Vibro-acoustic analysis of complex systems

A general method is presented for predicting the ensemble average steady-state response of complex vibro-acoustic systems that contain subsystems with uncertain, or random, properties. The method combines deterministic and statistical techniques to produce a non-iterative hybrid method that incorporates equations of dynamic equilibrium and power balance. The method is derived explicitly without reference to statistical energy analysis (SEA); however, it is seen that the wave approach to SEA can be viewed as a special case of the proposed method. The proposed method provides a flexible way to account for necessary deterministic details in a vibro-acoustic analysis without requiring that an entire system be modeled deterministically. The method therefore provides a potential solution to the mid-frequency problem (in which a system is neither entirely deterministic nor entirely statistical). The application of the method is illustrated with a numerical validation example.     

A WAVENUMBER APPROACH TO MODELLING THE RESPONSE OF A RANDOMLY EXCITED PANEL, PART I: GENERAL THEORY

Part I of this paper presents a self-contained analytical framework for determining the vibro-acoustic response of a plate to a large class of random excitations. The wavenumber approach is used, which provides an insight into the physical properties of the panel response and enables us to evaluate efficiently the validity of several simplifying assumptions. This formulation is used in Part II for predicting the statistical response of an aircraft panel excited by a turbulent boundary layer. In this paper, we first provide a general statement of the problem and describe how the spectral densities of the panel response can be obtained from an analysis of the system response to a harmonic deterministic excitation and a statistical model for the forcing field. The harmonic response of the system is then expanded as a series of the eigenmodes of the fluid-loaded panel and these fluid-loaded eigenmodes are approximated by a perturbation method. Then, we evaluate the conditions under which this series simplifies into a classical modal formulation in terms of the in vacuo eigenmodes.To illustrate the use of a wavenumber approach, we consider three examples, namely, the vibro-acoustic response of a panel excited by an incidence diffuse acoustic field, by a fully developed turbulent flow and by a pressure field which is spatially uncorrelated from one point to another. Convergence properties of the modal formulations are also examined.     

Virtual sensors for active noise control in acoustic-structural coupled enclosures using structural sensing: part II--Optimization of structural sensor placement

The work proposed an optimization approach for structural sensor placement to improve the performance of vibro-acoustic virtual sensor for active noise control applications. The vibro-acoustic virtual sensor was designed to estimate the interior sound pressure of an acoustic-structural coupled enclosure using structural sensors. A spectral-spatial performance metric was proposed, which was used to quantify the averaged structural sensor output energy of a vibro-acoustic system excited by a spatially varying point source. It was shown that (i) the overall virtual sensing error energy was contributed additively by the modal virtual sensing error and the measurement noise energy; (ii) each of the modal virtual sensing error system was contributed by both the modal observability levels for the structural sensing and the target acoustic virtual sensing; and further (iii) the strength of each modal observability level was influenced by the modal coupling and resonance frequencies of the associated uncoupled structural/cavity modes. An optimal design of structural sensor placement was proposed to achieve sufficiently high modal observability levels for certain important panel- and cavity-controlled modes. Numerical analysis on a panel-cavity system demonstrated the importance of structural sensor placement on virtual sensing and active noise control performance, particularly for cavity-controlled modes.     

Dynamic and acoustic response of bidirectionally stiffened plates with eccentric stiffeners subject to airborne and structure-borne excitations

An analytical method based on the modal expansion technique was developed to predict the vibro-acoustic response of both unidirectionally and bidirectionally stiffened flat panel. This paper presents the response to diffuse acoustic field (DAF) and turbulent boundary layer (TBL) excitations in terms of their joint acceptance. Numerical results for the dynamic and acoustic responses are compared with finite element method (FEM) and boundary element (BEM) results for stiffened panel with complex and eccentrically shaped stiffeners subject to point force excitation. A theoretical prediction of the transmission loss (TL) is also compared with laboratory measurements conducted on flat panels representing aircraft models as well as with hybrid statistical energy analysis (SEA)-FEM periodic model. The results confirm that the stiffened panel has the same acoustic response as the skin without stiffeners at frequencies where the structural wavelengths are equal to the spacing between the stiffeners. In addition, the transmission loss is lowered by the presence of the stiffeners at some particular region of frequencies below the critical frequency with respect to the unstiffened panel.     

Vibro-acoustic behaviors of an elastically restrained double-panel structure with an acoustic cavity of arbitrary boundary impedance

An analytical study on the vibro-acoustic behaviors of a double-panel structure with an acoustic cavity is presented. Unlike the existing studies, a structural–acoustic coupling model of an elastically restrained double-panel structure with an acoustic cavity having arbitrary impedance on sidewalls around the cavity is developed in which the two dimensional (2D) and three dimensional (3D) modified Fourier series are used to represent the displacement of the panels and the sound pressure inside the cavity, respectively. The unknown expansions coefficients are treated as the generalized coordinates and the Rayleigh–Ritz method is employed to determine displacement and sound pressure solutions based on the energy expressions for the coupled structural–acoustic system. The effectiveness and accuracy of the present model is validated by numerical example and comparison with finite element method (FEM) and existing analytical method, with good agreement achieved. The influence of key parameters on the vibro-acoustic behaviors and sound transmission of the double-panel structure is investigated, including: cavity thickness, boundary conditions, sidewall impedance, and the acoustic medium in the cavity.     

