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02.03 Numerical Methods in Vibro-Acoustics, Part 3

Tutorial on Acoustic Fluid Loading of Structures
Time: 2:00 pm

Author: Stephen Hambric

Abstract ID: 1002

Any vibrating structure is loaded by the fluid surrounding it.  Whether air, water, or something else, the fluid loading adds a spatially distributed resistance (in phase with the vibration) and reactance (out of phase with the vibration) over the structural surfaces.  The resistance absorbs energy, and damps structural vibrations.  The reactance is either mass-like, effectively adding to the structural density, reducing resonance frequencies and vibration amplitudes; or stiffness-like, increasing resonance frequencies.  Usually, mass-like reactance is caused by fluids external to a structure, and stiffness-like reactance is caused by enclosed volumes of fluids.  This tutorial uses analytic methods to compare and contrast external and internal fluid loading on a flat rectangular plate and demonstrates the effects of fluid loading on plate vibration and radiated sound.  The well-known stiffening effect of the internal Helmholtz resonance is demonstrated for a thin panel and a shallow entrained cavity.  The differences between heavy (water) and light (air) external fluid loading are also demonstrated, with significant reductions in resonance frequencies and peak vibration amplitudes for water loading.

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Predicting vibration transmission across junctions using diffuse field reciprocity
Time: 8:40 am

Author: Wannes Stalmans

Abstract ID: 1634

Predicting the sound insulation of an engineering system is a complex problem since not only the direct path through a separating element but also the flanking transmission paths can largely influence the sound insulation of the system. When conventionally analyzing flanking transmission, a diffuse field is assumed in the walls and floors, which are modelled as plates. The junction connecting the walls and floors is assumed to be of infinite extent and the transmission of vibration across the junction is calculated by integrating over all possible angles of incidence. Due to the limitations of the conventional approach, a new approach based on diffuse field reciprocity is proposed. The diffuse field reciprocity relationship relates the vibration transmission to the direct field of a diffuse subsystem to the direct field dynamic stiffness of the subsystem, i.e., the dynamic stiffness of the equivalent infinite subsystem as observed at the junction. The direct field dynamic stiffness matrices of thin, isotropic, elastic plates can be analytically derived. For more complex walls or floors a possible approach is to calculate the direct field dynamic stiffness using finite elements and perfectly matched layers. The perfectly matched layer surrounding the finite element model absorbs the wave propagating outwards from the bounded domain, thus simulating an infinite subsystem.

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Impact sound prediction of finite floor structures with the modal transfer matrix method
Time: 9:00 am

Author: Jasper Vastiau

Abstract ID: 1636

The transfer matrix method (TMM) is commonly employed for wave propagation analysis in layered media of fluid, elastic and porous nature. Up to now it has been used extensively to analyze airborne sound transmission and sound absorption. Its use for impact sound transmission has been investigated to a limited extent, i.e. for thick homogeneous elastic plates of infinite extent and for specific receiver points. This contribution aims to broaden the scope such that the global impact sound, radiated by finite floor structures containing elastic, fluid and/or porous layers, can be analyzed in a more robust way than previously available in literature. A disadvantage of the conventional TMM is that only floors of infinite extent can be implemented. It is possible to remove this drawback using a spatial windowing technique. Furthermore, the modal behavior of the floor is approximately taken into account by projecting the impact force onto the mode shapes and only allowing for the propagation of those waves, corresponding to modal wavenumbers, in the structure. Predictions of the radiated sound power are made for various bare floors and floating floor systems of both infinite and finite extent.

