Statistical Energy Analysis

The Energy analysis has become one of the important parameter of building design, and it gaining more importance as energy crises and global warming is becoming worst day by day. When energy analysis is started from the conceptual designing phase of the project it is the most useful as the design is in its most flexible stage and many changes can be done on the design and maximum energy can be saved. In Earlier days Energy analysis used to take too much time but now with BIM model it can be done very quickly. As In BIM model same model can be used directly for Energy analysis using Autodesk Energy analysis tools where there is no need to do complex calculations. This paper is the study done on residential building analysis using BIM technology.

Le comportement vibratoire d’une structure industrielle complexe est très bien représenté aux basses fréquences par ses modes propres de vibration. Cette représentation n’est en revanche plus du tout adaptée aux hautes fréquences, pour lesquelles la structure exhibe un comportement radicalement différent. Ces aspects ont une grande importance pratique notamment pour tout ce qui concerne l’acoustique interne et/ou externe des véhicules de transport : automobile, ferroviaire, domaine spatial... L’Analyse Statistique Énergétique, connue sous le signe SEA, est à ce jour la méthode la plus aboutie pour traiter de tels problèmes, bien qu’elle soit fondée sur une série d’hypothèses relativement restrictives. L’objectif de ce cours est double : d’une part présenter les principaux développements ayant conduit à la formulation SEA, d’autre part insister sur les hypothèses introduites au fur et à mesure et qui restreignent son champ d’application.
Plan
0. Introduction : vibration et vibro-acoustique hautes fréquences, exemples, besoins, stratégie de modélisation.
1. Aspects énergétiques de l’oscillateur simple, équivalences de moyenne.
2. Deux problèmes couplés élémentaires : système à 2 ddls, oscillateur immergé dans un fluide acoustique.
3. Vibrations aléatoires stationnaires des systèmes linéaires continus, aspects énergétiques.
4. Systèmes linéaires continus couplés.
5. Formulation de l’Analyse Statistique Énergétique.

Life is full of sounds and we want to hear the pleasant and vital ones; while shunning the unpleasant and dangerous variety. All told we are becoming steadily more sound conscious, as the relatively enormous growth of the telephone, radio, phonographic recording and talking motion picture industries sufficiently attests. Sounds touch people in different ways. Sound is extremely important to feel the taste of life. It is what we experience through the senses that make life meaningful. Sound is important for communication, signaling system, for finding depth of objects (SONAR) etc. Sound has been used to study earth’s history, to explore for oil and gas, to study undersea earthquakes and volcanic eruptions, to research wind energy, to measure temperature in ocean, to measure global climatic change. Even during the Cold War, the U.S. Navy allowed a small number of oceanographers to make use of the SOSUS (SOund SUrveillance System) for research in tracking soviet submarines at long ranges.

This paper is part of a PhD work, dealing with the acoustics of the public squares ('Agorá Acoustics'), especially when music (amplified or not) is played. Consequently, our approach will be to evaluate public squares using the same set of acoustics concepts for subjective evaluation and objective measurements as applied for concert halls and theatres. In this paper the acoustical

An efficient method is presented for the prediction of structure-borne sound transmission in large welded ship structures. SEA (Statistical Energy Analysis) is used, and the equations used for the SEA parameters are also presented. Traditionally, the SEA method requires a great deal of work when steel structures are modelled. It is almost impossible to prepare models manually for large structures

This paper presents the fundamentals of the numerical methods used in the acoustics and vibration fields as; Finite Elements Methods (FEM), Boundary Element Methods (BEM), Acoustic Rays, and Statistical Energy Analysis (SEA). State of arts and trend for future methods are presented. Some results are presented for practical study cases.

