Vibro-acoustic Systems Research Articles

Structure-preserving discretization and control of a two-dimensional vibro-acoustic tube

Abstract This paper deals with the structure-preserving discretization and control of a two-dimensional vibro-acoustic tube using the port-Hamiltonian framework. A discretization scheme is proposed, and a set of precise basis functions are given in order to obtain a structure-preserving finite-dimensional port- Hamiltonian approximation of the two-dimensional vibro-acoustic system. Using the closed-loop structural invariants of the approximated system an energy-Casimir controller is derived. The performance of the proposed discretization scheme and the controller is shown by means of numerical simulations.

Transmission loss of plates with multiple embedded acoustic black holes using statistical modal energy distribution analysis

Acoustic black holes (ABHs) in beams and plates have been extensively studied as a passive method for vibration attenuation and noise reduction. However, most research to date has focused on analyzing the behavior of a single ABH structural element, using numerical or semi-analytical deterministic approaches. If ABHs are to be exploited for practical industrial applications, there is a need to characterize their performance in complex built-up structures and to describe them with statistical methods in the mid-high frequency range. This paper presents a first step towards this goal by employing statistical modal energy distribution analysis (SmEdA) to evaluate the transmission loss of ABH panels separating two air cavities. SmEdA splits vibroacoustic systems into subsystems and establishes power balance equations between the modes belonging to different subsystems. This avoids the energy equipartition assumption of traditional statistical energy analysis (SEA) and extends it to low modal overlap systems. In this work, the benefits of embedding ABHs on plates for noise reduction between cavities are predicted with SmEdA. The role played by the size, shape and number of ABH indentations on the plate are inspected, as well as the influence of other parameters like the truncation thickness, ABH order and damping. The effects of added mass and stiffness of the damping layer are investigated and it is observed that, with proper design, ABH plates can exhibit substantial transmission loss improvement in the vicinity of the critical frequency of uniform plates. Both resonant and non-resonant transmission are considered in the analysis.

Efficient Acoustic Topology Optimization Using Vibro-Acoustic Coupled Craig–Bampton Mode Synthesis

Continuum topology optimization is an effective way to reduce vibration and noise of vibro-acoustic coupling system. However, the high computation time required for calculating the vibro-acoustic responses during topology optimization is a major obstacle for practical applications. In this paper, a novel symmetric method of vibro-acoustic system matrix is proposed, and a new model reduction method is developed based on Craig–Bampton mode synthesis method to compute dynamic responses with adequate efficiency and accuracy for topology optimization. The comparison results show that the proposed method substantially reduces the degrees of freedom (DOFs) and calculation time. The response values and eigenfrequencies calculated by the model reduction method are exactly the same as those of the full model. Furthermore, the bidirectional evolutionary structural optimization (BESO) algorithm is applied to solve the problem of minimizing the response of a single frequency and a certain range of frequency excitation at a specified target point. The results show that the optimization algorithm converges fast, the iterative process is robust and the response values of different coupling systems can be reduced to a higher extend, which indicates the applicability of the optimization algorithm.

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.

