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Estimating Vibration, Acoustic and Vibro-Acousticresponses using Transmissibility functions
Vasco Miguel Nascimento Martins
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Supervisor: Prof. Miguel António Lopes de Matos Neves
Examination Committee
Chairperson: Prof. Fernando José Parracho LauSupervisor: Prof. Miguel António Lopes de Matos Neves
Member of the Committee: Prof. Hugo Filipe Diniz Policarpo
July 2019
Para mim e para os meus.
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Acknowledgments
I would like to thank Prof. Miguel Neves for all his support and availability during the making of this
thesis.
I would also like to thank my dearest friends (they know who they are) and family, who were always
there if need be.
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Resumo
Neste trabalho e proposto um estudo sobre metodos destinados a estimar respostas Vibracionais,
Acusticas e Vibro-Acusticas atraves de funcoes de Transmissibilidade. Para o fazer, o autor propoe uma
extensao de metodologias ja existentes, utilizadas para transmissibilidade de deslocamentos dinamicos
e pressoes acusticas, para o caso da Vibro-Acustica. Ate a data, existe somente informacao de na-
tureza experimental para Transmissibilidade Vibro-Acustica escalar, presente na literatura disponıvel.
Comeca-se com uma verificacao de Transmissibilidade Vibracional e Acustica. Segue-se a criacao
de um metodo de elementos finitos 3D com uma interface fluıdo-estrutura, atraves do qual pressoes
e deslocamentos sao estimados e com funcoes de Transmissibilidade propostas atraves de Funcoes
de Resposta em Frequencia (FRFs) extraıdas do sistema acoplado. Primeiramente e proposta uma
transmissibilidade escalar, seguida de uma matricial que relaciona coordenadas de pressao com deslo-
camentos (fluıdo-estrutura). Isto e realizado para uma gama de frequencias, assumindo propagacao de
ondas harmonicas planas.
Em suma, o conceito de Transmissibilidade Vibro-Acustica e implementado, encontrando-se, no en-
tanto, ainda em desenvolvimento. Esta implementacao e descrita e discutida. E de notar que o processo
ainda e relativamente complexo e as simulacoes para elementos finitos acoplados sao relativamente
pesadas e ineficientes temporalmente. O procedimento e resultados apresentados sao considerados
uma contribuicao na direccao de uma resposta completa para o problema em questao.
Palavras-chave: Transmissibilidade Vibro-Acustica, Domınio da frequencia, Sistemas Acopla-
dos, Metodo de Elementos Finitos, Interface Fluıdo-Estrutura
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Abstract
With this work, it is proposed to study ways to numerically estimate Vibration, Acoustic and Vibro-
Acoustic (V-A) responses through Transmissibility functions. The author proposes to extend existing
methodologies of dynamic displacement transmissibility and acoustic pressure transmissibility to the V-
A case. So far, only experimental data for scalar V-A Transmissibility has been presented in available
literature.
The methodology and results of its’ implementation addresses initially the vibrational and acoustic
Transmissibility Verification. Then a 3D Finite Element Method (FEM) implementation created with a
fluid-structure interface, from which pressure and displacement response are calculated, and estimation
method proposed with Single and Multiple degrees of freedom (SDOF and MDOF) Transmissibility func-
tions obtained with Frequency Response Functions (FRFs) extracted from the coupled system. Primar-
ily a scalar Transmissibility is proposed, followed by a matricial one which relates sets of displacements
with pressures (structural-fluid). This is done for a range of frequencies, and assuming harmonic plane
waves.
In conclusion, the concept of V-A Transmissibility was implemented and is still in development. The
implementation is described and discussed. However, the process is still quite complex and the simula-
tions for coupled Finite Elements (FE) are relatively heavy and time costly. The procedure and results
presented are considered a contribution in the direction of a full answer to the challenge.
Keywords: Vibro-Acoustic Transmissiblity, Frequency domain, Coupled Systems, Finite Ele-
ment Method, Fluid-Structure Interface
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Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1 Introduction 1
1.1 Brief State of the Art: Vibration, Acoustic and Vibro-Acoustic Transmissibility . . . . . . . 2
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Vibration and Acoustic Transmissibility in the Aerospace Industry . . . . . . . . . . 4
1.2.2 Acoustic Induced Problem in Aerospace Structures . . . . . . . . . . . . . . . . . . 4
1.3 General Overview of Vibro-Acoustic Transmissibility . . . . . . . . . . . . . . . . . . . . . 5
1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Theoretical Background 9
2.1 Structural Dynamics Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Elastodynamic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1.1 Cauchy’s Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Euler-Bernoulli Beam Element (1D) . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2.1 Natural Frequencies of the Beam . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Spring Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Vibrations in MDOF Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Frequency Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Transmissibility in Solid Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3.1 Load Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3.2 Displacement Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.4 Finite Element Method - Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
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2.3.1 Types of Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Three-Dimensional Acoustic Wave Equation (Global Cartesian Coordinates) . . . 18
2.3.3 Harmonic Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.4 Determining the Speed of Sound in Fluids . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.5 Acoustic Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.5.1 Anechoic and Reflective Boundary . . . . . . . . . . . . . . . . . . . . . . 23
2.3.6 Helmholtz’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.7 Imposed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.8 Transmissibility in the Field of Acoustics . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.8.1 Dynamic Stiffness Method . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.8.2 Frequency Response/Receptance Method . . . . . . . . . . . . . . . . . 27
2.3.9 Finite Element Method - Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Coupled Vibro-Acoustic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Coupling Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1.1 Eulerian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1.1.1 Acoustic FE Model . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1.1.2 Structural FE Model . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.1.1.3 Coupled Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.2 Limitations of Coupled Finite Element Models . . . . . . . . . . . . . . . . . . . . . 31
2.4.3 Using FRF to compute Vibro-Acoustic transmissibility . . . . . . . . . . . . . . . . 31
3 Methodologies 33
3.1 Dynamic Force and Displacement Transmissibility Verification . . . . . . . . . . . . . . . . 33
3.1.1 Mass/Spring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1.1 Dynamic Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1.2 Receptance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1.3 Comparison of Nodal Reactions Between Both Methods . . . . . . . . . 35
3.1.2 Simply Supported Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.3 Beam Modelling in ANSYS APDL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Acoustic Pressure Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Transmissibility Using a Code in MATLAB . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 Transmissibility Using a Code in ANSYS From a 3D Model . . . . . . . . . . . . . 41
3.3 Vibro-Acoustic Transmissiblity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Results and Discussion 53
4.1 Force Transmissibility in a Mass/Spring System . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Transmissibility in a Simply Supported Beam . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.1 Using a Code in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.2 Using a Code in ANSYS APDL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Acoustic Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
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4.3.1 Modal Analysis of the tube (APDL) . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.2 Transmissibility Through the Tube Containing the Acoustic Fluid . . . . . . . . . . 66
4.3.2.1 1D Case Using a Code in MATLAB . . . . . . . . . . . . . . . . . . . . . 66
4.3.2.2 3D Case Using a Code in APDL . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.2.2.1 Scalar Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.2.2.2 MDOF Pressure Transmissibility from FRFs . . . . . . . . . . . . 73
4.4 Vibro-Acoustic Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.1 Pressure-Displacement Ratio for Scalar Vibro-Acoustic Transmissibility . . . . . . 76
4.4.2 Vibro-Acoustic Transmissibility Estimation Through FRFs . . . . . . . . . . . . . . 77
4.4.2.1 Frequency Response Sub-Matrices Ratio For Vibro-Acoustic Transmissi-
bility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.2.2 MDOF Vibro-Acoustic Transmissibility . . . . . . . . . . . . . . . . . . . . 78
5 Conclusions 79
References 81
A Finite Element Formulation for EB Beam 87
B Tables for Peak Representation Regarding the Beam Results 89
C The Problem of obtaining Harwell-Boeing Sparse Matrices from ANSYS APDL and convert-
ing to MATLAB 91
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List of Tables
4.1 Spring System Connectivity Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Natural Frequencies of the Considered Spring-Mass System . . . . . . . . . . . . . . . . 55
4.3 Beam Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Natural Frequencies of the Simply Supported Beam . . . . . . . . . . . . . . . . . . . . . 60
4.5 Tube Properties (FLUID30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Convergence Analysis For the Acoustic Tube . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Plate Properties (SHELL181 - ANSYS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.8 Modes of the Coupled System Tube+Plate (without constraints), with Corresponding Ele-
ment Type, referring to Fig. 4.34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
B.1 Comparison of Peaks (approx.), in Hz, From Fig. 4.6 . . . . . . . . . . . . . . . . . . . . . 89
B.2 Comparison of Peaks (approx.), in Hz, From Fig. 4.8 . . . . . . . . . . . . . . . . . . . . . 89
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List of Figures
1.1 One DOF Mass-Spring-Damper System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Vibro-acoustic Transmissiblity Inside a Car . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Simple Vibro-Acoustic Interaction Illustration . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Vibro-Acoustic Transmissiblity in a Landing Gear, with B and C located Inside the Fuse-
lage (Cavity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Spring Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Set of Generalized Coordinates K, U and C (source Y.E. Lage et al [23]) . . . . . . . . . . 14
2.3 Free elastic body with four sets of coordinates A, U, K, C (source Y.E. Lage et al [23]) . . 16
2.4 Unidimensional Sound Wave Reflecting on a wall . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Shell Coordinate System, as in [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Vibro-Acoustic Interaction depicted in Sets of Coordinates U , K and others C, for u impo-
