Statistical Energy Analysis Method Research Articles

Dynamic Receptance Analysis Combined with Hybrid FE-SEA Method to Predict Structural Noise from Long-Span Cable-Stayed Bridge in Urban Rail Transit

In this study, dynamic receptance analysis (DRA) is proposed and combined with hybrid finite element (FE)-statistical energy analysis (SEA) method to accurately predict structural noise from long-span cable-stayed bridge (LSCB) with steel box composite girder (SBCG) in urban rail transit (URT). To begin with, a vertical vehicle–track coupling model in frequency domain is established based on DRA, in which the rail is represented by an infinite Timoshenko beam supported by a series of fasteners that are regarded as springs with complex stiffness. The floating slab is regarded as the Euler beam with both ends free supported by steel springs. Using this model, the spectrums of the wheel–rail force and the forces transferred to the bridge can be efficiently obtained by taking rail roughness as the excitation. Due to the low modal density of the concrete deck, the hybrid FE-SEA method is introduced to establish the noise prediction model, in which the discontinuity caused by using distinct models for different frequency bands is avoided. Then the on-site noise tests of a LSCB with SBCG in URT are carried out to verify of the proposed method. Finally, based on the prediction method, the acoustic contributions of the bridge components are analyzed in detail. The force transfer characteristics as well as the noise reduction effects of different track structures are thoroughly investigated, so as to provide reference for the future research on bridge-borne noise control.

Assessing the potential to use structure-borne sound to detect survivors in collapsed buildings

When victims are trapped within a collapsed structure, it can be difficult to quickly locate survivors. Airborne sound tends to be highly attenuated hence there is greater potential to detect physical movement or signals from survivors by measuring the seismic response. Recent work has used Statistical Energy Analysis (SEA) and Finite Element Methods (FEM) to assess the feasibility of predicting structure-borne sound transmission across fragmented reinforced concrete structures. This has the potential to improve strategies for the detection of trapped survivors using structure-borne sound by search and rescue teams as well as seismic detection equipment. The main focus has been on vibration transmission between reinforced concrete elements that are in contact after the collapse. Laboratory experiments have been used to validate FEM models of reinforced concrete beam junctions with surface-to-surface contact conditions using Experimental Modal Analysis. This has given insights into the contact stiffness between reinforced concrete beams. FEM simulations of beams with surface-to-surface contacts as well as beam junctions with fragmentation at the junction indicate the potential to use SEA to give reasonable estimates of vibration transmission. This paper summarises the progress made to-date and considers the next steps that are needed in the research on this topic.

A unified hybrid Ritz-SEA acoustic vibration coupling method of a rectangular plate coupled with fast multipole boundary integration

The Ritz method, based on a segmental strategy, is known for its fast convergence and ability to handle arbitrary boundary conditions while maintaining accuracy comparable to benchmark solutions. It is particularly effective in calculating vibrations in composite laminates, functionally graded plates, and other novel plate types. However, this method faces challenges in accurately calculating the mid-frequency problems that arise between plate structures and acoustic cavities. To fill this gap, this paper proposes a Ritz and statistical energy analysis (Ritz-SEA) hybrid method for calculating rectangular plate acoustic vibration coupling in the mid-frequency range for both deterministic and random loads. This method utilizes the fast multi-pole boundary element integral equation and region segmentation mapping as the governing equation for the coupling boundaries. The power flow equation of the statistical energy analysis (SEA) method is incorporated to solve for the average sound pressure level and sound power level of the coupled acoustic cavity. A method that combines the CHIEF and Burton-Miller methods has been proposed to address the issue of non-unique solutions. Validation and parametric studies are conducted. The results demonstrate that this approach can effectively filter out random fluctuations in mid-frequency domains while exhibiting exceptional stability and precision.