SEA models are used to predict the performance of acoustic packages when assessing the performance of vehicle level
or body noise reduction targets. One of the challenges faced by CAE engineers is the ability to estimate the performance of
different materials used in the sound package at the design stage. Analysts can use measured data in the form of insertion
loss and random incidence absorption if available or can predict the performance of materials using a Biot type description.
The use of the full poro-elastic Biot model for materials requires knowledge of the fluid and elastic properties, however a
limp or rigid model can be used to describe the material based only on the fluid properties and this is often sufficient to
describe fibrous materials.
In this paper a method will be outlined which will allow the material properties of fibrous materials to be estimated
from basic normal incidence data that is provided by material suppliers. This approach will then allow the effect of
compression on the material properties to be predicted and the impact on the acoustic performance to be estimated. In
addition it will allow the analyst to estimate the effects of increasing or reducing the fiber weight. Predictions will be
compared with measured data.

In this paper, an optimization technique for medium-high frequency dynamic problems based on Statistical Energy Analysis (SEA) method is presented. Using a SEA model, the subsystem energies are controlled by internal loss factors (ILF) and coupling loss factors (CLF), which in turn depend on the physical parameters of the subsystems. A preliminary sensitivity analysis of subsystem energy to CLF's is performed to select CLF's that are most effective on subsystem energies. Since the injected power depends not only on the external loads but on the physical parameters of the subsystems as well, it must be taken into account under certain conditions. This is accomplished in the optimization procedure, where approximate relationships between CLF's, injected power and physical parameters are derived. The approach is applied on a typical aeronautical structure: the cabin of a helicopter.

Increase of sound transmission loss(TL) of the fuselage is vital to build a comfortable cabin environment. In this paper, to find a convenient and accurate means for predicting the fuselage TL, the fuselage is modeled as a composite cylinder, and its TL is predicted with the analytical, the statistic energy analysis (SEA) and the hybrid FE&SEA method. The TL results predicted by the three methods are compared to each other and they show good agreement, but in terms of model building the SEA method is the most convenient one. Therefore, the parameters including the layup, the materials, the geometry, and the structure type are studied with the SEA method. It is observed that asymmetric laminates provide better sound insulation in general. It is further found that glass fiber laminates result in the best sound insulation as compared with graphite and aramid fiber laminates. In addition, the cylinder length has little influence on the sound insulation, while an increase of the radius considerably reduces the TL at low frequencies. Finally, by a comparison among an unstiffened laminate, a sandwich panel and a stiffened panel, the sandwich panel presents the largest TL at high frequencies and the stiffened panel demonstrates the poorest sound insulation at all frequencies.

This paper presents a theoretical and experimental study of the transmission of vibration through a two element structure which consists of a cylindrical shell couple to an end plate. The first part of this paper deals with the derivation of coupling loss factor (CLF) for the study of vibration transmission using Statistical Energy Analysis (SEA). The derivation is based on travelling wave analysis and the assumption of equipartition of energy amongst resonant modes. Numerical results for three examples of cylinder/plate structures are presented and compared with those of their equivalent plate/plate structures. The results show that the CLF values of the cylinder/plate structures asymptote to their equivalent plate/plate structures above the ring frequency of the cylinder. In the second part of this paper, an experimental program for measuring the CLF of an example cylinder/plate structure is described. Experimental results of the cylinder/plate structure show good agreement with theoretical predictions and confirm the validity of the present formulation of CLF.

Intensity distributions of linear wave fields are, in the high frequency limit, often approximated in terms of flow or transport equations in phase space. Common techniques for solving the flow equations for both time dependent and stationary problems are ray tracing or level set methods. In the context of predicting the vibro-acoustic response of complex engineering structures, reduced ray tracing methods such as Statistical Energy Analysis or variants thereof have found widespread applications. Starting directly from the stationary Liouville equation, we develop a boundary element method for solving the transport equations for complex multi-component structures. The method, which is an improved version of the Dynamical Energy Analysis technique introduced recently by the authors, interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. We demonstrate that the method can be used to efficiently deal with complex large scale problems giving good approximations of the energy distribution when compared to exact solutions of the underlying wave equation.

