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.     

Regression curves for vibration transmission across junctions of heavyweight walls and floors based on finite element methods and wave theory

Sound insulation prediction models in European and International Standards use the vibration reduction index to calculate flanking transmission across junctions of walls and floors. These standards contain empirical relationships between the ratio of mass per unit areas for the walls/floors that form the junction and a frequency-independent vibration reduction index. However, calculations using wave theory show that there is a stronger relationship between the ratio of characteristic moment impedances and the transmission loss from which the vibration reduction index can subsequently be calculated. In addition, the assumption of frequency-independent vibration reduction indices has been shown to be incorrect due to in-plane wave generation at the junction. Therefore numerical experiments with FEM, SFEM and wave theory have been used to develop new regression curves between these variables for the low-, mid- and high-frequency ranges. The junctions considered were L-, T- and X-junctions formed from heavyweight walls and floors. These new relationships have been implemented in the prediction models and they tend to improve the agreement between the measured and predicted airborne and impact sound insulation.     

某稳瞄转塔结构的有限元模态分析与试验验证

结构动态效应是影响稳瞄系统稳定精度的关键因素。目前有限元模态分析与试验模态分析结合的方法是分析结构动力学特性的最有效途径之一。以某型稳瞄具两轴四框架结构为研究对象,首先利用有限元方法进行了模态预分析。根据仿真结果获取的振型特征,对两轴四框架的外环进行模态测试,给出了前8阶模态的固有频率、振型和模态阻尼比。模态仿真与测试的结果相互验证表明:有限元仿真可实现两轴四框架结构固有频率相对误差小于15%的精度要求,在测点密度足够的条件下振型相关性良好。     

SEA and contribution analysis for interior noise of a high speed train

This paper presents a detailed Statistical Energy Analysis (SEA) and contribution analysis of the interior noise of a high-speed train through extensive simulations and measurements. The SEA model was developed based on the actual geometrical parameters of a benchmark high-speed coach. Sound transmission loss levels of the structural components of the car body, which are required in the SEA model, were tested in a dedicated acoustic laboratory following international standard ISO 140-3:1995. Modal densities of these structural components were derived from measured frequency response functions using the modal counting method. Damping loss factors were obtained using the half-power bandwidth method and the vibration attenuation method. By considering the relationship between sound radiation and power transmission, coupling loss factors between structures and cavities were estimated. Source inputs to the SEA model were derived from field experiment data. Interior noise due to those sources was predicted using the SEA model and the prediction was generally in good agreement with measurement. Contribution analysis was then performed using this validated model through parametric study, and this analysis was further examined experimentally. In conclusion, for the coach that was investigated in this paper, the key factors for interior noise are sidewall vibration, bogie area noise, and floor sound transmission loss. Based on this and other engineering considerations, an interior noise control strategy can be defined.     

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.     

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.     

A damping mechanics model and a beam finite element for the free-vibration of laminated composite strips under in-plane loading

A theoretical framework is presented for predicting the nonlinear damping and damped vibration of laminated composite strips due to large in-plane forces. Nonlinear Green-Lagrange axial strains are introduced in the governing equations of a viscoelastic composite and new nonlinear damping and stiffness matrices are formulated including initial stress effects. Building upon the nonlinear laminate mechanics, a damped beam finite element is developed. Finite element stiffness and damping matrices are synthesized and the static equilibrium is predicted using a Newton-Raphson solver. The corresponding linearized damped free-vibration response is predicted and modal frequencies and damping of the in-plane deflected strip are calculated. Numerical results quantify the nonlinear effect of in-plane loads on structural modal damping of various laminated composite strips. The modal loss-factors and natural frequencies of cross-ply Glass/Epoxy beams subject to in-plane loading are measured and correlated with numerical results.     

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.     

