Modal Radiation Efficiency Research Articles

On the balance between the tonal and broadband noise of isolated propellers

This study presents a predominantly numerical and theoretical investigation into the balance of tonal and broadband noise due to an isolated propeller in uniform motion. The predicted trends in the balance between tonal and broadband noise radiation with varying blade number and speed of rotation is supported by preliminary experimental measurements. Here, we assume that the dominant noise generation mechanisms are the tones due to steady loading and blade thickness, while the broadband noise is due to boundary layer scattering at the trailing edge. The study also provides a detailed comparison between the tonal and broadband formulations to highlight their similarities and differences. In this paper, we show that the main differences in the behaviour and character of the tonal and broadband spectra and directivities are due to the number of acoustic modes that can be excited. This paper presents a parametric study in which the variation in tonal and broadband noise is investigated as a function of blade tip Mach number ( M t) and blade number ( B) whilst maintaining constant solidity and thrust. This study is repeated for three NACA airfoil profiles. It is found that tonal noise dominates at low blade number and low frequency and/or higher tip speeds, while broadband noise is the major contributor at high-frequencies and at high blade number and low tip speeds. The results show a clear distinction between the combinations of M t and B that are dominated by tonal and by broadband noise. These results are interpreted from fundamental principles relating to modal radiation efficiencies. We confirm this trend of balance between tonal and broadband noise with measured noise at different B and M t. The results of this paper will serve as useful guidelines for preliminary propeller design.

Modeling the acoustic radiation of plates using circular pistons

We propose an equivalent piston model (EPM) for estimating the modal radiation efficiency of complex-shaped plates. In this model, the structure is not discretized into small elementary radiators. Instead, it involves determining the minimum number, size, volume velocity, and position of vibrating pistons that radiate in accordance with the mode shape they emulate, i.e. one piston per lobe within the mode shape, regardless the plate’s shape. For shapes relatively close to rectangles, a specific version of the method is proposed by representing the shape only by two characteristic lengths (EPM-RD), and paving the virtual surface with pistons vibrating at alternate phases along a rectangular grid. The validity of this assumption is evaluated in terms of shape dissymmetry, which can cause residual radiators to appear, and where the dissymmetry can affect the spatial distribution of lobes. With a known spatial distribution mimicking the mode shapes, it is possible to estimate also average values of radiation parameters, such as the average radiation efficiency and time-average mean power. The average radiation efficiency and power follow the Equipartition Theorem and allow the calculation to be performed without cross-terms and using only the modal efficiencies and resonance frequencies. Due to their simplicity and ease of implementation, piston-based equivalent models have great potential for solving radiation problems from flat structures. Any mode shape can be modeled with vibrating pistons and analytical formulations can be easily derived, even for complex geometries. Moreover, with EPM-RD, average values can also be calculated, which provides an estimation of average radiated properties within seconds without requiring any integration or large matrices to be solved.

Radiation efficiency of a distribution of baffled pistons with arbitrary phases.

The radiation resistance and efficiency of a collection of circular pistons, randomly placed on a plane and vibrating with arbitrary phases, are expressed as a combination of the self- and mutual-radiation components. We use the first product or bridge theorem to construct the directivity pattern of this type of arrangement and the radiation properties are calculated according to Bouwkamp's impedance theorem. To illustrate the versatility of our approach, we refer to special cases for symmetric arrangements, for example, to compare with the modal radiation efficiency in structures having "regular" modal patterns.

A geometrical approach to estimate the modal radiation efficiency of complex structures

The radiation efficiency of plates has been the study of many works. The rectangular simply supported plate is the most studied case, followed by the circular one. Analytical formulas and approximations for multiple boundary conditions, baffled and unbaffled cases have been the subject of study for decades. For a structure with a complex shape, numerical methods are appropriated such as FEM and BEM approaches. Semi-analytical based on Rayleigh’s integral also appear in the literature to calculate the radiation, mixed with the surface velocity calculated by FEM; they are easy to implement but they are resource consuming as large matrices are presented. In this work, we present a method for estimating the modal radiation efficiency of simply supported and clamped complex plates, based on the theory of Greenspan for circular radiators and with a geometrical approach.

