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Universitat Polit` ecnica de Catalunya Programa de Doctorat d’Enginyeria Civil `lcul Num` Laboratori de Ca eric

Numerical modelling of sound transmission in lightweight structures

by Jordi Poblet-Puig

Doctoral Thesis Advisor: Antonio Rodr´ıguez-Ferran Barcelona, January 2008

Abstract Numerical modelling of sound transmission in lightweight structures Jordi Poblet-Puig The thesis deals with the numerical modelling of sound transmission. All the analyses are done in the frequency domain and assuming that the structures are linear and elastic. Linear acoustics is considered for the fluid domains. Thus, the fluid-structure interaction problems analysed here are governed by the vibroacoustic equations. The models are applied to the field of building acoustics, with especial interest on lightweight structures. A set of one-dimensional models for single and layered partitions considering finite acoustic domains is developed. Preliminary parametric analyses are done, considering aspects like the acoustic absorption, the structural damping, the separation between layers, the quality of absorbing material, or the influence of the eigenfrequencies of the problem in the isolation capacity. The analytical solution of these situations is available and it is used to test two and three-dimensional models. Numerical-based models for vibroacoustic problems lead to large system of linear equations. This is an important drawback for mid and high-frequencies where the computational costs become unaffordable. The block Gauss-Seidel algorithm has been applied for sound transmission problems. Its performance has been analysed by means of analytical expressions of the spectral radius obtained in one-dimensional situations. Moreover, a selective coupling strategy is developed in order to efficiently iii

solve problems where some acoustic domains are strongly coupled (i.e. double walls). In building acoustics, the acoustic domains are often cuboid-shaped rooms. Analytical expressions of the eigenfunctions are well known and can be used in order to obtain the pressure field by means of a modal analysis. A model that combines this with a more general finite element (FEM) description of the structure is presented. This mixed approach is more efficient (time and memory requirements) than a FEMFEM model. The most relevant aspects of the modal-FEM approach are analysed: computational costs, modelling of acoustic absorption, selection of the modal basis. The model is used in order to predict the isolation capacity of single and double walls. Heavy and lightweight structures are considered. The influence on the sound reduction index of parameters related with the wall environment like the room size, the position of the source, the correction due to acoustic absorption is shown. Wall properties such as its size, the damping or the boundary conditions are also considered, as well as more specific aspects related with double walls (cavity thickness, quality of the absorbing material). The effect of flanking transmissions on the sound reduction index is also taken into account. The vibration level difference and the sound transmission through several junction types (L, T and X-shaped) are calculated. In the X-shaped junction case, four rooms are analysed at the same time. The transmission between two of them is only caused by flanking paths. The possibilities of numerical models are illustrated with a case-study where the isolation capacities of a double wall with and without an accurate acoustic design are compared. The existence of a vibration transmission path between the floor and the leaves of the wall drastically reduces its performance. The response of double walls depends on the type of mechanical connections between leaves. The attention is focused here on the case of lightweight steel studs. They have been characterised by means of an equivalent spring. The value of the stiffness of the spring is obtained by comparison of the vibration level difference between leaves of a double wall obtained with two different models: i) considering the geometry of the stud; ii) using springs instead of studs. Spectral structural finite elements are used in order to increase the frequency range. iv

These steel studs also modifie the radiation properties of unsymmetrical floors. The different radiation efficiency between a planar face or a face with studs is calculated by means of a numerical model using boundary elements. Differences are significant. This is a clear example of how the details of the structure are important in order to perform accurate predictions of sound isolation.

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Acknowledgments I would like to thank my advisor Antonio Rodr´ıguez-Ferran, for his dedication, patience, and objective point of view. He has dedicated a lot of time and efforts in order to improve the contents of the research and the quality of the final document. This thesis has been done in an excellent research framework, the Laboratori de C` alcul Numèric (LaC` aN). I am very grateful to Antonio Huerta for the extra effort that suppose being at the head of this human group and giving me the opportunity to join the team. I appreciate very much the support and friendship of all the colleagues along these years. I would like to mention the help of Pedro D´ıez in order to obtain the grant and understand numerical errors, the numerous tricks revealed by the mestre Xevi Roca, and the comprehension and scientific conversations with Nati Pastor. It is thanks to the guideline provided by Alfredo Arnedo some years ago that I started Ph.D. studies. He convinced me to enter into the research world. I spent five months in the Centre Scientifique et Technique du Bˆ atiment (CSTB) in Saint Martin d’Hères. This period has been very important in my personal and academical education and decisive in the development of the thesis. The people that I found there made the adaptation easy and the stage very pleasant. I would like to thank Catherine Guigou for her sincere scientific opinions and for sharing her acoustic knowledge and modelling results with me, Michel Villot for providing ideas, proposing technological objectives and contributing with his experience in the field of acoustics and vibration, and Philippe Jean for the very productive discussions on the numerical modelling of vibroacoustic phenomena. The development of this work has been simultaneous in time with the ‘High quality acoustic and vibration performance of lightweight steel constructions (ACOUSVIvii

BRA)’ project. The technological challenges and the empirical information provided by the project have been important in order to enrich the thesis contents. I would like to thank the partners for the nice discussions and experiences. Many other people have helped me during this time with scientific discussions or sending papers and references. Among others, Dr. Bouillard, Dr. Brunskog, colleagues and teachers of Computational Aspects of Structural Acoustics and Vibration course at CISM and the members of the Laboratori d’Enginyeria Ac´ ustica i Mec` anica (LEAM, UPC). I have been mainly using free software like Gmsh, PETSc, LAPACK, GSL, OpenFEM or Code-Aster in a Linux environment during the thesis and the finite element code CASTEM for a long time. I would also like to thank Free Field Technologies for providing a free license for using ACTRAN and its user manual. Thanks a lot to my family, my parents Conxa i Albert, my sister M´ıriam, for unconditional support, understanding and respect and to tiet Coque for being always a technological reference. I appreciate very much the comprehension and opinions from all of my friends, such as Alfredo and Giuseppe. The financial support of the Generalitat de Catalunya i el Fons Social Europeu (2003 FI 00652), the Research Fund for Coal and Steel, el Departament de Matem` atica Aplicada III, Departament d’Infraestructura del Transport i del Territori and the Escola Tècnica Superior d’Enginyers de Camins Canals i Ports is gratefully acknowledged.

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Contents Abstract

iii

Acknowledgments

vii

Contents

xii

List of symbols

xiii

1 Introduction 1.1 Different models for sound transmission problems . . . . . . . . . 1.2 Lightweight structures and vibroacoustics . . . . . . . . . . . . . . 1.3 Acoustic standards and regulations . . . . . . . . . . . . . . . . . 1.4 Goals, scope and outline of the thesis . . . . . . . . . . . . . . . . 1.5 Review of vibroacoustics equations . . . . . . . . . . . . . . . . . 1.5.1 The acoustic problem . . . . . . . . . . . . . . . . . . . . . 1.5.2 Acoustic problem types . . . . . . . . . . . . . . . . . . . . 1.5.3 Analysis in the frequency-domain: the Helmholtz equation

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2 Review of numerical methods for vibroacoustics 2.1 Numerical methods for the low-frequency range . . . 2.1.1 The finite element method (FEM) . . . . . . . 2.1.2 The boundary element method (BEM) . . . . 2.1.3 Numerical techniques for the coupled problem 2.2 Numerical methods for the mid-frequency range . . . 2.2.1 Acoustic problems . . . . . . . . . . . . . . . 2.2.2 Structural dynamics . . . . . . . . . . . . . . 2.3 Concluding remarks . . . . . . . . . . . . . . . . . . .

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3 One-dimensional model for vibroacoustics 51 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 One-dimensional model for undamped vibroacoustics . . . . . . . . . . 52 3.2.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . 52 ix

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3.2.2 Analytical solution . . . . . . . . . . . . . 3.2.3 Application example . . . . . . . . . . . . One-dimensional model for damped vibroacoustics 3.3.1 Problem statement . . . . . . . . . . . . . 3.3.2 Analytical solution . . . . . . . . . . . . . 3.3.3 Application examples . . . . . . . . . . . . One-dimensional model of layered partitions . . . 3.4.1 Problem statement . . . . . . . . . . . . . 3.4.2 Analytical solution . . . . . . . . . . . . . 3.4.3 Application examples . . . . . . . . . . . . Validation of finite element models . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . .

4 The 4.1 4.2 4.3 4.4 4.5

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block Gauss-Seidel method in sound transmission problems Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block Gauss-Seidel algorithm . . . . . . . . . . . . . . . . . . . Review of block iterative solvers in acoustics . . . . . . . . . . . . . Influence of the degree of coupling . . . . . . . . . . . . . . . . . . . Analysis of the block Gauss-Seidel method . . . . . . . . . . . . . . 4.5.1 The convergence condition . . . . . . . . . . . . . . . . . . . 4.5.2 Physical interpretation of the convergence condition . . . . . 4.6 Application examples . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Influence of damping . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Influence of the fluid density . . . . . . . . . . . . . . . . . . 4.6.3 Influence of particular eigenfrequencies . . . . . . . . . . . . 4.7 The case of double walls: selective coupling of fluid domains . . . . 4.7.1 Validation: one-dimensional example . . . . . . . . . . . . . 4.7.2 Application: two-dimensional example . . . . . . . . . . . . 4.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Combined modal-FEM approach for vibroacoustics 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Formulation of the model . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Analysis of computational costs . . . . . . . . . . . . . . . . . . . . . 5.4 One-dimensional examples and sources of error . . . . . . . . . . . . . 5.5 Role of acoustic absorption and the size of the modal basis . . . . . . 5.5.1 Acoustic absorption . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Relationship of matrix bandwidth and the Robin boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Influence of frequency bandwidth . . . . . . . . . . . . . . . . 5.5.4 Selection of acoustic modes . . . . . . . . . . . . . . . . . . . x

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55 57 62 62 63 64 67 67 70 71 73 75

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5.6 Validation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6 Numerical modelling of sound transmission in single and double walls 135 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.3 Description of the problem analysed . . . . . . . . . . . . . . . . . . . . 141 6.4 Low-frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.5 Acoustic isolation of single walls . . . . . . . . . . . . . . . . . . . . . . 147 6.5.1 Influence of the absorption correction on R . . . . . . . . . . . . 147 6.5.2 Comparison between two-dimensional and three-dimensional models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.5.3 Influence of room size . . . . . . . . . . . . . . . . . . . . . . . . 153 6.5.4 Influence of sound source position . . . . . . . . . . . . . . . . . 154 6.5.5 Influence of window size . . . . . . . . . . . . . . . . . . . . . . 155 6.5.6 Influence of the mechanical properties and boundary conditions of the walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.6 Acoustic isolation of double walls . . . . . . . . . . . . . . . . . . . . . 158 6.6.1 Influence of the separation between leaves and the type of absorbing material . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.6.2 Effect of mechanical connections between leaves . . . . . . . . . 161 6.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7 The 7.1 7.2 7.3 7.4

role of studs in the sound transmission of double walls Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibration behaviour of steel studs . . . . . . . . . . . . . . . . Studs and leaves analysed . . . . . . . . . . . . . . . . . . . . Identification of the stiffness of studs . . . . . . . . . . . . . . 7.4.1 Cross-section structural vibration models . . . . . . . . 7.4.2 Influence of stud shape in the vibration level difference 7.4.3 Stud equivalent stiffness . . . . . . . . . . . . . . . . . 7.5 Using the stiffness values in a SEA model . . . . . . . . . . . . 7.6 Global response of double walls . . . . . . . . . . . . . . . . . 7.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . .

8 Numerical modelling of flanking transmissions 8.1 Introduction . . . . . . . . . . . . . . . . . . . . 8.2 The flanking transmission model of EN 12354 . 8.3 Sound transmission in rigid junctions . . . . . . 8.3.1 L-shaped junctions . . . . . . . . . . . . xi

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189 189 192 196 200

8.3.2 T-shaped junctions . . . . . 8.3.3 X-shaped junctions . . . . . 8.4 Case-study of flanking transmission 8.5 Concluding remarks . . . . . . . . .

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9 Numerical modelling of radiation efficiency 213 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 9.2 Role of beams in the radiation of a surface . . . . . . . . . . . . . . . . 215 9.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 10 Conclusions and future work 221 10.1 Conclusions and contributions of the thesis . . . . . . . . . . . . . . . . 221 10.2 Future developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Bibliography

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xii

List of symbols Latin symbols aj

Contribution of mode j

A

Acoustic admittance

A

Cross section area of a beam

A, B

BEM discretization matrices

B

Plate bending stiffness per unit width

c

Velocity of sound in air

C

Proportional damping coefficient of a single mass (Chapter 3)

C

Linear elastic constitutive tensor

Cac

Acoustic damping matrix (FEM)

Cs

Solid proportional damping matrix (FEM)

CSF , CF S

Fluid-structure coupling matrices (Chapter 4)

d

Separation between leaves in a double wall

dij

Vibration transmission factor

D

Sound level difference between acoustic domains

Dij

Vibration level difference

E

Young’s modulus

f

Frequency

fac

Acoustic nodal force vector (FEM)

mod fac

Acoustic force vector (Modal analysis)

fc

Coincidence (critical) frequency of a single wall

fF

Generic fluid force vector (Chapter 4)

fF S

Fluid force vector due to the interaction with a structure xiii

fgc

Geometrical coincidence frequency of a mode of a finite wall (Chapter 9)

fm−a−m

Theoretical mass-mir-mass resonance of a double fall (frequency)

fs

Solid nodal force vector (FEM)

fS

Generic solid force vector (Chapter 4)

fSF

Structure force vector due to the interaction with a fluid

F

Generic fluid matrix (Chapter 4)

g

Phasor of G

G

Injection/extraction of mass per unit time

G

Frequency factor in Chapter 3

G b h

Iteration matrix (Chapter 4) Dimensionless element size (2)

Hn i

Hankel function of second kind and order n √ Imaginary unit, −1

I

Modulus of acoustic intensity

I

Acoustic intensity

I

Inertia of a beam

Jn

Bessel function of first kind and order n

k

Wave number

K

Stiffness of a spring of a single mass

Kt

Translational stiffness of a spring

Kθ

Rotational stiffness of a spring

Kij

Vibration reduction index

Kac

Acoustic stiffness matrix

Ks

Solid stiffness matrix

`

Main dimension of an structural element (i.e. in the case of a beam its length)

`x , `y , `z

Dimensions of a cuboid

L

Sound pressure level

Lmod

Fluid-structure coupling matrix when modal analysis is used in the fluid part

xiv

LSF , LF S

Geometrical fluid-structure coupling matrices

M

Mass of a particle

Mac

Acoustic mass matrix

Ms

Solid mass matrix

Mψ

Modal analysis matrix

n

Outward unit normal

nac

Number of acoustic nodes

nmod

Number of acoustic modes

ns

Number of solid nodes

nsd

Number of space dimensions

nx , ny , nz

Number of half-waves in the x, y, and z directions of a cuboid

N

Number of waves (Chapter 3)

Nj

Shape function

Nj

Matrix shape function for the case of structural elements

p

Degree of interpolation

p

Phasor of the acoustic pressure

p

Vector of nodal values of acoustic pressure

p0

Reference value of pressure (usually 2 · 10−5

pd p

h

Pa)

Phasor of prescribed value of the acoustic pressure

Numerical solution for the phasor of the acoustic pressure

phom

Phasor of acoustic pressure (homogeneous solution)

pI

Interpolation of the exact solution

pn

Vector of nodal values of normal derivative of acoustic pressure

pp

Phasor of acoustic pressure (particular solution)

prms

Root mean square pressure

P

Acoustic pressure

P

Acoustic power

P0

Mean pressure

Pd

Prescribed value of pressure in the Dirichlet boundary

Pin

Incoming pressure

Pout

Outgoing pressure xv

Ps

Scattered pressure

PT

Total pressure

q

Phasor of the source-strength amplitude

Q

Source-strength amplitude of a punctual source

r

Radial coordinate

R

Sound reduction index (T L sound transmission loss in some countries)

RT60

Reverberation time

S

Surface

S

Generic solid matrix (Chapter 4)

S∗

Generic solid matrix taking into account some fluid domains (Chapter 4)

t

Thickness

t

Phasor of surface mechanical force vector

tC

T-complete system of functions

T

Period

T

Surface mechanical force vector

u

Phasor of a solid displacement

u

Phasor of the vector of solid displacements

u

Vector of nodal values of solid displacements (and rotations)

U

Vector of solid displacements

UD

Vector of imposed solid displacements

v

Phasor of the vector of acoustic velocity

vn

Phasor of normal acoustic air velocity

vrms

Root mean square velocity

V

Acoustic velocity

V0

Mean velocity

Vn

Normal acoustic air velocity

VT

Total velocity

W

Acoustic power

x

Vector of spatial coordinates

xvi

xF

Generic fluid unknowns vector (Chapter 4)

xS

Generic solid unknowns vector (Chapter 4)

Y

Punctual mobility

Yn

Bessel function of second kind and order n

Z

Acoustic impedance

Zw

Wall impedance

Greek symbols α

Acoustic absorption coefficient

αC

Storage cost of a complex variable

αI

Storage cost of an integer variable

ΓD

Dirichlet boundary

ΓF S

Fluid-solid interaction boundary

ΓIN F

Unbounded boundary

ΓN

Neumann boundary

ΓR

Robin boundary

δ

Dirac-delta

δij

Kronecker delta

∆

Determinant (Chapter 3)

η

Hysteretic damping coefficient (loss factor)

θ

Angle defining wave direction

ϑ b k

Ratio of speeds (Chapter 3) Dimensionless wave number

λ

Ratio of lengths (Chapter 3)

λair

Length of waves in air

λbending

Length of bending waves in a solid

µ

Ratio of masses (Chapter 3)

ν

Poisson’s ratio

Ξ

Acoustic energy

ρ

Acoustic air density xvii

ρ0

Mean air density

ρF

Fluid density (Chapter 4)

ρsolid

Volumetric solid density

ρsurf

Surface solid density

ρT

Total air density

%

Resistivity of an absorbent material

σ

Radiation efficiency

σ

Cauchy stress tensor

ˆ σ

Phasor of σ

ς

Integration constant depending on the boundary type (BEM)

τ

Sound transmission coefficient and arbitrary constant in Chapter 2

τav

Averaged sound transmission coefficient

ϕ

Acoustic test function

Υ

Robin boundary condition constant

ψ

Acoustic mode

Ψ

Fundamental solution (BEM)

ω

Angular frequency or pulsation

ωI

Imaginary part of the pulsation, Im {ω}

ωnat

Eigenfrequency of a single mass (Chapter 3)

ωm−a−m

Theoretical mass-air-mass resonance of a double fall (pulsation)

Ωac

Acoustic domain

Operators ∇n

Normal derivative

∇s

∇4 e < b

Symmetric gradient Biharmonic operator Time average of Complex conjugate of >

Spatial average of Dimensionless variable

(Chapter 3) xviii

char

Characteristic value (Chapter 3)

ρ( )

Spectral radius of

Re { }

Real part of

Im { }

(Chapter 4)

Imaginary part of

xix

Chapter 1 Introduction To satisfy acoustic requirements is nowadays important in many sectors, such as the automotive, aerospace and building industries, among others. A typical problem caused by poor acoustic designs is the high sound levels that final users have to bear. In this thesis, the problem of sound transmission is studied by means of numericalbased models. The interest is focused on the technological field of building acoustics with special emphasis in lightweight constructions. Most of the models used here are deterministic approaches.

1.1

Different models for sound transmission problems

In sound transmission problems a wide frequency range has to be considered. The human ear can hear sounds between 20 Hz and 20 000 Hz. However, the interest is mainly focused on the low-frequency part of the spectrum. Low-frequency sound tends to be transmitted while high-frequency sound is reflected. The main sources of low-frequency noise as well as their consequences in people have been studied by Berglund et al. (1996). The large number of models of sound transmission can be classified into deterministic and statistical approaches. None of them is able to correctly deal with the full fre1

2

Introduction

quency range of interest. The type of response is very variable with frequency. While for low frequencies the modal density is small and the response is modal-controlled, for high frequencies the modal density is very high and the variation of the outputs is smoother. Many different aspects can modify the obtained responses: geometrical data (room sizes, wall dimensions, thickness of layers, position of the sound source), mechanical properties (densities, stiffnesses of materials, damping) or acoustic parameters (absorption). For example, room sizes or wall boundary conditions can modify the response for low frequencies but are much less critical for higher frequencies. This variability of the relevant data of the problem is a demanding aspect for modelling techniques. Most of the existing models can only consider some of the problem variables. This determines the frequency range where the results can be considered as valid. Deterministic models can be divided between those dealing with the vibroacoustic equations without additional hypotheses and those that make extra simplifications, like considering infinite acoustic domains. The former often use numerical techniques in order to solve the equations (i.e. finite and boundary element methods, Atalla and Bernhard (1994)). They are very accurate since they can consider the exact geometry of the problem and use only basic data (densities, damping coefficients, Young modulus,...). Since the pressure and vibration fields are provided by these models, the detail level can be as high as necessary. All the different response types (modal, critical frequencies,...) are obtained in a natural way without modifying the model. These realistic models have two main drawbacks: (1) the computational cost and (2) the loss of meaning of a deterministic solution and the uncertainty of the validity of the dynamic properties (or even the governing equations) at high frequencies. The high computational cost is caused by the required discretisations of the geometry of the problem and the large number of situations to be analysed. The discretisation criteria depend on the ratio between the length of the expected waves (oscillations of acoustic pressure or structural vibrations) and the dimension of the studied domains (rooms, walls,...). These systems have to be successively solved in order to reproduce the reverberant pressure fields, to cover an acceptable frequency range or to check the

1.1 Different models for sound transmission problems

3

influence of some of the parameters of the model. The type of models developed in the thesis mainly belong to this first group: deterministic models without additional hypotheses. The most often used hypothesis is to consider unbounded acoustic domains and structures. This simplifies the equations and even analytical solutions can be obtained (i.e. the mass law, Fahy (1989)). The wave approach is a clear example of a deterministic model with an additional hypothesis. It is not expensive and can be phenomenologically enriched with experimental information. Nevertheless, it cannot be considered a general technique since different models have to be developed for each new situation: single walls, double walls, double walls with flanking paths between leaves,... Examples can be found in Guigou and Villot (2003). At high frequencies the response of the vibroacoustic system is diffuse. Any small modification in the geometry or the mechanical properties can completely change the detail of the solution (i.e. velocity field over a structure). Then, deterministic solutions become meaningless. The interest must not be focused on the detail because it is very variable. Statistical outputs like the averaged velocity on a part of the structure have physical meaning and are useful from an engineering point of view. They are often independent of small variations of the problem data. The more efficient way to obtain these outputs is by means of a statistical model. Most of them use the statistical energy analysis technique (SEA, Fischer (2006)). The whole domain is split in subsystems. Each of them is described by its energy. Averages in excitation and observation points, and in frequency bandwidths are considered. If the modal density is high and a large number of points are considered (i.e. rain-on-the-roof excitation) it leads to very simplified expressions. Dissipation of energy as well as power transfer between subsystems are also taken into account. The coupling factors are more difficult to obtain and so often deterministic models or experimental data are used to provide this important information. This can lead SEA models to be based in something more than the classical vibroacoustic equations and to incorporate phenomenological data. A general overview of SEA can be found in Fahy (1994). SEA has been used in order to study the sound transmission in buildings. See, for example, Craik (1996) and Koizumi et al. (2002).

