Advances in Acoustics and Vibration

Volume 2018, Article ID 5273280, 16 pages

https://doi.org/10.1155/2018/5273280

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

^{1}School of Mechanical Engineering, Shandong University, Jinan 250061, China^{2}Key Laboratory of High Efficiency and Clean Mechanical Manufacture (Shandong University), Ministry of Education, Jinan 250061, China

Correspondence should be addressed to Rui Huo; nc.ude.uds@iurouh

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.

#### Abstract

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

#### 1. Introduction

##### 1.1. Background

The work of this paper is backgrounded by prediction or evaluation and control of mechanical self-noise in sonar array cavity. The mechanical self-noise, which is caused by structural vibration of sonar cavity’s wall, might significantly weaken the detection performance of sonar at lower frequencies [1, 2]. The sources of mechanical self-noise might be multiple such as vibrating machines on the ship which diffuse vibration energy or second excitation of structure-borne sound. However, it is essential to comprehend the characteristics of interaction between the enclosed water sound field and its elastic boundary structures for the purpose of prediction, evaluation, and control of interior hydroacoustic noise [3].

The subjects of cabin noise in various flight vehicles and automobiles are more familiar in the investigation of fluid-structural coupled vibration of acoustoelastic enclosure, which mainly focus on characteristics of sound transmission through the elastic wall into interior sound field resulting from exterior air-borne sound [4]. In these cases, weak coupling has been commonly assumed because of the low density of air and high stiffness of cabin wall, which means that the cavity’s interior sound pressure would have little influence on the vibration of cavity wall, and modals of interior sound field would also be affected very lightly [5]. In contrast, a much stronger coupling might be present when a water sound field takes place of the air [6].

The sound pressure is most commonly used to represent the property of sound field in the study of acoustic-structural coupling of acoustoelastic enclosure. The ratio of sound pressure at the outside surface of the elastic cavity wallboard to that at internal surface, which is defined as “noise reduction,” is applied to evaluation of sound transmission characteristics [4, 7]. Since the sound pressure would change greatly at different points of the sound field, the value of noise reduction would also be very different, and a comprehensive measure, for example, power flow, would be expected for an overall evaluation of vibration level of the enclosed sound field. The power flow has been validated and widely utilized as a comprehensive measure for evaluation of overall level of vibration energy of vibration isolation systems mounted on flexible foundations [8], which could also be explained as average sound power when applied to sound field analysis.

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

##### 1.2. Theoretical Development

There has been a continuous effort for decades on investigation of fluid-structural mechanism of closed sound field with flexible boundaries. It has been recommended that the locally reactive acoustic normal impedance was the earlier theory to understand the sound absorption caused by the interaction between a reverberation room and its surrounding walls [9]. Later attention was paid to the modal coupling between the enclosed sound fields and the flexible walls to reveal the more complicated mechanism demonstrated by experimental results, which could not be interpreted by the locally reactive theory [10, 11].

The modal responses of acoustoelastic enclosures were first developed by Dowell et al. [12, 13] by applying Green’s function to the inhomogeneous wave differential equation of the enclosed sound field and applying the classical modal and eigenvalue theorem to the simultaneous fluid-structural differential equations to result in a resolution of coupling modals. There are still other resolution methods for the same acoustoelasticity equations which could be referred to, such as Laplace transformation [14] and Ritz series [15]. In general, Dowell’s method is based on the familiar uncoupled acoustic enclosure modes and structural modes, could be more easily implemented, and has been successfully applied to the investigation of variety of fluid-structural interaction systems [16, 17]. Beginning with the “modal coupling method,” Pan and Bies gave an insight analysis of the weak-coupled and well-coupled modals and their decay characteristics of a rectangular panel-cavity coupled system [18, 19]; Davis put forward a method for approximate estimation of the coupled natural frequencies of acoustoelastic enclosures by “coupling coefficient” [20].

Other important developments might lie in the field of discrete numerical techniques, such as FEM/BEM, for fluid-structural vibration analysis. However, these methods are usually preferred in the investigation of irregularly shaped cavities and targeting specific engineering problems. And that would be beyond the discussion of this paper, which would mainly focus on a general theoretical evaluation method for the overall vibration level of a fluid-filled enclosure through vibratory power flow calculation, especially based on Dowell’s modal coupling theory.

#### 2. Theory

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

Consider that a fluid-filled enclosure occupies a volume . Its boundary , where represents the flexible area of the surrounding wall and (might be zero) represents the acoustic rigid area.

The fluid inside the enclosure satisfied the wave equation and associated boundary condition.where is the sound pressure at point ; is the acceleration of the flexible wall in the normal direction (positive outward); and are the equilibrium fluid density and fluid volume stiffness, respectively.

