Complex Vibro-acoustic Systems Research Articles

An analytical dynamic stiffness method for efficient broadband vibro-acoustic analysis of complex built-up structures

An analytical modelling technique is presented, combining the dynamic stiffness method (DSM) with the spectral DSM (SDSM) for the broadband vibro-acoustic analysis of complex built-up structures. In particular, the structures are modelled exactly by the DSM based on particular solutions tailored for general acoustic or distributed force excitations; whereas the acoustic cavities are modelled by the SDSM where the boundary conditions are described by modified Fourier series with a rapid convergence rate. The vibro-acoustic coupling along the interfaces between the structures and the cavities is achieved by applying the normal velocity continuity condition, leading to a coupling matrix in an analytical manner. Both structural and acoustic cavity elements are assembled directly to model complex vibro-acoustic systems and no extra element discretization is required for both structural and acoustic domains. Consequently, the proposed method uses very few degrees of freedom (DoFs) but describes the broadband vibro-acoustic behaviour highly accurately. It is demonstrated that the proposed method exhibits a predominant advantage over the finite element package COMSOL in terms of accuracy and computational efficiency. This approach also has the potential to serve as the benchmark for a wide range of vibro-acoustic problems.

Quality criterion and errors of corrected negative SEA loss factors

Statistical Energy Analysis (SEA) is a well-known numerical method for predicting vibroacoustic phenomena in complex systems. The accuracy of SEA models relies on the precise determination of coupling loss factors and damping loss factors. Experimental SEA (E-SEA) methods, such as the Power Injection Method are commonly employed to measure these parameters. However, these techniques may yield negative loss factors, which are considered measurement errors. Monte Carlo Filtering (MCF) is one of the procedures, that allows the correction of negative loss factors, but the quality of the results remains unknown. The knowledge of the loss factors’ quality is directly related to the practical applications of SEA, where good quality of the input model parameters (coupling and damping loss factors) correspond to good quality and precise simulations of complex vibroacoustic systems (like trains, vehicle, airplanes, buildings) responses. In a previous study, a total loss factor (TLF) criterion was proposed as a quality indicator for the corrected loss factors. The current paper validates the TLF criterion through a comprehensive analysis of various numerical examples. By expanding the Monte Carlo sample’s value range (search area) and using different probability density functions, we intentionally introduced errors in the loss factors. The TLF criterion demonstrated resilience to increasing errors in certain scenarios, raising concerns about its sensitivity. Nevertheless, it seems, that the TLF criterion remains a good indicator of population stability and large error occurrence.

Vibroacoustics of a membrane-finite cavity resonator: Traditional and alternative approaches for analysis

This paper investigates a vibroacoustic model consisting of a flexible rectangular membrane backed by an acoustic air cavity of large lateral extent. A traditional methodology (termed Approach I) for analyzing the interaction between the acoustic airfield in the cavity and the flexible membrane is initially revisited and employed. In this approach, the transversal deformation of the mid-membrane is represented as an infinite series of eigenfunctions (normal modes) that satisfy the clamped boundary conditions for the in-vacuo membrane, regardless of the presence of the cavity. However, numerical challenges and limitations are encountered in obtaining direct insights into system characteristics, as this method heavily relies on matrix inversions and computing the roots of determinants. These operations can be computationally expensive and prone to inaccuracies and numerical instabilities, particularly when dealing with high-dimensional, non-sparse (dense) and nearly singular matrices. To overcome these limitations, an alternative methodology (termed Approach II) is proposed, which enables a precise analysis of the acoustically forced membrane and provides explicit expressions for the steady-state response, mode shapes, governed by a single transcendental characteristic equation for determining the vibroacoustic natural frequencies. Through a comparative analysis, we demonstrate the effectiveness and reliability of the proposed approach, showcasing its potential advantages over the traditional method in addressing numerical challenges. The influence of cavity depth on the vibroacoustic natural frequencies is examined across various levels of membrane stiffness, including low, intermediate, and high values. The analysis reveals the presence of both softening and hardening effects with respect to the in-vacuo case, suggesting that the air pressure plays a dual role in the system, i.e., influencing both the overall stiffness and inertia. The vibroacoustic membrane mode shapes exhibit significant deviations for soft and intermediate-stiff membranes but resemble in-vacuo mode shapes for stiff membranes, with the resemblance dependent on the selected cavity depth. Additionally, two qualitatively distinct mode families for the acoustic pressure, namely, propagating, and localized modes, are identified. The propagating modes are spatially extended and are characterized by the deformation of the volume of the medium, rendering them volume modes. In contrast, the localized modes are distinguished by localized vibrations occurring in the near field close the membrane surface, without significant energy transport or sound wave propagation in the cavity; as a result, they can be regarded as surface modes. The proposed methodology reveals features common to more complex vibroacoustic systems and offers a comprehensive and efficient approach for predictively designing the dynamics of such systems.

