Rayleigh Integral Research Articles

Sound transmission characteristics of a plate backed by an irregular cavity

The study of the coupling cavity provides an effective theoretical reference for the acoustic design. However, most of the previous studies are based on idealized models and there is little research on the sound radiation power about the irregular shapes. The work in this paper achieves the generalization of the model and explores the impact of irregular back cavities on the sound radiation of the plate structures. The influence on the results of the sound insulation performance testing of the plate is also investigated when irregular cavities involved. The displacement of the plate and the sound pressure inside the cavities are expressed as multidimensional Fourier series with additional terms, which will contribute to maintaining the continuity of pressure and velocity at the boundaries. The sound field of the irregular cavity is considered as a unit cubic one in a new coordinate system by coordinate transformation method. The arbitrary boundary of the cavities is realized by adding the energy terms of impedance into the Lagrange equation. The Hamilton’s principle is employed to realize the complete coupling between the plate and the sound fields on both sides. The sound radiation power of the plate can be obtained by using the Rayleigh integral. By using Fast Cosine Transform, the quadruple integrals of the radiation power are simplified to improve accuracy and efficiency of the calculation. Numerical results from the analytical approach are compared with those given by Finite Element Analysis (FEM) to guarantee the rightness and validity of the models. Moreover, experimental results are also given to discuss the reasons of the bias between theory and practice.

Reconstruction of the sound radiation field from flexural vibration measurements with multiple cameras

This paper presents the background theory and the experimental implementation of a new approach for the reconstruction of the sound radiation field produced by the flexural vibration of a distributed structure using video image acquisitions. The study is focused on tonal flexural vibration and sound radiation at the first five resonance frequencies of a baffled flat rectangular plate model-structure. The plate is divided into a regular mesh of rectangular elements whose centres are marked with small bullets. The transverse vibrations at the grid of target points are estimated via triangulation from the images acquired with six cameras unevenly spread along half of a circle located, with a small offset, parallel to the surface of the plate. The sound radiation field in free space is then reconstructed from the Rayleigh integral, which is approximated into a finite sum over the mesh of elements. Both the flexural vibration field and the sound radiation field derived from the cameras video acquisitions are contrasted with measurements taken respectively with a laser vibrometer and an array of microphones.

An Engineering Approach for Estimating the Radiation Efficiency of Orthogonally Stiffened Plates

AbstractA systematic investigation of the sound radiation of orthogonally stiffened plates is presented using a numerical procedure that combines the finite element method with the Rayleigh integral. Results are computed for stiffened plates with different numbers of stiffeners, stiffener depth, and plate thickness to investigate the dependence on the most important parameters. Differences between the radiation efficiency of stiffened plates and unstiffened panels are seen. In the monopole region, the result depends on the mode that dominates the response. For excitation within a bay, the radiation efficiency is reduced to that of the single bay if the stiffeners are stiff enough. If excited on a stiffener, the plate tends to radiate sound over its full surface area. In the short-circuiting region, on average, the radiation efficiency is equal to that of a smaller bay-sized panel with clamped edges, regardless of the excitation position. Results from the systematic study of 120 numerical cases are used to develop asymptotic formulae for the radiation efficiency of stiffened plates based on existing formulae for unstiffened panels. For all tested configurations, the average difference between the formulae and the numerical calculations was 0.3 dB over the whole frequency spectrum, with a standard deviation of ± 1.5 dB. Between the frequency bands, the mean value varied between −2 and 3 dB, with a standard deviation of up to ± 1.5 dB in the monopole region and up to ± 5 dB in the short-circuiting region.

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.

