Sound radiation from rectangular baffled and unbaffled plates

The radiation efficiency of a flat rectangular plate is often used as a basis on which to represent the sound radiation from more complex plate-like structures. The solution for a plate set in a rigid baffle is well known, including the radiation efficiency for multi-modal response of the plate. In this case the assumption is usually made that each mode within a given frequency band has equal modal vibration energy. This paper explores a number of limitations of this simple result. First, the extent to which the radiation efficiency for a particular forcing point deviates from the modal-average result is investigated. Second, the difference is shown between the results for a baffled plate and an unbaffled plate. For a multi-modal response, an empirical formula is also presented which allows the radiation efficiency to be estimated for the unbaffled case. Finally, the effect of different boundary conditions on both baffled and unbaffled results is demonstrated by comparing the results for guided boundaries with those for simply supported boundaries.     

Sound radiation from perforated plates

A practical engineering noise control measure that can often be used for plate-like structures is to construct them from perforates. This can dramatically reduce the sound radiation from such structures. Here, a prediction model is developed to quantify this effect. It is an extension of Laulagnet's model for the radiation from an unbaffled plate, which expresses the surface pressure difference as well as the plate velocity as a sum over plate modes. The perforation is included in terms of a continuously distributed surface impedance, which for moderately sized holes is predominantly inertial. Results show that the radiation efficiency reduces, not only as the perforation ratio increases but also as the hole size reduces for a given perforation ratio. Experimental validation is given which shows a good agreement with the predictions. An approximate formula is also proposed for the effect of perforation which corresponds well with the analytical calculations up to half the critical frequency and could be used for an engineering application to predict the noise reduction due to perforation. A model for the case of a perforated plate embedded in an equally perforated baffle is also discussed for comparison.     

The low frequency axisymmetric modal sound radiation efficiency of an elastically supported annular plate

This paper deals with the sound radiation efficiency of a vibrating, thin, elastically supported annular plate embedded into a flat rigid baffle. The free axisymmetric time harmonic vibrations have been considered for a single mode. It has been assumed that the influence of the air column above the plate on the plate's vibrations is negligible. First, the sound radiation efficiency has been formulated as an integral. Further, rigorous mathematical manipulations have been carried out based on the theory of summation of multiple expansion series containing the hypergeometric functions. As a result, the formulations have been expressed as some fast convergent expansion series containing only the Bessel and Struve functions of integer order and the spherical Bessel functions. The presented formulations of sound radiation efficiency of an elastically supported annular plate are useful for numerical calculations within the low frequency range what is important for practical reasons. The formulations are valid for axisymmetric boundary conditions and they enable changing the values of boundary stiffness constants. Consequently, the analysis of influence of the plate's edge attachment on the sound radiation efficiency has been performed. The limiting transitions have also been performed from formulations valid for the elastically supported annular plates to the formulations valid for annular plates with classical boundary conditions (clamped, simply supported and free) at one edge or at both edges.     

无障板圆形活塞换能器的辐射阻抗*

辐射阻抗是换能器的最重要辐射声场特性参数之一,影响着换能器的谐振频率、频带宽度、效率以及辐射声功率等。经典声学理论用一阶贝塞尔函数和斯特鲁夫函数给出了圆形活塞换能器辐射阻抗的数学表达式,但仅适用于无限大障板条件。该文用ANSYS有限元软件计算了无障板圆形活塞换能器的辐射阻抗,给出了数值解结果,通过与无限大障板条件下的理论结果相比较,显示了二者之间的偏离情况,尤其是在低频时差异显著。为了借助MATLAB工具模拟无障板圆形活塞换能器的辐射阻抗,并给出较准确的数学表达,选择了一系列初等函数以及它们的复合函数进行曲线拟合,最终给出了辐射阻和辐射抗关于ka的函数表达式,在ka值从0.08到20范围内与有限元计算结果很好吻合。     

Transient acoustic radiation from impacted beam-like structures

Transient acoustic radiation from transverse vibrations of beams and beam-like structures is obtained by modelling the structure as a series of contiguous dipoles. A time-dependent expression is developed for sound radiation from a dipole source by Fourier synthesis. Acoustic radiation from the beam is obtained by integrating the sound pressure from the differential dipole elements over the beam length. Time-dependent integration limits are used to account for the transient effects. An analogous discrete formulation is described for beams of arbitrary geometry and density. The radiation patterns of a uniform unbaffled beam are given for frequencies below and above the critical frequency. The results are applied to model the sound radiation from an impact-excited beam.     

