Hybrid Finite Element-statistical Energy Analysis Research Articles

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.

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.

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.

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.

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.

Stochastic hybrid perturbation technique-based smoothed finite element-statistical energy method for mid-frequency analysis of structure–acoustic systems with parametric and nonparametric uncertainties

At present, the most widely used mid-frequency method, i.e., the hybrid finite element-statistical energy analysis method (FE/SEA), still holds potential flaws, e.g., low accuracy and ignoring uncertain factors. In order to eliminate these problems, a generalized smoothing finite element statistical energy analysis method (SFE/SEA) is first proposed in this work, which is constructed based on the edge-based gradient smoothing technique, thus reducing the dispersion error in the deterministic calculations. Second, on the basis of the proposed SFE/SEA, the parametric uncertainty models will be introduced, combining the second-order random interval perturbation method and the random interval moment method, etc.; thus, a mid-frequency analysis method (SHPSFEM/SEA) considering both parametric uncertainties (both the random and interval variables) and nonparametric uncertainties is proposed, to effectively analyze the probability and interval characteristics of the random structural–acoustic coupled systems. Moreover, for the interval parameters with wide variation range, the subinterval perturbation method (S-SHPSFEM/SEA) is introduced by dividing the large interval parameters into small interval parameters. The proposed methods are compared with the Monte Carlo simulations of the hybrid FE/SEA model. The high accuracy and efficiency of the proposed methods are verified by two numerical examples.

Fast mean and variance computation of the diffuse sound transmission through finite-sized thick and layered wall and floor systems

A method is developed for computing the mean and variance of the diffuse field sound transmission loss of finite-sized layered wall and floor systems that consist of solid, fluid and/or poroelastic layers. This is achieved by coupling a transfer matrix model of the wall or floor to statistical energy analysis subsystem models of the adjacent room volumes. The modal behavior of the wall is approximately accounted for by projecting the wall displacement onto a set of sinusoidal lateral basis functions. This hybrid modal transfer matrix-statistical energy analysis method is validated on multiple wall systems: a thin steel plate, a polymethyl methacrylate panel, a thick brick wall, a sandwich panel, a double-leaf wall with poro-elastic material in the cavity, and a double glazing. The predictions are compared with experimental data and with results obtained using alternative prediction methods such as the transfer matrix method with spatial windowing, the hybrid wave based-transfer matrix method, and the hybrid finite element-statistical energy analysis method. These comparisons confirm the prediction accuracy of the proposed method and the computational efficiency against the conventional hybrid finite element-statistical energy analysis method.

Predicting the variance of the frequency-averaged energetic response in hybrid finite element—Statistical energy analysis

In this contribution, the hybrid finite element-statistical energy analysis method is extended such that not only the mean and the ensemble variance of the harmonic system response can be computed, but also the ensemble variance of the frequency band-averaged system response. The computed variance represents the uncertainty that is due to the assumption of a diffuse field in components of the hybrid system. The developments start with a cross frequency generalization of the diffuse field reciprocity relationship between the total energy in a diffuse field and the cross spectrum of the external loading. By making extensive use of this generalization in a first-order perturbation analysis, explicit expressions are derived for the variance of the vibrational energies in the diffuse components and for the variance of the cross spectrum of the response of the deterministic components. These expressions are extensively validated against Monte Carlo analyses of systems consisting of connected plates, in which diffuse fields are simulated by randomly distributing small concentrated masses, acting as wave scatterers, across the diffuse components.

