Alain Le Bot

Entropy in statistical energy analysis

In this paper, the second principle of thermodynamics is discussed in the framework of statistical energy analysis (SEA). It is shown that the “vibrational entropy” and the “vibrational temperature” of sub-systems only depend on the vibrational energy and the number of resonant modes. A SEA system can be described as a thermodynamic system slightly out of equilibrium. In steady-state condition, the entropy exchanged with exterior by sources and dissipation exactly balances the production of entropy by irreversible processes at interface between SEA sub-systems.

Radiative transfer equation for multiple diffraction

This paper aims to apply the radiative transfer method to acoustical diffraction by obstacles. Some fictitious sources are introduced at diffracting wedges and a transfer equation based on energy balance determines the diffracted powers. It leads to a set of linear equations on diffracted powers which can be solved in a finite number of steps. It is then possible to calculate the diffracted field anywhere. Some applications to diffraction by obstacles of various shapes are presented. Results of this method are compared with Geometrical Theory of Diffraction and BEM reference calculations. It is shown that this method is particularly efficient in case of multiple diffraction where the ray-tracing technique involves an infinite number of rays between a source and a receiver point.

Statistical Vibroacoustics and Entropy Concept

Statistical vibroacoustics, also called statistical energy analysis (SEA) in the field of engineering, is born from the application of statistical physics concepts to the study of random vibration in mechanical and acoustical systems. This article is a discussion on the thermodynamic foundation for that approach with particular emphasis devoted to the meaning of entropy, a concept missing in SEA. The theory focuses on vibration confined to the audio frequency range. In this frequency band, heat is defined as random vibration that is disordered vibration and temperature is the vibration energy per mode. Always in this frequency band, the concept of entropy is introduced and its meaning and role in vibroacoustics are enlightened, together with the related evolutionary equation. It is shown that statistical vibroacoustics is non-equilibrium thermodynamics applied to the audio range.

Energy exchange in uncorrelated ray fields of vibroacoustics

This paper is concerned with the basic equations governing energy and intensity in incoherent ray fields. Some fictitious sources are distributed on the boundary of the domain but also on diffracting wedges and peaks. Their powers are determined by some appropriate boundary integral equations. Once these powers are known, energy and intensity inside the domain are given by a simple superposition of contributions of these sources. All paths of propagation are taken into account including direct, reflected, refracted, transmitted, and diffracted rays, but also, radiation by surface, edge or corner modes, and the reciprocal paths for structural response. This theory unifies several fields from the “radiosity method” in room acoustics which determines the reverberation time to the “radiative transfer method” in structural dynamics which gives the repartition of vibrational energy inside subsystems of built-up structures. This is therefore a candidate for an alternative to statistical energy analysis when fiel...

Direct numerical simulation of the dynamics of sliding rough surfaces

The noise generated by the friction of two rough surfaces under weak contact pressure is usually called roughness noise. The underlying vibration which produces the noise stems from numerous instantaneous shocks (in the microsecond range) between surface micro-asperities. The numerical simulation of this problem using classical mechanics requires a fine discretization in both space and time. This is why the finite element method takes much CPU time. In this study, we propose an alternative numerical approach which is based on a truncated modal decomposition of the vibration, a central difference integration scheme and two algorithms for contact: The penalty algorithm and the Lagrange multiplier algorithm. Not only does it reproduce the empirical laws of vibration level versus roughness and sliding speed found experimentally but it also provides the statistical properties of local events which are not accessible by experiment. The CPU time reduction is typically a factor of 10.

Radiosity and radiative transfer in sound and vibration

At high frequencies in acoustics, the most popular method is ray-tracing and its variants including radiosity. In structural vibrations, the most known method is rather statistical energy analysis [1]. Both methods may be derived from a unique approach based on radiative transfer equations analogous to radiative exchanges of energy in thermics [2]. In this study, we present an overview of the radiative transfer equations in sound and vibration. We first show that radiosity is equivalent to ray-tracing with Lambertian reflection. In steady-state condition, radiosity is strictly equivalent to the view factor method in thermics. But in transient condition, radiosity provides an elegant solution to predict reverberation beyond the validity of Sabines law. The theory is also well suited for structural rays in built-up structures. The radiative transfer equations account for reflection and transmission at interfaces of structural components. Sound radiation may also be described in the limit of high frequencie...

Discussion on the validity domain of SEA

This paper is concerned with the validity domain of SEA which is defined in terms of four criteria. The mode count N and the modal overlap M must be high, the normalized attenuation factor m and the coupling strength gamma must be small. The application of the dimensional analysis on the governing equations of plates gives the space of dimensionless parameters in which the validity domain of SEA must be delimited. This domain is discussed on the basis of the geometric properties of the surfaces delimiting it. The diagrams of validity of SEA are introduced and proposed as a tool for a practical use of SEA.

A hybrid method for vibroacoustic coupling

A hybrid method is proposed to study the noise radiated in the high frequency range by a gearbox inside an engine compartment. Due to its elastodynamic characteristics, the vibratory behaviour of the transmission lies in the low frequency range. However, the noise radiated is rather in the high frequency range facing the size of the acoustical cavity. Thus the proposed hybrid method allows to couple a low-frequency vibratory approach to the radiative transfer method used for the high frequency acoustical calculation. The coupling is realized through equivalent energy sources introduced on the surface of the vibrating structure.