Rayleigh Integral Research Articles

Acoustic radiation characteristics of shark skin inspired surface modified plates

This paper aims to evaluate the acoustic radiation characteristics of thin plates featuring a layer of small-scale biomimetic shark skin type additive surface treatment. The shark skin dermal denticles are modelled as point masses arranged in a bi-directional pattern on both the upper and lower surfaces of the plate. The governing equations are obtained through a variational approach, incorporating the Dirac Delta function in the derivation of the proposed semi-analytical model for the shark skin layer. A semi-analytical method based on the Rayleigh–Ritz formulation is utilized to analyze the vibrations of these plates with surface modification. The sound radiation characteristics are then derived from the solution of the Rayleigh integral. A comprehensive investigation is performed on the influence of surface modification on different vibro-acoustic characteristics, using a continuous structural mode and power transfer matrix-based approach. Notable observations include a reduction in peak vibro-acoustic responses with dense denticle arrangements, especially at resonance, demonstrating a direct relationship with mass ratios, i.e., the ratio of denticle mass to plate mass. The study further reveals a shift of vibro-acoustic responses towards low frequencies with an increase in mass ratios. A thorough comparative study indicates that while additive surface modifications inspired by shark skin may weaken sound radiation characteristics at resonance frequencies, a reverse effect can be observed at intermittent operational frequencies.

A fast vibro-acoustic modeling method of plate-open cavity coupled systems

In this paper, a fast Chebyshev-Ritz method for vibro-acoustic analytical modeling of plate-open cavity coupled systems is developed for the first time. Based on the Chebyshev spectral method and the Rayleigh-Ritz solution procedure, the vibro-acoustic model of the open cavity coupled with a rectangular plate is established. The exterior acoustic field of the open cavity is expressed by the Rayleigh integral. Additionally, the Rayleigh integral is divided into a frequency-independent singular integral and a frequency-dependent non-singular integral, accelerating the calculation process. Furthermore, the Gauss-Chebyshev-Lobato sampling method is first developed for the plate-open cavity coupling model. By converting the integrals into tensor products, the method avoids complex quadruple integrals, increasing the efficiency of the entire integral operation. The vibration and acoustic responses from the proposed method agree well with existing literature and FEM analysis results, demonstrating the convergence and correctness of the current methodology. The mechanism of cavity depth on vibro-acoustic features of plate-open cavity systems is studied, which is less focused in the published literature. Other factors governing the plate-open cavity coupled model encompassed boundary conditions, fluid mediums, and plate thickness are fully examined. The results provide a theoretical foundation for the design and future research of plate-open cavity structures.

Soundbox-based sound insulation measurement of composite panels with viscoelastic damping

Sound transmission loss (STL), an essential index for assessing the sound insulation performance of composite laminated structures, typically relies on experimental methods to measure. The soundbox method (SBM), a straightforward technique for measuring the STL, is sensitive to microphones’ positions. Within the framework of the Chebyshev-Ritz method, a semi-analytical vibro-acoustic model extended to composite laminated panels with viscoelastic damping (VED) is proposed for the first time. Based on the developed simplified layer-wise theory, the panel is modeled using three layers: the top face layer, the VED layer, and the bottom face layer. A closed cavity is added to the model as the soundbox enclosure used in actual measurements. By employing the Hamilton's principle, the governing equation for the coupling system is derived, and the vibration and internal acoustic responses of the coupling system are calculated. A discretization strategy is introduced to address the frequency-dependent properties of the VED layer, avoiding the need to reconstruct the stiffness matrix at each frequency. To obtain the STL of the panel, sound pressures at external measurement points are calculated based on the Rayleigh integral. The proposed model is validated against numerical results from finite element analyses. The influences of the microphone position inside and outside the cavity on the measured STL are studied. Furthermore, parametric studies over the microphones' positions are performed to enhance the SBM-based evaluation of the composite panel's sound insulation performance. The optimal locations for two internal microphones and one external microphone are recommended. Finally, experimental studies are carried out to guide the implementation of the SBM.

