Baffled Rectangular Plate Research Articles

A fast Chebyshev spectral approach for vibroacoustic behavior analysis of heavy fluid-loaded baffled rectangular plates with general boundary conditions

A computationally efficient Chebyshev spectral approach is proposed to solve the vibroacoustic response of heavy fluid-loaded baffled rectangular plates. The governing equations for the displacements of motion in rectangular plates and the sound pressure in the Helmholtz integral are formulated using high-order, first-class Chebyshev polynomial expansions. This formulation is combined with Gauss-Chebyshev-Lobatto sampling. The quadruple integral encountered in solving the work done by the heavy fluid on the plate is reformulated into the configuration of a tensor product. The approach significantly reduces the computation time for solving the acoustic equations, bringing the vibration response of fluid-loaded plates closer to that of vacuum plates and limiting it to just a few seconds. The elastic boundary conditions of the rectangular plates are simulated using linear and rotational springs. Predictions of vibroacoustic behavior, including plate velocity, sound pressure, sound power, and radiation efficiency, are validated against literature results. The accuracy and efficiency of the Chebyshev spectral approach for the vibroacoustic coupling systems are demonstrated by the presence of exemplary agreements. Furthermore, this study investigates the effect of boundary conditions, geometric characteristics, and damping variables on the vibroacoustic behavior of rectangular plates.

Sound Radiation Estimate from Vibration Measurements With Multiple Cameras

This paper presents experimental results on the estimate of the sound radiation by a thin flat plate from flexural vibration measurements taken with multiple cameras. The study considers the sound radiation into free-field by a baffled rectangular plate fixed on a rigid frame. The plate is excited by a tonal point force exerted by a shaker at the first three resonance frequencies of the plate flexural response. The resulting vibration field is measured with a set of six cameras suitably synchronised. More specifically, 3D point tracking is implemented on a regular NxM grid of small circular points marked on the plate. The transverse displacements at these points are thus reconstructed from triangulation with multiple (i.e. more than 2) view-points. The transverse displacements are then used to derive the sound radiation from a direct boundary integral formulation. More specifically, for the flat rectangular plate at hand, the sound radiation is derived from the Rayleigh integral, which is approximated into a finite sum over the mesh of rectangular elements centred at the NxM grid of points depicted on the plate. The flexural vibration field and the sound radiation field derived from the camera measurements are contrasted with measurements taken with a laser vibrometer and microphone-scanner probe. The initial results presented in this paper demonstrate the feasibility of the proposed measurement approach at low audio frequencies.

On the acoustic radiation of all edges free rectangular thin plate

In this study, transverse vibrations and acoustic radiation of an all-edges-free baffled rectangular thin plate are examined by Rayleigh's formulation. While the Rayleigh-Ritz method is applicable for a wide variety of boundary condition combinations, in this study, the method is used for evaluating the modal characteristics of an all edges free plate. Acoustic pressure levels are calculated by Rayleigh's acoustic formula for a free field. This study provides a quick formula for those who are interested in calculating the sound pressure of a baffled plate on a specific point for a free field while only requiring the vibration of the surface to be expressed in terms of 2D power series. The results obtained from the formula are compared with the finite element analysis results.

Reconstructing the acoustic far-field using a semi-circular sensor array

A method to reconstruct the acoustic far-field using a semi-circular sensor array is proposed. Firstly, the pressures in the acoustic far-field are measured by a semi-circular sensor array; then, the acoustic far-field is reconstructed based on the theory of acoustic radiation modes. The numerical simulations for reconstructing the acoustic far-field of a baffled rectangular plate are conducted. The results show that the acoustic far-field of the baffled rectangular plate can be reconstructed accurately by a few sensors on the semi-circular array. The influence of noise to the reconstruction of acoustic field is analyzed, and the robustness of the proposed method is proved. This paper provides a method for the reconstruction of acoustic far-field through a simple measuring array. The method can be applied to the acoustic field reconstruction of large structures, and has good engineering application prospects.