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An isogeometric formulation of locally-conformal perfectly matched layer for acoustic radiation problems
Time: 6:20 am

Author: Yongzhen Mi

Abstract ID: 1660

This paper presents an isogeometric formulation of the locally-conformal perfectly matched layer (PML) for time-harmonic acoustic scattering problems. The new formulation is a generalization of the conventional locally-conformal PML, in which the NURBS patch supporting the PML domain is transformed from real space to complex space. This is achieved by complex coordinate stretching, based on a stretching vector field indicating the directions in which incident sound waves are absorbed. The performance of the isogeometric PML formulation is discussed through several acoustic scattering problems, spanning from one to three dimensions. It is found that the proposed method presents superior computational accuracy, high geometric adaptivity, and good robustness against challenging geometric features. The geometry-preserving ability inherent in the isogeometric framework could bring extra benefits by eliminating geometric errors that are unavoidable in the conventional PML. Meanwhile, these properties are not sensitive to the location of the sound source or the depth of the PML domain.

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3D shape optimization of loudspeakers
Time: 6:40 am

Author: Peter Risby Andersen

Abstract ID: 1708

Improving the performance of loudspeaker units and cabinet designs traditionally relies on a combination of trial and error, sometimes based on a lumped parameter modelling approach. During the last decades, however, large-scale numerical simulations are playing a growing role as a means of improving performance of complex engineering devices such as loudspeakers. However, a numerical model still relies on the experience of the operating engineer to make the appropriate design changes. This can be a difficult task. The use of numerical simulations combined with optimization has a huge potential for further guiding the design process of advanced industrial products where intuition alone is not sufficient. Nevertheless, broadband acoustic simulations are still very time consuming. In this work, we explore the efficiency of a newly proposed semi-analytical adjoint sensitivity approach based on the boundary element method in combination with a lumped parameter model. The sensitivity analysis is used to shape optimize the cabinet of a loudspeaker using free form deformation. The objective of the optimization is to improve frequency responses and directivity patterns.

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Vibrational energy distribution in plate excited with random white noise
Time: 11:00 am

Author: victor tyrode

Abstract ID: 1712

In Statistical Energy Analysis (SEA) and more generally in all statistical theories of sound and vibration, the establishment of diffuse field in subsystems is one of the most important assumption. Diffuse field is a special state of vibration for which the vibrational energy is homogeneously and isotropically distributed. For subsystems excited with a random white noise, the vibration tends to become diffuse when the number of modes is large and the damping sufficiently light. However even under these conditions, the so-called coherent backscattering enhancement (CBE) observed for certain symmetric subsystems may impede diffusivity. In this study, CBE is observed numerically and experimentally for various geometries of subsystem. Also, it is shown that asymmetric boundary conditions leads to reduce or even vanish the CBE. Theoretical and numerical simulations with the ray tracing method are provided to support the discussion.

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Solving linear systems from dynamical energy analysis – using and reusing preconditioners
Time: 12:00 pm

Author: Martin Richter

Abstract ID: 1740

Dynamical energy analysis (DEA) is a computational method to address high-frequency vibro-acoustics in terms of ray densities. It has been used to describe wave equations governing structure-borne sound in two-dimensional shell elements as well as three-dimensional electrodynamics. To describe either of those problems, the wave equation is reformulated as a propagation of boundary densities. These densities are expressed by finite dimensional approximations. All use-cases have in common that they describe the resulting linear problem using a very large matrix which is block-sparse, often real-valued, but non-symmetric. In order to efficiently use DEA, it is therefore important to also address the performance of solving the corresponding linear system. We will cover three aspects in order to reduce the computational time: The use of preconditioners, properly chosen initial conditions, and choice of iterative solvers. Especially the aspect of potentially reusing preconditioners for different input parameters is investigated.

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Band structure and defect states in acoustic phononic crystals using expansion and micro-perforated chamber mufflers
Time: 2:00 pm

Author: Adriano Mitsuo Goto

Abstract ID: 1775

The expansion and the micro-perforated chamber mufflers are acoustic silencers designed to attenuate the sound propagation at duct systems. These silencers can show interesting phononic crystals behavior when set periodically. The concept of phononic crystals still is an emerging topic in vibration and sound control. The periodic arrangement of acoustic silencers can provide a significant enhancement of the sound absorption due to the “wave filtering” property where the wave cannot propagate at certain frequency ranges, called stopbands or bandgaps. However, these properties may be affected by defects, like the break of the periodicity due to manufacturing errors. For the present work, the influence of some defects on the acoustic efficiency is investigated numerically for expansion and micro-perforated chamber mufflers. A direct and efficient approach is used to obtain the transfer and dynamic stiffness matrices. Simulated examples are used to calculate the forced response, transmission loss, and dispersion diagram, which are verified by other methods.