4

Introduction

1.2

Lightweight structures and vibroacoustics

Lightweight structures are characterised by an optimal performance with a minimum mass. The total mass of a lightweight structure is minimised by means of an accurate design that is oriented to the specific function of the structure. The frontier between the concepts of heavy and lightweight can be rather diffuse and depends on the technological field. Moreover, since the structure is designed depending on the final functions, the typology of lightweight structures is wide. We can be using the lightweight concept for a satellite, a cover constructed by means of textile membranes or even a competition car. In the context of this thesis, the concept of lightweight structures is applied to building constructions and more specifically to walls and floors. These lightweight construction elements can be made of very different materials like steel, wood, gypsum, mineral wool,... but rarely with concrete or masonry which are the typical options that lead to a heavy element. The former are more expensive and elaborate, but since inferior quantities are required the final structure can be cheaper. An important difference is that while heavy structures are often homogeneous and simple, the lightweight structures are heterogeneous and have a lot of construction details (i.e. connections between elements). A typical example of the lightweight structures studied here is a double wall (see Fig. 1.1). It is usually constructed by means of a wood or steel framework that provides stability and resistance and plasterboards and mineral wools that guarantee the other functions: to create a physical separation between contiguous rooms and provide thermal and acoustic isolation. The design is optimised in the sense that the steel framework is, in general, less resistant that a concrete or masonry wall but the minimal requirements are satisfied by both structural solutions. The materials used in the core of the double wall are chosen in order to be efficient in the heat and sound isolation. The lightweight wall has advantages in terms of economical costs and construction, transport and installation facilities. Lightweight structures require more elaborate modelling techniques. For example, their failure can often be caused by local and global bucking instead of plastification due to high stresses. Several relevant modelling aspects must be taken into account

1.3 Acoustic standards and regulations

5

when dealing with sound transmission problems. Considering construction details can be important. A typical case are the connections between the elements that compose the wall and the type of junctions with other walls and floors. Cold-formed steel profiles are often used and some of the structural assumptions valid for heavy structures can lead to unexpected results. Deformations (and vibrations) at cross section level are possible. The structure can be composed of different parts. This often leads to more elaborate models. The critical frequency of lightweight structures is higher than for the case of heavy structures. In the first case the forced transmission (caused by the geometrical coincidence of pressure and displacement waves) can be important while in the second case the sound is mainly transmitted due to resonant transmission (caused by the excitation of the closer modes in the structure and the acoustic domains). Moreover, the vibration wave lengths in a lightweight structure are shorter than in a heavy structure. This reduces the frequency range where the response is modal, but can accentuate the geometry effects of damping (i.e. in a lightweight structure the effects of a punctual force can be localised while for the same frequency and force, the displacements are large all around a heavy structure). In future chapters calculations on lightweight and heavy structures will be shown. The comparison between both type of responses enriches the discussions. The techniques exposed here are valid for both types of structures. However, some of the topics of the thesis are specific of lightweight structures: double walls, characterisation of the connecting elements, and modification in the radiation efficiency due to stiffeners (steel beams are not used in concrete or masonry walls).

1.3

Acoustic standards and regulations

Two general issues have to be considered in order to perform good acoustic designs of buildings: to control the quality of sound inside rooms and to isolate them from exterior noise (i.e. sound generated in contiguous rooms or coming from the exterior of the building). The key parameter in the first aspect is the reverberation time, which measures the

6

Introduction

(a)

(b)

Figure 1.1: Lightweight structures. (a) Detail of a double wall. (b) Construction of a lightweight house. Source: Kesti et al. (2006).

decay of sound in the room. A low reverberation time is important in order to ensure speech intelligibility. In small rooms it is mainly controlled by the amount of acoustic absorption. The problem is much more complicated in concert halls or auditoriums, where other aspects such as the room shape or the relative position between the sources of sound (i.e. loudspeakers, lecturers) and the audience is important. Several phenomena are distinguished in order to study the isolation capacity of floors and walls: • Impact noise is sound generated by fast mechanical excitations of structures

(e.g. footsteps). The sound is heard more clearly in other rooms different from

the one where it is produced. • The direct airborne sound transmission between contiguous rooms (i.e. the sound generated in a room is transmitted to another due to direct acoustic excitation of the separating element). • Flanking transmissions, that is the amount of sound not transmitted by the

1.3 Acoustic standards and regulations

7

direct path (through the wall) but through indirect paths in the structure. The thesis is focused on the analysis, development and application of prediction techniques of sound transmission. They have been mainly used in order to predict the direct sound transmission and flanking transmissions, and to provide acoustic data like the radiation efficiency or the vibration level difference between structural elements. Several parameters are defined by building acoustic regulations (e.g. the Spanish CTE (2007) or the European EN-12354 (2000) regulations) in order to ensure that structural elements have sufficiently good isolation capacities. These parameters are measured in the field or in the laboratory. The goal of modelling techniques is to predict the acoustic response of structural elements and provide tools in order to improve the acoustic design. The proposed parameters concerning direct airborne transmission between contiguous rooms and its limit value for some national European regulations are summarised in Table 1.1. Country Limitation

UK (DnT,w +Ctr,1003150 ) ≥ 45

France DnT A > 53

Spain1 RA > 45dBA

Spain2 DnT A > 45 − 55 dBA

Finland 0 Rw > 55

Country Limitation

Sweden (Rw with C50−3150 ) ≥ 52

Norway (Rw with C50−5000 ) ≥ 55

Denmark Rw ≥ 52 − 53

Iceland Rw ≥ 55

Netherlands Rw ≥ 51

Table 1.1: Summary of requirements for airborne sound insulation for separating walls and floors (dB) in several European countries. For Spain: 1 Old regulation NBE-CA-88, it can be used till October 2008. 2 New regulation CTE. Several parameters are used to measure the isolation capacity of a wall. The sound level difference D is the most intuitive one since it is the difference between the sound level in the sending room and the receiving room. It is a global measure in the sense that it includes all the aspects involved in the sound transmission problem: i) net isolation capacity of the wall; ii) effect of room characteristics like the size, the

8

Introduction

absorption, or the position of the source; iii) sound caused by flanking transmissions. Thus, the same wall has different values of sound level difference D depending on other parameters (laboratory where it is tested or design of the building including the furniture). In order to have a measure more related only with the wall, the sound reduction index R is defined. It is a logarithmic ratio between the incident acoustic power on the wall (face of the sending room) and the transmitted acoustic power. It is also known (with minor modifications) as transmission loss (ASTM E-90) in the US and other countries. Both D and R are frequency-dependent parameters and are often measured at each third octave band. More detailed definitions are given in Chapters 3 and 5 or Josse (1975); Beranek and Vér (1992). As shown in Table 1.1, there are other parameters like DnT , which is a sound level difference measured with empty rooms 0 and corrected with a normalised reverberation time or Rw (ISO 717-1), which is a

scalar value characterising the wall (instead of a frequency-dependent parameter). In the remainder of the thesis the sound level difference D and the sound reduction index R are used as output parameters. D is used in Chapters 3 to 5, where the discussion is not focused on the acoustic results but on the performance of methods. The discussion on the influence of the acoustic absorption is done in Sections 5.5.1 and 6.5.1. In the other sections, where the attention is more focused on the application of models, R is used as main output parameter.

1.4

Goals, scope and outline of the thesis

In Section 1.5 the vibroacoustic equations are presented with special emphasis on the acoustic part of the problem. A review of the numerical techniques currently used in order to solve these equations is done in Chapter 2. Classical methods that can be used in the low-frequency range as well as new techniques trying to extend the use of numerical methods to mid frequencies are considered. In Chapter 3 a one-dimensional model for vibroacoustics is presented. Since it deals with a very simplified situation, obtaining the analytical solution is possible.

1.4 Goals, scope and outline of the thesis

9

The model lies between the mass-law models that consider unbounded domains and models considering the exact geometry (bounded domains). Finite acoustic domains are considered and modal responses can be obtained. The analytical solutions are used in order to check the two and three-dimensional codes developed for the thesis (note that an exact check of a numerical solution cannot be done by means of wavebased approaches). Approximate solutions and fast parametric analyses in order to assess the sensitivity of sound isolation to each parameter can be done. The performance of the block Gauss-Seidel algorithm for the linear system of equations obtained after the discretisation of the vibroacoustic equations is analysed in Chapter 4. Fluid-structure interaction problems often lead to matrices with a particular block structure. Moreover, in the case of sound transmission, the coupling between the fluid and the structure can be weak. These two aspects are exploited in order to efficiently solve the systems of equations. The solver is modified in order to deal with double walls and other structures where some of the acoustic domains are strongly coupled. A selective coupling strategy is presented. Some of the results presented in the thesis have been obtained by means of the finite element method, the boundary element method or spectral methods. However, they can be time-consuming for sound transmission problems (especially if the problem is three-dimensional). It also represents a limitation in the maximum frequency analysed. A model where the cuboid-shaped acoustic domains are solved by means of modal analysis (with available analytical solution) is presented in Chapter 5. The analytical solutions are combined with finite elements (for the structure). The main improvements done with respect to similar models already used are the generalisation to more elaborate situations (it is not restricted to the study of sound transmission through single walls and it is extended to the use in double walls, and flanking transmissions) and the use of the solver presented in Chapter 4 (which allows an adequate treatment of strongly coupled situations). An analysis of the computational costs of the model as well as the influence of some of the parameters is done. This model is used in Chapter 6 in order to predict sound transmission in single and double walls. In Chapter 7, the steel studs often used between the leaves of a

10

Introduction

double wall are characterised. Three-dimensional finite element models of laboratory tests as well as two-dimensional models using spectral finite elements are used in order to obtain values of spring stiffnesses characterising the transmission of vibrations through the steel studs (at cross section level). In Chapter 8 flanking transmissions are modelled. Comparisons of the predictions done by the EN-12354 model and numerical results for the cases of L-shaped, T-shaped, and X-shaped junctions are presented. The results shown have been obtained by means of the three-dimensional version of the model presented in Chapter 5. In Chapter 9 the radiation efficiency of floors with stiffeners is studied. Finally, the conclusions of the thesis and proposals for future work are exposed in 10.

1.5

Review of vibroacoustics equations

1.5.1

The acoustic problem

The linear theory of sound will be used in order to describe the fluid (acoustic) domains (Pierce (1981), Kinsler et al. (1990)). Air is considered as a perfect, compressible and adiabatic fluid. Its weight is neglected and it is assumed that acoustic perturbations cause small changes in displacement and velocity of fluid particles. The basic variables of a fluid from a constitutive point of view are the total pressure PT and the total density ρT . Both are described by means of a steady or mean value (P0 and ρ0 ) and small variations (acoustic variables: P and ρ). The total acoustic velocity V T can also be described in the same way x, t) = P0 (x x ) + P (x x, t); ρT (x x, t) = ρ0 (x x ) + ρ(x x, t); V T (x x , t) = V 0 (x x) +V V (x x , t) (1.1) PT (x Conservation of mass and linear momentum laws can be rewritten in terms of acoustic variables

x , t) ∂ρ(x + ρ0 ∇ · V = 0 ∂t

(1.2)

1.5 Review of vibroacoustics equations

11

x, t) = −ρ0 ∇P (x

V (x x , t) ∂V ∂t

(1.3)

An equation describing the physics between pressure and density is required. A linear relationship between acoustic pressure and air density is assumed. Only the first term in P (ρ) =

∂PT ∂ρT

1 ρ+ 2 ρ0

∂ 2 PT ∂ρ2T

ρ2 + . . .

(1.4)

ρ0

is considered. The constitutive equation for the acoustic fluid can be written as x, t) = c2 ρ(x x , t) P (x

(1.5)

It can be shown (see for example Pierce (1981)) that c is the velocity of propagation of waves in the fluid. It is a constant value for linear fluids. The constitutive relation (1.5) can be introduced in Eqs. (1.2) and (1.3) in order to obtain the governing equation of an acoustic fluid: x, t) = 4P (x

x, t) 1 ∂ 2 P (x c2 ∂t2

(1.6)

This is the wave equation, which is also the governing equation of other physical phenomena. Boundary conditions are required to have a well-posed boundary value n is the fluid normal problem. Defining the normal derivative as ∇n (•) = n · ∇ (•) (n

vector), the Neumann boundary condition can be written as x, t) = −ρ0 ∇n P (x

dVn dt

(1.7)

It is used in order to impose a known value of normal velocity (Vn ) in the contour. The fluid velocity is related to the normal derivative of the pressure field by multiplying Eq. (1.3) by the normal vector. The physical meaning of this boundary condition is to have a vibrating surface in the acoustic domain. The Robin boundary condition can be written as x, t) = −ρ0 ∇n P (x

x, t) d ΥP (x dt

(1.8)

12

Introduction

In that case, the normal velocity in the boundary is an unknown value. Υ is a parameter defining the Robin contour. Its physical meaning, when it is invariable with time, is the relationship between the normal derivative of the pressure field and the normal velocity. The Robin boundary condition is used in order to introduce attenuation into the model. For the case of acoustic it is the absorption. Details and discussions will be done in following chapters. The Dirichlet boundary condition x, t) = Pd P (x

(1.9)

which is very usual in other physical problems, is rarely used in acoustics. A special boundary condition (Sommerfeld radiation condition) lim

r→∞

r

nsd−1 2

∂P 1 ∂P + ∂r c ∂t

=0

(1.10)

which ensures the uniqueness of the solution has to be imposed in the unbounded acoustic domains. r is a radial coordinate and nsd the number of space dimensions. The results presented in following chapters deal with bounded acoustic domains and these last two boundary conditions have not been used. The acoustic energy per unit volume is defined as 1 1 P2 V ·V )+ Ξ = ρ0 (V 2 |2 {z } |2 ρ{z0 c } kinetic

(1.11)

potential

It can be interpreted as the kinetic energy of the fluid particles due to their oscillatory movement and the potential energy due to the compression of the fluid (like if it was an elastic spring). The conservation equation of acoustic energy is ∂Ξ +∇·I = 0 ∂t

Z

Ξ dΩ + Ω

Z

∂Ω

I · n dS = 0

(1.12)

V is the acoustic Both differential and integral forms have been written. I = PV intensity. It is a measure of the power flow per unit of surface. It is a vectorial


13

variable since the flow is different in each direction.

1.5.1.1

Punctual sound sources. Non-homogeneous wave equation

Acoustic sources will not be modelled in detail (i.e. the shape of a loudspeaker and how it can modify the radiation in several directions). Their effect in the acoustic fluid is modelled by means of punctual sound sources. The parameter characterising an acoustic sound source is the source-strength amplitude (Q). For the case of a small sphere (radius r) vibrating in an unbounded medium, Q = 4πr 2 vr (t). It can be interpreted as the volume of air displaced by the source per unit time. Q=

Z

Vn dΓ

(1.13)

∂Ω

The sound source has to be introduced in the differential equation (1.6). This leads to the non-homogeneous wave equation (see for more details Kinsler et al. (1990)) x, t) − 4P (x

x, t) x , t) 1 ∂ 2 P (x ∂G(x =− 2 2 c ∂t ∂t

(1.14)

x, t) is the injection/extraction of mass per unit time (its magnitude is where G(x x, t)] = M/TL3 ). For a punctual sound source, G can be defined with the aid [G(x x, t) = G(t)δ(x x 0 , x ). of the Dirac delta function: G(x Integrating the Eq. (1.14) over a very small domain (which must include the sound source) we obtain ∂ −ρ0 ∂t

Z

1 Vn dΓ − 2 c ∂Ω

Z

Ω

∂2P ∂G(t) dΩ = − 2 ∂t ∂t

(1.15)

The domain of integration Ω can be as small as necessary. In the limit the volume integral vanishes and G can be expressed as G(t) ' ρ0 Q ([G(t)] = M/T).

14

Introduction

1.5.2

Acoustic problem types

The acoustic problems are often classified depending on their goals and boundary conditions (Ochmann and Mechel (2002)). Every situation can model a wide group of practical applications. The set of equations to be solved as well as its boundary conditions are different in every problem type. This influences the choice of the adequate numerical method. The acoustic problems can be classified as follows: 1. Exterior problem 1: Radiation The study of the sound field generated by a sound source in an unbounded domain is known as the radiation problem (Fig. 1.2(a)). The Sommerfeld boundary condition (1.10) has to be imposed at ΓIN F in all the problems involving infinite domains. The wave equation (1.6) will be solved in ΩEXT . The sound source is usually modelled by means of a Neumann boundary condition (1.7), considering a known imposed velocity. The radiation efficiency (σ) of a sound source is a typical output of interest of the radiation problem. It can be defined as 2 ρ0 cS < Vf n > σ = P with P =

I

I dS

(1.16)

2 P is the power of the source, S is the surface of of the radiating body and < Vf n >

is the space-average (along the vibration surface) value of the time-average vibration velocity.

2. Exterior problem 2: Scattering The response of a body inside an unbounded domain to an incoming sound wave (Fig. 1.2(b)) is the solution of an scattering problem. The incoming wave has to be known a priori and then the wave equation is solved in terms of the scattered (reflections of the incident, Pin , wave in the body) wave: Ps = Ptotal − Pin . The

scattered pressure has to satisfy the Sommerfeld boundary condition (1.10) in the exterior contour. In the interior contour the radiation boundary conditions


15

have to be reformulated in terms of the scattered pressure. For the case of an infinitely rigid body x, t) = 0 ∀ x ∈ Γint ⇒ ∇n Ps = −∇n Pin ∇n Ptotal (x

x ∈ Γint ∀x

(1.17)

The target strength which is a ratio between the scattered and the incoming intensities at a distance of 1 m is a typical result of the scattering problem.

3. Interior problem An interior problem (Fig. 1.2(c)) is an acoustic problem solved inside a bounded domain. Only Neumann, Robin and Dirichlet boundary conditions can be imposed. The main characteristic of the interior problems is their modal behaviour.

4. Vibroacoustic problem Vibroacoustic problems are those with acoustic (fluid) and solid domains. The acoustic domain is modelled by the governing equations presented in Section 1.5.1. The solid domain is usually considered as a linear elastic solid with small strains and displacements (this hypothesis will be sufficient for our purposes). The vibration behaviour of the solid (due to pressure waves for example) is studied. Continuity of normal velocities and pressures in the interface are imposed in order to couple the solid and acoustic domains. The set of equations governing

16

Introduction

the vibroacoustic problem are Acoustic domain: x, t) 1 ∂ 2 P (x x, t) − 2 = 4P (x c ∂t2 X ∂(Gs (t)δ(x x s , x )) − ∂t s

in Ωac

(1.18)

on ΓN

(1.19)

on ΓR

(1.20)

on ΓD

(1.21)

on ΓIN F

(1.22)

on ΓF S

(1.23)

in Ωs

(1.24)

in Ωs

(1.25)

x, t) · n = T (x x , t) σ(x

on ΓsN

(1.26)

x, t) = U D U (x

on ΓsD

(1.27)

on ΓsF S

(1.28)

dVn dt x, t) d ΥP (x x, t) = −ρ0 ∇n P (x dt x, t) = PD P (x nsd−1 1 ∂P ∂P 2 =0 + lim r r→∞ ∂r c ∂t U · n) d2 (U x, t) = −ρ0 ∇n P (x dt2 x, t) = −ρ0 ∇n P (x

Solid domain: x, t) = ρsolid ∇ · σ(x

d 2U dt2

σ = C : ∇sU

x, t) · n = −P (x x, t)n n σ(x

where U is the solid displacement (measured like in the case of acoustic pressure as a variation from a reference configuration), σ is the Cauchy stress tensor, ρsolid is the density of the solid, T is the vector of solid forces and U D are the imposed displacements in the solid, and the deformation of the solid is: ∇sU =

1 UT (∇U 2

+ U ∇T ). n is the exterior normal for every domain (see

Fig. 1.2(d)). The weight of the solid is neglected because vibrations around the deformed shape (due to self weight) are considered.


17

5. Transmission problem A particular case among all the vibroacoustic problems is the study of sound transmission. A special mention to this physical situation is done because one of the main goals of the present work is the study of sound transmission between acoustic domains and through solids.

Γ INF

Γ INF Γ INT

Γ INT

pS

Ω INT

Ω INT Ω EXT

Ω EXT

pIN

(a)

(b)

ΩINT

Ω INT

Γ

n

FS

t

ΓR (c)

D

N

ΓD

ΓN

Γ

Γ

Γ R

ΩS n

(d)

Figure 1.2: Problem types in acoustics. (a) Radiation (b) Scattering (c) Interior problem (d) Vibroacoustic problem

1.5.3

Analysis in the frequency-domain: the Helmholtz equation

Physical phenomena like acoustic control, structural dynamics, marine engineering, seismic engineering and electrical engineering have a common feature: its oscillatory

18

Introduction

time dependence. Temporal data is often post-processed in terms of Fourier series which introduce the concepts of amplitude and frequency. The solution methods for time-dependent phenomena can be classified in two categories: time-domain and frequency-domain approaches. The choice depends on several factors. In general, processes with transient time-dependence and with short duration are studied in the time-domain. On the contrary, physical phenomena which tend to be steady-harmonic and prolonged in time are usually studied in the frequencydomain. Vibroacoustics and noise control are, in general, studied in the frequencydomain. It will be the choice in this work. For example, an imposed normal velocity (with periodicity T ) can be described as Vn (t) =

+∞ X

jπ

vn(j) ei T t

(1.29)

j=−∞

The temporal function Vn (t), has been transformed to discrete values of amplitude (j)

vn and pulsation, jπ/T . It can be done by means of a Fourier series for periodic signals or a Fourier transform for arbitrary excitations. Details of these techniques can be found in Smith (1997). The main advantages of working in the frequency-domain are: • Better understanding of physical phenomena. Parameters and responses are frequency-dependent.

• The initial time-domain problem can be decomposed in several frequency-domain problems which are simpler to solve. The time-domain solution can be recovered

by combining all the frequency-domain solutions. • As discussed in Sections 1.5.1 and 1.5.2, we will deal with linear problems. Superposition can be done without problems due to linearity.

We will assume all the variables of the problem to be steady-harmonic. Pressure and displacements can then be expressed as x, t) = Re p(x x)eiωt P (x

x, t) = Re u (x x)eiωt U (x

(1.30)


19

x) ∈ C is the spatial variation of pressure (or its Fourier transform), and where p(x

x ) ∈ C is the spatial variation of solid displacements (phasor). u (x

The angular frequency (pulsation) of the steady-harmonic dependence is ω = ωR + ωI i, where ωR = 2πf , being f the frequency. It has to be interpreted as an harmonic oscillation and an exponential attenuation: iωR t eiωt = e|{z}

harmonic

−ωI t e|{z}

·

(1.31)

attenuation

The split decomposition for the basic unknowns (1.30) can now be used in order to reformulate the vibroacoustic problem (Eqs. (1.18) to (1.28)) in the frequency-domain. Acoustic domain: x) + k 2 p(x x) = − 4 p(x

X

xs , x ) iωgsδ(x

in Ωac

(1.32)

on ΓN

(1.33)

x) = −iρ0 ωAp(x x) ∇n p(x

on ΓR

(1.34)

x) = pd (x x) p(x nsd−1 ∂p + ikp =0 lim r 2 r→∞ ∂r

on ΓD

(1.35)

on ΓIN F

(1.36)

on ΓF S

(1.37)

ˆ x) = −ρsolid ω 2u ∇ · σ(x

in Ωs

(1.38)

ˆ x) = C : ∇su σ(x

in Ωs

(1.39)

x, t) ˆ x) · n = t (x σ(x

on ΓsN

(1.40)

x) = u d u (x

on ΓsD

(1.41)

on ΓsF S

(1.42)

x) = −iρ0 ωvn ∇n p(x

x) = ρ0 ω 2 (u u · n) ∇n p(x

s

Solid domain:

x )n n ˆ x) · n = −p(x σ(x

The new governing equation (1.32) for the acoustic domains is the Helmholtz equation. k = ω/c is the wave number. A is the admittance of the Robin contour and Z the impedance: A = 1/Z = vn /p. A is a known frequency-dependent parameter. Note

20

Introduction

that the frequency ω is a known data of the problem. vn and q are the coefficients of the Fourier decomposition and they are known values too. The only unknown of the x ). problem is p(x The discussion on the type of solutions obtained in the frequency domain is limited for simplicity to the acoustic problem. The general problem Eqs. (1.32) to (1.35) is split into an homogeneous problem

lim

r→∞

r

x) + k 2 phom (x x) = 0 4phom (x

x ∈ Ωac ∀x

x) = 0 ∇n phom (x

x ∈ ΓN ∀x

x ) = −iρ0 ωAphom (x x) ∇n phom (x

x ∈ ΓR ∀x

x) = 0 phom (x + ikphom =0

x ∈ ΓD ∀x

nsd−1 2

∂phom ∂r

(1.43)

x ∈ ΓIN F ∀x

and a particular problem x ) + k 2 pp (x x) = −iρ0 ω 4pp (x

lim

r→∞

r

X

x, x s ) qs δ(x

x ∈ Ωac ∀x

x ) = −iρ0 ωvn ∇n pp (x

x ∈ ΓN ∀x

x ) = −iρ0 ωApp (x x) ∇n pp (x

x ∈ ΓR ∀x

x ) = pd (x x) pp (x ∂pp + ikpp =0 ∂r

x ∈ ΓD ∀x

nsd−1 2

s

(1.44)

x ∈ ΓIN F ∀x

x ) = pp (x x) + phom (x x ). The solution is split too: p(x If the given pulsation ω is not an eigenfrequency of the problem (1.43), the homox) ≡ 0) and only the particular problem has to be geneous solution will be null (phom (x

solved.