Vibro-acoustic response and sound transmission loss analysis of functionally graded plates

This paper presents analytical studies on the vibro-acoustic and sound transmission loss characteristics of functionally graded material (FGM) plates using a simple first-order shear deformation theory. The material properties of the plate are assumed to vary according to power law distribution of the constituent materials in terms of volume fraction. The sound radiation due to sinusoidally varying point load, uniformly distributed load and obliquely incident sound wave is computed by solving the Rayleigh integral with a primitive numerical scheme. Displacement, velocity, acceleration, radiated sound power level, radiated sound pressure level and radiation efficiency of FGM plate for varying power law index are examined. The sound transmission loss of the FGM plate for several incidence angles and varying power law index is studied in detail. It has been found that, for the plate being considered, the sound power level increases monotonically with increase in power law index at lower frequency range (0–500 Hz) and a non-monotonic trend is appeared towards higher frequencies for both point and distributed force excitations. Increased vibration and acoustic response is observed for ceramic-rich FGM plate at higher frequency band; whereas a similar trend is seen for metal-rich FGM plate at lower frequency band. The dBA values are found to be decreasing with increase in power law index. The radiation efficiency of ceramic-rich FGM plate is noticed to be higher than that of metal and metal-rich FGM plates. The transmission loss below the first resonance frequency is high for ceramic-rich FGM plate and low for metal-rich FGM plate and further depends on the specific material property. The study has found that increased transmission loss can be achieved at higher frequencies with metal-rich FGM plates.     

高斯波束对界面附近离轴球形粒子的轴向声辐射力

从声波的散射理论出发,利用级数展开法得到高斯波束的波束因子,推导其对阻抗边界下离轴球形粒子声辐射力.针对刚性球与液体球两种球形粒子进行数值模拟,与自由空间的情况进行比较.讨论边界反射系数、粒子与边界距离、束腰半径以及离轴角度与距离等对声辐射力的影响.仿真结果表明：边界反射系数的增大会引起声辐射力的增加,但不改变峰值的位置;在合适的频率处,可以产生负向声辐射力;声辐射力随粒子与边界距离呈周期性变化;束腰半径的影响主要体现在中高频;随着粒子偏离传播轴的距离和角度增大,声辐射力明显衰减.该研究为利用高斯波束实现对粒子的操纵提供理论基础.     

Theoretical analysis of sound radiation of an elastically supported circular plate

This study focuses on the analysis of the active and reactive sound power of the axisymmetric modes of free vibrations of elastically supported circular plates embedded in a rigid baffle. Some linear and time-harmonic processes have been considered. It is assumed that the plate radiates some acoustic waves into a hemisphere filled with a lossless gaseous medium. The integral formulations for the active and reactive sound power have been derived and expressed in their Hankel representations. They have been used to derive some elementary formulations in the form of some high-frequency asymptotes valid for frequencies higher than the successive coincidence frequencies of the plate. Therefore, the discussion on some sample numerical results mostly covers the sound power radiated at those frequencies. The asymptotes are easy to express in a computer code and they do not need great processor capacity. They are therefore useful for engineering use.The main benefit of the analysis presented in this paper is that the sound power for all the possible boundary configurations of the boundary stiffnesses, i.e., classical clamped, guided, simply supported or completely free boundaries as well as all the intermediate situations, has been described using the same formulae. This is possible simply by changing the two values of stiffnesses associated with the boundary conditions, whose influence on the radiated sound power has been discussed. The solution of the problem of sound power radiated by a vibrating elastically supported circular plate presented herein is essentially more general than the solutions presented earlier for the classical boundary configurations, such as clamped, simply supported, guided or completely free circular plates.     

暗声学超材料研究

由于普通材料的固有耗散在低频区域的微弱性,长久以来,低频声波的衰减一直都是一个颇具挑战性的任务.为了能够在100—1000Hz范围内完全吸收某些频率的低频声波,文章作者设计了一种薄膜型的暗声学超材料样品:它是由在弹性薄膜上镶嵌有一些非对称性的硬质金属片而制成.实验表明,该样品在低频区域几乎能够百分之百地吸收声波,而在共振吸收频率处,空气中的声波波长要比薄膜的厚度大3个数量级以上.当共振发生时,硬质金属片的"拍动"导致很大的弹性曲率能量聚集在金属片的边界附近.由于薄膜的拍动模式与声波的辐射模式仅存在微弱的耦合作用,而弹性薄膜的整体能量密度又比入射声波的能量密度大2—3个数量级,该样品本质上是一个开放的共振腔,这也是它能够高效地吸收低频声波的原因所在.     