sition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Euler-Bernoulli beam element, based on [29] . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 BEAM3 element (source [41]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Sets of Coordinates U , K and C for an Acoustic Enclosed Domain . . . . . . . . . . . . . 40
3.4 Base element with 3 nodes per width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 fluid30 - 3D Acoustic Fluid Element (source [40]) . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 Vibro-Acoustic Interaction depicted in Sets of Coordinates U , K and others C, with im-
posed pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Mass/Spring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Comparison of T14 (from the receptance method), obtained in this work (a) with the one
in [1] (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 T14 (receptance and dynamic) obtained through H and Z (a) and receptance between
nodes 1 and 4 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Comparison of reactions obtained in node 1, obtained from the Receptance and Dynamic
stiffness methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5 16 Finite Element Beam, generated in ANSYS APDL . . . . . . . . . . . . . . . . . . . . . 57
4.6 a) Transmissibility T1,7 Obtained in This Work; b) From [23] . . . . . . . . . . . . . . . . . 58
4.7 H1,7 plotted in the frequency spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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4.8 a) Transmissibility T17,7, obtained in this work ; b) From [23] . . . . . . . . . . . . . . . . . 59
4.9 H17,7 plotted in the frequency spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.10 a) Transmissibility T1,7 (same as figure 4.6a ) ; b) The one obtained through APDL . . . . 60
4.11 FRF and Transmissibility Plots (for odes 1 and 7) . . . . . . . . . . . . . . . . . . . . . . . 61
4.12 Superposition of results obtained with different methods from figure 4.10, and relative
deviation calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.13 Analytic Wave Equation Solution for: a) Reflective and b) Anechoic End . . . . . . . . . . 62
4.14 Acoustic tube simulated in APDL with N = 12 (a) and N = 24 (b), evidencing a greater
difference between mesh refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.15 Pressure along the tube for N=12 (a) and N=24 (b) with a reflective end, extracted from
APDL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.16 Pressure along the tube for N=12 (a) and N=24 (b) with an anechoic end, extracted from
APDL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.17 Pressure plot for reflective top (z = −L). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.18 Error plot between the first six natural frequencies for increasing N and the analytic solution 65
4.19 1D acoustic medium for case illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.20 Transmissibility obtained from [3], using the FRF ratio with imposed pressure in x = 0 and
from pressure ratio with a source in x = 0, for N=12 (a) and (b) N=24, with an anechoic
top. The transmission is done to a node in x = 2 m . . . . . . . . . . . . . . . . . . . . . . 67
4.21 Transmissibility obtained from [3], using the FRF ratio with imposed pressure in x = 0 and
from pressure ratio with a source in x = 0, for N=12, 25 nodes (a) and (b) N=24, with an
reflective top. The transmission in done to a node in x = 2 m . . . . . . . . . . . . . . . . 67
4.22 FRF plot for the DOF in the middle of the 1D tube (x = 2) and N =12, for a reflective (a)
and anechoic end (b), with a source in x = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.23 Side View of the 3D tube model (z0y plane) with N = 24 elements per λ, 56 elements
along the length and a reflective end at z = −L . . . . . . . . . . . . . . . . . . . . . . . . 68
4.24 Results obtained based on [2, 3] compared against the 3D model developed in APDL,
for a) 6; b) 10 and c) 24 elements per wavelength, having as reference f=200 Hz, and a
reflective end. Pressure ratio in black, for 1D, and FRF ratio in blue for 3D . . . . . . . . . 69
4.25 Z matrix condition number for (a) N=6 ,(b) N=10, (c) N=12 and (d) N=24, having as
reference f=200 Hz, and a reflective end (no damping) . . . . . . . . . . . . . . . . . . . 70
4.26 3D tube model with N = 20, 48 elements along the length and a reflective en at z = −L . 71
4.27 Pressure measured in point (0.05;0.05;-2) of the 3D model, with a reference pressure im-
posed at z=0 of 1 Pa, for (a) N=6 ,(b) N=10, (c) N=12 and (d) N=20, having as reference
f=200 Hz, and a reflective end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.28 Pressure measured in point (0.05;0.05;-2) of the 3D model, with a reference pressure im-
posed at z=0 of 1 Pa, for (a) N=6 ,(b) N=10, (c) N=12 and (d) N=20, having as reference
f=200 Hz, and an anechoic end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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4.29 a) ”Measured” pressure at the centre of the tube with N = 36, in APDL; b) Transmissibility
for N = 36, in the manner of [3], with MATLAB, except for the fact that in this case, only
the real part of the Transmissibility was regarded . . . . . . . . . . . . . . . . . . . . . . . 72
4.30 Pressure Imposition on z = 0 in a tube model with N = 36, 84 elements along its’ length,
and a reflective boundary at z = −L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.31 Plane Pressure wave along the tube at 200 Hz and a reflective end in (a) and anechoic in
(b), with P= 1 Pa applied at z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.32 Pressure measured at the centre of the tube (black) and calculated with T aKU (green) with
an imposed pressure of 1 Pa at z=0, and a reflective top, for N = 12 (a) and N = 36 (b) . 74
4.33 Pressure measured at the centre of the tube (black) and calculated with T aKU (green) with
an imposed pressure of 1 Pa at z=0, and an anechoic top, for N = 36 . . . . . . . . . . . 74
4.34 Model for the coupled System, with the plate at the end, in red. Mesh generated with 567
DOFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.35 Pressure Profile along the tube (plane wave), with displacement excitation applied to the
plate, for a reflective (a) and anechoic boundary (b) . . . . . . . . . . . . . . . . . . . . . . 76
4.36 Pressure/displacement Ratio results from the model in fig.4.34. Model with a reflective
end in z = 0. The results were plotted for an imposed load at the center coordinates of
the plate, and a ”measured” pressure at the midsection of the tube. N = 12, or 315 DOFs
(a), and N = 24 or 567 DOFs (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.37 Pressure/displacement Ratio results from the model in fig.4.34. Model with an anechoic
end in z = 0. The results were plotted for an imposed load at the center coordinates of
the plate, and a ”measured” pressure at the midsection of the tube. N = 12, or 315 DOFs
(a), and N = 24 or 567 DOFs (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
xvii
xviii
Nomenclature
APDL ANSYS Parametric Design Language.
ASL Acoustic Source Localization.
CLE Constitutive Law Error.
DOF Degree-of-Freedom.
EB Euler-Bernoulli.
FEA Finite Element Analysis.
FE Finite Element.
FSI Fluid-Structure Interaction.
FEM Finite Element Method.
FR Frequency Response.
FRF Frequency Response Function.
HB Harwell-Boeing.
MDOF Multiple Degree-of-Freedom.
OAMA Operational Acoustic Modal Analysis.
OMA Operational Modal Analysis.
SDOF Single Degree-of-Freedom.
SEA Statistical Energy Analysis.
SPL Sound Pressure Level.
V-A Vibro-Acoustic.
Greek symbols
β Damping Coefficient.
γ Specific Heat Ratio.
xix
λ Solution of Eigenvalue Problem; Wavelength.
µe Effective Cinematic Viscosity.
Ω Domain Boundary.
ω Angular Frequency.
Φ Scalar Function (Velocity Potential).
ψ Complex Amplitude in Harmonic Wave Propagation With no Time Dependence.
Ψr Eigenvector.
ρ Density.
ρ′ Density Perturbation.
ρ0 Equilibrium Density.
σji Cauchy’s Stress Tensor.
θ Beam Deflection (Slope).
Roman symbols
A Cross-sectional Area.
a Acoustic Resistance.
r Acoustic Reactance.
c Sound Speed.
FU ,FK ,FC ,FA Harmonic Loads (vectors) applied in K, U , C and A.
F1, F2 Nodal Forces in Spring Element.
Fi Volumic Force Vector.
fn Acoustic Tube Natural Frequencies (1D Propagation).
fm,n,l Acoustic Tube Natural Frequencies (3D Propagation).
f Transversely Distributed Load along a Beam; Frequency.
gx, gy, gz Gravity Acceleration Components.
H Frequency Response Matrix.
h Element Step in Acoustic FEM.
i Imaginary Constant.
k Spring Stiffness; Wave Number.
xx
K,U,C,A Known, Unknown, other Coordinate Sets inside a Domain. A is Additional and is on the
Domains’ Boundary.
kx, ky, kz Wave Number Components in 3D Wave Equation.
[K], [M], [C] Global Stiffness, Mass and Damping Matrices.
le Beam Element Length.
M Bending Moment in Beam.
m Mass.
nj Normal Direction.
O,G Grouped Coordinate Sets.
PK ,PU ,PC Imposed pressure in K, U and C.
p Acoustic Pressure.
P (ω) Pressure Vector in Acoustic Medium.
p′i Incident Acoustic Wave Pressure.
p′r Reflected Acoustic Wave Pressure.
p′ Acoustic Pressure Perturbation.
p0 Equilibrium Pressure.
Q(ω) Volume Acceleration Vector.
q External Acoustic Source Distribution.
r Cartesian Coordinates Vector.
RHU Reaction in Nodes (Spring/Mass System), obtained from [H].
RZU Reaction in Nodes (Spring/Mass System), obtained from [Z].
R Universal Gas Constant.
r Reflection Coefficient.
Rx, Ry, Rz Distributed Resistance Components.
(s1, s2, sn) Coordinate System at the Center of an Elastic Shell’s Surface.
S Surface.
Tf1−→2 SDOF Force Transmissiblity between Nodes 1 and 2 in Mass-Spring-Damper System.
xxi
TFSKU Transmissiblity from Fluid-Structure Interaction that aims to convert Acoustic Fluid Pressure to
Structural Displacement.
TSFKU Transmissiblity from Structure-Fluid Interaction that aims to convert Structural Displacement to
Acoustic Fluid Pressure.
T(d)UK Displacement Transmissibility Matrix between Sets U and K.
T(f)UK Load Transmissibility Matrix between Sets U and K.
T Temperature.
t Time.
Tni Stress Vector in a Normal Direction.
Tx, Ty, Tz Losses Due to the Effect of Viscosity.
T kir(ω) Scalar Pressure Transmissibility from Literature.
T aKU MDOF Acoustic Transmissiblity between Sets K and U .
u Acoustic Velocity.
u0 Equilibrium Velocity.
ux, uy, uz Nodal Displacement in Beam (Chapter 3).
u′ Acoustic Velocity Perturbation.
u1, u2 Nodal Displacement in Spring Element.
V Volume.
v Weight Function (Galerkin).
vx, vy, vz Flow Velocity Components.
w Transverse Displacement in Beam.
YK ,YU ,YC ,YA Dynamic Displacements (vectors) in K, U , C and A.
Z Dynamic Stiffness Matrix.
Z Acoustic Impedance.
Z0 Specific Acoustic Impedance, for Plane Waves.
Subscripts
ai Acoustic-Interface.
e Element.
xxii
i, j, k Computational Indexes.
ir Between Node i and r.
m,n, l Wave Propagation Indexes.
n Normal Component.
ref Reference Condition.
si Structure-Interface.
x, y, z Cartesian Components.
Superscripts
T Transpose.
(d) From Displacement.
(f) From Force.
FS Fluid-Structure.
k Reference Node k.
+ Pseudo-Inverse.
SF Structure-Fluid.
xxiii
xxiv
Chapter 1
Introduction
The topic of vibrations is quite present everywhere as every single thing that can be perceived anywhere
has an inherent vibration, being that the engine of a car, a stimulated spring or even the paddles of a
turbine in an A380 plane. Being a part of reality, it needs to be thoroughly grasped and analyzed.
Above all else, it is relevant to know how these vibrations are transmitted through structures whenever
there are certain kinds of stimuli like harmonic and steady-state loads/displacements being applied to the
aforementioned. One, in this field of Engineering, has to be able to estimate how imposed displacements
and loads will propagate along a considered solid structure. And not just that, but also how these will
act upon certain specified points (might also be considered nodes, for the sake of further argument).