Broadband acoustic shape optimization of studs in double-leaf walls

Design optimization of vibro-acoustic systems over a wide frequency band is challenging. It does not only require a computationally efficient numerical prediction model of sufficient accuracy, but the optimization scheme itself should also be computationally efficient and the design space should be limited by all relevant manufacturing and performance constraints. In this paper, a methodology is presented for the shape optimization of components in a complex wall system, with the aim of achieving an optimized sound insulation of the overall system across the entire building acoustics frequency range. As an example, the cross-sectional shape of studs in a double-leaf wall is first parameterized and subsequently optimized for broadband sound insulation with a gradient-based optimization scheme. A recently developed sound insulation prediction model that has the required balance between accuracy and computational efficiency is adopted and validated for a range of plasterboard walls with acoustic studs. The model is further complemented with a novel sensitivity analysis, such that the sensitivities of the predicted sound insulation to the cross-sectional stud shape parameters can be obtained in a semi-analytic way. This approach reduces the computation cost related to broadband acoustic design optimization significantly. Furthermore, inequality constraints that are necessary for obtaining a feasible design in terms of material usage and manufacturing limitations are identified and incorporated in the optimization procedure. The relevant constraints related to strength and stiffness of the wall are very mild and therefore verified after optimization. As an example of the proposed methodology, the cross-sectional shape of flexible metal studs in double-leaf plasterboard walls is optimized for the overall A-weighted sound reduction under pink noise excitation. A range of combinations in stud depths and number of sheets is analyzed. Walls containing the optimized studs have an airborne sound insulation that is close to that of walls with fully decoupled leafs. Their sound insulation is on average 11.8 dB higher than when they would contain conventional C-shaped studs, and 5.1 dB higher than when they would contain acoustic studs that are presently available.

A Multidomain Spectral Method for Analysis of Interior Vibroacoustic Systems with Segmented Boundaries

Spectral methods have previously been applied to analyze a multitude of vibration and acoustic problems due to their high computational efficiency. However, their application to interior structural acoustics systems has been limited to the analysis of a single plate coupled to a fluid-filled cavity. In this work, a general multidomain spectral approach is proposed for the eigenvalue and steady-state vibroacoustic analyses of interior structural-acoustic problems with discontinuous boundaries. The unified formulation is derived by means of a generalized variational principle in conjunction with the spectral discretization procedure. The established framework enables one to easily accommodate complex systems consisting of both a structure assembly and a built-up cavity with moderate geometric complexities and to effectively analyze vibroacoustic behaviors with sufficient accuracy at relatively high frequencies. Two practical examples are chosen to demonstrate the flexibility and efficiency of the proposed formulation: a built-up cavity backed by an assembly of multiple connected plates with arbitrary orientations and a thick irregular elastic solid coupled with a heavy acoustic medium. Comparison to finite element simulations and convergence studies for these two examples illustrate the considerable computational advantage of the method as compared to finite element procedures.

User-friendly nonlinear nonparametric estimation framework for vibro-acoustic industrial measurements with multiple inputs

This paper introduces a user-friendly estimation framework for industrial measurements of vibro-acoustic systems with multiple inputs. Many mechanical structures are inherently nonlinear and there is no unique solution for modeling nonlinear systems. This is especially true when multiple-input, multiple-output (MIMO) systems are considered. This paper addresses the questions related to the user-friendly semi-automatic processing of MIMO measurements with respect to the design of experiment and the analysis of the measured data. When the proposed framework is used, with a minimal user interaction, it is easily possible a) to decide, if the underlying system is linear or not, b) to decide if the linear framework is still accurate (safe) enough to be used, and c) to tell the inexperienced (non-expert) user how much can be gained using an advanced nonlinear framework. The proposed nonparametric industrial framework is illustrated on various real-life measurements.

Simplified Analysis for Multiple Input Systems: A Toolbox Study Illustrated on F-16 Measurements

This paper introduces a nonparametric, nonlinear system identification toolbox called SAMI (simplified analysis for multiple input systems) developed for industrial measurements of vibro-acoustic systems with multiple inputs. It addresses the questions related to the user-friendly (semi-)automatic processing of multiple-input, multiple-output measurements with respect to the design of an experiment and the analysis of the measured data. When the proposed toolbox is used, with minimal user interaction, it is easily possible (a) to decide whether the underlying system is linear or not, (b) to decide whether the linear framework is still adequate to be used, and (c) to tell an inexperienced user how much can be gained using an advanced nonlinear framework. The toolbox is illustrated on openly accessible F-16 ground vibration testing measurements.