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Finite Element Method and Dynamical Energy Analysis in Vibroacoustics – A Comparative Study
Time: 12:20 pm

Author: Sebastian Zettel

Abstract ID: 1906

Future aircraft concepts utilizing innovative lightweight structures and novel propulsion concepts are a necessity for long term sustainable air travel. These concepts pose new challenges for the vibro-acoustic assessment of cabin structures and the associated noise impact on passengers. Finite Element (FE) models derived from aircraft pre-design data are not optimized for use in acoustic analyses, i.e. the mesh is too coarse to provide meaningful results while setting up Statistical Energy Analysis models for this specific purpose is adding another time-consuming step. A possible alternative, Discrete Energy Analysis (DEA), is evaluated. This method allows to calculate the acoustic behavior of thin-walled structures in higher frequency ranges simply using existing FE meshes. In this paper an experimental lightweight aluminum structure and its respective FE model is investigated for a frequency range up to 5000 Hz. A comparison in terms of vibrational energy between DEA, FE and measurement results are presented. Finally, a lower-bound frequency range is identified in which DEA and FEM correlate and thus allow a substitution for further simulations at higher frequencies.

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Subtractive modeling using the reverse condensed transfer function method: influence of the numerical errors
Time: 11:20 am

Author: Florent Dumortier

Abstract ID: 2186

Decoupling procedures based on substructuring methods allow to predict the vibroacoustic behaviour of a given system by removing a part of an original system that can be easily modelled. The reverse Condensed Transfer Function (rCTF) method has been developed to decouple acoustical or mechanical subsystems that are coupled along lines or surfaces. From the so-called condensed transfer functions (CTFs) of the original system and of the removing part, the behaviour of the system of interest can be predicted. The theoretical framework as well as a numerical validation have been recently published. In the present paper, we focus on the influence of numerical errors on the results of the rCTF method, when the CTFs are calculated using numerical models for the original system and/or the removed part. The rCTF method is applied to a test case consisting in the scattering problem of a rigid sphere in an infinite water domain and impacted by an acoustic plane wave. Discrete green formulation and finite element method are used to estimate the CTFs. Numerical results will be presented in order to evaluate the sensitivity of the method to model errors and the potential promises and limitations of the method will be highlighted.

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A hybrid method for broadband vibroacoustic simulations
Time: 2:40 pm

Author: Scott Sommerfeldt

Abstract ID: 2236

Many methods for simulating acoustic responses of vibrating systems are only suitable for limited frequency ranges, providing either an accurate low frequency or high frequency response. A hybrid method is presented to combine a low frequency modal response and a high frequency statistical energy response to obtain a unified broadband response. The method is designed to produce an auralizable response. An experimental setup is used to validate the method. Listening tests are conducted to assess the realism of the auralizations compared to measurements. The listening tests confirm that the method is able to produce realistic auralizations, subject to a few limitations.

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Development of a hybrid SmEdA/SEA model for predicting the power exchanged between low and high modal density subsystems
Time: 11:40 am

Author: Guang ZHU

Abstract ID: 2535

Statistical modal Energy distribution Analysis (SmEdA) was developed from classical Statistical Energy Analysis (SEA). It allows computing power flow between coupled subsystems from the deterministic modes of uncoupled subsystems without assuming the SEA modal energy equipartition. SmEdA is well adapted in mid-frequency when the subsystems have not a very high modal density. However, for some systems e.g. the plate-cavity system, one subsystem can exhibit a low modal density while the other one a high one. The goal of the paper is then to propose an extension of SmEdA formulation that allows describing one subsystem by its deterministic modes, and the other one as a diffuse field statistically supposing modal energy equipartition. The uncertain subsystem is then characterized by sets of natural frequencies and mode shapes constructed based on Gaussian Orthogonal Ensemble matrix and the cross-spectrum density of a diffuse field, respectively. This formulation permits not only the computation of mean noise response but also the variance generated by the uncertainties and furthermore without bringing in much computation. It is demonstrated that the obtained analytical results from the proposed hybrid SmEdA/SEA are consistent with numerical results computed by FEM with an appropriate degree of uncertainty.

