Non resonant transmission modelling with statistical modal energy distribution analysis

Statistical modal Energy distribution Analysis (SmEdA) can be used as an alternative to Statistical Energy Analysis for describing subsystems with low modal overlap. In its original form, SmEdA predicts the power flow exchanged between the resonant modes of different subsystems. In the case of sound transmission through a thin structure, it is well-known that the non resonant response of the structure plays a significant role in transmission below the critical frequency. In this paper, we present an extension of SmEdA that takes into account the contributions of the non resonant modes of a thin structure. The dual modal formulation (DMF) is used to describe the behaviour of two acoustic cavities separated by a thin structure, with prior knowledge of the modal basis of each subsystem. Condensation in the DMF equations is achieved on the amplitudes of the non resonant modes and a new coupling scheme between the resonant modes of the three subsystems is obtained after several simplifications. We show that the contribution of the non resonant panel mode results in coupling the cavity modes of stiffness type, characterised by the mode shapes of both the cavities and the structure. Comparisons with reference results demonstrate that the present approach can take into account the non resonant contributions of the structure in the evaluation of the transmission loss.     

双层板腔结构声传输及其有源控制研究

利用子系统模态综合方法,结合阻抗-导纳矩阵法,建立了双层板腔结构向自由空间声传输及其在入射板PZT控制、辐射板PZT控制,和腔中次级声源作动等多种控制策略下,系统物理模型的统一的分析模型,导出了系统模态响应及最优次级源强度的统一的阻抗-导纳矩阵表达式。该模型表达式各部分物理意义清晰、明确,便于进行系统耦合理论、有源控制及其机理的分析和数值研究。然后,在此基础上对双层板腔结构声传输有源控制进行了全面深入的数值计算和分析研究,重点探讨了控制方法策略及系统参数对有源控制效果的影响及其对应的控制机理。结果表明:入射板PZT作动辐射声功率最小控制策略是通过入射板、声腔和辐射板三个子系统的模态抑制或重组达到消声的目的,涉及多种复杂控制机理,对入射板、辐射板和声腔模态均有效,但对入射板模态更有效;在低频段声腔(0,0,0)模态在系统耦合响应中起主导作用,因此利用腔中次级声源作动能获得较理想的控制效果,是一种较好的控制策略;由于声腔模态与结构模态间复杂的耦合关系,使得某些频率处腔中声势能一定程度上的降低并不一定导致系统声传输损失的增加,因此,腔中声势能最小控制策略不一定能够获得理想的声传输控制效果。      

Extension of SEA model to subsystems with non-uniform modal energy distribution

In order to widen the application of statistical energy analysis (SEA), a reformulation is proposed. Contrary to classical SEA, the model described here, statistical modal energy distribution analysis (SmEdA), does not assume equipartition of modal energies.Theoretical derivations are based on dual modal formulation described in Maxit and Guyader (Journal of Sound and Vibration 239 (2001) 907) and Maxit (Ph.D. Thesis, Institut National des Sciences Appliquées de Lyon, France 2000) for the general case of coupled continuous elastic systems. Basic SEA relations describing the power flow exchanged between two oscillators are used to obtain modal energy equations. They permit modal energies of coupled subsystems to be determined from the knowledge of modes of uncoupled subsystems. The link between SEA and SmEdA is established and make it possible to mix the two approaches: SmEdA for subsystems where equipartition is not verified and SEA for other subsystems.Three typical configurations of structural couplings are described for which SmEdA improves energy prediction compared to SEA: (a) coupling of subsystems with low modal overlap, (b) coupling of heterogeneous subsystems, and (c) case of localized excitations.The application of the proposed method is not limited to theoretical structures, but could easily be applied to complex structures by using a finite element method (FEM). In this case, FEM are used to calculate the modes of each uncoupled subsystems; these data are then used in a second step to determine the modal coupling factors necessary for SmEdA to model the coupling.     