Limited driving characteristics of combustion instability in a swirling flame: An experimental study

Regarding combustion instability, heat-release pulsation leads to pressure pulsation. Conversely, the driving force of the pulsating flame restricts the oscillation equilibrium point of combustion instability. Thus, in this paper, we developed a flame limited driving model explaining the finiteness of the driving energy of flame pulsation. Our research indicates that the limit cycle state of combustion instability was affected by the limitations of heat-release pulsation and Rayleigh integration. The coupling intensity of the heat-release pulsation and oscillating pressure decreases gradually with the increase in the oscillating pressure amplitude. In addition, an increase in flame pulsation amplitude presents finiteness similarly. When the acoustic amplitude exceeds a certain threshold, the amplitude of heat-release pulsation stops increasing until the flame approaches the oscillation blow-out limit. The further strengthening of the oscillation intensity of combustion instability is simultaneously restricted by the limitations of heat-release pulsation and Rayleigh integration, and thus the combustion instability finally stabilizes at the oscillation equilibrium point.

The Influence of Curved Profiles on the Vibro-Acoustic Characteristics of Light-Weighted Sandwich Shell: An Analytical Solution

The overall aircraft noise is significantly minimized using curved light weighted sandwich shells as an airframe element. However, there is a lack of studies on how the curvature can influence vibro-acoustic characteristics. To the best of our knowledge, this paper presents an analytical solution that predicts the vibro-acoustic characteristics of simply supported light-weighted sandwich shells with various curved profiles for the first time. Natural frequencies, displacement and velocity responses, sound pressure level (SPL), sound transmission loss (STL), and radiation efficiency are the critical parameters when analyzing vibro-acoustic characteristics of any structure and are presented in this paper. The generalized kinematic equations are established for the sandwich shells, and the governing differential equations are derived using Hamilton’s principle. The solutions are obtained using Navier’s method, and the vibration response is determined by solving the first five unknown mode coefficients. The Rayleigh Integral is used to extract the acoustic responses and is validated. In addition, the effect of face sheet to core thickness ratio [Formula: see text] of curved sandwich shells on vibro-acoustic characteristics is critically examined and presented. It is found that introducing the positive curvature to the sandwich shell reduces the bending moment, thus improves improving the vibro-acoustic characteristics in the lower frequency range. An increase in the [Formula: see text] enhances the shell’s vibro-acoustic characteristics until a threshold value [Formula: see text], beyond which the mass contribution deteriorates the characteristics. For a hyperbolic paraboloid shell [Formula: see text], this effect was observed at a lower-frequency value of 425[Formula: see text]Hz.

The Rayleigh integral is always positive in steadily operated combustors

The Rayleigh index has been used for decades by a large number of researchers as an indicator to determine if a flame is driving or damping thermoacoustic interaction mechanisms. The use of the Rayleigh criterion has found applications in rocket combustors, gas turbine combustion technology and basic combustion research. The global Rayleigh index or integral is obtained by integrating the product of heat release rate and pressure fluctuations over space and time. Depending on the phase between pressure oscillations and heat release rate response, the oscillations can be enhanced or damped. It is commonly assumed in literature that the sign of the Rayleigh index from steady state data can be used to determine if the thermoacoustic feedback loop is stabilizing or destabilizing. However, we show in this paper that under fairly general conditions, a correctly measured Rayleigh index is always positive if evaluated from statistically stationary data. This proves to be true even if the heat release rate response to pressure fluctuations is in phase opposition to those pressure fluctuations. This is shown in a straightforward manner by substituting the wave equation with a heat release rate source term into the Rayleigh index. This was verified experimentally on a fully premixed combustion system by measuring the flame chemiluminescence using a photo multiplier and pressure fluctuations using a microphone placed sufficiently close to the flame to ensure acoustic compactness for the frequency range of interest. A large range of operating conditions have been tested, spanning linearly stable and unstable stationary thermoacoustic states, respectively corresponding to resonance or a limit cycle driven by the inherent stochastic forcing from the turbulent combustion noise. The experimental results corroborated the analytic finding: the Rayleigh index is found to be positive for all frequencies and all operating conditions.

