A Chebyshev–Lagrangian method for acoustic analysis of a rectangular cavity with arbitrary impedance walls

A general Chebyshev–Lagrangian method is proposed to obtain the analytical solution for a rectangular acoustic cavity with arbitrary impedance boundary conditions. The originality of the present paper is the successful attempt of applying orthogonal polynomials, such as Chebyshev polynomials of the first kind, to the analysis of a rectangular sound field with general wall impedance. The sound pressure is uniformly expressed as triplicate Chebyshev polynomial series which is independent in each direction. The Chebyshev polynomial series solution is obtained using the Rayleigh–Ritz procedure after considering the influence of boundary impedance on the cavity as the work done by the impedance surfaces in the Lagrangian function. The accuracy and reliability of the proposed method are validated against the analytical solutions and some numerical results available in the literature. Excellent orthogonality and complete properties of the Chebyshev polynomials ensure the rapid convergence, numerical stability, high accuracy of the current solution. The simplicity and low computational cost of the present approach make it preferable to obtain the results of complex models even in the relative high frequency range by choosing enough truncated terms in the sound pressure expression. Numerous cases with various uniform or non-uniform impedance boundary conditions are analyzed numerically and some of the results can be used as benchmark. It is shown that the impedance boundary condition can effectively influence or modify the acoustic characteristics and response of a cavity.     

Acoustic characteristics analysis of slender cavity system with arbitrary impedance end conditions

A modeling method for the dynamic characteristics analysis of a slender acoustical cavity with impedance end conditions is established. In order to satisfy the continuity requirement at impedance ends for the first order differential of sound pressure, field function is constructed as the standard Fourier series supplemented by boundary smoothed auxiliary polynomials. System characteristic equation is derived by solving the governing differential equation and impedance acoustic boundary of slender acoustical cavity system simultaneously,relevant acoustical modal information is obtained via the state space solution procedure. In numerical simulation, various acoustic variables, such as acoustical modal frequency, sound pressure modal shape, sound pressure response and the particle velocity, are presented for the slender acoustical cavity system with different boundary conditions and compared with those results in the existing literature. The correctness and effectiveness of the proposed method are then fully validated.     

Transmission loss of orthogonally rib-stiffened double-panel structures with cavity absorption

The transmission loss of sound through infinite orthogonally rib-stiffened double-panel structures having cavity-filling fibrous sound absorptive materials is theoretically investigated. The propagation of sound across the fibrous material is characterized using an equivalent fluid model, and the motions of the rib-stiffeners are described by including all possible vibrations, i.e., flexural displacements, bending, and torsional rotations. The effects of fluid-structure coupling are account for by enforcing velocity continuity conditions at fluid-panel interfaces. By taking full advantage of the periodic nature of the double-panel, the space-harmonic approach and virtual work principle are applied to solve the sets of resultant governing equations, which are eventually truncated as a finite system of simultaneous algebraic equations and numerically solved insofar as the solution converges. To validate the proposed model, a comparison between the present model predictions and existing numerical and experimental results for a simplified version of the double-panel structure is carried out, with overall agreement achieved. The model is subsequently employed to explore the influence of the fluid-structure coupling between fluid in the cavity and the two panels on sound transmission across the orthogonally rib-stiffened double-panel structure. Obtained results demonstrate that this fluid-structure coupling affects significantly sound transmission loss (STL) at low frequencies and cannot be ignored when the rib-stiffeners are sparsely distributed. As a highlight of this research, an integrated optimal algorithm toward lightweight, high-stiffness and superior sound insulation capability is proposed, based on which a preliminary optimal design of the double-panel structure is performed.     