While the problem of estimating load and displacement transmissiblity in SDOF systems is relatively
simple in general, where an imposed constant amplitude load in a point is directly related, through a
scalar unit, to the one felt in the other point (assuming a two point system, like a simple mass-spring-
damper system). However, when considering an MDOF system where can be applied as many loads
and displacements needed and wanted, the case differs. As it is referred by Maia in [1], MDOF Trans-
missibility functions are obtained through FRFs which relate, within a steady-state regime (stationary
and periodic), displacements among the structure with applied loads (conjugate variables), as it will be
analyzed more in depth further ahead into this work (chapter 2).
Before advancing any further, it is of relevance to refer that the Transmissibility studies that will be
done in this work are limited to the frequency domain. Despite the fact that a time based study would
also be of most relevance for the case of a transient regime (where variables like displacement and loads
would be studied through time and not through excitation in only certain spectra).
Now, a parallel analogy could be established where acoustics are considered, as it is proposed by
Guedes in [2] and Devriendt in [3] (that will be addressed later on in this work).
Acoustics is also a part of reality, being it outside a vehicle, inside an exhaustion pipe or even inside
a vehicle, and it is important knowing how it originates, propagates and how it influences the many
obstacles it surpasses in its track. Variables such as the acoustic pressure and the volumetric speed
of particles are regarded as transmittable when an acoustic field is considered. In acoustics it is not
a solid structure that is being dealt with, concepts like Transmissibility are still regarded all the same,
1
but now assuming different but mathematically equivalent parameters, such as acoustic loads, pressure
disturbances which might result either from sources or from imposed pressure boundary conditions.
For instance, regarding now the vibrational topic, there is the example of a single mass, single spring
and single damper system, like in the following picture:
M
2
1
k β
F1 = F1eiωt
F2 = F2eiωt
T f1→2 = F2
F1
X1 = X1eiωt
Node 2 is fixed
Figure 1.1: One DOF Mass-Spring-Damper System
In this case there are two nodes (let us assume that node 1 is free and node 2 is fixed, as it is
depicted in figure 1.1, where an harmonic steady-state load F1 can be applied with a certain frequency
ω, resulting in the dynamic displacement X1. In this case, one can analyze how the force applied to, let
us assume, the first node (where the massM is) will manifest in the second node, i.e. will be transmitted.
This is expressed by T f1→2 which allows the computation of F2. The nodes are connected by a spring
with stiffness k and a damper with damping coefficient β. Lastly, the barred variables represent the
amplitudes.
In the same line of thinking, one can revert now to the field of Acoustics and try and figure out how
this would work when there is solely an already mentioned acoustic fluid, with a certain source. Here
the actual relevance is to analyze how a certain number of sources (or even none) would create and
distribute pressure waves along the acoustic fluid.
However, the main question of this work is how to estimate vibrational and acoustic Transmissibility,
and how these two fields interact in order to compute Transmissibility. Essentially, the debate sets on how
the vibrational behaviour (one can assume a set of harmonic forces) of a solid structure is transmitted
to the fluid and how these vibrations manifest as acoustic pressures, and vice-versa.
1.1 Brief State of the Art: Vibration, Acoustic and Vibro-Acoustic
Transmissibility
The concept of Transmissibility is not new in engineering, for there is already a good amount of work/lit-
erature in this specific area, being it in Vibrations or Acoustics.
2
In [4], the authors proposed a relationship between FRFs for MDOF systems with diagonal mass
matrices and tri-diagonal stiffness ones, through Transmissibility functions, for Vibrations.
In [5], it is discussed the prediction of motion transfer through single-point and multi-point FRFs, with
a standard scalar approach and a more complex one through transformation matrices.
Fontul et al [6] studied Transmissibility in MDOF systems for coupled structures, developing condi-
tions to calculate Transmissibility matrices valid for both the main structure and the coupled ones.
Additionally to the ones already mentioned for Acoustics [2, 3], in [7] is proposed an analytic deriva-
tion of Transmissibility of the pressure transducer-in-capsule arrangement used in experiments to mea-
sure wall pressure spectra in a bundle of cylinders with application to the vibration of nuclear reactor
fuel rods and heat exchanger tubes. The way Acoustic modelling and analysis through finite element
methods (FEMs) is done embodies some specific nuances (like pollution) towards result feasibility, as
in [8] and [9], where the author mentioned the effects of noise attenuation with resort to acoustic filters,
in an array of excitation frequencies. In these works there was sensitivity towards effects of pollution
(difference between the interpolation error and the total error) when considering computational acoustic
analysis, as explained in [10–13], for instance. The first three articles study the mesh number and wave
number’s influence in the convergence of acoustic (and subsequently ) FEM solutions, namely h-FEM
(where the error is modelled by C1hk + C2k3h2), based on the Galerkin method (typical FEM) and the
last one even proposes mixed boundary conditions updating through CLE (constitutive law error) as a
means of validating an acoustic model. This sensitivity is also considered when actual V-A transmissi-
blity is calculated and proposed a priori, since a good part of the enclosure (structural+acoustic) will be
acoustic, and a proper refining is needed for the best results.
When it comes to transmissibility computation in MDOF systems, this subject takes on a new mean-
ing and approach.
Until this point, there have been theoretical projections regarding the ways of actual V-A coupling
in the frequency domain, as observed in [14], which serve as a basis for some commercial FEM/FEA
(Finite Element Analysis) softwares in order to analyze Fluid-Structure Interaction (FSI). ANSYS ME-
CHANICAL APDL is the one employed in this work. With that being said, V-A Transmissibility already
exists in the literature, verified in the frequency domain, as presented in [15], but by the usage of trans-
fer functions (employed in certain transfer paths), with MDOF V-A Transmissibility not being actually
present. In this article, the author relates, inside said domain, outputs (receiver position for volume ve-
locity source strength estimation) with inputs (volume velocity source position for sound pressure, and
force) through transfer functions (also nominated FRF), by using microphones (two microphone method)
and accelerometers, in a much simpler approach that does not actually involve a wide coverage area
within the considered domain. In more recent years, with [16] the author conducted an acoustic and
structural characterization of a wooden cavity, in terms of natural frequencies and mode shapes to al-
low for a proper characterization and consequent vibration Transmissibility analysis in terms of V-As (if
possible). The nature of this work was based around numerical simulation (numerical measuring) and
experimental results, so there was still no progress with regards to estimating actual Transmissibility
Lastly, in [17], a method to estimate V-A Transmissibility is announced, based on FRFs, but the model
3
was not developed.
1.2 Motivation
Since the literature on V-A Transmissibility is scarce, almost non-existent among the scientific community
(at least in the frequency domain), and a full answer has yet to be achieved towards actual behaviour
prediction, there is a considerable interest towards estimation of Transmissibility between variables of
different nature. Indeed, it would be of interest to further deepen the relation between pressure distur-
bances in an acoustic medium and displacements (and subsequent loads) existing in a certain structure,
if the interaction is properly established and both mediums perform an enclosure.
So, by understanding the ways of structural and acoustic dynamics, one can realize how to numeri-
cally establish V-A Transmissibility, and by this way answer the main question that motivates this work.
1.2.1 Vibration and Acoustic Transmissibility in the Aerospace Industry
Specifically in the Aerospace industry there have been several studies with respect to vibration and
acoustic responses of aerospace structures. These have been mostly done with regard to specific ele-
ments that comprise these structures, for instance in [18], where the authors evaluate structure-borne
Transmissibility by analysis of an aircraft’s vibration dampers through Statistical Energy Analysis (SEA)
and FEA, or in [19], where the vibration and acoustic behaviour in composite aerospace structures
(sandwich panels) is studied, essentially uncovering which specific material combinations and core ge-
ometry ensure the best results.
More recently, Guedes in [2] proposed a 2D acoustic source localization (ASL) through MDOF Trans-
missibility, through a commercial FEM software, for further application inside the cabin of a mock-up
aircraft, more precisely where the seats for passengers would be located. A simpler model for source
localization inside a mock up A-340 had already been developed by Weber in [20], but using purely
scalar Transmissibility, in an experimental approach.
1.2.2 Acoustic Induced Problem in Aerospace Structures
The existence of acoustic sources (i.e. by volume acceleration or pressure imposition) and pressure
propagation within the innings/outings of an array of Aerospace structures (vehicles, satellites,etc) with
a structural or acoustic origin, gives rise to a necessity towards studying and predicting the regions
where this propagation is more heavily concentrated.
There already exist studies and articles dedicated towards analyzing how these structures are af-
fected and how to mitigate the damage, while ensuring the survivability and healthiness of the parts that
constitute the whole, for instance in [18, 21, 22].
In [22] the author proposes a V-A behaviour analysis, through SEA, of a fairing placed around launch
vehicles during flight missions, in order to protect any damage done to electronics and the like. During
these missions, such vehicles are submitted to dynamic pressure loading, as well as aero-acoustic and
4
structure borne excitation. Hence, it would be most beneficial if the design of these fairings would be
optimal towards where (acoustic) fatigue is most likely to appear and degrade the material faster.
It is also plausible to consider that there are specific points in interior cavities distributed along these
structures with extremely difficult access (some points inside the fuselage of a plane for instance). In
such cases, a more in-situ approach is quite often not possible, and the existence of a model which tries
to emulate with most accuracy the pressure levels (SPL) in these places, would be of some importance
in the industry and could contribute towards a better planning for sections (closed, tight spaces) that
typically are exposed to excitation or noise and often neglected, precisely because there are no easy
ways of reaching them.
1.3 General Overview of Vibro-Acoustic Transmissibility
Several previous works essentially define Transmissibility as an input/output relation between variables
of the same nature (normally), which manifest in sets of points and are measured through an enclosed
volume (and affect it), being it a load or a displacement within a structure or pressure (excitation, volume
acceleration) inside an acoustic medium.
On a first approach, within a steady-state/harmonic regime in Vibrations, for example in [1], the au-
thors propose the identification/estimation of applied loads in a mass/spring MDOF system, by the uti-
lization of Transmissibility matrices which relate, either from the dynamic stiffness or the FR (frequency
response) matrix, load input/output between sets of points. In [23], following similar principles, it is pro-
posed, verified and validated that in an MDOF system (beam) the displacement Transmissibility matrix
can be obtained from the load one, under certain conditions, which will be clarified further ahead in
chapter 2. Also, in [24], a Transmissibility based damage detection approach is proposed. This is done
through an Operational Modal Analysis (OMA), from which the Transmissibility coherence is extracted
and analyzed. Afterwards, an indicator of damage severity is developed and compared with already
developed ones, in order to verify any faults detected.
In acoustics, in works like [3], an Operational Acoustic Modal Analysis (OAMA) is done through
Transmissibility measurements defined as a ratio between pressure values in specific points, originated
by applying volume acceleration sources in multiple places within the defined volume (along a center-
line). By doing this, the author was able to identify acoustic parameters from output-only Transmissibility
measurements, already inspired by previous works like [25], where the same OMA logic was applied but
in a more general way, for structural parameters like damping ratios and damped natural frequencies.
