 (𝜎)(wave frequency 𝑓 = 500 Hz, velocity 𝑐 = 344 m/s, and incident angle 𝜃=𝜋/3 ). Pressure (nodal values).1 Occurrence 540 (./G ) 64.9 36.5 8.05 −20.4 −48.8 −77.2 −50 −106 −134 −162 −191 −100 −219 On Boundary −150 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz) matlab VL (b) Cloud picture of boundary sound pressure by VL (a) Comparison of sound pressure solutions (540 Hz) Figure 3: Comparison of sound pressure solutions between the modal coupling approach and FEM. coupling between the enclosed water sound field and its unneglectable departure of coupled natural frequencies from elastic surrounding structures. If the u fl id in the cavity was every natural frequency of the uncoupled flexible structures and cavity. air (𝜌 ≈ 1.29 kg/m ; 𝑐 ≈ 344 m/s), it could be found that 0 0 Except 𝑓 =0, the natural frequencies of the water the coupled natural frequencies 𝑓 would approximately be 𝐴0 sound efi ld are much higher than those of the simply equal to either some uncoupled structural natural frequencies supported plate, and the uid-s fl tructural coupled natural 𝑓 or some uncoupled cavity acoustical natural frequencies frequencies are inclined to come to be rather lower. It 𝑓 , and that is a situation of weak coupling. eTh motion seemed that one might carry out a comparison between of each subsystem in a weakly coupled system will not be the coupled natural frequencies and those of plate, and the essentially different from that of the uncoupled systems. change regulation of differences of adjacent fluid-structural However, if the density of the medium in the cavity is much natural frequencies 𝑓 −𝑓 is similar to those 𝑓 −𝑓 denser than air, such as water, the coupling may turn out to (𝑘+1) 𝐶 𝐵(−𝑗 1) of the plate. However, the frequency distribution of coupled be strong, and big deformation of the resulting modes from naturalfrequencieswouldturntobelowerandmorecrowded the uncoupled panel and the cavity modes may be expected as the order 𝑘 increased. [18]. In this sense, the strong coupling could be judged by any Sound pressure (dB) 𝐵𝑗 𝐶𝑘 𝐴𝑟 𝐵𝑗 𝐶𝑘 8 Advances in Acoustics and Vibration 0.8 0.5 0.6 0.4 −0.5 0.2 0 −1 80 80 60 60 50 50 40 40 40 40 30 30 y y 20 20 20 20 x x 10 10 0 0 0 0 (a) 𝑊 = sin(𝜋𝑥/𝐿 ) sin(𝜋𝑦/𝐿 ),𝑓 ≈108.0 Hz (b) 𝑊 = sin(𝜋𝑥/𝐿 ) sin(3𝜋𝑦/𝐿 ),𝑓 ≈373.4 Hz 𝐵1 𝑥 𝑦 𝐵1 𝐵4 𝑥 𝑦 𝐵4 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 80 80 60 60 50 50 40 40 40 40 30 30 y y 20 20 20 20 x x 10 10 0 0 0 0 (c) 𝑊 = sin(3𝜋𝑥/𝐿 ) sin(𝜋𝑦/𝐿 ),𝑓 ≈705.9 Hz (d) 𝑊 = sin(𝜋𝑥/𝐿 ) sin(5𝜋𝑦/𝐿 ),𝑓 ≈905.2 Hz 𝐵8 𝑥 𝑦 𝐵4 𝐵11 𝑥 𝑦 𝐵11 Figure 4: Some uncoupled plate modals. In order to reveal modal coupling characteristics, Figures the plane 𝑧=𝐿 /2, and Figure 6(b) illustrates the appur- 4 and 5 show several uncoupled plate modals 𝑊 (𝜎)which tenant participant coefficients of the uncoupled acoustical are expressed by (30) and uid-s fl tructural coupled plate cavity modals in the constitution of coupled acoustical cavity modals 𝑊 (𝜎)which are determined by (29) through modal