考虑任意阻抗壁面条件管腔结构声场特性分析

针对任意阻抗壁面条件一维管腔声学系统建模,对系统动力学特性进行预报。为了满足阻抗边界条件对声压一阶导数连续性要求,管腔声压函数通过在标准傅里叶级数端点位置引入边界光滑辅助多项式进行构建。结合壁面阻抗声学边界和管腔声学Helmholtz控制微分方程得到强形式标准特征值问题,获得相应的声学模态信息。在数值仿真中,通过算例给出各种边界条件下管腔声学模态频率、声压振型、声压和质点振速频率响应曲线,与现有文献中相关结果进行对比,充分验证了本文求解方法的正确性和有效性,证明该方法可对任意阻抗壁面条件管腔系统声学特性进行准确预报。     

Effectiveness of T-shaped acoustic resonators in low-frequency sound transmission control of a finite double-panel partition

Double-panel partitions are widely used for sound insulation purposes. Their insulation efficiency is, however, deteriorated at low frequencies due to the structural and acoustic resonances. To tackle this problem, this paper proposes the use of long T-shaped acoustic resonators in a double-panel partition embedded along the edges. In order to facilitate the design and assess the performance of the structure, a general vibro-acoustic model, characterizing the interaction between the panels, air cavity, and integrated acoustic resonators, is developed. The effectiveness of the technique as well as the optimal locations of the acoustic resonators is examined at various frequencies where the system exhibits different coupling characteristics. The measured optimal locations are also compared with the predicted ones to verify the developed theory. Finally, the performance of the acoustic resonators in broadband sound transmission control is demonstrated.     

Analysis of structural - acoustic coupling of elastic rectangular enclosure with arbitrary boundary conditions

The structural acoustic coupling characteristics of a rectangular enclosure con- sisting of two elastic supported flexible plates and four rigid plates are analyzed.A general formulation considering the full coupling between the plates and cavity is developed by using Hamiltonian function and Rayleigh-Ritz method.By means of continuous distributions of ar- tificial springs along boundary of flexible plates,a wide variety of boundary conditions and structure joint conditions are considered.To demonstrate the validity of the analytical model, the responses of sound pressure in the cavity and plate velocity are worked out.The analytical results coincides well with Kim's experimental results.The result is satisfactory.Finally,an- alytical results on the structure vibration and the sound field inside the cavity are presented. These results indicate that the coupling of the combined structure is relatively weak,so the internal cavity sound is controlled by plate directly excited,and the translational stiffness affects the sound more than the rotational stiffness does.     

Vibro-acoustic analysis of a rectangular cavity bounded by a flexible panel with elastically restrained edges

A coupled system consisting of an acoustic cavity and an elastic panel is a classical problem in structural acoustics and is typically analyzed using modal approaches based on in vacuo structural modes and the rigidly walled acoustic modes which are pre-determined based on separate component models. Such modeling techniques, however, tend to suffer the following drawbacks or limitations: (a) a panel is only subjected to ideal boundary conditions such as the simply supported, (b) the coupling between the cavity and panel is considered weak, and (c) the particle velocity cannot be correctly predicted from the pressure gradient on the contacting interface, to name a few. Motivated by removing these restrictions, this paper presents a general method for the vibro-acoustic analysis of a three-dimensional (3D) acoustic cavity bounded by a flexible panel with general elastically restrained boundary conditions. The displacement of the plate and the sound pressure in the cavity are constructed in the forms of standard two-dimensional and 3D Fourier cosine series supplemented by several terms introduced to ensure and accelerate the convergence of the series expansions. The unknown expansions coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz procedure based on the energy expressions for the coupled structural acoustic system. The accuracy and effectiveness of the proposed method are demonstrated through numerical examples and comparisons with the results available in the literature.     