Layout optimization of viscoelastic damping for noise control of mid-frequency vibro-acoustic systems

This paper presents an integrated, new, and generic framework for layout optimization of viscoelastic damping for noise control of mid-frequency vibro-acoustic systems. The method is developed based on the concept of power balance among different modal energies between coupled structural and acoustic subsystems and is formulated within the framework of a statistical modal energy distribution analysis (SmEdA). In the novel optimization formulation, the total energy of the acoustic subsystem is chosen as the objective function for minimizing the internal acoustic response in the vibro-acoustic system; and the relative material volume densities for viscoelastic element groups are selected as design variables using a volume-preserving Heaviside function. A new sensitivity analysis formulation is developed in a semi-analytical form via a SmEdA for solving the vibro-acoustic optimization problem. Two numerical examples are presented to demonstrate the efficiency and effectiveness of the present method. The present numerical results reveal two important findings: (a) the total acoustic energy of the chosen vibro-acoustic system can be significantly reduced; and (b) the optimum viscoelastic material layouts not only decrease the peak values of the modal coupling strengths between structural and acoustic subsystems but also create relatively more uniform acoustic modal energy distribution.

Shaping the acoustic radiation of a vibrating plate

A new method is proposed for precise shaping of the acoustic radiation of a vibrating plate. The potential applications of this method are: to optimize acoustic radiation of a plate employed as a sound source; and to enhance transmission loss of a plate used as a noise barrier, both in passive or active control solutions. The proposed method is based on mounting several additional ribs and masses (passive elements) to the plate surface at locations determined by an optimization process. This optimization involves a model of the vibroacoustic system, a cost function representing the objective under consideration, and an optimization algorithm. The method is described, and results of both numerical simulation and experimental validation are presented. Analysis of results shows that the transmission loss introduced by a passive noise barrier can be enhanced by more than 7 dB in frequency bands of interest. Furthermore, these enhancements can be combined with optimized arrangements of actuators and sensors, resulting in a hybrid passive-active noise control system.

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.

Comments on the validity of transfer matrix based models for the prediction of the effect of curved sound packages

In this paper, the validity of using the Transfer Matrix Method (TMM) to account for the effect of practical curved sound packages (i.e. spring-mass treatment) in vibroacoustic systems is assessed. For this purpose, a sub-structuring method that employs a Patch Transfer Functions (PTF) approach is used to efficiently couple standard Finite Element Method (FEM) of the structure and the acoustic cavity with the sound package model. Three models of the noise control treatment are compared, namely (i) a locally reacting model, (ii) a non-locally reacting (NLR) model based on the TMM, and (iii) a FEM. The FEM model considers the actual curved geometry of the SP in order to estimate its dynamic behavior while the simplified TMM based models assume the SP to be flat (i.e. unwrapped if curvature is present), homogeneous and of infinite lateral extent. In addition, these results are systematically compared to a full Finite Element/Boundary Element Methodology. The accuracy and the limitations of adopting the unwrapped approximation in TMM based models are shown through two parametric studies. The first considers the radius of curvature of the system as a parameter while the SP thickness is constant. The second considers the thickness of the SP as a parameter while the radius of curvature of the system is held fixed. For the considered configurations, the first parametric study allows to conclude that the curvature of the SP does not affect the dynamic behavior of the system due to its large dissipation and softness and it can be practically assumed flat if the NLR model is used to model the SP. However, the second parametric study shows that the thickness of the curved SP may affect the efficiency of the NLR model. Moreover, the limitations of locally reacting models are confirmed.

Orthocomplement reduced order models for strongly coupled structural-acoustic systems

Despite the enormous strides in computing power, reduced-order models (ROMs) remain essential for design optimization and early stage design. The energetically optimal basis for ROMs of vibroacoustic systems derives from the coupled Eigen problem: no basis of smaller dimension contains a greater amount of system energy. While vital, this L2-type framework is also insufficiently accurate on a pointwise basis for many problems. A simply supported beam with a point load is an obvious thought experiment: the smooth modal functions converge to the correct displaced shape very slowly. The well-known “residual mode” or orthocomplement concept overcomes this problem, by augmenting the space of modal functions with static solutions to the point loads. These new residual modes are orthogonalized against the modal space, and used otherwise conventionally. This work describes the development of ROMs of submerged structures by extending the residual mode approach to the strongly coupled structural-acoustic case. We derive methods to use these ROMs in substructures. Numerical examples show that the structural acoustic residual mode formulation is very effective in improving the solution method in the low-frequency limit, especially in the fluid, and also at the high-frequency limit, where response quantities tend to the average of the direct-solve FEM response.