A potential alternative to Statistical Energy Analysis that is gaining increasing interest in recent years is the ''thermal'' energy flow approach. Its advantage is represented by the possibility of modeling the spatial distribution of energy density at high frequencies, thus yielding a more effective estimate of the system behaviour than the average constant value given by SEA. However, the thermal analogy proposed by the energy flow approach is questionable for any type of wave in any type of structure. To make the analysis more clear, exact equations for power balance in continuous structures are derived. The investigation confirms the questionability of the thermal approach and shows whether and when it is possible to determine exact equations for the energy density. Physical considerations are developed to explain some critical points. For one-dimensional systems a transmission potential is defined in analogy to the temperature in heat conduction problems.

We establish the basic SEA (Statistical Energy Analysis) equations for coupled structural-acoustic systems from the results derived in the previous lectures. The underlying hypotheses are carefully reviewed, and we subsequently discuss some modeling issues. We focus on the definition of sub-systems in SEA and elaborate on the determination of the input parameters: modal densities, loss factors, coupling loss factors, and input powers.

The traditional approach for the prediction of noise radiated from induction motors is usually based on a deterministic approach in which the motor is modeled using finite-element and/or boundary-element methods (FEM/BEM). A great deal of structural details has to be modeled using the deterministic approach. While these deterministic methods have been shown to provide reasonable estimates of the radiated acoustic noise at low frequencies (usually, well below 2 kHz) and for small motors, the demands on computing memory and time render calculations at high frequencies and for large motors almost impossible to make. In this paper, a statistical method especially suited for calculations at high frequencies is introduced. The statistical approach does not require the same amount of details to be modeled as in the FEM/BEM because only the averaged noise within a given frequency band is sought. Furthermore, it utilizes the analytical results obtained from simple structural elements that make up the motor structure. Results demonstrate the feasibility of the method, its ability to discern the contribution of various components of a motor to the overall radiated noise, and substantial savings in the computational effort compared with FEM/BEM. Index Terms-Acoustic noise prediction, boundary-element analysis, finite-element analysis, statistical energy analysis, variable-speed electric motors.

As the manufacturing technologies for composites materials are advanced, the use of composites for engineering applications has increased significantly. The purpose of this work is to study the response of the idealized subsystem made up of composite material. To study the vibrational response theoretically, Statistical Energy Analysis (SEA) method is used. The S.E.A. techniques are used to predict the vibrational parameters of coupled structural elements and acoustic volumes and it depends largely on an accurate estimate of three parameters: the modal densities, damping loss factors, and coupling loss factors of the subsystems. In this paper the modal density of rectangular composite plate made up of fiberglass is obtained by experimentally and theoretically. Also experimentally analyzed the effect of changes in fiber orientations on modal density the same sized plates, with different fiber orientations.

In wireless sensor networks, adversaries can easily launch multiple attacks on the application layer and the network layer, such as false report injection attacks and wormhole attacks. These attacks drain finite energy resources through false reports of compromised nodes, and devastate the constructed routing paths through the illegal messages of adversary nodes. Statistical en-route filtering (SEF) is proposed in order to drop the false reports against the false report injection attack, and a localized encryption and authentication protocol (LEAP) is proposed to detect the illegal messages to protect against wormhole attacks. When these attacks occur at the same time, SEF and LEAP should be operated simultaneously to confront the false reports and the illegal message in the sensor network. In this paper, we propose a method to improve the energy efficiency and the security level as compared to the simultaneous application of SEF and LEAP. In the proposed method, three types of new keys are designed to effectively detect these attacks, identifying and eliminating the redundancies. The effectiveness of the proposed method is verified through experimentation, and the results are compared to the simultaneous application of SEF and LEAP when the two attacks occur. The experiment results show that our proposed method saves up to 8% more energy, improves the security level of the compromised nodes up to 10%, and maintains the same detection power of the adversary nodes.