If no Robin boundary condition is used, the eigenfrequencies of problem (1.43) can be pure real values. If our given pulsation ω (in general a pure real value) coincides with an eigenfrequency, more than an unique solution exists. Nevertheless, a correct physical modelling always includes a Robin boundary condition (it has to be


21

interpreted as the attenuation of the system). In that case the eigenfrequencies are always complex values and the general problem always has a unique solution for the pure real values of pulsation. In the results presented in following chapters only the frequency response of the vibroacoustic systems will be analysed. The most important outputs shown here (sound reduction index, sound levels, radiation efficiencies, vibration levels) can be understood in the frequency-domain. Time-dependent results are difficult to interpret and very often meaningless. Moreover, some of the vibroacoustic parameters are frequency-dependent (i.e. wall admittances). It would be different for the case of transient phenomena like the reverberation time of rooms. The eigenfrequencies of the problem (1.43) have been used in Chapter 5 in order to solve the acoustic problem by means of modal analysis. The spatial description of the solution is done by means of a basis composed of eigensolutions. This has physical meaning and some orthogonality properties simplifies the solution of the problem. Details on the modal analysis applied to acoustics can be found in Pierce (1981), Kuttruff (1979) and Davidsson (2004).

Chapter 2 Review of numerical methods for vibroacoustics Numerical methods are a precise and rigorous tool in order to solve the vibroacoustic equations. However, wave problems require a fine discretisation of the domains and the resolution of multiple frequencies in order to obtain valid engineering results. This causes numerical methods to be computationally expensive. Wave phenomena were considered by Zienkiewicz (2000) as one of the open problems in the field of numerical methods. A very important aspect is the relationship between the expected wave lengths in the solution and the physical dimensions of the studied domains. We distinguish then between low-frequency and mid or high-frequency problems. In the first case there is a small number of waves in the physical domain. The modal density is low and the eigenfrequencies of the problem can be clearly distinguished. On the contrary, for mid and high-frequency problems the wave length is small when compared with the characteristic length of the problem and the modal density is high. This classification is important from both a numerical and a modelling point of view. The physical response is also different depending on the frequency. This chapter is a review of numerical techniques for the vibroacoustic equations. The discussion is slightly oriented to the field of sound transmission. Thereby more 23

24

Review of numerical methods for vibroacoustics

emphasis is put in problems formulated in the frequency domain and the extension of numerical techniques to the mid-frequency range. The numerical methods discussed here can be classified into two categories: the more consolidated methods mainly used for low frequencies (Atalla and Bernhard (1994)) and the new methods under development for mid and high frequencies (Desmet (2002)). The distinction between methods used for acoustics (Helmholtz equation) and for structural dynamics is also done.

2.1

Numerical methods for the low-frequency range

2.1.1

The finite element method (FEM)

2.1.1.1

Acoustics

Two main FEM formulations are used for acoustic problems. On the one hand the mixed formulation where the acoustic medium is characterised by means of the acoustic pressure and velocity fields. On the other hand the classical formulation where the unknown is only the acoustic pressure. Detailed descriptions of the available options can be found in Stifkens (1995), Everstine (1997) and Junger (1997). Modifications of the mixed formulation and formulations in which only displacements are employed for both the acoustic and the solid domains can be found in Bathe et al. (1995), Berm´ udez and Rodr´ıguez (1999) and Berm´ udez et al. (2000). In the mixed formulation the coupling with solids is simpler due to the use of the velocity in the acoustic domain as variable. Moreover, the acoustic intensity can be obtained as a direct postprocess. However, it is more expensive because both the pressure and the velocity fields have to be solved while in the classical formulation the only unknown is the pressure. Velocity is a vectorial variable and two (2D) or three (3D) degrees of freedom per node have to be added. The classical formulation of acoustics will be considered from now on. By applying the usual weighted residual approach, the strong form (1.32) is trans-

2.1 Numerical methods for the low-frequency range

25

formed into the weak form Z

Ω

∇p · ∇ϕdΩ +

Z

iρ0 ωApϕdΓ − ΓR Z

Z

k 2 pϕdΓ = Ω

iωρ0 Ω

X s

x, x s )ϕdΩ − qs δ(x

Z

iρ0 ωvn ϕdΓ (2.1) ΓN

where ϕ is the test function. A typical FEM interpolation is used (Zienkiewicz and Taylor (2000), Mathur et al. (2001)), x) = p(x

nac X

x ) pj Nj (x

;

pj ∈ C ;

j=1

Nj : Rnsd 7→ R

(2.2)

Using the Galerkin formulation, the discretised form of the acoustic problem can be written as

Kac + iωCac − ω 2 Mac p = fac

(2.3)

where Mac , Cac , Kac are the mass, absorption and stiffness matrices defined as (Kac )ij =

Z

(Cac )ij =

Z

Ω

∇Ni · ∇Nj dΩ

(2.4)

ρ0 ANi Nj dΓ

(2.5)

Z

(2.6)

ΓR

1 (Mac )ij = 2 c

Ni Nj dΩ Ω

Kac and Mac are the usual, real-valued, stiffness and mass matrices (with the only difference that Mac is multiplied by 1/c2 and Kac can be multiplied by 1 + ηi if hysteretic damping is considered). Cac makes necessary the use of complex arithmetics because A is a complex value. The force vector takes into account two sound sources: vibrating panels (non-homogeneous Neumann boundary condition)

26


and punctual sources (fac )i = iρ0 ω

XZ s

2.1.1.2

Ω

x, x s )Ni dΩ − qs δ(x

Z

iρ0 ωNi vn dΓ

(2.7)

Γn

Structural dynamics

Finite elements have been widely used for solid mechanics and structural problems. Details on the use of FEM for solid and structural dynamics can be found in Argyris and Mlejnek (1991), Clough and Penzien (1993), Hughes (1987), Bathe (1996) and Zienkiewicz and Taylor (2000).

2.1.2

The boundary element method (BEM)

2.1.2.1

Acoustics

The other low-frequency method considered here is the boundary element method (BEM). A general overview of the method can be found in Chen and Zhou (1992), Brebbia and Dom´ınguez (1992), Hunter and Pullan (1997) and Bonnet (1999). More specific descriptions of the BEM oriented to acoustic problems can be found in Ciskowski and Brebbia (1991), Von Estorff (2000), Kirkup (2007) and Shaw (1988). Even if a complete discretisation of the domain is not required by BEM (only boundaries are discretised), FEM is more popular and more widely used (especially for structural problems). The use of BEM for nonlinear problems and in heterogeneous domains can be quite complicated (with respect to FEM). However, BEM has very interesting properties for the acoustic problem. It seems to be a numerical method especially designed to deal with the Helmholtz equation. The direct version of the BEM can be formulated beginning with the integral equation Z

Ω

p4ϕdΩ −

Z

ϕ4pdΩ + Ω

Z

∂Ω

ϕ∇n pdΓ −

Z

p∇n ϕdΓ = 0

(2.8)

∂Ω

For the case of the Helmholtz equation it can be obtained by means of a double


27

application of the Green-Gauss theorem. BEM requires a fundamental solution. It is the Green function of the problem in an unbounded domain. This is the solution of the governing equation when a punctual force is acting in our domain. The force can be a mechanical force, a heat source, a punctual sound source,. . . depending on the problem type. Boundary conditions do not have to be satisfied by the fundamental solution. For the case of the Helmholtz equation the fundamental solution Ψ satisfies x, x 0 ) + k 2 Ψ(x x, x 0 ) = δ(x x − x0) 4Ψ(x

(2.9)

The expression of Ψ for the two-dimensional Helmholtz problem is (2)

H (kr) x, x 0 ) = 0 Ψ(x 4i (2)

(2.10)

(2)

where H0 is the Hankel function of second kind (Hn (z) = Jn (z)−iYn (z)). Jn and Yn are the n-order Bessel functions of first and second kind. And for a three-dimensional Helmholtz problem x, x 0 ) = Ψ(x

e−ikr 4πr

(2.11)

where r is the distance between x and x 0 . In order to reduce the formulation of the problem to the boundary, the test function ϕ in Eq. (2.8) is replaced by the fundamental solution The following property can x , x 0 )4p(x x) − p(x x)4Ψ(x x, x 0 ) = δ(x x − x 0 ). It can be obtained by the now be used: Ψ(x

addition of Ψ times the homogeneous Helmholtz equation plus −p times Eq. (2.9). Finally the boundary integral equation for the homogeneous Helmholtz equation is Z

∂Ω

x )∇n Ψ(x x, x 0 )dΓ − p(x

Z

x, x 0 )∇n p(x x)dΓ = Ψ(x ∂Ω

Z

Ω

x)δ(x x − x 0 )dΩ p(x

(2.12)

The volume integral can be computed analytically: Z

Ω

x )δ(x x − x 0 )dΩ = ς(x x 0 )p(x x0 ) ∀x x∈Ω p(x

(2.13)

x 0 ) depends on the location of x 0 (inside the domain or at the boundary) The value of ς(x

28


and on the type of boundary (smooth or if x 0 is a corner in an angular boundary). Eqs. (2.1) and (2.12) are the basis of the discretisation process in FEM and BEM respectively. Note that while in Eq. (2.1) there are volume and surface integrals, Eq. (2.12) only contains surface integrals. In FEM the unknown variable is p in the whole domain while in BEM we have p and ∇n p at the boundaries. Finally, the role

of the FEM test function is assumed by the fundamental solution Ψ in BEM.

Once the weak formulation and the fundamental solution are known, a discretisation of the boundary can be done. The discretisation procedure is similar for FEM and BEM. In BEM, not only the variable p is discretised but also its normal derivative

x) = p(x

nod X

x)pj Nj (x

;

j=1

x) = ∇n p(x

nod X

x ) (pn )j Nj (x

(2.14)

j=1

Considering x 0 of Eq. (2.12) to be every node in the boundary a linear system of equations can be obtained (A − ςI) p = Bpn where (A)ij =

Z

(B)ij =

∂Ω

Z

(2.15)

x , x i )Nj (x x ) dS ∇n Ψ(x

(2.16)

x, x i )Nj (x x) dS Ψ(x

(2.17)

∂Ω

xi ) (ς)ij = δij ς(x

(2.18)

It should be noted that matrix coefficients are complex numbers (like the fundamental solutions). In addition, matrices are full and non-symmetric. Some usual linear solvers for banded matrices (typical of FEM) cannot be used. Harari and Hughes (1992) carried out a comparative study of the computational costs of solving both types of matrices (large, symmetric, banded matrices in FEM versus small, non-symmetric, full matrices in BEM). They concluded that although matrices derived from FEM are larger, numerical solvers can be faster and sometimes it is more efficient to deal with


29

a larger matrix but that has an a priori known structure (banded in that case). In any case, the choice between FEM and BEM depends on the type of problem (where other aspects different from the numerical cost of solving the linear system of equations will be considered) and especially on the personal preference. All these considerations have to be modified for vibroacoustic problems. Due to the fluid-structure interaction, FEM matrices are no longer banded. The choice of the more adequate solver depends on other factors (i.e. degree of coupling, type of formulation of the problem, ...) A critical issue in the BEM is the computation of the integrals in Eqs. (2.16) and (2.17). Singularities due to the fundamental solutions used can be found (especially for diagonal coefficients). Moreover, for the case of Helmholtz equation fundamental solutions are oscillatory. Several options are available. The first option is to use Gauss quadratures of adaptive order. The number of Gauss points is changed depending on the distance to the singularity and the wave length. This option is not the most efficient one but can be implemented without major difficulties. Another option is to develop specific quadratures to deal with the singularities of the fundamental solutions. This has been done in Ozgener and Ozgener (2000). For more general methods on integration of singular and oscillatory functions, see Milovanovic (1998). Finally, the fundamental solutions can be approximated by an analytical expression around the singularity (i.e. by means of Taylor series). The integration in that case is done analytically. An example of this technique can be found in Ramesh and Lean (1991). The complete solution of the problem is split in two steps. First the boundary is solved and once variables on the boundary are known the values of the solution inside the domain can be evaluated. This is one of the main advantages of BEM. Only the boundary and the interesting points in the domain (and not all the domain) have to be solved. Only in the first step a system of linear equations has to be solved. The evaluations of the variable in the interior of the domain requires as major task the computation of integrals (2.16) and (2.17). x i is now the position of the interior point where the solution will be evaluated. The modifications to be done in Eq. (2.15) in order to take into account the boundary conditions are simpler than in other problems because Dirichlet boundary

30


conditions are rarely used in acoustics. Then, p is always unknown. It is in general more difficult to solve non-homogeneous equations with BEM (i.e. x) = g(x x)). The force term in the integral formulation has to be Poisson problem 4ψ(x evaluated by means of a volume integral which is an important break down within the philosophy and the general organisation of the method. However, it is not a problem for the case of punctual sound sources. The integration of a Dirac-delta is directly done. The new force term in the integral (2.13) is x0 )p(x x0 ) − ς(x

X s

xs , x 0 ) ∀x x∈Ω iρ0 ωqs Ψ(x

(2.19)

BEM is less adequate than FEM for the eigenfrequency analysis of the acoustic problem. In FEM, the problem can be formulated in terms of mass, stiffness and absorption matrices. The pulsation ω of the problem can be isolated. On the contrary, in BEM, the fundamental solutions include the frequency parameter and the classical matrix formulation of the eigenvalue problem is not obtained. Special eigenvalue techniques must be used. An example of a purely numerical technique for transcendental eigenproblems

1

is found in Zhaohui et al. (2004). A review of the techniques based in the

modification of BEM formulations for eigenvalue analysis is done in Ali et al. (1995). Two of them are the Internal cell method (Ciskowski and Brebbia (1991)) and the Dual reciprocity method (Partridge et al. (1992)). In the first case the Helmholtz problem is considered as a Laplace problem. The main difficulty is that the mass contributions require volume integrals (all the other parts of the problem can be reduced to the boundary using the fundamental solution of the Laplace problem). The dual reciprocity method reduces the eigenvalue problem to the boundary by interpolating the solution (eigenfunctions) by means of harmonic shape functions.

1

An eigenvalue problem where the eigenvalue parameter is included in the formulation of matrices and cannot be isolated analytically.


2.1.2.2

31

Structural dynamics

BEM has also been used for solid mechanics in Dom´ınguez (1993) and structural dynamics (see for example Providakis and Beskos (1989) and Palermo (2007)). However, the most common option is to use FEM for the structural part of the problem. Formulations are simpler and the use of structural finite elements better established. There are also more software and libraries available.

2.1.3

Numerical techniques for the coupled problem

Two possibilities will be considered in this short review of vibroacoustic formulations for low frequencies. On the one hand, using FEM for both the acoustic and the structural domain and on the other hand using FEM for the structural domain and BEM for the acoustic domain. 2.1.3.1

FEM-FEM

Assuming that the structural problem is linear elastic and can also be formulated in the frequency domain, it can be written in matrix form as Ks + iωCs − ω 2 Ms u = fs

(2.20)

where Ms , Cs and Ks are the mass, damping and stiffness matrices. Their particular expression depends on the specific structural formulation. fs is the vector of structural forces and u the vector of structural displacements (and rotations). If this structural problem is coupled with the FEM formulation of the acoustic problem presented in Section 2.1.1, the global system is "

ω

2

−Ms

0

ρ0 LF S −Mac

!

+ iω

Cs

0

0

Cac

!

+

Ks −LSF 0

Kac

!#(

u p

)

=

(

fs

)

fac (2.21)

LF S and LSF are the coupling matrices. LF S takes into account the effect of the structure over the acoustic fluid. The structural vibration causes an imposed normal

32


velocity in the acoustic contour (boundary condition of Eq. (1.37)). Only normal continuity of displacement is imposed and the tangential effects are not taken into account due to the lack of viscosity of the acoustic fluid. The interaction with the structure adds an integral in the left-hand-side of the weak formulation of the acoustic problem (2.1) 2

x) = ρ0 ω (u u · n) ∇n p(x

Z

=⇒

ΓF S

u · n )dΓ ρ0 ω 2 ϕ(u

(2.22)

note that n is here the outward unit normal with respect to the acoustic domain. If the structural displacement field is interpolated as u(x x) =

ns X j

x ) · uj Nj (x

(2.23)

where Nj are the shape functions and a Galerkin formulation is also used for the structure, the expression of the acoustic nodal forces due to the vibroacoustic coupling is (fF S )i = −ρ0 ω

2

ns Z X j

|

ΓF S

n −Niac (n

· Nj )dΓ uj {z }

(2.24)

(LF S )ij

where (LF S )ij is the sub-matrix that links the acoustic force in node i with the structural displacements of the structural node j. LSF takes into account the effect of the acoustic fluid over the structure. The contribution of the acoustic pressure of node j to the mechanical force of the structural node i can be expressed as (fSF )i =

nac Z X j

Note that LF S = LTSF .

|

ΓF S

Ni ·

n )Njac dΓ (−n {z

(LSF )ij

pj

(2.25)

}

The first examples of the use of finite elements for acoustic and vibroacoustic problems are usually related to automotive applications. Car cabins are much smaller


33

than rooms in laboratories or apartments, so the computational cost of the simulation is also much smaller. Car designers have always been interested in the control of the resonances in the car cabin at low frequencies, which is a problem where deterministic models based in the numerical resolution of vibroacoustic equations can contribute with more detailed information than other modelling approaches (i.e. pressure distribution in a cabin for frequencies closer to a resonance). In Craggs (1972) the eigenfrequencies of a car cabin are calculated. The sound transmission between the motor cavity and the car cabin is modelled in Craggs (1973). Flanking or indirect paths are included in the analysis. The same author performs a vibroacoustic modelling in the time domain of a plate radiating sound inside a room (Craggs (1971)). Later, Ramakrishnan and Koval (1987) use finite elements to study the sound transmission through a plate. The excitation pressure wave is imposed by means of an analytical formula while the receiving room is modelled with finite elements. Kang and Bolton (1996) use two-dimensional finite element models to study sound transmission through single and layered structures. Absorbent materials are also modelled. The dimensions of the studied systems are small. The modelling of absorbent materials is a field where finite elements have often been used. In Panneton and Atalla (1996) and Panneton and Atalla (1997) the modelling effort is concentrated only in the solid, a piece of absorbent material with or without solid layers. More recently, in Maluski and Gibbs (2000), finite elements have been used for building acoustics. The sound transmission through partitions is modelled. The attention is focused on low frequencies and the modal response. Validation with a reduced scale model is done. In Maluski and Gibbs (2004) the effect of room sizes and room absorption in the sound reduction index is checked. The effect of the room absorption is also studied in Melo et al. (2002). The reported results are in the low-frequency range and have been obtained using the commercial software Sysnoise.

The courage and effort of the first users of finite elements for vibroacoustic problems is admirable. The computational resources in the 70’s or even in the beginning of the 90’s cannot be compared with today’s computers.

34


2.1.3.2

FEM-BEM

The other coupling possibility that is often considered for vibroacoustic problems is the use of BEM for the acoustic part and FEM for the structural part. The idea is to combine the interesting properties of BEM for acoustics (facilities to deal with unbounded domains and reduction of the dimension of the problem to the boundary) with the well established FEM formulations, solid/beam/shell elements and implementations for the structural part of the problem. Moreover, the disadvantage of BEM with respect to FEM due to the type of matrices obtained is not so important in a vibroacoustic problem, where other aspects influence the structure of the global matrix (i.e. type of variables chosen for the description of each domain, and the use or not of a block solver or a semi-decoupled approach). The two matrices modelling the coupling between a FEM structural domain and the BEM acoustic fluid have to be reformulated. In order to take into account the force of the fluid over the solid the correct shape functions for the acoustic part have to be considered: while in Eq. (2.25) the finite element shape functions reduced to the boundary are used, here the boundary element interpolation of the pressure in the boundary NjBEM is considered (fSF )i =

ns Z X j

|

ΓF S

Ni ·

n)NjBEM dΓ (−n {z

(LBEM )ij SF

pj

(2.26)

}

On the other hand the acoustic force in node i due to the structural displacement of node j can be obtained by means of (fF S )i =

Z

x, x i ) (∇n p(x x)) dS = ρ0 ω Ψ(x ∂Ω

2

ns Z X j

|

ΓF S

x, x i )(n n · Nj )dΓ uj Ψ(x {z }

(2.27)

(LBEM )ij FS

Note that while in Eq. (2.24), Ni was the acoustic test function here the fundamental solution Ψ has to be used.


35

The system of linear equations to be solved for FEM-BEM coupling is BEM (ω 2 Ms + ωCs + Ks ) −LSF

−ρ0 ω 2 LFBEM S

A

   ( ) !   u   fs 0 = p  −B  fac    p  n

(2.28)

Examples of FEM-BEM coupling applied to vibroacoustics can be found in Wu and Dandapani (1994). The sound transmission between two rooms and from the interior to the exterior of an sphere is modelled. In Suzuki et al. (1989) the numerical model is used in order to predict the response of car, aircraft or train cabins. Interesting numerical details are also shown. On the one hand, boundaries with vibroacoustic coupling (fluid-solid interaction) and absorption (Robin boundary condition) are considered. On the other hand comparison between coupled and uncoupled results are shown. Finally, in Coyette (1999) layered solids as well as absorbent materials placed inside are modelled by means of finite elements. The rectangular structure is baffled and the radiation is calculated by means of BEM.

The criteria in order to decide the element size of the meshes for the fluid and structure domains depend on the length of waves in the medium. The wave lengths depend on the physical properties of the medium. Then, in both types of couplings reviewed here, meshes for the structural and for the acoustic part of the problem can be different. The possibility of working with nonconforming meshes is desirable. On the one hand, having different element sizes at each side of the coupling contour minimise the total number of elements used (avoid a size transition zone around the coupling boundary). On the other hand the meshing construction procedure is simplified and it can be done faster. To deal with nonconforming meshes has to be taken into account in the implementation of the coupling matrices as well as in the integration procedures of them.