modal. coupling calculation. The figure shows that 𝑊 is almost the 𝐶0 same as 𝑊 , while it is true according to 𝑊 ’s expression 3.3. Power Flow Transmission. In the simulation model of 𝐵1 𝐶0 and that is just an example of weak-coupled modal. It seemed Figure 1, the vibratory power flow inputted by exterior that the coupled modal shape 𝑊 is similar to the uncoupled excitation into the whole uid-s fl tructural coupled system and 𝐶3 modal shape 𝑊 ; however, they are quite different in fact the enclosed water sound efi ld, that is, 𝑃 calculated by (24) 𝐵4 in according to the expression of 𝑊 ,and that is astrong and 𝑃 calculated by (26), is dissipated by system damping. 𝐶3 tr coupled modal. Phenomena of strong modal coupling are And thus the higher or lower power flow level would be a obvious when inspecting 𝑊 and 𝑊 shown in Figure 5. comprehensive indicator to measure the vibration level or 𝐶7 𝐶8 The similar couplings have also happened to the acous- energy level of the panel-cavity coupled system and water tical cavity modals. And, moreover, except that the coupled sound efi ld. acoustical modal 𝐹 is practically equal to 𝐹 ,thatis,the Figure 7showsthespectrumofinput powerflow 𝑃 and 𝐶0 𝐴0 in rigid body modal of the uncoupled cavity sound efi ld, all transmitted power flow 𝑃 ,inwhichthedropbetween 𝑃 and tr in other uid-s fl tructural coupled acoustical cavity modals are 𝑃 is the dissipation power of the plate’s damping. Because the tr composed in a strong coupling manner; that is, they are linear exterior excitation is symmetric (uniform 𝑃 as mentioned combinations of several 𝐹 ,theuncoupledacousticalmodals in Section 3.1), only symmetric modals are present, and the of the rigid wall cavity. As 𝐹 (𝜎)is defined in three-dimen- spectrum peaks at 0 Hz, 217.6 Hz, 451.8 Hz, and 537.2 Hz could sional space and it is inconvenient to plot it by a planar figure, be associated with the modals shown in Figures 5 and 6. Figure 6(a) illustrates one coupled acoustical modal shape in There is some similarity between 𝑃 or 𝑃 and the water in tr B8 W B1 W B4 C11 𝐶𝑟 𝐴𝑟 𝐶𝑘 𝐵𝑗 Advances in Acoustics and Vibration 9 f ≈ 217.6 Hz C3 f =0 Hz C0 0.025 0.02 0.02 0.01 0.015 0.01 −0.01 0.005 −0.02 80 80 50 50 40 40 40 40 30 30 y 20 20 20 10 10 0 0 0 0 (a) 𝑊 ≈0.0248(𝑊 +0.05𝑊 +0.02𝑊 +0.01𝑊 ) (b) 𝑊 ≈0.0132(−0.21𝑊 +𝑊 +0.04𝑊 +0.05𝑊 +0.03𝑊 ) 𝐶0 𝐵1 𝐵4 𝐵8 𝐵11 𝐶3 𝐵1 𝐵4 𝐵8 𝐵11 𝐵12 f ≈ 451.8 Hz f ≈ 537.2 Hz C7 C8 0.01 0.01 0.005 0.005 0 0 −0.005 −0.005 −0.01 −0.01 50 50 40 40 40 40 30 30 y 20 y 20 x x 10 10 0 0 0 (c) 𝑊 ≈0.0074(−0.28𝑊 −0.26𝑊 +𝑊 +0.14𝑊 +0.07𝑊 ) (d) 𝑊 ≈0.0062(−0.11𝑊 −0.16𝑊 −𝑊 +0.18𝑊 +0.08𝑊 ) 𝐶7 𝐵1 𝐵4 𝐵8 𝐵11 𝐵12 𝐶8 𝐵1 𝐵4 𝐵8 𝐵11 𝐵12 Figure 5: Some coupled plate modals. ×10 ×10 8.8 4 8.75 8.7 8.65 −2 8.6 80 −4 −6 y 20 0 10 20 30 40 50 60 70 80 90 100 0 Modal serial number (a) 𝐹 (𝑥,𝑦,𝐿 ),𝑓 ≈537.2 Hz (b) Participant coefficients of the uncoupled cavity modals 𝐶8 𝑧/2 𝐶8 Figure 6: Simple illustration of one coupled cavity modal shape. C7 C0 F (x,y,L /2) C8 z C3 C8 Modal participant