The sound field in train compartments, treated as a series of connected air cavities, is modelled using statistical energy analysis, SEA. For the case under study, with five cavities in series and the source in the second cavity, a closed--form solution is obtained. An adjusted SEA model is used to predict the rate of spatial decay within a cavity. The SEA model is validated using results from a ray tracing method and from scale model measurements. For the octave bands 500-4000 Hz, good agreement is shown between the results from measurements, the ray tracing and the SEA model, for the two saloons closest to the source cavity (a vestibule). The SEA model was shown to slightly underestimate the rate of spatial decay within a cavity. It is concluded that a reasonable cause is the additional diffusion due to the seating.

In the automotive industry, vehicle interior acoustics has become an important design criterion. Both legal restrictions and the growing demand for comfort, force the car manufacturers to optimize the vibro-acoustic behaviour of their products. The tendency to minimise the number of physical prototypes and to reduce the overall development cost and time, as well as the high performance of modern computer resources, clear the path for numerical prediction techniques to drive the noise, vibration and harshness optimization. The most commonly used numerical prediction techniques for vibro-acoustics are the element based methods, such as the finite element method and the boundary element method, and the statistical energy analysis method. However, the element based methods are only applicable in the lower frequency range and the statistical energy analysis method in the higher frequency range, leaving a frequency band, which is called the mid-frequency gap, for which no effective prediction technique is available at this moment. The wave based prediction technique has the potential to narrow this frequency gap. Like the element based methods, the wave based prediction technique is a deterministic technique, which is however computationally more efficient and thus suited for the mid-frequency range. This paper discusses the application of the wave based prediction technique for the vibro-acoustic analysis of a three-dimensional car-like cavity. Both uncoupled acoustic and coupled vibro-acoustic analyses are performed. A comparison with the finite element method is made with respect to convergence rate and computational efforts to show the efficiency of the new technique.

In the automotive industry, vehicle interior acoustics has become an important design criterion. Both legal restrictions and the growing demand for comfort, force the car manufacturers to optimize the vibro-acoustic behaviour of their products. The tendency to minimise the ...

Complex built-up structures often consist of structural elements of a periodic nature. Examples of these include plate or shell elements reinforced by stiffeners at regular intervals. This paper describes a theoretical method for evaluating the coupling loss factor (CLF) of a coupled periodic structure for use in the application of statistical energy analysis. The structure considered consists of a plate with periodic stiffeners coupled at right angles to a uniform plate and the method of travelling wave analysis was used for the investigation. An experimental program was conducted to verify the CLF of the structure. Results from the measurements of input power and vibrational energy were used in a numerical procedure to determine the internal loss factors and CLF. The experimental results show good agreement with theoretical predictions at low frequencies. However, some discrepancies between the theoretical and experimental results were observed at high frequencies, possibly due to the frequency dependent nature of the assumptions involved in the theoretical analysis.

For a number of years, a model well suited to medium and high frequencies in structures, and called Energy Flow analysis, has been studied in order to generalize Statistical Energy Analysis. This model is based on a thermal analogy: a law analogous to Fourier's law for heat flow is involved. This relationship, which relates the energy flow to the energy density, leads to a differential equation similar to the heat conduction equation in steady state conditions. The aim of this study is to generalize previous works on one-dimensional structures. A wave approach is adopted. It is shown that Fourier's law is valid for one symmetric propagation mode (one group velocity). However this law has to be modified for non-symmetric propagation modes or multi-mode propagation. In each case, the wave approach determines the relationship betwen energy density and energy flow. Finally, the theoretical models are illustrated with several examples of waveguides: an Euler-Bernoulli beam on an elastic support, pipes carrying moving fluid and a Timoshenko beam.

In statistical energy analysis, coupling loss factor is the essential parameter for vibro-acoustic analysis of complicated structures. The
coupling loss factors have been estimated using energy-level difference method. The tightening torque applied at structural junction
has been varied. Higher values of coupling loss factor have been observed for higher tightening torque on bolted junction. The
coupling loss factors have been determined for various structural junctions of rectangular composite plates. The riveted and bolted
junctions have been examined for composite plates in same plane and size. The coupling loss factors for bolted junction are relatively
higher than that for riveted junction of composite plates. The values of coupling loss factors are found to increase with increasing
tightening torque applied at structural junctions of composite plates. It is also noted that the experimental results of coupling loss
factors for point junctions vary with changes in fiber orientations of composite plates. It is firmly believed that the various findings of
the coupling loss factors in this article help for vibro-acoustic analysis of complicated structures.

