Table Of ContentCISM COURSES AND LECTURES
Series Editors:
The Rectors of CISM
Sandor Kaliszky -Budapest
Mahir Sayir - Zurich
Wilhelm Schneider -Wien
The Secretary General of CISM
Giovanni Bianchi -Milan
Executive Editor
Carlo Tasso -Udine
The series presents lecture notes, monographs, edited works and
proceedings in the field of Mechanics, Engineering, Computer Science
and Applied Mathematics.
Purpose of the series is to make known in the international scientific
and technical community results obtained in some of the activities
organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECfURES -No. 396
FLUID-STRUCTURE INTERACTIONS
IN ACOUSTICS
EDITED BY
DOMINIQUE HABAULT
C.N.R.S. - L.M.A., FRANCE
~ Springer-Verlag Wien GmbH
This volume contains 135 iIIustrations
This work is subject to copyright.
AII rights are reserved,
whether the whole or part of the material is concerned
specifically those of translation, reprinting, re-use of illustrations,
broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks.
© 1999 by Springer- Verlag Wien
Origina11y published by CISM, Udine in 1999.
SPIN 10728749
In order to make this volume available as economically and as
rapidl)' as possible the authors' typescripts have been
reproduced in their original forms. This method unfortunately
has its typographical limitations but it is hoped that they in no
way distract the reader.
ISBN 978-3-211-83147-2 ISBN 978-3-7091-2482-6 (eBook)
DOI 10.1007/978-3-7091-2482-6
PREFACE
Nowadays, environmental pollution by noise and vibration is an
important nuisance. The necessity of reducing this pollution has increased
during the last decades. Silence is now one of the main qualities for a car, a
train, a plane, a building ...
A great number of studies are conducted (in universities and in industry)
on the noise radiation phenomena which are responsible for this kind of
pollution. Generally speaking, noise radiation is caused by the coupling between
a structure which vibrates because or mechanical and/or flow excitations and
the surrounding fluid. The expressions "fluid-structure interactions" and
"Vibro-Acoustics" are also used to refer to these phenomena.
The course presents an advanced overview on interaction phenomena
between a structure and a fluid, including nonlinear aspects.
Chapters 1 and 2 are mainly dedicated to the description of the
phenomena and provide the basic equations. Chapter 1 is concerned with the
description or the vibrations or thin bodies immersed in a fluid and submitted to
deterministic or random excitations. Chapter 2 is concerned with the
des.cr_iption or the fluid flow (turbulent flows, turbulent boundary layer,
wall-pressure fluctuations) and or the vibro-acoustic response or a flexible
structure to the adjacent turbulent wall-pressure field.
Chapters 3 to 5 are dedicated to analytical and numerical methods to
compute the displacement on the structure and the sound pressure radiated in
the fluid. They include the description or methods such as transform techniques,
the Wiener-Hop/ method, integral equations, perturbation techniques and finite
element and finite difference methods.
Chapter. 6 is dedicated to the modeling or nonlinear dynamics of
structures excited by flows (quasi-steady fluid cases). Examples such as
pendulum structures in a windfield and a suspension bridge are examined in
detail.
Chapter 7 describes some or the noise radiation phenomena in the
car-industry. Several metheds such as structural acoustic analysis and noise
transfer path analysis are presented and applied to these noise and vibration
problems.
It is my pleasure to thank again the C.J.S.M. board and secretariat who
gave us the opportunity to organise the summer school in September 1998 and
publish this book. Thanks again also to all the lecturers who have carefully
prepared the texts published in this book.
Dominique Habault
CONTENTS
Page
Preface
Chapter 1
Modelling of Fluid/Structure Interactions
by P.J. T. Filippi ..................................................................................................................................................................................... I
Chapter 2
Vibroacoustics of Flow-Excited Structures
by D. Juve, Ch. Bailly, Ch. Durant and G. Robert ......................................................................... 51
Chapter 3
Some Analytical Methods for Fluid-Structure Interaction Problems
by N. Peake ............................................................................................................................................................................................... 87
Chapter 4
Some Computational Methods for Sound Radiation Problems
by D. Habault ........................................................................................................................................................................ 135
Chapter 5
Finite Difference and Finite Element Methods
by U.R. Kristiansen, M. Dhainaut and T.F. Johansen .......................................................... 179
Chapter 6
Nonlinear Dynamics of Structures Excited by Flows:
Quasi-Steady Modelling and Asymptotic Analysis
by A.H.P. van der Burgh .................................................................................................................................................. 221
Chapter 7
Acoustic Applications in Vehicle Engineering
by R. Freymann ............................................................................................................................................................................... 261
CHAPTER 1
MODELLING OF FLUID/STRUCTURE INTERACTIONS
P.J.T. Filippi
Laboratoire de Mecanique et d' Acoustique, Marseille, France
Abstract
This chapter is devoted to the basic equations of Vibro-Acoustics. Only
thin elastic bodies are considered.
First, the approximate equations governing the linear vibrations of thin
plates, thin circular cylindrical shells and spherical shells are established.
The approximation is based on the hypothesis that the elastic body has
one dimension which is small compared to the other two ones and to the
wavelengths of the vibrations.
Then, the theory of 'tn vacuo thin plates and cylindrical shells under har
monic excitations is rapidly summarized (resonance mode series expansion,
boundary integral representation).