36


2.2

Numerical methods for the mid-frequency range

Errors in Helmholtz equation differ from errors in static linear elasticity or heat propagation problems. In these problems the error is local and concentrated around singularities of the domain (i.e. corners, stiffeners of a structure,...) or boundary conditions (i.e. supports of a structure or punctual sources). Very often the solution has high gradients only in some parts of the domain and the error does not propagate from one zone to another. In the Helmholtz problem it is completely different. The error is global and it propagates all around the domain due to the nature of the differential operator. Geometrical singularities or boundary conditions can also be a source of error, especially at low frequencies. The study, prediction and estimation of error in Helmholtz equation must differ from the techniques used for elastostatics or heat propagation problems. In the field of numerical acoustics, a usual rule of thumb is that six linear finite elements per wave length are enough in order to obtain accurate results. It is a frequency-dependent criterion since wave length depends on the frequency of the problem and the physical properties of the studied media. The wave length can be calculated as the length of a wave propagating in an unbounded medium or the wave length of the nearest eigenfrequency. Using this criterion it can be seen that the computational costs soon become unaffordable. The maximum calculable frequency is often too small. More detailed analyses (Ihlenburg (1998), Ihlenburg and Babuska (1997)) predict that the interpolation error for linear elements can be estimated as |p − pI |1 = Clocal b kb h |p|1

(2.29)

k = k`char and b h = h/`char are dimensionless wave number and element size, where b

`char is a characteristic length of the problem, Clocal is a constant that depends on each particular situation but it is independent of b k and b h, p is the exact solution and

2.2 Numerical methods for the mid-frequency range

37

pI is an interpolation of p. The six-elements-per-wave-length rule of thumb only takes into account this local interpolation error. Nevertheless, two additional phenomena make this criterion insufficient: the dispersion effect (k-singularity) and the existence of eigenfrequencies of the equation (λ-singularity) (Bouillard and Ihlenburg (1999)). The wave number of the numerical solution has been proved to be different from the exact wave number in the Helmholtz equation (for a linear one-dimensional finite b3 b 2 element solution of Helmholtz equation we have b knumerical = b k − k h + o(b k 5b h4 )). 24

This causes an increasing phase shift of the discrete solution. The error affects all the domain. Due to dispersion, a numerical solution obtained by keeping constant b kb h

would have more error for high frequencies. The increase of error due to the increase of dimensionless wave number b k is known as pollution effect (Deraemaeker et al. (1999)). Introducing the numerical wave number (affected by dispersion error) in the anal-

ysis, an expression for the total error can be obtained as |p − ph |1 ≤ C1 |p|1

b kb h 2p

!p

+ C2 b k

b kb h 2p

!2p

b kb h

(2.35)

With this basis, the exact solution is obtained in 1D problems. For two-dimensional problems the following basis has been used: x) =< 1, cos(k cos(θ)x + k sin(θ)y), sin(k cos(θ)x + k sin(θ)y), P T (x

(2.36)

cos(−k cos(θ)x + k sin(θ)y), sin(−k cos(θ)x + k sin(θ)y) > This leads to frequency-dependent shape functions that can interpolate plane waves (the direction depends on the angle θ). The number of plane waves in the basis can be increased in order to reduce errors. In a first approach, a finite number of directions was used. This idea has been generalised in Lacroix et al. (2003) where now the basis is defined as x) =< 1, cos(θ ∗ (x x)), sin(θ ∗ (x x)) > P T (x

(2.37)

x) is an unknown angle and can be different for every position of the domain. where θ ∗ (x x ) that An iterative procedure is proposed in order to chose the correct function θ ∗ (x minimises the dispersion error. Results are successful but the procedure can be quite expensive.

2.2.1.3

Trefftz methods

The original Trefftz method can be used in order to solve boundary value problems. It is less popular than FEM or BEM and also less flexible. However, it has interesting properties and can be more efficient than FEM and BEM for mid and high frequencies. A mesh of the domain is not required, so this approach can be regarded as a boundary method. An overview of the Trefftz method can be found in Kita and Kamiya (1995). A set of functions that satisfy the homogeneous differential equation is required


43

(note that nothing is said about the boundary conditions). They are called T-complete systems and examples of them can be found in Kita and Kamiya (1995) or Cheung et al. (1991). This functions are different for each differential equation. For the two-dimensional Helmholtz equation C

t =

Jn (kr)e

inθ

n = 0, 1, 2, ...

(2.38)

where Jn are the Bessel functions of the first kind and order n, and r and θ polar coordinates. T-functions have to be independent of the geometry. A particular solution of the non-homogeneous equation is also required. In the indirect version of the Trefftz method, the numerical solution is interpolated by means of the set of T-complete functions and the particular solution. The discrete solution exactly satisfies (by construction) the differential equation inside the domain, but not the boundary conditions. An error function can be defined with the residuals for the Dirichlet, Neumann and Robin contours. The unknown coefficients of the discrete function can be determined by minimisation of this error function. Punctual collocation, least-square and Galerkin versions of the minimisation procedure have been used. A direct version of the Trefftz method also exists. It is more similar to BEM and variables are only interpolated at boundaries. An element-based version of the Trefftz method is included in the overview by Jirousek and Wrólewski (1995). A partition of the domain with softer requirements than a finite element mesh is needed. The subregions can be distorted but they must cover all the domain. T-elements can be used to solve both structural mechanics problems (Jirousek and Wrólewski (1996)) and Poisson, Laplace or Helmholtz equations (Jirousek and Stojek (1995)). T-complete functions using local polar coordinates are used in order to interpolate the variable inside the element. Continuity between elements and boundary conditions must be imposed. The most common technique are the so-called hybrid T-elements where an interface variable is defined in order to impose continuity (Jirousek and Zieli´ nski (1997)). However, it can also be imposed by means of least-squares. T-elements imposing continuity in that way have been

44


successfully used for Helmholtz equation in Stojek (1998). A more specific approach called the wave based prediction technique has also been used for vibroacoustic problems. The physical domains should be inserted in a box being as small as possible. It is used in order to define a set of plane waves travelling in different directions and that are solutions of the acoustic or structural dynamic problems. They are used as interpolating functions. The unknown coefficients are obtained after the minimisation of the errors at boundaries (using for example leastsquares minimisation or a Galerkin formulation). The method has been compared with finite element results in Pluymers et al. (2003) and with experimental results in Pluymers et al. (2003), with satisfactory agreements. It has been also used in vibroacoustic problems with infinite domains and coupled with finite elements in Pluymers et al. (2004), and for the study of shell vibrations in Desmet et al. (2002) and Vanmaele et al. (2003). 2.2.1.4

Spectral methods

The use of finite elements with enriched shape functions has often been considered an alternative in order to improve the performance. In this kind of problems having oscillatory solutions, the inclusion of high-order polynomials or even trigonometric functions in the interpolation basis can reduce the numerical errors. We can see the influence of the degree of the polynomials in the interpolation space in Eq. (2.30). The expected FEM error for elements with interpolations based on higher order polynomials is smaller than for linear elements. A general approach to the spectral concept can be found in Boyd (1999). In Babuska et al. (1981) the mathematical basis of the p-version of the FEM is discussed. Several options exists in order to create element families of arbitrary degree of the polynomials used for interpolation. A review of the most frequently used as well as a comparison of the convergence depending on the polynomial family can be found in Petersen et al. (2006). Besides convergence, other important aspects must be considered. Among them, the condition number of the mass and stiffness matrices obtained.


45

The most classical options are to use Lagrange or hierarchic polynomials. The use of triangular and quadrilateral Lagrangian elements can be found in Warburton et al. (1999). How to make an efficient implementation is shown in Hesthaven and Warburton (2002), where tetrahedra are also used in three-dimensional problems. The main advantage of using Lagrangian polynomials is that the unknown variables keep a physical meaning. The correct location of the nodes inside the element in order to avoid ill-conditioned matrices is a very important aspect in their use. In the hierarchic elements, the shape functions of degree p can be obtained by adding a new function to the p − 1 family. Moreover the matrices of order p − 1 are

included in the larger matrices of order p. This allows a reduction in the number of operations to be done during the p refinement if the code is correctly optimised. However, to perform a correct implementation and guarantee continuity between elements is not simple and requires additional topological information. Details on the correct use of hierarchic elements are shown in Ainsworth and Coyle (2001) for problems using quadrilateral and triangular elements, and in Ainsworth and Coyle (2003) for the more difficult case of tetrahedra in three-dimensional problems.

2.2.2

Structural dynamics

The computational cost of the structural part of the vibroacoustic problem is often smaller than the cost of the acoustic part. Shells or plates (two-dimensional elements) are often used in three-dimensional situations and beams (one-dimensional elements) for two-dimensional vibroacoustic problems. Finally, while the acoustic waves are nondispersive, the bending waves in plates are dispersive. Nondispersive waves have the same wave velocity for every frequency and the relationship between the wave number and the frequency is linear. On the contrary, dispersive waves have a different wave velocity for every frequency and the relationship between the wave number and the frequency is nonlinear. For low frequencies structural waves are shorter than acoustic waves. The element size in the structure is smaller than in the acoustic domain. However, after the critical frequency the acoustic wave length become smaller while the bending wave length is almost constant. The size of finite elements in the

46


acoustic part is then smaller than in the structure. In any case, some techniques have also been used in order to improve the efficiency of numerical methods in the structural part of the vibroacoustic problem. The same idea of the partition of unity method (generalised FEM) that is used for Helmholtz equation (see Section 2.2.1.1), has been applied to structural problems, by Bouillard et al. (2002) and Bouillard et al. (2004) for beam elements, and De Bel et al. (2004) for plates. High-order polynomial shape functions, have also been used for solid mechanic problems. In D¨ uster (2001) p-FEM is used for modelling structures with anisotropy. Implementation details can be found in Campion and Jarvis (1996). However, for structural finite elements the interpolation space has more frequently been enriched by using trigonometric shape functions than with high order polynomials. Both EulerBernoulli beam elements and plate elements use cubic polynomials as shape functions and it is not easy to generate shape functions of arbitrary order due to the continuity requirements between elements (C 1 ). The p-FEM has also been used in Bardell (1996) for static problems and for composite panels in Bardell et al. (1997) where the interpolation basis has been enriched with trigonometric functions. In Beselin and Nicolas (1997) the bending eigenfrequencies of a rectangular plate with arbitrary boundary conditions have been calculated. A trigonometric functional basis is used instead of a polynomial basis. Due to the simple geometry of the problem, a pure spectral approach is considered (it is not an element formulation). However, it is valid in order to see the efficiency of spectral methods for dynamic problems and to compare the performance of polynomial-based versus trigonometric-based functional spaces. It has been shown that the polynomial basis leads to large numerical errors and ill-conditioned matrices for very high interpolation orders. On the contrary, trigonometric basis have a better numerical performance and can deal correctly with the eigenfrequencies of the plate in a higher frequency range. This idea of generating functional basis by means of trigonometric functions is generalised by Houmat (2005) in order to obtain an element-based formulation that is useful for more complicated geometries. A very versatile triangular element of arbitrary degree is formulated by


47

combining polynomial and trigonometric functions using the area coordinates. The resulting element inherits properties from the Lagrangian elements and also from the hierarchic elements. On the one hand shape functions can easily be associated with a node of the element and vanish on the other nodes. On the other hand, the new shape functions of higher degree can be obtained without modifying the previous ones. The examples shown in Houmat (2005) deal with membranes also governed by the Helmholtz equation (these elements could be used for acoustics). The same author has extended the idea to quadrilateral elements in Houmat (2006). Again an eigenvalue analysis is used in order to compare the efficiency of the new elements against standard Lagrangian elements of low order. Leung et al. (2004) propose a similar formulation for trapezoidal elements in plane dynamic problems. Continuity is not achieved automatically and additional topological information is required. Due to the small number of elements that can be used (increasing the interpolation degree), the geometry is often not properly described. In that situations some technique like blending functions is necessary (D¨ uster (2001)). The family of spectral elements described in Doyle (1997) is also a powerful tool in order to deal with high-frequency structural vibration problems. They were firstly formulated in order to solve time-domain problems in the frequency domain by using fast Fourier transforms. They differ from the p-element formulations mentioned before because now the weak formulation of the problem is not necessary. The chosen interpolation field must exactly satisfy the differential equation inside the element and the interpolation constants are obtained by dynamic equilibrium. In previous spectral elements the exact solution inside the elements cannot be satisfied by the interpolation field and minimisation of errors is afterwards imposed by means of the weak formulation. The exact solution is obtained in one-dimensional structural elements (beams) by using the following interpolation of the displacement field: u(x) = C1 ek1 x + C2 ek2 x + C3 e−k1 x + C4 e−k2 x + up (x) where Ci

(2.39)

i = 1, 2, 3, 4 are constants, up (x) is a function taking into account the

48


distributed loads and

k1 =

sr

ω2ρ

solid A

EI

k2 =

s r −

ω 2 ρsolid A EI

(2.40)

ρsolid , A, E and I are the density, the cross section area, the Young modulus and the inertia of the beam. The general procedure to derive dynamic stiffness matrices of structural elements is detailed in Banerjee (1997). These elements have been used for framed structures in Igawa et al. (2004) and Lee (2000), for the case of Timoshenko beams in Ahmida and Arruda (2001), Rayleigh beams (taking into account dynamic influence of rotational inertia), rods and beams with elastic support in Yu and Roesset (2001). Very small changes (with respect to a standard finite element formulation) are needed in the equations and implementation in order to take into account all these beam theories in the same finite element code. Moreover modifications of the method which take into account contraction of cross section in beams can be found in Yu and Roesset (2001) and in Gopalakrishnan (2000). The generalisation for multi-dimension structural elements is not trivial. Since the analytical solution of the problem is required, the two-dimensional problems analysed are often infinite or semi-infinite. A spectral plate formulation can be found in Lee and Lee (1999), Danial et al. (1996), Danial and Doyle (1992), Danial and Doyle (1995) and Rizzi and Doyle (1992). More complex geometries discretised by means of shell spectral elements have been solved in Solaroli et al. (2003). In Gopalakrishnan and Doyle (1995) the spectral solution has been combined with two-dimensional structural elements. In Doyle (2000), the application of these spectral elements to a vibroacoustic problem with curved beam and shell elements can be found. An interesting property is that the solution inside the element is not interpolated and can be exactly known by evaluation of exponential or trigonometric shape functions. It is an interesting property in order to calculate the coupling terms. The spectral element methods lead to dynamic stiffness matrices where the mass and stiffness matrices cannot be separated and the pulsation of the problem is implicitly included. A linear eigenvalue problem cannot be obtained and especial techniques must be used Zhaohui et al.

2.3 Concluding remarks

49

(2004). Finally, the hybrid approach for structures at high frequencies presented in Ohayon and Soize (1998) must be mentioned. It is more than a finite element formulation or an efficient numerical method. The theory exposed there is based in the viewpoint that an statistical understanding of the problem is required in the high-frequency range. The concept of fuzzy structure is introduced. In general, the variables are described by a mean value and an uncertain variation of it.

2.3

Concluding remarks

This review has been focused only on numerical-based techniques. In the examples shown in the following chapters, finite elements (Chapters 4 to 9) and boundary elements (Chapter 9) have been used for the low-frequency range. For the twodimensional vibrational problems of Chapter 7, the structural spectral method proposed in Doyle (1997) has been used in order to avoid discretisation errors and extend the analysis to higher frequencies. For vibroacoustic problems in the mid-frequency range, a model combining finite elements and the analytical solution of some of the acoustic domains has been used in order to reduce the numerical cost (the detailed description is done in Chapter 5). FEM and BEM (Section 2.1) are consolidated techniques that have been used and tested for vibroacoustic problems. Moreover, even some open and commercial codes (EDF (2007), Free-Field-Technologies (2007)) and libraries (INRIA and SDTools (2007)) exists which is a proof of maturity and a guarantee that engineering results can be obtained (at least for low frequencies). Most of the main problems that can be found (numerical errors, meshing criteria, implementation tricks) have already been documented. On the contrary, most of the techniques presented in Section 2.2 are very recent. Their use is not generalised beyond the research groups that have developed them. All of them have better performance than FEM when analysed in terms of meshing requirements or degrees of freedom per frequency. However, most of them also require non-standard tools or numerical techniques: specific integration

50


rules, additional geometrical data (connectivities), to perform interpolations of pressure fields... i.e. it is well known that an efficient implementation of a mesh-free software is more difficult that and efficient implementation of FEM. The possible advantages of mesh-free methods can then be hidden if a non-efficient implementation is used. The same conclusion is also valid for some of the other new numerical techniques.

Chapter 3 One-dimensional model for vibroacoustics 3.1

Introduction

An usual simplification in sound transmission models is to deal with unbounded acoustic domains and structures. This hypothesis, where the modal behaviour is completely neglected, is more adequate for high frequencies and makes it less difficult to obtain analytical solutions. On the contrary, numerical methods and deterministic models based on the vibroacoustic equations deal with acoustic domains and structures of finite size and have been mainly applied in the low frequency range. In this chapter, a one-dimensional model where the vibroacoustic equations are solved analytically will be presented. Solutions can be obtained by solving only a linear system of equations and the model can deal with sound transmission through layered partitions. The model also considers the various relevant sources of damping. Two main applications are envisaged. On the one hand, to have an analytical solution is an important tool to test elaborate numerical models (two and threedimensional models must be able to solve one-dimensional situations too). On the other hand, the most important properties of deterministic solutions can be understood through the one-dimensional model. How to obtain classical acoustic outputs 51

52

One-dimensional model for vibroacoustics

(sound reduction index or sound intensity) with a model that determines the total pressure (instead of a wave splitting of the pressure field) will be shown. The model can also be used to perform simple parametric analyses. The chapter is structured as follows. In Section 3.2 the main features of a simple version (undamped) of the model are presented. A first example where the modal behaviour of a vibroacoustic response is shown will be presented and the most important parameters will be defined. In Section 3.3 the model is enriched with acoustic absorption and structural damping effects. Modifications in the model caused by the consideration of “double walls” and acoustically absorbing materials are done in Section 3.4. The ability of the modified model to describe the influence of the air gap between double walls and the improvement of the vibroacoustic response due to absorbing materials is shown in two new examples. In Section 3.5 a comparison of the analytical model with some numerical solutions with one-dimensional behaviour is carried out. Some two-dimensional vibroacoustic examples solved by means of numerical models that have been tested with the one-dimensional solution proposed here are presented. Finally, the conclusions of the work are summarised in Section 3.6.

3.2

One-dimensional model for undamped vibroacoustics

3.2.1

Problem statement

In its simpler version, the model for vibroacoustics is based on the conceptual device of Fig. 3.1. The structure (e.g. partition wall) is represented by a particle of mass M connected to a spring of stiffness K. It separates two acoustic domains (e.g. rooms) Ω1 and Ω2 of lengths `1 and `2 . The sound source is the vibrating panel in the left edge, while the right edge represents a pure reflecting contour. Note that, for convenience, a different coordinate is used in each acoustic domain (i.e. x1 ∈ (0, `1 ) for Ω1 and

x2 ∈ (0, `2 ) for Ω2 ).

The unknowns are the acoustic pressure P (x, t) in domains Ω1 and Ω2 and the par-

3.2 One-dimensional model for undamped vibroacoustics

M

Ω1

x1

53

Ω2

x2 K

l1

l2

Figure 3.1: Conceptual model of undamped vibroacoustics

ticle displacement U (t). Assuming steady-harmonic solutions, they can be expressed as P (x, t) = Re p(x)eiωt U (t) = Re ueiωt

(3.1) (3.2)

where p(x) ∈ C is the spatial variation of pressure, u ∈ C is the phasor of the displacement of the mass particle and ω = 2πf is the angular frequency (f frequency).

For simplicity, the case of real frequencies ω (steady-harmonic excitation with no attenuation) is considered here. 3.2.1.1

Dimensional equations

The one-dimensional differential model for undamped vibroacoustics is summarised in Table 3.1. In each acoustic domain, a Helmholtz equation governs the spatial variation of pressure, Eqs. (3.3) and (3.4), with k = ω/c (k: wave number; c: speed of sound in air). The particle moves according to the undamped equation of dynamics (3.5), with the external force given by the pressure difference in its two interfaces (times a surface S required for dimensionality). By replacing expressions (3.1) and (3.2) into Eq. (3.5), the particle displacement can be solved in closed form as u=

S [p1 (`1 ) − p2 (0)] . K − ω 2M

(3.10)

54


Differential equations Acoustic domain Ω1 :

Acoustic domain Ω2 :

Particle:

d2 p1 + k 2 p1 = 0 dx2

in 0 < x < `1

d2 p2 + k 2 p2 = 0 dx2

in 0 < x < `2

(3.3)

(3.4)

M U¨ + KU = S [P1 (`1 ) − P2 (0)]

(3.5)

Boundary conditions Vibrating panel: Interface Ω1 –particle: Interface Ω2 –particle: Reflecting contour:

dp1 dn dp1 dn dp2 dn dp2 dn

dp1 = −iρ0 ωvn dx dp1 ≡ = ρ0 ω 2 u dx dp2 ≡− = −ρ0 ω 2 u dx dp2 ≡ =0 dx ≡−

at x = 0

(3.6)

at x = `1

(3.7)

at x = 0

(3.8)

at x = `2

(3.9)

Table 3.1: One-dimensional differential model for undamped vibroacoustics

The boundary conditions (BC) for the Helmholtz equations are also shown in Table 3.1. The vibrating panel is represented by a non-homogeneous Neumann BC, Eq. (3.6), where vn is the prescribed normal velocity of the panel. An homogeneous Neumann BC, Eq. (3.9), models the reflecting contour. The interfaces between the acoustic domains and the particle are coupling BC, Eqs. (3.7) and (3.8), that couple the two acoustic domains. In these BC, ρ0 is the density of air and n denotes the outer unit normal.

3.2 One-dimensional model for undamped vibroacoustics

3.2.1.2

55

Dimensionless equations

The dimensionless version of the differential model will give insight into the relevant factors that control the vibroacoustic response. The characteristic values of length, frequency (i.e. time) and pressure (i.e. force) and the corresponding dimensionless variables, denoted by a b, are shown in Table 3.2. Magnitude Length Frequency Pressure

Characteristic value `char := `1 ωchar := ω pchar := ωM c/S

Ddimensionless variable x b := x/`char ω b := ω/ωchar = 1 pb = p/pchar

Table 3.2: Characteristic values for adimensionalisation

With these new variables, the dimensionless version of the model shown in Table 3.3 is obtained. Note that the equation of dynamics does not longer appear explicitly in the model. Instead, its closed-form solution, Eq. (3.10), has been used in the dimensionless Robin BC, Eqs. (3.14) and (3.15). Five dimensionless numbers appear naturally during the adimensionalisation process. All these numbers, see Table 3.4, have a clear physical interpretation. N/2π is the number of waves that fit into acoustic domain Ω1 . λ is the ratio of lengths of the two acoustic domains. µ is the ratio of masses between the acoustic domain Ω1 (mass of air: density times length times surface) and the particle. ϑ is a ratio of speeds (speed of vibrating panel and speed of sound). Finally, G is a factor that depends on the ratio of frequencies (frequency of acoustic wave and natural frequency of particle).

3.2.2

Analytical solution

Due to its one-dimensional nature, the model of Table 3.3 is a very simple system of ordinary differential equations (ODE), with analytical solution pb1 (b x) = C1 cos(N x b) + C2 sin(N x b) ;

pb2 (b x) = C3 cos(N x b) + C4 sin(N x b)

(3.17)

56


Differential equations d2 pb1 + N 2 pb1 = 0 db x2

Acoustic domain Ω1 :

d2 pb2 + N 2 pb2 = 0 2 db x

Acoustic domain Ω2 : Boundary conditions Vibrating panel: Interface Ω1 –particle: Interface Ω2 –particle: Reflecting contour:

db p1 db x db p1 db x db p2 db x db p2 db x

= iµϑ

in 0 < x b 4ρ0 c

(5.16)

where prms is the root mean square pressure and < • > means spatial averaged inside

the room.

Some acoustic (not vibroacoustic, structures are not considered) problems have been solved in order to verify if the proposed model can reproduce these hypotheses (diffuse pressure field and correct amount of acoustic absorption introduced in the acoustic domains). If the absorbed acoustic energy in the room is known, the mean absorption coefficient of the boundaries can be expressed as α=

Pabs SRobin Iabs = SRobin Iinc Pinc

(5.17)

Pabs and Pinc are the total absorbed and incident acoustic powers. As shown in Section 5.4, the obtained pressure field is, at least in a broad part of the room, an acceptable approximation of the exact solution. Unfortunately, it would be incorrect to calculate local outputs at boundaries. A large error is made in the calculation of acoustic intensities at the contours of the room. This is a drawback of the presented model, caused again by the lack of normal velocity at contour of the modal basis. The same thing can be said in the nearest surroundings of the sound source. However, the acoustic intensity can be quite correctly calculated in an intermediate region between the source and the boundaries.