Performances of coupled the partition of unity finite element method and the discrete shear gap method for the analysis of 3D vibro-acoustic problems

In 3D vibro-acoustic analysis, the accuracy of the vibro-acoustic FE model degrades with increasing excitation frequency due to the “dispersion error”. In this paper, an efficient hybrid method, namely the PUFEM/DSG, is proposed for improving the computational performance of vibro-acoustic analysis, which uses the partition of unity finite element method for the acoustic cavity and the discrete shear gap method for the flexible shell. The acoustic cavity is discretized into tetrahedral elements of large size, and a plane wave function is introduced in the interpolation space to construct the PUFEM. Triangular elements are adopted for the shell, and the DSG eliminates the “shear locking”. At the interface with non-matching meshes, this paper realizes the coupling of two field variables by dividing virtual elements and implementing coordinate transformation and applies the exact integration method for the PUFEM system matrix and coupling matrix to improve the efficiency of the proposed method. The proposed method not only retains the FEM’s ability to model complex vibro-acoustic systems but also effectively reduces the “dispersion error” and the computational effort. It also can improve the accuracy by adding plane waves at the nodes of the acoustic cavity without remeshing.

Acoustic simulation of cavities with porous materials using an adaptive model order reduction technique

Porous materials are often used in the context of passive noise control applications due to their low cost and good sound absorption properties. As the most commonly used predictive tool, the finite element (FE) method is often applied in the design phase to obtain detailed information on the performance of complex vibro-acoustic systems with porous damping treatments. In this case, however, its use inevitably leads to very large, complex-valued and frequency-dependent numerical models, which makes original full-order model evaluation intractable due to time and memory limitations. In this paper, an adaptive model order reduction strategy is presented to reduce the number of degrees of freedom involved in the associated problems such that the required computational cost can be largely alleviated while a desired high accuracy can be achieved in a controllable way. This technique consists of making use of Taylor expansion to the multiple frequency-dependent scalar functions coming from the complex material behavior, and then performing the structure-preserving second-order Arnoldi algorithm based on a robust error indicator to solve the underlying FE model in the frequency domain. A further improvement in terms of the approximation accuracy of the constructed compact reduced-order model is also proposed to incorporate all available moment-matching information. Two widely used model types for porous materials are investigated, i.e. the equivalent fluid description and the Biot theory. Mono-physical poroelastic-poroelastic coupling and multi-physical acoustic-porous coupling are considered in order to demonstrate the simplicity, versatility and efficiency of the approach.

A novel hybrid deterministic-statistical approach for the mid-frequency vibro-acoustic problems

It is difficult to predict precisely the frequency response of a complex vibro-acoustic system in mid-frequency region. To overcome this deficiency, a novel hybrid stable node-based smoothed finite element method/statistical energy analysis model is proposed in this work. The whole vibro-acoustic system can be divided into a combination of a structural subsystem with statistical behavior and an acoustic subsystem with deterministic feature. The recently developed stable node-based smoothed finite element method is employed here to simulate the deterministic subsystem, and the statistical energy analysis is utilized to deal with the statistical subsystem. Based on the so-called diffuse field reciprocity relation, these two subsystems can be easily connected and coupled. Due to the introducing of gradient smoothing and compensation operation, our algorithm can significantly reduce the dispersion error compared with the traditional finite element method. Thus, it is expected that the present coupled model can provide ultra-accurate results. Numerical examples, including both benchmark and practical engineering cases, demonstrate that our algorithm works very well in mid-frequency vibro-acoustic analysis.

