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.     

On the diffuse field reciprocity relationship and vibrational energy variance in a random subsystem at high frequencies

A recent paper has shown that under certain conditions the cross-spectral matrix of the forces exerted by a vibrational or acoustic wave field on its surrounding boundaries can be expressed in terms of (i) the energy of the wave field, and (ii) the direct field dynamic stiffness matrix of the boundary. This "diffuse field reciprocity relation" was derived using wave mechanics, and it is not immediately clear how the required wave field properties translate to conditions on the vibrational modes of the system or the applied forcing. This issue is addressed here by deriving an extended version of the reciprocity relation using modal methods, and the conditions required for the extended version to reduce to the existing relation are delineated. It is shown that the existing diffuse field reciprocity relation leads to an anomalous result when used to predict the energy variance of a subsystem, and that this anomaly is resolved by using the present extended version of the relation. A supplementary result arising from the analysis is that for systems with a sufficient degree of randomness the ensemble average of the dynamic stiffness matrix of a random subsystem is equal to the inverse of the ensemble average of the receptance matrix.     

A unified approach for predicting sound radiation from baffled rectangular plates with arbitrary boundary conditions

This paper discusses sound radiation from a baffled rectangular plate with each of its edges arbitrarily supported in the form of elastic restraints. The plate displacement function is universally expressed as a 2-D Fourier cosine series supplemented by several 1-D series. The unknown Fourier expansion coefficients are then determined by using the Rayleigh-Ritz procedure. Once the vibration field is solved, the displacement function is further simplified to a single standard 2-D Fourier cosine series in the subsequent acoustic analysis. Thus, the sound radiation from a rectangular plate can always be obtained from the radiation resistance matrix for an invariant set of cosine functions, regardless of its actual dimensions and boundary conditions. Further, this radiation resistance matrix, unlike the traditional ones for modal functions, only needs to be calculated once for all plates with the same aspect ratio. In order to determine the radiation resistance matrix effectively, an analytical formula is derived in the form of a power series of the non-dimensional acoustic wavenumber; the formula is mathematically valid and accurate for any wavenumber. Several numerical examples are presented to validate the formulations and show the effect of the boundary conditions on the radiation behavior of planar sources.     

Radiation efficiency of unbaffled and perforated plates near a rigid reflecting surface

The accurate prediction of sound radiation from plate-like structures remains a challenging problem. Although the case of a plate set in a rigid baffle can be solved analytically, when the plate radiates sound into free space the problem is more difficult to solve; nevertheless, several approaches have been proposed to determine the sound radiation from an unbaffled plate. The present study extends the consideration to the situation of an unbaffled plate which is located close to a rigid reflecting surface. For this purpose, Laulagnet's model for the radiation efficiency of an unbaffled plate is extended by modifying the Green's function to include an image source due to the reflecting surface. The results show that, depending on the distance between the plate and the rigid surface, the radiation efficiency is considerably reduced at low frequencies. Additional reduction of sound radiation can be achieved by introducing perforation to the plate. However, at higher frequencies, the radiation efficiency is amplified relative to that for the plate in the absence of the rigid surface, both with and without perforation. These results have also been validated experimentally.     

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。     

Hybrid tool for quickly estimating the radiated acoustic power from a vibrating structure in a multiple-source environment

This paper presents a new hybrid method for predicting overestimating and underestimating indicators of the acoustic power radiated by a vibrating surface even in the presence of other surrounding acoustic sources. This method is applicable to plates or low curvature surfaces radiating in open acoustic fields. The method is hybrid in the sense that the vibration field is measured and the parietal pressure field is predicted considering two extreme academic cases "baffled" and "unbaffled." Many simplifications are made and justified in order to save running time. The method is successfully validated in comparison with experimental results on both laboratory and real life structures. This method has led to a quick tool, allowing one to obtain a good approximation of the radiated power in a few minutes. It provides a natural extension of a classical analyzer for vibroacoustics engineering.     

Reduction of sound radiation by using force radiation modes

The location of a vibration source within a machine is sometimes found to have a significant effect upon its radiated acoustic power. It is known that a simple reduction of vibration cannot always reduce the radiated acoustic power, so that treatments based on analysis of a structure’s vibration modes are not always effective. At the same time, radiation mode analysis is known to be a powerful tool for interpreting sound radiation since those modes are independent of a structure’s surface vibration. However, knowledge of the radiation modes alone cannot be used directly to understand the relationship between vibration source location and acoustic power radiation. In this paper, it is shown that the radiation mode concept can be extended to understand the relationship between acoustic power and driving force distribution by considering the product of the structure’s mobility matrix and the radiation modes: the resulting functions are here defined to be force radiation modes (f_rad-modes). An example is presented in which the acoustic power radiated by a simply-supported, baffled beam is reduced by using guidance provided by the structure’s force radiation modes. The results demonstrate that the force radiation modes can be used to guide the reduction of radiated acoustic power by changing the driving force location without the need to perform additional calculations or experiments.     