Mid-to-high frequency piecewise modelling of an acoustic system with varying coupling strength

Mid-to-high frequency vibro-acoustic modelling has always been a challenging topic. The previous study shows the promise of a piecewise calculation scheme based on the Condensed Transfer Function (CTF) approach in a frequency range where the modal overlap factor roughly exceeds one. The piecewise scheme has shown its capability in terms of balancing the accuracy, efficiency and the wealth of information for the modeling of lightly coupled vibroacoustic systems. This paper extends the method beyond the weak-coupling assumption. A coupling strength factor is first proposed to quantify and adjust the coupling strength between two sub-systems. Two mutually connected sub-cavities are then used as an example to validate the piecewise scheme in relation to the changes in the coupling strength level, as well as the variations in the coupling strength factor itself. The effect of the coupling strength on the computational error of the piecewise scheme is systematically studied. Finally, the applicability of the piecewise calculation scheme is experimentally validated.

Prediction of the airborne sound transmission through the front end of a vehicle

Acoustics represents a quality feature of a passenger vehicle. During the first development phases, prediction tools of the acoustic performance are required to assess in the design process. A combination of numerical and experimental data provides a good compromise between accuracy and modeling effort. A key part of the airborne noise transmission chain are the different passive acoustic treatments applied to minimize the noise impact. These treatments usually include one or more layers of poroelastic media. They exhibit a highly dissipative behavior, making them suitable as noise barriers but, at the same time, complex to model. This article aims to identify the parameters that define the appropriate numerical modeling approach for vibroacoustic systems with poroelastic acoustic packages. Two different concepts for the noise reduction based on poroelastic materials have been investigated, namely the insulation through a spring–mass component and the absorption of a porous layer. The essential principles of the theory of poroelasticity are recalled and three material formulations are derived. The application range of each material model is examined with the help of two configurations: a flat plate and a simplified model of a vehicle front end. The acoustic response of the system is solved with the help of the finite element method using the different material formulations for the description of the poroelastic layers, and the results are compared to measurements conducted in a window test bench. Finally, the main findings are summarized and recommendations towards a more realistic representation of the complete transmission chain are presented.

Uncertainty propagation of the energy flow in vibro-acoustic system with fuzzy parameters

Abstract Uncertainties to varying degrees often exist in the analysis and design problems of vibro-acoustic systems. This paper investigates the high-frequency vibro-acoustic analysis problem with large levels of fuzzy parameters. By employing the level-cut strategy, the energy balance equations with fuzzy parameters are firstly transformed into the interval forms. Then, by combining the perturbation method, the Sherman-Morrison-Woodbury formula, and the statistical energy analysis, a modified perturbation statistical energy analysis (MPSEA) is proposed to predict the uncertainty propagation of the energy flow in vibro-acoustic systems with fuzzy parameters. Further, the sub-interval technique is introduced in MPSEA to improve its accuracy for the systems with large levels of fuzzy parameters. Numerical simulations of a plate-cavity system and a launcher fairing model with fuzzy parameters are presented to demonstrate the efficiency and accuracy of proposed method. Results indicate that the proposed MPSEA with a suitable number of sub-intervals is capable of predicting the energy flow in vibro-acoustic system with large fuzzy parameter accurately and efficiently.