Prediction of the transient energy response for complex vibro-acoustic systems

As research works of the transient statistical energy analysis (TSEA) and transient local energy approach (TLEA) mostly focus on simple structures, TSEA and TLEA are adopted to quantify the transient response of a complex vibro-acoustic system at the mid-high frequency range in this paper. Numerical examples of a coupled oscillator system, an L-shaped plate, and a launch vehicle fairing model are conducted to demonstrate the effectiveness and accuracy of TSEA and TLEA. The computational precision of TSEA and TLEA is verified by the analytical solution and finite element method. Furtherly, the transient energy responses of subsystems with different coupling ratios between subsystems are investigated. Results show that TLEA has a better performance than TSEA. With the increasing coupling ratio between subsystems, the rise time and peak energy of transient energy response of subsystems decrease gradually. Both ratios of rise time and peak energy predicted by TLEA to these of the TSEA increase as the rising of the coupling ratio.

Prediction of the far-field sound radiation for complex system with pseudoinverse transfer function and operational recognition

Prediction of the far-field sound radiation has been a major concern in many applications, in particular for complex vibro-acoustic systems due to practical issues of the complicated structure, boundary condition and sensor deployment. In this paper, a novel predication method based on the pseudoinverse transfer function in combination with operational recognition is proposed for far-field sound radiation of complex systems. In the method, the pseudoinverse transfer function between the near field and far field is first deduced by using the operational data. Besides, the independent calculation method of the pseudoinverse transfer functions is investigated and performed to reduce an error in the prediction caused by interferences among the operations. A hybrid model with the principal component analysis and k-means is subsequently presented in the independent operation recognition of the complex system. Experimental work is conducted on a double-layer cylindrical shell in an anechoic pool to validate the proposed method.

Analysis and experimental validation of the middle-frequency vibro-acoustic coupling property for aircraft structural model based on the wave coupling hybrid FE-SEA method

The finite element (FE) method is suitable for low frequency analysis and the statistical energy analysis (SEA) for high frequency analysis, but the vibro-acoustic coupling analysis at middle frequency, especially with a certain range of uncertainty system, requires some new methods. A hybrid FE-SEA method is proposed in this study and the Monte Carlo method is used to check the hybrid FE-SEA method through the energy response analysis of a beam-plate built-up structure with some uncertainty, and the results show that two kinds of calculation results match well consistently. Taking the advantage of the hybrid FE-SEA method, the structural vibration and the cabin noise field responses under the vibro-acoustic coupling for an aircraft model are numerically analyzed, and, also, the corresponding experiment is carried out to verify the simulated results. Results show that the structural vibration responses at low frequency accord well with the experiment, but the error at high frequency is greater. The error of sound pressure response level in cabin throughout the spectrum is less than 3dB. The research proves the reliability of the method proposed in this paper. This indicates that the proposed method can overcome the strict limitations of the traditional method for a large complex structure with uncertainty factors, and it can also avoid the disadvantages of solving complex vibro-acoustic system using the finite element method or statistical energy analysis in the middle frequency.

On the modeling of sound transmission through a mixed separation of flexible structure with an aperture.

Modeling sound transmission among acoustic media through mixed separations, consisting of both rigid/flexible structures with apertures, is a challenging task. The coexistence of both structural and acoustic transmission paths through the same coupling surface adds system complexities, hampering the use of existing sub-structuring modeling techniques when the system configuration becomes complex. In the present work, a virtual panel treatment is proposed to model thin apertures involved in such complex vibroacoustic systems. The proposed virtual panel considers an aperture as an equivalent structural component, which can be integrated with the solid/flexible structure to form a unified compound interface. This allows handling the entire compound interface as a pure structural element, thus providing an efficient and versatile tool to tackle system complexities when using sub-structuring techniques. The accuracy and convergence of the method are investigated and validated, and the effective thickness range allowing for the virtual panel treatments is determined. The capability and the flexibility of the proposed formulation are demonstrated through several numerical examples, with underlying physics being explored.

Assessment of a hybrid finite element-transfer matrix model for flat structures with homogeneous acoustic treatments.