Statistical energy analysis, energy distribution models and system modes

Expressions for the energy influence coefficients of a built-up structure are found in terms of the modes of the whole structure. These coefficients relate the time and frequency average energies of the subsystems to the subsystem input powers. Rain-on-the-roof excitation over a frequency band Ω is assumed. It is then seen that the system can be described by an SEA model only if a particular condition involving the mode shapes of the system is satisfied. Broadly, the condition holds if the mode shapes of the modes in the frequency band of excitation are, on average, typical enough of all the modes of the system in terms of the distribution of energy throughout the system. If this condition is satisfied then the system can be modelled using an “quasi-SEA” approach, irrespective of the level of damping or of the strength of coupling. However, the resulting model need not be of a proper SEA form, and in particular the indirect coupling loss factors may not be negligible.     

Validation of a medium-frequency computational method for the coupling between a plate and a water-filled cavity

The aim of this paper is the validation of a computation by a numerical method, normally adapted to the medium-frequency (MF) domain, of the strong coupling between an elastic structure (plate) and a cavity entirely filled with an internal acoustic viscous dense fluid (water). The method of computation does not lie in a modal approach (no need to extract the modal basis of the system) but it directly computes the frequency response of the system using a specific algorithm called the “Onera-MF method”.This method was developed to accurately calculate the response of complex systems in an MF broad band at a lower cost than standard modal method or direct step-by-step frequency method. The method can also be used for the low-frequency (LF) domain where modal densities of systems are low. The computation of the vibroacoustic response of the system lies in a finite element modelling of the overall system (structure and fluid) in which the coupling between the structure and the fluid (light or heavy) is directly taken into account within the formulation of the finite elements.A simple and well-known experimental case was chosen for validation: a parallelepipedic cavity entirely filled with water contained in a box defined by five rigid faces and closed at its end by an elastic clamped homogeneous plate.The validation is based on a comparison between measurement, the numerical computation and an analytical approach in the frequency band . Both the vibratory response of the plate and the acoustic pressure were compared. This study shows that the MF-method of computation used herein for this case also works in a modal domain.     

The effects of design modifications on the apparent coupling loss factors in SEA-like analysis

At high frequencies it is often desirable to describe the behaviour of a structure in terms of subsystem energies. The most important method used for high frequency analysis is statistical energy analysis (SEA). Recently, the frequency range in which finite element analysis is applied is being extended to higher frequencies resulting in SEA-like analysis. Methods such as energy distribution modelling can be used to obtain the matrix of energy influence coefficients (EICs); the EIC matrix can be inverted to estimate SEA-like “apparent” coupling loss factors (ACLFs). The ACLFs so estimated depend on details of global modal properties, especially at low and moderate modal overlap. This has implications for design modifications, for example by adding damping treatment to one subsystem, since generally all the EICs change and hence so do all the ACLFs. In principle a full re-analysis is required; this is in contrast to classical SEA. This paper describes these problems and their causes and approximations to the SEA-like parameters of the modified system are proposed. Estimates of the response of the structure after modifications can be found without full re-analysis, leading to a computationally efficient method. The case studies show good agreement between the estimates based on the proposed approaches and the ones based on full re-analysis. The net outcome is that the ACLFs can be estimated after the modification has been made in a manner similar to conventional SEA.     

矩形封闭声腔壁板轻量化对声固耦合特性的影响

基于模态耦合法建立了考虑对侧柔性壁板的矩形封闭腔体声固耦合模型,导出了耦合系统声振响应随柔性板材料属性及厚度变化的近似关系.通过数值仿真计算,详细分析了单柔性板-单侧激励、双柔性板-双侧激励及双柔性板-单侧激励三种情况下板的轻量化设计对系统声固耦合特性及腔内声压响应的影响.结果表明:对于单柔性板-单侧激励及双柔性板-双侧激励情况,低频段腔内声压主要取决于板厚,其次是杨氏模量,而与板密度关系较弱.因此使用适当增厚的轻质板可以同时达到减重和低频区降噪的目标.对于双柔性板-单侧激励情况,适当的轻量化设计可以显著增强两板间低频段振动的耦合,从而降低该频段的声腔模态响应及腔内噪音.在中高频段,增加板厚和材料密度对降低三种情况的腔内声压均有利。      