无障板圆形活塞换能器波束特性建模

针对ka<π时无障板活塞声源波束特性不能用无限平面障板活塞换能器理论进行描述的问题,提出一种基于有限元方法结合曲线拟合的简便数学建模方法.模型采用了无限平面障板活塞换能器理论的基本函数表达,通过增加修正因子函数来改变模型的细节部分,用曲线拟合方法给出了较准确的两个修正因子函数的数学表达式。最终建立了无障板圆形活塞换能器波束特性经验公式模型,模型在ka为0.03~5.2范围内与有限元方法给出的波束曲线相一致,并且在ka取更大值时与无限平面障板活塞换能器理论公式等效.      

Reduction of acoustic radiation by impedance control with a perforated absorber system

A model of sound radiation from an infinite plate with an absorptive facing is proposed and investigated theoretically from the viewpoint of acoustic power. Acoustic characteristics on the plate surface are represented by impedance derived from iso-absorption curves. A parametric study is carried out to clarify the effect of the impedance on the acoustic power. Results derived from this model show that acoustic radiation depends on change in impedance as well as the absorption coefficient, and there is a possibility of reducing the radiation from vibrating surface by introducing an appropriate impedance surface. In order to realize this effect, a model using a perforated board with a back cavity attached to the vibrating surface is proposed, in which the motion of the perforated board is made equal to that of the vibrating surface. To obtain fundamental data, a theoretical study is performed under a simplified condition, assuming an infinite plane piston. The calculated results are compared to experimental data measured by using an acoustic tube. The results, which are in good agreement in the reduction effect, show that this system can achieve the reduction of radiated sound power at arbitrary frequencies.     

Sound radiation from a double-leaf elastic plate with a point force excitation: effect of an interior panel on the structure-borne sound radiation

The sound radiation from a double-leaf elastic plate subjected to a point force excitation is investigated theoretically, to gain a fundamental insight into the sound radiation from an interior panel of a double-leaf structure in buildings. The effects of the interior panel on the sound radiation, which show a negative effect at low frequencies due to the mass-air-mass resonance, are discussed in detail. The theory is validated experimentally. As a measure of the efficiency of the interior leaf in reducing noise radiation, the radiation reduction is defined in this study, and it is found useful for predicting the sound radiation due to the structure-borne sound in building elements. Parametric studies through theoretical results are made to clarify the effects of the parameters of the sound radiation system, and to gain a fundamental insight into the control of structure-borne noise radiation. It is shown that it is difficult to reduce the radiated sound power by an interior panel alone, even if its mass is increased.     

The effect of different combinations of boundary conditions on the average radiation efficiency of rectangular plates

The boundary conditions of a vibrating plate are known to have an influence on its sound radiation for frequencies below the critical frequency. To investigate this effect in a systematic way, the average radiation efficiency and radiated power are calculated for a rectangular plate set in an infinite baffle using a modal summation approach. Whereas analytical expressions exist for simply supported boundary conditions, a numerical approach is required for other cases. Nine combinations of boundary conditions are considered, consisting of simply supported, clamped and free edges on different plate edges. The structural vibration is approximated by using independent beam functions in orthogonal directions allowing simple approximate formulae for mode shapes and natural frequencies. This assumption is checked against a finite element model and shown to give reliable results. It is shown that a free plate has the lowest radiation efficiency and a clamped plate the highest for most frequencies between the fundamental panel natural frequency and the critical frequency. Other combinations of boundary condition give intermediate results according to the level of constraint introduced. The differences depend on frequency: excluding the extreme case of a fully free plate all the other boundary conditions give results within a range of 8 dB in the middle part of the short-circuiting region, decreasing towards the critical frequency. At low frequency the differences can be even greater, in some cases up to 20 dB. These conclusions are shown to hold for a range of plate thicknesses and dimensions.     