Modeling complex vibroacoustic systems including poroelastic materials using finite element based methods can be unfeasible for practical applications. For this reason, analytical approaches such as the transfer matrix method are often preferred to obtain a quick estimation of the vibroacoustic parameters. However, the strong assumptions inherent within the transfer matrix method lead to a lack of accuracy in the description of the geometry of the system. As a result, the transfer matrix method is inherently limited to the high frequency range. Nowadays, hybrid substructuring procedures have become quite popular. Indeed, different modeling techniques are typically sought to describe complex vibroacoustic systems over the widest possible frequency range. As a result, the flexibility and accuracy of the finite element method and the efficiency of the transfer matrix method could be coupled in a hybrid technique to obtain a reduction of the computational burden. In this work, a hybrid methodology is proposed. The performances of the method in predicting the vibroacoutic indicators of flat structures with attached homogeneous acoustic treatments are assessed. The results prove that, under certain conditions, the hybrid model allows for a reduction of the computational effort while preserving enough accuracy with respect to the full finite element solution.

Efficient parametric uncertainty analysis within the hybrid Finite Element/Statistical Energy Analysis method

Abstract This paper is concerned with the development of efficient algorithms for propagating parametric uncertainty within the context of the hybrid Finite Element/Statistical Energy Analysis (FE/SEA) approach to the analysis of complex vibro-acoustic systems. This approach models the system as a combination of SEA subsystems and FE components; it is assumed that the FE components have fully deterministic properties, while the SEA subsystems have a high degree of randomness. The method has been recently generalised by allowing the FE components to possess parametric uncertainty, leading to two ensembles of uncertainty: a non-parametric one (SEA subsystems) and a parametric one (FE components). The SEA subsystems ensemble is dealt with analytically, while the effect of the additional FE components ensemble can be dealt with by Monte Carlo Simulations. However, this approach can be computationally intensive when applied to complex engineering systems having many uncertain parameters. Two different strategies are proposed: (i) the combination of the hybrid FE/SEA method with the First Order Reliability Method which allows the probability of the non-parametric ensemble average of a response variable exceeding a barrier to be calculated and (ii) the combination of the hybrid FE/SEA method with Laplace's method which allows the evaluation of the probability of a response variable exceeding a limit value. The proposed approaches are illustrated using two built-up plate systems with uncertain properties and the results are validated against direct integration, Monte Carlo simulations of the FE and of the hybrid FE/SEA models.

A hybrid modeling approach for vibroacoustic systems with attached sound packages

Modeling complex vibroacoustic systems including poroelastic materials using finite element (FE) based methods can be computationally expensive. Several attempts have been made to alleviate this drawback, such as high order hierarchical basis and substructuring approaches. Still, these methods remain computationally expensive or limited to simple configurations. On the other hand, analytical approaches, such as the Transfer Matrix Method (TMM), are often used, thanks to the lower computational burden. However, since the geometrical flexibility of the FE method is always needed in the low/mid-frequency range, attempts have been made to couple the FE model of the master system with a TM model of the sound package. Although these hybrid approaches seem promising, the open literature is not comprehensive. The aim of this work is to present a hybrid FE-TMM approach based on a Green's function formulation. The idea is to account for the sound package by approximating the effects over the treated surface using fundamental solutions (i.e., Green's functions) obtained by the TMM. A benchmark representative of typical applications is used to illustrate the capabilities of the presented methodology in terms of efficiency and accuracy in comparison to other classical methods.

A mode count procedure for mid-frequency analysis of complex vibro-acoustic systems

Satellites and space equipment are exposed to high diffuse acoustic fields during the launch process that are able to excite the structure. The use of adequate techniques to model the structure response to the acoustic loads is a fundamental task during the design and verification phases. Currently, the efforts are focused in the mid-frequency range that in this paper is defined as the frequency range in which a combination of models is required. To select the most accurate modelling technique in each frequency band, this paper proposes a procedure based on the modal count of each subsystem. The procedure is applied in an ordinary satellite case and the frequency response continuity and agreement with experimental results are shown.