Active structural acoustic control using the remote sensor method

Active structural acoustic control (ASAC) is an effective method of reducing the sound radiation from vibrating structures. In order to implement ASAC systems using only structural actuators and sensors, it is necessary to employ a model of the sound radiation from the structure. Such models have been presented in the literature for simple structures, such as baffled rectangular plates, and methods of determining the radiation modes of more complex practical structures using experimental data have also been explored. A similar problem arises in the context of active noise control, where cancellation of a disturbance is required at positions in space where it is not possible to locate a physical error microphone. In this case the signals at the cancellation points can be estimated from the outputs of remotely located measurement sensors using the “remote microphone method”. This remote microphone method is extended here to the ASAC problem, in which the pressures at a number of microphone locations must be estimated from measurements on the structure of the radiating system. The control and estimation strategies are described and the performance is assessed for a typical structural radiation problem.

Analyzing Acoustic Radiation Modes of Baffled Plates With a Fast Multipole Boundary Element Method

In the analysis of an acoustic radiation mode of a baffled plate, Rayleigh integral with free space Green's function is involved. The boundary element method (BEM) is one of the approaches to compute its modes and radiation efficiencies. In this paper a fast multipole BEM in conjunction with an iterative solver based on the implicit restart Arnold method is proposed to efficiently and accurately evaluate acoustic radiation modes and efficiencies. Even though a 3D free space Green's function is used here, a quad tree is used for the hierarchical tree structure of the boundary mesh instead of an oct tree, which can speed up the fast multipole BEM. Similar to the analytical integration of moment evaluations, the analytical integration is also employed to compute the local expansion coefficients which further improves the efficiency of the fast multipole BEM for the analysis of an acoustic radiation mode of baffled plates. Comparison between numerical and theoretical radiation efficiencies of a baffled circular plate vibrating as a piston shows that the fast multipole BEM proposed here can give results with very good accuracy. The computation of the eigenvalues and eigenvectors of a baffled rectangular plate further reveals the efficiency in CPU time, smaller memory size, and accuracy of the fast multipole BEM in the analysis of an acoustic radiation mode.

Dynamic analysis of rectangular plates with a single side crack and in contact with water on one side based on the Rayleigh–Ritz method

A computational method is proposed based on the Rayleigh–Ritz method for the analysis of dynamic characteristics of a baffled rectangular plate with an arbitrary side crack and in contact with an infinite water domain on one side. Based on the hypothesis that the water is incompressible, inviscid and the motion is irrotational, the added mass density expressed by the displacement trial functions is obtained by means of Green function method. The behavior of crack is described by the corner functions, and the natural frequencies and the corresponding modes are calculated through the Rayleigh–Ritz technique. The validity and the accuracy of the proposed method are demonstrated based on the comparison with the results obtained by ANSYS and those published in other literature, and the influences of water on the natural frequencies and mode shapes of the rectangular plates with side cracks are investigated. It is illustrated that the differences between dry and wet mode shapes increase with increasing mode order on the whole, except for the completely free plates, and the differences are evident for some higher modes.

Sound radiation from point-excited structures: Comparison of plate and sphere

Acoustic radiation from a structure can be expressed in terms of “modal radiation” and “modal coefficients”. This paper investigates the contributions of these two modal properties to radiation excited by a point force. Sound radiation from two basic structures is considered: a baffled rectangular plate and a closed spherical shell. The plate behaviour is familiar, and governed by the relation between the natural frequency of a mode and its coincidence frequency. For a closed spherical shell, there are either zero or two “critical frequencies”, depending on the radius and thickness. When there are two the shell radiates well both above and below the two frequencies, and poorly in the frequency range between them.

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.

Enhancing accuracy in reconstruction of vibro‐acoustic responses of a complex structure using Helmholtz equation least squares based nearfield acoustical holography.