Now, having mentioned the fact that certain studies have already been done regarding this sole topic,
is not exactly inferring that it can only be analyzed as it has been so far, as an adimensional quantity.
5
E
TEB
Figure 1.2: Vibro-acoustic Transmissiblity Inside a Car
The image presented in figure 1.2 can be taken as an example towards applicability, namely in the
aeronautic and automobile ones. For instance, as it can be seen, whenever a steady-state harmonic load
is applied in the car’s front wheel, the suspension of the car reacts and vibrations are induced in point
D. The same can be said about the exhaust’s vibration in point E, which will propagate through the main
body and enter the acoustic cavity (to points B or C). From this point, it can also be stated that the are
several other points with which interaction is established (existing therefore Transmissibility), for instance
A, B and C. These propagations (transmissibilities) are represented by T, besides the respective sets.
Let us assume that A, B and C are residing inside an acoustic medium, and inside this medium
every interaction can be dictated by pure acoustic Transmissibility (pressure disturbances), so, if there is
periodicity in the loads applied in the suspension of the car, and the exhaust’s vibration, one can make
the assumption there will also be periodicity in the pressure waves inside the car. A proposed reason for
this to happen, and as it will be studied in this work, would be the existence of a certain flexible interface
which connects and converts displacements to pressure disturbances and vice-versa.
AuA
Fluid-Structure Interface
x
y
Structure
B
C
PB
PC
Acoustic Medium
Figure 1.3: Simple Vibro-Acoustic Interaction Illustration
Figure 1.3 ends up simplifying what is suggested in figure 1.2, where the interactions are varied
and where there are multiple sources of noise as well as multiple points of interest to measure either
displacements or pressure fluctuations. Essentially, whenever there is a harmonic steady-state dis-
6
placement u imposed onto the structure (with whichever origin), some displacement vibration will travel
to the fluid-structure interface (FSI) through vibrational Transmissibility, where it will be ”converted” into
acoustic pressure P that will be travelling (and acoustically transmitted as described in [2, 3]) through
the acoustic medium. Afterwards it is measured at points of interest B and C.
A
B C
D
TADSt
TDBAc
TDCAc
Figure 1.4: Vibro-Acoustic Transmissiblity in a Landing Gear, with B and C located Inside the Fuselage(Cavity)
In figure 1.4, an illustration is presented for V-A Transmissibility on a landing gear, just like previously
for a car. In this case, the interface in D models displacement-pressure interaction and the Transmis-
sibility. For the case of Transmissibility within the structure TSt was used, and for the acoustic cavity,
TAc.
For this particular case, the rectangular box would be representing the fuselage of a certain plane,
as the considered acoustic cavity.
1.4 Objectives
The purpose of this work, to put it simply, is to answer a question, indicated in the following.
Through various works ([26, 27], besides the ones already mentioned) within the fields of Acoustics
and Structural Mechanics, the subject of Transmissibility is already quite developed, with an already
somewhat deep understanding of how it works and how it is presented in the frequency domain of a
steady-state, harmonic analysis. But, this depth still does not reach Transmissibility, considering the
already mentioned, almost non existence of published works in the area.
This lack of answers towards obtaining V-A Transmissibility in a range of excitation frequencies,
being it point-by-point or through matrices, outlines what is to be achieved with this thesis, as well as it’s
hardships.
A model is proposed, which aims to predict with a degree of certainty, how an enclosed volume
7
Structure-Fluid interaction converts certain variable to another (pressure to displacement or load, or
vice-versa) when there is an imposition on either side, being it inside a car with the engine running, or
inside a plane when the landing gear is active. This will be done for:
• A simpler single point (scalar) comparison, where displacement/pressure ratio (and vice-versa) are
calculated for specific coordinates inside the simulated model;
• A more complex approach, following the obtention of Transmissibility matrices which relate sets of
displacements within a structure, with pressure disturbances in a region inside an acoustic medium,
whose relation is generated through an interface.
It might be of relevance to mention that for the sake of semantics, while V-A Transmissibility is not adi-
mensional (hence it typically relates same nature variables), it is still considered Transmissibility nonethe-
less.
Now, to get back to the first paragraph of this subsection, can there be a verified numerical method-
ology that dictates how Transmissiblity can be calculated (estimated) in the frequency domain, like the
ones that already exist either for Vibrations or Acoustics?
1.5 Thesis Outline
This work is comprised by five distinct chapters. The first one, the Introduction is where the author gives
a brief description (introduction) towards the concept of Transmissiblity, vibrational and acoustic, gives
it context and industrial relevance while proposing a solution for the Transmissibility problem. In the
second chapter, the Theoretical Background, the most relevant theoretical aspects behind this work are
clarified and explained in a more detailed manner, mostly regarding structural and acoustic dynamics, as
well as how Transmissiblity comes to be and can be obtained, among others. In chapter 3, the method-
ologies followed to compute vibrational (verification and model computation), acoustic (verification and
1D/3D model computation) and (method/model proposition) Transmissibilities are presented, following
the background presented in the previous chapter. In chapter 4, all the results are presented and dis-
cussed for vibrational, acoustic (1D and 3D models) and Transmissiblity (3D) for single input/output as
well as MDOF. Finally, in chapter 5, the Conclusion, the main conclusions from this work are made
known and some suggestions for possible future work are presented.
8
Chapter 2
Theoretical Background
In this chapter, the theoretical fundamentals for this work, regarding the fields of Vibrations, Acoustics
and V-A coupling, as well as the respective notions of Transmissibility in each field.
2.1 Structural Dynamics Aspects
Before delving deeper into this work, a brief explanation on how solid mechanics, and in a more detailed
way, how the elastodynamic regime in solids is presented. So to say, there needs to be a clear under-
standing on how structures (and solids in general) behave when they are subjected to sets of loads or
imposed displacements.
Now this is quite a dense topic, so in this work, for the sake of not losing track of the actual objective,
only the essentials will be revised.
2.1.1 Elastodynamic Problem
2.1.1.1 Cauchy’s Law of Motion
When a certain body is submitted to external loading, it develops a distribution of internal forces, which
can either be surface or body forces (stresses). Since the field in study is Continuum Mechanics, these
forces are continuously distributed within the solid body and through it’s surface. This indicates the
existence of a stress state inside the body that can be quantified with resource to a stress tensor σji
(Cauchy’s stress tensor), which gives representation in a fixed reference frame. Generalizing, this state
of stress can be obtained for general coordinates, with the help of a stress (or traction) vector, along with
a normal direction nj , as in [28]:
Tni = σji.nj (2.1)
The law of motion provided by Cauchy in the field of finite Elasticity (Continuum Mechanics Theory) is
considered to be staple and essential.
Assuming a domain with surface S and volume V (certain mass, by relating with mass density ρ),
with surface tractions Tni and body forces Fi, in a state of equilibrium, obtains, through the expression
9
of equilibrium in terms of stress, that:
∫ ∫S
Tni dS +
∫ ∫ ∫V
FidV =
∫ ∫ ∫V
ρudV (2.2)
Through the substitution of equation (2.1) in (2.2), and applying the divergence theorem to the equation,
converting an enclosed surface integral to a volume one, it is obtained:
~∇.(σji) + Fi = ρu (2.3)
where σji is the Cauchy stress tensor, that encompasses direct and shear stress applied to the domain,~∇
the gradint operator, and Fi is the volumic force vector applied to the same domain and ρu is the inertial
term resulting from the body’s mass (with u being the acceleration term).
2.1.2 Euler-Bernoulli Beam Element (1D)
For rectiline structural members with cross section dimensions much smaller its length (from two orders
of magnitude), one can assume a 1D treatment [29]. In this model, it is assumed that the main axis of
the beam x remains perpendicular to the plane cross-sections even after the deformation. Under this
premise, assuming a homogeneous distributed downwards load applied along it’s length, it is possible
to establish a governing fourth order equation, that represents the relation between transverse loads
across the beam V , bending moments M , transverse displacement (deflection) w and slope θ DOFs:
ρA∂2w
∂t2+
∂2
∂x2
(EI
∂2w
∂x2
)= f(x, t) (2.4)
where E [Pa] is the Young’s Modulus of the material, I [m4] the Second Moment of Area, f [N] the
transversely distributed load, A [m2] the cross-sectional area and finally ρ [kg/m3] the mass density. This
expresses the strong form when combined with its four boundary conditions
Since this beam element is harmonically vibrating, one assumes:
w(x, t) = W (x)e−iωt (2.5)
where ω is the excitation frequency of transverse motion and W (x) is the mode shape (transversal
diplacement).
Now, from (2.4), one can obtain the weak form for the 1D beam element through integration along
the length. Firstly equation (2.5) is substituted in (2.4), then the partial derivative terms with relation to
time are removed through variable separation, and the remaining position dependant ones will no longer
10
be partially derived, but derived with respect only to position. This is done as follows:
∫ xe+1
xe
v
ρA∂2w
∂t2+
∂2
∂x2
(EI
∂2w
∂x2
)− f(x, t)
dx = 0⇔
⇔∫ xe+1
xe
(EI
d2v
dx2
d2W
dx2− ω2ρAvW
)dx+
v d
dx
(EI
d2W
dx2
)xe+1
xe
−
[(dv
dx
)EI
d2W
dx2
]xe+1
xe
= 0
(2.6)
where the functions v are auxiliary functions (weight function). The FEM for this beam element is pre-
sented in Appendix A.
Since equation (2.4) is a fourth order differential equation, after integration there will be four de-
pendant constants. Hence there will a need for four boundary conditions (at xe and xe+1) to close the
problem, some will be essential, corresponding to the deflection w and to the slope dWdx , and others
natural, corresponding to the bending moment EI d2Wdx2 and shear force
(ddxEI
d2Wdx2
).
2.1.2.1 Natural Frequencies of the Beam
For determining the natural frequencies on beams we assume,
w(x) = C1cosh(λx) + C2sinh(λx) + C3cos(λx) + C4sin(λx) (2.7)
By applying the (free-free) boundary conditions, and solving the indeterminate system, one obtains [29]:
λ4 = ω2 ρA
EI(2.8)
which results in:
ωnatural = (nπ)2
√EI
ρL4(2.9)
where n essentially represents the mode of vibration, from 1 to N, so to say from the fundamental to the
Nth natural frequency.
2.1.3 Spring Element
The spring element is a FE that is usually represented as illustrated in figure 2.1.
11
Figure 2.1: Spring Element
Following Hooke’s law, where the load applied on a spring depends on the stiffness and on the
displacement (F = kδ), one obtains:k(u1 − u2) = F1
k(u2 − u1) = F2
⇔
k −k
−k k
u1
u2
=
F1
F2
(2.10)
In equation (2.10) one can observe the two node spring element stiffness matrix K.
2.2 Vibrations in MDOF Systems
For this work and in reality in general, SDOF systems are not what is most widely present, therefore
existing only to establish MDOF assembled ones. Thereby single dimension variables will no longer be
regarded, but rather matrices representing systems.