传递矩阵法预报时空随机激励下任意薄壳腔体内部噪声

利用结构有限元结合声有限元及边界元方法,建立了任意薄壳腔体弹性壳板振动与内外声场的耦合模型,并计算了激励力与壳板振动和内部声场之间的传递矩阵;湍流边界层脉动压力具有时空随机面激励特性,引入整体形状函数矩阵,进一步推导弹性壳板广义节点力功率谱密度函数矩阵与随机面分布激励力功率谱密度函数的关系,再利用声振耦合传递矩阵,得到弹性壳板振动和内部声场功率谱密度函数与广义节点力功率谱密度函数矩阵的关系,形成随机分布激励下任意薄壳腔体结构振动及内部声场的计算方法。以典型的内外均有声介质且一面为弹性矩形板的矩形腔声振耦合模型为例,计算了弹性壳板振动和内部声场功率谱密度函数,并与解析方法进行了比较,两者基本吻合,偏差分别为1 dB和2 dB左右。传递矩阵法不受腔体结构及其内部区域形状的制约,具有良好的适用性。      

双层加筋结构有源隔声物理机制研究

深入分析双层加筋结构有源隔声的物理机制有助于控制系统的优化设计。用模态叠加法和声振耦合理论对双层加筋有源隔声结构建模。在辐射加筋板声功率最小的控制条件下,从模态耦合的角度对有源隔声的物理本质进行了详细阐述。分析结果表明,由于双层加筋板中筋的耦合作用影响,系统的声能量传输规律及有源隔声机理与现有的模态分析结论相比均发生改变。结合筋的耦合影响,对空腔声场的模态抑制与重构机理进行修正和补充,清晰解释了双层加筋结构有源隔声的物理本质。      

一种预测扩散声场在非规则形状刚性壁面附近干涉图样的方法

扩散声场会在反射边界附近形成干涉图样,研究方法包括平面波模型、简正模态分析、渐进模态分析等,但仅适用于尺度远大于声波波长的矩形声腔。提出一种预测扩散声场在非规则刚性壁面结构附近形成的干涉图样的数值方法,表明结构附近“受挡”声压的互谱矩阵取决于:(1)假定该结构在自由空间中振动辐射声音时其表面法向振速到表面及场点声压的边界元系数矩阵;(2)假定结构置于自由空间中且表面刚性时,点声源辐射声波入射到结构表面上产生的散射声场的边界元系数矩阵;(3)扩散声场均方声压。仿真表明,该途径预测的干涉图样与理论值完全吻合。该预测方法还可用于混响环境下声源附近直达声压均方值的空间分布估计,为混响环境下设备的声源定位提供帮助。      

Comparison of various decentralised structural and cavity feedback control strategies for transmitted noise reduction through a double panel structure

This paper compares various decentralised control strategies, including structural and acoustic actuator–sensor configuration designs, to reduce noise transmission through a double panel structure. The comparison is based on identical control stability indexes. The double panel structure consists of two panels with air in between and offers the advantages of low sound transmission at high frequencies, low heat transmission, and low weight. The double panel structure is widely used, such as in the aerospace and automotive industries. Nevertheless, the resonance of the cavity and the poor sound transmission loss at low frequencies limit the double panel's noise control performance. Applying active structural acoustic control to the panels or active noise control to the cavity has been discussed in many papers. In this paper, the resonances of the panels and the cavity are considered simultaneously to further reduce the transmitted noise through an existing double panel structure. A structural–acoustic coupled model is developed to investigate and compare various structural control and cavity control methods. Numerical analysis and real-time control results show that structural control should be applied to both panels. Three types of cavity control sources are presented and compared. The results indicate that the largest noise reduction is obtained with cavity control by loudspeakers modified to operate as incident pressure sources.     

Theoretical Model and Experimental Study of an Efficient Resonance-Type Sound Absorber for the Middle Frequency Region

A model of an efficient sound absorber with an internal structure consisting of several compact resonators with different parameters is considered. Maximum efficiency is achieved by matching the impedance characteristics of the sound absorber and the acoustic environment. Experimental results on determining the maximum absorption coefficient in the 100–1000 Hz frequency range for test samples made with 3D printing technology are reported.     