coefficients 10 Advances in Acoustics and Vibration −150 has been predicted by Table 3. u Th s the alteration of elasticity modulus of plate would lead to a phenomenon of “frequency −200 shifting.” eTh 𝑃 spectrum with smaller plate’s elasticity mod- tr ulus could be regarded as a contraction of that with greater −250 elasticity modulus toward lower frequencies, and the peaks of 𝑃 would occur at relatively lower frequencies and become −300 tr more crowded. However, since the variation range of Young’s −350 modulus would be limited in practice, its influence on power flow transmission might not be very serious, and it could also −400 be observed that reduction of Young’s modulus might bring about a benefit of slight reduction of 𝑃 ’s peak valleys. −450 tr In order to make an inspection of the alteration of differ- −500 ences between 𝑃 and 𝑃 with different plate elasticity mod- in tr ulus, Figure 9(b) shows the spectra of power flow ratio PR = −550 10 log(𝑃 /𝑃 ).ThepeaksofPR wouldalwaysappeararound 0 200 400 600 800 1000 1200 1400 1600 1800 2000 in tr the resonance frequencies except at 𝑓 ,which couldbeeasily 𝐶0 Frequency (Hz) explained by the fact that the plate’s damping consumes more CH energy when system resonances take place. In the lowest NL frequency range around 𝑓 =0,the plateconsumeslittle 𝐶0 energy, and therefore 𝑃 ≈𝑃 and PR ≈0.Itshouldbenoted tr in Figure 7: Spectrum of 𝑃 and 𝑃 . in tr that greaterPRs mightnot implylowerlevelsoftransmitted power flow 𝑃 ;thefactisprobablyjust theoppositebecause tr the power flow input 𝑃 might be in much higher levels at in thesametime.In this sense, minority of PR peaknumbers 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 𝑃 attenuation, which could tr be discovered through a synthesized analysis of the figures shown. In Figure 10(a), different mass density values were given −50 to the elastic plate, in which the value of 2770 kg/m is by reference to aluminium. A signicfi ant feature of the spectra in thefigureisthat changesinplate’s massdensity wouldappar- −100 ently change the average level of transmitted power flow 𝑃 . tr According to (29), the increase of plate’s mass density would −150 cause decreasing of plate’s natural frequencies and “frequency 0 200 400 600 800 1000 1200 1400 1600 1800 2000 shifting” of uid-s fl tructural coupled modals, as shown in Fig- Frequency (Hz) ure 10(a), similar to the situation of decreasing Young’s mod- P(L /2, L ,0) x y ulus in Figure 9. However, the increase of plate’s mass density P(L /2, L ,L /2) x y z would increase its modal masses at the same time. And that is the reason why 𝑃 ’s average level is cut down even tr Figure 8: Spectrum of water sound pressure. though more resonance modals would come into being in the relative lower frequency band. It could also be explained by thefactthataheaviervibrating masswouldgenerate sound pressure at the center of the plate’s interior surface, a greater reduction in dynamic force (or pressure) trans- that is, 𝑃((𝐿 /2, 𝐿 /2, 0), 