The finite layer method for modelling the sound transmission through double walls

The finite layer method (FLM) is presented as a discretisation technique for the computation of noise transmission through double walls. It combines a finite element method (FEM) discretisation in the direction perpendicular to the wall with trigonometric functions in the two in-plane directions. It is used for solving the Helmholtz equation at the cavity inside the double wall, while the wall leaves are modelled with the thin plate equation and solved with modal analysis. Other approaches to this problem are described here (and adapted where needed) in order to compare them with the FLM. They range from impedance models of the double wall behaviour to different numerical methods for solving the Helmholtz equation in the cavity. For the examples simulated in this work (impact noise and airborne sound transmission), the former are less accurate than the latter at low frequencies. The main advantage of FLM over the other discretisation techniques is the possibility of extending it to multilayered structures without changing the interpolation functions and with an affordable computational cost. This potential is illustrated with a calculation of the noise transmission through a multilayered structure: a double wall partially filled with absorbing material.     

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.     

The prediction of airborne sound transmission between two rooms using first-order flanking paths

Airborne sound transmission between adjacent rooms can be predicted using the Standard EN 12354-1 (ISO 15712-1), which is equivalent to a first-order approximation of statistical energy analysis (SEA). This paper analyses airborne sound transmission between adjacent rooms in a masonry building, by comparing results obtained from EN 12354-1 to SEA predictions and measurements. It is shown that the restriction of the Standard to first-order flanking paths can lead to large errors in predictions when compared to measurements and SEA results taking into account all transmission paths. This is observed both for individual flanking paths and overall transmission between rooms, for which the Standard provides results similar to those obtained by the first-order approximation of SEA. The paper also looks at possible reasons why previous studies using the approach in EN 12354 have generally shown good agreement with measurements.     

Investigation of structural response and noise reduction of an acoustical enclosure using SEA method

Noise reduction capability of acoustical enclosures has been extensively investigated and is commonly quantified by noise reduction. Although much work has been done in predicting the noise reduction by using statistical energy analysis (SEA) method, discrepancy often exists between the predicted and measured results, especially at the low and intermediate frequencies. This is largely due to the fact that some physical mechanisms which affect the prediction of noise reduction were not addressed in the available SEA models. In this paper, the sources of the discrepancy were identified by investigating the limitation of SEA for energy transfer in the entire frequency range and the effect of structural–structural coupling, acoustical–structural coupling on the prediction of noise reduction. An improved SEA model including the non-resonant response and more accurate transmission coefficient of finite panels is presented. Finally, the predicted structural response and noise reduction of an acoustical enclosure were compared with measured results, and the improved agreement between the prediction and experiment was observed.     

含周期性空腔结构吸声机理的研究*

首先建立并验证了含轴对称空腔周期性结构吸声特性计算的简化有限元仿真方法。在水下环境,用简化的有限元模型结合遗传算法对含周期性圆柱空腔结构的吸声性能进行了优化设计。从能量耗散、变形和模态的角度分析了含周期性空腔结构的吸声机理。空腔结构谐振包括表层的弯曲振动和空腔附近粒子的径向运动,且径向运动随吸声结构厚度方向也是变化的。相对低频区主要激起表层振动模态,高频区激起径向运动模态,且径向振动对声学性能影响很大,其更有利于促进纵波转化为能量更易消散的横波。     

住宅建筑中相邻房间的侧向传声预测

侧向传声作为建筑中声传递的组成部分,对住宅的整体隔声效果具有重要的影响,通过将建筑中相邻房间的各建筑构件划分为若干子系统,应用统计能量分析(Statistical Energy Analysis,SEA)理论,从系统的声功率平衡的角度建立侧向传声的预测模型,在描述各路径的传声规律的同时确定主要传声路径。研究结果表明:当外围护结构为重质结构,且为匀质单一材料构造时,(1)在低频处,全程通过两相邻房间的侧墙或楼板的非通过隔墙的侧向路径成为主要侧向传声路径;(2)在中高频,各侧向路径的声压级差趋于一致,此时的建筑隔声性能取决于通过隔墙的直接路径上的声传递;(3)采用重质隔墙可以缩小侧向传声影响的频率范围。本研究为改善住宅的声环境质量及建筑隔声设计提供了理论依据。      