This paper is concerned with the derivation of SEA equations from structural ray equations. Rays are assumed to be uncorrelated leading to the additivity of energy. Inside all subsystems, the energy density is the sum of a direct field from driving forces, a reflected field from the boundary and a transmitted field from adjacent subsystems. Assuming a "rain-on-theroof" excitation and a compact shape for subsystems, actual and fictitious sources on the boundary are found to be constant. Furthermore, if the attenuation of rays during a mean free path (normalized attenuation factor) is light, the field becomes diffuse i.e. homogeneous and isotropic. The net exchanged power between two adjacent subsystems is then proportional to the difference of energy densities and therefore, to the difference of modal energies. The derived proportionality coefficient is consistent with the well-known formula for coupling loss factor in terms of transmission factors. These results are illustrated by a numerical simulation for a multi-plate system. Finally, the validity domains of SEA and the ray theory are discussed and particularly the diffuse field assumption.

We propose a new approach towards determining the distribution of mechanical and acoustic wave energy in complex built-up structures. The technique interpolates between standard Statistical Energy Analysis (SEA) and full ray tracing containing both these methods as limiting case. By writing the flow of ray trajectories in terms of linear phase space operators, it is suggested here to reformulate ray-tracing algorithms in terms of boundary operators containing only short ray segments. SEA can now be identified as a low resolution ray tracing algorithm and typical SEA assumptions can be quantified in terms of the properties of the ray dynamics. The new technique presented here enhances the range of applicability of standard SEA considerably by systematically incorporating dynamical correlations wherever necessary. Some of the inefficiencies inherent in typical ray tracing methods can be avoided using only a limited amount of the geometrical ray information. The new dynamical theory -Dynamical Energy Analysis (DEA) -thus provides a universal approach towards determining wave energy distributions in complex structures.

As customer awareness of product sound grows, the
need exists to ensure that product sound quality is
maintained in the manufacturing process. To this end in process controls that employ a variety of traditional
acoustical and alternate sound quality metrics are
utilized, usually partly or wholly housed in a test
enclosure. Often times these test cells are required to
attenuate the background noise in the manufacturing
facility so that the device under test can be accurately
assessed. While design guidelines exist the mere size
and cost of such booths make an iterative build and test
approach costly in terms of materials as well as
engineering and testing time. In order to expedite the
design process and minimize the number of confirmation
prototypes, SEA can be utilized to predict the
transmission loss based upon material selection and
booth construction techniques. Additionally, SEA can be
used to identify those components of the booth that are
likely to be problematic from either a transmission loss
or a leakage perspective.

The composite materials having low weight to strength ratio and easy to manufacture due to which the usage of composites for different applications has aggrandized remarkably. Statistical energy analysis is used to predict the vibrational response of the complex structural systems. In the present work rectangular composite specimens have been used as an idealized subsystem. To analyse theoretically the vibra-tional response of these fabricated composite specimens employing glass fibers and different epoxy resins, Statistical Energy Analysis (SEA) method is implemented. The SEA response majorly depends on three parameters i.e. damping loss factors, modal densities, and coupling loss factors of the subsystems. In the present study, the damping loss factors of fabricated rectangular composite laminates have been estimated by employing both theoretical and experimental approach. Furthermore an experimental investigation has been employed to observe the effect of different fiber orientations on damping loss factor of the identical composite specimens. From the obtained results it has been observed that the damping loss factor for a composite plate with unidirectional fiber orientation is greater than the plates with quasi isotropic and cross ply orientations.

