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.     

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.     

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.     

A hybrid approach for predicting the distribution of vibro-acoustic energy in complex built-up structures

Finding the distribution of vibro-acoustic energy in complex built-up structures in the mid-to-high frequency regime is a difficult task. In particular, structures with large variation of local wavelengths and/or characteristic scales pose a challenge referred to as the mid-frequency problem. Standard numerical methods such as the finite element method (FEM) scale with the local wavelength and quickly become too large even for modern computer architectures. High frequency techniques, such as statistical energy analysis (SEA), often miss important information such as dominant resonance behavior due to stiff or small scale parts of the structure. Hybrid methods circumvent this problem by coupling FEM/BEM and SEA models in a given built-up structure. In the approach adopted here, the whole system is split into a number of subsystems that are treated by either FEM or SEA depending on the local wavelength. Subsystems with relative long wavelengths are modeled using FEM. Making a diffuse field assumption for the wave fields in the short wave length components, the coupling between subsystems can be reduced to a weighted random field correlation function. The approach presented results in an SEA-like set of linear equations that can be solved for the mean energies in the short wavelength subsystems.     

A quantitative criterion validating coupling power proportionality in statistical energy analysis

The response of two general spring-coupled elements is investigated to develop a unifying approach to the weak coupling criterion in Statistical Energy Analysis (SEA). First, the coupled deterministic equations of motion are expressed in the bases given by the uncoupled elements’ eigenmodes. Then, an iterative solution is expressed as a succession of exchanges between elements, where uncoupled motion provides the start approximation, converging if the ‘coupling eigenvalue’ is less than unity, in which case coupling is said to be weak. This definition is related to whether response is ‘local’ or ‘global’, encompassing a number of previously defined coupling strength definitions, applying for deterministically described structures. A stochastic ensemble is defined by that its members are equal to the investigated structure but the elements have random frequencies. It is required that the coupling eigenvalue be less than unity for all members of the ensemble. This requirement generates the title subject of the article: ‘the modal interaction strength’. It is similar to the previously defined coupling strength criterion characterising the ensemble average energy flow in uni-dimensional waveguides. Finally, SEA models are formulated in terms of the uncoupled elements’ modal data.     

A novel hybrid ES-FE-SEA for mid-frequency prediction of Transmission losses in complex acoustic systems

The widely-used numerical modeling approaches such as the finite element method (FEM) and statistical energy analysis (SEA) often have limited applicability to the transmission loss prediction in mid-frequency range. In this paper, a novel hybrid edge-based smoothed FEM coupled with statistical energy analysis (ES-FE-SEA) method is proposed to further improve the accuracy of “mid-frequency” transmission loss predictions. The application of ES-FEM will “soften” the well-known ‘‘overly-stiff’’ behavior in the standard FEM solution and reduce the inherent numerical dispersion error. While the SEA approach deals with the physical uncertainty in the relatively higher frequency range. The plate of interest is appropriately described by an ES-FEM model, due to its relative robustness to perturbations. Its adjacent reverberation cavities are modeled by employing the SEA approach, because of their high model density. The coupling and interaction between SEA subsystems and the FE subsystem is governed by the “reciprocity relationship” theorem. A standard numerical example for benchmarking is examined and excellent agreement was achieved between the prediction and reference results. The proposed ES-FE-SEA is also verified by various numerical examples. The method is finally applied to the modeling a complicated engineering problem–acoustic fields on both sides of the front windshield in a passenger car.     

STATISTICAL ENERGY ANALYSIS OF COUPLED PLATE SYSTEMS WITH LOW MODAL DENSITY AND LOW MODAL OVERLAP

Finite element methods, experimental statistical energy analysis (ESEA) and Monte Carlo methods have been used to determine coupling loss factors for use in statistical energy analysis (SEA). The aim was to use the concept of an ESEA ensemble to facilitate the use of SEA with plate subsystems that have low modal density and low modal overlap. An advantage of the ESEA ensemble approach was that when the matrix inversion failed for a single deterministic analysis, the majority of ensemble members did not encounter problems. Failure of the matrix inversion for a single deterministic analysis may incorrectly lead to the conclusion that SEA is not appropriate. However, when the majority of the ESEA ensemble members have positive coupling loss factors, this provides sufficient motivation to attempt an SEA model. The ensembles were created using the normal distribution to introduce variation into the plate dimensions. For plate systems with low modal density and low modal overlap, it was found that the resulting probability distribution function for the linear coupling loss factor could be considered as lognormal. This allowed statistical confidence limits to be determined for the coupling loss factor. The SEA permutation method was then used to calculate the expected range of the response using these confidence limits in the SEA matrix solution. For plate systems with low modal density and low modal overlap, relatively small variation/uncertainty in the physical properties caused large differences in the coupling parameters. For this reason, a single deterministic analysis is of minimal use. Therefore, the ability to determine both the ensemble average and the expected range with SEA is crucial in allowing a robust assessment of vibration transmission between plate systems with low modal density and low modal overlap.     