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Metamaterial plate with an arrangement of different resonators
Time: 4:20 pm

Author: Giovanna Pisicchio Zanoni

Abstract ID: 2571

Local resonant metamaterials have been widely studied for vibration suppression in the last 20 years. They produce band gaps, which are frequency regions where the wave is not allowed to propagate. They are an alternative to reduce vibration levels at lower frequencies when compared to phononic crystals, which require larger periodic cells to create band gaps at lower frequencies. The most common configuration for a local resonant metamaterial is a periodic cell of a known structure with one attached resonator. In this study, a plate with a periodic cell using two different resonators is analyzed. Some configurations of mass and stiffness for the two resonators will be discussed to pursue the best compromise between a wider band gap and a more considerable vibration attenuation. The dispersion relation for the proposed metamaterial unit cell will be calculated using the Wave Finite Element Method to evaluate these configurations. The frequency response function for a finite structure with the proposed arrangement will also be calculated using the Finite Element Method to compare the results.

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Computing Radiated Sound Power using Quadratic Power Transfer Vector (QPTV)
Time: 2:20 pm

Author: Rajendra Gunda

Abstract ID: 2643

Pressure Acoustic Transfer Functions or Vectors (PATVs) relate the surface velocity of a structure to the sound pressure level at a field point in the surrounding fluid. These functions depend only on the structure geometry, properties of the fluid medium (sound speed and characteristic density), the excitation frequency and the location of the field point, but are independent of the surface velocity values themselves. Once the pressure acoustic transfer function is computed between a structure and a specified field point, we can compute pressure at this point for any boundary velocity distribution by simply multiplying the forcing function (surface velocity) with the acoustic transfer function. These PATVs are usually computed by application of the Reciprocity Principle, and their computation is well understood. In this work, we present a novel way to compute the Velocity Acoustic Transfer Vector (VATV) which is a relation between the surface velocity of the structure and fluid particle velocity at a field point. To our knowledge, the computation of the VATV is completely new and has not been published in earlier works. By combining the PATVs and VATVs at a number of field points surrounding the structure, we obtain the Quadratic Power Transfer Vector (QPTV) that allows us to compute the sound power radiated by a structure for ANY surface velocity distribution. This allows rapid computation of the sound power for an arbitrary surface velocity distributions and is useful in designing quiet structures by minimizing the sound power radiated.

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Vibrational response of a fluid-loaded baffled plate near the free surface of a fluid
Time: 7:40 am

Author: Jamie Kha

Abstract ID: 2721

An analytical model to predict the vibrational response of a simply supported rectangular plate embedded in an infinite baffle with an upper free surface under heavy fluid loading and excited by a point force is presented. The equations of motion of a thin plate are solved using­­­­­­ modal decomposition technique by employing admissible functions for an in-vacuo plate and by directly solving the Helmholtz equation for acoustic waves in a fluid. The vibrational response for a flat plate in an infinite baffle and unbounded domain (semi-infinite domain) using analytical formulation available in literature is initially computed. These results are then compared against present results to observe the effect of a free surface. Predictions from analytical models are validated by comparison with results obtained by numerical models. The proposed analytical approach presents a novel formulation to describe a fluid-loaded flat plate in a waveguide and an efficient method for predicting its vibrational response.