𝜔)(refer to (21)), which is shown in mission. Figure 10(b) is about the power flow ratio PR. PR 𝑥 𝑦 Figure 8. However, the power flow is of evaluation of sound could not effectively reveal the apparent 𝑃 reduction at tr power. nonresonance frequencies, such as 800 Hz∼1200 Hz, because As a theoretical investigation, hypothesize that the mate- the transmitted power flow 𝑃 is very close to the power flow tr rial property parameters of the plate, that is, Young’s modulus input 𝑃 at those frequencies. in 𝐸, mass density 𝜌, Poisson’s ratio 𝜇,and dampinglossfactor, Another important influential factor that should be paid could be altered independently. In Figure 9(a), transmitted attention is the damping loss factor of the plate. Figure 11(a) power flows are compared under different Young’s modulus demonstrates a conflictive situation where smaller damping loss factorwouldincreasethepeaksoftransmitted power oftheelasticplate, wherethevalueof 7.24 × 10 Pa is by reference to aluminium. When the plate’s elasticity modulus flow at resonance frequencies, while at the broadband non- decreases, the plate’s natural frequencies would decrease resonance frequencies smaller damping would be beneficial simultaneously,andthiswouldmovethefluid-structuralcou- to reduction of transmitted power flow. To explain this result, pled natural frequencies into lower frequency ranges, which one might make an analogy with the vibration isolation P and P (dB) Sound pressure (dB) CH NL Advances in Acoustics and Vibration 11 −200 80 −250 −300 −350 −400 −450 −500 −550 −10 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Frequency (Hz) 11 −3 11 −3 E = 2.0 × 10 0; × (1 + 10 D) E = 2.0 × 10 0; × (1 + 10 D) 10 −3 10 −3 E = 7.24 × 10 0; × (1 + 10 D) E = 7.24 × 10 0; × (1 + 10 D) 10 −3 E = 2.0 × 10 0; × (1 + 10 D) (a) Transmitted power flow (b) Power flow ratio Figure 9: Effect of plate’s elasticity modulus on power flow transmission. −200 90 −250 −300 −350 −400 −450 −500 −550 −10 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Frequency (Hz)  = 7800 EA/G  = 7800 EA/G = 2770EA/G = 2770EA/G 4 3 = 1 × 10 EA/G (a) Transmitted power flow (b) Power flow ratio Figure 10: Effect of plate’s mass density on power flow transmission. theory. If the elastic plate was considered as some kind of be favorable in the nonresonance frequency ranges, which elastic isolator which was specially designed to attenuate the would not be counted in power flow ratio. transmission of exterior excitation energy into the water- Poisson’s ratio 𝜇 would aeff ct the power flow transmission filledcavity, thedampingwouldincreasethepowertrans- the same way Young’s modulus does as shown in Figure 9. mission and would not be expected except for attenuation Referring to (29), increasing Young’s modulus could be equiv- of resonance peak. Through illustration of power flow ratio alent to increasing Poisson’s ratio. However, the variation PR as that in Figure 11(b), it could be confirmed that greater scope of Poisson’s