116

Combined modal-FEM approach for vibroacoustics

Using a closed surface that includes the punctual source, the outgoing power flow can be calculated as P=

I

Ie · n dS

with

1 Ie = Re {pvv } 2

(5.18)

where Ie is the time average of the acoustic intensity I . p is the pressure phasor and can be directly obtained using the modal basis and v is the acoustic velocity phasor

(when the control surface is a sphere, vr = v · n is the radial velocity). The acoustic velocity can be obtained by means of numerical differentiation of the pressure field. The differentiation procedure must be adapted to the acoustic wave length. While in wave-based models incident, absorbed and reflected waves can be dis-

tinguished in a natural way, in the models considering bounded domains this is not possible. All variables (in particular, the pressure phasor) are global variables. This means that the calculated acoustic power is the net acoustic power flow. For the case of the single room, acoustic energy is destroyed only in Robin boundaries. It means that the calculated power is equal to the absorbed power, P = Pabs . The averaged absorption coefficient α can be calculated using Eqs. (5.16) and (5.17) and the calculated value of absorbed power Pabs . The averaged absorption coefficient has been calculated using the described procedure in different rooms with a punctual sound source. The dimension of the rooms are: `x = 2.5 m, `y = 2.2 m, `z = 2.8 m and `x = 6.35 m, `y = 4.2 m, `z = 3 m. The punctual sound source is placed in a corner of the room and separated 0.5 m from each contour. Different values of admittance have been used. They are listed in Table 5.3. Three different configurations of the absorbing boundaries have been considered: 1. All the walls of the room are absorbing boundaries and acoustic energy can be absorbed. The same admittance is used for all of them. 2. All the walls of the room except one of them (its surface is `y × `z ) are absorbing boundaries. This situation is like in the receiving room of a sound reduction

index test.

5.5 Role of acoustic absorption and the size of the modal basis

117

3. Only one of the walls (its surface is `y × `z ) is considered to be a absorbing boundary. Less acoustic energy is absorbed in that case. α(%)

Z/ρa c

Z (Ns/m3 )

A (m3 /Ns)

5 10.098 20.353 30.306 50.136 86.263 97.817

78.4 38 18.0 11.5 6.17 2 + 0.8i 1.5 + 0.2i

3.1454 · 104 1.52456 · 104 7.2216 · 103 4.6138 · 103 2.4754 · 103 8.024 · 102 + 3.2096 · 102 i 6.018 · 102 + 80.24i

3.1792 · 10−5 6.55927 · 10−5 1.38473 · 10−4 2.16741 · 10−4 4.03974 · 10−4 1.07436 · 10−3 − 4.29745 · 10−4 i 1.63266 · 10−3 − 2.17688 · 10−4 i

Table 5.3: Values of admittance used. The absorption coefficient α is calculated by simplified expressions. In Fig. 5.4, the absorbed power and the averaged pressure level obtained for the same room changing the admittance of the Robin boundaries have been plotted. The numerical power flow has been compared with the theoretical expression of a monopole radiating in an unbounded domain P=

2π|S|2 ρ0 c

with S =

−iωρ0 Qs 4π

(5.19)

S is the monopole amplitude and Qs is the source strength amplitude. The variation of the power flow due to room absorption is small. However, for cases with nearly nonabsorbent walls, the modal behaviour of the room can be seen. Values of radiated power in finite acoustic domains are smaller than in an unbounded domain (α = 100 %). Differences in the mean pressure level of the room are more relevant. The sound level in the room decreases for higher values of the absorption. The difference between the prediction of the absorption using simplified expressions (see Pierce (1981) and Bell and Bell (1993)) and the value calculated by means of the numerical model has been plotted in Fig. 5.5 (∆α = αnumerical − αsimplif ied ). The

differences between the three different configurations of absorbing walls described above are small and almost constant with frequency (especially for high frequencies).


1

100

0.1

10

2

Power (N•m/s)

118

0.01

1

0.001

0.1

1e-04

0.01

1000

800

630

(a)

500

num. (α = 30 %)

400

exact, α = 100 % num. (α = 10 %)

315

250

200

160

125

100

1000

800

630

500

400

315

250

200

160

125

100

f (Hz)

f (Hz) (α = 10 %) (α = 20 %)

(α = 30 %) (α = 50 %)

(b)

Figure 5.4: Punctual sound source inside a room. Effect of room absorption: (a) acoustic power flow; (b) averaged pressure level.

The largest differences are found at low frequencies. This is caused by the poor modal density. It does not satisfy the hypothesis of a diffuse field and a uniform distribution of incidence angles for the pressure waves. This uniformity in incidence angles is assumed by the analytical formulations of the absorption coefficient. The difference is larger for more absorbent rooms. The modal approach has better performances for low absorption situations due to the type of basis employed. For a given value of the admittance, the numerical absorption is, in general, larger than the absorption predicted by simplified expressions. This difference has consequences in the calculation of the sound reduction index by means of numerical methods (if the simplified expressions of the absorption coefficient are used instead of the systematic calibration of the room absorption shown here). The correction term due to acoustic absorption in the receiving room of Eq. (5.15) takes into account the averaged absorption coefficient as −10 log10 (α). With the differences shown in Fig. 5.5, it can

∆α (%)

∆α (%)


6 4 2 0 -2 -4 -6 6 4 2 0 -2 -4 -6

119

Expected absorption: 20 %



Expected absorption: 10 % 800

630

500

400

315

250

200

160

125

100

800

630

500

400

315

250

200

160

125

100

f (Hz)

f (Hz)

Figure 5.5: Differences between the absorption coefficients ∆α = αnumerical −αsimplif ied obtained by means of the numerical model (punctual sound source in a room) or by simplified expressions. The case 1 for the configuration of absorbing walls is plotted. Cases 2 and 3 are very similar. Note that they are signed absolute differences and not relative differences.

be said that the differences in the pressure measurement of the sound reduction index due to the absorption modelling are not larger than ±1 dB. The model will tend in general to predict slightly higher values of sound reduction index.

5.5.2

Relationship of matrix bandwidth and the Robin boundary condition

In Eq. (5.6), it can be seen that the bandwidth of matrix Mψ depends on the treatment of Robin boundaries. If there are no Robin boundaries, the matrix is diagonal and if all the eigenfunctions are considered in Robin integrals, the matrix is full. Nevertheless the main advantage of using this model is the fast resolution of acoustic domains (with diagonal matrix). In Fig. 5.6 the influence of the bandwidth is shown. The Kundt’s tube has been solved using modal analysis for different bandwidths of the acoustic matrix. The modal solution has been compared with the available analytical solution and the

120


relative error in the mean pressure level of the acoustic domain calculated. The value of absorption used for the example is high (A = 4.03974 · 10−4 m3 /Ns, α = 50 %).

The errors in the modal approach are larger for higher values of absorption, this is an unfavourable situation for the modal approach. 6

%

3

e

5

2 1 0 50

100

150

200

250

f (Hz) No Absorption

BW = 0 (Diag.)

Full

BW = 5

Figure 5.6: Influence of the bandwidth of matrix Mψ (i.e. number of modes considered to represent Robin boundary conditions). 4 m length Kundt tube with an admittance A = 4.03974 · 10−4 m3 /Ns. The eigenfunctions in the modal basis are ordered by eigenfrequency and not by geometrical affinity at absorbing contours. A measure of the affinity could be R

R

x)ψj (x x)dΓ ψi (x R x )dΓ ΓR ψj (x x)dΓ ψ (x ΓR i ΓR

(5.20)

This information cannot be known a priori with a reasonable cost. In addition, the ordering of the eigenfunctions in that way would only be interesting for this part of the problem (frequency ordering is more important in order to choose significant eigenfunctions, Section 5.5.3). This means that considering a matrix bandwidth is a random process, in the sense that the off-diagonal terms are chosen with no physical criterion. In the results shown in Fig. 5.6 four different situations are considered: i) diagonal matrix Mψ where absorption is not modelled (‘No absorption’); ii) diagonal matrix Mψ where the absorption is considered by means of the diagonal coefficients (bandwidth = 0, this is the option used in the three-dimensional calculations); iii) all the off-diagonal terms have been taken into account (‘full’); iv) an intermediate


121

situation with matrix bandwidth = 5. In the case i), the modal solution is very bad but the error is concentrated around eigenfrequencies. This shows that the main role of acoustic absorption is to attenuate the resonant responses. In the case ii), the error is almost constant and bounded within an acceptable range (below 4 %). It has been the option used in the model implementation. The other options studied reduce the error between eigenfrequencies but the improvement is not clear at resonances (even for the case of full matrix).

5.5.3

Influence of frequency bandwidth

As shown in Section 5.3, the number of modes used to interpolate the pressure field is an important parameter. It controls the computational efficiency of the method and also the quality of the obtained solution. The modal basis is defined with the modes whose eigenfrequency belongs to a given bandwidth ∆f . The most relevant modal contributions are found around the excitation frequency. Modes falling outside a reasonable frequency bandwidth (in this case ±200 Hz) have

a poor modal contribution.

The influence of the frequency bandwidth in the solution of a one-dimensional sound transmission problem can be seen in Fig. 5.7. The relative error between the analytical solution and the modal solution is plotted. The chosen outputs are the averaged root mean square pressure in sending and receiving domains. The error is reduced when the bandwidth is increased. The improvement is more important in the sending domain.

5.5.4

Selection of acoustic modes

Two different mechanisms of sound transmission have been historically distinguished: resonant transmission and forced transmission. In the first case vibrations in the structure are mainly caused by direct excitation of modes whose eigenfrequency is close to the analysed frequency (resonant modes). This kind of vibration is only found in structures of finite size. In the second mechanism, the structure vibrates in order


10

1

1 %

10

2 rms >

0.1

e a

Ra

k

kc Fa --> s

Fs --> a

Rs

Fa --> s

Rs

Ra f1s

f1

f1a

f

fc

f2a

f2

f2s

Figure 5.8: Selection of acoustic modes. Conceptual plot with the mechanisms of sound transmission depending on the excitation frequency and the wave numbers in the acoustic domains and the structure. R means resonance excitation and F forced excitation (geometrical coincidence between structural and acoustic modes), ‘a’ means air and ‘s’ structure.

In the proposed modal-FEM approach, only the acoustic modes have to be chosen.

124


20

f = 200 Hz, sending f = 200 Hz, receiving f = 200 Hz, f = 654.8 Hz, sending f = 654.8 Hz, receiving f = 654.8 Hz,

e%

15 10 5 0

0 40

0 35

0 30

0 25

0 20

0 15

0 10

50

0

∆f (Hz)

Figure 5.9: Convergence of the outputs of interest depending on the bandwidth considered in the definition of the modal basis. Relative error by comparison with a reference value (modal basis with ∆f = 600 Hz).

l x2

l x1

l y1

Sending room

S

Receiving room

l y2

Figure 5.10: Sketch of a sound transmission problem between two rooms.

However, these considerations should be also taken into account to define meshing criteria for the FEM part of the problem. The actual importance of both transmission paths has been studied by means of the two-dimensional example of Fig. 5.10. Since it is a two-dimensional problem, the geometrical coincidences can be exactly controlled. In all the situations analysed the dimensions of rooms are `x1 = 4 m, `y1 = 3.5 m, `x2 = 3 m, `y2 = 3.5 m, S = 3 m. The single wall, modelled with beam elements, is simply supported with allowed rotations at the endings. With these dimensions it is ensured that structural modes with 3 waves (n = 6) and acoustic modes with 6.5


125

waves in the vertical direction (ny = 7, nx = 0, 1, 2, . . .) are geometrically coincident.

Meaning Young’s modulus Solid density Wall thickness Hysteretic damping coefficient Critical frequency

Symbol E ρsolid t η fc

Wall 1 Value 4.8 · 109 N/m2 913 kg/m3 0.03 m 3% 926.54 Hz

Wall 2 Value 2.94 · 1010 N/m2 2500 kg/m3 0.1 m 3% 371.51 Hz

Table 5.4: Material and geometrical properties of two single walls. Two different single walls have been analysed (geometric and mechanical properties in Table 5.4). In the first one (plasterboard) the eigenfrequency of the wall with 3 waves (n = 6) is 124.95 Hz. In the second one (concrete) this eigenfrequency is 654.83 Hz. The first problem is solved for a frequency of 124.90 Hz (below the critical frequency fc ) in order to force the wall to have a vibration shape similar to the eigenshape of mode n = 6. This structural vibration is geometrically coincident (2)

in the coupling surface with the modes of the receiving room having ny

= 7 and

(2)

nx = 0, 1, 2, 3, . . .. The eigenfrequencies for this family of modes are: 340.0, 344.7, 358.4, 380.1, 408.6, 442.6, 480.83, 522.44, 566.66, 612.9, 660.8, 710.0, . . . Hz. The frequency chosen for the second wall is 654.80 Hz. It is also close to the eigenfrequency of the wall n = 6. In Fig. 5.11, the modal contributions (frequency spectrum) of the sending and receiving acoustic domains have been plotted. In all the situations, the dominating part of the frequency spectrum is around the studied frequency (resonant modes). However, in the first wall where this frequency is under the critical frequency, we can distinguish around 340 Hz a group of modes generated by the forced path of sound transmission. Nevertheless, the contribution of these modes to the total pressure level in the room is small. The same calculations have been performed with different modal bases. They are created considering the eigenfunctions with eigenfrequency in the band [f0 − ∆f, f0 + ∆f ]. f0 is the excitation frequency and ∆f is a variable param-

eter. It can be seen in Fig. 5.9 that variations are smaller for values of ∆f larger

126


|aj|

1 Sending room f = 124.9 Hz, fc = 371.51 Hz

0.1 0.01 0.001

0 80

0 63

0 50

0 40

5 31

0 25

0 20

0 16

5 12

0 10

80

63

fj (Hz)

(a)

|aj|

1 Sending room f = 654.8 Hz, fc = 926.54 Hz

0.1 0.01 0.001

00

10

0

80

0

63

0

50

0

40

5

31

0

25

0

20

0

16

5

12

0

10

80

63

fj (Hz)

(b)

1 Receiving room f = 124.9 Hz, fc = 371.51 Hz 0.1 |aj|

Acoustic modes: ny = 7, nx = 0,1,2,... 0.01

0.001 0

80

0

63

0

50

50

40

0

40

5

31

0

25

0

20

0

16

5

12

0

10

80

63

fj (Hz)

(c)

0.01

Receiving room f = 654.8 Hz, fc = 926.54 Hz

|aj|

0.001 1e-04 1e-05

00

10

0

80

0

63

0

0

5

31

0

25

0

20

0

16

5

12

0

10

80

63

fj (Hz)

(d)

Figure 5.11: Modal contributions of two-dimensional acoustic domains in two different sound transmission problems: (a) heavy single leave, sending room; (b) lightweight single leave, sending room; (c) heavy single leave, receiving room, the geometrically coincident acoustic modes are indicated; (d) lightweight single leave, receiving room.


127

than 200 Hz. This means that only resonant modes are important because the modal contribution of the modes with frequency that is very different from the excitation frequency is small. A residual error (under 5 %) can never be eliminated. It is due to the basis used. We know a priori that with a basis composed of eigenfunctions the exact solution is never reached because it is a vibroacoustic problem and the acoustic absorption is 10 %. An study of the factors involved in the acoustic modal analysis has also been done in order to find some additional guideline to determine the parameter ∆f (especially for the case of geometrically coincident modes that are non resonant). The modal contribution ai of an uncoupled acoustic domain that is only excited by means of an structural vibration field can be expressed as ai = Z

−ρ0 ω 2

x)ψi (x x)dΩ ψi (x | {z } Ω

(Int 1 )

1 2 (k − ki2 )

Z |

ΓF S

x)un dΓ ψi (x {z }

(5.21)

(Int 2 )

This expression can be obtained by simplification of Eq. (5.4). For a constant frequency of excitation, the variable contributions to ai are the factors marked with Int1 , Int2 and 1/ k 2 − kj2 . Int1 has the same value for most of the modes. The effect of resonance is included in the factor 1/ k 2 − kj2 , see Fig. 5.12(a). It represents a factor 100 if a very resonant eigenfrequency is compared with an

eigenmode which differs 100 Hz from the excitation frequency. The contribution of the geometrical coincidence is included in Int2 . The value of this integral depends on two characteristics of the pressure and vibration waves: similarity of wave number and phase. The following numerical experiments have been done in order to quantify the importance of each factor: 1. Pressure and displacement fields with the same wave number but a difference in phase. It would be the case of the two-dimensional example presented before if the vertical dimensions of the rooms were `y1 = `y2 = 3. It would never be exact coincidence between both wave types due to the boundary conditions in

128


the structure. The parameter controlling this phenomenon is R yb ya

sin(kbending y) sin(ka y +

ξπ )dy 16

yb − y a

with ka = kbending

(5.22)

kbending is the wave number of bending waves and ka the wave number over the structure of acoustic waves. ξ controls the shifting of the pressure and velocity waves. 2. Pressure and displacement fields having different wave length R yb ya

sin(kbending y) sin(ξka y)dy yb − y a

with ka = kbending

(5.23)

The variation of these two factors depending on the value of ξ (ξ = 0 implies exact coincidence in Eq. (5.22) and ξ = 1 for the case of Eq. (5.23)) can be seen in Fig. 5.12(b) and Fig. 5.12(c). Except for very particular situations the ratio between geometrically coincident modes and the others is not larger than 10. The parameter 1/ k 2 − kj2 is more important than the two analysed integrals.

Thus, geometrically coincident modes are only important if they are close to the problem frequency. According to the presented analysis, good results can be obtained with ∆f = 200 Hz.

5.6

Validation example

The model has been used in order to calculate the sound reduction index of a single plasterboard. The material properties are those of Table 5.4. The dimensions of the sending and receiving rooms are: `x = 5.7 m, `y = 4.7 m, `z = 3.7 m (sending); and `x = 6.35 m, `y = 5 m, `z = 4 m (receiving). The dimensions of the plasterboard are `y = 4 m and `z = 3 m. The wall is supported. The acoustic absorption of the rooms is 10 %. Since the dimensions of the rooms are large, the modal behaviour is reduced to the very low frequency range and comparisons with classical prediction formulas of

5.6 Validation example

129

0.55

100

0.5

10

0.45

1/(k(f)2 - k(f+∆f)2)

1 0.1 0.01

0.4 0.35 0.3 0.25 0.2 0.15 0.1

0.001

0.05

f = 50 Hz f = 100 Hz f = 500 Hz f = 1000 Hz

1e-04 1e-05 1

10

0 10

100

100 ∆f (Hz)

ξ=3 ξ=4 ξ=5

ξ=0 ξ=1 ξ=2

1000

(a)

1000 f (Hz) ξ=6 ξ=7

(b)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

00

00

50

50

40

00

31

00

25

00

20

50

16

00

12

0

10

0

80

0

63

0

50

5

40

0

31

0

25

0

20

5

16

0

12

10

80

63

50

.5

40

31

25

20

16

f (Hz) ξ = 1.0

ξ = 0.5

ξ = 0.1

(c)

Figure 5.12: Evolution of parameters controlling the modal contribution (line integrals): (a) resonance parameter; (b) shifting of waves that have the same wave number; (c) coincidence of waves with different wave number.

130


sound transmission are quite adequate (more details can be found in Fahy (1989) and Beranek and Vér (1992) as well as in Chapter 6). The critical frequency of the plasterboard is 2138 Hz. The numerical model cannot be used in this frequency range. Meaning Young’s modulus Poisson’s ratio Solid density Wall thickness Hysteretic damping coefficient

Symbol Value E 4.8 · 109 N/m2 ν 0.2 ρsolid 913 kg/m3 t 0.013 m η 0.5 %

Table 5.5: Geometric and material properties of the plate. Results are shown in Fig. 5.13. The pressure field in each room and the vibration field of the structure have been plotted in Fig. 5.14 for a frequency of excitation of 150 Hz. Four different numerical simulations have been included. Each of them has been generated with a different scale factor of the laboratory: ξ = 0.3, 0.5, 0.7, and 1 (actual size). The influence of modal behaviour is more important for higher frequencies if the laboratory size is small. 40 35

R (dB)

30 25 20 15 10 5 0

16

20

25

31

.5

40

50

63

80

10

0

12

5

16

0

20

0

25

0

31

5

40

0

50

0

63

0

f (Hz) Field incidence Diff. field (f > fc)

ξ = 0.3 ξ = 0.5

ξ = 0.7 ξ = 1.0

Figure 5.13: Sound reduction index of a single plasterboard. Different dimensions of the same problem (ξ is the scale factor).

The slope of the curve is similar to the slope of the field incidence mass law,


131

which predicts too low values of isolation for the lower frequencies. In this zone, the presented model provide higher values of sound reduction index. The low-frequency laboratory measurements presented in Fausti et al. (1999) are also over the mass-law predictions. The mass law supposes infinite structures and unbounded acoustic domains while the presented modal approach supposes finite rooms and bounded structures. Masslaw expressions provide the same result independently of boundary conditions and dimensions of the structure and rooms. Differences between theories assuming finite and infinite structures has already been found in Takahashi (1995) and Kernen and Hassan (2005). This can explain why numerical results are consistently over the mass law predictions.

5.7

Concluding remarks

A model for sound transmission problems has been presented. The vibroacoustic equations are solved by means of a hybrid approach. Finite elements are combined with truncated modal basis of cuboid acoustic domains. The main differences with existing similar models in the literature are the block organisation of the code in order to deal with coupling with the block Gauss-Seidel strategy presented in Chapter 4, and the extension of the model to situations different from a single or double wall. In Chapter 8 it is used for flanking transmission problems. The computational costs of the model have been analysed. Memory requirements can be smaller than the cost of a FEM analysis. It depends on the type of problem analysed and the adequate definition of the modal basis. The model is more efficient (in terms of storage costs) for heavy walls with large rooms. On the contrary for lightweight walls between small rooms the improvement with respect to FEM is small. The critical parameter for efficiency is to choose the adequate bandwidth in which the modal basis is defined. In any case the computation times are largely reduced since in the acoustic part of the problem only a diagonal system of equations has to be solved.

132


Figure 5.14: Sound reduction index of a single plasterboard. Sound pressure level in the rooms of the laboratory in dB and velocity field over the tested plasterboard 2 vrms (m2 /s2 ). Scale factor of the plot 0.5 and frequency, 150 Hz.

The obtained solutions have localised errors (around vibrating surfaces, absorbing surfaces or sound sources). They are mainly caused by the lack of normal velocity of the eigenfunctions at boundaries and the finite number of eigenfunctions considered in the modal basis. Their effect on the averaged outputs is not important. The modelling of the absorption done by means of the presented model has been compared with the predicted values obtained with simplified formulations. Differences are large for low frequencies but the (∆α) is not larger than 2 % above 300 Hz (for α = 10 − 20 %).


133

The importance of the bandwidth considered in the definition of the modal basis has also been analysed. This has been done by direct observation of the variation of the outputs (sound levels and sound reduction index) or by means of an analysis of several critical parameters. In both cases the conclusion is that with values of ∆f = 200 Hz, acceptable solutions are obtained. The predictions of sound reduction index of a plasterboard obtained by means of the modal-FEM model have been compared with the results obtained by means of the field incidence mass law. The slope of both R − f curves is almost the same. The values of R obtained by the model are slightly higher than the values provided by

the field incidence mass law. The approximation in the modelling of the absorption can cause a difference of 1 dB. The finite dimensions of the wall and the damping can explain the other differences. Note that the field incidence mass law provides the same results for walls having different dimensions and damping. More comparisons will be done in Chapter 6.

Chapter 6 Numerical modelling of sound transmission in single and double walls 6.1

Introduction

Predicting the acoustic isolation capacity of room partitions is of great interest in order to perform correct acoustic designs of buildings. The influence on the sound isolation of the wall properties (materials, dimensions, construction type) and environmental parameters (like room and wall dimensions or room absorption) has been studied here. The model presented in Chapter 5 has been used for the prediction of sound transmission through single and double walls. Two-dimensional and three-dimensional versions of the model have been considered. In Section 6.2 a literature review of basic formulations for the prediction of the sound reduction index of single and double walls is done. The situations analysed are described in Section 6.3. The results are divided in three parts. In the first one, in Section 6.4, the low-frequency response illustrates the type of output obtained from a numerical model and the effect of each parameter in the sound level difference. Afterwards, in Section 6.5, the influence of some aspects more related with the envi135

136

Numerical modelling of sound transmission in single and double walls

ronment of the wall like the size of rooms or the sound source position is discussed. In Section 6.6 attention is focused on some particular aspects related with double walls like the effect of absorbing material or the role of steel connections. The concluding remarks of Section 6.7 close the chapter.

6.2

Literature review

The airborne sound isolation of walls is quantified by means of the sound reduction index

1 Iin R = 10 log10 = 10 log10 τ Iout

(6.1)

which is a measure of the ratio between incoming (Iin ) and transmitted (Iout ) intensities. These intensities are caused by incoming and outgoing pressure waves, pin and pout . τ is the transmission coefficient. The sound reduction index is a parameter characterising only the isolated wall. Only three different waves are considered: incoming and reflected (sending side), and transmitted (receiving side). This ideal situation is only found in unbounded acoustic domains.

pin

p−

pout θ

pin

pout

p+

Figure 6.1: Sketch with the incoming (and reflected) pressure wave pin and the pressure wave generated by the radiation of the wall pout . The pressure fields in the receiving side p− and sending side p+ are generated by the combination of these waves.