自由阻尼复合板的模态密度研究

模态密度是统计能量分析（SEA）的一个重要参数,尽管有关阻尼复合板振动特性的文献很多,便至今为止,研究其模态密度及变化规律的论文尚未见到,为此本文利用弹性最小势能原理和变分法,并考虑振动阻尼的影响,导出了自由阻尼复合板的弯曲振动模态密度计算公式,系统地分析了模态密度随阻尼层厚度、温度和频率而变化的规律。     

A hybrid SEA/modal technique for modeling structural-acoustic interior noise in rotorcraft

This paper describes a hybrid technique that combines Statistical Energy Analysis (SEA) predictions for structural vibration with acoustic modal summation techniques to predict interior noise levels in rotorcraft. The method was applied for predicting the sound field inside a mock-up of the interior panel system of the Sikorsky S-92 helicopter. The vibration amplitudes of the frame and panel systems were predicted using a detailed SEA model and these were used as inputs to the model of the interior acoustic space. The spatial distribution of the vibration field on individual panels, and their coupling to the acoustic space were modeled using stochastic techniques. Leakage and nonresonant transmission components were accounted for using space-averaged values obtained from a SEA model of the complete structural-acoustic system. Since the cabin geometry was quite simple, the modeling of the interior acoustic space was performed using a standard modal summation technique. Sound pressure levels predicted by this approach at specific microphone locations were compared with measured data. Agreement within 3 dB in one-third octave bands above 40 Hz was observed. A large discrepancy in the one-third octave band in which the first acoustic mode is resonant (31.5 Hz) was observed. Reasons for such a discrepancy are discussed in the paper. The developed technique provides a method for modeling helicopter cabin interior noise in the frequency mid-range where neither FEA nor SEA is individually effective or accurate.     

基于高级统计能量分析的周期加筋板振动特性研究

统计能量分析(statistical energy analysis, SEA)是复杂耦合系统中、高频动力学特性计算的有力工具. 本文以波传播理论和SEA的基本原理为基础, 研究周期加筋板中弯曲波传播特性. 分析了周期结构的频率带隙特性和加强筋对板上弯曲波的滤波特性对SEA计算结果的影响规律, 发现经典SEA由于忽视了加筋板中物理上不相邻子系统间存在的能量隧穿效应, 而导致响应预测结果产生最高近 40 dB的误差. 为了解决这一问题, 本文应用高级统计能量分析(advanced statistical energy analysis, ASEA)方法, 考虑能量在不相邻子系统间的传递、转移和转化的物理过程, 从而大幅提高子系统响应的预测精度, 将误差在大部分频段降低至小于5 dB. 设计了模拟简支边界条件的加筋板振动测试实验装置, 实验测试结果与有限元结果符合较好, 对理论模型进行了验证.     

Response variance prediction in the statistical energy analysis of built-up systems

In the statistical energy analysis (SEA) of high frequency noise and vibration, a complex engineering structure is represented as an assembly of subsystems. The response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem, and these energies are found by employing the principle of power balance. Strictly the computed energy is an average taken over an ensemble of random structures, and for many years there has been interest in extending the SEA prediction to the variance of the energy. A variance prediction method for a general built-up structure is presented here. Closed form expressions for the variance are obtained in terms of the standard SEA parameters and an additional set of parameters alpha(k) that describe the nature of the power input to each subsystem k, and alpha(ks) that describe the nature of the coupling between subsystems k and s. The theory is validated by comparison with Monte Carlo simulations of plate networks and structural-acoustic systems.     