Unified Multi-Phase Modeling and Topology Optimization for Complex Vibro-Acoustic Systems consisting of Acoustic, Elastic and Poroelastic Media

This talk introduces the UMP (Unified Multi-Phase) modeling method, recently developed for the streamlined analysis and systematic design of complex vibro-acoustic systems. The main difference between the conventional method and the UMP method is that the latter uses the same governing equations for all acoustic components while the former, different equations for different acoustic components. The UMP approach also treats all interface conditions by a unified formulation unlike the conventional approach. Consequently, physically different components and various interface conditions are treated by a unified multi-phase model along with unified interface conditions. The key in this approach is to realize that Biot’s equations for poroelastic media can degenerate to the Helmholtz equation or the elastodynamic equations if some of poroelastic material coefficients are properly controlled and that the interface conditions for poroelastic-poroelastic media can also degenerate to those for interfaces of any similar/dissimilar media. Since any reduction in the degrees of primary field variables in degenerated systems can cause numerical singularity, some of the poroelastic coefficients are slightly perturbed from the exact limit values in representing the degenerated systems. The unified approach becomes extremely useful when an iterative design process requiring interface evolutions at every iteration step is implemented as in the topology optimization of complex vibro-acoustic systems.

Hybrid deterministic-statistical analysis of vibro-acoustic systems with domain couplings on statistical components

Recently a hybrid analysis method has been developed in which the various components of a complex vibro-acoustic system can be modeled either deterministically or statistically. The coupling between these two types of component is effected by using a diffuse field reciprocity relation, which relates the cross-spectrum of the forces at the boundary of a statistical component to the vibrational energy level of the component. In practical applications, components may be coupled over a domain, rather than just at the component boundaries—for example, the coupling between a plate and an acoustic volume involves the surface of the plate. It is shown in the present work that the reciprocity relation and the hybrid method can be extended to this situation, so that, for example, it is possible to couple a statistical model of a plate to a finite element model of an acoustical cavity, or to a statistical acoustical cavity. The coupling of statistical acoustical and structural components is not new, but rather is done routinely within statistical energy analysis (SEA); this part of the present work demonstrates that the relevant coupling loss factors can be found by using the hybrid method. The analysis is illustrated by various numerical examples.

A statistical energy analysis subsystem formulation using finite element and periodic structure theory

In statistical energy analysis (SEA), a complex vibro-acoustic system is represented as an assembly of coupled subsystems. The parameters of an SEA model are typically derived by considering wave propagation within each subsystem. Analytical formulations exist for describing wave propagation in many commonly encountered subsystems (for example, flat plates, curved shells, ribbed panels, etc.). However, problematic subsystems are often encountered that are not well represented by existing analytical formulations. Examples include composite fuselages, isogrid fairings, complex ribbed panels and extruded train floors. This paper describes a SEA subsystem formulation based on a combination of finite elements (FEs), component mode synthesis (CMS) and periodic structure theory. The method enables the SEA parameters to be efficiently computed for very general structural panels. Expressions are derived for the modal density, damping loss factor and ‘engineering units’ response of the panel. By making use of an efficient Fourier transform approach, the resonant radiation efficiency, non-resonant transmission loss and acoustic input power are also obtained. The method is derived and a number of analytical and experimental validation cases are presented.

Modal criteria for optimizing the acoustic behaviour of a coupled fluid–structure system based on a systemic approach

AbstractThis paper proposes modal criteria to represent various noise sources within a complex structure, such as an automobile. By optimizing a complex system using criteria linked to modal mass and stiffness matrices, different modes of noise propagation can be investigated separately. Several criteria are thus suggested, each related to a vibrational propagation path. Since the system is studied using modal analysis, criteria can be found based on modes associated with the structure's hollow parts, plates, and cavities. These different criteria are analysed based on the assumption of a complex vibroacoustic system. It is shown that by analysing such criteria, one can determine which part of the structure needs to be optimized. The optimization of such a system could constitute a research topic in its own right, and is beyond the scope of the present paper. Copyright © 2007 John Wiley & Sons, Ltd.

Vibro-acoustic analysis of complex systems

A general method is presented for predicting the ensemble average steady-state response of complex vibro-acoustic systems that contain subsystems with uncertain, or random, properties. The method combines deterministic and statistical techniques to produce a non-iterative hybrid method that incorporates equations of dynamic equilibrium and power balance. The method is derived explicitly without reference to statistical energy analysis (SEA); however, it is seen that the wave approach to SEA can be viewed as a special case of the proposed method. The proposed method provides a flexible way to account for necessary deterministic details in a vibro-acoustic analysis without requiring that an entire system be modeled deterministically. The method therefore provides a potential solution to the mid-frequency problem (in which a system is neither entirely deterministic nor entirely statistical). The application of the method is illustrated with a numerical validation example.