Analytical solution for a focused vortex beam radiated by a Gaussian source

Current interest in focused vortex beams is motivated by the ability to trap particles axially and laterally using the resulting radiation force. A simple closed-form solution is obtained in the Fresnel approximation for a sound beam radiated by a Gaussian source with time dependence e−iωt, focal length d, amplitude distribution exp(−r2/a2), and azimuthal phase dependence einθ, where θ is the angle in the plane perpendicular to the beam axis, and the integer n is the topological charge, referred to here as the vorticity. The solution is in good agreement with the pressure field predicted in the paraxial region by numerical evaluation of the Rayleigh integral. Of interest in optics as well as acoustics is the distance from the minimum along the beam axis to the first local maximum, referred to as the vortex ring. The present solution yields rn = ηnd/ka for the vortex ring radius in the focal plane, where k is the wavenumber, and ηn = 1.69 + 0.965(n−1) for vorticities in the range 1 ≤ n < O(20). Within this range the radius rn thus increases linearly with the vorticity. Results are also presented for dependence of the focusing gain on the vorticity. [CAG was supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

An exact solution for vibro-acoustic response of MEE composite plate

To the best of our knowledge, this is the first endeavor to present an exact solution to predict vibro-acoustic characteristics of Magneto-electro-elastic (MEE) composite plate. The transverse and in-plane fields are considered based on thin-plate conditions. The variation of electric and magnetic potentials is determined according to electromagnetic boundary conditions and the Maxwell equation. The stress resultants and mass inertias are used in Hamilton’s principle to generate the governing equation. Here the mathematical formulation is developed using third-order shear deformation theory. Also in this work, the dynamic displacement responses are provided by finding five undetermined mode coefficients relevant to u, v, w, ϕx, and ϕy to predict forced vibration response. The forced vibration response obtained based on the developed governing equation is used to calculate the acoustic characteristics using the Rayleigh integral. The effect of magnetic and electric potential is shown in the acoustic responses. From the results, it is understood that the acoustic responses are highly influenced by the applied magnetic and electric potential. The radiation efficiency of the MEE plate did not show any variations in the lower frequency and it shows the variation near the resonant frequencies on the application of electric and magnetic potential.

Inversion of Infrasound Time Series for Seismoacoustic Source Parameters Produced by a Buried Chemical Explosion at the Source Physics Experiment Phase II: Dry Alluvium Geology

ABSTRACT We invert infrasonic timeseries produced by a single buried chemical explosion detonated as part of the Source Physics Experiment phase II Dry Alluvium Geology (DAG). The key to our inversion is that we use a three dimensional, fully coupled, linear, elastic-to-acoustic forward model to predict the recorded acoustic wavefield. We assume that the fourth buried chemical explosion in the series, DAG-4, produces an atmospheric acoustic wavefield due to a combination of direct elastic-to-acoustic coupling at the air–earth interface and ground upheaval, or spall, at the Earth’s surface. In our linear model we approximate these two phenomena with distinct source terms separated in space and time: (1) a buried point source consisting of six source time functions, each corresponding to a single component of a generalized moment tensor and (2) a vertically directed, time-variable force applied to the Earth’s surface. Inverting the infrasonic data with this linear forward model results in estimated source parameters that accurately predict the observed infrasound signals. To test our estimated buried source, we use the Rayleigh integral (RI) to model the infrasound signal from the measured and predicted acceleration surfaces. First, we simulate the infrasound signal waveforms using vertical accelerometer data and find that these closely match the observed infrasound. Next, using the estimated buried seismic source model without spall, we estimate the acceleration of the Earth’s surface using a linear approximation. When using the predicted surface acceleration in the RI, we simulate infrasound signals that reproduce pulse shape characteristics but not the amplitude of the observed infrasound, indicating our linear model does not completely account for nonlinear spall effects. Based on these results, we argue that for the scale of this experiment, purely linear models can estimate seismoacoustic source attributes and predict the far-field infrasonic signal, but the primary contribution to the infrasonic signal is the spall source.