简支矩形板模态辐射抗的一种快速计算方法分析

在模态辐射阻快速计算方法的基础上,进一步推导出简支矩形板模态辐射抗快速计算方法的相关计算公式,给出了相应的计算步骤,并与传统模态辐射阻抗的数值计算方法进行了详细的对比研究。与以往的二重积分方法相比,该方法为一重数值积分,因而其计算速度远高于以往方法。而且,与以往方法不同,该方法的计算速度不随模态数的增加而迅速减小。该方法可在结构与重流体的声振耦合问题上得到广泛的应用。      

Frequency average loss factors of plates and shells

This paper describes a thorough investigation of the measurement of frequency band average loss factors of structural components for use in the statistical energy analysis (SEA) method of computation of vibration levels. The “traditional” method of measurement is to excite the structure by a random force having a flat spectral density in the frequency band of interest. The force is then cut off and the decay of the modes excited in the band is noted. The average loss factor is deduced from the decay curve. The alternative energy method is the subject of this study. In this test the power input from the band limited random force is measured and the spatial average vibration level of the structure is estimated from several surface accelerometers. It is shown that when the modes in the band have similar loss factors (as is usually the case) the energy method gives a result which is very close to that obtained from the decay method. These in turn are close to the arithmetic average of the loss factors of the individual modes in the band. It is shown that only when the band contains one or two very lightly damped modes amongst several more heavily damped modes is there a difference between the two methods. In this case it is better to use the energy result in the SEA calculations.     

Extension of the statistical modal energy distribution analysis for estimating energy density in coupled subsystems

The present article deals with an extension of the Statistical modal Energy distribution Analysis (SmEdA) method to estimate kinetic and potential energy density in coupled subsystems. The SmEdA method uses the modal bases of uncoupled subsystems and focuses on the modal energies rather than the global energies of subsystems such as SEA (Statistical Energy Analysis). This method permits extending SEA to subsystems with low modal overlap or to localized excitations as it does not assume the existence of modal energy equipartition. We demonstrate that by using the modal energies of subsystems computed by SmEdA, it is possible to estimate energy distribution in subsystems. This approach has the same advantages of standard SEA, as it uses very short calculations to analyze damping effects. The estimation of energy distribution from SmEdA is applied to an academic case and an industrial example.     

Analytical research on structural-acoustic coupling of a cavity surrounded by flexible panel

The structural-acoustic coupling characteristics, mechanisms, effect of structural-acoustic coupling on natural mode and natural frequencies of the system are analyzed theoretically and numerically. Formulae for the natural frequencies of the coupled system are derived. Some new conclusions are obtained. Analytical results demonstrate that the strongly coupled system indicates obvious closed-loop feedback characteristics, whereas the weakly coupled system indicates obvious feedforward characteristics, and it is because of the presence of the feedback loop that the natural characteristics and natural frequencies are changed. Cluster coupling characteristic between the structural and acoustic modes for the regular cavity and panel system is found, which determines the coupling interaction between the flexible panel and cavity. Any mode in one mode cluster only interferes the modes and the modal natural frequencies in the same cluster independently. The modal cluster coupling changes not only the natural frequencies of the system but also the modal order and structural mode shape.     

Modeling and Measurement of a Tunable Acoustoelastic System

Acoustoelastic coupling occurs when a hollow structure’s in-vacuo mode aligns with an acoustic mode of the internal cavity. The impact of this coupling on the total dynamic response of the structure can be quite severe depending on the similarity of the modal frequencies and shapes. Typically, acoustoelastic coupling is not a design feature, but rather an unintended result that must be remedied as modal tests of structures are often used to correlate or validate finite element models of the uncoupled structure. Here, however, a test structure is intentionally designed such that multiple structural and acoustic modes are well-aligned, resulting in a coupled system that allows for an experimental investigation. First, coupling in the system is identified using a measure termed the magnification factor. Next, the structural-acoustic interaction is measured. Modifications to the system demonstrate the dependency of the coupling on changes in the mode shape and frequency proximity. This includes an investigation of several practical techniques used to decouple the system by altering the internal acoustic cavity, as well as the structure itself. These results show that acoustic absorption material effectively decoupled the structure while structural modifications, in their current form, proved unsuccessful. Readily available acoustic absorptive material was effective in reducing the coupled effects while presumably adding negligible mass or stiffness to the structure.     