In industries, the use of appropriate junctions between components is of paramount interest. Coupling loss factor is one of the important parameters in statistical energy analysis for vibroacoustic analysis of complicated structures in drawing board stage. The values of coupling loss factor were calculated and compared for different junctions. The screwed and bolted junctions were examined for thin rectangular plates of same size. The energy level difference method was used to find coupling loss factors because of its simplicity. These experimentally found coupling loss factors were later compared with analytical solutions. It is noticed that the analytical results are in good agreement with experimental results. It is also observed that coupling loss factor for bolted junction are relatively high than that for screwed junction.

This paper presents an energy flow post-processing tool for structural dynamic analysis which is based on conventional dynamic finite element (FE) analysis. Based on the global dynamic behaviour of a system and on local subcomponent descriptions, the energy flow between the various components can be predicted, as can the energy dissipation within individual subcomponents. This information can assist a design engineer in making design modifications and holds potential in optimising high-frequency energy based models, such as statistical energy analysis (SEA) models. The developed approach is implemented using a commercially available FE code and is validated for both simple academic cases and for an industrial vehicle body-in-white.

8 A limit of Statistical Energy Analysis (SEA) is that of providing only the mean 9 values of the mechanical energy of a vibrating system. In this paper, the variability 10 of SEA solution under uncertain SEA parameters (coupling loss factors, internal 11 loss factors and injected powers) is investigated by comparing a sensitivity approach 12 and a Design of Experiment (DoE) approach. Uncertainties of the SEA parameters 13 depend on uncertainties in the physical properties of the considered mechanical 14 system. Numerical results are derived using a benchmark structure made by three 15 aluminum plates with a common junction and a launcher fairing. The analysis of 16 the effect of uncertainties of SEA parameters can be used for design purposes, i.e. to 17 identify which are the most effective areas to modify in order to control the energy 18 level of a given subsystem. 19 SEA estimates the mean value of the energy stored in the modal groups con-30 stituting the studied system. The mean value provided by SEA equations is 31 in principle the average response of a set of similar systems. However, SEA 32 equations are represented by a linear system of equations for each frequency, 33 or better for each frequency band. The solution of each linear system gives the 34 energy of each subsystem in a given frequency band. No average operation is 35 explicitly performed, but all the statistics is not visible to the user. In general, 36 this is not a problem because many simple relationships used by physicists 37 and engineers are the result of more complicated mathematical procedures. 38 Unfortunately, in this case this simple model holds only under many strong 39 hypotheses, listed in Section 2. 40 The linear system results from some mathematical manipulations, that include 41 also averages on frequency bands, on the classical equations of motion of multi 42 degrees of freedom systems, and the observance of the strong hypotheses men-43 tioned before. The coefficients of the linear system, named coupling loss factor 44 (CLF) and internal loss factor (ILF), are the result of these average processes 45 and account for the parameters of the native physical system [3]. Therefore, 46 SEA gives the energy of each modal group belonging to the studied system. 47 This energy is the most representative sample of a statistical population of 48 similar systems and on a frequency band. No information is given about the 49 dispersion of the data around the result. In order to provide a true statistic 50 solution it is necessary, at least, to know the variance of the result. A cor-51 rect procedure should calculate the variance of the energy by starting from 52 the equation of motion following a similar procedure like that performed for 53 the mean calculation. Lyon [1,2] estimates the variance by using a particular 54 probability distribution of natural frequencies. Therefore this procedure ne-55 glects a direct dependence of the modal parameters on the randomness of the 56 physical properties. However, he concludes that "There is a considerable area 57 of interesting research work that needs to be done in analysing variance of 58 interacting systems." 59 Radcliffe and Huang [4] study the problem by introducing a stochastic per-60 turbation in SEA equations, by using a first order approximation of them in 61 terms of this random perturbation and by calculating the variance of the new 62 linear equations. They state that the lack of information of SEA solution may 63 be filled by calculating the solution variance due to the random perturbation 64 of SEA parameters (coupling loss factor, injected power, ...). Langley and Co-65 toni [5] follow the Lyon's approach and study the problem by considering the 66 Gaussian orthogonal ensemble, detailed in Weaver [6]. Some other authors [7,8] 67 2 assume that uncertainties lie on the system parameters and they achieve re-68 sults not exhibiting the same tendency of the Lyon's prediction. Bussow et 69 al. [9] propose an analytical approach for the investigation of the problem of 70 uncertain system parameters. The analysis is performed by partial derivatives 71

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 behaviour 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 which are treated by either FEM or SEA depending on the local wavelength. Subsystems with relative long wavelengths are modelled 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 which can be solved for the mean energies in the short wavelength subsystems.