Analysis of the vibro-acoustic behaviours of underwater reinforced rectangular plates

In this paper, the vibration and sound radiation of underwater stiffened plates with arbitrary boundary condition have been studied. The vibration of the structure is described by an improved Fourier series, the water-load acts on the structure is solved by Rayleigh integral, and the dynamic response is solved by an energy method. The results show that the increase of plate thickness and boundary constraint can enhance the vibration and sound transformation ability of underwater rectangular plate, and the mechanism is that the increase of vibration mode frequency increases the radiation efficiency of the dominant acoustic radiation mode. By analyzing the effect of stiffening mode and quantity on acoustic radiation, it can be seen that for the plate (1,1) mode, the longitudinal stiffening mode has the lowest vibration energy conversion rate, and the main mechanism is that the dominant acoustic radiation mode has the lowest radiation efficiency, while the transverse stiffening mode has the highest radiation efficiency. In addition, when the reinforcement is distributed, the natural frequency and the radiation efficiency of the corresponding dominant acoustic radiation modes of the stiffeners are reduced, the vibration- acoustic conversion of the plate showed a decreasing trend.

Transient vibration and sound radiation analysis of simply supported functionally graded sandwich plates

Vibroacoustic response of simply supported functionally graded sandwich plates under transient loading is reported in this work. Simply supported square sandwich plate configurations of different thickness ratios and material property gradation in the core and face sheet are analyzed. The effective material properties of the sandwich plate are derived by considering an approximate laminate model where the Poisson’s ratio of the constituents of the sandwich are assumed to be constant and the plate kinematics is modelled using the first-order shear deformation theory. The structural system matrices of the functionally graded sandwich plate and plate deformations are computed using the finite element method. The time history of the structural response is obtained using the Newmark Beta time marching scheme. The radiated acoustic pressure in the far-field is determined using the time domain Rayleigh integral that is solved numerically by an elementary radiator approach. A detailed parametric study investigates the influence of the volume fraction index and thickness ratio on the transient vibroacoustic response of several sandwich configurations.

Comparison of computer simulations and clinical treatment results of magnetic resonance-guided focused ultrasound surgery (MRgFUS) of uterine fibroids.

PurposeMagnetic resonance‐guided focused ultrasound surgery (MRgFUS) can be used to noninvasively treat symptomatic uterine fibroids by heating with focused ultrasound sonications while monitoring the temperature with magnetic resonance (MR) thermometry. While prior studies have compared focused ultrasound simulations to clinical results, studies involving uterine fibroids remain scarce. In our study, we perform such a comparison to assess the suitability of simulations for treatment planning.MethodsSonications (N = 67) were simulated retrospectively using acoustic and thermal models based on the Rayleigh integral and Pennes bioheat equation followed by MR‐thermometry simulation in seven patients who underwent MRgFUS treatment for uterine fibroids. The spatial accuracy of simulated focus location was assessed by evaluating displacements of the centers of mass of the thermal dose distributions between simulated and treatment MR thermometry slices. Temperature–time curves and sizes of 240 equivalent minutes at 43°C (240EM43) volumes between treatment and simulation were compared.ResultsThe simulated focus location showed errors of 2.7 ± 4.1, −0.7 ± 2.0, and 1.3 ± 1.2 mm (mean ± SD) in the anterior–posterior, foot–head, and right–left directions for a fibroid absorption coefficient of 4.9 Np m–1 MHz–1 and perfusion parameter of 1.89 kg m–3 s–1. Linear regression of 240EM43 volumes of 67 sonications of patient treatments and simulations utilizing these parameters yielded a slope of 1.04 and a correlation coefficient of 0.54. The temperature rise ratio of simulation to treatment near the end of sonication was 0.47 ± 0.22, 1.28 ± 0.60, and 1.49 ± 0.71 for 66 sonications simulated utilizing fibroid absorption coefficient of 1.2, 4.9, and 8.6 Np m–1 MHz–1, respectively, and the aforementioned perfusion value. The impact of perfusion on peak temperature rise is minimal between 1.89 and 10 kg m–3 s–1, but became more substantial when utilizing a value of 100 kg m–3 s–1.ConclusionsThe results of this study suggest that perfusion, while in some cases having a substantial impact on thermal dose volumes, has less impact than ultrasound absorption for predicting peak temperature elevation at least when using perfusion parameter values up to 10 kg m–3 s–1 for this particular array geometry, frequencies, and tissue target which is good for clinicians to be aware of. The results suggest that simulations show promise in treatment planning, particularly in terms of spatial accuracy. However, in order to use simulations to predict temperature rise due to a sonication, knowledge of the patient‐specific tissue parameters, in particular the absorption coefficient is important. Currently, spatially varying patient‐specific tissue parameter values are not available during treatment, so simulations can only be used for planning purposes to estimate sonication performance on average.