Vibro-acoustic modeling of the turbulent excited cavity-plate-exterior space coupled system

To mitigate flow-induced noise in sonar self-noise, this study introduces a vibro-acoustic modeling technique for sonar cabins. This technique couples a turbulent-flow-excited plate with the interior cavity and the exterior field within a unified model. Employing the Rayleigh-Ritz method, the Chebyshev spectral method, and the Rayleigh integral, the model accounts for general boundary restraints, wall impedances, and reinforcement configurations. Flow-induced responses are determined through the Monte Carlo quadruple integration of frequency response functions and the cross-spectral density model, which characterizes turbulent pressure fluctuations. The convergence and reliability of the present method are verified, achieving excellent agreement with numerical methods and previous results. The study reveals that periodic impedance arrangements in the cavity significantly attenuate sound wave propagation, especially when periodic arrangements are added at low frequencies. The reinforcement configuration perpendicular to the flow direction minimizes low-frequency vibrations. However, the influence of reinforcement on acoustic energy within the cavity is marginal and could potentially lead to increases under simply supported boundary conditions. This method effectively addresses vibro-acoustic challenges in sonar cabins, offering substantial practical value for the development of low-noise sonar systems.

Study on the radiation directivity of a ring-excited thin circular plate with a fixed boundary

Air-coupled transducer with a flat plate structure have many applications in the fields of ground weather observation, ultrasonic defoaming, and directional strong acoustic radiation. In this paper, an analytical equation of the far-field radiation directivity of a fixed boundary ring-excited thin circular plate (RTCP) is deduced using Rayleigh integration method. A finite element model of the RTCP is established, and the relationship between the far-field radiation directivity and the excitation position, excitation area and working frequency is studied by considering the third-order axisymmetric flexural vibration of the RTCP. Computation results show that, for a RTCP, the excitation position has more effect on its radiation directivity. When the plate is excited at the positions between first two nodes, the directivity can be enhanced. When the excitation position is in the trough of the normal displacement curve along radius direction, the side lobes of the radiation directivity of the RTCP are minimized. The area of excitation region has smaller influence on the frequency and radiation directivity of the RTCP. However, working frequency has a great influence on the radiation directivity of the RTCP. When the working frequency is close to the vibration frequency of the circular plate, the sound radiation directivity is the best. A prototype fixed boundary circular plate excited by a longitudinal sandwich transducer was designed and manufactured. For comparison, its finite element model was also setup to simulate its acoustic radiation directivity. Experimental results were found to be in agreement with the theoretical calculations and finite element simulation results.

Comparison of Acoustic Radiation Characteristics in the Near and Far Fields Due to the Vibration of a Circular Disk Placed on an Infinite Rigid Baffle Using the Rayleigh Integral Method and Lumped Parameter Method

Comparison of Acoustic Radiation Characteristics in the Near and Far Fields Due to the Vibration of a Circular Disk Placed on an Infinite Rigid Baffle Using the Rayleigh Integral Method and Lumped Parameter Method

Free vibration analysis of hybrid CNT/GPL-reinforced Porous composite plates under fluid-loading

The present study aims to explore free vibration analysis of hybrid reinforced porous composite plates subjected to fluid loading. The hybrid composite comprises a porous core layer covered by two reinforced porous face sheets. Graphene Platelets (GPLs) and Carbon Nanotubes (CNTs) are considered as reinforcement materials. The linear three-dimensional piezo-elasticity theory is employed to model the structural part of the system, while the classical Rayleigh integral is used to obtain the fluid interaction part. The accuracy and reliability of the proposed model are validated by comparing the calculated natural frequencies of the plate in vacuum and water-loaded conditions with those reported in the literature. For the first time, several parametric studies are performed to investigate the effects of dimension ratios, porosity distributions, reinforcement dispersion pattern, porosity coefficient, and weight fraction of nanofiller on the natural frequency of hybrid reinforced porous composites. The results indicate that adjusting the porosity and weight fraction of the face sheets can enhance stiffness without changing the mass, emphasizing the importance of hybrid composites for achieving optimal stiffness while maintaining a constant mass. Furthermore, the influence of core layer porosity distributions on the composite behavior differs from that of porosity distributions in the face layers. Additionally, the effectiveness of the reinforcement pattern depends on the type of porosity distribution in the face layers.

