Vibration transmission between coupled plates using finite element methods and statistical energy analysis. Part 2: The effect of window apertures in masonry flanking walls

Part 1 of this paper demonstrated the validity of predictions of vibration transmission across junctions of masonry walls using Finite Element Methods (FEM). Part 2 uses numerical experiments with FEM to calculate the vibration transmission between masonry walls with window apertures at different positions in the flanking wall(s). Results from the numerical experiments are used to assess a simple “rule-of-thumb” estimate for calculating the change in the coupling parameters due to the introduction of an aperture into a flanking wall. Conclusions are drawn concerning use of the “rule-of-thumb” estimate for the coupling loss factor in Statistical Energy Analysis and the vibration reduction index in European standard EN 12354.     

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.     

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.     

A coupled edge-/face-based smoothed finite element method for structural-acoustic problems

The edge-based smoothed finite element method (ES-FEM) and the face-based smoothed finite element method (FS-FEM) developed recently have shown great efficiency in solving solid mechanics problems with triangular and tetrahedral meshes. In this paper, a coupled ES-/FS-FEM model is extended to solve the structural-acoustic problems consisting of a plate structure interacting with the fluid medium. Three-node triangular elements and four-node tetrahedral elements are used to discretize the two-dimensional (2D) plate and three-dimensional (3D) fluid, respectively, as they can be generated easily and even automatically for complicated geometries. The field variable in each element is approximated using the linear shape functions, which is exactly the same as that in the standard FEM. The gradient field of the problem is obtained particularly using the gradient smoothing operation over the edge-based and face-based smoothing domains in 2D and 3D, respectively. The gradient smoothing technique can provide a proper softening effect to the model, effectively solve the problems caused by the well-known “overly-stiff” phenomenon existing in the standard FEM, and hence significantly improve the accuracy of the solution for the coupled systems. Intensive numerical studies have been conducted to verify the effectiveness of the coupled ES-/FS-FEM for structural-acoustic problems.     

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.     

Integrating the original face images and “symmetrical faces” to perform face recognition

Using the original and ‘symmetrical face’ training samples to perform representation based face recognition was first proposed in [1]. It simultaneously used the original and ‘symmetrical face’ training samples to perform a two-step classification and achieved an outstanding classification result. However, in [1] the “symmetrical face” is devised only for one method. In this paper, we do some improvements on the basis of [1] and combine this “symmetrical faces” transformation with several representation based methods. We exploit all original training samples, left “symmetrical face” training samples and right “symmetrical face” training samples for classification and use the score fusion for ultimate face recognition. The symmetry of the face is first used to generate new samples, which is different from original face image but can really reflect some possible appearance of the face. It effectively overcomes the problem of non-sufficient training samples. The experimental results show that the proposed scheme can be used to improve a number of traditional representation based methods including those that are not presented in the paper.     

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.     

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.     

Comparison of 1D transfer matrix method and finite element method with tests for acoustic performance of multi-chamber perforated resonator

Multi-chamber perforated resonator (MCPR) is a kind of typical silencer element which can both attenuate broadband noise and satisfy specific installation requirements. The one-dimensional transfer matrix method (TMM) and finite element method (FEM) are widely used to predict the transmission loss of the resonators. This paper mainly focuses on the comparison between 1D TMM and FEM in which detailed perforation modeling is applied for the acoustic modeling of MCPRs. Five resonators with different acoustic attenuation frequency ranges are built for simulation and test. In order to verify the results of the above methods, a transmission loss test facility is designed based on two-load method. Through adjusting the distance between microphones, the facility’s effective measurement frequency can be changed. The results show that despite of the complex modeling and calculation, FEM with detailed perforation modeling shows good consistency with test results in both frequency and amplitude within entire frequency range. In comparison, TMM is limited by the cut-off frequency when calculating transmission losses. Besides, accuracy of TMM in low frequency range is also affected by perforation conditions. However, TMM is time-saving in calculation and structure optimization. In MCPRs’ development process, TMM can be used to quickly design and optimize structure parameters while FEM can be used to verify the acoustic performance before prototyping.     

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.     

Fast algorithms for spherical harmonic expansions,III

We accelerate the computation of spherical harmonic transforms, using what is known as the butterfly scheme. This provides a convenient alternative to the approach taken in the second paper from this series on “Fast algorithms for spherical harmonic expansions”. The requisite precomputations become manageable when organized as a “depth-first traversal” of the program’s control-flow graph, rather than as the perhaps more natural “breadth-first traversal” that processes one-by-one each level of the multilevel procedure. We illustrate the results via several numerical examples.     