ratio is much smaller than that of Young’s damping could be used to obstruct energy transmission modulus, and the power flow transmission would be affected at resonance frequencies, whereas smaller damping would more by Young’s modulus than by Poisson’s ratio. And, for this P (dB) P (dB) NL NL PR (dB) PR (dB) 12 Advances in Acoustics and Vibration −200 160 −250 −300 −350 −400 −450 −500 −550 −600 −20 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Frequency (Hz) 11 −3 11 −3 E = 2.0 × 10 0; × (1 + 10 D) E = 2.0 × 10 0; × (1 + 10 D) 11 −4 11 −4 E= 2.0 × 10 0; × (1 + 3.310 D) E= 2.0 × 10 0; × (1 + 3.310 D) (a) Transmitted power flow (b) Power flow ratio Figure 11: Eeff ct of damping loss factor of plate on power flow transmission. reason, no additional repetitive gfi ures would be put forward reduction of peak valleys of transmitted power flow could also here. be expected on the other hand. (4) Denser material could be beneficial to an apparent attenuation of average level of power flow transmission 4. Conclusions into the water-filled enclosure, even though it would be Backgrounded by evaluation or control of mechanical self- accompanied with a decrease of system natural frequencies. noise in sonar array cavity, transmitted power flow or sound (5) Smaller inner damping of enclosure’s thin-wall struc- power input calculation is carried out by modal coupling tures, which were distinguished from that specially designed analysis on the uid-s fl tructural vibration of the uid-fil fl led for sound absorption destination and only for the purpose enclosure with elastic boundaries. Power flow transmission of suppression of resonance peaks, might be proposed to analysis is presented through a numerical simulation exam- attenuate the average level of power flow transmission. ple of water-filled rectangular panel-cavity coupled system. Detailed discussion is carried out about power flow transmis- Appendix sion characteristics affected by variation of material property parameters of cavity’s elastic boundary structure, aiming at A. Derivation Procedures of Related reduction of water sound level inside the cavity. From the Matrices Equations results, one could draw the following conclusions. (1) Power flow or sound power transmission analysis A.1. Eigenvalue Problem. Equations (14) and (15) could be could be a valuable method for evaluation or prediction of easilyveriefi dtobeequivalenttothe followingsimultaneous water sound level in dealing with strong coupled vibration modal differential equations, which are fundamental in Dow- problems of water-filled acoustoelastic enclosure systems. ell’s modal coupling method: (2) eTh uid-s fl tructural coupled natural frequencies of a water-filledacoustoelasticenclosurewouldbegreatlyaeff cted by interaction between the hydroacoustic field and its sur- ̈ ̈ 𝑀 𝑃 +𝑀 𝜔 𝑃 =−∑𝐿 𝐵 𝑟 = 0, 1, 2, . . . ( ) 𝑟 𝑟 𝑗 rounding elastic boundaries. There would be a tendency that 𝑗=1 the uid-s fl tructural coupled natural frequencies turn to