Energy transmission in a mechanically-linked double-wall structure coupled to an acoustic enclosure

The energy transmission in a mechanically linked double-wall structure into an acoustic enclosure is studied in this paper. Based on a fully coupled vibro-acoustic formulation, focus is put on investigating the effect of the air gap and mechanical links between the two panels on the energy transmission and noise insulation properties of such structures. An approximate formula reflecting the gap effect on the lower-order coupled frequencies of the system is proposed. A criterion, based on the ratio between the aerostatic stiffness of the gap cavity and the stiffness of the link, is proposed to predict the dominant transmitting path, with a view to provide guidelines for the design of appropriate control strategies. Numerical results reveal the existence of three distinct zones, within which energy transmission takes place following different mechanisms and transmitting paths. Corresponding effects on noise insulation properties of the double-wall structure are also investigated.     

Vibro-acoustic behaviour of laminated double glazing enclosing a viscothermal fluid cavity

This paper presents a new numerical model to investigate the vibro-acoustic behaviour of two laminated glass plates enclosing a thin viscothermal fluid cavity. The aim of this work is to develop an original five layer (two skins plies, two adhesive films and a core ply) laminated plate finite element by mixing Kirchhoff and Mindlin plate’s theory. The formulation is based on the theory that accounts for the transverse shear in the adhesive films and in the core. The acousto-elastic model is established in dimensionless appropriate form including the effects of viscosity and thermal conductivity of fluid and by taking into account the fluid-structure interaction. The discretization of the energy functional by finite element method gives after minimisation a symmetrical coupled matrix system in which the acoustic matrices are frequency dependent. Therefore, an iterative procedure is derived to determine the eigenmodes of the coupled system. The modal approach is adopted to determine the vibro-acoustic system’s response. Then, the validation of the new laminate finite element model is achieved by comparing the sandwich plate results against data obtained from literature. Subsequently, predicted responses, such as the vibration transmissibility and the transmission loss of the coupled system, for a given laminated double glazing under an imposed homogeneous pressure are presented and discussed. Numerical results show the importance of both lamination and viscothermal fluid effects on double glazing vibro-acoustic behaviour.     

Vibro-acoustic analysis of a rectangular cavity bounded by a flexible panel with elastically restrained edges

A coupled system consisting of an acoustic cavity and an elastic panel is a classical problem in structural acoustics and is typically analyzed using modal approaches based on in vacuo structural modes and the rigidly walled acoustic modes which are pre-determined based on separate component models. Such modeling techniques, however, tend to suffer the following drawbacks or limitations: (a) a panel is only subjected to ideal boundary conditions such as the simply supported, (b) the coupling between the cavity and panel is considered weak, and (c) the particle velocity cannot be correctly predicted from the pressure gradient on the contacting interface, to name a few. Motivated by removing these restrictions, this paper presents a general method for the vibro-acoustic analysis of a three-dimensional (3D) acoustic cavity bounded by a flexible panel with general elastically restrained boundary conditions. The displacement of the plate and the sound pressure in the cavity are constructed in the forms of standard two-dimensional and 3D Fourier cosine series supplemented by several terms introduced to ensure and accelerate the convergence of the series expansions. The unknown expansions coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz procedure based on the energy expressions for the coupled structural acoustic system. The accuracy and effectiveness of the proposed method are demonstrated through numerical examples and comparisons with the results available in the literature.     

Estimating ensemble average power delivered by a piezoelectric patch actuator to a non-deterministic subsystem

Engineering systems such as aircraft, ships and automotive are considered built-up structures. Dynamically they are taught of as being fabricated from many components that are classified as ‘deterministic subsystems’ (DS) and ‘non-deterministic subsystems’ (Non-DS). Structures' response of the DS is deterministic in nature and analysed using deterministic modelling methods such as finite element (FE) method. The response of Non-DS is statistical in nature and estimated using statistical modelling technique such as statistical energy analysis (SEA). SEA method uses power balance equation, in which any external input to the subsystem must be represented in terms of power. Often, input force is taken as point force and ensemble average power delivered by point force is already well-established. However, the external input can also be applied in the form of moments exerted by a piezoelectric (PZT) patch actuator. In order to be able to apply SEA method for input moments, a mathematical representation for moment generated by PZT patch in the form of average power is needed, which is attempted in this paper. A simply-supported plate with attached PZT patch is taken as a benchmark model. Analytical solution to estimate average power is derived using mobility approach. Ensemble average of power given by the PZT patch actuator to the benchmark model when subjected to structural uncertainties is also simulated using Lagrangian method and FEA software. The analytical estimation is compared with the Lagrangian model and FE method for validation. The effects of size and location of the PZT actuators on the power delivered to the plate are later investigated.     