声辐射阻测量系统设计及测量精度影响因素分析*

在噪声控制领域,声辐射阻是一个重要的声学变量,在产品的设计阶段,可以通过改变结构表面形状来改变结构的声辐射阻抗,从而达到降噪的目的。本文基于集中参数模型,设计了测量设备,搭建了实验测量系统,并通过对无限大障板上圆形活塞声源的声辐射阻的测量对影响实验测量结果的因素进行了研究,之后对消音器入口处辐射阻进行了测量。测量结果表明:该实验测量系统具有操作简单,测量方便快捷,测量结果精度高等特点,可用于确定结构表面声辐射阻矩阵。     

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

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Dynamic acoustic radiation force acting on cylindrical shells: theory and simulations

An object placed in an acoustic field is known to experience a force due to the transfer of momentum from the wave to the object itself. This force is known to be steady when the incident field is considered to be continuous with constant amplitude. One may define the dynamic (oscillatory) radiation force for a continuous wave-field whose intensity varies slowly with time. This paper extends the theory of the dynamic acoustic radiation force resulting from an amplitude-modulated progressive plane wave-field incident on solid cylinders to the case of solid cylindrical shells with particular emphasis on their thickness and contents of their hollow regions. A new factor corresponding to the dynamic radiation force is defined as Y(d) and stands for the dynamic radiation force per unit energy density and unit cross sectional surface. The results of numerical calculations are presented, indicating the ways in which the form of the dynamic radiation force function curves are affected by variations in the material mechanical parameters and by changes in the interior fluid inside the shell's hollow region. It was shown that the dynamic radiation force function Y(d) deviates from the static radiation force function for progressive waves Y(p) when the modulation frequency increases. These results indicate that the theory presented here is broader than the existing theory on cylinders.     

An approximate method to acoustic radiation problems: element radiation superposition method

An approximate method is brought forward to predict the acoustic pressure based on the surface velocity. It is named Element Radiation Superposition Method （ERSM）. The study finds that each element in Acoustic Transfer Vector （ATV） equals the acoustic pressure radiated by the corresponding surface element vibrating in unit velocity and other surface elements keep still, that is the acoustic pressure radiated by the corresponding baffled piston vibrating in unit velocity. So, it utilizes the acoustic pressure radiated by a baffled piston to establish the transfer relationship between the surface velocity and the acoustic pressure. The total acoustic pressure is obtained through summing up the products of the surface velocity and the transfer quantity. It adopts the regular baffle to fit the actual baffle in order to calculate the acoustic pressure radiated by the baffled piston. This approximate method has larger advantage in calculating speed and memory space than Boundary Element Method. Numerical simulations show that this approximate method is reasonable and feasible.     

Influence of circumferential partial coating on the acoustic radiation from a fluid-loaded shell

Acoustic compliant coatings are a common approach to mitigate the radiation and scattering of sound from fluid-loaded submerged structures. An acoustic compliant coating is a coating that decouples an acoustic source from the surrounding acoustic medium; that is, it provides an acoustic impedance mismatch (different density and speed of sound product). Such a coating is distinct from an ordinary compliant coating in that it may not be resilient in the sense of low stiffness, but still provides an acoustic impedance mismatch. Ideally, the acoustic coating is applied uniformly over the entire surface of the fluid-loaded structure to minimize the acoustic radiation and scattering. However, in certain instances, because of appendages, it may not be practically possible to completely cover the surface of a fluid-loaded structure to decouple it from the adjacent acoustic medium. Furthermore, there may be some inherent advantages to optimizing the distribution of the coating around areas from which the acoustic radiation appears to be dominant. This would be analogous to the application of damping treatment to a vibrating structure in areas where the vibration levels are highest. In the case of the acoustic radiation the problem is more complex because of the coupling between the acoustic fluid and the structure. In this paper, the influence of a partial coating on the acoustic radiation from a fluid-loaded, cylindrical shell of infinite extent and excited by either a line force or an incident plane acoustic wave is examined. The solution to the response and scattered pressure is developed following the procedure used by the authors in previous work on the scattering from fluid-loaded plates and shells. The coating is assumed to be normally reacting providing a decoupling layer between the acoustic medium and the structure; that is, it does not add mass or stiffness to the base structure. The influence of added mass or stiffness of the coating can be included as an added inhomogeneity and treated separately in the solution.     

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。      

Partial pole placement in structures by the method of receptances: Theory and experiments

The theory and practical application of the receptance method for vibration suppression in structures by multi-input partial pole placement is described. Numerous advantages of the receptance method over conventional matrix methods such as state-space control based on finite elements have been demonstrated, in particular there is no need to know or to evaluate the structural matrices M, C, K and in practical experimentation the measurement of ‘receptance’ may be generalised so that explicit modelling of actuator dynamics becomes unnecessary. Active vibration control is demonstrated experimentally using two test rigs. In the first set of experiments partial pole placement is applied to a lightweight glass-fibre beam using macro fibre composite (MFC) actuators and sensors. In the second set of experiments active vibration control is implemented on a heavy modular test structure representative of systems of differing dynamic complexity using electromagnetic actuators and piezoelectric (ICP) accelerometers. It is demonstrated that chosen poles may be assigned to predetermined values without affecting the position of other poles of interest.     