The derivation and solution of the modal flux equations for backscattering in a duct in the presence of random space and time fluctuations

Abstract

In this paper we first derive the equations governing the energy fluxes propagating in each of the modes of a duct. In each mode there is a forward and backward component and the equations are intended to treat ducts in which backscattering plays a major role. The modal fluxes are coupled since there is transfer of energy between the modes that occurs as a result of random time and space sound-speed fluctuations in the medium in the duct. Since the fluctuations are both space and time dependent the governing equations are radiation transport equations. This is not the case if the fluctuations depend only on space. The basic method is to develop a coupled set of equations for the energy spectra in the modes and then to integrate over the frequency to obtain the fluxes. In the second section of this paper the modal flux equations are solved. A numerical result is presented to show how energy is transferred between modes. It is also shown how the reflected energy varies as a function of duct length.     

Deterministic-statistical analysis of a structural-acoustic system

The purpose of this paper is to develop an efficient approach for vibro-acoustic analysis. Being simple and representative, an exited plate-acoustic system is selected as a validation case for the vibro-acoustic analysis as the system presents one two-dimensional statistical component (modal dense structure panel—plate) connected to the other component (deterministic acoustic volume—cavity) through the area junction over a surface domain, rather than at a line boundary. Potential industrial applications of the system vibro-acoustic analysis would be in acoustic modelling of vehicle body panels such as the cabin roof panel, and door panels for the boom noise analysis.A new deterministic-statistical analysis approach is proposed from a combination or hybrid of deterministic analysis and statistical energy analysis (SEA) approaches. General theory of the new deterministic-statistical analysis approach is introduced. The main advantage of the new deterministic-statistical analysis approach is its possibility in place of the time consuming Monte Carlo simulation. In order to illustrate and validate the new deterministic-statistical analysis approach, three approaches of the deterministic analysis, the statistical energy analysis and the new deterministic-statistical analysis are then applied to conduct the plate-acoustic system modelling, and their results will be compared. The vibro-acoustic energy coupling characteristic of the plate-acoustic system will be studied. The most suitable frequency range for the new approach will be identified in consideration of computational accuracy, information and speed.     

Reduced models for the medium-frequency dynamics of stochastic systems

In this paper, a frequency domain vibration analysis procedure of a randomly parametered structural system is described for the medium-frequency range. In this frequency range, both traditional modal analysis and statistical energy analysis (SEA) procedures well-suited for low- and high-frequency vibration analysis respectively, lead to computational and conceptual difficulties. The uncertainty in the structural system can be attributed to various reasons such as the coupling of the primary structure with a variety of secondary systems for which conventional modeling is not practical. The methodology presented in the paper consists of coupling probabilistic reduction methods with dynamical reduction methods. In particular, the Karhunen-Loeve and Polynomial Chaos decompositions of stochastic processes are coupled with an operator decomposition scheme based on the spectrum of an energy operator adapted to the frequency band of interest.     

Plane wave propagation and frequency band gaps in periodic plates and cylindrical shells with and without heavy fluid loading

Free plane wave propagation in infinitely long periodic elastic structures with and without heavy fluid loading is considered. The structures comprise continuous elements of two different types connected in an alternating sequence. In the absence of fluid loading, an exact solution which describes wave motion in each unboundedly extended element is obtained analytically as a superposition of all propagating and evanescent waves, continuity conditions at the interfaces between elements are formulated and standard Floquet theory is applied to set up a characteristic determinant. An efficient algorithm to compute Bloch parameters (propagation constants) as a function of the excitation frequency is suggested and the location of band gaps is studied as a function of non-dimensional parameters of the structure's composition. In the case of heavy fluid loading, an infinitely large number of propagating or evanescent waves exist in each unboundedly extended elasto-acoustic element of a periodic structure. Wave motion in each element is then presented in the form of a modal decomposition with a finite number of terms retained in these expansions and the accuracy of such an approximation is assessed. A generalized algorithm is used to compute Bloch parameters for a periodic structure with heavy fluid loading as a function of the excitation frequency and, similarly to the previous case, the location of band gaps is studied.