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Power balance analysis of nonperiodic structural components from a model converted from FEM to SEA.
Time: 3:20 pm

Author: Mathias Hinz

Abstract ID: 2791

Using Statistical Energy Analysis (SEA) to characterize the power flow within a vibroacoustic system is a challenging task when the subsystems have irregular shape and complex construction. Retrieving analytical solutions for the ordinary SEA parameters is nearly impractical without restricting simplifications and periodicity is usually not exploitable due to the lack of repetition patterns. A promising option to perform the power balance for such cases is to filter part of the information contained in a Finite Element Method (FEM) model of the system, in order to convert it into a SEA model. In this paper, the Lorentzian Frequency Average and the Nonparametric Random Matrix Theory are applied to randomize the dynamic stiffness matrix of the FEM components from a system of industrial application. The obtained direct field dynamic stiffness matrices are employed along the diffuse field reciprocity relationship as a general framework to determine the energetic content of each component. The results obtained with this procedure are evaluated against the ones from classical SEA and Monte Carlo techniques.

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Efficient prediction of construction equipment exterior and cabin interior noise over broad frequency range using novel SEA method
Time: 6:00 am

Author: Hiromitsu Emoto

Abstract ID: 3124

Statistical Energy Analysis (SEA) is commonly used for the prediction of interior cabin noise from construction equipment such as excavators, dump trucks, or graders. While traditional SEA method is computationally efficient and effective for the prediction of total radiated noise, it isn't suitable for prediction of sound diffraction around machinery and evaluation of spatial variations in sound field. As a result, prediction of cabin airborne interior noise transmission using SEA method typically requires experimental measurements in order to estimate incident sound field over the exterior boundary of the cab which makes it unsuitable for use in early stage design where test data isn't available. A novel SEA method that accounts for spatial gradients in the reverberant field has been developed and is introduced in this paper. It's usage for prediction of both exterior and cab interior noise over broad frequency range is demonstrated along with experimental validation for construction equipment under operating conditions.

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Study on target energy transfer of 3D acoustic cavity – plate coupling system with the membrane nonlinear energy sink
Time: 8:00 am

Author: Jinmeng Yang

Abstract ID: 3986

The target energy transfer (TET) between a membrane nonlinear energy sink (NES) and the acoustic medium inside a rectangular cavity is studied. The acoustic medium is interacted with a plate and multi-order modes coupling of the 2 structure is considered. Based on the modal expansion approach, with Green's function, Helmholtz equation and the boundary conditions of the acoustic medium and the plate, the coupling coefficient matrix of the mode of 2 structures is derived. The equations of the membrane NES, multi-order modes of the acoustic medium and multi-order modes of the plate are established, and numerical analysis is used to investigate the TET phenomenon. The results show that in condition of a single-point excitation to the plate, under a certain range of excitation levels, the membrane can be seen as a kind of NES, and the energy in the acoustic medium can be unidirectionally transmitted to the membrane NES and attenuated, reducing the sound pressure level in the cavity. At the same time, it is found that the NES can suppress multi-order sound pressure of the acoustic medium at the same time, and realize the control of cascaded resonance noise.

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Joint modeling for the analytical estimation of dynamic behaviors of beam-coupled structures
Time: 8:20 am


Abstract ID: 2164

In this study, analytical method is applied for the estimation of dynamic behaviors of beam-coupled structures. Mathematical expressions are given with terms of shape factors, material information and assembly angles of each sub-component. Based on Euler-Bernoulli beam theory, entire formulation is built with compatibility of system dynamics. The coupled structures are divided into two types, point coupling and mass coupling, related with the properties of coupling points. Point coupling is commonly used assumption that two sub-components are combined with lumped spring or damping, and mass coupling has undeformable rigid joint which has mass and inertia like welded structures. Dynamic properties of coupled structures are predicted in forms of frequency response functions and spectral responses about given forces. The verification process is conducted for assessing the accuracy of the estimation formula by using modal frequencies and mode shapes of beam-coupled structures. Extracted modal parameters from experimental modal analysis and finite element method are adopted as reference values for verification.

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