Laurent Maxit

Extension of SEA model to subsystems with non-uniform modal energy distribution

Abstract In order to widen the application of statistical energy analysis (SEA), a reformulation is proposed. Contrary to classical SEA, the model described here, statistical modal energy distribution analysis (SmEdA), does not assume equipartition of modal energies. Theoretical derivations are based on dual modal formulation described in Maxit and Guyader (Journal of Sound and Vibration 239 (2001) 907) and Maxit (Ph.D. Thesis, Institut National des Sciences Appliquees de Lyon, France 2000) for the general case of coupled continuous elastic systems. Basic SEA relations describing the power flow exchanged between two oscillators are used to obtain modal energy equations. They permit modal energies of coupled subsystems to be determined from the knowledge of modes of uncoupled subsystems. The link between SEA and SmEdA is established and make it possible to mix the two approaches: SmEdA for subsystems where equipartition is not verified and SEA for other subsystems. Three typical configurations of structural couplings are described for which SmEdA improves energy prediction compared to SEA: (a) coupling of subsystems with low modal overlap, (b) coupling of heterogeneous subsystems, and (c) case of localized excitations. The application of the proposed method is not limited to theoretical structures, but could easily be applied to complex structures by using a finite element method (FEM). In this case, FEM are used to calculate the modes of each uncoupled subsystems; these data are then used in a second step to determine the modal coupling factors necessary for SmEdA to model the coupling.

Patch Transfer Functions as a Tool to Couple Linear Acoustic Problems

A method to couple acoustic linear problems is presented in this paper. It allows one to consider several acoustic subsystems, coupled through surfaces divided in elementary areas called patches. These subsystems have to be studied independently with any available method, in order to build a database of transfer functions called patch transfer functions, which are defined using mean values on patches, and rigid boundary conditions on the coupling area. A final assembly, using continuity relations, leads to a very quick resolution of the problem. The basic equations are developed, and the acoustic behavior of a cavity separated in two parts is presented, in order to show the ability of the method to study a strong-coupling case. Optimal meshing size of the coupling area is then discussed, some comparisons with experiments are shown, and finally a complex automotive industrial case is presented.

Modeling of micro-perforated panels in a complex vibro-acoustic environment using patch transfer function approach

A micro-perforated panel (MPP) with a backing cavity is a well known device for efficient noise absorption. This configuration has been thoroughly studied in the experimental conditions of an acoustic tube (Kundt tube), in which the MPP is excited by a normal incident plane wave in one dimension. In a more practical situation, the efficiency of MPP may be influenced by the vibro-acoustic behavior of the surrounding systems as well as excitation. To deal with this problem, a vibro-acoustic formulation based on the patch transfer functions (PTF) approach is proposed to model the behavior of a micro-perforated structure in a complex vibro-acoustic environment. PTF is a substructuring approach, which allows assembling different vibro-acoustic subsystems through coupled surfaces. Upon casting micro-perforations and the flexibility of the MPP under transfer function framework, the proposed PTF formulation provides explicit representation of the coupling between subsystems and facilitates physical interpretation. As an illustration example, application to a MPP with a backing cavity located in an infinite baffle is demonstrated. The proposed PTF formulation is finally validated through comparison with experimental measurements available in the literature.

Prediction of flow induced sound and vibration of periodically stiffened plates

Stiffened structures excited by the turbulent boundary layer (TBL) occur very frequently in engineering applications; for instance, in the wings of airplanes or the pressure hulls of submarines. To improve knowledge of the interaction between stiffened structures and TBL, this paper deals with the modeling of infinite periodically stiffened plates excited by TBL. The mathematical formulation of the problem is well-established in the literature. The originality of the present work relies on the use of a wavenumber-point reciprocity technique for evaluating the response of the plate to convected harmonic pressure waves. It follows a methodology for estimating the vibro-acoustic response of the plate excited by the TBL from the wall pressure spectrum and its displacements in the wavenumber space due to point excitations located at the receiving positions. The computing process can be reduced to the numerical integration of an analytical expression in the case of a periodically stiffened plate. An application to a naval test case highlights the effect of Bloch-Floquet waves on the vibrations of the plate and its radiated pressure in the fluid.

A condensed transfer function method as a tool for solving vibroacoustic problems

Substructuring approaches are nowadays widely used to predict numerically the vibroacoustic behavior of complex mechanical systems. Some of these methods are based on admittance or mobility frequency transfer functions at the coupling interfaces. They have already been used intensively to couple subsystems linked by point contacts and enable to solve problems at higher frequency while saving computation costs. In the case of subsystems coupled along lines, a Condensed Transfer Function method is developed in the present paper. The admittances on the coupling line are condensed in order to reduce the number of coupling forces evaluated. Three variants are presented, where the transfer functions are condensed using three different functions. After describing the principle of the CTF method, simple structures will be given as test cases for validation.