An adaptive MOR method for vibro-acoustic analysis of dynamic systems with viscoelastic damping

Viscoelastic materials have the capability of dissipating vibrational energy which makes them very effective in the passive control of structural vibration and noise radiation. Efficient modeling-and simulation strategies for vibro-acoustic systems with these frequency-dependent damping materials are necessary. One way of doing that is to utilize the finite element (FE) method supported by a damping model that describes the frequency-dependency of the complex elastic modulus. However, in many applications, very large-scale FE models are required to obtain reliable predictions, which makes full-order system evaluation often intractable. Model order reduction (MOR), however, cannot be directly applied to models including viscoelastic materials due to the frequency-dependent property which exactly makes that the equations of motion are not of a standard second-order form as that for regular FE models. In this paper, a transformation technique based on Taylor’s theorem is introduced to treat the non-standard form of the equations of motion. After transforming the problem into a second-order system equation with an associated remainder term, an adaptive MOR procedure is applied to reduce the computational complexity of the vibro-acoustic systems. A numerical example is provided to demonstrate the simplicity and efficiency of the proposed approach.

Validation of a uniaxial structure-borne sound benchmark with emphasis on power and phase accuracy

The evaluation of energetic quantities by means of structure-borne sound power is an attractive approach in analyzing coupled vibroacoustic systems and material characterization. When cyclic mean power quantities are determined for non-resonant frequencies of a system, the accuracy of the relative phases between force and acceleration signals can be crucial for both measurement and simulation-based procedures. In particular, in the case of light damping, the frequency response of the mean power is sensitive to small errors in the phase. This paper first discusses the problem of the phase-correct determination of power quantities from a theoretical perspective. Secondly, a benchmark structure is introduced in order to validate corresponding experimental and numerical methods. For this purpose, the benchmark has a finite element representation, which correlates to hardware realization. All the data required to perform the validation are provided. Based on this benchmark structure, the numerical solution for structure-borne power is first reviewed and generalized. Thereafter, for uniaxial vibration an experimental methodology to determine power quantities is developed in the frequency domain. It is observed that filtering the measurement data with a frequency-dependent phase correction, which includes the predetermined phase error of the individual measurement chain, can lead to physically significant and accurate results of structure-borne sound power. Based on these experimental results, damping values are extracted for any resonant and non-resonant frequency. Additionally, the benchmark case aids in explaining effects, such as apparent non-passive system behaviors that are observed in the measurements with impedance heads.

Parametric model order reduction without a priori sampling for low rank changes in vibro-acoustic systems

Abstract A parametric model order reduction scheme is presented for second order systems that does not require a priori sampling of the parameter space. The proposed scheme transfers the parameter dependence to the throughput matrix of the dynamic system by using an auxiliary input matrix. The resulting model can thus be reduced with non-parametric model reduction techniques. Furthermore, it allows for independent low rank changes in the stiffness, damping and mass matrix of the system. It is shown that in combination with the tangential iterative rational Krylov algorithm, a high number of low rank changes can be parametrized, while keeping the reduced model accurate and of moderate size. Also, a scheme is proposed to further reduce the model size, by a frequency limited post-processing step. The methodology is illustrated with two numerical examples: a purely structural example that simulates an unknown defect by locally reducing the stiffness and damping, and a fully coupled vibro-acoustic example that demonstrates how the method can be used to simulate added mass loading, due to for instance the placement of sensors/actuators.

Random vibro-acoustic control of internal noise through optimized Tuned Mass Dampers

Abstract The present work deals with the passive control of interior sound pressure induced by randomly vibrating structure coupled to an enclosure filled with air. The control in the low frequency range is achieved using a Tuned Mass Damper (TMD) device. In order to guarantee an optimal performance of the TMD device, a stochastic acoustic optimization strategy is proposed. In the later, the objective function is the root mean square acoustic pressure measured at a given location inside the cavity. Assuming linear behavior of the vibro-acoustic system, the modal interaction approach is used and the objective function is evaluated by means of spectral analysis. The effectiveness of the proposed strategy is investigated and compared with others from the open literature. The capability of the proposed optimization strategy to deal with multimodal control of interior sound has also been addressed. To this end, multiple TMDs have been used. The optimal TMD parameters have been obtained using a multi-objective optimization approach and the results that have been gathered show significant attenuations in all target resonant modes.