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.     

Sound transmission using statistical energy analysis

Statistical energy analysis is used to predict the sound transmission loss, the radiation resistance and the vibration amplitude of a partition. Agreement between theory and experiment is shown to be good. The “mass-law” sound transmission is seen to be due to non-resonant modal vibration while the increased transmission in the coincidence region is seen to be due to resonant modal vibration. The observed vibration amplitude is also shown to be due to resonant modes. The previously observed discrepancy between the values of vibration amplitude derived from the mass law and those observed experimentally which has been described in the literature [1 ] is thus satisfactorily explained.     

Finding the dominant energy transmission paths in statistical energy analysis

A key issue for noise, vibration and harshness purposes, when modelling the vibroacoustic behaviour of a system, is that of determining how energy is transmitted from a given source, where external energy is being input, to a target where energy is to be reduced. In many situations of practical interest, a high percentage of the transmitted energy is driven by a limited set of dominant paths. For instance, this is at the core of the existence of transmission loss regulations between dwellings. In this work, it is shown that in the case of a system modelled with statistical energy analysis (SEA), the problem of ranking dominant paths can be posed as a variation of the so-called K shortest path problem in graph theory. An algorithm for the latter is then modified and adapted to obtain the sorted set of K dominant energy transmission paths in a SEA model. A numerical example to show its potential for practical applications is included.     

Non-resonant sound transmission through double walls using statistical energy analysis

Although SEA is a suitable framework for predicting sound transmission through double walls it has been found that the standard method of computing the non- resonant coupling loss factor from a room to cavity underestimates the coupling. A revised model for computing this coupling loss factor is presented which gives much better agreement with measured data. This allows better predictions to be made of sound transmission through lightweight double walls.     

The prediction of sound transmission through buildings using statistical energy analysis

Measurements were carried out on a building to evaluate the uses of statistical energy analysis for determining sound transmission performance. Coupling loss factors were measured and compared with predicted values. It was found that, in general, good agreement was obtained. The coupling loss factors were also used to calculate the sound pressure level, or surface velocity, of each subsystem in the building for a number of different sources. Comparison with the measured results gave an average error of 4 dB. Some large errors were obtained but these were due mainly to the omission of airborne flanking paths from the SEA model or due to the breakdown of the theory for specific coupling loss factors.     

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.     

Validity diagrams of statistical energy analysis

This paper is concerned with the validity domain of statistical energy analysis (SEA) which is defined in terms of four criteria. The mode count N and the modal overlap M must be high, the normalized attenuation factor and the coupling strength γ must be small. The application of dimensional analysis on the governing equations of plates gives the space of dimensionless parameters in which the validity domain of SEA must be delimited. This domain is discussed on the basis of geometry of the surfaces delimiting it. The diagrams of validity of SEA are introduced and discussed. A numerical simulation on a couple of rectangular plates coupled along one edge illustrates the theoretical approach.     

Rigorous electromagnetic analysis of resonant sub-wavelength metallic gratings by parametric Fourier modal method

Electromagnetic TM modes of metallic lamellar reflection or transmission gratings are analyzed by means of the parametric fourier modal method. We show that the resonant optical response of such devices is due to the excitation of guided waves inside the slits. Our rigorous methods allows us to define a resonance condition number and to calculate the field everywhere inside the grating layer.     

Analysis of the modal energy distribution of an excited vibrating panel coupled with a heavy fluid cavity by a dual modal formulation

This paper describes the modal interaction between a panel and a heavy fluid cavity when the panel is excited by a broad band force in a given frequency band. The dual modal formulation (DMF) allows describing the fluid–structure coupling using the modes of each uncoupled subsystem. After having studied the convergence of the modal series on a test case, we estimate the modal energies and the total energy of each subsystem. An analysis of modal energy distribution is performed. It allows us to study the validity of SEA assumptions for this case. Added mass and added stiffness effects of the fluid are observed. These effects are related to the non-resonant acoustic modes below and above the frequency band of excitation. Moreover, the role of the spatial coupling of the resonant cavity modes with the non-resonant structure modes is also highlighted. As a result, the energy transmitted between the structure and the heavy fluid cavity generally cannot be deduced from the SEA relation established for a light fluid cavity.     