be smaller in value and more crowded in frequency distribution (A.1) 𝑀 𝐵 +𝑀 𝜔 𝐵 = ∑𝐿 𝑃 +𝑄 𝑗 𝑗 𝑟 than those of elastic boundary structures and inner water 𝑟=0 sound field in rigid boundary condition. (3) Decreasing elasticity modulus or Poison’s ratio of (𝑗 = 1,2,...), water sound cavity’s thin-wall structures would cause the frequency distribution of system modals to contract toward where 𝐿 =∬ 𝐹 𝑊 d𝐴, 𝑄 =−∬ 𝑊 𝑝 d𝐴,and all lower frequency ranges and result in more power flow 𝐷 𝐷 𝐹 𝐹 transmission peaks in the lower frequency band, while slight other symbols have been mentioned previously. P (dB) NL PR (dB) 𝐵𝑗 𝐵𝑗 𝐵𝑗 𝐴𝑟 𝑟𝑗 𝐵𝑗 𝐵𝑗 𝑟𝑗 𝐵𝑗 𝐵𝑗 𝑟𝑗 𝐴𝑟 𝐴𝑟 𝐴𝑟 Advances in Acoustics and Vibration 13 󸀠 2 −1 −1 𝑇 −1 2 󸀠 Through some simple algebraic algorithm, (14) and (15) 𝜒 Ω + M LM L −M LΩ 𝜒 𝐴 𝐴 𝐵 𝐴 𝐵 2 𝐴 𝐴 𝜔 [ ]=[ ][ ] couldbegrouped togetherandwrittenas 󸀠 󸀠 −1 𝑇 2 𝜒 −M L Ω 𝜒 𝐵 𝐵 𝐵 𝐵 (A.3) 2 −1 −1 𝑇 −1 2 P Ω + M LM L −M LΩ P 𝐴 𝐴 𝐵 𝐴 𝐵 󸀠 [ ]+[ ][ ] = A [ ]. −1 𝑇 2 B 𝜒 B −M L Ω 𝐵 𝐵 (A.2) −1 −1 The above equation raises an eigenvalue problem, but it −M LM Q 𝐴 𝐵 𝐵 =[ ]. is nonstandard because the matrix A is asymmetric. eTh −1 M Q transformation expressed by (16) would result in a standard eigenvalue problem, which has been expressed by (17), where While carrying out a modal analysis, one could simply let the matrix A is symmetric. And, moreover, one could prove 󸀠 󸀠 P = 𝜒 ⋅ exp(𝑗𝜔𝑡), B = 𝜒 ⋅ exp(𝑗𝜔𝑡),and Q =0;thus that 𝐴 𝐵 −1 −1 −1 −1 (√M ) L (√M ) Ω (√M ) L (√M ) −Ω 𝐴 𝐵 𝐴 𝐵 𝐴 𝐵 A =[ ][ ] −Ω O Ω O 𝐵 𝐴 (A.4) −1 −1 −1 −1 (√M ) L (√M ) Ω (√M ) L (√M ) Ω 𝐴 𝐵 𝐴 𝐴 𝐵 𝐴 =[ ][ ] , −Ω O −Ω O 𝐵 𝐵 where O denotes a zero matrix with proper dimension rank. = T ⋅ X ⋅ 𝛼 ( 𝑡 ), Before any damping factors are introduced, (A.4) proves (A.5) that the matrix A is positive semidefinite; that is, all its eigenvalues would be nonnegative. That would provide a where 𝛼(𝑡) isacolumnvectorvariabletotakeplace of P(𝑡) mathematical guarantee that the eigenvalue calculation of and B(𝑡). matrix A would not fail at any occurrences of negative square Aeft r substituting the above equation into (A.2), multiply natural frequencies. both sides of the resultant equation with a le-ft multiplication 𝐻 −1 By substituting any 𝑘th eigenvalue of matrix A and its matrix X T ;one has accompanying eigenvector, which have been symbolled with 2 (𝑘) 𝑇 (𝑘) 𝑇 (𝑘) 𝑇 𝜔 and [𝜒 ] =[{𝜒 } ,{𝜒 } ],respectively,into(16) 𝐻 −1 𝐻 −1 󸀠 𝐴 𝐵 X T T X ⋅ 𝛼 ̈ + X T A T X ⋅ 𝛼 0 0 0 0 (𝑘) (𝑘) (i.e., let 𝜔=𝜔 and 𝜒 = 𝜒 and 𝜒 = 𝜒 ), the 𝑘th 𝐴 𝐴 𝐵 𝐵 −1 −1 uid-s fl tructural coupled modal of the cavity sound field and (A.6) −M LM Q 𝐴 𝐵 𝐻 −1 = X T [ ]. its flexible boundary structures, which have been symbolled −1 M Q 𝐵 𝐵 with 𝐹 and 𝑊 , respectively, are obtained by taking out the time-independent part of P(𝑡)and B(𝑡), that is, (18). −1 󸀠 𝐻 𝐻 Take notice that T A T = A,and X X and X AX are all diagonal matrices according to the theory of orthogonality A.2. Harmonic Solution. In the situation of forced vibration, of eigenvectors. In fact, the