These systems will exhibit an harmonic displacement through time x(t) = Xe−iωt (amplitude X),
under a free or forced regime, f(t) = Fe−iωt, where load amplitude F will be zero in the first case.
Primarily, in this subject, an undamped, freely vibrating body (where f(t) = 0) will be analyzed.
So to say, a body whose behaviour is solely described through inertia and stiffness. This body will be
described as having a stationary (yet dynamic) movement along N DOF.
The announced body (or system) can be described through the following equation:
([K]− ω2[M]) X = 0 (2.11)
Now, if the body has damping and has an applied force, the dynamic response is a different [30]:
([K] + iω[C]− ω2[M]) X = F (2.12)
which essentially comes from an initial equation of movement, after derivation with respect to time:
[K]x(t)
+ [C]
x(t)
+ [M]
x(t)
=f(t)
(2.13)
12
2.2.1 Frequency Response Functions
FRFs essentially describe behavioural responses of systems, in the frequency domain, from an initially
established movement equation. These responses tipically relate conjugate variables, for instance pres-
sure response to volume acceleration input.
An example of these would appear through a receptance matrix designated H, which encompasses
stiffness, mass and damping, as already seen in equation (2.12), and relates dynamic displacements
with excitation loads [30]:
X = H F (2.14)
This receptance matrix can also be interpreted as the inverse of the dynamic stiffness matrix [Z]:
[Z] = [K] + iω[C]− ω2[M] = [H]−1 (2.15)
2.2.2 Modal Analysis
From equation (2.11), there are non-trivial solutions to be obtained for a problem of either eigenvalues
and eigenvectors. This can be done with the help of:
det∣∣∣[K]− ω2[M]
∣∣∣ = 0 (2.16)
The solutions of the system of equations that arises are the N squared undamped natural frequencies
ω2r of an N DOFs system.
By substituting these values back into equation (2.11), one obtains a correspondent set of values for
X (said eigenvector, also represented as Ψr), which represents the mode shape of vibration of the
system, corresponding to certain natural frequency.
Ultimately, the modal model can be represented through the eigenmatrices [ω2r ] and [Ψr].
2.2.3 Transmissibility in Solid Structures
SDOF Transmissibility can be found clarified in an array of textbooks, for instance [31], where this relation
can be quite evident and linear as a displacement or load is transmitted immediately and predictably from
point A to point B in the system, following a uni-dimensional scalar input/output mathematical relation .
Just recently, methods have been developed to predict how randomly applied dynamic loads (and
again, forced dynamic displacements) are transmitted in MDOF to also randomly chosen points, [1,
23]. With MDOF systems the mentioned relation is not so simple anymore, therefore needing matricial
representation, where random multiple inputs will have multiple responses in an array of outcomes.
2.2.3.1 Load Transmissibility
In accordance with [23], the definition of a set of generalized coordinates for a generic MDOF system
needs to be done. For this to be achieved, some assumptions have to be made.
13
Firstly, it is established a set of coordinates, that will be called K, where the external (known) loads
are to be applied.
Figure 2.2: Set of Generalized Coordinates K, U and C (source Y.E. Lage et al [23])
Then, there is U , which defines another set where the unknown reaction forces will appear, and
ultimately the C set which encompasses all the other coordinates, as illustrated by fig. 2.2.YK
YU
YC
=
HKK HKU
HUK HUU
HCK HCU
FK
FU
(2.17)
From the receptance frequency response (FR) matrix H (where the receptance is defined by other sub-
matrices), which relates, in steady-state conditions, the dynamic displacement amplitudes Y (at dis-
cretized nodes of the structure), with the force amplitudes F, one describes a free body moving through
space. This is all expressed in (2.17).
Taking into account that the supports at U constrain the displacements, one can consider YU = 0,
which implies the following:
HUKFK + HUUFU = 0 (2.18)
which is equivalent to:
FU = −(HUU )−1HUKFK (2.19)
and ultimately comes to:
T(f)UK = −(HUU )−1HUK (2.20)
from which, comes the load transmissibility matrix:
(T(f)UK)+ = −(HUK)+HUU (2.21)
Besides being obtainable from the Receptance Matrix, the Transmissibility, linked to MDOF, can also be
14
obtained from the Dynamic Stiffness Matrix, just like in [1].
As it is done for FRFs, also in steady-state conditions, one can relate dynamic displacements and
external loading through the dynamic stiffness matrix [Z], which is defined in the eigenvalue/eigenvector
problem presented in section 2.2.2. In this case, as it was done in Maia, Fontul and Ribeiro et al [32], the
relation between known loads and unknown ones can be written like (and keeping the same designations
as for the receptance case): FK
FC
FU
=
ZKK ZKC ZKU
ZCK ZCC ZCU
ZUK ZUC ZUU
YK
YC
YU
(2.22)
with FC being a fictitious load.
Furthermore, coordinates K and C can be grouped into a new one that will henceforth be named G
(grouped). This changes equation (2.22) into: FG
FU
=
ZGG ZGU
ZUG ZUU
YG
YU
(2.23)
If one assumes that YU becomes a null vector, this will result in: FG
FU
=
ZGG
ZUG
YG
(2.24)
Equation (2.24) is essentially a system of equations that can be solved with respect to YG,
FU = ZUG(ZGG)−1FG (2.25)
which therefore will result in a load transmissibility matrix:
T(f)UG = ZUG(ZGG)−1 (2.26)
Since the C coordinate is fictitious, one can assume, for the sake of simplicity that the harmonic loads
applied on it can be zero, which is basically equivalent to saying that equation (2.26) will revert back to
having a K (known) as an index. This change will result in the disregarding of the column and row that
will be present in the final transmissibility matrix, which refer to the fictitious coordinate.
2.2.3.2 Displacement Transmissibility
In this segment, just like in the load transmissibility one, there is a need to generate a set of coordinates
in a structure (free elastic body). There is a small difference in this case though. Now there are no
constraints applied by supports, so the structure presents no reaction forces, where there used to be
ones. But just like for the previous case, there will firstly be an A set where the known external loads
15
F are being applied, then a K set which has the known Y responses, followed by a U set, where the
unknown Y responses exist, and finally a C set, where all the remaining coordinates of the structure are.
The FRF receptance matrix H relates dynamic displacement and external dynamic loading according
to the following equation:
YA
YU
YK
YC
=
HAA
HUA
HKA
HCA
FA (2.27)
Figure 2.3: Free elastic body with four sets of coordinates A, U, K, C (source Y.E. Lage et al [23])
While displacement transmissibility is considered, it is of extreme relevance to take into account that
the body is not constrained in any way, so that reactions are not developed in said supports (which is not
the case for load transmissibility), just like it is shown in figure 2.3. If the displacements in coordinates
U and K are caused by an harmonically applied load, and if equation (2.27) is added to the matter, one
comes by: YU
YK
=
HUA
HKA
FA (2.28)
which utimately, if the external loads are disregarded, comes to:
YU = T(d)UKYK (2.29)
where the displacement transmissibility matrix can be defined as:
T(d)UK = HUA(HKA)+ (2.30)
Considering that there are no restrictions to how the A set is constructed, it makes sense to assume
16
that U and A can coincide, converting equation (2.30) into:
T(d)UK = HUU (HKU )+ (2.31)
After some mathematical, from equations (2.21) and (2.31) one can relate the displacement transmissi-
bility with the load one [23]:
T(d)UK = ((T(f)
UK)T )+ =⇒ T(f)UK = ((T(d)
UK)+)T (2.32)
where T is for transposed and + is for pseudo-inverse.
Now, only under the condition that #U=#K can both relations be applied, because only in this sit-
uation is the possibility of properly inverting both displacement and load transmissibility matrices, as
explained in [23].
2.2.4 Finite Element Method - Vibrations
The theoretical deductions for the FEM in vibrating structures are shown in appendix A, namely the Hat
Functions, the boundary conditions, the Galerkin approximation and the element Matrices. This is done
for an Euler-Bernoulli (EB) 1D beam.
2.3 Acoustic Waves
The field of acoustics is often associated with the propagation of sound, through disturbances trans-
lated into waves, travelling along fluids. These disturbances (fluctuations or perturbations) convey the
propagation of acoustic waves, absent of mass transportation. As it will be further explained ahead,
perturbations as such are typically analyzed regarding pressure variations or fluctuations.
Among the various kinds of pressure fluctuations that can be observed in compressible fluids, acous-
tics waves can be seen as having various ways of travelling.
2.3.1 Types of Acoustic Waves
There exist a set of different ways a sound (acoustic) wave can propagate, there are some specific
terminologies, through which, these can be addressed, as in [33]. The said terminologies are next
mentioned by order of relevance in this work:
• Plane Wave - A plane sound wave exists when corresponding wavefronts of a sound wave propa-
gate parallel to each other;
• Progressive Wave - A wave whose direction of propagation is associated with a transfer of energy;
• Diverging Wave - one where the sound energy is spread over a continuously greater area as the
wave propagates away from the sound source;
17
• Standing Wave - produced by the constructive interference of two or more sound waves which
gives rise to a pattern of pressure maximum and minimum which is stable with time;
• Spherical Wave - Characterized by having a source that radiates equally along it’s surroundings.
2.3.2 Three-Dimensional Acoustic Wave Equation (Global Cartesian Coordinates)
Since the matter is being dealt in fluctuations/perturbations (as it usually is in the field of acoustics), from
this point on, being it either pressure (p around p0), velocity (u around u0) or density (ρ around ρ0), the
said variable will have an equilibrium value and, as mentioned, a perturbation [34].
u(x, y, z, t) = u0 + u′(x, y, z, t) (2.33)
p(x, y, z, t) = p0 + p′(x, y, z, t) (2.34)
ρ(x, y, z, t) = ρ0 + ρ′(x, y, z, t) (2.35)
In the case of velocity fluctuations u′, it can be named the velocity of a set of particles in a previous
state of equilibrium. This is illustrated, respectively for velocity, pressure and density, in equations (2.33)
to (2.35):
∂ρvx∂t
+∂ρvxvx∂x
+∂ρvyvx∂y
+∂ρvzvx∂z
= ρgx−∂p
∂x+Rx+
∂
∂x
(µe∂vx∂x
)+∂
∂y
(µe∂vx∂y
)+∂
∂z
(µe∂vx∂z
)+Tx
(2.36a)
∂ρvy∂t
+∂ρvxvy∂x
+∂ρvyvy∂y
+∂ρvzvy∂z
= ρgy−∂p
∂y+Ry +
∂
∂x
(µe∂vy∂x
)+∂
∂y
(µe∂vy∂y
)+∂
∂z
(µe∂vy∂z
)+Ty
(2.36b)
∂ρvz∂t
+∂ρvxvz∂x
+∂ρvyvz∂y
+∂ρvzvz∂z
= ρgz−∂p
∂z+Rz +
∂
∂x
(µe∂vz∂x
)+
∂
∂y
(µe∂vz∂y
)+∂
∂z
(µe∂vz∂z
)+Tz
(2.36c)
with u(x, y, z, t)=(vx, vy.vz).