In traditional nearfield acoustical holography, the acoustic pressures in the near field are measured, and the entire acoustic field including the normal component of the surface velocity are reconstructed. This type of approach may be appropriate for reconstructing the acoustic pressure field, but not for reconstructing normal surface velocity in practice because not enough nearfield information is collected to yield satisfactory reconstruction of the normal surface velocity. In this paper, we propose to supplement the acoustic pressure measurements with a few normal surface velocity measurements, which can be used as benchmarks in optimizing the reconstruction of normal surface velocity. These data are taken as input to the Helmholtz equation least squares method based NAH to reconstruct the acoustic pressure and normal velocity on the surface of a vibrating structure. The accuracy in reconstruction is examined experimentally on a baffled rectangular plate with clamped boundary conditions subject to random excitations. To validate the results, the reconstructed field acoustic pressures are compared with those measured by microphones, and the normal surface velocities were compared with the benchmark values collected by using a scanning laser vibrometer under the same condition.

Experimental validation of vibroacoustic reconstruction of a rectangular plate with different boundary conditions.

This paper provides an experimental validation of vibroacoustic responses of a rectangular plate reconstructed using Helmholtz equation least squares (HELS) method. Experimental studies were conducted on baffled rectangular plates of different aspect ratios with free as well as clamped boundary conditions under random excitation. The radiated acoustic pressures were measured using a planar array of microphones at a very close distance to the plate surface and taken as input for the HELS codes. The normal surface velocity distributions were then reconstructed, and the results were compared against the benchmark data obtained using a laser vibrometer. The structural modes of the plate with different boundary conditions were compared with those obtained by experimental modal analysis. Good agreements were obtained for both clamped and free boundary conditions.

Vibroacoustic Modes and Active Control of Both Vibration and Sound

Vibration reduction and sound reduction are not necessarily compatible. The sum of the structural kinetic energy and acoustic power in free space (acoustic potential energy in an enclosed space) is an error criterion to control both vibration and sound. The orthogonal contributors with respect to the sum, and the frequency dependent spatial which can estimate the amplitudes of the vibroacoustic modes, modal filters, are formulated. Vibroacoustic modal filters allow a controller to coordinate both vibration and sound reduction and achieve order reduction. However, the implementation is not very easy because of the frequency dependence. The frequency independent vibroacoustic modal filters are formulated by omitting the frequency dependence. The omission of the frequency dependence mathematically proves to be valid for baffled structures in free space. Moreover, the developed theories are applied to baffled rectangular plates in free space, and the simulation results, for example, the contributions of the vibroacoustic modes to the sum of the structural kinetic energy and acoustic power, the shapes of the vibroacoustic modes, and the performance of the feedforward control using the frequency independent vibroacoustic modal are presented.

A new rule of vibration sampling for predicting acoustical radiation from rectangular plates

Structural surface velocity distribution is often used to predict structural borne sound radiation. However the sampling interval of velocity should be chosen carefully to increase the prediction accuracy and to reduce the system cost. In this paper, several factors affecting the sampling interval are theoretically analyzed and discussed for a vibrating baffled rectangular plate. A new rule for the determination of the sampling interval is formulated. Using this rule, the results from both numerical simulations and experiments may be explained well.

The Sound Radiation Characteristic from Un-baffled Rectangular Plates

The sound radiation characteristic from un-baffled rectangular plates is studied in this paper. Both the pressure jump and the transverse displacement of vibrating rectangular plates are also presented. The sound power level radiated from un-baffled plates is numerically calculated by using a double layer integral representation of the sound radiation pressure and the modal coupling coefficients. Rectangular plates with clamped support and elastically support encountered frequently are calculated as numerical experiments. The sound power level radiated from baffled rectangular plates with the same boundary conditions are also calculated in numerical experiments for comparison. From the viewpoint of energy, average velocity square of un-baffled and baffled rectangular plates with␣different boundary conditions is also calculated respectively. The relationship of sound radiation efficiency between un-baffled and baffled rectangular plates is also deduced.