The sound transmission loss of a single panel measured with the two-microphone and the conventional method-comparison with the statistical energy analysis model

The sound transmission loss of a single metal panel obtained with the sound intensity technique and the conventional method have been compared with the theoretical model of Statistical Energy Analysis (SEA). This method of comparison with a theoretical model is useful in order to explain small systematic deviations between the results of both experimental methods.It happens that the difference between the so-called residual reactivity level and the measured reactivity gives a good estimate of the accuracy of the intensity measurement. Even in very simple cases this quantity can disturb the measuring results due to the complex sound field close to the sound radiating panel. By correcting the sound transmission loss results with the well known Waterhouse correction and by choosing an appropriate probe distance to the sound radiating construction, good agreement is obtained between the results of both experimental methods as well as with the SEA model.     

Sound transmission loss of foam-filled honeycomb sandwich panels using statistical energy analysis and theoretical and measured dynamic properties

Comparisons between the experimental and predicted sound transmission loss values obtained from statistical energy analysis are presented for two foam-filled honeycomb sandwich panels. Statistical energy analysis (SEA) is a modeling procedure which uses energy flow relationships for the theoretical estimation of the sound transmission through structures in resonant motion. The accuracy of the prediction of the sound transmission loss using SEA greatly depends on accurate estimates of: (1) the modal density, (2) the internal loss factor, and (3) the coupling loss factor parameters of the structures. A theoretical expression for the modal density of sandwich panels is developed from a sixth-order governing equation. Measured modal density estimates of the two foam-filled honeycomb sandwich panels are obtained by using a three-channel spectral method with a spectral mass correction to allow for the mass loading of the impedance head. The effect of mass loading of the accelerometer is corrected in the estimations of both the total loss factor and radiation loss factor of the sandwich panels.     

非保守耦合系统的统计能量分析原理

传统的统计能量分析(SEA)理论不能解决非保守耦合系统的能量分析问题。本文在非保守耦合振子的能量分布与功率流特征的研究基础上,推导了互不相关随机激励条件下非保守耦合系统的功率平衡方程式及各有关功率项的计算式,建立了非保守耦合系统的统计能量分析理论。研究结果表明,保守耦合仅是非保守耦合的一个特例,耦合阻尼对非保守耦合系统的能量分布和功率流的特征有着显著的影响,只有在耦合阻尼远小于系统内阻尼时这种影响才可近似忽略。作为理论的一个应用实例,本文对非保守耦合板的能量问题进行了理论和实验研究。     

The time-frequency characteristics of violin vibrato: modal distribution analysis and synthesis

A high-resolution time-frequency distribution, the modal distribution, is applied to the study of violin vibrato. The analysis indicates that the frequency modulation induced by the motion of the stopped finger on the string is accompanied by a significant amplitude variation in each partial of that note. Amplitude and frequency estimates for each partial are extracted from the modal distribution of ten pitches that span the range of the violin instrument. The frequency modulation is well-represented by a single sinusoid with a mean rate of 5.9 Hz and a mean excursion of +/- 15.2 cents. A spectral decomposition of the amplitude envelopes of the partials shows that the peaks lie primarily at integer multiples of the vibrato rate. These amplitude and frequency estimates are used in an additive synthesis model to generate synthetic replicates of violin vibrato. Simple approximations to these estimates are created, and synthesized sounds using these are evaluated perceptually by seven subjects using discrimination, nonmetric multidimensional scaling (MDS), and sound quality scoring tasks. It is found that the absence of frequency modulation has little effect on the perceptual response to violin vibrato, while the absence of amplitude modulation causes marked changes in both sound quality and MDS results. Low-order spectral decompositions of the amplitude and frequency estimates also occupy the same perceptual space as the original recording for a subset of the pitches studied.     

A deterministic model of impact noise transmission through structural connections based on modal analysis

Vibration transmission through structural connections is modelled in a deterministic way by means of modal analysis. This model is used first to study the effect of elastic joints across the floor in the transmission of impact noise. They are an effective means of reducing impact noise propagation, and can almost eliminate it for small values of the joint stiffness. The method is also used to study the acoustic relevance of studs in lightweight floor transmission. Different ways of modelling the studs are presented and compared. For the examples developed, the best option is to use springs for modelling the studs rather than more complex models involving springs and beams. Also the different behaviour of point and line connections is verified, as well as the influence of the position of the studs.     

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.     

A direct transmissibility formulation for experimental statistical energy analysis with no input power measurements

An experimental statistical energy analysis method is proposed that requires neither input power measurements nor strong restrictive hypothesis on the nature of coupling between subsystems. Coupling loss factors are obtained from direct or blocked energy transmissibilities provided that total or internal loss factors are known, for instance, from application of the energy decay rate method. In turn, direct transmissibilities are computed from standard measurable energy transmissibilities, following the ideas of the direct transfer approach to transmission path analysis. The theoretical background of the proposed formulation is presented together with a numerical example to indicate how it could be applied in a practical case.