Finite Element Statistical Energy Analysis Research Articles

Similitude laws for dynamic response of beam-plate coupled structure

The beam-plate coupled structure is a typical structure in spacecraft solar wings, and the prediction of its mechanical environment is crucial for overall design, structure and mechanism subsystem design. However, it is usually difficult to perform ground simulation tests on spacecraft solar wings with full-size structure. Therefore, a scaled model is established based on similarity theory to forecast the dynamic response of the full-size structure. In this article, finite element (FE) and hybrid finite element-statistical energy analysis (FE-SEA) models of the beam-plate coupled structure are established according to the band division principle. Finite element model allows for detailed analysis of structural behaviour, while hybrid models incorporate statistical energy analysis to account for high-frequency dynamics. By integrating these approaches, a comprehensive understanding of the coupled structure’s response can be achieved, facilitating efficient design and optimisation processes. The scaling laws of the dynamic response are obtained by combining the dimensional analysis and equation analysis methods, respectively. The scaled models are designed for verification, and the natural characteristics and vibration responses predicted by the scaled models are basically consistent with those of the prototype. The findings indicate that the prediction accuracy derived from the FE-SEA model is higher, which provides a novel approach for the dynamic similarity of complicated structures such as the solar wings of spacecraft.

Research on modeling joining and joining modification method of hybrid FE-SEA model

Since the connection method at the boundary of the hybrid FE-SEA (finite element-statistical energy analysis, FE-SEA) model affects the overall calculation accuracy of the model, the missing or inaccurate connection relation will lead to a large error in the model calculation. In order to effectively solve the calculation error caused by the connection problem of hybrid FE-SEA model, this paper starts from the modeling methods and connection relations of “hybrid-point” connection, “hybrid-line” connection and “hybrid-surface” connection. In order to solve the modeling method and correction problems of the “hybrid-point” connection in the hybrid FE-SEA model, the “out-of-plane” wave motion equation at the “hybrid-point” connection was established by using the superposition principle of plane waveforms in polar coordinate system. The “hybrid-point” connection and wave-number relation in Cartesian coordinate system are studied. The radiation radius correction method of “hybrid-point” connection is proposed. An example is given to verify the effectiveness of the method. In order to solve the modeling and correction problem of “hybrid-line” in hybrid model, a “hybrid-line” connected triangular waveform function model was established by using the method of linear difference. The direct field dynamic stiffness matrix of “hybrid-line” connection in node coordinate system is studied. According to the shape function of the “hybrid-line” connection wavenumber space, a method to correct the shape function of the “hybrid-line” connection is proposed, and the validity of the method is verified.

Topology optimization of single and double panels for improved sound insulation in specific frequency bands

Single and double panels are commonly employed in noise control, e.g., as partitioning elements between rooms and to enclose noisy machines. In presence of narrowband disturbances, related for example to rotating machines, further improvements in sound insulation can be obtained by properly distributing the material within the area of the panels. This allows maximizing the sound Transmission Loss (TL) in the frequency band of interest by controlling structural resonances and antiresonances. In order to achieve this goal, in this work, we apply numerical topology optimization to find the optimal material thickness distribution for single and double panels. In the iterative design process, single panels are schematized through a simple mechanical Finite Element (FE) model, while double panels are modelled considering the vibroacoustic coupling between the two mechanical plates and the internal acoustic air cavity. The sound insulation properties of the designed solution are evaluated by TL computations through hybrid Finite Element-Statistical Energy Analysis (FE-SEA) simulations. The method is applied targeting different frequency bands in the audible range and focusing on practically relevant design cases, such as single and double PMMA and glazing panels.

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.

Extension of a hybrid FE-SEA approach for vibro-acoustic analysis with prescribed displacements

A hybrid Finite Element-Statistical Energy Analysis (FE-SEA) method has been developed and improved over the past fifteen years to analyse complex vibro-acoustic built-up systems, comprised by both deterministic and statistical components, and the FE-SEA equations have been derived considering that the power input to the structure is due to prescribed forces applied to the degrees of freedom of either the deterministic and/or the statistical components. In this paper, it is presented a generalised set of FE-SEA equations to estimate the response of a system when prescribed displacements contribute to the power input to the complex system. It was found that the equations are not necessarily reversible, meaning that if a set of displacements estimated from the hybrid approach with prescribed forces are used as input to the same randomised structure, the response will be altered due to its statistical nature. The generalised set of FE-SEA equations is validated against numerical data of FE Monte Carlo simulations of two complex built-up structures, where the input is given to the statistical system and to a deterministic resonant structure, respectively.