Acoustic response analysis of periodic orthogonal stiffened composite sandwich structure with pyramidal truss cores

This paper presents analytical studies on the sound transmission loss behavior of functionally graded carbon nanotube reinforced composite (FG-CNTRC) truss core sandwich structure plates subjected to plane sound wave excitation, wherein three different functionally graded patterns are introduced for the carbon nanotubes through the thickness direction of structures. The effective CNTs material properties for each type are determined by the extended rule of mixture and the governing equations are based on the first order shear deformation theory. The compatibility of displacements on the interface between the plate and the stiffeners is employed to derive the governing equation of each stiffener. By using the modal expansion approach and Rayleigh integral, the sound transmission loss is described analytically. Since no existing results of sound insulation can be found for such composite material plate structure, experimental measurements of composite stiffener sandwich plate which is composed of carbon fiber reinforced composite material are subsequently carried out to validate the theoretical model, and good agreement is achieved. Based on the developed theoretical model, the influences of the volume fractions of CNT, truss core type, distribution type of CNT, stiffener spacing, and truss core height on sound transmission loss are subsequently presented.

Vibroacoustic response of mode localized thin functionally graded plates using physical neutral surface

This paper investigates an analytical solution to the sound radiation behavior of vibrating thin mode localized Al/Al2O3 functionally graded plates (FGM) with varying boundary conditions utilizing its physical neutral surface through analysis and simulation. For the study, simply-supported, clamped–clamped, and clamped–free FGM plates with power-law index k = 0,1,5,10 are examined. 20% of the plate’s weight is used to constraint the plate with point and distributed mass. Matlab formulation is used to compute the modal participation factors by analytically solving the governing equation. Furthermore, modal participation factors are used to compute the normal velocity and far-field sound radiation fields by solving the Rayleigh integral using the elemental radiator approach. Sound power level results are verified using ANSYS’s finite element method and good agreement is observed. The effects of boundary conditions and power law indexes on the sound radiation fields of mode localized FGM plate are correspondingly discussed. Cross-comparison of sound radiation by mode-localized FGM plate with point and distributed mass are discussed. Sound power level in terms of “A” weighted scale(dBA) illustrates that the sound power level increases with an increased power-law index. Moreover, mode localized FGM plate with distributed mass reported a lower sound power level for all power-law indexes.

Free-field sound radiation measurement with multiple synchronous cameras

This paper investigates the possibility of using photogrammetry to obtain the sound radiation into free-field from flexural vibrations of distributed thin structures. A simple model problem is considered, where multiple (i.e. two or more) cameras are used to track at a regular grid of target points the flexural vibration of a baffled thin rectangular flat plate structure. The sound radiation into free-field is then derived from the Rayleigh integral, which is approximated into a finite sum assuming the plate is discretised into a regular mesh of elements whose centres correspond to the grid of points where the flexural vibrations are measured with the cameras. The paper shows that the vibration measurement errors produced by the image pixelisation in the cameras compensate over the grid of target points so that the estimate of the radiated sound field with the Rayleigh integral formulation is comparatively more accurate.

Optimum Multinozzle Configuration for Minimizing the Rayleigh Integral During High-Frequency Transverse Instabilities

Abstract This paper develops a formalism for optimizing nozzle location/configuration with respect to combustion stability of high-frequency transverse modes in a can combustor. The stability of these acoustically noncompact flames was assessed using the Rayleigh integral (RI). Several key control parameters influence RI—flame angle, swirling strength, nozzle location, as well as nozzle location with respect to the acoustic mode shape. In this study, we consider a N-around-1 configuration such as typically used in a multi-nozzle can system and study the overall stability of this system for different natural transverse modes. Typically, such nozzles are distributed in a uniformly circular manner for which we study the overall RI, and for cases where RI > 0, we optimize the nozzle distribution that can reduce and minimize RI. For a fixed geometry such as a circular configuration, the analysis shows how the flame's parameters must vary across the different nozzles, to result in a relatively stable system. Additionally, for a fixed set of flame parameters, the analysis also indicates the noncircular distribution of the N nozzles that minimizes RI. Overall, the analysis aims to provide insights on designing nozzle locations around the center nozzle for minimal amplification of a given transverse mode.