Effectiveness of T-shaped acoustic resonators in low-frequency sound transmission control of a finite double-panel partition

Double-panel partitions are widely used for sound insulation purposes. Their insulation efficiency is, however, deteriorated at low frequencies due to the structural and acoustic resonances. To tackle this problem, this paper proposes the use of long T-shaped acoustic resonators in a double-panel partition embedded along the edges. In order to facilitate the design and assess the performance of the structure, a general vibro-acoustic model, characterizing the interaction between the panels, air cavity, and integrated acoustic resonators, is developed. The effectiveness of the technique as well as the optimal locations of the acoustic resonators is examined at various frequencies where the system exhibits different coupling characteristics. The measured optimal locations are also compared with the predicted ones to verify the developed theory. Finally, the performance of the acoustic resonators in broadband sound transmission control is demonstrated.     

Vibro-acoustic modelling of a railway bridge crossed by a train

This paper presents a vibro-acoustic modelling of a railway bridge excited by a moving train. The modelling of the bridge-train system is carried out by the modal superposition method taking into account the train mass, the viscoelastic suspension of the vehicles with an unlimited number of trucks and spans of the bridge. The numerical resolution of the coupled equations of motion is carried out by the Newmark’s method with an iterative process. we studied on one hand the influence of the track irregularities on the dynamic behaviour of the bridge-train system on the other hand the noise radiated by the bridge due to the passage of the train. The acoustic pressure is obtained by solving the wave equation which has as excitations source the bridge acceleration, they are considered as acoustic monopoles.     

基于声压场描述的扩散声场多自由度互易原理研究

应用于复杂结构中频声振分析的扩散场多自由度互易原理采用位移变量描述系统,实质为弹性波场互易原理,应用于声波场时会造成模型自由度数不必要的增加。建立基于声压描述的扩散声场受挡模型,利用声辐射模态描述扩散声场中结构的表面受挡声压;据此提出基于声压描述的扩散声场多自由度互易原理,发现扩散声场中结构表面受挡声压的互谱矩阵与该结构在自由空间中振动辐射声波的声阻矩阵成正比。该互易原理与传统的单自由度互易原理表达形式相似,但适用于任意自由度结构。该互易原理可用于扩散声场中复杂结构的表面受挡声压的自谱及相关分析,仿真研究表明当边界元网格尺寸小于声波波长的1/6(线性单元)或1/3(二次单元)时,数值解与理论解完全吻合。      

Noise control of an acoustic enclosure covered by a double-wall structure with shallow gap: Design of resonator-like cavities

The present study addresses the possibility of using resonator-like cavities for noise attenuation of an acoustic enclosure covered by a double-wall structure with shallow gap. Different from the conventional design of using Helmholtz resonators, a set of coupled equation describing resonator-like cavity is integrated into the vibro-acoustic model, under which the modes in a broad frequency range will be controlled. Based on the tuned weighting coefficient and the acoustic potential energy, an objective function is developed to optimize parameters (number and location) of resonator-like cavities. The effect of shallow gap on the first two dominant modes of the coupled system and energy transmission is investigated. Simulation results indicate the effectiveness of the proposed method for noise attenuation, which might be of direct benefit to engineering applications.     

Vibro-acoustic analysis of a rectangular-like cavity with a tilted wall

In this paper, a fully coupled vibro-acoustic model is developed to characterize the structural and acoustic coupling of a flexible panel backed by a rectangular-like cavity with a slight geometrical distortion, which is introduced through a tilted wall. The combined integro-modal approach is used to handle the acoustic pressure inside the irregular-shaped cavity. Based on the model proposed, the distortion effect on the vibro-acoustic behavior of the coupled system is investigated using the averaged sound pressure level inside the enclosure and the averaged quadratic velocity of the vibrating plate. Simulations are conducted to examine the distortion effect on acoustic natural frequencies, acoustic pressures and structural responses. Effects of the wall inclination on coupling coefficients are also assessed, and an index is proposed to quantify the degree of variation of coupling strength.     

ANALYSIS OF SOUND TRANSMISSION THROUGH PERIODICALLY STIFFENED PANELS BY SPACE-HARMONIC EXPANSION METHOD

An analysis method is developed to study sound transmission characteristics of a thin plate stiffened by equally spaced line stiffeners. The dynamic equation that describes the vibro-acoustic response of the system is derived by expanding the structural and acoustic responses in terms of the space harmonics and by using the virtual energy method. The series solution can be considered as the exact solution because the structural and acoustic-structural coupling effects in the system are fully considered and the solution converges. Parameter studies are conducted for major design parameters to understand the characteristics of the system.     