A comprehensive vibroacoustic investigation of a cross laminated timber floor

In recent years, the use of sustainable construction materials has increased significantly. It is of paramount importance to increase the general knowledge of acoustic behaviour of these materials. This research presents the results of a comprehensive acoustic investigation of a 200 mm timber slab. Excitations using different acoustic sources are performed. The results to this investigation are reported in this paper, focusing on the sound reduction index, impact noise reduction, static and frequency-dependant Young’s modulus, vibration and modal mapping and radiation efficiency. Finally, a correlation between impact noise levels caused by the tapping machine and the rubber ball is provided.

Vibroacoustic response of stiffened thin plates to incident sound

A model was developed and applied for predicting the vibroacoustic response of stiffened plates excited by different incident sound fields. A relationship between the Joint Acceptance Function (JAF) and the modal radiation efficiency is derived. This allows for a direct calculation for both the vibration response and the sound transmission loss (TL) in the presence of a diffuse field whilst accounting for full vibroacoustic coupling. The response for different structural configurations, plane versus diffuse acoustic excitation and modelling approximations are highlighted, so presenting a systematic quantification of the factors influencing the vibroacoustic response. It is found that vibroacoustic coupling is enhanced by the presence of stiffeners, with the effects for different types of acoustic excitation and the contribution from the modal cross terms being most significant for the TL results. A coupled FE-BE numerical model of stiffened plate and other empirical formulae for unstiffened plate are subsequently used for the validation and verification of the proposed model. The results show that the computational cost of simulation can be significantly reduced with a small loss of accuracy when using the proposed model in comparison to the fully coupled FE-BE numerical model.

Radiation modes and sound radiation from a flexible structure mounted in a duct.

The concept of the radiation modes, originally proposed for free-field problems, has found its widespread use in sound radiation analyses of vibrating structures and their active control applications. In this paper, the sound radiation of a flexible structure, flush-mounted inside a duct in both 2D and 3D configurations, is investigated via an energy-based formulation in conjunction with the near-field integration technique. The structural radiation characteristics are first discussed in terms of modal radiation efficiency, which exhibits obvious oscillating behavior with respect to frequencies, in which symmetric patterns are dominant with smooth variations for small acoustic wavenumbers. Then the interior sound radiation modes are investigated. It is shown that, as compared with their free-space counterparts, the lower-order radiation modes in a duct are more sensitive and prone to be affected by the duct starting from its cut-on frequency. Drastic changes in the radiation mode shapes are observed around the cut-on frequency and each of its multiples/harmonics. Finally, analyses are extended to a coupled panel-duct system. It is observed that, contrary to the free-space case, lower-order radiation modes exhibit predominant variations along the duct length direction, suggesting a possible simplification of the 3D configuration into a 2D one.

Prediction of Sound Radiation from Submerged Cylindrical Shell Based on Dominant Modes

A sound radiation calculation method by using dominant modes is proposed to predict the sound radiation from a cylindrical shell. This method can provide an effective way to quickly predict the sound radiation of the structure by using as few displacement monitoring points as possible on the structure surface. In this paper, modal analyses of a submerged cylindrical shell are carried out by taking the vibration mode of a cylindrical shell in a vacuum, as a set of orthogonal bases. The modal sound radiation efficiency and modal contributions to sound radiation power are presented, and comparison results show that a few modes dominantly contribute to the sound radiation power at low frequencies. These modes, called dominantly radiated structural modes in this paper, are applied to predict the sound radiation power of submerged cylindrical shells by obtaining the modal participant coefficients and sound radiation efficiency of these dominant modes. Aside from the orthogonal decomposition method, a method of solving displacement modal superposition equations is proposed to extract the modal participant coefficients, because few modes contribute to the vibration displacement near the resonant frequencies. Some simulations of cylindrical shells with different boundaries are conducted, and the number of measuring points required are examined. Results show that this method, based on dominant modes, can well predict the low-frequency sound radiation power of submerged cylindrical shells. In addition, compared with the boundary element method, this method can better reduce the number of required measuring points significantly. The data of these important modes can be saved, which can help to predict the low-frequency sound radiation of the same structure faster in the future.

Acoustic radiation of rotating and non-rotating finite length cylinders

In this paper, a semi-analytical solution is presented for the problem of modal acoustic radiation efficiency of a finite-length rotating cylinder. The cylinder is immersed in an irrotational inviscid compressible fluid. Assuming the cylinder to vibrate sinusoidally in transverse direction and rotate at constant angular velocity, the scalar equation of sound propagation is decomposed into a coupled set of time independent equations. Using Galerkin method and employing spherical eigen-functions as a complete set of solution, modal acoustic pressure and acoustic radiation efficiency of the cylinder are calculated. Numerical solutions are presented to discuss modal radiation efficiency corresponding to forward and backward circumferential waves.