diagonal elements of matrix that is, Q =0 ̸ , there is a standard decoupling procedure 𝐻 𝐻 𝐻 M = X X are just the modal masses of the u fl id-structural for (A.2) by applying an eigenmatrix X =[X ,X ] 𝐴 𝐵 coupled modals, and the diagonal elements of matrix Ω = which is composed of the series of A’s eigenvectors, where −1 𝐻 the detailed definitions of X and X have been mentioned M (X AX)arethecomplexfluid-structuralcouplednatural 𝐴 𝐵 previously in Section 2.2 aer ft (19) and (20), and note that frequencies; these two matrices have already been defined in Hermitian transposition of matrices has been employed here Section 2.2 aer ft (20). With this knowledge, (A.2) or (A.6) is in consideration of the introduction of damping loss factors. turned into its decoupled expression: Replacing the real elasticity modulus of the flexible bound- −1 −1 ary structures or real volume stiffness of the cavity sound −M LM Q 𝐴 𝐵 2 −1 𝐻 −1 field with a complex elasticity modulus or complex volume 𝛼 ̈ + Ω 𝛼 = M X T [ ]. (A.7) 𝐶 𝐶 0 −1 M Q stiffness is a regular method when damping factors are to be 𝐵 involved in analysis, which has been declared in Sections 2.2, Obviously, the transient or steady solution of the above 3.1, and 3.3; the method has also been adopted by many other equation under arbitrary deterministic Q (𝑡) could be authors. obtained by the method of convolution integral. Only har- Now let monic solution is concerned here. Referring to (12), when −1 𝑝 (𝜎, 𝑡) is harmonic, Q (𝑡)is harmonic, too. And suppose 𝐵 𝐵 P ( 𝑡 ) (√M ) O X [ ] [ ]= [ ]𝛼 ( 𝑡 ) that Q (𝑡) =P ⋅ exp(𝑗𝜔𝑡) and 𝛼(𝑡) = 𝛼 exp(𝑗𝜔𝑡);bysub- 𝐵 𝐵 −1 B ( 𝑡 ) X √ 𝐵 O (Ω M ) stituting them into (A.7), the time-dependent part exp(𝑗𝜔𝑡) 𝐵 𝐵 [ ] 𝐶𝑘 𝐶𝑘 𝐶𝑘 𝐶𝑘 14 Advances in Acoustics and Vibration −1 would be eliminated, and the time-independent part of 𝛼(𝑡) 𝑇 ∗ ⋅ X [(Ω M ) ] W ( 𝜎 )⋅𝑃 ( 𝜎 )d𝐴} . 𝐵 𝐵 𝐵 𝐵 is obtained; that is, (A.10) −1 −1 −M LM P −1 2 2 −1 𝐻 −1 𝐴 𝐵 𝛼 =(−𝜔 I + Ω ) M X T [ ] 𝐶 𝐶 0 −1 Note that in the last expression of the right hand of M P the above equation, the matrices P , T, H(𝜔), X , Ω ,and 𝐵 𝐵 𝐵 −1 2 2 M are all 𝜎-independent; only W(𝜎) ⋅ 𝑃 (𝜎)would join in 𝐵 𝐵 =(−𝜔 M + Ω M ) 𝐶 𝐶 𝐶 ∗ the integral in (A.10). The integral would result in (P),in −1 −1 which P has been den fi ed in the beginning paragraph of √M O −M LM P 𝐴 𝐴 𝐵 𝐵 𝐻 𝐻 Section 2.3, and the superscript “∗” denotes conjugation of ⋅[X X ][ ][ ] 𝐴 𝐵 −1 (A.8) O Ω √M M P 𝐵 𝐵 𝐵 𝐵 complex numbers. Other matters need attention; H(𝜔),Ω , andM are all diagonal matrices; that is, [H(𝜔)]= H(𝜔)and = H ( 𝜔 ) −1 𝑇 −1 [(Ω√M ) ] =(Ω √M ) . Aeft r taking into account all 𝐵 𝐵 𝐵 𝐵 −1 −1 above factors, (24) is realized. 