On the sound radiation of a rolling tyre

The sound radiation from rolling tyres is still not very well understood. Although details such as horn effect or directivity during rolling have been investigated, it is not clear which vibrational modes of the tyre structure are responsible for the radiated sound power. In this work an advanced tyre model based on Wave Guide Finite Elements is used in connection with a contact model validated in previous work. With these tools the tyre vibrations during rolling on an ISO surface are simulated. Starting from the calculated contact forces in time the amplitudes of the modes excited during rolling are determined as function of frequency. A boundary element model also validated in previous work is applied to predict the sound pressure level on a reference surface around a tyre placed on rigid ground as function of the modal composition of the tyre vibrations. Taking into account different modes when calculating the vibrational field as input into the boundary element calculations, it is possible to identify individual modes or groups of modes of special relevance for the radiated sound power. The results show that mainly low-order modes with relative low amplitudes but high radiation efficiency in the frequency range around 1 kHz are responsible for the radiated sound power at these frequencies, while those modes which are most strongly excited in that frequency range during rolling are irrelevant for the radiated sound power. This fact is very essential when focusing on the design of quieter tyres.     

On acoustic radiation by a rigid object in a fluid flow

A problem of sound radiation by an absolutely rigid object, moving with respect to the surrounding fluid, is considered on the basis of the Lighthill's equation for aerodynamic sound. An integral representation of the radiated acoustic field is utilized, where the field is characterized as the sum of three fields, generated by a volume distribution of monopoles and by distributions of monopoles and dipoles on the surface of the rigid object. It is shown that, due to a discontinuity of Lighthill's stress tensor on the rigid boundary, a layer of surface divergence of hydrodynamic stresses on the boundary must be taken into account when evaluating the volume integral over Lighthill's quadrupole sources. When the contribution of the surface divergence is included in the solution of Lighthill's equation, amplitudes of the monopole and dipole sound radiated by the rigid object are shown to depend on the potential components of the normal velocity and the pressure on the rigid surface. The obtained solution is compared with Curle's solution for this problem, which establishes that the sound radiation by a rigid object is determined by the force exerted by the object upon the fluid. Both solutions are applied to two known problems of sound scattering and radiation by a rigid sphere in variable pressure and velocity fields. It is shown that predictions based on the obtained solution are equivalent to the results known from literature, whereas Curle's solution gives predictions contradicting the known results. It is also shown that the Ffowcs Williams and Hawkings equation, which coincides with Curle's equation for an immoveable rigid object, does not lead to the correct predictions as well.     

Experimental research on vibrating objects and its radiation field by using nearfield acoustic holography

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Sound transmission loss of metamaterial-based thin plates with multiple subwavelength arrays of attached resonators

This paper is concerned with sound transmission loss of metamaterial-based thin plates consisting of multiple subwavelength arrays of spring–mass resonators attached to an unbounded homogenous thin plate. Two analytical wave approaches are developed for the calculation of diffuse field sound transmission loss of such metamaterial-based thin plates. Numerical results show that a metamaterial-based plate can result in much higher sound transmission loss than a bare plate (with the same surface mass density) at frequencies within the mass-law region and the coincidence region. It is also demonstrated that by using an extremely thin plate to form a metanaterial-based plate, the construction can be implemented as a potential sound insulation material with good performance at low frequencies.     

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。      

Flexural vibration of perforated plates and porous elastic materials under acoustic loading

This paper presents a method of theoretical treatment of acoustic coupling due to flexural vibration of perforated plates and plates of porous elastic materials. The analytical model is developed by introducing flow continuity at the plate surface in a spatially mean sense and air-solid interaction within the plate material. To demonstrate the method of application, some fundamental acoustic problems based on a classical thin-plate theory are analyzed and discussed in relation to the interactive effect of flexural vibration and plate permeability. For acoustic radiation from a vibrating plate excited by a harmonic point-force, the attenuation effect of power radiation appears at frequencies below the critical frequency of coincidence. In the problem of sound absorption of a perforated plate or a plate of porous elastic material backed by an air layer, as permeability decreases, the effect of plate vibration increases. For perforated absorber systems including plate vibration effects, the trend of variation from ordinary theory depends on plate thickness.     

变压器典型工作环境对声功率测量结果的影响

变压器两侧常建有高大的防火墙,其从声学角度可视为刚性反射壁面,会改变变压器的辐射声场,进而影响变压器声功率测量结果。本文利用有限元、边界元等数值计算方法建立了变压器声辐射的仿真模型用以分析反射壁面对变压器声功率测量结果的影响,并通过实际测量验证了仿真计算所得结论。结果表明,反射壁面对变压器声功率测量结果的影响程度随反射壁面到变压器箱体距离增加而减弱,且当反射壁面距变压器箱体5 m以上时,其对变压器声功率基频及各谐波成份测量结果的影响均在2 dB以内。另外反射壁面对变压器噪声高频成份声功率测量结果的影响较大,而对100 Hz、200 Hz等低频成份测量结果影响较小,基本低于3 dB。