Application of FE-SEA approach in investigation of track properties influences on railway rolling noise generation

One of the main concerns in the railway transportation is environmental pollutions caused by railway noise. The most important source of this noise is the rolling contact between wheel and rail, and therefore, prediction of the rolling noise is a key factor in the management and reduction of the noise. Despite numerous studies carried out on the rolling noise prediction in the available literature, the role of track components on the rolling noise has not been taken into account. That is, the power dissipation within the track superstructures has not been given sufficient consideration. This is addressed in this research by investigating the influences of the track components on the level of railway rolling noise. For this purpose, a new noise prediction approach is developed in this research using finite element method and the statistical energy analysis. This approach has an advantage of considering power dissipation along the track components compared with the previous approaches. The results obtained from the new approach were compared with field measurements carried out in this research. It is shown that the proposed finite element–statistical energy analysis approach is effective in accurate predictions of the ballasted railway track noise.

Equivalent linearisation in a hybrid FE-SEA approach for nonlinear vibro-acoustic modelling

Hybrid techniques, such as a Finite Element-Statistical Energy Analysis (FE-SEA), have been widely employed to model vibro-acoustic systems in order to estimate the ensemble averaged response due to external inputs, assuming that the governing equations of motion of the comprising structures and connections are linear. However, nonlinear behaviour can be found in the interfaces between the point where the input is applied and the system itself, resulting in the force being transmitted through a nonlinear path, that might affect the structural or acoustic response of the system depending on the amplitude of the input. With the aim of improving the estimation of the mean squared response of the system subjected to random inputs through a nonlinear interface, an equivalent linearisation approach has been adopted to linearise the equations of an existing FE-SEA method that supports prescribed displacements as inputs rather than forces. Results from numerical simulations of a nonlinear single degree of freedom system demonstrate the ability of the equivalent linearisation approach to estimate the mean square response with a minimum error, even for systems with strong nonlinearities. The method here developed has been validated against experimental data collected from a vibro-acoustic system excited through a nonlinear massless and undamped spring.

The Shock Response Prediction of Spacecraft Structure Based on Hybrid FE-SEA Method

An improved method based on the Hybrid Finite Element-Statistical Energy Analysis (FE-SEA) method and quasi-steady state theory is proposed to predict the response of spacecraft structure during the process of pyrotechnics separation. Firstly, the amplitude–frequency value of shock load is obtained by using time-frequency conversion technology. Then, according to the frequency response characteristics of each part of the spacecraft structure, a more accurate hybrid FE-SEA model is established. The piecewise loading method is used to predict the response of the hybrid model. Finally, the time domain response results are reconstructed, and the shock response spectrum (SRS) is calculated. Based on the test system of simulating pyroshock, the shock experiment of spacecraft structure is conducted. The high frequency and high velocity character of pyroshock could be effectively simulated, and an accurate shock force function could be obtained through the experiment. This indicates that the numerical results are in line with the ones of the experiment. The SRS obtained from experiments and calculations have similar trends and amplitudes. This conclusion verifies the rationality and sufficient accuracy of the novel method in this paper. The novel method presented in this paper greatly improves the computational efficiency. At the same time, it provides theoretical guidance for shock response prediction of spacecraft structure by steady-state methods.

Study on sound transmission loss modeling through simplified sealing specimens and an automotive door sealing system

An imperfect sealing system would be the main path of the noise propagating into the interior compartment of a high-speed vehicle. The study on the sound insulation modeling of automotive door sealing systems, which involve complicated threedimensional sealing structures and gap cavities, has attracted more and more attention. This study employs hybrid finite element–statistical energy analysis (FE-SEA) models to predict the sound transmission loss of three simplified sealing specimens and an actual automotive door sealing system. For the actual sealing system under compression, a three-dimensional FEmodel is built to simulate the nonlinear compression, which can acquire the compressed geometries and pre-stress modal results of the seals for further prediction of the sound transmission loss. The hybrid FE-SEA method is firstly verified by the experimental result of a double plate vibro-acoustic system and another numerical method. Several factors concerning the modeling, including the boundary conditions, the equivalent elastic modulus for the hyper-elastic rubber, the specimen length, and the structural grid size, are considered to study their impacts on the sound transmission loss. The effects of using shell elements and using solid elements to model the sealing rubber layers are also compared. The results of this study can provide guides regarding the trade-off between the modeling efficiency and accuracy, so that it has significance for engineering modeling, as well as the design and optimization of automotive door sealing systems.