Investigation of statistical energy analysis coupling loss factors of vibro-acoustic systems under thermal environment

Vibro-acoustic systems in aviation engineering could operate in a thermal environment and suffer from high-frequency dynamic excitations. The high-frequency vibro-acoustic problems are commonly solved by the Statistical Energy Analysis method, in which the Coupling Loss Factor is the key parameter for modeling vibro-acoustic systems. This work investigates the CLFs of vibro-acoustic systems under thermal environment. First, based on the modal data considering different thermal effects, the modal CLFs between structural modes and acoustic modes are predicted by the Statistical modal Energy distribution Analysis. Then, the CLFs between the structural subsystem and the acoustic subsystem are calculated. The simulation models employed in this research include the coupled plate-cavity system and the cylindrical shell-cavity system. The influence of thermal effects on the CLFs is discussed. For the investigated cases, in which the boundaries of structural susbsystem are set to simply supported, the results demonstrate that the CLFs decrease with increasing temperature when only the temperature-dependent material properties are considered. However, when only the additional stress stiffness is considered, the CLFs experience a more dramatic decrease compared to when only the temperature-dependent material properties are taken into account. Moreover, when both effects are considered, the influence of the additional stress stiffness has a leading effect on the CLFs. The proposed method can be conveniently applied to various vibro-acoustic systems operating in the thermal environment.

Investigation of sound transmission through a finite length cylindrical shell in the presence of an exterior turbulent boundary layer

This paper studies sound transmission through a thin cylindrical shell of finite length. The structure is subjected to an exterior turbulent boundary layer (TBL) excitation and the boundary conditions of its two ends are considered as simply-supported. The classical shell theory (CST) is employed to derive wave propagation in the cylindrical shell, and the Corcos model is applied to describe pressure fluctuation due to TBL. According to the existence of two axial and radial modes, a complete convergence study is presented. To examine sound transmission through the structure, a comprehensive parameters study is provided on the resulting power spectral density (PSD) of the kinetic energy. To validate the results of the present method, the statistical energy analysis (SEA) method is used. For this purpose, modeling is performed by SEA in VA One software, which shows the high accuracy of the results of the present method. The analytical predictions suggest that the sound insulation performance of such a finite length structure is enhanced considerably by increasing the thickness of the shell in a broad-band frequency range. Similar effects on sound transmission are observed for Young’s modulus and density of the shell particularly in the low and high-frequency ranges, respectively. Whereas, enhancing the turbulent pressure and freestream velocity adversely affects sound radiation into the structure.

A simulation study on the influence of aircraft panel thickness on the cabin sound quality

Currently, noise control aims to optimise the psychoacoustic characteristics of aircraft interior noise to improve comfort and security during a flight. This study investigates the influence of transmission loss of aircraft panels on the sound quality of the cabin. We established an aircraft cabin acoustic model based on the statistical energy analysis method. The psychoacoustic parameters (i.e., loudness, sharpness, roughness, psychoacoustic annoyance, and articulation index) were simulated with the varying thickness of the aluminium alloy layer. Results showed that loudness and roughness significantly reduced, whereas sharpness increased with increasing thickness. The noise annoyance also declined as the thickness increased. However, the articulation index appeared a sudden drop when the thickness increased.

Effects of installing concrete deck on noise radiation of steel box girders: Mechanisms, modeling, and analysis of influential factors

Steel–concrete composite box girder (SCC BG) is widely used in engineering because it can make full use of the tensile properties of steel and the compressive properties of concrete. However, research on the vibrational transmission and vibroacoustic response of the SCC BG is still relatively rare. In this study, a laboratory hammering test was performed to obtain the dynamic characteristics of a scaled specimen of an SCC BG before and after installing a concrete deck. The experimental results revealed that the vibroacoustic response was greatly reduced after the installation of concrete deck due to a reduction in the transmission of on-deck vibrations to the steel box girder. Following this, we introduce a model to predict the structural vibroacoustic response of the SCC BG based on the hybrid finite element–boundary element–statistical energy analysis method, and verify its accuracy through experiments. The accurately calibrated model is then used to study the characteristics of vibration transmission between plates of the SCC BG. The influences of the thickness of the concrete deck, concrete material, and degree of shear connection on vibroacoustic responses of the SCC BG are also discussed.