Passive and active interior noise control of box structures using the structural intensity method

This paper presents passive and active vibro-acoustic noise control methods for attenuating the interior noise level in box structures which can be an analogy of cabins of vehicle and aircraft. The structural intensity (SI) approach is adopted to identify the predominant vibration panels and interior noise sources for box structures. In the study, the finite element method is used to determine the structural vibration and structural intensity in the box surfaces. According to structural intensity vectors plot and structural intensity stream lines presentation, the possible effective control positions where the dampers may be attached and the active control forces may act to reduce vibration and interior noise, are identified. From the study, it can be demonstrated that the structural intensity approach and stream line presentation are possible methods for identifying the vibro-acoustic interior noise source and predominant panels which may be modified to reduce the interior noise level. The structural intensity methodology, passive and active noise control results can be extended to the further study of the vibration and interior noise control of actual cabins of vehicles and aircraft.     

Boundary element dynamical energy analysis: A versatile method for solving two or three dimensional wave problems in the high frequency limit

Dynamical energy analysis was recently introduced as a new method for determining the distribution of mechanical and acoustic wave energy in complex built up structures. The technique interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. As such the applicability of the method is wide ranging and additionally includes the numerical modelling of problems in optics and more generally of linear wave problems in electromagnetics. In this work we consider a new approach to the method with enhanced versatility, enabling three-dimensional problems to be handled in a straightforward manner. The main challenge is the high dimensionality of the problem: we determine the wave energy density both as a function of the spatial coordinate and momentum (or direction) space. The momentum variables are expressed in separable (polar) coordinates facilitating the use of products of univariate basis expansions. However this is not the case for the spatial argument and so we propose to make use of automated mesh generating routines to both localise the approximation, allowing quadrature costs to be kept moderate, and give versatility in the code for different geometric configurations.     

Thickness optimization of a multilayered structure on the coupling surface between a structure and an acoustic cavity

This paper describes a new design method to optimize thickness distribution of a multilayered structure which is located on the coupling surface between a structure and an acoustic cavity. The design method is based on the concept of the density approach in topology optimization incorporating a transfer matrix for a multilayered structure that includes a poroelastic media layer. The one-dimensional transfer matrix adopted here is an approximate representation addressing vibro-acoustic effects inherent in a multilayered structure, and balances calculation resources and desired accuracy. Applying the transfer matrix representation as boundary conditions on the coupling surface between a structure and an acoustic cavity, the modified equilibrium equation of the vibro-acoustic system is derived which is approximately but efficiently solved by the modal approach. In this study, the problem of minimizing the acoustic pressure within the cavity over the prescribed frequency range is formulated under the volume constraint of the poroelastic media layer. The continuous approximation of thickness distribution is assumed, and the thickness of the poroelastic media layer at each nodal point is chosen as design variables. Numerical results show that an acoustic response is significantly reduced by the optimal thickness distribution having a total weight equal to or less than that in the initial uniform thickness. These demonstrate that the proposed method is effective to design the optimal thickness distribution of a multilayered structure.     

Random focusing of sound into spatially coherent regions

Abstract

Oceanographic variability creates a weak random propagation medium for acoustic waves. The impact on acoustic transmission is becoming increasingly appreciated as the deterministic modelling of sound propagation in the ocean has become tractable and better understood. Beyond the near field (where phase fluctuations are weak) and the far field (where the scintillation index becomes saturated) multiple-scattering theory predicts that random focusing will greatly enhance the acoustic energy density over small volumetric regions, which we call ‘ribbons’. In 1986 an experiment was carried out in the eastern Mediterranean to test this prediction using acoustic propagation along distinct, resolvable ray paths. This experiment is one of the few to map the spatial structure of acoustic intensity with such a large vertical aperture, and as far as the authors are aware it remains the only experiment to attempt to detect the two-dimensional structure of the predicted focused ribbons for individual energy paths. Renewed impetus to publish the results has been provided by the recent focus on moderate- to high-frequency acoustics in near-shore and shallow-water environments. The experiment is described and high-intensity regions consistent with the theoretical predictions are reported. A 3.5 kHz pulsed signal was transmitted over ranges of 11–23 km and sampled over a vertical aperture of 250–350 m and horizontal apertures of 4–4.5 km. The acoustic signals travelling along individual ray paths developed randomly focused regions of 6–18 dB over regions with a vertical dimension of about 20 m and whose horizontal length could possibly be up to 1 km. The understanding of these features allows system limitations to be estimated quantitatively and opens up the way to their constructive tactical use. The results are applicable to many systems from towed array sonars to high-frequency bathymetric sidescan and minehunting.