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.     

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.     

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.     

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.     

Numerical and experimental validation of variance prediction in the statistical energy analysis of built-up systems

In the statistical energy analysis (SEA) approach to vibration modeling, a complex system is represented as an assembly of coupled subsystems, and the method leads to the prediction of the vibrational energy level of each subsystem. The averaging procedures implicit in the technique imply that the predicted energy is the mean value taken over an ensemble of random structures, such as a set of vehicles leaving a production line. Recently, a new method has been developed to allow the ensemble variance, in addition to the mean, to be predicted within the context of SEA, and the present paper concerns further extension and validation of this work. The theoretical extension concerns the variance of the energy density at a single point in any of the subsystems, and the validation includes both simulation and experimental studies. The simulation results concern plate assemblies, while experimental results are presented both for a single-plate and for a cylinder-plate structure. In each case an ensemble of random structures has been generated by adding small point masses at random locations on the structure. In general, good agreement between the predictions and the validation results is observed.     

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.     

Vibration of L-shaped plates under a deterministic force or moment excitation: a case of statistical energy analysis application

Analytical and closed form solutions are presented in this paper for the vibration response of an L-shaped plate under a point force or a moment excitation. Inter-relationships between wave components of the source and the receiving plates are clearly defined. Explicit expressions are given for the quadratic quantities such as input power, energy flow and kinetic energy distributions of the L-shaped plate. Applications of statistical energy analysis (SEA) formulation in the prediction of the vibration response of finite coupled plate structures under a single deterministic forcing are examined and quantified. It is found that the SEA method can be employed to predict the frequency averaged vibration response and energy flow of coupled plate structures under a deterministic force or moment excitation when the structural system satisfies the following conditions: (1) the coupling loss factors of the coupled subsystems are known; (2) the source location is more than a quarter of the plate bending wavelength away from the source plate edges in the point force excitation case, or is more than a quarter wavelength away from the pair of source plate edges perpendicular to the moment axis in the moment excitation case due to the directional characteristic of moment excitations. SEA overestimates the response of the L-shaped plate when the source location is less than a quarter bending wavelength away from the respective plate edges owing to wave coherence effect at the plate boundary.     

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。     

Random response of identical,one-dimensional coupled subsystems

A precise statistical analysis is carried out for the steady-state response of a one-dimensional wave-bearing system formed from two identical subsystems coupled together at an end with a general coupling. The statistics correspond approximately to averaging in frequency. Cross-correlation between the wave fields incident on the coupling in the two subsystems (neglected in most statistical analyses) strongly affects the magnitude of the coupling power and, for dissipative couplings, the dissipative power. Asymmetry of the wave field in each subsystem (also usually neglected) can have a significant additional effect even when the dissipative loss factor is small. These effects are evaluated, and their analogs in analyses based in modal theory are discussed.     

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.     

Hybrid power flow analysis using coupling loss factor of SEA for low-damping system—Part II: Formulation of 3-D case and hybrid PFFEM

The hybrid power flow analysis (PFA) is an analytic method proposed for the effective prediction of vibrational and acoustic responses of low-damping system in the medium-to-high frequency ranges by using the PFA algorithm and statistical energy analysis (SEA) coupling concepts. This paper presents the hybrid boundary condition on 3-D case for hybrid PFA in addition to 1-D and 2-D cases which are derived in the other companion paper, and formulates the hybrid power flow finite-element method (PFFEM) including coupling loss factor (CLF) of SEA to extend the application area of hybrid PFA to built-up structures. To verify the derived boundary condition and hybrid PFFEM, numerical analyses were successfully performed for various analytic models and reverberance factors.     

Numerical and experimental validation of a hybrid finite element-statistical energy analysis method

The finite element (FE) and statistical energy analysis (SEA) methods have, respectively, high and low frequency limitations and there is therefore a broad class of "mid-frequency" vibro-acoustic problems that are not suited to either FE or SEA. A hybrid method combining FE and SEA was recently presented for predicting the steady-state response of vibro-acoustic systems with uncertain properties. The subsystems with long wavelength behavior are modeled deterministically with FE, while the subsystems with short wavelength behavior are modeled statistically with SEA. The method yields the ensemble average response of the system where the uncertainty is confined in the SEA subsystems. This paper briefly summarizes the theory behind the method and presents a number of detailed numerical and experimental validation examples for structure-borne noise transmission.