Improving the Patch Transfer Function Approach for Fluid-Structure Modelling in Heavy Fluid

The vibro-acoustic behaviour of elastic structures coupled with cavities filled with a heavy fluid can be modelled by using the Finite Element Method. In order to reduce computing time, the Patch Transfer Function (PTF) approach is used to partition the global problem into different sub-problems. Different types of problem partitioning are studied in this paper. Partitioning outside the near field of structures to reduce the number of patches of the coupling surface for frequencies below the critical frequency is of particular interest. This implies introducing a non standard modal expansion to compute the PTF accurately enough to guarantee the convergence of the PTF method and reduce computation time in comparison to a direct Finite Element resolution. An application on a submarine structure illustrates the interest of this approach.

Non resonant transmission modelling with statistical modal energy distribution analysis

Statistical modal Energy distribution Analysis (SmEdA) can be used as an alternative to Statistical Energy Analysis for describing subsystems with low modal overlap. In its original form, SmEdA predicts the power flow exchanged between the resonant modes of different subsystems. In the case of sound transmission through a thin structure, it is well-known that the non resonant response of the structure plays a significant role in transmission below the critical frequency. In this paper, we present an extension of SmEdA that takes into account the contributions of the non resonant modes of a thin structure. The dual modal formulation (DMF) is used to describe the behaviour of two acoustic cavities separated by a thin structure, with prior knowledge of the modal basis of each subsystem. Condensation in the DMF equations is achieved on the amplitudes of the non resonant modes and a new coupling scheme between the resonant modes of the three subsystems is obtained after several simplifications. We show that the contribution of the non resonant panel mode results in coupling the cavity modes of stiffness type, characterised by the mode shapes of both the cavities and the structure. Comparisons with reference results demonstrate that the present approach can take into account the non resonant contributions of the structure in the evaluation of the transmission loss.

SmEdA Vibro-Acoustic Modeling of a Trimmed Truck Cab in the Mid-Frequency Range

The City Lightweight and Innovative Cab (CLIC) project was a scientific collaboration gathering public and private organizations. The aim was to propose an innovative lighten truck cab, where a high strength steel was used. As long as it could affect directly the acoustic environment of the cab, it was necessary to be able to simulate the vibroacoustic behavior of the truck cab in the mid frequency range. The dissipative treatments used for noise and vibration control such as viscoelastic patches and acoustic absorbing materials must then be taken into account in the problem. A process based on the SmEdA (Statistical modal Energy distribution Analysis) method was developed and is presented in this paper. SmEdA allows us substructuring the global problem, to study the interaction between the floor and the interior cavity. The process consists in building finite element models (FEM) of each subsystem (floor, internal cavity), including the dissipative material (damping layer, poroelastic material). Standard modal FEM calculations are then performed for each uncoupled subsystem. From the spatial mode shapes, and the modal strain -kinetic energies, the modal loss factors of both subsystems are estimated. Finally, the pressure levels inside the cavity are deduced from the resolution of the SmEdA equations. To validate this process, a truck cabin has been excited mechanically on a rail of the floor and the pressure levels at different positions inside the cabin were measured for different configurations of dissipative treatment. Comparisons between SmEdA and experimental results allows us to assess the accuracy of the proposed method.

Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles

This paper aims at developing an experimental method to characterize the vibroacoustic response of a panel to a diffuse acoustic field (DAF) excitation with a different laboratory setup than those used in standards (i.e., coupled rooms). The proposed methodology is based on a theoretical model of the DAF and on the measurement of the panels sensitivity functions, which characterize its vibroacoustic response to wall plane waves. These functions can be estimated experimentally using variations of the reciprocity principle, which are described in the present paper. These principles can either be applied for characterizing the structural response by exciting the panel with a normal force at the point of interest or for characterizing the acoustic response (radiated pressure, acoustic intensity) by exciting the panel with a monopole and a dipole source. For both applications, the validity of the proposed approach is numerically and experimentally verified on a test case composed of a baffled simply supported plate. An implementation for estimating the sound transmission loss of the plate is finally proposed. The results are discussed and compared with measurements performed in a coupled anechoic-reverberant room facility following standards.

Simulation of the pressure field beneath a turbulent boundary layer using realizations of uncorrelated wall plane waves.

This paper investigates the modeling of a vibrating structure excited by a turbulent boundary layer (TBL). Although the wall pressure field (WPF) of the TBL constitutes a random excitation, the element-based methods generally used for describing complex mechanical structures consider deterministic loads. The response of such structures to a random excitation like TBL is generally deduced from calculations of numerous Frequency Response Functions. Consequently, the process is computationally expansive. To tackle this issue, an efficient process is proposed for generating realizations of the WPF corresponding to the TBL. This process is based on a formulation of the problem in the wavenumber space and the interpretation of the WPF as uncorrelated wall plane waves. Once the WPF has been synthesized, the local vibroacoustic responses are calculated for the different realizations and averaged together in the last step. A numerical application of this process to a plate located beneath a TBL is used to verify its efficiency and ability to reproduce the partial space correlation of the excitation. To further illustrate the proposed method, a stiffened panel modeled using the finite element method is finally examined.