If , (1) has modal solutions , where is the th acoustical natural frequency in the condition of rigid boundary and is the corresponding natural mode with orthogonality as follows:where is the acoustic velocity of the fluid, is the th acoustical modal mass in the condition of rigid boundary, and is the column gradient vector of modal function .

Consider the solution of (1) with being in the form of modal superposition; that is,where and are column vectors of modal function and its corresponding modal coordinate , respectively; that is, and .

After substituting (5) into (1), multiply both sides of the resultant equation with a left-multiplication matrix (vector) and finally integrating the equation over volume , one obtains

By applying Green’s theorem to the first term of above equation, one haswhere is the gradient matrix of modal function : .

Now substitute the boundary condition equations (2)~(3) and orthogonality equation (5) into (7):where and are diagonal matrices of acoustical modal masses and natural frequencies, respectively; that is, and .

The flexible boundary of the cavity is assumed to be thin-wall structures, where linear partial differential equations would be adopted to fit the thin-wall structures’ vibration, such thatwhere is a linear differential operator representing structural stiffness; is structural mass per unit area; and are excitations on the surface of the thin-wall structures due to the cavity acoustics and external dynamical forces (intensity of pressure), respectively; is the displacement response of the thin-wall structures, which is defined in the normal direction of .

The solution of (9) could be expressed aswhere is the th modal function that is defined on and concerned with the property of the thin-wall structures in vacuo and is the modal coordinate corresponding to ; and are column vectors of and , respectively; that is, and .

By substituting (10) into (9) and using the orthogonality of , there would be a modal differential function as follows:where and , expressed as and , are diagonal matrices of the modal masses and natural frequencies respectively, and and represent the th natural frequency and modal mass of the thin-wall structures in vacuo, respectively. and are column vectors of the general forces due to and loaded on the thin-wall structures in vacuo, respectively, and

The right-hand term of (8) and the first term on the right hand of (11) are of the fluid-structural interaction between the sound field inside the cavity and its flexible walls. Substituting (10) into the right-hand term of (8) and taking notice of at and at , one could define a coupling matrix as follows:where denotes the element of the coupling matrix at the th row and the th column.

And (8) turns into

Dealing with , one could express in (12) with (5), and (11) would become

##### 2.2. Modal Analysis

In order to carry out a modal analysis about the fluid-structural vibration system governed by (14) and (15), let , and suppose that there exist vibration solutions as follows:where and are square roots of the diagonal acoustical modal matrix and the diagonal structural modal matrix , respectively. and are column vectors of fluid-structural coupled modal shape coefficients related to the cavity sound field and the flexible boundary structures, respectively.

Substituting (16) into (14) and (15), an eigenvalue problem could be obtained aswhere could be named as vector of fluid-structural coupled modal shape coefficients and is a symmetric characteristic matrix, and ’s partitioned matrices could be calculated by , , and .

Equation (17) would give eigenvalues of matrix , that is, , and the accompanying eigenvectors , , where corresponds to the th fluid-structural natural frequency. is the loss factor associated with the th damped normal mode, which might be resulting from the introduction of a complex stiffness of the flexible boundary or a complex volume stiffness of the fluid in consideration of the damping properties of the fluid-structural system. It should also be noticed that [] might be complex vectors when are complex numbers.

The fluid-structural coupled modal functions of the cavity’s sound field and the flexible boundaries would be expressed as

##### 2.3. Vibratory Power Flow Transmission

In the condition that the flexible boundary structures of the cavity are subjected to a harmonic exterior excitation, that is, , let ; indicates that the amplitudes of general forces belong to the uncoupled flexible structures. The steady responses of the fluid-structural coupling cavity would bewhere and denote the amplitudes of the harmonic sound pressure in the cavity and harmonic displacement of the thin-wall structures. and are matrices composed of arrays of eigenvectors of the characteristic matrix ; that is, and . Andwhere could be named as the complex frequency response matrix of the fluid-structural coupled cavity and is a transformation matrix to transform the general force into its fluid-structural expression (the derivation of the matrices and has been explained via (A.5)~(A.8) in Appendix A.2). is a diagonal matrix of the fluid-structural coupled modal masses, and is a diagonal matrix of the fluid-structural coupled natural frequencies. The superscript “” denotes Hermitian transposition of matrices.

The power flow (density) inputted by exterior excitation into the fluid-structural system is

The total power flow input iswhere the superscript “” denotes conjugation of complex numbers.

The power flow (density) transmitted through the fluid-structural interaction boundary of the cavity into the enclosed sound field is

The total transmission power flow iswhere might be named as a power transmission matrix and denotes the element of matrix at the th row and the th column.

#### 3. Numerical Simulation and Analysis

##### 3.1. Simulation Model

A panel-cavity coupled system shown in Figure 1 consists of a rectangular water-filled room with five rigid walls and one simply supported plate subject to exterior harmonic distributed force (pressure) .