Acoustic radiation from a shell-encapsulated baffled cylindrical cap

An exact study of radiation of an acoustic field due to radial/axial vibrations of a baffled cylindrical piston, eccentrically positioned within a fluid-filled thin cylindrical elastic shell, into an external fluid medium is presented. This configuration, which is a realistic idealization of a liquid-filled cylindrical acoustic lens with a focal point inside the lens when used as a sound projector, is of practical importance with a multitude of possible applications in underwater acoustics and ocean engineering. The formulation utilizes the appropriate wave field expansions along with the translational addition theorems for cylindrical wave functions to develop a closed-form solution in the form of an infinite series. Numerical results reveal the key effects of excitation frequency, cap angle, radiator position (eccentricity), dynamics of the elastic shell, and cap surface velocity distribution on sound radiation.     

Identification of substructure properties of railway tracks by dynamic stiffness measurements and simulations

A new vehicle has been developed to measure dynamic vertical track stiffness while in motion. This technique allows the resonance behaviour of the track below 50 Hz to be measured. Soft soils like clay and peat are the main causes of resonance below 20 Hz. By means of simulation studies with the software VibTrain, soft soil resonance behaviour may be characterized using a few key parameters originating from track stiffness measurements, such as the minimum phase delay and corresponding frequency of the receptance transfer function. Statistical models are built to relate these key parameters with substructure properties, such as embankment thickness, shear wave velocity and thickness of the soft soil layer using pattern recognition methods. Two case studies are used to show the methodology, and the results are verified using Ground Penetration Radar (GPR) measurements and borehole investigations. Models are also developed from the statistical relationship between GPR-data and stiffness measurements. It is shown that embankment thickness is the easiest quantity to estimate, but indicative results are also presented for the other quantities (shear wave velocity and thickness of soil layer).     

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.     

Modeling and optimization of local constraint elastomer treatments for vibration and noise reduction

This paper presents the vibroacoustic study of a constrained elastomer treatment used in the industry for reducing noise. It can be trimmed and bonded conveniently to vibrating structures for reducing radiated noise. First, an identification of the elastomer viscoelastic characteristics is carried out with a program that models damped vibrations, a conjugate gradient search technique and experimental data extracted rom two contact-free modal analyses. The first modal analysis, adapted to dissipation characterization, is made on a partially covered suspended plate. The second modal analysis, adapted to identifying the elastomer stiffness behavior, concerns a cantilever beam that has almost been covered by a large treatment. The complete dynamic characterization is finally deduced from an iterative procedure that combines information from both experiments. The procedure highlights the influence of the treatment bonding quality on the achieved elastomer damping. Second, practical rules are deduced from a number of parametric studies on beams with baffled radiation conditions. In particular, a design criterion is introduced to help positioning patches where the elastomer damping can be maximized. A threshold, for which an optimal acoustic performance with a minimum of elastomer can be fulfilled, is also identified.     

Transient nearfield acoustic holography based on an interpolated time-domain equivalent source method

Transient nearfield acoustic holography based on an interpolated time-domain equivalent source method (ESM) is proposed to reconstruct transient acoustic fields directly in the time domain. Since the equivalent source strengths solved by the traditional time-domain ESM formulation cannot be used to reconstruct the pressure on the source surface directly, an interpolation function is introduced to develop an interpolated time-domain ESM formulation which permits one to deduce an iterative reconstruction process. As the reconstruction process is ill-conditioned and especially there exists a cumulative effect of errors, the Tikhonov regularization is used to stabilize the process. Numerical examples of reconstructing transient acoustic fields from a baffled planar piston, an impulsively accelerating sphere and a cube box, respectively, demonstrate that the proposed method not only can effectively reconstruct transient acoustic fields in the time domain, but also can visualize acoustic fields in the space domain. And, in the first numerical example, the cumulative effect of errors and the validity of using the Tikhonov regularization to suppress the errors are described.     

Acoustic damping enhanced by gaps in baffled injectors in an acoustic chamber

Acoustic damping enhanced by gap width in baffled injectors is investigated numerically, which are installed to suppress pressure oscillations in a model acoustic chamber. The previous experimental works reported that baffled injectors show larger acoustic damping with gaps between adjoining injectors than baffled injectors without the gap or conventional baffles. Acoustic-damping behaviors of baffled injectors are simulated numerically and the damping mechanism is examined. Damping factors are calculated as a function of baffle gap and it is found that the maximum acoustic damping is observed at a gap of 0.1-0.2 mm. The enhanced damping by gaps is attributed to viscous dissipation on the surfaces of the injectors or baffle blades. The optimum gap for maximum damping depends on the viscosity of the medium in the chamber and it increases with the viscosity. As a quantitative parameter of viscous dissipation, the dissipation rate of kinetic energy is calculated as a function of baffle gap. The parameter shows its maximum value at a specific gap and especially, the viscous dissipation rate has the same profile as that of damping factor. It verifies that the enhanced damping by gaps is attributed to the viscous dissipation of acoustic energy increased by gaps in baffled injectors.