Mathematical model for characterizing noise transmission into finite cylindrical structures

This work presents a theoretical study of the sound transmission into a finite cylinder under coupled structural and acoustic vibration. Particular attention of this study is focused on evaluating a dimensionless quantity, "noise reduction," for characterizing noise transmission into a small cylindrical enclosure. An analytical expression of the exterior sound pressure resulting from an oblique plane wave impinging upon the cylindrical shell is first presented, which is approximated from the exterior sound pressure for an infinite cylindrical structure. Next, the analytical solution of the interior sound pressure is computed using modal-interaction theory for the coupled structural acoustic system. These results are then used to derive the analytical formula for the noise reduction. Finally, the model is used to predict and characterize the sound transmission into a ChamberCore cylindrical structure, and the results are compared with experimental data. The effects of incidence angle and internal acoustic damping on the sound transmission into the cylinder are also parametrically studied.     

Acoustic analysis of a rectangular cavity with general impedance boundary conditions

A Fourier series method is proposed for the acoustic analysis of a rectangular cavity with impedance boundary conditions arbitrarily specified on any of the walls. The sound pressure is expressed as the combination of a three-dimensional Fourier cosine series and six supplementary two-dimensional expansions introduced to ensure (accelerate) the uniform and absolute convergence (rate) of the series representation in the cavity including the boundary surfaces. The expansion coefficients are determined using the Rayleigh-Ritz method. Since the pressure field is constructed adequately smooth throughout the entire solution domain, the Rayleigh-Ritz solution is mathematically equivalent to what is obtained from a strong formulation based on directly solving the governing equations and the boundary conditions. To unify the treatments of arbitrary nonuniform impedance boundary conditions, the impedance distribution function on each specified surface is invariantly expressed as a double Fourier series expansion so that all the relevant integrals can be calculated analytically. The modal parameters for the acoustic cavity can be simultaneously obtained from solving a standard matrix eigenvalue problem instead of iteratively solving a nonlinear transcendental equation as in the existing methods. Several numerical examples are presented to demonstrate the effectiveness and reliability of the current method for various impedance boundary conditions, including nonuniform impedance distributions.     

Reduction of sound transmission into a circular cylindrical shell using distributed vibration absorbers and Helmholtz resonators

A modal expansion method is used to model a cylindrical enclosure excited by an external plane wave. A set of distributed vibration absorbers (DVAs) and Helmholtz resonators (HRs) are applied to the structure to control the interior acoustic levels. Using an impedance matching method, the structure, the acoustic cavity, and the noise reduction devices are fully coupled to yield an analytical formulation of the structural kinetic energy and acoustic potential energy of a treated cylindrical cavity. Lightweight DVAs and small HRs tuned to the natural frequencies of the targeted structural and acoustic modes, respectively, result in significant acoustic and structural attenuation when the devices are optimally damped. Simulations show that significant interior noise reduction can only be achieved by adding damping to both structural and acoustic modes, which are resonant in the frequency bandwidth of interest. In order to be independent of the azimuth angle of the excitation and to avoid unwanted modal interactions, the devices are distributed evenly around the cylinder in rings. This treatment can only achieve good performance if the structure and the acoustic cavity are lightly damped.     

板厚对无障薄板声辐射特性影响的分析

分析无障薄板的声辐射特性时通常忽略板厚对格林函数的影响而采用双层势计算.本文考虑板厚引起的声辐射阻抗,采用混合势计算结构表面声压与振速,并分析板厚对声辐射参数的影响。根据交界相容性条件,采用边界积分方程分别表示平板上下表面的声压和振速,并合并同类项.进一步将结构的动力方程代入混合势形式的振速方程中,离散声压差值和板的位移为振动模态叠加的形式,获得二重积分形式的声辐射阻抗,从而求解振动模态系数,确定声辐射特性参数.以水下简支矩形板为例计算对比了声辐射参数,并讨论了其对板厚的敏感性。结果表明:板厚引起的声辐射阻抗对声辐射参数的大小影响较小,但随着频率的增加致使共振频率发生较大偏移;在相同阶数的共振频率范围内,板厚度越大,采用双层势计算的误差越大。      

Sound transmission across lightweight all-metallic sandwich panels with corrugated cores