Increasing Radiation Efficiency Using Antenna Shape Optimization Approach

From the characteristic mode point of view of electrically small antennas, radiation efficiency normally decreases significantly if multiple modes are well excited. Among all these well-excited modes, the one with the lowest modal radiation efficiency determines how much actual radiation efficiency is guaranteed. Based on this idea, this letter proposes a method for improving antenna efficiency using shape optimization approach. The optimized shape can give rise to a high efficiency and an implied high gain after being excited using a single port whose location is easily selected by observing characteristic-mode current distributions.

The poles and minima of radiated acoustic power from plate vibration

The poles and minima of radiated acoustic power of vibrating plate are discussed and a novel formulation for obtaining the poles and minima is presented. The poles of radiated acoustic power are coincident with the poles of vibration response, the minima of radiated acoustic power can be obtained with knowledge of the location of exciting force, modal radiation efficiencies, poles and modal shapes of structure. An example of a simply supported plate is also presented to illustrate the formulation and provides the comparison between the minima of radiated acoustic power and the corresponding antiresonances. In addition, some applications of poles and minima of radiated acoustic power are discussed in this study.

Sound Radiation Characteristics of a Rectangular Duct with Flexible Walls

Acoustic breakout noise is predominant in flexible rectangular ducts. The study of the sound radiated from the thin flexible rectangular duct walls helps in understanding breakout noise. The current paper describes an analytical model, to predict the sound radiation characteristics like total radiated sound power level, modal radiation efficiency, and directivity of the radiated sound from the duct walls. The analytical model is developed based on an equivalent plate model of the rectangular duct. This model has considered the coupled and uncoupled behaviour of both acoustic and structural subsystems. The proposed analytical model results are validated using finite element method (FEM) and boundary element method (BEM). Duct acoustic and structural modes are analysed to understand the sound radiation behaviour of a duct and its equivalence with monopole and dipole sources. The most efficient radiating modes are identified by vibration displacement of the duct walls and for these the radiation efficiencies have been calculated. The calculated modal radiation efficiencies of a duct compared to a simple rectangular plate indicate similar radiation characteristics.

Acoustic transmission properties of pressurised and pre-stressed composite structures

This work was focused on the examination of the effect of the pre-stress, namely tension and pressure, on the wave propagation and acoustic behaviour of composite laminates. The dispersion characteristics of two dimensional layered and sandwich structures were predicted using Wave Finite Element Method (WFEM). The structures were examined in non-stressed and pre-stressed scenarios. After extracting the mass and stiffness matrix of a small periodic segment of the structure using commercially available Finite Elements software, a polynomial eigenvalue problem was formed, the solutions of which consisted of the propagation constants of the waves of the structure. This way the wavenumbers and eigenvectors of the out of plane structural displacements were extracted. These wave propagation magnitudes were then used to calculate important Statistical Energy Analysis (SEA) quantities, such as modal density and radiation efficiency. The effect of pre-stress on these quantities, along with its effect on loss factor of the structure were examined.

Broad-Band Noise Mitigation in Vibrating Annular Plates by Dynamic Absorbers

Steady-state and transient acoustic radiation characteristics of clamped-free annular plate with tuned mass damper (TMD) device is studied. Galerkin’s procedure is employed to obtain the transverse vibration of the annular disk. Based on the Rayleigh integral approach, acoustic pressure radiation is obtained and subsequently the modal sound power and modal radiation efficiency are obtained. A new formulation for the transient acoustic pressure in Laplace domain is presented for the first time in this paper. Durbin’s numerical Laplace transform inversion scheme is employed to obtain the response spectrum. The optimum parameters of vibration absorbers are proposed for suppressing the dynamic vibration and acoustic pressure. A parametric study is carried out and the effects of vibration absorber characteristics are investigated using the analytical procedure. Limiting cases are considered and good agreements with the finite element solution, as well as with those available in the literature, are achieved.

Radiation efficiency of plates subjected to turbulent boundary layer fluctuations.

The modal averaged radiation efficiency of a rectangular plate subjected to turbulent boundary layer fluctuations is discussed in this letter. Numerical results demonstrate that below the hydrodynamic coincidence frequency, the modal averaged radiation efficiency will be larger with the increase of the convection velocity, and an increase of the structural damping will increase the modal averaged radiation efficiency at sub-critical frequencies.