Low frequency sound insulation using stiffness control with honeycomb panels

A new honeycomb core design has been used to increase the stiffness of the panel and applied to improve the noise transmission loss at frequencies between 100 and 200 Hz. A model is presented to predict the transmission loss of the honeycomb panels based on the structural modal parameters. A new test specimen with fiber reinforced plastic cores and face sheets had been used to investigate the effect of stiffness and damping on noise transmission loss. The measurements of noise transmission loss have been compared with data for common structural panels. The results show that the new core fabrication techniques using moulding to improve the noise transmission are effective. In comparison to a cement panel of the same mass, the honeycomb panels have higher TL at low frequencies between 100 and 200 Hz due to higher stiffness and damping. The honeycomb panels have more significant vibration responses above 500 Hz but these are limited by damping.     

Identifying multiple axial load patterns using measured vibration data

Multiply redundant frames possess as many linearly independent axial load patterns as their degree of static indeterminacy. Any state of equilibrium can be described as the linear sum of these axial load patterns. As a frame is loaded in the multiple patterns, the changes in member axial loads affect the frame frequencies and mode shapes in a complicated way through geometric stiffening. By representing this behaviour in a finite element model (FEM), it is possible to measure the dynamic characteristics of a physical frame, update the axial loads in the FEM until the difference between measured and model frequencies is minimised and thus infer the member axial loads. What is updated is the factors on the axial load patterns and this is done iteratively using traditional model updating (Newton's method). Thus, all member axial loads can be identified using just a set of updating parameters equal in number to the degree of static indeterminacy. A numerical simulation and, for the first time, an implementation on a physical realisation of a multiply redundant frame using measured vibration data are presented herein and show that member loads can be identified to an encouraging degree of accuracy. Beneficial strategies such as formulating the problem on an orthonormal basis for the axial load patterns and a ‘modal tagging’ scheme, which helps to conserve the strength of the necessary, but otherwise typically compromised, pairing of mode shapes of the FEM to those measured, are discussed.     

Calculation methods for the transmission loss of single,double and triple partitions

Methods of calculating the transmission loss for single and double walls are presented. These methods are developed using a statistical energy analysis (SEA). In principle the methods are based on separate calculations of resonant and non-resonant transmission for frequencies less than the critical frequency, $\text{f}{}_{\mathrm{c}}$, of the panels and calculation of only resonant transmission for frequencies equal to, and greater than, $\text{f}{}_{\mathrm{c}}$.Comparisons between calculated and measured results show good agreement.A calculation method is also presented for the transmission loss of triple panels for frequencies greater than the cut-off frequency for the cavities. For frequencies less than the cut-off frequency for the smallest cavity depth it is shown that, in most cases, the effect of the middle panel is very slight.     

Analytical modeling and vibration analysis of internally cracked rectangular plates

This study proposes an analytical model for nonlinear vibrations in a cracked rectangular isotropic plate containing a single and two perpendicular internal cracks located at the center of the plate. The two cracks are in the form of continuous line with each parallel to one of the edges of the plate. The equation of motion for isotropic cracked plate, based on classical plate theory is modified to accommodate the effect of internal cracks using the Line Spring Model. Berger?s formulation for in-plane forces makes the model nonlinear. Galerkin?s method used with three different boundary conditions transforms the equation into time dependent modal functions. The natural frequencies of the cracked plate are calculated for various crack lengths in case of a single crack and for various crack length ratio for the two cracks. The effect of the location of the part through crack(s) along the thickness of the plate on natural frequencies is studied considering appropriate crack compliance coefficients. It is thus deduced that the natural frequencies are maximally affected when the crack(s) are internal crack(s) symmetric about the mid-plane of the plate and are minimally affected when the crack(s) are surface crack(s), for all the three boundary conditions considered. It is also shown that crack parallel to the longer side of the plate affect the vibration characteristics more as compared to crack parallel to the shorter side. Further the application of method of multiple scales gives the nonlinear amplitudes for different aspect ratios of the cracked plate. The analytical results obtained for surface crack(s) are also assessed with FEM results. The FEM formulation is carried out in ANSYS.     

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.