The transmission of sound through all-metallic sandwich panels with corrugated cores is investigated using the space-harmonic method. The sandwich panel is modeled as two parallel panels connected by uniformly distributed translational springs and rotational springs, with the mass of the core sheets taken as lumped mass. Based on the periodicity of the panel structure, a unit cell model is developed to provide the effective translational and rotational stiffness of the core. To check the validity of the model, it is used first to study the sound insulation properties of double-panel structures with air cavity, and the analytical predictions agree well with existing experimental data. The model is then employed to quantify the influence of sound incidence angle and the inclination angle between facesheet and core sheet on sound transmission loss （STL） across sandwich panels with corrugated cores. The results show that the inclination angle has a significant effect on STL and it is possible to avoid STL dips by altering the inclination angle. Moreover, it is found that sandwich panels with corrugated cores are more suitable for the insulation of sound waves having small incidence angles.     

Acoustic transmission analysis on cavity resonance sound in a cylindrical cavity system: application to a Korean bell

This paper presents an analytical model for acoustic transmission characteristics of a cylindrical cavity system representing the acoustic resonance conditions of a Korean bell. The cylindrical cavity system consists of an internal cavity, a gap, an auxiliary cavity, and a rigid base. Since the internal cavity is connected to the external field through a gap, determination of the acoustic transmission characteristics becomes a coupling problem between the internal cavity and external field. The acoustic field of the internal cavity is considered by expanding the solution method of the mixed boundary problem, and the external field is addressed by modifying the radiation impedance model of a finite cylinder. The analytical model is validated by comparison with both experiment and a boundary element method. Using the analytical model, the resonance conditions are determined to maximize the resonance effect. Thus, the resonance frequencies of the bell cavity system are investigated according to the gap size and auxiliary cavity depth. By adjusting gap size or auxiliary cavity depth, the cavity resonance frequency is tuned to resonate partial tones of the bell sound. Finally, the optimal combination of gap size and auxiliary cavity depth is determined.     

Measurement of the resistivity of porous materials with an alternating air-flow method

Air-flow resistivity is a main parameter governing the acoustic behavior of porous materials for sound absorption. The international standard ISO 9053 specifies two different methods to measure the air-flow resistivity, namely a steady-state air-flow method and an alternating air-flow method. The latter is realized by the measurement of the sound pressure at 2 Hz in a small rigid volume closed partially by the test sample. This cavity is excited with a known volume-velocity sound source implemented often with a motor-driven piston oscillating with prescribed area and displacement magnitude. Measurements at 2 Hz require special instrumentation and care. The authors suggest an alternating air-flow method based on the ratio of sound pressures measured at frequencies higher than 2 Hz inside two cavities coupled through a conventional loudspeaker. The basic method showed that the imaginary part of the sound pressure ratio is useful for the evaluation of the air-flow resistance. Criteria are discussed about the choice of a frequency range suitable to perform simplified calculations with respect to the basic method. These criteria depend on the sample thickness, its nonacoustic parameters, and the measurement apparatus as well. The proposed measurement method was tested successfully with various types of acoustic materials.     

Numerical analysis of the 2D acoustic field with epistemic uncertainty

Aiming at the problem that the epistemic uncertain parameters exist in an acoustic field,an evidence theory-based finite element method(ETFEM) is proposed by introducing the evidence theory,in which the focal element and basic probability assignment(BPA) are used to describe the epistemic uncertainty.In order to reduce the computational cost,the interval analysis technique based on perturbation method is adopted to acquire the approximate sound pressure response bounds for each focal element.The corresponding formulations of intervals of expectation and standard deviation of the sound pressure response with epistemic uncertainty are deduced.The sound pressure response of a 2D acoustic tube and a 2D car acoustic cavity with epistemic uncertain parameters are analyzed by the proposed method.The proposed method is verified through the comparison of the analysis results of random acoustic field with that of epistemic uncertain acoustic field.Numerical analysis results show that the proposed method can analyze the 2D acoustic field with epistemic uncertainty effectively,and has good prospect of engineering application.