A hybrid meshless–statistical energy analysis method for complex structure vibration analysis

A new hybrid deterministic–statistical energy analysis (SEA) formulation is presented by introducing a meshless method for modeling the deterministic components. Moving least square Ritz (MLSR) meshless method is applied, in which MLS is used to build the discrete model and the Ritz method allows to obtain variational formulation of the deterministic components of the governing equations. Such governing equations can be formulated via boundary conditions by penalty method and Lagrange multipliers. The hybrid model by penalty method keeps a similar formulation with the framework of the finite element SEA, while the model by the latter increases the size of the dynamic stiffness matrix and the expanded components are determined by the constraints. For validation purpose, three case studies are provided, including beam–coupled plates and plate–coupled plates built-up structure. The results by the hybrid MLSR-SEA model are compared with those by FE-SEA and Monte Carlo simulation. Good agreements of responses between the methods demonstrate the reliability of the MLSR-SEA formulation.

Study on Interior Aerodynamic Noise Characteristics of the High-Speed Maglev Train in the Low Vacuum Tube

As the next-generation high-speed transportation system, the low vacuum tube high-speed maglev system combines the tube with a certain degree of vacuum with the high-speed maglev train, which can realize high-speed operation under low aerodynamic resistance and noise mode. In order to study the interior aerodynamic noise characteristics of the high-speed maglev train in the low vacuum tube, a computational model of the external flow field of the high-speed maglev train in a low vacuum tube was established, and the computational model of the interior aerodynamic noise of the high-speed maglev train was established using the statistical energy analysis method; then the interior aerodynamic noise characteristics of the high-speed maglev train in the low vacuum tube were studied. The research results show that in the low vacuum tube, the distribution of the interior aerodynamic noise of the high-speed maglev train shows the characteristics of large head car and tail car and small middle car, and the aerodynamic noise on the top of the car is smaller than that on the floor. With the increase in frequency, the sound pressure level of the interior aerodynamic noise of the high-speed maglev train has the tendency of increasing first and then decreasing, and the main energy of the interior aerodynamic noise is distributed in the range of 200–1000 Hz. From the perspective of the total sound pressure level of the interior aerodynamic noise, the interior aerodynamic noise of the tail car is the greatest, followed by the head car, and the interior aerodynamic noise of the middle car is the smallest. As the direction of the travel of the maglev train will change, the optimization design of the interior aerodynamic noise of the head and tail cars should be emphasized.

A hybrid modal/statistical formulation for predicting the energy response of vibroacoustic systems in the mid frequency range

The Statistical modal Energy distribution Analysis (SmEdA) method predicts the power flow between coupled subsystems excited by random excitations from a deterministic modal description of the uncoupled subsystems. As the modes can be computed by Finite Element Method (FEM) for complex subsystems, it can be seen as an extension of FEM to the mid frequency range where the modal densities of subsystems are not too high. Conversely, the Statistical Energy Analysis (SEA) method is a statistical approach predicting the mean power flow of a population of similar structures presenting manufacturing uncertainties. Assuming a diffuse field within each subsystem, it is dedicated to the high frequency range where modal densities of subsystems are high. However, in many applications, subsystems with low and high modal densities can coexist in the mid frequency range and in that case neither SmEdA nor SEA is well adapted. The purpose of this article is then to propose a hybrid SmEdA/SEA formulation allowing some subsystems with low modal densities to be described by SmEdA and other ones by SEA. For the SEA-described subsystems, the vibratory field of the statistical population is supposed to be diffuse. These subsystems are then characterized by sets of natural frequencies and mode shapes constructed from the Gaussian Orthogonal Ensemble matrix and the cross-spectrum density of a diffuse field, respectively. In another hand, the SmEdA-described subsystems are represented by their modes that can be extracted by usual computer codes. In order to couple the two models, Monte Carlo simulations are used for generating samples of the stochastic modes of the SEA-described subsystems. From the distribution of the estimated energy response of the coupled subsystems, the ensemble average and the confidence interval can finally be estimated. For validation purpose, the results of the proposed hybrid SmEdA/SEA approach are compared to the numerical results computed with the finite element method (FEM) on a population of plate-cavity systems having similar properties. A good agreement is observed whereas the computation time of the proposed approach is much less important than the one of the FEM which can be up to several days for each element of the population.