These three equations, from (2.36a) to (2.36c) , are known as the Navier-Stokes equations, which
essentially depict conservation of momentum, taking into account an homogeneous flow in cartesian
coordinates. They encompass:
• The density of the acoustic fluid ρ;
• The flow’s velocity in every direction vx , vy , vz;
• The acceleration of gravity in each component gx , gy , gz;
• The pressure p;
• The effective cinematic viscosity µe;
18
• The losses due to the effect of viscosity in every direction Tx , Ty , Tz;
• The components of distributed resistance Rx , Ry , Rz ;
• Time t.
Simplifications for these equations can be made when regarding acoustics, as it is done in [35], being
those:
• Inviscid fluids, which, when compared to solids, show fewer constraints to deformations (so to
say, the propagation of a wave is made possible with pressure changes through compression and
decompression);
• Compressible fluids, which implies that density variations are caused by pressure changes;
• Mean flow is non-existant;
• Uniform average pressure and density.
Before advancing any further it would be relevant mentioning that the calculations made in order
to get to the final Acoustic Wave formula are done through an approach, where it is assumed a fixed
infinitesimal control volume δV through which there is a passing homogeneous flow (for the sake of
linearity, a cube is assumed).
This model provides the tools to establish the equation of mass conservation, also known as the con-
tinuity equation, which essentially dictates that whatever mass goes into a control volume with a certain
shape has to be equal to whichever resides inside plus what comes out. So to say, the connection be-
tween expansion, compression and motion is created this way, and since it is considered, as previously
mentioned, a compressible fluid (air), this connection is mostly between particle velocity ~u and density
ρ. This equation is presented as such:∂ρ
∂t+ ~∇.(ρ~u) = 0 (2.37)
being ~∇ the gradient operator, which encompasses the partial derivatives in every taken Cartesian frame
direction (in this case, since no coordination transformation as been done).
~∇ =∂
∂x~x+
∂
∂y~y +
∂
∂z~z (2.38)
It is known from equations (2.33) to (2.35) that the density has an equilibrium value with an acoustic
perturbation (acoustic density). This is joined to the fact that the equilibrium velocity u0 is zero (u=u′).
With this, equation (2.37) can be seen as an equation of acoustic perturbations, through the sheer
substitution of variables, as follows [35]:
∂ρ′
∂t+ ~∇.(ρ~u) = 0 (2.39)
Now, by making an Eulerian assumption, where the control volume moves with the fluid, it is going
to be taken an infinitesimal force applied to the said mentioned control volume. This translates into a
19
Newtonian force situation as described by:
d~f = −~∇pdV + ~gρdV (2.40)
which essentially can be interpreted as:
− ~∇p+ ~gρ = ρ
(∂~u
∂t+ (~u.~∇)~u
)(2.41)
considering d~f = ~a ρdV .
Since the acceleration in a fluid is calculated from the partial derivative of its velocity through time
and through the variation of each of its components in the correspondent coordinate, the final equation
obtained whose name is the linear Euler’s (conservation of momentum) equation, is represented by:
ρ0∂~u
∂t= −~∇p′ (2.42)
Almost now getting to the final linear wave equation, one final step is required, being it, applying
a gradient operator in both sides of the equation (2.42), deriving equation (2.39) in time, substituting
variables and generating:~∇2p′ =
∂2ρ′
∂t2(2.43)
with ~∇.~∇=~∇2.
In order to obtain the lossless wave equation it is needed to establish an isentropic relationship
between pressure and density fluctuations. This will be done resorting to a Taylor expansion about an
equilibrium density, as in [35]:
p = p0 +
(∂p
∂ρ
)ρ=ρ0
(ρ− ρ0) + ... (2.44)
Higher order terms are disregarded since this is a linear problem. It is also considered that a change in
density translates into a change in acoustic pressure, hence:
c2 =
(∂p
∂ρ
)ρ=ρ0
(2.45)
Equation (2.42) can be rewritten as:~∇2p′ =
1
c2∂2p′
∂t2(2.46)
which represents the mentioned loss-less wave equation.
An acoustic fluid is fundamentally different from a typical fluid. One of the most relevant differences is
that, while modelling the later as a group of particles enclosed to a certain volume (as it has been done
before), these particles will exhibit movement described through translations and rotations, whereas the
typical acoustic fluid particle does not show any kind of rotation when disturbed. This is assumed as
an irrotational fluid. Besides the typical compression and expansion, there is no presence of rotation,
therefore being able to call the said particles irrotational where their velocity, as the perturbed velocity
20
can be seen as:~∇× ~u = 0 (2.47)
Equation (2.47) is only valid because the fluid was at an equilibrium (u0=0), and an initially irrotational
fluid remains resting throughout the whole time spectrum. Essentially there are no vortices forming
(absence of vorticity), because forces are only being applied in the center of mass of the mentioned
particles [34].
Therefore, the velocity can be expressed as the gradient of a certain scalar function (a potential), as
in:
~u = ~∇Φ (2.48)
By replacing equation (2.48) in (2.42), the following is obtained:
p′ = −ρ0∂Φ
∂t(2.49)
2.3.3 Harmonic Plane Waves
One solution that can be obtained from the previous wave equation is of a certain wave that travels
harmonically (without angular frequency variations) in a certain direction, through a plane. This wave
travels along a homogeneous fluid, where the velocity of wave propagation c is invariant. Essentially,
every property of the fluid along the direction of propagation remains unchanged, something that does
not happen with diverging waves, when short distances are considered. Considering now propagation
in a specific direction, let us say, along the Ox axis. Equation (2.46) is now reduced to the 1D case:
∂2p′
∂x2=
1
c2∂2p′
∂t2(2.50)
which has a complex solution described by:
p′ = C1ei(ωt−kx) + C2e
i(ωt+kx) (2.51)
where:
• ω, which is the angular frequency of the wave related with the frequency f itself through:
ω = 2πf (2.52)
• k, which is the repentency, also known as the acoustic wavenumber and is obtained by:
∥∥∥~k∥∥∥ = k =ω
c(2.53)
When only unidirectional propagation is taken into account, k is simply taken for a scalar unit. But
this is not always the case. It is actually a vector (propagation vector) with dimension in every
cartesian coordinate, and equation (2.53) is only referring to its magnitude.
21
• −x and +x, which indicate the axis and the direction of propagation;
• t for time;
• C1 and C2, which designate the amplitude of the waves propagating on either direction, and which
can differ or be the same.
Equation (2.51) can also be represented as:
p′(x, t) = f(x− ct) + g(x+ ct) (2.54)
which indicates propagation from a certain point,through a certain direction (with said orientation), in the
taken case, x.
From equations (2.42) and (2.51), one can obtain the formulation:
~u =
[(C1
ρ0c
)ei(ωt−kx) −
(C2
ρ0c
)ei(ωt+kx)
]~x (2.55)
also consideringOx propagation. A new term appears in here, known as the specific acoustic impedance
Z0 = ρ0c (for the case of plane waves), which will be clarified further ahead.
In the case that an arbitrary direction is assumed, slight adjustments will have to be made, namely in
equation (2.51), since the propagation is no longer restricted to a singular direction, thereby getting [35]:
p′ = A1ei(ωt−kxx−kyy−kzz) (2.56)
with ~k = (kx, ky, kz).
2.3.4 Determining the Speed of Sound in Fluids
As it was already mentioned, the thermodynamic sound speed can be determined with the help of
equation (2.45).
When sound is travelling a said perfect fluid, and considering a nearly isentropic process, where the
temperature gradients and thermal conductivity of the fluid are such that there is no heat transmission
to neighboring fluids, one gets to the following pressure/density relationship:
p
ργ=p0
ργ0(2.57)
where p0 [Pa] and ρ0 [kg/m3] are respectively the pressure and density in a state of equilibrium, and γ is
the isentropic expansion factor (γ=Cp
Cv,specific heat ratio, which is equal to 1.402 in the case of the fluid
being air).
Now resorting to (2.45), it is secured:
c2 = γp0
ρ0(2.58)
22
From the equation of state, one knows that:
p0
ρ0= RT (2.59)
where R [J/(K.mol)] is the universal gas constant and T [K] is the absolute gas temperature. Since
R typically remains constant within the same gas, c [m/s] will essentially depend upon T . It is finally
realized that c can be determined from equations (2.58) and (2.59), as follows:
c = (γRT )12 (2.60)
The speed of a sound wave can also be determined via:
c = λf (2.61)
with λ being the wavelength and f the frequency of the wave.
2.3.5 Acoustic Impedance
The acoustic impedance Z [Pa.s/m] is essentially the ratio of acoustic pressure to the particle speed in
a medium. It translates into the resistance/opposition of the medium to the perturbations that form an
acoustic flow, and is defined by [35]:
Z =p′
u′(2.62)
For the case of plane waves, equation (2.62) is reduced to (see [34]):
Z0 = ρ0c (2.63)
Just like it is verified for electronics, the same happens with acoustics, where this impedance can be
interpreted as a complex function, where there will be a acoustic resistance a as well as acoustic reac-
tance b.
Z = a+ ib (2.64)
This does not happen for progressive plane waves, where it is simply a real value, but does happen for
diverging and standing plane waves.
2.3.5.1 Anechoic and Reflective Boundary
The relation between a wave reflected on a wall and the wave that originated it can be interpreted as a
ratio r of acoustic pressure equations, that define either incident or reflected wave propagation. Thus,
from equation (2.51) and [36]:
r =p′rp′i
=C2
C1(2.65)
23
x = −L x = 0
Incident Wave
Reflected Wave
Figure 2.4: Unidimensional Sound Wave Reflecting on a wall
From equation (2.62), (2.51) and (2.55), and considering the wall coordinate (x=0):
Z(x=0)
Z0=
1 + r
1− r(2.66)
Now, if the wall does not reflect any wave (anechoic boundary), r = 0 and the impedance will just be the
specific acoustic impedance. If this is not the case, and the wall is completely reflective (r = 1), then the
impedance has an infinite limit value (Z(x=0)
Z0=∞)
2.3.6 Helmholtz’s Equation
As far as an acoustic wave is concerned, one can consider an harmonic perturbation with a certain
spatial pressure distribution. Having said this, the scalar function in equation (2.48) can be defined as
an harmonic propagation with a certain complex amplitude, as in:
Φ(x, y, z, t) = ψ(x, y, z)e−iωt (2.67)
where the mentioned complex amplitude (with no time dependence) is defined by ψ.