The other sections deal with the response of fluid-loaded plates and
shells, excited either by deterministic forces (harmonic or transient) or by
random forces. The fluid load is represented by a boundary integral. Dif
ferent representations of the solution are developed: boundary integral rep
resentation of the structure displacement and of the sound pressure field;
fluid-loaded eigenmode series and fluid-loaded resonance mode series. These
different theoretical aspects are developed on canonical examples with in
creasing complexity.
2 P.J.T. Filippi
1.1 Introduction
Vibro-Acoustics phenomena occur in many real-life situations. In most cases, thin
elastic or visco-elastic bodies are involved; the generation or the transmission of sound
by three-dimension bodies is much less common.
Vibro-Acoustics phenomena can be classified into two reciprocal groups: the gener
ation of sound due to the vibrations of an elastic body and the vibrations of an elastic
body due to an incident acoustic wave. The combination of these two aspects results
in the transmission of acoustic energy through an elastic structure.
Noise pollution comes from house equipments like washing machines, coffee grinders,
vacuum cleaners, etc ... in which the motor induces vibrations of the external struc
tures, creating so an acoustic (mainly noisy) wave. The origin of outdoor noise pollution
is the same phenomenon: the hull of a car or a truck, being set into vibrations by the
engine and by the contact wheels/road, generates noise. Inside an enclosure, like a
house, a car, a plane, etc ... , noise can be due to external force sources which set into
vibrations a part of the enclosure boundary (windows, ceilings and floors; the elastic
shell and the windows of a vehicle, . . . ) : the external force can have any sort of origine
(the motor in a car or a plane, for example) or a wall pressure (incident acoustic wave,
wall pressure due to a turbulent or a vortex flow). Lots of other noise sources which
imply Vibro-Acoustics phenomena could be cited. It can be remarked that most of the
classical noise sources involve the coupling between a fluid and a thin body (plate or
shell). This is the motivation for paying attention in this course to Vibro-Acoustics
phenomena which involve thin plates and shells. It must be mentioned that the ba
sic results and methods which are presented here remain valid for three-dimensional
bodies.
In the next section, approximate equations governing the vibrations of thin struc
tures (plates, circular cylindrical and spherical shells) are established. The method
is first based on the following hypothesis: a structure is said to be thin if one of its
dimensions - called its thickness - is small compared to both the other two ones and
the vibration wavelengths involved. The second hypothesis is that every mechanical
quantity which describes the structure motion can be expanded into a formal Taylor
like series of the transverse coordinate. By keeping a finite number of such terms in
the strain/stress law and in the subsequent energy density, one gets an approximated
equation of energy conservation, or, equivalently, an approximated equation of motion.
Here, the lowest order terms are kept: this leads to the plate equation (as it has been
developed in particular by Kirchhoff) and the Donnell and Mushtari shell equations.
In section 3, elementary results of the theory of in vacuo plates and cylindrical
shells are recalled: resonance mode and resonance frequencies; resonance mode series
representation of the forced vibrations; Boundary Integral Equations equivalent to the
initial boundary value problem.
Section 4 starts with a general presentation of the load that a fluid exerts on a vi-
Modelling of Fluid/Structure Interactions 3
brating structure: various integral representations of the fluid influence are proposed,
which depend on the complexity of the structure. This is illustrated by several exam
ples.
First, baffled plates and baffled cylindrical shells are considered: for both cases, the
fluid load is expressed by a Green's formula and it is shown how the initial boundary
value problem can be reduced to a system of Boundary Integral Equations. The eigen
modes and resonance modes of the fluid-loaded structure are then defined. Finally,
the response of the system to either a deterministic excitation (harmonic or transient)
or a random one (mainly, a turbulent wall pressure) is expanded into a series of the
resonance modes.
Then an example of structure immersed in a first fluid and containing a second
fluid is examined: the importance of the internal resonance modes is pointed out.
The more complex structure is a Line-2 shell (finite length circular cylinder with
semi-spherical end-caps). The radiated acoustic pressure- and, consequently, the fluid
load - is expressed as a boundary integral, but no simple Green's representation of
the shell displacement exists: we are, thus, left with a system of integra-differential
equations to solve.
The numerical solutions of these problems, together with comparisons between
predictions and experiments, are presented in chapter 4.
1.2 Equations governing the vibrations of thin
bodies
We are interested in the vibrations of an elastic (or visco-elastic) thin body: by thin
we mean that one of the dimensions of the domain occupied by the elastic solid, which
is called its thickness, is small compared with the other two ones, and equally small
compared with the wavelengths of the vibration waves. It must be expected that the
various mechanical quantities which describe the motion of such a solid -in particular
the stress and the strain tensors - do not vary very much within the thickness of the
solid. This suggests that the general three-dimensional equations of motion can be
simplified and reduced to two-dimensional equations defined over a mean surface.The
aim of this section is to show on three examples - a plate, a circular cylindrical shell
and a spherical shell - how such equations can be established.
1.2.1 Overview of the method proposed here
Let I:, with boundary 8I:, be a surface which can be parameterised by a coordinate
G
system (6, 6). It is assumed that a unit vector exists everywhere on this surface. A
point in the neighborhood of I: can be defined by local coordinates (6, 6, 6), where
6 is counted along the normal vector G. Let h(6,6) be a function varying between
Description:The subject of the book is directly related to environmental noise and vibration phenomena (sound emission by vibrating structures, prediction and reduction, ...). Transportation noise is one of the main applications. The book presents an overview of the most recent knowledge on interaction phenomen