An equivalent modelling approach for vibroacoustic characteristics analysis of railway aluminium extrusion

For railway carbody structures with irregular cross-sectional shapes and great longitudinal lengths, it is difficult to obtain their vibroacoustic characteristics using traditional computational methods. Moreover, traditional methods are time intensive and not conducive to the optimisation design of such complex and large structures. This study introduces an equivalent modelling approach for vibroacoustic analysis, i.e., sound transmission loss and sound radiation analysis, of railway aluminium extrusion. The aluminium extrusion is equivalent to a homogeneous orthotropic plate, and the equivalent mechanical properties of the plate are obtained via static analysis based on the small-deflection theory. The vibroacoustic characteristics are obtained using the hybrid method of finite element and statistical energy analysis. First, the equivalent modelling approach is validated using the results of a detailed prediction model and experimental test. Subsequently, the applicability of the equivalent approach is analysed by comparing it to the predicted results of the detailed model with different dimensions and boundary conditions. Ultimately, the minimum longitudinal length that needs to be considered for the analysis of railway aluminium extrusion is discussed using the equivalent approach and the computational efficiency is compared. The results reveal that the equivalent approach can effectively simulate the vibroacoustic performances of the aluminium extrusion in 1/3 octave bands centred below 1000 Hz and reduce the calculation time by 90% compared with the detailed prediction models of aluminium extrusion, which provided a foundation for vibroacoustic optimisation design of railway carbody structures.

Development of High-Fidelity Numerical Methodology for Prediction of Vehicle Interior Noise Due to External Flow Disturbances Using LES and Vibroacoustic Techniques

A pleasant and quiet cabin in driving a car is one of the most critical factors affecting a customer’s choice in a market. As the traditional noise sources such as power trains become less, the relative contribution of aerodynamic noise to the interior noise of a road vehicle becomes even more critical. In this study, a high-fidelity numerical methodology is developed for the reliable prediction and analysis of the interior transmitted noise due to external flow disturbance. The developed numerical methodology is based on the sequential application of the high-resolution LES technique, wavenumber–frequency transform, and vibroacoustic model. First, the compressible LES techniques with high-resolution grids are employed to accurately predict the external turbulent flow and aeroacoustic fields due to the turbulent flow, at the same time, of a vehicle running at a speed of 110 km/h. Second, surface pressure fluctuations on the front windshield and side windows, obtained from the LES simulation, are decomposed into incompressible and compressible ones using the wavenumber–frequency transform. Lastly, the interior sound pressure levels are predicted using the vibroacoustic model, which consists of the finite element (FE) and statistical energy analysis (SEA) methods. For the efficient computation of the vibroacoustic interaction between the vibration of the vehicle windows and the acoustic field inside the cabin room, the FE and SEA methods are applied in low- and high-frequency ranges, respectively. The predicted interior sound pressure spectral levels agree well with the measured ones. In addition, although the magnitudes of the compressible pressure components are generally lower than those of the incompressible ones, the compressible field is found to contribute more to the interior noise in high-frequency bands. The physical mechanism of the higher transmission is shown to be related to the coincident effect between the compressible pressure field and the structural vibration of the vehicle window.