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 behaviour 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 which 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 which can be solved for the mean energies in the short wavelength subsystems.

A limit of Statistical Energy Analysis (SEA) is that of providing only the mean values of the mechanical energy of a vibrating system. In the proposed paper, the variability of SEA solution under uncertain SEA parameters (coupling loss factors and internal loss factors) is investigated by comparing a sensitivity approach and a Design of Experiment (DoE) approach. Uncertainties of the SEA parameters depend on uncertainties in the physical properties of the considered mechanical system (Young modulus, material density, geometry, ...). Numerical results are derived using a benchmark structure made by three aluminum plates with a common junction.

There are well-established techniques by which the coupling loss factors (CLFs) of statistical energy analysis (SEA) can be estimated from finite element analysis (FEA). These are typically based on a single, selected system. A slightly different choice of system would give different estimates. There is a need for robust methods that give good estimates of the SEA average CLFs, independent of the details of the chosen system. Estimates of variance, confidence limits, are also of interest. Two approaches to this problem are discussed. These involve attempts to randomise the properties of the system and averaging the resulting estimates, but without repeating the full FEA. The first involves perturbation of component modal properties which can be related to perturbations in the modes of the assembled structure and hence to the energies and CLFs. In the second approach, it is assumed that the statistics of the modal properties of the system analysed are a fair representation, when taken over a wide enough frequency range, of the statistics of the modes of the SEA ensemble. The modes of the system are then randomly sampled to provide robust estimates. Numerical examples are presented. The methods are computationally very cheap. r

Complex built-up structures often consist of structural elements of a periodic nature. Examples of these include plate or shell elements reinforced by stiffeners at regular intervals. This paper describes a theoretical method for evaluating the coupling loss factor (CLF) of a coupled periodic structure for use in the application of statistical energy analysis. The structure considered consists of a plate with periodic stiffeners coupled at right angles to a uniform plate and the method of travelling wave analysis was used for the investigation. An experimental program was conducted to verify the CLF of the structure. Results from the measurements of input power and vibrational energy were used in a numerical procedure to determine the internal loss factors and CLF. The experimental results show good agreement with theoretical predictions at low frequencies. However, some discrepancies between the theoretical and experimental results were observed at high frequencies, possibly due to the frequency dependent nature of the assumptions involved in the theoretical analysis.

This paper presents a new and efficient method to calculate point mobilities from subcomponents of a full structure. Subcomponent modelling is a commonly used method to gain information on the dynamic behaviour of complex assembly structures using smaller and more efficient models. For instance, point mobility calculations on subcomponent level are used to obtain more accurate input parameters for statistical energy analysis (SEA) models. A full system analysis is often too computationally expensive, so normally only individual subcomponents of the structure are extracted and analysed. This procedure yields a large reduction in computational effort, but also often results in a substantial loss of accuracy. This is due to the use of an approximation of boundary conditions to represent the eliminated remainder part of the structure, i.e. the full structure except the subcomponent at hand. Commonly, these boundary conditions are simplified by assuming clamped, free or simply supported edges. However, this is a huge simplification and may introduce large errors, especially in the low-and mid-frequency ranges. Earlier work has shown that a more accurate description of the boundary conditions can be achieved by describing the interface dynamics by a combination of so-called dynamic waves. In this paper, the method is developed further and a more robust and efficient wave extraction procedure is presented. An industrial body in white BIW is used as a test case and results are presented for three different cases. The results show that the wave-based boundary condition for point impedance calculations from a subcomponent model gives more accurate results than the results obtained with free or clamped boundary conditions.