Sound radiation from a baffled rectangular plate under a variable line constraint

Sound radiation from plates in baf.es, and shells is a well-studied subject. More recently, researchers are trying to optimize these structures for minimizing sound radiation [1]. A few typical parameters which are optimized include material tailoring [2], location and number of point masses attached to the structure [3], dynamic compliance [4], and damping layer placements [5]. These techniques include numerical methods such as BEM and FEM.

PIEZOELECTRIC ARRAY SENSING OF VOLUME DISPLACEMENT: A HARDWARE DEMONSTRATION

This paper presents the results from a hardware demonstration of a noise radiation sensor for a baffled rectangular plate. The sensor consists of an array of independent piezoelectric patches connected to a linear combiner. The coefficients of the linear combiner are computed in order to reconstruct (in real time) the volume displacement (or velocity). The robustness of the volume displacement sensor with respect to the location of the disturbance source is investigated.

Acoustic radiation from an elastic baffled rectangular plate covered by a decoupling coating and immersed in a heavy acoustic fluid

The vibroacoustic behavior of an elastic, simply supported rectangular plate covered by a locally reacting decoupling layer supporting thickness deformation is presented. The model simulates the vibration and acoustic response of the system immersed in water and subjected to a point force disturbance. A simplified version of the theory is derived in the limiting case of a large decoupling (low mechanical impedance of the layer/high frequency). An appropriate vibratory indicator, representative of the acoustic attenuation provided by the decoupling treatment, and independent of the structure dimensions, is also investigated from the perspective of small-scale laboratory characterization.

Comparison of two structural sensing approaches for active structural acoustic control

A numerical study comparing the use of two structural sensing approaches for sound radiation control is performed on a baffled rectangular plate. The first sensing approach implements an array of accelerometers whose outputs are filtered to construct an estimate of the sound pressure radiated at given angles in the far field. The second method uses the same array of point sensors to estimate the net volume acceleration of the plate. Results show the improved performances of the sensor observing far-field sound radiation over a volume acceleration based on sensor.

A TWO-DIMENSIONAL TCHEBYCHEFF COLLOCATION METHOD FOR THE STUDY OF THE VIBRATION OF A BAFFLED FLUID-LOADED RECTANGULAR PLATE

The vibration and the acoustic radiation of a baffled rectangular plate in contact with a dense fluid is considered. By using Green's representation theorem, the original system of differential equations governing both the displacement and the acoustic sound pressure is transformed into a system of coupled boundary integral equations. The unknown functions (the displacement of the plate, the pressure jump at the surface of the plate and the boundary sources) are expanded as series of Tchebycheff polynomials. The new unknowns (the coefficients of the series) are calculated by using a collocation method. Some results, both theoretical and numerical, are given concerning the resolution properties of the Tchebycheff approximation. The numerical difficulties are presented, and solutions of these are proposed. Numerical examples are given.

Vibration and sound radiation of fluid-loaded stiffened plates with consideration of in-plane deformation

A Ritz method is presented for computing the vibration and sound radiation of baffled rectangular plates reinforced by eccentric stiffeners. The model incorporates both bending, transverse shear and in-plane deformation of the plate, and bending, torsional, and extensional deformation of the stiffeners. The in-plane deformation of the plate, resulting from the eccentric stiffener deformations, has not been considered in previous studies. Relevant expressions of the kinetic and strain energies of the mechanical system are detailed. Numerical results of the in-vacuo natural frequencies for various configurations of plate thickness, stiffeners, and boundary conditions show the importance of in-plane displacements of the plate, especially when thick plates are involved. The comparisons in terms of in-vacuo natural frequencies between the present approach and available published results as well as FEM calculations show a good agreement. An example of structural response and sound radiation in water is also presented.