Through sheer substitution of equation (2.48) and (2.49) in (2.39), one gets:
(ω
c
)2
ψ + ~∇2.ψ = 0 (2.68)
From equation (2.53), the final Helmholtz equation takes the following shape [37]:
k2p′(x, y, z) + ~∇2.p′(x, y, z) = 0 (2.69)
24
where ψ can be interpreted as p′, which can also be known from equations (2.67) and (2.49), where it
is shown that the time dependence is irrelevant, while by deriving the complex amplitude remains the
same.
This final equation will be relevant again in this work, when the FEM is regarded. Namely in the initial
strong form (and conversion to weak), described in equation (2.69).
If a source is considered, equation (2.69) will change into [14]:
k2p′(x, y, z) + ~∇2.p′(x, y, z) = −iωρ0q(x, y, z) (2.70)
where the only still unknown term is the external source distribution q and i =√−1.
2.3.7 Imposed Boundary Conditions
Equation (2.70) governs a steady-state acoustic pressure propagation wave within a enclosed fluid do-
main V and a certain source spatial distribution q. Adding just a small remark, for the sake of simplicity,
from now on the condition p′ = p will be applied, since the only pressure needed is the acoustic one
(perturbation).
To define the pressure field in V , one needs to define a boundary condition for each position on the
closed boundary surface Ω = Ωp ∪ Ωv ∪ ΩZ :
• Imposed Pressure:
p = p ,on Ωp (2.71)
• Imposed Normal Velocity:
vn =i
ρ0ω
∂p
∂n= vn ,on Ωv (2.72)
• Imposed Normal Impedance:
p = Z.vn =iZ
ρ0ω
∂p
∂n,on ΩZ (2.73)
2.3.8 Transmissibility in the Field of Acoustics
The notion of transmissibility has been, for some years now, already quite well established in the field of
Structural and Solid Mechanics. While this remains true, this notion has been rising and getting a good
dimension in the field of Acoustics as well.
Typically this concept would be invoked whenever there where loads (distributed or punctual) being
applied to a system or even displacements. In these cases, these kinds of stimuli would be ”transmit-
ted” to certain chosen points, also inside the system, where there was the need to uncover how said
forces/displacements would be acting. But when it comes to Acoustics, this whole discussion goes to
another level. Now there are no, so to say, forces (said loads) or displacements being applied to a struc-
ture since there is no longer a solid one, but there is in fact, in it’s place an acoustic fluid where there
are pressure, velocity and density fields (with disturbances about equilibrium points). Because of this,
25
the inputs and outputs of the transmissibility functions will now be imposed pressures values, as it is de-
scribed in [3]. So, whenever there is a pressure disturbance (or excitation source, volume acceleration)
imposed in a certain point in an acoustic fluid, a transmissibility model will indicate how this disturbance
is manifested in another point in the said field, by conveying a pressure response. This model can be
described with the helping hand of the following equation:
T kir(ω) =pi(ω)
pr(ω)=Hik(ω)qk(ω)
Hrk(ω)qk(ω)(2.74)
where for a single acoustic source present at a known DOF k, T kir(ω) is the ratio between the fixed
pressure measured at a reference-output DOF r and a measured one at DOF i. Since both the reference
and the source points or locations are fixed, and while the reference is well chosen, one can assure that
the transmissibility functions are established with most certainty.
Similarly to a dynamic structural system, which in steady-state regime can be modelled in the fre-
quency domain by equation (2.12) (with damping or not), so can a dynamic acoustic one (where the
damping would be the acoustic impedance Z), through a similar equation, as introduced in [3]:
([K]− ω2[M] + iω[C])
P (ω)
=Q(ω)
⇔
Z(ω)
P (ω)
=Q(ω)
(2.75)
where:
• [K] is the Acoustic Global Stiffness Matrix;
• [M] is the Acoustic Global Mass Matrix;
• [C] is the Acoustic Global Damping Matrix;
• P (ω) is the pressure vector (equivalent of the displacement vector Y);
• Q(ω) is the volume acceleration vector (equivalent of the mechanical load vector F).
Having stated the formerly, one can infer that either Dynamic Stiffness Matrix Z(ω) and Receptance
Matrix H = Z−1 can be used to predict transmissibility in an acoustic fluid (for this work the air is
considered, as in [2]).
2.3.8.1 Dynamic Stiffness Method
Considering three sets of DOFs, K, U and C, then equation (2.75) may be rewritten to (similar to the
described in section 2.2.3.2):ZKK ZKC ZKU
ZCK ZCC ZCU
ZUK ZUC ZUU
PK
PC
PU
=
FK
FC
FU
(2.76)
For the remaining exposition, the volume acceleration will be treated as an acoustic load vector F.
Since the dynamic stiffness of the fluid is an intrinsic property of it (as the transmissibility as well), in
order to determine pressure transmissibility, the acoustic loads imposed in every set of coordinates U
26
(where the pressure will be imposed), K (where the pressure is known) and C (the rest of the coordi-
nates) are null, so F = (0, 0, 0)T , resulting in:
ZKK ZKC ZKU
ZCK ZCC ZCU
ZUK ZUC ZUU
PK
PC
PU
=
0
0
0
(2.77)
Similarly to the structural systems, through equations (2.22-2.23), sets K and C will be grouped into a
new set O, which turns equation (2.77) into:ZOO ZOU
ZUO ZUU
PO
PU
=
0
0
(2.78)
where PU is a known imposed pressure in the set of coordinates U . From equation (2.78), one gets:
ZUOPO + ZUU PU = 0 (2.79)
After some algebraic manipulations, equation (2.79) turns into:
PO = −(ZOO)−1ZOU PU −→ TZOU = −(ZUO)+ZUU (2.80)
2.3.8.2 Frequency Response/Receptance Method
PK
PU
PC
=
HKK HKU
HUK HUU
HCK HCU
FK
FU
(2.81)
The acoustic loads in C are assumed to be zero FC=0.
From the first two equations of the system described in (2.81)
PK = HKKFK + HKUFU (2.82a)
PU = HUKFK + HUUFU (2.82b)
Now, by defining FU as a function of PU , one gets:
FU = (HUU )−1(PU − HUKFK) (2.83)
If one substitutes (2.83) into (2.82a), and while considering FK = 0 the result is:
PK = HKU (HUU )−1PU (2.84)
27
which will consequently give out the corresponding transmissibility matrix, between the sets:
T aKU = HKU (HUU )−1 (2.85)
2.3.9 Finite Element Method - Acoustics
The FEM (h-FEM, Galerkin) used in some of the acoustics in this work, from the Helmholtz Equation,
was based on the description in [38], which was taken from [14].
2.4 Coupled Vibro-Acoustic Problem
Much like an acoustic field (or a fluid pressure field), which is comprised by a boundary Ω inside a said
volume V , as described in section 2.3.7, a V-A one will also be defined inside this volume, but with a
slight difference. In this case, the boundary enclosing the volume, Ω, will have another segment which
will correspond to the coupling with a vibrational field or a structure. So, this new boundary surface will
be composed of Ω = Ωp ∪ Ωv ∪ ΩZ ∪ Ωs, where the s refers to an elastic structure.
As it can be seen in [14], in this new segment Ωs, a new boundary condition, with respect to the
continuity of normal velocity, will be imposed. This condition will represent the V-A coupling through the
equality between normal fluid velocity and normal structural velocity:
• normal velocity continuity:
vn =i
ρ0ω
∂p
∂n= iωwn ,on Ωs (2.86)
where wn is the normal displacement imposed unto the structure.
Once again, there is relevance in referring that these boundary conditions are imposed in equation
(2.70).
In a V-A system, there exists an array of elastic structures that could be a part of it. Nevertheless,
a shell-type elastic structure is chosen, since this type has a relatively small thickness dimension and
subsequent mass, and V-A effects occur mainly for these kinds of conditions.
Figure 2.5: Shell Coordinate System, as in [14]
For shell-type structures, as the one in figure 2.5, the displacement field is characterized through
consideration of coordinate system (s1,s2,sn) at the centre of the shell’s surface and composed of
(ws1 ,ws2 ,wn), where n has a positive orientation with respect to the fluid. These displacements are
28
regulated by a certain linear relation, represented as follows:
([Ls]− ω2[Ms]).
ws1(r)
ws2(r)
wn(r)
=
fs1(r)
fs2(r)
fn(r)
+
0
0
p (r)
, r ∈ Ωs (2.87)
with [Ls] being a matrix of differential operators of the shell structure governing elastic and damping
forces, and [Ms], the inertial parameters of the same structure.
The V-A coupling can be done with resource to three distinct formulations. These are the Eulerian
one, the Lagrangian one and finally, the mixed one.
2.4.1 Coupling Formulations
2.4.1.1 Eulerian Formulation
In this formulation, the acoustic and structural responses are usually defined by a single scalar function,
for instance the pressure for the acoustic case, and the displacement vector for the structural one.
2.4.1.1.1 Acoustic FE Model
The FE approximation of the steady-state pressure p in the enclosed fluid domain with volume V , is an
expansion p in terms of a set of global functions Ni, as it is represented in:
p(x, y, z) =
na∑i=1
Ni(x, y, z).pi +
np∑i=1
Ni(x, y, z).pi ⇔
p(x, y, z) = [Na]. pi+ [Np]. pi , (x, y, z) ∈ V
(2.88)
where np is the number of DOFs where a pressure pi is imposed, i.e. the number of constrained DOFs.
These nodes are located on the pressure section of the curve Ω enclosing V , Ωp, and are essentially
represented by vector pi. The global shape functions associated with them are comprised into vector
[Np]1xnp. As for the unconstrained DOFs, they are represented by pi, number na and are comprised into
vector pi. The global shape functions associated with them are comprised into vector [Na]1xna .
The unconstrained DOFs FE model is as follows,
([Ka] + iω[Ca]− ω2[Ma]) pi = Fa (2.89)
where Fanax1 contains the contributions from the external acoustic sources in the fluid domain V , the
terms from the constrained DOFs and lastly, the contributions from the dictated normal velocity vn which
is imposed upon Ωv and has positive orientation relative to the domain V . These last contributions are
defined as: ∫Ωv
(−iρ0ω[Na]T vn).dΩ (2.90)
29
2.4.1.1.2 Structural FE Model
The FE approximation for dynamic displacements on an elastic shell’s surface Ωs, in a steady-state
regime and global cartesian coordinates is represented as follows:wx(x, y, z)
wy(x, y, z)
wz(x, y, z)
= [Ns] wi+ [Nw] wi (2.91)
where the matrix [Ns]3xnscomprises the global shape functions corresponding to the nS unconstrained
translational and rotational DOFs wi1xnsin the shell structure. The [Nw]3xnw
matrix is essentially the
same, given it also contains the global shape functions but, in this case, for the constrained rotational
and translational DOFs wi1xns. The FE model for unsconstrained DOFs on the shell presents itself as
the following,
([Ks] + iω[Cs]− ω2[Ms]) wi = Fs (2.92)
The contributions from all the shell elements within the plate is represented by:
nse∑e=1
(∫Ωse
([Ns]. ne .p
)dΩ
)(2.93)
2.4.1.1.3 Coupled Model
The force loading of the acoustic pressure on the elastic shell structure along the fluid-structure coupling
interface in an interior coupled V-A system may be regarded as an additional normal load. This new
normal load will force the appearance of an additional term to be added to the structural FE element:
([Ks] + iω[Cs]− ω2[Ms]) wi+ [Kc] pi = Fsi (2.94)
with [Kc] having dimension nsxna, corresponding respectively to the number of structural and acoustic
unconstrained degrees of freedom. The dimension of Fsi will be 1xns.