As explained in Section 1.1, different techniques have been considered in the prediction of sound transmission. Among them, useful analytical solutions of the sound transmission through a single wall can be found in Fahy (1989), Josse (1975) or Be-

6.2 Literature review

137

ranek and Vér (1992). Thesa analytical expressions can be obtained because the unbounded situation is assumed (the wall and the rooms on each side are infinite). Otherwise it is very difficult to find analytical solutions of the sound transmission problem. The sound reduction index can be calculated by means of mass-law type expressions like R = 10 log10

2 ! Z cos θ 1 + 2ρ0 c

(6.2)

where ρ0 is the air density, c is the speed of sound in air, θ is the angle of incidence of acoustic waves (θ = 0 for normal incidence) and Zw is the wall impedance defined as Zw =

p− − p + vn

(6.3)

vn is the phasor of normal wall velocity. p+ is the complex amplitude (phasor) of the sound pressure wave on the receiving side. It is fully caused by the radiation of the wall that generates an outgoing wave pout . p− = 2pin −pout is the complex amplitude of the pressure wave on the source-side of the wall. This is caused by blocked pressure field

and the pressure wave radiated by the wall into the sending domain. The blocked pressure field is composed by the incoming wave (pin ) and its reflected wave on a perfectly rigid boundary, see Fig. 6.1. Zw usually deals with mass, bending and shear effects. The mass law is obtained when the impedance only includes mass effects, Zw = iρsurf ω (ρsurf is the surface density of the wall). Attenuation in the structure can be introduced by means of hysteretic damping, where the Young’s modulus of the material is E ∗ = E (1 + ηi). η is the damping coefficient. More details on the modelling of damping can be found in Nashif et al. (1985). An expression of the transmission coefficient of a single wall including mass and stiffness effects is (see for example Beranek and Vér (1992)) 2 2 2 ω 2 B sin4 θ ω B sin4 θ ωρsurf cos θ ωρsurf cos θ 1 1− 4 = 1+η + τ (θ) 2ρ0 c c4 ρsurf 2ρ0 c c ρsurf (6.4) where ω = 2πf is the pulsation of the problem and B the bending stiffness per unit

138


width. Eq. (6.4) has to be averaged. This technique was used in London (1950) and Mulholland et al. (1967) and reproduces the effect of a reverberant pressure field. It is done by considering multiple incoming waves from all the directions (in the wall plane) and having incidence angles θ between 0 (normal incident waves) and θlim (oblique incidence). θlim is a value varying from 70◦ to 85◦ . This angle is measured in planes that are orthogonal to the wall plane. The following equation is obtained after the double average (more details can be found in Pierce (1981)) τav =

R θlim 0

τ (θ) cos θ sin θdθ R θlim cos θ sin θ dθ 0

(6.5)

The theoretical sound reduction index curve for a single wall can be split in three zones, see Fig. 6.2(a): low frequencies, coincidence frequency and high frequencies. The smallest values of sound reduction index are found at low frequencies. The response can be modal dependent or mass controlled. In the modal zone, there are large variations of sound reduction index caused by the resonances. The values can even be negative for certain frequencies. Between the modal zone and the coincidence frequency, the increase of isolation is around 6 dB per octave. The pressure waves make the vibration shape of the wall to be similar to the pressure field in the surroundings. This causes forced sound transmission. Sound reduction index can then be predicted by the random incidence mass law Rrandom = R (θ = 0) − 10 log10 (0.23R (θ = 0))

(6.6)

R (θ = 0) is the normal incidence mass law, obtained by considering only mass effects and orthogonal incoming pressure waves in Eq. (6.2). Eq. (6.6) is the result of the average proposed in Eq. (6.5) with θlim = π/2 using an expression of τ that takes only into account the mass effects. Discrepancies with laboratory measurements are found and a new expression is proposed, the field incidence mass law Rf ield = R (θ = 0) − 5

(6.7)

6.2 Literature review

139

which can be obtained in the same way as Eq. (6.6) but using θlim = 4π/9. An important drop of the sound reduction index is found around the coincidence frequency. It is caused by the geometrical matching between the shape of pressure and bending waves. For frequencies below the coincidence, the length of bending waves is smaller than the length of pressure waves. Expressions (6.6) and (6.7) cannot predict the dip because stiffness effects are not considered. For the case of a beam (single wall of a two-dimensional problem) the coincidence frequency is c2 fc = 2π

r

ρsolid A EI

(6.8)

where I is the inertia, A is the cross section area and ρsolid is the volumetric density of the wall. If unbounded domains are considered, an exact coincidence between waves can always be found. However, if models considering bounded domains are used (i.e. the model presented in Chapter 5), it depends on the geometry of the problem and the boundary conditions. In those cases, perfect coincidence is rarely found. Around the critical frequency the response is controlled by the damping of the structure. For frequencies over the critical frequency the sound reduction index can be predicted with the following formula proposed in Josse (1975) R = 10 log10

1+

ωρsurf 2ρ0 c

2 !

+ 10 log10

f fc

1 − 10 log10 −3 η

(dB) (6.9)

Similar expressions are found in Fahy (1989) and Schmitz et al. (1999). For high frequencies the stiffness becomes the important parameter and the slope of the curve increases. It can be around 18 dB per octave. Numerical results presented in this chapter have been compared with some of these analytical expressions. The notation used in the plots is ‘Field incidence’ for the case of Eq. (6.7) and ‘Diff. field (f > fc )’ for the case where R is obtained using Eq. (6.9). Similar models have been developed for the case of double walls. In Hongisto (2006), seventeen models have been analysed. They cover a complete range of situations: double walls with and without absorbing material inside the cavity, with


lle

ss

Modal Ma

ro nt

tave

R (dB)

R (dB)

/ dB

co

Coincidence

18 dB /Oc

ta

Oc

6 d(

) ve

Modal

140

12

led rol nt ve) co cta s O s Ma dB/ (6

e

av

ct

/O

dB

Resonances Coincidence Mass−Air−Mass

fc

(a)

log 10(f)

f m−a−m

f c log (f) 10

(b)

Figure 6.2: Expected evolution of the sound reduction index with frequency for: (a) single walls; (b) double walls.

and without mechanical connections between leaves and considering or not averaged incoming pressure waves. Only models that provide a set of analytical expressions in order to evaluate the sound reduction index have been considered. Numerical-based models or SEA models are not covered in the study. The conclusions of the paper are not hopeful at all since differences larger than 20 dB between comparable models and between models and experimental data have been found. Some other models of double walls not appearing in the study are also interesting. In Kropp and Rebillard (1999) and Wang et al. (2005) the possibility of considering mechanical devices connecting the two leaves of a double wall has been considered. In Brunskog (2005), the influence of the mechanical path (stud) and the cavity path in the sound reduction index of double walls have been studied. The dynamic equations have been simplified due to the periodicity of structures. Afterwards they have been solved by means of wave approach. The cavities between leaves have been described by means of a modal basis expansion. The expected evolution of sound reduction index with frequency has been plotted in Fig. 6.2(b), according to the model for double walls proposed in Fahy (1989). In spite of the discrepancies found in the analytical models for double walls, most of

6.3 Description of the problem analysed

141

them define the mass-air-mass resonance as

fm−a−m

v u ρ 0 c2 1u t = 2π d

(1)

(2)

ρsolid t(1) + ρsolid t(2) (1)

(2)

ρsolid t(1) ρsolid t(2)

!

(6.10)

(1)

(2)

where d is the separation between two leaves with volumetric densities ρsolid , ρsolid and thicknesses t(1) , t(2) . This resonance is the eigenfrequency of a system composed by two masses (the two leaves of the double wall) separated by an air gap of length d that acts as spring. For frequencies below the mass-air-mass resonance, the isolation of the double wall is equivalent to the isolation provided by a single wall with the (1)

(2)

same mass, R(ρsolid t(1) + ρsolid t(2) ). For higher frequencies the isolation of a double (1)

(2)

wall with an empty cavity can reach R(ρsolid t(1) ) + R(ρsolid t(2) ) + 6 dB. This optimum behaviour is altered by two resonance phenomena. On the one hand, the resonances of the air in the cavity. Analytical formulations only take into account resonances in the shortest direction of the cavity (thickness d). They are found at high frequencies and can be attenuated if absorbing material is placed between leaves. On the other hand the coincidence frequencies of each single leave. This effect is more important if both leaves are equal.

6.3

Description of the problem analysed

The model problem considered here can be seen in Fig. 5.10. Sound is transmitted through a single or double wall of finite dimensions. The wall separates two rooms. Rooms 1 and 2 are the sending and receiving rooms respectively. Results presented in Section 6.4 have been obtained using only finite elements in order to solve vibroacoustic equations. The results in other sections have been obtained with the combined modal-FEM model presented in Chapter 5. A punctual sound source is placed in the sending room. Its source-strength amplitude is Q = 4.2 · 10−2 m3 /s (see Kinsler et al. (1990) for more details). Since the

problem is linear, this value is only important in order to guarantee a minimum value

of sound pressure level in the sending room. For the examples presented here, it is

142


always around 100 dB. The partition can be a single or a double wall. In the case of a double wall the cavity between leaves is often full with absorbing material. The fluid equivalent model proposed by Delany and Bazley (1970) is used. Three different wall types have been considered: heavy single walls, lightweight single walls and lightweight double walls, see Table 6.1. The heavy single wall is made of concrete and the mechanical properties of the other structural elements are typical of gypsum plasterboards. The dimensions for the sending and receiving rooms in the (1)

(1)

(2)

(2)

(2)

(2)

single wall two-dimensional examples are lx = 4 m, ly = 3 m, lx = 3.5 m, ly = 2.8 (1)

(1)

m; and for the double wall examples lx = 6.35 m, ly = 4 m, lx = 5.7 m, ly = 3.7 m. In the three-dimensional problems analysed, the notation for the third dimension will be lz .

Meaning Thickness

Symbol t

Young’s mod. Density Damping Length Critical freq.

E ρsolid η ` fc

Single walls Heavy Lightweight 0.05 m 0.02 m (2D) / 0.013 m (3D) 10 2 2.943 · 10 N/m 4.5 · 109 N/m2 3 2500 kg/m 900 kg/m3 2% 2% 2.5 m 2.5 m 371.5 Hz 1425.1 Hz

Double walls Leave 1 Leave 2 0.013 m 0.009 m 4.8 · 109 N/m2 913 kg/m3 2% 3m 2138.16 Hz

3.8 · 109 N/m2 806 kg/m3 2% 3m 3261.38 Hz

Table 6.1: Geometrical and mechanical properties of the two single walls and the leaves of the double wall used in the examples. All the boundaries of the acoustic domains that are not in contact with the structure are considered absorbing boundaries. The acoustic absorption of rooms is related to the impedance Z of these boundaries (see for example Bell and Bell (1993))

αθ =

Iabs−θ Iinc−θ

2 Z − 1 ρ0 c cos θ = 1 − Z + 1 ρ c cos θ 0

The values of absorption used here are can be found in Table 6.2.

(6.11)

6.4 Low-frequency response

Z ρ0 c α

3

143

4

6

10

20

40

70

0.79 0.67 0.51 0.34 0.18 0.1 0.06

Table 6.2: Values of normalised impedance and averaged absorption for the Robin boundary condition.

6.4

Low-frequency response

Discussion in this section is focused on the low-frequency response. All the results presented here have been obtained using only finite elements. They are presented without averaging in frequency bands in order to preserve the detail. The study of the low-frequency response is useful in order to understand the phenomena of sound transmission in models considering bounded domains. A wave-based interpretation does not lead to a correct understanding in this context (mainly due to the modal behaviour). Low-frequency results also provide useful information in order to decide the value of some parameters required by the numerical model (i.e. the number of situations per Hz to be calculated). SEA, wave approaches, or phenomenological models cannot be used to perform this kind of analysis. Two different situations have been considered: sound transmission through a single heavy wall and sound transmission through a double lightweight wall. The heavy (rather than the lightweight) single wall is chosen in order to have a lower modal density and to be able to differentiate the vibration modes. In Fig. 6.3 the sound level difference between two rooms separated by the single heavy wall has been plotted. Each curve has been obtained with a different value of acoustic absorption in the rooms (see Table 6.2). The hysteretic damping of the wall has been kept constant (η = 2 %). The eigenfrequencies of the undamped vibroacoustic problem are also shown. It is not necessary here to consider the damped eigenproblem that leads to a quadratic eigenvalue problem (Tisseur and Meerbergen (2001)). This hypothesis is based in the results shown in Chapter 3, where the real part of the eigenfrequencies of the damped and undamped eigenvalue analysis were very similar. Only qualitative information is required here in order to understand the

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D (dB)

sound level difference curve. 60 55 50 45 40 35 30 25 20 15 10 5 50

100

150

200

250

300

f (Hz) α=0% Z/(ρac) = 20, α = 18 %

Z/(ρac) = 10, α = 34 % Z/(ρac) = 4, α = 67 %

Eig. structure (bending) Eig. sending room

Eig. receiving room

Figure 6.3: Heavy single wall. Sound level difference for the low-frequency range showing the modal behaviour.

The response of the system is modal-dependent. The system is weakly coupled, and the value of the coupled eigenfrequencies is very similar to the values of the in vacuo eigenfrequencies of each part: sending room, receiving room and wall. In Table 6.3, the lower eigenfrequencies (except 0 Hz) of each part have been compared with the eigenfrequencies obtained from the coupled analysis. This weak coupling is often found in sound transmission problems and justifies simplifying assumptions. The coupled and uncoupled eigenfrequencies are similar. However, the associated eigenfunctions have different support. The modal shapes associated with an eigenfrequency of the sending room have high values of pressure in this room but also in the receiving room, as well as important displacements in the wall. Similar conclusions have been obtained in Chapter 3 with the one-dimensional model. Dips in the sound level difference are found around eigenfrequencies corresponding to resonances in the receiving room or modes of vibration of the structure. The sound level difference around these frequencies can even be negative as predicted in Mulholland and Lyon (1973). Acoustic absorption attenuates the first type of dips while the others are controlled by structural damping. The low values of D caused by structural resonance remain unchanged here because the damping in the structure is constant along this analysis. On the contrary, the dips caused by acoustic resonances

6.4 Low-frequency response

Wall uncoupled coupled 12.45 12.73 49.82 49.28 112.18 111.23

145

Sending room uncoupled coupled 42.5 42.64 56.67 56.83 70.83 71.03

Receiving room uncoupled coupled 48.57 48.73 60.71 60.89 77.75 77.99

Table 6.3: Heavy single wall. Comparison between the eigenfrequencies of the coupled vibroacoustic problem and the eigenfrequencies of the isolated parts of the system. are attenuated by acoustic absorption. Acoustic absorption and structural damping are necessary in order to perform an accurate physical modelling. Moreover, they are positive from a numerical point of view since they can improve the efficiency of numerical solvers (see Chapter 4 and Poblet-Puig and Rodr´ıguez-Ferran (2008)) and attenuate the oscillations of the sound reduction index curve. This last aspect is very important in order to decide the number of calculations to be done in every frequency band. The oscillations are found between modes. It is necessary to perform some calculations between these modes in order to accurately reproduce the sound level difference curve. The modal density grows with frequency (for the present case, two modes per Hz are expected at 500 Hz, eight at 2000 Hz, and twelve at 3000 Hz), so the number of calculations to be done should also grows with frequency. This is an important drawback because the cost of solving a single calculation also increases with frequency, as shown in Chapter 5. However, looking at Fig. 6.4 we can affirm that this restrictive reasoning is only completely true for undamped situations. Sound level difference curves become smoother if attenuation is considered in the model. The number of calculations per Hz can then be decreased. In the examples presented here, one frequency per Hz has been analysed for frequencies below 500 Hz and 1.5 for frequencies over 500 Hz. A similar analysis has been done for the case of a lightweight double wall without absorbing material in the cavity (0.175 m thick). The sound level difference curve and the eigenfrequencies of the system are presented in Fig. 6.4. All the comments on modal analysis as well as on acoustic absorption and structural damping done for the

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30

D (dB)

20 10 0 -10 -20 -30 20

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f (Hz) α=0% Z/(ρac) = 20, α = 18 % Z/(ρac) = 6, α = 51 %

Eig. leave 1 Eig. leave 2 Eig. sending room

Eig. receiving room Eig. cavity Coupled eig.

Figure 6.4: Lightweight double wall. Sound level difference for the low-frequency range showing the modal behaviour.

case of the single wall are still valid. Several phenomena increase the complexity of the problem. On the one hand, new modes of vibration appear. The structural modes can be caused by vibrations in both leaves and the air cavity between them can also resonate. On the other hand, the system is not weakly coupled and new mode types are found. These are mass-air-mass type resonances where the two leaves and the air cavity inside have an important interaction. They cannot be related to any in vacuo mode. The first of them is found at 83.70 Hz by means of a vibroacoustic eigenvalue problem. The two leaves vibrate in their first mode. The value of the mass-air-mass frequency predicted by Eq. (6.10) (66.22 Hz for this double wall) is obtained with a numerical-based model if the eigenvalue analysis of a double wall with free leaves is performed. In Table 6.4 low frequency eigenvalues obtained in three different ways are shown: i) complete eigenvalue analysis where the sending room, the receiving room and the double wall with air cavity are considered at the same time (‘cou.’ in Table 6.4) ; ii) uncoupled eigenfrequencies (‘unc.’ in Table 6.4); iii) analysis of only the double wall (two leaves and air cavity) without considering the sending and receiving rooms (‘DW’ in Table 6.4). The agreement between the three analyses is quite correct except for the first modes and the very coupled resonances. It can be seen that the modal behaviour

6.5 Acoustic isolation of single walls

147

of the double wall (even the very coupled eigenfrequencies) is correctly obtained in case iii) where rooms are not considered. The first coupled mode is almost the same in both modal analyses: 83.70 Hz for the complete case i) and 83.75 Hz for the isolated situation iii). The modal density is high even for low frequencies due to room dimensions and leave characteristics. The coupled modes are between a lot of another modes. cou. 2.9 5.63 12.64 22.92

Leave 1 unc. 1.5 6.01 13.52 24.53

DW 1.32 5.53 12.79 23.22

cou. 5.34 7.73 14.52 23.2

Leave 2 unc. DW 3.94 2.99 8.86 7.8 15.75 14.77 24.62 23.51

Send. Room cou. unc. 27.4 26.77 42.68 42.5 50.41 50.23 52.68 53.54

Recei. cou. 30.37 46.22 55.27 61.37

Room unc. 29.82 45.95 54.78 59.65

cou. 57.89 110.51

Cavity unc. 56.67 113.33

DW 61.6 117.47

Table 6.4: Comparison between the eigenfrequencies obtained by means of a vibroacoustic eigenvalue problem (cou.), the eigenfrequencies of the isolated parts of the system (unc.) and the eigenfrequencies of the double wall (DW).

6.5

Acoustic isolation of single walls

The increase of modal density makes the analysis done in Section 6.4 impossible for higher frequencies. In order to obtain useful engineering outputs, the numerical results must be post-processed by means of space and frequency averages. From now on the results will be presented averaged in third frequency bands. This is a clearer way to understand the behaviour of a vibroacoustic system and obtain useful conclusions. The discussion is focused here on environmental aspects, which are independent of the wall type analysed. Thus, simple walls have been used in order to avoid additional and unnecessary complications that could mask the relevant discussion of each section.

6.5.1

Influence of the absorption correction on R

The results in Section 6.4 have been presented in terms of the sound level difference D. It is a global measure that not only contains information about the studied wall but also about the test environment. D is the direct output obtained from a pressure

148


field measurement or from the models used here, but its value highly depends on the acoustic absorption. On the contrary, the sound reduction index R defined in Eq. (6.1) is a parameter that only depends on the wall. Two main techniques are used in order to measure R in the field or the laboratory (see Hongisto (2000) for more details and references). On the one hand, intensity measurements which are laborious but provide interesting information on the spatial distribution of sound transmission (i.e. leakages or sound bridges can be detected). The required intensities cannot be easily obtained from the numerical results (it is difficult to split the pressure field into incoming, reflected and radiated fields). On the other hand, the more often used pressure measurements. The sound reduction index is obtained indirectly by means of R = L1 − L2 +10 log10 | {z } D

S SRobin α

(6.12)

where L1 and L2 are the mean sound pressure levels in the sending and receiving rooms, S is the surface of the wall, SRobin the absorbing surface (the Robin boundary in the numerical model) and α the averaged absorption coefficient. Eq. (6.12) is valid for steady harmonic states where the acoustic power radiated by the wall is in dynamic equilibrium with the absorption of acoustic energy by room contours (it is the case of the used model). It has to be assumed that the incoming acoustic intensity on the wall can be expressed as I ∝< p2rms > /ρ0 c. prms is the root mean square pressure, supposing that the acoustic field is reverberant. The assumption of reverberant field in the numerical model is quite reasonable due to the frequency average. In this section, the correction proposed in Eq. (6.12) in order to transform D into R is used for the numerical results. D can be obtained post-processing the pressure fields. The absorption α can be obtained by means of Eq. (6.11) using the impedances Z of Robin boundaries. The expected result is that several sound level difference curves of the same wall obtained with different values of acoustic absorption provide the same sound reduction index curve. The equivalent situation in a laboratory would be to modify the absorption of the receiving room (the reverberation time) and compare


149

the sound reduction index measurements in each case. Both two-dimensional (heavy single wall) and three-dimensional (heavy and lightweight single walls) examples are shown. The dimensions for the two-dimensional rooms are (1)

those listed in Section 6.3. For the three-dimensional rooms they are: lx = 2.8 m, (1)

(1)

(2)

(2)

(2)

ly = 4 m, lz = 3 m, lx = 3.3 m, ly = 4 m, lz = 3 m (heavy single wall); and (1)

(1)

(1)

(2)

(2)

(2)

lx = 2.8 m, ly = 3 m, lz = 2.5 m, lx = 3.2 m, ly = 3 m, lz = 2.5 m (lightweight wall). The thickness of the lightweight wall for the three-dimensional example is 0.013 m. In Fig. 6.5 the sound level difference D and the sound reduction index R for the case of the two-dimensional heavy single wall have been plotted. The three-dimensional results can be found in Fig. 6.6 (for the heavy wall) and in Fig. 6.7 (for the 0.013 m thick lightweight wall). In the two-dimensional analysis the differences between sound level difference curves are large (around 15 dB). After the transformation to sound reduction index by means of Eq. (6.12), the differences are reduced to 5 dB. At very low frequencies and around coincidence frequency, the differences are larger. Similar comments are valid for the three-dimensional examples. Differences in the sound level difference curves are larger than for the two-dimensional case (they are around 25 dB). After the calculation of R they are not larger than 5 dB. This is not valid for cases with very large values of absorption (α > 50 %). The absorption correction provided by Eq. (6.12) is quite satisfactory. Differences between curves corresponding to values of absorption not larger than 50 % are smaller than 5 dB. This is not true for very low frequencies where it would be better not to use the correction and around the coincidence frequencies where differences in the R curves are larger than in the other zones. The same value of absorption α can be obtained with different values of impedance Z. It has been checked that using these different values of impedance, equivalent sound level difference curves are obtained. Thereby, sound reduction index curves are also equal. The values of R predicted by the numerical models are slightly higher than those

150


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60

50

50 R (dB)

70

D (dB)

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30

30

20

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1250 1000 800 630 500 400 315 250 200 160

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2000 1600

1250 1000 800 630 500 400 315 250 200 160

125 100 80 63 50 40 31.5 25 20

f (Hz)

Field incidence, θlim = 4π/9 Diff. field (f > fc)

f (Hz) Z/(ρac) = 3, α = 79 % Z/(ρac) = 4, α = 67 %

(a)

Z/(ρac) = 10, α = 34 % Z/(ρac) = 70, α = 6 %

(b)

Figure 6.5: Two-dimensional analysis of a heavy single wall: (a) sound level difference D; (b) sound reduction index R.

predicted by mass-law type expressions. The agreement is better with the field incidence mass-law, Eq. (6.7), than for the random incidence mass-law, Eq. (6.6). These differences occur because the compared models are very different (i.e. finite vs. infinite). Differences are especially important for low frequencies where the mass law cannot correctly reproduce the modal behaviour and underestimates the sound reduction index of the walls. The modal zone depends on the room dimensions but also on the wall type. While for the three-dimensional heavy single wall case it is found for frequencies under 450 Hz, for the case of lightweight single wall, modal behaviour is only found under 80 Hz. R is also underestimated in the coincidence frequency of the heavy single walls. As justified in Section 6.2, pure coincidence will never be found in that case due to the geometry and boundary conditions of the model. In the following examples of the chapter, an acoustic absorption α = 18 % has


151

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60

55

55

50

50 R (dB)

D (dB)

been considered.