Evaluation and optimization of sound quality in high-speed trains

The rapid development of high-speed railways has been tempered by various technical challenges. As more high-speed lines carrying more high-speed trains have opened in China, abnormal train noise has become an intermittent problem, affecting the quality of high-speed trains and the subjective comfort of passengers and crew. To mitigate the abnormal noise in the conductor room of a high-speed train, this paper proposes a method for evaluating and optimizing the sound quality (SQ) inside a high-speed train. First, the vibrational noise in the train was tested and analyzed under passenger-carrying operating conditions. Second, the SQ was objectively evaluated by quantifying the interior noise with one traditional physical acoustic parameter and six psychoacoustic parameters. In a statistical analysis of a subjective listening questionnaire, the low-frequency noise was best described by a verbal expression (the Dichen descriptor in Chinese), which serves as an evaluation index in jury tests. A subjective jury (30 participants) scored the values of various sound samples. Although the A-weighted sound pressure level (A-W SPL) in the conductor room was relatively small, it was perceived as extremely irritating. The auditory perception of the human ear was not objectively described by the traditional A-W SPL, but was quantified by the psychoacoustic parameter evaluation. In a correlation analysis of the subjective and objective evaluations, the correlation coefficient between tonality and the Dichen sensation was 0.983. The tonality is the key parameter of perceived SQ in high-speed trains. Therefore, an objective quantitative model of the subjective Dichen sensation was established through multiple linear regression theory with tonality as a variable. To determine the source of the discomforting noise, the vibro-acoustic characteristics in the conductor room were analyzed and a finite element analysis of the end wall was performed. The abnormal noise was traced to a high-intensity 40-Hz pure tone caused by an externally sourced structural modal excitation near the end wall. Based on hybrid finite element–statistical energy analysis, a vibro-acoustic radiation simulation model of the end wall was established. After optimizing the end-wall structure, the SQ in the high-speed train was significantly improved. The optimization effect was verified in a noise test conducted in the interior of a real high-speed train, and in a vibro-acoustic radiation simulation model. Our results can guide the evaluation and optimization of SQ in high-speed trains.

A Linearised Hybrid FE-SEA Method for Nonlinear Dynamic Systems Excited by Random and Harmonic Loadings

The present paper proposes a linearised hybrid finite element-statistical energy analysis (FE-SEA) formulation for built-up systems with nonlinear joints and excited by random, as well as harmonic, loadings. The new formulation was validated via an ad-hoc developed stochastic benchmark model. The latter was derived through the combination of the Lagrange-Rayleigh-Ritz method (LRRM) and the Monte Carlo simulation (MCS). Within the build-up plate systems, each plate component was modelled by using the classical Kirchhoff’s thin-plate theory. The linearisation processes were carried out according to the loading-type. In the case of random loading, the statistical linearisation (SL) was employed, while, in the case of harmonic loading, the method of harmonic balance (MHB) was used. To demonstrate the effectiveness of the proposed hybrid FE-SEA formulation, three different case studies, made-up of built-up systems with localized cubic nonlinearities, were considered. Both translational and torsional springs, as joint components, were employed. Four different types of loadings were taken into account: harmonic/random point and distributed loadings. The response of the dynamic systems was investigated in terms of ensemble average of the time-averaged energy.