There are well-established techniques by which the coupling loss factors (CLFs) of statistical energy analysis (SEA) can be estimated from finite element analysis (FEA). These are typically based on a single, selected system. A slightly different choice of system would give different estimates. There is a need for robust methods that give good estimates of the SEA average CLFs, independent of the details of the chosen system. Estimates of variance, confidence limits, are also of interest. Two approaches to this problem are discussed. These involve attempts to randomise the properties of the system and averaging the resulting estimates, but without repeating the full FEA. The first involves perturbation of component modal properties which can be related to perturbations in the modes of the assembled structure and hence to the energies and CLFs. In the second approach, it is assumed that the statistics of the modal properties of the system analysed are a fair representation, when taken over a wide enough frequency range, of the statistics of the modes of the SEA ensemble. The modes of the system are then randomly sampled to provide robust estimates. Numerical examples are presented. The methods are computationally very cheap. r

There are well-established techniques by which the coupling loss factors (CLFs) of statistical energy analysis (SEA) can be estimated from finite element analysis (FEA). These are typically based on a single, selected system. A slightly different choice of system would give different estimates. There is a need for robust methods that give good estimates of the SEA average CLFs, independent of the details of the chosen system. Estimates of variance, confidence limits, are also of interest. Two approaches to this problem are discussed. These involve attempts to randomise the properties of the system and averaging the resulting estimates, but without repeating the full FEA. The first involves perturbation of component modal properties which can be related to perturbations in the modes of the assembled structure and hence to the energies and CLFs. In the second approach, it is assumed that the statistics of the modal properties of the system analysed are a fair representation, when taken over a wide enough frequency range, of the statistics of the modes of the SEA ensemble. The modes of the system are then randomly sampled to provide robust estimates. Numerical examples are presented. The methods are computationally very cheap. r

The problem of determining the ensemble averages of vibration energy flow in harmonically driven trusses with random parameter variations is considered. The mass, elasticity, damping and length of the truss members are modelled as random variables. The uncertainty associated with member lengths results in randomness in truss geometry. Four different linear damping models for truss members, namely, strain rate dependent viscous/hysteric models and velocity dependent viscous/hysteric models are considered. ...

A statistical energy analysis (SEA) approach is used to predict the sound transmission loss (STL) of sandwich panels numerically. Unlike conventional SEA studies of the STL of sandwich panels, which consider only the antisymmetric (bending) motion of the sandwich panel, the present approach accounts for both antisymmetric and symmetric (dilatational) motions. Using the consistent higher-order sandwich plate theory, the wave numbers of the waves propagating in the sandwich panel were calculated. Using these wave numbers, the wave speed of the propagating waves, the modal density, and the radiation efficiency of the sandwich panels were determined. Finally, the sound transmission losses of two sandwich panels were calculated and compared with the experimentally measured values, as well as with conventional SEA predictions. The comparisons with the experimental data showed good agreement, and the superiority of the present approach relative to other approaches is discussed and analyzed.

This paper presents a theoretical and experimental study of the transmission of vibration through a two element structure which consists of a cylindrical shell coupled to an end plate. The first part of this paper deals with the derivation of coupling loss factor (CLF) for the study of vibration transmission using Statistical Energy Analysis (SEA). The derivation is based on travelling wave analysis and the assumption of equipartition of energy amongst resonant modes. Numerical results for three examples of cylinder/plate structures are presented and compared with those of their equivalent plate/plate structures. The results show that the CLF values of the cylinder/plate structures asymptote to their equivalent plate/plate structures above the ring frequency of the cylinder. In the second part of this paper, an experimental program for measuring the CLF of an example cylinder/plate structure is described. Experimental results of the cylinder/plate structure show good agreement with theoretical predictions and confirm the validity of the present formulation of CLF. 7 1997 Academic Press Limited