45 40

45 40

35

35

30

30

25

25

20

20 1250 1000 800 630 500 400 315 250 200 160 125 100 80 63 50 40 31.5 25

1250 1000 800 630 500 400 315 250 200 160 125 100 80 63 50 40 31.5 25

f (Hz)

f (Hz)

Field incidence Diff. field (f > fc)

(a)

α=5% α = 10 %

α = 30 % α = 50 %

(b)

Figure 6.6: Three-dimensional analysis of a heavy single wall. Influence of acoustic absorption: (a) sound level difference, D; (b) sound reduction index, R.

6.5.2

Comparison between two-dimensional and three-dimensional models

In Section 6.5.1, results obtained with a two-dimensional or a three-dimensional version of the model have been shown. The frequency range analysed with the twodimensional model is larger since the computational costs are smaller. The sound reduction index of the heavy wall of Section 6.5.1 has been calculated now with the three-dimensional version of the model. Different values of the third dimension lz have been considered. The comparison has been plotted in Fig. 6.8. Differences are not large between the two-dimensional and three-dimensional models. Two-dimensional predictions of R are slightly over the three-dimensional results. The variations of R due to the increase of lz are not important.


45

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R (dB)

D (dB)

152

25

25

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15

15

10

10

5

5 630 500 400 315 250 200 160

125 100 80 63 50 40 31.5 25

630 500 400 315 250 200 160

125 100 80 63 50 40 31.5 25

f (Hz)

f (Hz) α=5% α = 10 %

Field incidence Diff. field (f > fc)

α = 30 % α = 50 %

(a)

(b)

Figure 6.7: Three-dimensional analysis of a lightweight single wall. Influence of acoustic absorption: (a) sound level difference, D; (b) Sound reduction index, R.

65 60

Field incidence Diff. field (f > fc) 2D lz = 2 m (3D) lz = 4 m (3D)

55

R (dB)

50 45 40 35 30 25

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800

630

500

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315

250

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125

100

80

63

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31.5

25

20

16

f (Hz)

Figure 6.8: Sound reduction index of the heavy single wall. Comparison between the two-dimensional model and three-dimensional solutions with different problem width lz .


153

This analysis has been done in a problem where all the elements of the threedimensional model (geometry, boundary conditions, properties of the structure, the excitation on the structure caused by a diffuse pressure field,...) can be obtained by extrusion of the two-dimensional model. In general, three-dimensional problems cannot be easily reduced to a two-dimensional situation (i.e. impact sound where the excitation is a punctual force on a floor). On the one hand, modes can exist in the third dimension. On the other hand, the modal density evolution with frequency of a two-dimensional structure (shell in the three-dimensional problem) and a one-dimensional structure (beam for the two-dimensional problem) is different. This difference comes from the nature of the governing equations. For the former case the modal density increases with frequency and for the latter it decreases. In any case, for the presented situation, a two-dimensional model provides a good approximation to the three-dimensional case. This means that two-dimensional models can provide more than a qualitative description of a problem. In the present case a first approximation to the numerical result is obtained. An important part of the models and formulations for sound transmission found in the literature are two-dimensional and have been used in three-dimensional problems.

6.5.3

Influence of room size

The room size can be a source of discrepancies between measurements of sound reduction index obtained in different laboratories and in field measurements. Here nine different situations combining three different types of rooms have been analysed. The

room

Sending

dimensions of the rooms and their combination can be found in Table 6.5.

Small Medium Big

Receiving room Room size Small Medium Big `x (m) `y (m) 1 2 3 3 2.8 4 5 6 5 4 7 8 9 7 6

Table 6.5: Rooms employed in every case (cases 1 to 9) and room dimensions.

154


Some of the results are presented in Fig. 6.9. It is possible to see that differences due to the room size are only important for low frequencies but not for mid and high frequencies (especially for frequencies over the critical frequency of the wall). The sound reduction index is more dependent on the size of the receiving room than on the size of the sending room. As discussed in Sections 3.2.3 and 6.4 the dips in sound reduction index are more influenced by the modes due to resonances in the receiving

65

65

60

60

55

55

50

50

45

45

R (dB)

R (dB)

room than modes due to resonances in the sending room.

40

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25

25

20

20 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100 80 63 50 40 31.5 25 20

2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100 80 63 50 40 31.5 25 20

f (Hz)

f (Hz)

Field incidence Diff. field (f > fc) Case 1

Case 4 Case 7

Field incidence Diff. field (f > fc) Case 3

Case 6 Case 9

Figure 6.9: Heavy single wall. Influence of room size in the sound reduction index: (a) small receiving room; (b) large.

6.5.4

Influence of sound source position

The influence of sound source position has been also studied. Results are presented in Fig. 6.10. The source has been placed in two corners of the room (positions 1 and 4), in the centre (position 2) and just in front of the wall (position 3). The differences in sound reduction index are important. The largest differences are found in the low-frequency range. However, above the critical frequency they are


155

also around 5 dB. It justifies the usual practise done in laboratory measurements of sound reduction index. The final result is the average between the measurements obtained for several positions of the sound source in the sending room. It is more representative of a real situation where the sound source (television, music device, industry machinery) can be placed everywhere. 70 60 R (dB)

50 40

4m

30 20

Position 2 Position 3 Position 4

2000

Field incidence Diff. field (f > fc) Position 1

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63

50

40

31.5

25

20

f (Hz)

3m

10

2 1

3

4

Figure 6.10: Heavy single wall. Influence of the sound source position.

6.5.5

Influence of window size

Sometimes the dimensions of the tested element are smaller than the laboratory wall sizes, see for example Fig. 6.11. There also are laboratories that have a window in a wall that separates two rooms. The tested element has to be placed there. In this section, the effect of this window size on the sound reduction index has been checked. The results presented here have been obtained with a three-dimensional model. (1)

The dimensions of the rooms for the heavy and lightweight cases are lx = 2.8 m, (1)

(1)

(2)

(2)

(2)

ly = 4 m, lz = 3 m, lx = 3.3 m, ly = 4 m, lz = 3 m. The dimensions of the wall (1)

(1)

are βly m × βlz m. The same problem has been solved by changing the dimension

of the tested wall (window): β = 0.2, 0.4, 0.5, 0.6 and 0.8. Results are presented in Fig. 6.12.

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(1)

lx

(1) ly

(2)

lx

(1)

lz

(2)

β lz

(2)

lz

(2)

β ly

(2)

ly

(a)

(b)

Figure 6.11: Influence of window size: (a) in the Acoustical and Mechanical Engineering Laboratory (UPC), the tested wall is placed in a window between two rooms. This is a double wall where the plasterboard has been removed and the absorbing material can be seen; (b) sketch and notation for the numerical model.

50 80

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40 35 R (dB)

R (dB)

60 50 40

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f (Hz) Field incidence Diff. field (f > fc) β = 0.2

f (Hz) β = 0.5 β = 0.8

Field incidence Diff. field (f > fc) β = 0.2

β = 0.5 β = 0.8

Figure 6.12: Influence of the window size in the sound reduction index. Threedimensional calculations: (a) heavy single wall; (b) lightweight single wall.


157

The size of the tested wall has several effects in the sound reduction index curve. For small walls the frequency range where the behaviour is modal is larger. There exists a frequency under which the sound reduction index is very high. This is caused by the poor radiation capacity of a wall under its first eigenfrequency. This frequency is higher for smaller walls. Finally, in the high frequency zone the size of the wall can modify the sound reduction index around ±5 dB (better isolation for small walls). It can be seen for the case of the plasterboard for frequencies over 250 Hz. It has also

been found in Kernen and Hassan (2005) using analytical formulations that take into account the finite size of walls. A similar analysis has been carried out with the two-dimensional model. The influence of wall length on the isolation is shown in Fig. 6.14(a). The sound reduction index for three different heavy walls (2.5 m, 1.5 m and 0.5 m in length) has been plotted. The first eigenfrequencies are 12.45 Hz, 34.59 Hz and 312.45 Hz respectively. It can be seen that the sound reduction index is high below this frequency because it is difficult for the wall to radiate sound into the receiving room. It is a clear example that shows how the low and high frequency concepts depend a lot on the geometry. For the last wall, 1000 Hz is still in the low-frequency range.

6.5.6

Influence of the mechanical properties and boundary conditions of the walls

For models considering bounded domains, the boundary conditions of the tested wall are required. Their effect on sound reduction index and also the importance of the structural damping will be discussed in this section. In Fig. 6.13 the effect of boundary conditions for the cases of a heavy and a lightweight single wall can be seen. Several combinations have been considered (supported, clamped and free). For both types of wall, a low-frequency zone where boundary conditions are important can be distinguished. For the case of the heavy wall, the limit frequency is around 400 Hz while for the case of the lightweight wall it is around 50 Hz. In this limit frequency the vibration wave length is equal to the length of the structure divided by 2.5.

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40

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R (dB)

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25 20

f (Hz)

10

f (Hz) Field Incidence Diff. Field (f > fc) s-s s-c

c-c c-f f-f

(a)

Field Incidence Diff. Field (f > fc) s-c

c-c c-f f-f

(b)

Figure 6.13: Influence of the boundary conditions (s = simply supported, c = clamped, f = free) of the wall in the sound reduction index: (a) heavy single wall; (b) lightweight single wall.

The effect of structural damping is shown in Fig. 6.14(b). It has no importance in the low-frequency range where the response is controlled by room modes. However, around and above the critical frequency it can improve the acoustic isolation and attenuate the coincidence effect.

6.6

Acoustic isolation of double walls

Double walls are a common type of construction, widely used in practice. Quite good acoustic performance can be reached by means of a reasonable low use of material. The vibration mechanisms are very different from those of a single wall. As described in Section 6.2, the interaction between leaves and the cavity inside is very important. From a numerical point of view, there also are differences between vibroacoustic problems dealing only with single walls and those dealing with double walls. The formulation of the problem remains unchanged but some procedures have to be modified due to the strong coupling of the cavity (i.e. the solver presented in Chapter 4 and

6.6 Acoustic isolation of double walls

159

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25 20

f (Hz) Field incidence Diff. field (f > fc) l = 0.5 m

f (Hz) l = 1.5 m l = 2.5 m

(a)

η = 10% η = 20%

Field incidence Diff. field (f > fc) η = 1%

(b)

Figure 6.14: Heavy single wall. Influence of wall properties on the sound reduction index: (a) length; (b) damping.

the selective coupling strategy). The discussion is focused here on two particular aspects of double walls: the effect of the type of absorbing material placed inside and the influence of mechanical connections in the sound reduction index. The other parameters of the problem are kept constant as described in Section 6.3. The effect of these two parameters has also been studied in Novak (1992) and Hongisto et al. (2002) where laboratory measurements can be found.

6.6.1

Influence of the separation between leaves and the type of absorbing material

A double wall is rarely used without absorbing material placed in the cavity. The effect of the quality of the absorbing material and the importance of the cavity thickness have been studied by means of two-dimensional numerical simulations. The Delany and Bazley (1970) model has been used in order to take into account the absorbing material as an equivalent fluid. Thus, it is characterised by the resistivity. The

160


equations corresponding to this part of the model as well as the two leaves have been solved by means of finite elements. If no absorbing material is placed in the cavity, finite elements have also been used due to its reduced dimensions. On the contrary, the rooms have been modelled by means of the modal approach. Results obtained have been plotted in Fig. 6.15. The double wall described in Section 6.3 has been analysed for two different cavity thicknesses (0.07 m and 0.175 m) and also for different resistivity values of the absorbing material (% = 104 , 3·104 and 5·104 Ns/m4 ).

100

100 air

90

air 4

90

σ = 10000 N•s/m

4

σ = 30000 N•s/m

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σ = 50000 N•s/m4

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(a)

σ = 30000 N•s/m4

50

40

f (Hz)

4

σ = 10000 N•s/m

f (Hz)

(b)

Figure 6.15: Influence of the type of absorbing material on the sound reduction index of double walls, for two wall thicknesses: (a) 0.07 m; (b) 0.175 m.

The effect of the absorbing material is important at very low and at high frequencies. Around the mass-air-mass frequency, the sound reduction index is considerably increased when absorbing material is placed inside the cavity. The quality of the material is also important. Between this resonance and the coincidence frequency of the leaves the increase of R is smaller. Finally, the effect of absorbing material is very important for high frequencies and especially around the coincidence frequency. Increasing the cavity thickness improves the isolation capacity of the double wall for all the frequencies. Moreover, it also emphasises the effect of the quality of the

6.6 Acoustic isolation of double walls

161

absorbing material. The differences between R curves corresponding to different values of resistivity are more relevant for double walls with a larger separation between leaves. The general tendency and the influence of each parameter found in Fig. 6.15 is similar to the prediction done by the one-dimensional model (see the example in Section 3.4.3). This confirms that even if the one-dimensional model cannot be used in order to perform very accurate predictions of the sound reduction index, it can give a first approximation of the influence of each parameter.

6.6.2

Effect of mechanical connections between leaves

Mechanical connections between leaves have to be often used in order to provide a minimum structural stability to the double wall. However, they create a new path of sound transmission. This new path can be modelled by means of translational or rotational springs that establish a link between points in both leaves. Numerical-based models can deal with these springs with minor modifications in their implementation (i.e. modification of the stiffness matrix or use of Lagrange multipliers). A double wall with a 0.175 m thick cavity full of absorbing material (% = 8000 N/ (s · m4 )) has been used in order to illustrate the effect of mechanical connections

(i.e. springs) in the sound reduction index. This is an adequate example because the isolation of the wall without mechanical connections (with only cavity path) is high (see results in Section 6.6.1). The springs have been placed each 0.6 m (four springs per 3 m length double wall). Four different values of translational stiffness and five of rotational stiffness have been considered: Kt = 105 , 106 , 107 and 108 N/m2 ; Kθ = 10, 103 , 104 , 105 and 106 N· m/rad · m. These ranges of variation are very related with the type of wall considered

(i.e. lightweight or heavy) and the isolation capacity of the cavity path. Thresholds of stiffness over and under which the isolation capacity is not modified can be found. Results are shown in Fig. 6.16. In the low-frequency range (with the dimensions and material properties of the problem, frequencies below 200 Hz) the value of stiffness is not important. It can be seen that almost the same sound reduction index curve is obtained for all the studied

162


spring stiffnesses. On the contrary, for higher frequencies, the sound reduction index of the double wall is completely dependent on the stiffness of the springs placed between leaves. Differences larger than 40 dB are found. The translational stiffness has more influence in the sound reduction index curves than the rotational stiffness. It can be seen that variations in the translational stiffness always cause modifications in the isolation (for all the studied values of rotational stiffness). On the contrary, the value of rotational stiffness is irrelevant if high values of translational stiffness are considered (i.e. for Kt = 108 N/m2 , where the same sound reduction index is obtained for all the values of rotational stiffness). The minimum frequency above which the rotational stiffness is an important parameter depends on the value of translational stiffness. For Kt = 105 N/m2 , the influence of rotational stiffness begins at 315 Hz while for Kt = 107 N/m2 it begins at 1000 Hz. The translational stiffness is more important for mid frequencies. This changes the slope of the curve. Rotational stiffness is more important for high frequencies and especially around the critical frequency. The rotational stiffness also controls the maximum value of sound reduction index reached. The different importance in the frequency range is explained by the type of vibration waves found in the structure. For low and mid frequencies, the vibrations of the structure are of large wave length. The displacements are more translational than the rotational. On the contrary, for high frequencies the displacements of the structures are very small and the rotations large. These results will be used in Chapter 7 in order to model the role of steel studs in the transmission of vibrations between leaves.

6.7

Concluding remarks

The sound reduction index of single and double walls has been predicted using numericalbased models. The frequency range analysed is quite reasonable and interesting information for engineers is obtained. The vibroacoustic equations have been solved without excessive simplifications and bounded domains and structures have been con-


100

100

Kt = 105 N/m2 7

Kt = 10 N/m 90

163

2

Kt = 108 N/m2 90

1

Kθ = 10 N•m/rad•m 3

Kθ = 104 N•m/rad•m

80


Kθ = 104 N•m/rad•m Kθ = 105 N•m/rad•m


70

R (dB)

R (dB)

70

Kθ = 101 N•m/rad•m Kθ = 103 N•m/rad•m

Kθ = 10 N•m/rad•m 80

Kt = 106 N/m2

60

60

50

50

40

40

30

30

20

20 3150 2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100 80 63 50 40 31.5 25 20 16

3150 2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100 80 63 50 40 31.5 25 20 16

f (Hz)


f (Hz)

Figure 6.16: Influence of the translational and rotational stiffness of mechanical connections (i.e. springs) between leaves in the sound reduction index of double walls.

sidered. This allows the analysis of environmental parameters (i.e. room sizes or boundary conditions of the wall) besides the intrinsic parameters of the wall (i.e. density or thickness). A low-frequency study of the sound level difference of single and double walls has been done. Two main conclusions can be obtained. On the one hand, the causes of poor isolation at some particular frequencies can be the resonances of the wall or the receiving acoustic domain. These dips in the sound level difference curve are attenuated by structural damping and acoustic absorption. On the other hand, the number of analysed situations per Hz depends on the attenuation of the problem (structural damping and acoustic absorption reduce the number of required calculations). Two and three-dimensional results of sound transmission for the same wall have been compared. In spite of the differences in the modal density evolution of each model version, the results are similar. Thus, with two-dimensional models, correct

164


approaches of the sound reduction index of prismatic situations (geometry, excitation and boundary conditions) can be obtained. The sound level difference curves of the same wall have been obtained for different values of room absorption. The correction for absorption proposed by the pressure method, Eq. (6.12), has been used in order to obtain sound reduction index curves. Differences between R curves corresponding to different absorptions are around 5 dB (0 dB would be a perfect result in this analysis). Discrepancies are larger for low frequencies and around coincidence frequency. The predicted values of sound reduction index by the numerical model are always slightly higher than those predicted by mass law type expressions. The modelling of double walls has also been considered. It is more demanding for the numerical model mainly due to the strong coupling between the leaves and the cavity and the modelling of absorbing material. This second aspect is important around eigenfrequencies, where sound reduction index can be significantly increased. The separation between leaves is important in the whole frequency range.

Chapter 7 The role of studs in the sound transmission of double walls 7.1

Introduction

Double walls are a common solution in lightweight structures. They are typically constructed by means of two thin leaves (plasterboards, wood plates or similar) and some kind of absorbing material placed inside the air cavity to improve the acoustic isolation capacity of the system. In order to satisfy construction requirements and to give a certain stiffness to the wall, some kind of connection has to be employed. Wood beams or steel studs are examples of actual solutions. The studied double walls can be seen in Fig. 1.1. Stud path Cavity path

Figure 7.1: Sound transmission paths in a double wall with studs.

165

166

The role of studs in the sound transmission of double walls

These connecting elements cause the actual acoustic response of the wall to be worse than that of an ideal double wall without connections between leaves. A new vibration transmission path (besides the airborne or cavity path) is created, see Fig. 7.1. Studs act as sound bridges between the two leaves. The decrease in the sound reduction index of the double walls highly depends on the mechanical properties (mainly stiffness) of the connecting elements. An ideal one would be so flexible that does not transmit vibrations from one leave to the other. In this chapter, the attention is focused on the study and characterisation of lightweight cold-formed steel studs. Their effect is very influenced by the cross-section shape. Two direct applications can be mentioned. On the one hand, stud manufacturers are interested in knowing which stud is better from an acoustic (vibration transmission) point of view. On the other hand, wave approach or statistical energy analysis models cannot reproduce the exact geometry of the stud and require some parameters describing its mechanical response. Some of these parameters can be provided by a numerical model because it can deal with accurate geometry descriptions. Numerical models can also deal with the whole problem. However, working with two different levels of detail (rooms - double wall and stud shape) increases the meshing tasks and the computational cost. Thereby, it is also interesting for a numerical approach to simplify the modelling of the stud. Several models dealing with double walls with connections can be found in the literature. In the simplest cases the studs are considered as infinitely rigid connections between the leaves (Fahy (1989)). Such models can be quite correct for rigid studs (i.e. wood studs) but underestimate the isolating capacity of lightweight double walls by neglecting the benefits of using steel studs, which are more flexible. In Wang et al. (2005) or Kropp and Rebillard (1999) the two leaves of the double walls are supposed to be connected by means of springs. Both translational and rotational springs are considered. The value of the stiffness is considered to be constant in the frequency range and it is typically taken from an elastic measurement (i.e. elastic stiffness of the flange of the stud). In the model proposed by Davy (1991), each transmission path is described by a different expression. They are piecewise-defined in the frequency

7.1 Introduction

167

domain and based on ad-hoc considerations, not in any governing equation. This contrasts with Wang et al. (2005) and Kropp and Rebillard (1999), which solve the vibroacoustic equations of the double wall by means of a wave approach. All the models cited above predict the acoustic isolation of the double wall with studs, but the characterisation of the connecting element is not their main goal. A review of the more referenced models of sound transmission in double walls is done in Hongisto (2006). It shows that only five of the seventeen models considered take into account the possible existence of studs. Moreover only two of them allow these studs to be flexible. There are not many references dealing with the characterisation of the connecting element. Some laboratory measurements of the effect of the studs in the sound reduction index can be found in Green and Sherr (1982a) and Green and Sherr (1982b). The same type of steel studs studied in this chapter has been characterised in Hongisto et al. (2002). Measurements of the dynamic stiffness of the isolated stud and its effect on the sound reduction index of double walls have been done. In Larsson and Tunemalm (1998), studs with non-conventional cross-section shape have been tested in order to check the improvement in sound isolation. However, studies dealing with a deterministic approach to the problem (exact descriptions of stud geometry and solution of the problem by means of analytical or numerical methods) can be rarely found. No practical rule on how to choose the correct value of the stud stiffness has been found. It is a necessary parameter in most of the models mentioned before. The aim of our research is to study in detail how to characterise the studs. The vibration response of small pieces of studs has been studied. Laboratory measurements of the point mobility of the studs due to the application of a punctual force in the upper flange have been compared with numerical models. The results presented in Section 7.2 illustrate the main phenomena found in the vibration response of this kind of structural elements. The situations analysed (types of studs and double walls) are presented in Section 7.3. Section 7.4 deals with the characterisation of the studs. Two different models are employed. One reproduces the geometry of the stud while the other uses

168


translational and rotational springs instead of studs. They are presented in detail in Section 7.4.1. Differences in the performance of a double wall depending on the stud type are shown in Section 7.4.2. A strategy based on the equivalence of transmission of vibrations between leaves is employed in Section 7.4.3 in order to identify the values of the spring stiffness. These values are employed in Section 7.5 as input data for an statistical energy analysis model reproducing the same situation. In Section 7.6 the sound reduction index between two rooms has been calculated for different stud and wall types. The concluding remarks of the study are presented in Section 7.7.

7.2

Vibration behaviour of steel studs

Several small pieces of steel studs have been tested in acoustique et éclairage department of the Centre Scientifique et Technique du Bâtiment (CSTB) at Saint Martin d’Hères (France), see Fig. 7.2. A punctual force is applied on the upper flange while the lower flange is glued over a nut. The punctual mobility (Y = v/F , v is the velocity and F the force) is measured in the point of application of the load. A finite element model, using shell elements, has been developed. Several details have been considered: • Exact shape of the support zone. • Geometry of the stud, taking into account the thermal slots in the web. • Inclusion or not of the laboratory devices (vibrating machine which is in contact with the top flange of the stud).