A hybrid finite element-statistical energy analysis approach for the dynamic response of built-up systems with nonlinear joints

The present article deals with the development and validation of a hybrid finite element-statistical energy analysis (FE-SEA) formulation employed to obtain the ensemble-average of the time-averaged vibrational energy response of dynamic systems with nonlinear joints. The proposed FE-SEA formulation is validated via a nonlinear stochastic benchmark model. The theoretical formulation related to the latter, is entirely derived by employing a variational approach. The weak-form of the governing equations, for each of the sample in the ensemble, is based on Kirchhoff’s thin-plate assumptions and is restricted to the out-of-plane motion only. The classical Lagrange-Rayleigh-Ritz method (LRRM), combined with the Monte Carlo simulation (MCS), is used as solution technique. An appropriate degree of uncertainty is introduced into the model in order to break the system symmetries ensuring transition from an exponential to a Rayleigh distribution of the modal spacing. Both in the hybrid FE-SEA and in the LRRM+MCS the localised nonlinearities are linearised by means of the method of harmonic balance. Various built-up plate systems, consisting of rectangular isotropic, homogeneous and linear elastic plates, elastically coupled by virtue of nonlinear translational and/or torsional springs and subjected to harmonic point loads are investigated.

Research on Noise of Tractor Cab Based on FE-SEA Method

The mid-frequency noise in a tractor cab is predicted, the FE-SEA hybrid model of tractor cab is established, the cab structure modal density, internal loss factor and coupling loss factor are obtained by theoretical calculation and test method, and the finite element-statistical energy analysis is simulated after loading vibration and noise excitation. Comparing the cab acoustic pressure level obtained by the simulation with the measured data, the analysis and comparison show that the result of the model using FE-SEA hybrid method in the mid-band is highly fitting with the test value, and the prediction absolute error is less than 2dB, which can be used for further analysis.

A Chebyshev convex method for mid-frequency analysis of built-up structures with large convex uncertainties

PurposeThis paper aims to develop an efficient numerical method for mid-frequency analysis of built-up structures with large convex uncertainties.Design/methodology/approachBased on the Chebyshev polynomial approximation technique, a Chebyshev convex method (CCM) combined with the hybrid finite element/statistical energy analysis (FE-SEA) framework is proposed to fulfil the purpose. In CCM, the Chebyshev polynomials for approximating the response functions of built-up structures are constructed over the uncertain domain by using the marginal intervals of convex parameters; the bounds of the response functions are calculated by applying the convex Monte–Carlo simulation to the approximate functions. A relative improvement method is introduced to evaluate the truncated order of CCM.FindingsCCM has an advantage in accuracy over CPM when the considered order is the same. Furthermore, it is readily to consider the CCM with the higher order terms of the Chebyshev polynomials for handling the larger convex parametric uncertainty, and the truncated order can be effectively evaluated by the relative improvement method.Originality/valueThe proposed CCM combined with FE-SEA is the first endeavor to efficiently handling large convex uncertainty in mid-frequency vibro-acoustic analysis of built-up structures. It also has the potential to serve as a powerful tool for other kinds of system analysis when large convex uncertainty is involved.

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.

Research progress on the medium frequency expansion method based on statistics energy

The SEA medium frequency extension methods are described. Then, medium frequency extension methods (FE-SEA (finite element-statistical energy analysis), CI-FE/SEA (CHEBYSHEV interval finite-element/statistical energy analysis), ES-FE-SEA (Edge smoothing -finite element- statistical energy analysis)) are compared. In FE-SEA method, neutron structure assembly and modeling method, interval parameter uncertainty and hybrid interval parameter analysis, interval variables and the hybrid interval variable response method are emphatically expounded. And on this basis, the existing methods are compared and analyzed. Based on the differences among the CHEBYSHEV expansion CI-FE/SEA method, the edge smooth domain ES-FE-SEA method and the Taylor series expansion FE-SEA method in solving parameter interval range and solution precision, the hybrid and wide range interval algorithm combined existing methods is expected to develop.

Investigation of train-induced vibration and noise from a steel-concrete composite railway bridge using a hybrid finite element-statistical energy analysis method

In this study a hybrid finite element-statistical energy analysis (FE-SEA) method is used to investigate the structure-borne noise of a steel-concrete composite railway bridge. The rail is represented by an infinite Timoshenko beam connected to the sleepers which are regarded as finite Timoshenko beams supported in ballast. The fasteners and ballast are simplified as a series of springs with complex stiffness. This model allows the receptance of the track to be determined. The wheel-rail forces are computed in the frequency domain from the contact-filtered roughness and the receptances of the wheel, track, and contact. The forces transmitted to the bridge are determined by substituting the wheel-rail forces into the equation of motion for the track. This model could also be applied to a slab track mounted on a bridge. A hybrid FE-SEA method is introduced in which FE is used to model the concrete deck and SEA is used to model the steel girders. This enables the computation of the vibration and noise of the composite railway bridge. The proposed method is verified by comparing its predictions with field measurements. The structure-borne noise level of the bridge is found to increase with train speed v by approximately 20lg(v). It is shown that the adjacent spans in a multi-span bridge can be ignored in deriving the bridge-borne noise at receiver points in the middle of the main span, provided that the distance to the track centreline is less than 0.3 times the length of the main span.