Correlation of finite element models of multi-physics systems

The modal assurance criterion (MAC) and normalized cross-orthogonality (NCO) check are widely used to assess the correlation between the experimentally determined modes and the finite element model (FEM) predictions of mechanical systems. Here, their effectiveness in the correlation of FEM of two types of multi-physics systems, namely, viscoelastic damped systems and a shunted piezoelectric system are investigated using the dynamic characteristics obtained from a nominal FEM, that are considered as the ‘true’ or experimental characteristics and those obtained from the inaccurate FEMs. The usefulness of the MAC and NCO check in the prediction of the overall loss factor of the viscoelastic damped system, which is an important design tool for such systems, is assessed and it is observed that these correlation methods fail to properly predict the damping characteristics, along with the responses under base excitation. Hence, base force assurance criterion (BFAC) is applied by comparing the ‘true’ dynamic force at the base and inaccurate FEM predicted force such that the criterion can indicate the possible error in the acceleration and loss factor. The effect of temperature as an uncertainty on the MAC and NCO check is also studied using two viscoelastic systems. The usefulness of MAC for the correlation of a second multi-physics FEM that consists of a shunted piezoelectric damped system is also analyzed under harmonic excitation. It has been observed that MAC has limited use in the correlation and hence, a new correlation method – current assurance criterion – based on the electric current is introduced and it is demonstrated that this criterion correlates the dynamic characteristics of the piezoelectric system better than the MAC.     

Dissipative heat and exchange dispersion on metastability of Gd5Si4

The response of the magnetic and crystal structure of Gd₅Si₄ to both ‘isothermal’ and ‘thermo-mechanical’ process is investigated by the dispersion exchange fluctuation, which can be the cause of the compound's metastability. The X-ray diffraction DTA, SEM and magnetization at room temperature are measured to explore the effect of annealing process (latent heat as isothermal) at different annealing temperatures. The discharge heat (exchange heat) by the annealing process, which is manifested by the thermal loop of DTA, is caused by the reduction in the lattice constant as well as in magnetization behavior measured at room temperature with no observable crystal phase transition. The exchange fluctuation (entropy), which is related to the DTA thermal loop, is increased by the sample's heat absorption (discharge heat) in the direction to lower the free energy of the crystal. The crystallographic slip is investigated by the dissipative heat through the high ball milling energy (HBME) to provide the nature and strength of the exchange. It is shown that the DTA thermal loop and SEM-crystallographic slip should be the character of the “exchange heat” (discharge heat) and “heat exchange” (absorption of heat through the wet environment of ball milling).     

“Step-up” versus “step-down” scattering asymmetry in the neutralization of H on free-electron vicinal metal surfaces

The charge transfer between H⁻ and a free-electron vicinal metallic surface is studied using a wave-packet propagation method. We apply a statistical Thomas-Fermi-von Weizsäcker model with a local density approximation for the exchange-correlation energy to compute the ground-state electronic structure of the substrate. The long-range image charge effects in the electron transfer are included on a phenomenological level. We obtain the ion-survival probability from a rate equation for a set of realistic scattering trajectories of projectiles that are incident with a kinetic energy of 50 eV. Our calculations reveal a pronounced substrate orientation dependence of the charge transfer dynamics expressed in a “left-right” (or “step-up-step-down”) scattering asymmetry in the final ion-survival probability, which is caused by an enhancement of electron loss on the outgoing part of those ion trajectories which approach steps from below.     

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.     

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.     

Measuring statistical evenness: A panoramic overview

Motivated by the question “how equal is the distribution of wealth within a given human population?” economics devised an impressive toolbox of quantitative measures of societal egalitarianism including the Lorenz curve and the following indices: Gini, Pietra, Hoover, Amato, Hirschman, Theil and Atkinson. These quantitative measures-considered in the broader context of general data-sets with positive values-are, in effect, general gauges of statistical evenness. While the application of Gini’s index grew beyond economics and reached diverse fields of science, the aforementioned “evenness toolbox” has largely remained within the confines of the social sciences. The aim of this Paper is to expose this “evenness toolbox” to the physics community by presenting a comprehensive evenness-based approach to a fundamental problem in science—the measurement of statistical heterogeneity.     

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.     

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.     

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.