Моделювання акустичного поля на основі дисперсії звуку при відбитті трасуючих хвиль на відкритих майданчиках

A method of numerical modeling of acoustic fields in open areas with the possibility of parallelization of calculations is proposed. This method is part of a developed software solution that allows you to perform physical field modeling in various subject areas, being scalable in the sense of using an arbitrary set of parallel calculators. The use of existing modeling systems is associated with great difficulties in solving complex problems with a high degree of detail of the simulated object. Greater accuracy implies a high degree of discretization, a greater number of elementary model calculations performed. Parallel and distributed computing systems have a much better ratio of accuracy-approximation and time and cost costs compared to single-processor systems. Modern general purpose modeling systems use a simplified ray model of sound propagation, which neglects diffractional and interference effects, which are often critical in industrial acoustics. The article proposed a method based on the approximation of the principle of superposition of sound fields. It is accurate, while the linearity of the equations of acoustics is relevant. The basis is the Rayleigh integral and the approximation of reflective surfaces by flat point radiators. A parallel form of such a method is presented, as well as an analysis of its properties, both in sequential and parallel forms.

Analysis of vibro-acoustic characteristics of functionally graded sandwich microplates under thermal-electric effects

Based on the hyperbolic tangent parabola mixed shear deformation theory, the paper studied the sound radiation and the sound insulation of the functionally graded (FG) sandwich microplates under thermal-electric effects. The sandwich plate is composed of piezoelectric skin layers and FGM core layer. By using Hamilton’s principle, a size-dependent model considering the thermal-piezoelectric effects are established. The acoustic response and sound insulation are calculated with Rayleigh integral and solid-fluid coupling conditions. The accuracy of the presented method is verified by the numerical simulations. Then the effects of various parameters on the vibro-acoustic characteristics are analyzed and discussed. Numerical results show that the temperature variations, the length scale parameter to thickness ratio and the gradient index have great impacts on the FG sandwich microplates.

Treatment Planning and Aberration Correction Algorithm for HIFU Ablation of Renal Tumors.

High-intensity focused ultrasound (HIFU) applications for thermal or mechanical ablation of renal tumors often encounter challenges due to significant beam aberration and refraction caused by oblique beam incidence, inhomogeneous tissue layers, and presence of gas and bones within the beam. These losses can be significantly mitigated through sonication geometry planning, patient positioning, and aberration correction using multielement phased arrays. Here, a sonication planning algorithm is introduced, which uses the simulations to select the optimal transducer position and evaluate the effect of aberrations and acoustic field quality at the target region after aberration correction. Optimization of transducer positioning is implemented using a graphical user interface (GUI) to visualize a segmented 3-D computed tomography (CT)-based acoustic model of the body and to select sonication geometry through a combination of manual and automated approaches. An HIFU array (1.5 MHz, 256 elements) and three renal cell carcinoma (RCC) cases with different tumor locations and patient body habitus were considered. After array positioning, the correction of aberrations was performed using a combination of backpropagation from the focus with an ordinary least squares (OLS) optimization of phases at the array elements. The forward propagation was simulated using a combination of the Rayleigh integral and k-space pseudospectral method (k-Wave toolbox). After correction, simulated HIFU fields showed tight focusing and up to threefold higher maximum pressure within the target region. The addition of OLS optimization to the aberration correction method yielded up to 30% higher maximum pressure compared to the conventional backpropagation and up to 250% higher maximum pressure compared to the ray-tracing method, particularly in strongly distorted cases.

Vibro-acoustic response and sound transmission loss of sandwich plates with tunable Poisson's ratio core in a hygrothermal environment

In this study, vibro-acoustic response and sound transmission loss of sandwich plates with tunable Poisson's ratio (TPR) core in a hygrothermal environment are investigated. The sandwich plate is taken as a moderately thick plate and various plate theories are used to examine the vibroacoustic behavior. First, the equations of motion of a sandwich plate in the hygrothermal environment are derived based on Hamilton's principle. Then, the solution to free vibration for a simply supported sandwich plate is presented. Next, we obtain the sound power level (SPL) for the point harmonic force exerted on the surface of the sandwich plate using the Rayleigh integral and the sound transmission loss (STL) as the harmonic sound wave is incident on the plate surface. Finally, numerical examples are conducted and discussed. The effects of the core-to-thickness ratio, plate theories, parameters of the TPR core, hygrothermal environment, and incident elevation angle on the SPL and STL of the sandwich plate are analyzed. This study will contribute to the design and manufacturing of lightweight structures in the aviation industry.