Feasibility investigation of rail monitoring by wheel-mounted acoustic emission technology using two-dimensional analytical contact acoustic nonlinearity model

In an attempt to analyze the feasibility of a rail monitoring scheme with a wheel-mounted acoustic emission measurement method, a two-dimensional analytical spring model was proposed to interpret the interactions between acoustic waves and stationary wheel–rail contact interfaces. The spring model represents the coupling strength of the interfaces with their stiffness. It accommodates the environmental information of contact stiffness and the acoustic source information, such as the feature frequency and the incident angle in the analysis of interface transmissibility. The fractal dimension of the interface is introduced into the spring model to interpret the effect of the axle load on the interface stiffness and the variation in transmissibility. Discrete Rayleigh integration is further combined to obtain the directivity of the interface and estimate the acoustic field intensity in the entire wheel. A wheel–rail contact rig was designed to simulate the actual contact conditions in a railway. Experimental data acquired from this test rig were utilized to validate the model, with regard to the interfacial stiffness in normal incidence and amplitudes of the transmitted waves in oblique incidence. After verifying the reliability of the model, the acoustic field intensity in the wheel was visualized under the assumed environmental conditions. Finally, a discussion is presented to determine a proper angular separation for the wheel-mounted sensor system, and substantiate the feasibility of the system.

Multi-view videogrammetry for the estimate of plate sound radiation

This paper explores the possibility of estimating the acoustic pressure field resulting from the vibrations of thin structures using photogrammetric techniques. In particular, the performance of measurement systems using two or more cameras to estimate the displacement of 3D markers placed on the surface of a plate are analyzed by simulation. The projection of the markers on the image plane of each camera is calculated assuming a perfectly calibrated system, in which the errors in the 3D reconstruction of the marker positions are substantially determined by the intrinsic pixellation intrinsic to the measurement process. The acoustic pressure in space is finally calculated, starting from the position of the markers, by means of the Rayleigh integral. The accuracy of the estimate is evaluated by varying the geometry of the cameras setup, their resolution and their number. The results presented show that the estimation of the acoustic radiation is substantially more accurate than the accuracy with which the vibration is estimated. The effect is due to the smoothing operation intrinsically linked to the Rayleigh integration procedure. This study demonstrates the effectiveness of optical systems with multiple, relatively low-cost low-resolution cameras for the measurement of the acoustic pressure field generated by flexural vibrations of distributed structures.

Vibroacoustic analysis of a thin laminated composite plate with surface-boned piezoelectric patches and subjected to general boundary conditions

In this paper, a semi-analytical model is proposed to deal with the vibroacoustic problems of laminated composite plates with surfaced-boned piezoelectric patches and subjected to general boundary condition using the modified Fourier series method. Based on Kirchhoff plate theory, the dynamic equation of the laminated composite plate is derived using Hamilton's principle. In order to satisfy general boundary conditions, the displacement solution of the plate is expressed in the form of two-dimensional Fourier series and serval auxiliary functions. The acoustic response of the laminated composite plate due to a harmonic concentrated force is obtained with the Rayleigh integral. Besides, the mass and stiffness contribution of the piezoelectric patch are taken into consideration in the present study. Through enough convergent studies and comparative studies, the convergence, accuracy and universality of the proposed method are validated. The developed semi-analytical model can be used for efficient and accurate analysis and design of laminated composite plates equipped with shunted piezoelectric patches. Finally, the effects of the resistor and inductor shunt damping circuits on the vibration and acoustic response is discussed.