Parametric Study of Sound Radiation Properties of Complex Vibration Patterns in Rectangular Plates using an Analytical Model

Complex vibration modes, which can be characterised by the superposition of standing and travelling bending waves, occur for instance in structures with inhomogeneous damping distribution. The existence of complex structural modes raises the question of their influence on sound radiation properties. In this paper an analytical model of a simply supported rectangular plate is introduced. It allows a quick calculation of complex vibration patterns with different properties, avoiding time consuming and sometimes unstable damped modal analysis in finite-element-software. Using this model it is possible to generate vibration patterns with varying level and spatial distribution of travelling waves as well as different symmetry of amplitude distributions, in order to reveal the important influence factors on sound radiation. Despite the fact that the analytical model generates idealised vibration patterns, the presented consideraton significantly improves phenomenological understanding of acoustic effects in complex vibration modes. The accuracy of the presented model is validated by results of finite element modal analysis and is evaluated in terms of resulting amplitude and phase distributions of the vibrating surface. The results of both suggested approaches are compared using modal radiation efficiencies, spatial distributions of far field acoustic intensity and sound field directivity. Some additional conclusions are made using the analytical model concerning important acoustic metrics of structural vibration patterns. It is shown, that vibration shapes of even order are very sensitive to the presence of travelling waves regarding their radiation capability and character of sound pressure field.

Evaluating radiation efficiency from characteristic currents

This study describes an effective technique for calculating modal radiation efficiency calculation based on decomposition into characteristic modes. The key assumption is that the current distribution on the perfect electric conductor is almost the same as in the case of a very good conductor, for example, metals such as copper, aluminium and silver. This assumption is verified against the conventional technique, the impedance boundary condition (IBC). The proposed approach does not require any modification of the formulation of method of moments for perfectly conducting surfaces, which is assumed for the modal decomposition. Modal efficiencies provide an additional insight that is useful especially for the design of small antennas. Taking the feeding into account, the modal losses can be summed up to obtain the total efficiency. The technique works perfectly for common metals, is fully comparable with the IBC, and can easily be incorporated into any present-day in-house solver. A numerical analysis of three antennas is presented to demonstrate the merits of the approach. Radiation efficiency of coupled dipoles, an electrically small meandered dipole, and PIFA were investigated by the presented method. The results are in perfect agreement with the reference commercial package.

Audio frequency response of glass flat panel distributed mode loudspeakers

The development of glass flat panel distributed mode loudspeakers (DML’s) will enable compelling new applications, such as flat panel displays that double as loudspeakers, or in architectural and automotive applications, window glass that can produce sound or actively cancel environmental noise. In this paper we discuss the frequency response of glass flat-panel distributed mode loudspeakers. The density of panel bending modes, their quality factors (Q’s), and the modal radiation efficiencies determine a DML’s frequency response. For example, a 0.55 mm thick cover glass for a 55 in. television has more than 10 000 bending modes in the audible frequency range. The density of modes increases with frequency and above a threshold frequency, determined by the panel dimensions and the mode Q’s, the mode spacing becomes less than the width of the individual modes, so the overall frequency response approaches the smooth, flat response of an ideal pistonic loudspeaker. However, below this frequency, discrete modes produce prominent peaks in the frequency response. We present simulated and measured glass panel frequency responses, including the effect of a thin layer of trapped air behind a DML panel, and we discuss the need for high internal friction (low Q) glass.

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

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

Optimization of piezoelectric patch positioning for passive sound radiation control of plates

The purpose of this article is the optimization of piezoelectric patch positioning for reducing the radiated sound power of thin plates. To this end, an aluminium plate equipped with a set of piezoelectric patches connected to a passive circuit is considered. The difficulties in designing a smart structure are not only related to the conception of the electric circuit used as controller, but also to the choice of how the circuit itself is coupled with the structure. The selection of the number of transducers to be used, and their positioning, is a crucial step in the designing process. In this work the entire design process of a smart structure is proposed, with the goal of obtaining the best efficiency in terms of reduction of radiated sound power, while using a lightweight optimization procedure in terms of computational costs. To this end, classical instruments of vibrations mechanics are used together with acoustic concepts, such as the modal radiation efficiency. The introduction of this acoustic characterization of the structure in the optimization process is done by using a new utility function, the acoustic controllability, which will be used for the optimization of transducer positioning. The proposed optimization process is applied to the case study of an aluminium plate with nonstandard boundary conditions, and experimental results confirm the validity of this novel procedure.