𝐻 −1 𝐻 √ √ ⋅[−X ( M ) LM + X Ω ( M ) ]P 𝐴 𝐵 𝐵 𝐵 𝐴 𝐵 𝐵 The obtainment of (25) is similar to that of (23), in which the cavity sound pressure at the interior surface of the thin- = H 𝜔 ⋅ T ⋅ P , ( ) wall structures takes place of the exterior excitation. And also the integral of 𝑝 (𝜎, 𝜔) is processed similar to that of 𝑝 (𝜎, 𝜔) tr in at the beginning by quotation of (22) and (25). where I isaunitmatrixofproperdimension rank. A.3. Power Flow Formulation. eTh basic formulation for 𝑃 ( 𝜔 )= ∬ Re {𝑗𝑊 ( 𝜎, 𝜔 )⋅𝑃 ( 𝜎, 𝜔 ) | }d𝐴 tr 𝜎∈𝐷 power flow computation is = ∬ Re {𝑗𝑃 ( 𝜎, 𝜔 ) 2𝜋/𝜔 𝑃 = ∫ 𝐹 cos ⋅ 𝑉 cos (𝜔𝑡 + 𝜑) d𝑡 2𝜋 󵄨 𝜔 ⋅ 𝑊 𝜎, 𝜔 󵄨 } d𝐴= [ ( ) ] 2𝜋/𝜔 󵄨𝜎∈𝐷 = ∫ Re {𝐹 exp (𝑗𝜔𝑡)} 2𝜋 0 −1 ⋅ Re { ∬ 𝑗 [F ( 𝜎 ) ] ( M ) (A.9) 𝐷 ⋅ Re {𝑉 exp (𝑗𝜔𝑡 + 𝑗𝜑)} d𝑡= 𝐹𝑉 cos 𝜑 −1 𝐻 𝐻 𝐻 𝐻 ∗ ⋅ Χ HTP P T H X [(Ω √M ) ] 𝐴 𝐵 𝐵 𝐵 𝐵 𝐵 (A.11) = Re {[𝐹 exp (𝑗𝜔𝑡)] [𝑉 exp (𝑗𝜔𝑡 + 𝑗𝜑)]} 1 𝜔 = Re {[𝐹 exp (𝑗𝜔𝑡)] [𝑉 exp (𝑗𝜔𝑡 + 𝑗𝜑)] }, ⋅ W ( 𝜎 )d𝐴} = Re { ∬ [F ( 𝜎 ) ] 2 2 −1 where 𝐹 cos or 𝐹 exp(𝑗𝜔𝑡) is a harmonic excitation force, ⋅[𝑗 (√M ) 𝑉 cos or 𝑉 exp(𝑗𝜔𝑡 + 𝑗𝜑) is the harmonic velocity response at theactionpointof theexcitationforce,and 𝜑 is the phase −1 𝐻 𝐻 ∗ 𝐻 ∗ difference between the excitation force and velocity response. ⋅ Χ HTP P T H X (Ω √M ) ] 𝐴 𝐵 𝐵 𝐵 𝐵 𝐵 By applying the above formulation, (23) would be appar- ent. And because 𝑃 (𝜎)is of 𝜎-dependent distribution over 𝐷 , 𝑝 (𝜎, 𝜔) in (23) is of distribution of power flow density. ⋅ W ( 𝜎 )d𝐴} . 𝐹 in Equation (25) is an integral of 𝑝 over 𝐷 to count up the in 𝐹 total power flow. The integral is initiated by quotation of (22) The basis of the above transformation includes the fol- and (23) as follows: lowing: W(𝜎)is real, H and Ω are diagonal, and M is real 𝐵 𝐵 anddiagonal.Itcouldbeexaminedthatthe productofthe matrices between [F(𝜎)]and W(𝜎)in the last expression of 𝑃 ( 𝜔 )=− ∬ Re {𝑗𝑊 ( 𝜎, 𝜔 )⋅𝑃 ( 𝜎 ) }d𝐴 in 𝐵 the right hand of the above equation, which has been den fi ed as Λ(𝜔)in (26), is 𝜎-independent. And the expansion of the 𝑇 ∗ =− ∬ Re {𝑗 𝑊 𝜎, 𝜔 ⋅𝑃 𝜎 }d𝐴 [ ( ) ] ( ) 𝐵 product [F(𝜎)]Λ(𝜔)W(𝜎)is 𝑇 𝑇 𝑇 𝜆 ( 𝜔 ) 𝜆 ( 𝜔 ) ⋅⋅⋅ 𝑊 ( 𝜎 ) =− Re { ∬ 𝑗 T [H ( 𝜔 ) ] 11 12 𝐵1 ∞ ∞ 𝑇 [ ] [ ] 𝜆 ( 𝜔 ) 𝜆 ( 𝜔 ) ⋅⋅⋅ 𝑊 ( 𝜎 ) 21 22 𝐵2 F ΛW =[𝐹 ( 𝜎 ) 𝐹 ( 𝜎 ) 𝐹 ( 𝜎 ) ⋅⋅⋅][ ] [ ] = ∑ ∑𝜆 ( 𝜔 )⋅𝐹 ( 𝜎 )⋅𝑊 ( 𝜎 ). (A.12) 𝐴0 𝐴1 𝐴2 . . . 𝑟=0 𝑗=1 . . . . . d . [ ] [ ] 𝐵𝑗 𝐴𝑟 𝑟𝑗 𝜔𝑡 𝜔𝑡 𝐹𝑉 𝜔𝑡 Advances in Acoustics and Vibration 15 Table 4: Dimension ranks of related matrices with R cavity acoustical modals and N thin-wall structural modals. Matrices Dimension rank Matrices Dimension rank Matrices Dimension rank ∇𝐹 3×1 F 𝑅×1 P 𝑅×1 ∇F 3×𝑅 M 𝑅×𝑅 Ω 𝑅×𝑅 𝐴 𝐴 W 𝑁×1 B 𝑁×1 M 𝑁×𝑁 Ω 𝑁×𝑁 Q 𝑁×1 Q 𝑁×1 𝐵 𝐴 𝐵 (𝑘) 󸀠 (𝑘) 󸀠 L 𝑅×𝑁 𝜒 ,𝜒 , 𝜒 𝑅×1 𝜒 ,𝜒 ,𝜒 𝑁×1 𝐴 𝐴 𝐴 𝐵 𝐵 𝐵 (𝑘) 󸀠 𝜒, 𝜒 (𝑅 + 𝑁) × 1 A,A (𝑅 + 𝑁) × (𝑅 + 𝑁) A 𝑅×𝑅 A 𝑅×𝑁 A 𝑁×𝑅 A 𝑁×𝑁 12 21 22 X 𝑅×(𝑅 +𝑁) X 𝑁×(𝑅 +𝑁) X (𝑅 + 𝑁) × (𝑅 + 𝑁) 𝐴 𝐵 S (𝑅 + 𝑁) × 1 H (𝑅 + 𝑁) × (𝑅 + 𝑁) T (𝑅 + 𝑁) × 𝑁 S 𝑁×1 M (𝑅 + 𝑁) × (𝑅 + 𝑁) Ω (𝑅 + 𝑁) × (𝑅 + 𝑁) 𝑄 𝐶 𝐶 P 𝑁×1 Λ 𝑅×𝑁 T (𝑅 + 𝑁) × (𝑅 + 𝑁) 𝐵 0 𝛼, 𝛼 (𝑅 + 𝑁) × 1 Since 𝜆 (𝜔) is 𝜎-independent, when the integral of [5] G. B. Warburton, “Vibration of a cylindrical shell in an acoustic 𝑇 medium,” Journal of Mechanical Engineering Science,vol.3,no. [F(𝜎)]Λ(𝜔)W(𝜎)over 𝐷 is proceeding, only 𝐹 (𝜎)𝑊 (𝜎) 1, pp.69–79,1961. is involved, and that would result in 𝐿 ,whichhasbeen [6] J.-M. David and M. 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