Electronic Focus Steering Capabilities of a Diagnostic-Type Linear Ultrasound Array Designed for High Power Therapy and Its Visualization

The focus steering capabilities of a 1 MHz linear phased array transducer (64 rectangular elements, 14.8 × 51.2 mm aperture) intended for drug delivery applications in abdominal organs were assessed and compared with its design-stage computer model. Acoustic fields generated by the transducer and predicted by the models of an ideal array with uniformly vibrating elements and either a plane or a cylindrically focused surface were simulated using the Rayleigh integral and angular spectrum methods. The boundary conditions for the transducer were reconstructed from acoustic holography measurements performed for selected focusing configurations of the array and also synthesized from holography data measured for each of its individual elements. It was shown that the transducer field with electronic focus steering can be accurately synthesized based on the holography data of its elements, which significantly simplified acoustic field characterization. Variability of the power and directivity patterns of the array elements were analyzed. A twofold smaller range of electronic steering in the transverse direction for the transducer compared to its computer model is discussed.

An accurate coupled method for analysis of transcranial magneto-acoustic-electrical stimulation

In this paper, a numerical method for simulating transcranial magneto-acoustic-electrical stimulation (TMAES) is presented. The smoothed finite element method (SFEM) is applied to calculate the magnetic field and the Rayleigh integral (RI) is used for solving the ultrasonic field. Three-node triangular and four-node tetrahedral elements are set as the background cells for constructing the smoothing domains (SDs). A gradient smoothing technique (GST) is applied over each SD to obtain the magnetic flux density. The discretized system equations are obtained by using the generalized smoothed Galerkin weakform. To solve the sound pressure, the ultrasonic transducer is discretized into a series of micro-elements. The acoustic pressure in the problem domain is then equal to the superposition of that generated by the individual micro-element. Finally, the current density can be calculated by the magneto-acoustic coupling effect. Numerical examples demonstrate that SFEM can improve accuracy, convergence rate, and efficiency to 10, 1.8, and 4 times those of FEM, respectively. In the calculation of coupled fields, SFEM-RI also improves the accuracy by 2.5 times. Additionally, the presented method possesses the property of insensitivity to mesh distortion.

Numerical Analysis of Vibroacoustic Response of Timber Floor Panels Damped with Porous Materials

This research aims to investigate the impact of incorporating porous materials on reducing noise and vibration in wooden floor panels, and to analyze the vibroacoustic performance of the assembled panel under different types of excitation and boundary conditions, particularly in the lower frequency range. The study begins with an experimental investigation and numerical modeling to determine the mechanical properties of the orthotropic wood material used in the floor panels. Subsequently, a finite element formulation, based on a variational approach, is presented to study the vibroacoustic response of an elastic structure coupled with a porous material exhibiting realistic behavior. The porous material is characterized by two phases: solid and fluid, represented in the formulation through the displacement field for the solid phase and the pressure for the fluid phase. This formulation offers the advantage of reduced computation cost and simplifies the coupling between all domains. To calculate the acoustic radiation of the structure, the Rayleigh integral is employed. Utilizing the proposed numerical approach, a comprehensive study is conducted to analyze the reduction in vibration–acoustic response of the floor with the incorporated porous layer, taking into account different types of excitation and boundary conditions applied to the system.

On the Influence of Scattered Errors over Full-Field Receptances in the Rayleigh Integral Approximation of Sound Radiation from a Vibrating Plate

Spatially dense operative deflection shapes and receptances, acquired in broad frequency bands, increase the detail in the spatial and frequency domains of the responses of parts in actual dynamic loading, manufacturing and mounting conditions. This work remarks the potential benefits of greater spatial resolution in the Rayleigh integral approximation of sound pressure—here reformulated to exploit the increased quality output from experiment-based optical full-field technologies in contactless structural dynamics—radiated by a vibrating surface in a broad frequency band. But in some cases the noise that is scattered over the estimated receptance maps might be heavier, or with different patterns, than expected, with potential repercussions on the sound pressure simulations that come thereof. This work covers this specific latter issue with insight over examples from experiment-based receptances of a lightweight vibrating plate. The effects of error spreading are analysed in the space and frequency domains, with special attention to the contribution of the experiment-based full-field receptance maps to the accuracy of the vibro-acoustic frequency response function maps.