Study on Sound Radiation Characteristics and Sound Insulation of Extruded Aluminium Plate

The use of extruded aluminium plates in the subway car body can reduce the weight of the car body, but it will bring about the negative result of increased noise in the car. The extruded aluminium panels belong to rib-stiffened plate structure in which both local short wave deformation and whole long wave deformation coexist. Studying the vibro-acoustic characteristics of this structure helps to effectively control the noise in the car. In this paper, Acoustic model of extruded aluminium plate and equivalent plate were both built up based on the hybrid method of finite element and statistical energy analysis(FE-SEA) and the statistical energy method (SEA) respectively. According to the theory of statistical energy analysis(SEA), the equivalent modelling method of the aluminium extrusion was presented using a single plate This paper focuses on the sound radiation characteristics and sound insulation of extrude and equivalent structures. Results show that no matter what method is used, the calculation of the sound power and sound insulation of the two structures has the same trend. However, the results of the FE-SEA method can capture the natural frequency of the board and be more accurate reflect vibro-acoustic characteristics of extruded aluminium panels in the middle and low frequency. Equivalent plate can well predict the sound radiation characteristics and sound insulation in the mid and high frequency by using the SEA parameters of the aluminium extrusion.

Sound transmission paths through a statistical energy analysis model of mechanically linked aircraft double-walls

Sound transmission loss (TL) through mechanically linked aircraft double-walls is studied with a statistical energy analysis method. An overview of the method is given with details on acoustic and structural transfer path analysis. The studied structure is composed of a thick composite sandwich panel representative of a skin panel, lined with an acoustic insulation layer (glass wool), and structurally connected via vibration isolators to a thin composite sandwich lining panel representative of a trim panel. Two types of vibration isolators are considered: a soft and rigid mechanical link. Various experimental methods were used to assess the accuracy of this model. This study shows the robustness of the simple four-pole modeling of isolators, which depends mainly on the importance of correctly determining the experimental dynamic stiffness of typical aircraft vibration isolators. The prediction of the TL while acceptable was, however, found less satisfactory for the soft configuration. This is traced to the uncertainties on the used coupling loss factor. Finally, a transfer path analysis is performed to identify the contribution of each transmission path in the entire frequency range of interest. Results show that non-resonant airborne transmission dominates in low frequencies, the airborne radiation is significant in the critical frequency region of the panels, while the structure-borne radiation increases the noise transmitted in the mid- and high-frequency ranges.

On the analysis of coupling strength of a stiffened plate-cavity coupling system using a deterministic-statistical energy analysis method

This paper combines the Impedance Mobility Compact Mothed (IMCM) with Statistical Energy Analysis (SEA) method to develop analytical structural-acoustic approach for analysis of a plate-cavity coupling system. The coupled system is composed of an acoustic cavity covered with a rectangular flexible plate that is stiffened by a beam integrated to the inner surface of the plate to modify the coupling strength between the plate and the air inside the cavity. This paper discloses the relationship between the coupling strength and coupling quotient. The results of the study indicate that the coupling strength between the plate and acoustic cavity can be determined by the natural frequency proximity and the matching degree of the mode shapes of the plate and acoustic cavity. Addition of stiffener onto the plate, increase of the plate thickness or decrease of the cavity volume will reduce the coupling strength. The newly proposed coupling coefficient can better quantify the coupling characteristics of the plate and cavity system in the low-middle frequency ranges. The results assessed by the proposed coupling coefficient have been verified by those assessed by the previously defined coupling coefficient and the coupling quotient in the SEA model in the high frequency range.

Acoustic Packet Optimization of Tire Noise Based on SEA Method

As the size of the engine decreases and its performance increases, its noise also decreases, and tire noise has become an important source of vehicle noise. Based on the Statistical Energy Analysis (SEA) method, use Hypermesh and VA-One software to establish a SEA model; through the vehicle PBNR simulation analysis and test benchmarking, the analysis model was calibrated and the accuracy of the simulation analysis was improved. In this research, a new TPNR calculation method is proposed to evaluate the noise reduction effect of tire noise excitation to the car. The acoustic performance is evaluated based on the calibrated SEA analysis model by optimizing the thickness, leakage rate and coverage of the acoustic bag at the luggage compartment floor, front and rear floors, and front and rear wheel covers. It is found that the optimization scheme for the front and rear floors and the luggage compartment floor has better effect, but the optimization effect for the rear wheel cover is not obvious.