A review of mid-frequency vibro-acoustic modelling for high-speed train extruded aluminium panels as well as the most recent developments in hybrid modelling techniques

The present paper reviews the vibro-acoustic modelling of extruded aluminium train floor structures including the state-of-the-art of its industrial applications, as well as the most recent developments on mid-frequency modelling techniques in general. With the common purpose to predict mid-frequency vibro-acoustic responses of stiffened panel structures to an acceptable accuracy at a reasonable computational cost, relevant techniques are mainly based on one of the following three types of mid-frequency vibro-acoustic modelling principles: (1) enhanced deterministic methods, (2) enhanced statistical methods, and (3) hybrid deterministic/statistical methods. It is shown that, although recent developments have led to a significant step forward in industrial applicability, mature and adequate prediction techniques, however, are still very much required for solving sound transmission through, and radiation from, extruded aluminium panels used on high-speed trains. Due to their great potentials for predicting mid-frequency vibro-acoustics of stiffened panel structures, two of recently developed mid-frequency modelling approaches, i.e. the so-called hybrid finite element-statistical energy analysis (FE-SEA) and hybrid wave-based method-statistical energy analysis (WBM-SEA), are then recapitulated.

Hybrid finite element/statistical energy method for mid-frequency analysis of structure−acoustic systems with interval parameters

The hybrid Finite Element/Statistical Energy Analysis (FE-SEA) model for mid-frequency analysis consists of two parts: the FE components and the SEA components, which are called as the master system and the subsystem respectively. The hybrid Finite Element/Statistical energy analysis approach is established on the assumption that the FE components have fully deterministic properties, while the SEA subsystems have a high degree of randomness. This method has been recently extended to allow the properties of the FE components of built-up structures to be uncertain like interval or probabilistic, and the hybrid model with parametric and non-parametric uncertainties is obtained. By dealing with the non-parametric uncertainty analytically and the parametric uncertainty with Monte Carlo Simulations, the distribution of the responses of the hybrid model can be obtained. In this paper, the interval parametric uncertainty is introduced into the hybrid FE–SEA framework for structure−acoustic systems, and the interval dynamic equilibrium equations are established, thus a hybrid model with non-parametric and interval parametric uncertainties for structure−acoustic systems is obtained; to improve the computational efficiency, the second-order interval perturbation finite element method is introduced into the hybrid FE-SEA framework and the second-order interval perturbation finite element/statistical energy analysis (SIPFEM/SEA) method is proposed. For the structure−acoustic system with interval parameters modeled by FE-SEA, the parameters of FE components are considered as interval parameters instead of deterministic ones, thus the expectations of the second order quantities such as the vibrational energy and the response cross-spectrum in the FE-SEA framework become interval variables, too. In the SIPFEM/SEA, the expectations of the second order response quantities of a structure−acoustic system are expanded with the second-order Taylor series at the nominal values of interval parameters, and for the sake of simplicity and efficiency, the non-diagonal elements of the Hessian matrices are neglected; then by searching the target positions of interval parameters that maximize or minimize the objective functions, the bounds of the vibrational energy and the response cross-spectrum can be obtained. Because of the neglect of the higher order terms of Taylor series, SIPFEM/SEA is limited to the interval analysis with narrow parameter intervals. For larger parameter intervals, the sub-interval perturbation method based on the SIPFEM/SEA is introduced. The proposed methods are illustrated by applications to two example structure−acoustic systems in which the acoustic cavities are modeled by using the interval FE method and the structures are modeled by using the SEA. Reference comparisons are made with the Monte Carlo simulations of the hybrid FE/SEA model. The numerical results verify the accuracy and efficiency of the proposed methods.

Numerical and experimental validation of a hybrid finite element-statistical energy analysis method

The finite element (FE) and statistical energy analysis (SEA) methods have, respectively, high and low frequency limitations and there is therefore a broad class of "mid-frequency" vibro-acoustic problems that are not suited to either FE or SEA. A hybrid method combining FE and SEA was recently presented for predicting the steady-state response of vibro-acoustic systems with uncertain properties. The subsystems with long wavelength behavior are modeled deterministically with FE, while the subsystems with short wavelength behavior are modeled statistically with SEA. The method yields the ensemble average response of the system where the uncertainty is confined in the SEA subsystems. This paper briefly summarizes the theory behind the method and presents a number of detailed numerical and experimental validation examples for structure-borne noise transmission.