An Improved MSA Model for Evaluating the Sound Transmission Loss of a Rectangular Plate for Diffuse Field Incidence

This paper presents an approximate analytical model for estimating the transmission loss (TL) of a finite rectangular plate in the low frequency range, which is based on the modal summation approach (MSA) taking into account the modal radiation impedance and fluid loading. The mode-dependent radiation resistance is calculated using the Rayleigh integral. The fluid loading is taken into account through the natural frequency modified by the added mass. The results are compared with the ones of Statistical Energy Analysis (SEA) coupled with FEM and FEM coupled with BEM. In addition, the effects of the various vibration modes and the fluid loading on TL, and a way for reducing the calculation time are discussed.

Acoustic Response of an Isotropic Beam Under Axially Variable Loads Using Ritz and Rayleigh Integral Methods

Acoustic Response of an Isotropic Beam Under Axially Variable Loads Using Ritz and Rayleigh Integral Methods

Thermo-vibro-acoustic analysis of pavement under a harmonically rectangular moving load

This study presents an analytical framework based on the third-order shear deformation theory (TSDT) to conduct a comprehensive thermo-vibro-acoustic evaluation of a multi-layered asphalt system subjected to a harmonically rectangular moving load. The research aims to investigate the dynamic response and sound radiation characteristics of the pavement under various operating conditions. The materials and methods employed in this study are as follows: The temperature variations along the thickness axis of the layers are considered, and Hamilton's principle is adopted to derive the governing equations in the time-based domain. The equations are then solved in the Laplace domain using the Fourier series and the Laplace transform. The acoustic pressure generated by the moving load is investigated by applying Durbin's numerical Laplace transform inversion technique and analyzing the Rayleigh integral. The accuracy and flexibility of the proposed approach are demonstrated through comparisons with previous studies and finite element (FE) simulations using COMSOL Multiphysics®. Finally, a comprehensive analysis is conducted to examine the impact of various factors, including the frequency and velocity of the moving excitation load, the viscoelastic medium, and variations in temperature, on both the dynamic response and sound radiation of the pavement. Some of the major findings of this research are as follows: An increase in load velocity leads to a higher frequency of pressure waves and closer spacing between them, resulting in elevated sound pressure amplitude. Higher pavement temperatures cause increased softness and rutting depth under moving loads. This research offers a novel and practical contribution to the field of pavement systems by presenting an analytical framework for a comprehensive thermo-vibro-acoustic evaluation and providing valuable insights into the dynamic response and sound radiation characteristics of multi-layered asphalt systems under various operating conditions.

Acoustic Radiation of a Beam Subjected to Transverse Load

In this paper, the dynamic response of a Euler–Bernoulli beam subjected to transverse harmonic forces is calculated. The method of separation of variables combined with the mode shape superposition method, which includes the determination of eigenvalues, is used to define the velocity field of the beam surface. The Rayleigh integral was used to calculate the sound radiation and the beam was placed in an infinite baffle. Additional actuators are introduced in order to minimize the sound radiation, or, more specifically, the total sound power level of the vibrating beam, and their optimal position and force amplitude are determined; the conclusions were drawn from the optimization results. This paper proposes a method for faster determination of the optimal actuator parameters in order to achieve the minimum total sound power level. The validity of the obtained results is demonstrated with examples, whose solutions are compared to the results in the published literature.

Effect of truss core on sound radiation behavior of sandwich plate structures under structure borne excitation

Compared with traditional honeycomb and stiffened sandwich structures, lattice-based structures are considered an ideal substitute for traditional core structures due to their high stiffness and other potential multifunctional application characteristics. Based on that, the sound radiation characteristics of the sandwich plate with different truss cores under structure borne excitation were investigated in the current paper, wherein three truss cores, including pyramidal, tetrahedral and 3D-kagome cores types, are considered. The governing equations for the vibration and sound radiation analysis of the sandwich plate structure are carried out using the static equilibrium method, and then solved via the Navier approach and Rayleigh integral. By comparing the experimental results, the validity of the proposed theoretical model is verified. The theoretical model is subsequently used to investigate the influences of truss core geometric parameters and types, face sheet-core face sheet thickness ratio and aspect ratio on sandwich plate sound radiation characteristics. The results indicate that, in the range [ 5 ∘ ≤ θ ≤ 52 ∘ ] and [ 52 ∘ < θ ≤ 85 ∘ ] , the tetrahedral and 3D-Kagome core types have the maximum natural frequency values, respectively, while the resonance peaks of sound pressure level curves move to lower frequency region with the increase in truss core radius and face sheet-core face sheet thickness ratio, and the sandwich plate with the 3D-kagome core shows better sound radiation characteristics in the wide frequency range.