An improved hybrid FE-SEA model using modal analysis for the mid-frequency vibro-acoustic problems

Based on the hybrid finite element (FE)-statistical energy analysis (SEA) method, an improved hybrid model with less time-consuming is proposed for the mid-frequency vibro-acoustic problems. Mode shapes are used to reduce the order of the deterministic acoustic component modeled by the finite element method in this paper. The governing equation of the residual response, which is derived to correct the modal truncation error, can be reduced by constructing a set of basis vectors. The model order reduction (MOR) equation of the acoustic component is formulated through a combination of the truncated mode shapes equation and the reduced residual response equation. The dimension of the overall equation of motion is efficiently decreased. The statistical structural component is modeled by SEA, and the responses of a mid-frequency vibro-acoustic system can be solved after introducing the diffuse field reciprocity relationship. Based on the efficient correction of the modal truncation error, the proposed hybrid MOR-SEA method promotes the reduction of order using mode shapes in mid-frequency vibro-acoustic systems. Compared with the traditional hybrid FE-SEA method, the present method has an evident advantage in computational efficiency, as shown in two numerical examples.

Simulations and Ground Experiments of Interior Noise for Space Station Module

China Space Station (CSS) module is currently under construction. Considering astronauts will be working and living on orbit for several months, a comfortable acoustic environment in manned space station must be provided. This paper investigates the interior noise effects for CSS using two typical vibro-acoustic analysis methods, that is, coupled finite element/finite element method (FE/FEM) and statistical energy analysis (SEA) method. The simulation models are validated through ground experiments by the noise test for space station simulation module. The combination of the above two methods can well predict the acoustic levels for both low and high frequencies. A ground experiment scheme for interior noise of space station module is also proposed on the basis of its noise characteristics. Sound power equivalent method is utilized to simulate a single noise source and multi-input multi-output (MIMO) closed-loop control method is employed for multiple noise sources. It is shown that the sound field simulation results are in perfect agreement with the simulation requirements of noise sources.

Experimental Studies of the Elastic-Dissipative Properties of Structural Materials and the Calculation of Sound Insulation of Building Partitions Based on Refined Characteristics by the SEA Method

An analytical solution to the problem of sound and vibration transmission through structures, taking into account the energy exchange with adjacent structures, requires determination of the elastic-dissipative properties of structural materials: the elastic modulus and the loss factor. In this work, the parameters of the elastic modulus and the loss factor of three known building materials are investigated and experimentally established in comparison with previously obtained data of other authors. For the automation and accuracy of measuring the internal loss factor, a new measurement technique was transferred using Zetlab software and digital equipment. The calculation of sound insulation of a single-layer enclosing structure is performed based on the method of statistical energy analysis (SEA) using the measured data of the elastic modulus and loss factor. In order to determine the reliability of the data obtained for the elastic modulus and the coefficient of internal loss of structural materials, a comparison was made of the values of sound insulation of a single-layer panel made of gypsum fiber board, obtained by means of experimental research, using the calculated values of sound insulation calculated using the elastic modulus and the coefficient of internal losses from literature sources and using the calculated sound insulation values calculated using the measured values of the modulus of elasticity and the internal loss factor.

A similitude for the middle-frequency vibration response of satellite solar array based on the wave coupling hybrid finite element–statistical energy analysis method

A similitude is proposed for the dynamic response analysis of a satellite solar array. It is defined by invoking the wave coupling hybrid finite element–statistical energy analysis method which is considered as an effective way to solve the wide frequency (especially middle-frequency) vibration response of complex structures. The scaling laws are then derived by looking for equalities in the structural responses. A numerical case involving an assembly of two plates is discussed to verify the proposed scaling laws. Moreover, the new similitude is applied to a simplified satellite solar array model. If the analysis frequency step is selected appropriately, it is shown that a complete similitude can be defined: it allows predicting the vibration response of satellite solar array and provides the possibility for the model test of complex structures.