Optimization of mid-frequency vibration for complex built-up systems using the hybrid finite element–statistical energy analysis method

This article deals with the sensitivity analysis of dynamic response and optimal size design of complex built-up systems in the mid-frequency range. A complex built-up system may be fabricated from many components which often differ greatly in materials and sizes. It may be subjected to many different wavelength structural deformations and may typically exhibit mixed mid-frequency behaviour which is very sensitive to uncertainties at higher frequencies. To perform optimization on the mid-frequency vibration of complex built-up systems, the hybrid finite element (FE)–statistical energy analysis (SEA) method, in which the deterministic and statistical subsystem are respectively modelled using FE and SEA, is implemented in this work. In this context, an efficient direct differentiation method for sensitivity analysis is derived. Two numerical examples illustrate the efficiency and effectiveness of the proposed optimization model.

A study on acoustic characteristics of automotive magnesium composite dash panel

A dash panel is a plate that separates the engine compartment from the cabin and mainly serves to block engine noise. The effect of automotive lightweight design on sound insulation performance of the dash panel needs to be studied. In this paper, the study of acoustic characteristics of automotive magnesium composite dash panel was conducted in four steps. Firstly, the sound insulation performance of a basic magnesium alloy plate was investigated using finite element method/indirect boundary element method (FEM/IBEM) and was verified with corresponding sound insulation test. The result proved that the structural-acoustic coupling based FEM/IBEM was feasible for calculating the sound insulation. Secondly, finite element models of two bare (steel and magnesium alloy) dash panels were created and the FE model of the steel dash panel was verified with a constrained modal test. The sound insulation performance of the two bare dash panels of different materials was then predicted by developing their FEM/IBEM models, and the modal contribution was examined based on the partially-coupled modal assumption. It was found that although the weight was lightened the sound insulation performance did not deteriorate, and the performance was even better in some frequency bands after replacing steel with magnesium alloy. Thirdly, the sound insulation performance of composite dash panel applied with acoustic packages was studied. It was found that the sound insulation performance deteriorated in a low frequency band (0–1000 Hz) for certain acoustic packaging materials, and it was not always true that the thicker the acoustic package, the better the sound absorbing performance. Finally, the finite element-statistical energy analysis (FE-SEA) method was used to extend the sound insulation research of the composite dash panel to the full-frequency band. The results showed that acoustic packaging design of key areas could have significant acoustic optimization effects. Relevant discussions on the results were provided and conclusions were made.

Shock response prediction of the typical structure in spacecraft based on the hybrid modeling techniques

In order to solve the failure of traditional methods in the response prediction for the complex spacecraft under the wide frequency shock load, this paper provides the FE-SEA (Finite Element-Statistical Energy Analysis) hybrid modeling technique to improve the virtual mode synthesis and simulation (VMSS). Taking the L-shape typical structure as an example, the research is developed through the numerical analysis and experiment verification. In the FE-SEA hybrid model of the typical structure, the two honeycomb boards which have a higher modal density are built as the SEA model, while the connection beam that has a larger stiffness is established as FE model. And then the hybrid model is computed with the VMSS code, giving the transient response results. A shock experiment for the typical structure is carried out to verify the correctness of the methodology above. The results indicate that the numerical results have the similar trend and magnitudes with the ones of the experiment in time domain, and the shock response spectrum (SRS) of the numerical results can realize the ±6 dB tolerance limit compared with the experiment results. This paper finds that the FE-SEA hybrid modeling techniques for the typical structure can effectively overcome the problem of the dynamic differences between the substructures under the excitation of shock load, and the combination of FE-SEA and VMSS has the ability to predict the shock responses of the typical structure accurately. The research methods and conclusions achieved in this paper are expected to further extend to the shock response calculation of the complex spacecraft structures.