A typical result can be seen in Fig. 7.3. The measurement in laboratory of the point mobility in a C-shaped section is compared with several numerical models. In the first numerical model, only the steel stud has been considered. In the second one, both the steel stud and the laboratory device have been included. For all the numerical models the hysteretic damping is η = 3 %. This is a realistic value and causes the attenuation of resonance peaks.

7.2 Vibration behaviour of steel studs

169

F

M K

R (a)

(b)

(c)

Figure 7.2: Vibration behaviour of isolated studs: (a) experimental set-up; (b) finite element mesh; (c) mechanical device: the reaction R is different from the applied force F.

The agreement between the laboratory measurements and the numerical results is better in the second case. This indicates that, when measuring the point mobility (and the dynamic stiffness) of a steel stud, the boundary conditions are very important. The laboratory device gives an additional stiffness to the steel piece that modifies the value of point mobility. Various factors make the accurate agreement between the laboratory measurements and the predicted response difficult. When testing an isolated stud piece, it is too free to move. Some of the parameters are not well known, like the correct amount of damping, the degree of constraint between solids or the characteristics of the laboratory equipment. In the low frequency range (f < 1000 Hz for this type of problem), a modal behaviour zone can be seen. On the contrary, for high frequencies, the measured curves become smoother. If the stud is isolated, it is difficult to distinguish between translational and rotational effects. Moreover, it is not clear that all the required information is well

170


represented by the punctual mobility. It is a local measurement and the transmission of vibrations from the upper to the lower flange is produced along the stud. The reaction at the bottom flange can differ a lot from the applied force. Consider as an example the device of Fig. 7.2(c). The reaction in the base R (t) = Re {reiωt } can be expressed in terms of the applied force F (t) = Re {ϕeiωt } as r=

−Kϕ −K + ω 2 M

(7.1)

where K is the stiffness of the device and M its mass. This equation shows that the reaction can differ a lot from the applied force depending on the frequency. The same effect can be observed for some sections and it is difficult to know a priori when this is important or not in order to predict the vibration transmission. For these reasons and because of the importance of boundary conditions, we will consider from now on the entire package leave-studs-leave, see Fig. 7.4. This situation is closer to the actual use of studs in the double wall. 10

measured C + lab. device only C Eigenfrequencies

1

|Y| [m/(s N)]

0.1

0.01

0.001

1e-04

1e-05 0

1000

2000

3000

4000

5000

6000

f (Hz)

Figure 7.3: Comparison of the point mobility of a C-shaped stud measured in the laboratory and the results obtained with two different numerical models. While one of them considers only the stud piece the other takes into account laboratory devices.

7.3 Studs and leaves analysed

171

Figure 7.4: Laboratory tests of the entire package (studs and leaves).

7.3

Studs and leaves analysed

The analysis has been extended to several stud sections. The results presented here are by default obtained using double walls with a separation between leaves (stud height) of 0.07 m. Two other stud heights (0.125 m and 0.175 m) have been considered in the analysis. For every height, several cross-section shapes have been studied. Five of them will be employed in order to illustrate the most interesting aspects of the study. They are plotted in Fig. 7.5, and the dimensions for the 0.07 m series can be found in Table 7.1. Both conventional studs (TC, S, O) and acoustic studs (AWS, LR) are analysed. Acoustic studs have a similar stiffness in the beam direction but are more flexible at cross-section level than conventional studs. Experimental studies of the influence of cross-sectional shape on acoustic performance can be found in Hongisto et al. (2002) and Larsson and Tunemalm (1998). Besides the changes at cross-section level, three different floor typologies have been considered, see Tables 7.2 and 7.3. For all the situations, the length of the double wall is 3 m and the separation between studs is 0.6 m (four studs employed in every double wall). The leaves are supported at the beginning and ending points. Efforts are focused on the characterisation of the studs by means of simple pa-

172

The role of studs in the sound transmission of double walls d2

d2 d4

d4 d5

d3

d4

d5

TC

AWS

S

LR

d1

O

Figure 7.5: Sketch of the stud cross sections.

Section TC LR S O AWS

d1 [m] 0.07 0.07 0.07 0.07 0.07

d2 [m] 0.04 0.04 0.04 0.04 0.04

d3 [m] 0.01 0.01 0.01 – 0.01

d4 [m] – 0.7·d2 0.2·d1 – 0.0241

d5 [m] – 0.2·d1 – – 0.013

e [mm] 0.47 0.47 0.47 0.47 0.47

Table 7.1: Dimensions of the cross sections according to the notation in Fig. 7.5. e is the thickness of the section.

rameters like the translational stiffness Kt and the rotational stiffness Kθ of springs. They are useful in order to describe vibration transmission, and provide useful data for global models and manufacturers (decide which stud is better or if it is necessary or not to improve it). Another important aspect to be checked is the need of considering frequency-dependent parameters.

Case Upper leave Test 1 GN Test 2 GN Test 3 GEK

Lower leave GN GEK GN

Connection ux , uy and θ ux , uy and θ ux , uy and θ

` [m] 3 3 3

Table 7.2: Description of the three floor typologies.

no. studs 4 4 4

7.4 Identification of the stiffness of studs

Meaning Thickness Young’s modulus Density Damping

Symbol t E ρsolid η

173

Value, GN 13 mm 2.5 · 109 N/m2 692.3 kg/m3 3%

Value, GEK 13 mm 4.5 · 109 N/m2 900 kg/m3 3%

Table 7.3: Geometrical and mechanical properties of the leaves

7.4

Identification of the stiffness of studs

7.4.1

Cross-section structural vibration models

In order to study the vibration transmission path, the two structures of Fig. 7.6 are considered. It is assumed that the transmission of vibrations between leaves of a double wall can be studied at cross-section level. The required parameter for the studs in sound transmission models is the cross-section stiffness or line stiffness. In addition, the main difference between double walls using different stud types is found at cross-section level. This is represented by the models of Fig. 7.6. The output of interest is the vibration level difference between the upper and the lower leaves Dij = −10 log10 (dij )

with dij =

2 < vrms,j > 2 < vrms,i >

(7.2)

where < • > is the spatial average of •, vrms,i is the root mean square velocity in

leave i (upper), where the force is applied, j is the receiving leave (lower) and dij is

the vibration reduction factor. Dij can be calculated from the data of the numerical model as Dij = 10 log10

< |uupper |2 > < |ulower |2 >

(7.3)

where uupper and ulower are the phasors of displacements for the upper and lower leaves. The spatial average is done along the leave. A larger value of Dij means a better vibration isolation. In the detailed model of Fig. 7.6(a), the actual geometry of the stud is discretised. In the connection between the stud flanges and the leaves, continuity of displacements

174


and rotations is imposed. In the simplified model of Fig. 7.6(b), the stud is replaced by a translational spring, a rotational spring and concentrated masses. The equivalence between both models has been established by comparison of the vibration level difference Dij . For all the results presented here, four load configurations have been considered, see Fig. 7.6(b). The positions of punctual loads have been chosen in order to be representative: load applied over the stud or between studs, and at the centre of the double wall or on the side. The study is done in terms of averaged responses: average of the load position, average in time (over a period), average in frequency (the results are given in third frequency bands) and average in space. Two different regions, the upper and lower leaves, have been considered in order to perform space averages. One analysis per Hz is carried out in order to describe the response spectrum; this implies 20 000 elastic calculations per case (case means, analysed section or analysed value of stiffness in one of the three tests). The structural spectral element method (see Doyle (1997) or Yu and Roesset (2001)) has been chosen. In two-dimensional situations, the exact solution is reached using only the necessary elements in order to describe the geometry (i.e. 5 elements are required for the case of the TC cross-section, 9 for the AWS and 4 for the O). This is a very important advantage since typical mesh requirements of the finite element method can be forgotten. Results presented in Sections 7.4.2 and 7.4.3 are obtained using this model.

7.4.2

Influence of stud shape in the vibration level difference

Fig. 7.7 is an example of the results obtained when comparing several sections. For all of them, the isolation of vibrations in the low frequency range is really poor. It cannot be improved by changing the stud shape. Note that standard sections like TC, O or S provide an almost constant level of vibration isolation. On the contrary, acoustic sections like LR or AWS improve the isolation of vibrations in the mid and high frequency range (but they can be worse than the others for some frequencies in the low frequency range).


175

F

F1 F2 M

relationship: u x

uy

Kt

F3 F4

M/2

Kθ

θ)

(

(a)

(b)

Figure 7.6: Models of the leave-stud-leave package: (a) detailed model with the actual geometry of the studs; (b) simplified model with studs modelled as a translational stiffness Kt , a rotational stiffness Kθ and two concentrated masses M/2. F i indicates the four load positions considered.

35

70

30 60 25 50

Dij

Dij

20

15

40

10 30 5 20 0

-5

10 6300

5000

4000

f (Hz) LR

(a)

3150

S TC

2500

2000

1600

1250

1000

1000

800

630

500

400

315

250

200

160

125

100

80

63

50

40

31.5

25

20

16

f (Hz) AWS O

AWS O

S TC

LR

(b)

Figure 7.7: Comparison of the vibration level difference for several studs with different cross-section: (a) low-frequency range; (b) mid-frequency range. Note the different vertical scales.

176


The vibration level difference Dij is a useful parameter in order to compare the performance of different studs used in the same double wall. However, it is an environment-dependent parameter: the values of Dij do not only depend on the stud type. They also depend on other variables, such as leave properties and boundary conditions. Thus, Dij is not a parameter characterising the stud. As shown in the following sections, the stiffness of the stud is a better parameter, less dependent on each particular situation and more related with every stud type.

7.4.3

Stud equivalent stiffness

A set of admissible values of rotational and translational stiffness can be obtained by comparing the two deterministic models presented in Fig. 7.6. The key is to find pairs of values that provide, for a given frequency, the same vibration level difference. This requires to generate surfaces of vibration level difference in the plane Kt Kθ for every third frequency band. The surfaces obtained for 200, 500, 1000 and 3150 Hz and for test 2 are shown in Fig. 7.8. They have been generated analysing 36 different situations combining values of Kt = 104 , 105 , 106 , 107 , 108 , 109 N/m2 and Kθ = 101 , 102 , 103 , 104 , 105 , 106 N · m/ (rad · m). Once the surfaces have been generated, the admissible values of rotational and translational stiffness can be obtained by imposing the same value of vibration level difference for both models of Fig. 7.6: (simplif ied)

Dij (simplif ied)

Both the surface Dij (detailed)

value Dij

(detailed)

(Kt , Kθ ) = Dij

(7.4)

(obtained with the simplified model, see Fig. 7.8) and the

(obtained with the detailed model) are known. The equality provides

a set of admissible values (Kt , Kθ ), which yield the same vibration isolation in the simplified model and the detailed model. The values for the TC section and test 2 can be seen in Fig. 7.9. Three sets of figures like these ones have been obtained for every section (one for each test). Several important aspects have to be highlighted. On the one hand, an equivalent


177

f = 0200.0 Hz

0

10

20

30

40

50

f = 0500.0 Hz

60

70

80

90

Dij [dB]

0

10

20

30

40

50

60

70

80

Dij [dB]

100 80 60 40 20 0

1e+06 100000

100 80 60 40 20 0

1e+06 100000

10000 10000

10000

1000 Kθ [N•m/rad•m] 100000

10000

100

1e+06

1e+07 2

Kt [N/m ]

1e+08

1000 Kθ [N•m/rad•m] 100000

10 1e+09

1e+06

10

20

30

40

50

100 1e+07

Kt [N/m2]

f = 1000.0 Hz

0

1e+08

10 1e+09

f = 3150.0 Hz

60

70

80

90

Dij [dB]

0

10

20

30

40

50

60

70

80

90

Dij [dB]

100 80 60 40 20 0

1e+06 100000

100 80 60 40 20 0

1e+06 100000

10000 10000

90

1000 Kθ [N•m/rad•m] 100000

100

1e+06

1e+07 2

Kt [N/m ]

1e+08

10 1e+09

10000 10000

1000 Kθ [N•m/rad•m] 100000

1e+06 Kt [N/m2]

100 1e+07

1e+08

10 1e+09

Figure 7.8: Vibration level difference Dij in test 2 as a function of translational stiffness Kt and rotational stiffness Kθ , for various frequencies. effect in terms of Dij can be obtained using: i) only a translational spring; ii) only a rotational spring; iii) and adequate combination of both. The option chosen here is to use Kθ = 0 and a frequency-dependent Kt . On the other hand, a frequency dependence of the parameters is observed. For low frequencies the values of stiffness are smaller (around the typical values of elastic measurements), and they generally increase with frequency. For a given section, similar results are obtained for the three different tests. This indicates that a steel stud can be characterised by parameters that not depend on the environment (boundary conditions, leaves, symmetry of the double wall,...). In


1e+05

1e+05

1e+04

1e+04 Kθ [N•m/rad•m]

Kθ [N•m/rad•m]

178

1e+03

1e+03

1e+02

1e+02

1e+01

1e+01

1e+04

1e+05

1e+06

1e+07

1e+04

1e+05

Kt [N/m2] f = 200 f = 250 f = 315 f = 400

Hz Hz Hz Hz

1e+06

1e+07

Kt [N/m2] f = 500 Hz f = 630 Hz f = 800 Hz f = 1000 Hz

f = 1250 f = 1600 f = 2000 f = 2500

Hz Hz Hz Hz

f = 3150 f = 4000 f = 5000 f = 6300

Hz Hz Hz Hz

Figure 7.9: Admissible values of rotational and translational stiffness for the TC section obtained by comparison of deterministic models. Test 2. Fig. 7.10, the results of tests 1, 2 and 3 for the AWS section are compared. The agreement is not perfect; however, the tendencies are correct. The lines corresponding to the same frequency are close together. The dispersion is more important for low frequencies, where the modal behaviour governs the response and the number of modes in the third frequency band is smaller. In the low-frequency range (frequencies under (simplif ied)

200 Hz) it is difficult to obtain this kind of laws. The values of Dij

oscillate

between 0 dB and 5 dB depending on the value of the stiffnesses and the mechanical (simplif ied)

and geometrical characteristics of the leaves. Thus, the intersection of the Dij (detailed)

surface with a constant value Dij

does not provide smooth curves like in Fig. 7.9.

In Fig. 7.11 a typical final output for this kind of analysis can be seen. Frequencydependent translational stiffness laws are presented for every section.

7.5

Using the stiffness values in a SEA model

The values of stiffness characterising the sections can be used as input data for other modelling techniques. A clear example is its use as input data for Statistical Energy

7.5 Using the stiffness values in a SEA model

179

Kθ [N•m/rad•m]

1e+05 1e+04 1e+03 1e+02 1e+01 1e+04

1e+05

1e+06

1e+07

2

Kt [N/m ] f = 200 Hz T1 f = 200 Hz T2 f = 200 Hz T3

f = 1250 Hz T1 f = 1250 Hz T2 f = 1250 Hz T3

f = 4000 Hz T1 f = 4000 Hz T2 f = 4000 Hz T3

Figure 7.10: Comparison of the admissible values of rotational and translational stiffness of section AWS obtained by means of different tests.

KT (N/m2)

1e+08

1e+07

1e+06

1e+05

6300

5000

4000

3150

TC

2500

S

2000

O

1600

AWS

1250

1000

800

630

500

400

315

250

200

160

125

100

f (Hz)

LR

Figure 7.11: Laws of translational stiffness for Kθ = 0. Averaged values between tests 1, 2 and 3.

Analysis (SEA). The transmission of vibrations between two beams (i and j) connected by means of a spring has been studied by Craik (1996). A mechanical load is applied to the upper beam (i = 1). The SEA vibration level difference can be written

180


as Dij = D12 = 10 log10

|Y1 + Y2 + Yt |2 m2 η 2 ω nRe {Y2 }

(7.5)

where m2 is the mass per unit length, η2 is the loss factor, n is the number of connections (springs) per unit length, Y1 and Y2 are the mobility of the two beams and Yt is the mobility of a point connecting tie. The last parameter can be related to the dynamic stiffness of this tie, Yt = v/F = iω/Kt , where Kt is the stiffness (in this SEA model, only translational stiffness is considered), and v is the rate of length change of the spring. For the mobility of the two beams, the formulas given in Cremer et al. (1973) can be used (1 − i) 1 Y∞beam = Ysemi-∞beam = 4 4m` cB

with

ω cB = p . 4 ω 2 m` /E I

(7.6)

m` is the density per unit length of the beam, E the Young’s modulus and I the inertia. Note that two reference cases have been considered: infinite and semi-infinite beams. The former provides better approximations for high frequencies where the influence of boundaries is not important and the vibrations are localised due to damping. The latter is a better approximation for low frequencies or situations where the mechanical load is close to the boundary. We have considered first the case of two beams connected by means of springs, see Fig. 7.6(b). In this case the value of the stiffness is known and a ‘pure’ comparison between the SEA model and the numerical method (SFEM) can be established without additional errors caused by the uncertainties due to the characterisation of the stud shape. The results are presented in Fig. 7.12. The transmission of vibrations in test 2 has been calculated with the numerical model and with the SEA model considering two situations: infinite and semi-infinite beams. The agreement is correct. The numerical results are closer to the infinite beam curve. Differences are smaller for mid and high frequencies. The modal behaviour at low frequencies is affected by the value of the stiffness. For the smaller values of Kt , the two leaves are weakly connected and vibrations are

7.5 Using the stiffness values in a SEA model

100

80

90

70

80

60

70

50

181

60

50

40

Dij

Dij

Dij

30 60

40

20 50

30

40

20

30

10

20

0

10

0

-10

6300 5000 4000 3150 2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100

6300 5000 4000 3150 2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100

6300 5000 4000 3150 2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100

f (Hz)

f (Hz)

f (Hz)

Kt = 104 N/m2

Kt = 105 N/m2

Kt = 106 N/m2

40

16

14

35

14

12

12

30

10

10 25

8 8 Dij

Dij

6

Dij

20 6

15

4 4

10

2 2

5

0

0

0

-2

-2

-5

-4

-4

Kt = 107 N/m2

6300 5000 4000 3150 2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100

6300 5000 4000 3150 2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100

6300 5000 4000 3150 2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100 f (Hz) SFEM

f (Hz) SEA inf

Kt = 108 N/m2

f (Hz) SEA semi-inf

Kt = 109 N/m2

Figure 7.12: Two leaves connected with springs. Comparison of the vibration level difference obtained by means of a numerical model (SFEM) and statistical energy analysis. Note the different scale for Dij .

182


developed all along the span length (3 m). Nevertheless, for larger values of Kt the link between leaves becomes stronger, and each leave cannot be considered as a 3 m long beam. Due to the connections it behaves like a group of short cells (space between springs). Oscillations in the response curve are then important for frequencies below 400 Hz (Kt = 107 N/m2 ), 2000 Hz (Kt = 108 N/m2 ) and 4500 Hz (Kt = 109 N/m2 ). Results presented here are also important in order to understand the type of laws obtained for the translational and rotational stiffnesses. The values of Dij for some of the studied sections have an small variation range. See for example the variation of O, TC and S studs in Fig. 7.7(b), where the values of Dij are between 15 and 35 dB. On the contrary, the variation range of Dij for some cases of constant spring stiffness is very large (see Fig. 7.12, Dij ∈ [25, 90] for Kt = 104 N/m2 , Dij ∈ [10, 80]

for Kt = 105 N/m2 , Dij ∈ [0, 60] for Kt = 106 N/m2 . . .). This means that the studied

sections behave like a spring of variable, rather than constant, stiffness. If the required

stiffness was constant, the variation of Dij obtained with the model considering the geometry of the stud would be larger. This is not the case. The same method has been used for the case where the two leaves are connected by means of steel studs, see Fig. 7.6(a). In this case, two different errors are possible. On the one hand the agreement between a SEA model and a numerical model (shown with the previous example). On the other hand, the correct characterisation of steel studs (Kt − f laws). The results for AWS and TC studs are presented in Fig. 7.13. Again the vibration level difference calculated by means of a numerical model and by means of the SEA model is compared. Now, for the case of the SEA model, the value of stiffness is variable with frequency. The laws obtained in Section 7.4.3 have been used as input data. This example shows how the double wall behaviour predicted by a model that considers the geometrical detail of the studs can be reproduced by means of a SEA model, where the geometrical complexity has been reduced to the use of a frequencydependent stiffness law.

7.6 Global response of double walls

183

60

60

50

50

40

40 Dij

70

Dij

70

30

30

20

20

10

10

0

0

6300 5000 4000 3150 2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100

6300 5000 4000 3150 2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100

f (Hz)

f (Hz)

SFEM

SEA infinite

(a)

(b)

Figure 7.13: Comparison of vibration level difference obtained by i) a detailed numerical model where the actual geometry of the studs is discretised (SFEM) and ii) a SEA model, which uses a frequency-dependent stiffness provided by the numerical model: (a) acoustic stud AWS; (b) standard stud TC.

7.6

Global response of double walls

In previous sections the effort has been focused on the characterisation of flexible steel studs and the study of the vibration transmission path (vibration level difference). It is only one of the parts of the problem of sound transmission. The performance of the studs and the validity of results obtained in Section 7.4 is now verified in a twodimensional vibroacoustic problem. This means that both the stud path and the cavity path are considered at the same time, see Fig. 7.1. The relevance of acoustic design of studs depends on the type of double wall. If the cavity path is very insulating, the stud path will be the critical path and then the type of stud used is very important. On the contrary, if the isolation of the cavity path is poor, the type of stud used will not be relevant because most of the sound is not transmitted through the stud.

184


The effect of mechanical connections between leaves has been studied in Section 6.6.2. The main conclusion is that there are limit values of stiffness below and above which the effect of the stud is not important. These limit values depend on the type of double wall (separation between leaves, use or not of absorbing material, type of leaves). If the modifications in the stud (shape, thickness, materials, damping) can change its stiffness in this frequency range the optimisation is possible, otherwise the type of stud used is not an important variable of the problem. The model presented in Chapter 5 has been used. The studs have been considered now, see the finite element mesh of Fig. 7.14. Two rectangular acoustic domains, the two leaves connected by means of a steel stud and the cavity between leaves can also be seen. The dimensions of the rooms are 5.7 m × 4.7 m and 6.35 m × 5 m. The double

wall is 3 m long. For some cases absorbing material (resistivity % = 8000 Pa · s/m 2 ) is

placed inside the air cavity. The cavity has been considered continuous through the

studs. The opposite situation where small cavities between the studs are modelled instead of a large single cavity can also be reproduced with the numerical model. However, the thermal slots in the studs web establish a continuity in the air between leaves that justifies the use of a single air cavity. A sound source has been placed in the lower left corner of the sending room (separated 0.5 m from each wall). The value of impedance in Robin contours is Z/ (ρ0 c) = 19.03. In Fig. 7.15 the sound level difference for the 0.07 m thick double wall with and without absorbing material in the cavity can be seen. Several values of translational stiffness have been considered. The limit values for this double wall are 105 N/m2 and 108 N/m2 . It can be seen that the improvement by the use of flexible studs is larger if there is absorbing material in the cavity (high cavity path isolation) than in the case of air cavity (poor cavity path isolation). Fig. 7.16 shows the results obtained from the vibroacoustic model where the springs have been replaced by studs. A set of TC studs with increasing value of thickness (0.00047 m, 0.001 m and 0.003 m) have been considered. The translational stiffness laws obtained by means of the analysis presented in Section 7.4 for these TC studs and the sound level difference of every section can be seen in Fig. 7.16(a). The value

7.6 Global response of double walls

185

(a)

(b)

80

80

70

70

60

60 D (dB)

D (dB)

Figure 7.14: Finite element mesh used to solve the vibroacoustic problem: (a) general view of the two rooms and the double wall; (b) detail in the double wall zone.

50

50

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10

2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100 80 63 50 40 31.5 25 20 16

2500 2000 1600 1250 1000 800 630 500 400 315 250 200 160 125 100 80 63 50 40 31.5 25 20 16 f (Hz) 5

f (Hz) 7

2

2

Kt = 10 N/m Kt >= 108 N/m2

Kt = 108 N/m2

2

Kt

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