The continuity of the normal shell velocities and the normal fluid’ velocities at the fluid-structure
coupling interface may be regarded as an additional velocity input on the part Ωs of the boundary surface
of the acoustic domain. The resulting modified acoustic FE model consequently becomes:
([Ka] + iω[Ca]− ω2[Ma]) pi − ω2[Mc] wi = Fai (2.95)
By combining the modified structural and acoustic FE, one obtains [14]:Ks Kc
0 Ka
+ iω
Cs 0
0 Ca
− ω2
Ms 0
−ρ0KTc Ma
wi
pi
=
Fsi
Fai
(2.96)
The resulting matrices are no longer symmetric (unlike the acoustic ones, without damping). This is due
30
to the fact that the force loading of the fluid on the structure is proportional to the pressure which results
in a cross-coupling matrix Kc in the coupled stiffness matrix, while the force loading of the structure
on the fluid is proportional to the acceleration, resulting in a cross-coupling matrix in the coupled mass
matrix −ρ0KTc .
2.4.2 Limitations of Coupled Finite Element Models
Coupled FE/FE present inadvertently some limitations, such as:
• In order to incorporate the V-A coupling effects, the structural and the acoustic problem must be
solved simultaneously, whereas for uncoupled V-A systems, the structural and the acoustic problem
may be solved in a sequential procedure;
• Coupled models have a much lower efficiency, being at the same time much larger in size, because
of new DOFs;
• Reduced efficiency of model size reduction techniques.
2.4.3 Using FRF to compute Vibro-Acoustic transmissibility
The deductions ahead are based off the logic used in section 2.3.8.2 [2].
Two approaches are proposed here, one based on pressure imposition (and structure response with
dynamic displacement), and the other on dynamic load imposition in a structure (causing displacement
in structure, and acoustic response in the form of pressure in the fluid). This is done in the announced
order, with U and K being the set of imposition and response respectively.. uK
PU
=
HKK HKU
HUK HUU
FK
FU
(2.97)
From the system described in (2.97)
uK = HKKFK + HKUFU (2.98a)
PU = HUKFK + HUUFU (2.98b)
Now, by defining FU as a function of PU , one gets:
FU = (HUU )−1(PU − HUKFK) (2.99)
If one substitutes (2.99) into (2.98a), and while considering FK = 0 the result is:
uK = HKU (HUU )−1PU (2.100)
31
which will consequently give out the corresponding transmissibility matrix, between the sets:
TFSKU = HKU (HUU )−1 (2.101)
Now considering the other idea, where the imposition is done through dynamic displacements on the
plate. uU
PK
=
HUU HUK
HKU HKK
FU
FK
(2.102)
From the system described in (2.102),
uU = HUUFU + HUKFK (2.103a)
PK = HKUFU + HKKFK (2.103b)
and defining FU as a function of uU , one gets:
FU = (HUU )−1(uU − HUKFK) (2.104)
If one substitutes (2.104) into (2.103a), and while considering FK = 0 the result is:
PK = HKU (HUU )−1uU (2.105)
and consequently
TSFKU = HKU (HUU )−1 (2.106)
U K
FSI FAI C
z = 0 z = −L
Structure
Int.
Acoustic Fluid
u↔ P
Figure 2.6: Vibro-Acoustic Interaction depicted in Sets of Coordinates U , K and others C, for u imposi-tion
Figure 2.6 describes the latter approach, as it includes both the strcutural and the acoustic sets of
coordinates, as well as the modelled interface, and the interaction forces.
32
Chapter 3
Methodologies
To answer the question on how to estimate Vibration, Acoustic and V-A responses through transmissi-
bility functions, this work follows a sequence.
Firstly, it starts with an initial verification of results already present in the literature [1] and [23]. The
methodology to be presented is shown and justified accordingly, following the theoretical background
introduced in the previous Chapter.
The main idea is first to verify both the acoustic and vibrational transmissibility and after try to extend
to V-A by using a V-A model for V-A point by point and matricial transmissiblity estimation.
The methodologies to do this are presented in this chapter.
3.1 Dynamic Force and Displacement Transmissibility Verification
To verify results present in Neves and Maia et al [1], for force transmissibility matrices obtained either
through the Receptance Matrix H, or through the Dynamic Stiffness one Z, as explained in section 2.2.3,
the following methodology is presented. This verification was done firstly for a specific mass/spring
system, and then for a simply supported beam.
3.1.1 Mass/Spring System
Just like it was aforementioned, a simple discretized mass/spring system will be assembled in order to
verify already existing results regarding force transmissibility in MDOF systems.
An MDOF system is considered (in this case the number of DOFs is coincident with the number
of nodes), with an harmonic set of known loads applied in certain nodes, and unknown reactions in
some others. The stiffness of the system is simply defined by the multiple springs positioned in between
specific nodes and represented through a Global Stiffness matrix [K]NxN , whose assemblage is done
through a connectivity table, as typically done with FEMs. This final [K] matrix is brought upon with
assembled equations, which contain the sum of coefficients and source terms at all nodes.
As for the Global Mass matrix [M]NxN , it is a diagonal one.
33
Each spring element has two local nodes (1 and 2) and stiffness of k. The stiffness of this element
is defined through a 2x2 matrix:
[K]e2x2 =
k −k
−k k
(3.1)
So, a MATLAB (see manual [39]) algorithm is made in order to calculate the assembled global ma-
trices of a considered mass/spring system, followed by transmissibility matrices, using the dynamic
stiffness as well as the Receptance/FR matrices. This algorithm will look as such:
• Initially, the mass and stiffness constants are defined for mass and spring elements, as well as the
already assembled (with equation 3.1, for K) global mass and stiffness matrices are defined as
inputs to be manipulated (depending on the relative arrangement of each spring and mass node,
which is set on a connectivity table, as it will be seen further ahead in the results chapter regarding
this part). The load vector to be applied in the system is also declared in this step;
• Then, a cycle is run, where in each iteration (depending on the excitation frequency range), a
number of procedures occur:
– Obtention of the dynamic stiffness for the applied frequency Z(ω);
– Definition of sub-matrices of Z(ω), according to (2.26), which relate the sets of coordinates
(see (2.22)), between which, the transmissibility matrix will be computed;
– Then Z is inverted, originating H, subsequent sub-matrices will also be declared, following
eq. (2.17). The Transmissibility will then be computed, following equation (2.21);
• Finally, a specific value for transmissibility, obtained from both methods, will be chosen and plotted
in the excitation frequency range for comparison.
3.1.1.1 Dynamic Stiffness
After the FE model is built and the global matrices are assembled, from equation (2.12) one can establish
a dynamic stiffness matrix Z that changes with incrementing excitation frequency (inside the code) ω:
[Z(ω)] = [K]− ω2[M] (3.2)
To obtain the transmissibility matrix, one will follow the theoretical principles introduced in section 2.2.3
for the dynamic stiffness part.
The transmissibility force matrix is thereby obtained through equation (2.26). Since that, for the sake
of calculations, the known K-set and other coordinates, C-set, were grouped in the set (G), the only rel-
evant values inside the matrix will be those referring to the region of application of the harmonic external
load (the lines in the matrix corresponding to these sets), as well as the unknown load (constraining/re-
acting) region, so K and U , respectively.
34
3.1.1.2 Receptance Matrix
The approach to be taken in this section is quite similar to the previous one in terms of applicability.
Firstly, the FRF which relates dynamic displacement with applied dynamic loads does it so, with the
help of the so called receptance matrix H(ω), present in equation (2.14), which can also be represented
as the inverse of the previous dynamic stiffness matrix Z, as in equation (2.15).
Next, the force transmissibility matrix will be obtained through some algebra with the help of equation
(2.21), where the explicit relationship between receptance sub-matrices in known and unknown load
coordinates, is expressed.
3.1.1.3 Comparison of Nodal Reactions Between Both Methods
As a way of further validating the methods used to determine the Transmissibility matrix, one might
desire to show results regarding the reactions at the constrained nodes, which is done with the help of
equations (2.19) and (2.25):
RZU = T(f)UGFG (3.3a)
RHU = T(f)UKFK (3.3b)
3.1.2 Simply Supported Beam
The methodology for obtaining force transmissibility functions in a discretized spring/mass system can be
generalized since the principles announced in section 2.2.3 are applicable to elastodynamic problems.
This implies that it can therefore be applied in systems like a simply supported beam.
Figure 3.1: Euler-Bernoulli beam element, based on [29]
Much like the spring/mass system, the beam has to be discretized into small EB 2D beam elements,
with two nodes each, but in this case, each element will have four DOFs, two for translational vertical dis-
placement and the other two for rotation about Oy, as described in section 2.1.2. The fact that four DOFs
are considered implies that the mass and stiffness element matrices are 4x4 matrices, represented in
equation (3.4), whose deduction is shown in appendix A, [29]:
35
[Ke4x4] =EI
l3e
12 6le −12 6le
6le 4l2e −6le 2l2e
−12 −6le 12 −6le
6le 2l2e −6le 4l2e
, [Me4x4] =
ρAle420
156 22le 54 −13le
22le 4l2e 3le −3le
54 13le 156 −22le
−13le −3l2e −22le 4l2e
, (3.4)
where E [Pa] is the Young’s Modulus, I [m4] is the second moment of area, A [m2] the cross-sectional
area of the beam (taking into account that the element used does not change, neither transversely nor
longitudinally), ρ [kg/m3] the density and finally le [m] is the length of the beam element used. These
matrices are obtained through a FEM formulation, as described in section 2.1.2, and are based on [29].
The final Global Stiffness and Global Mass matrices ([K]NxN and [M]NxN , respectively) are assem-
bled, with it’s connectivity table. After the assemblage, dynamically imposed loads and displacements in
certain nodes of the final beam structure will enable the study and comparison of force transmissibility
versus displacement transmissibility, under certain needed conditions, much like it was already done in
the literature [23] (where this implementation was based on), and whose theory used was clarified back
in chapter 2. This final analysis is to be done in a spectrum of excitation frequencies ω, following again,
equations (2.21) and (2.26).
3.1.3 Beam Modelling in ANSYS APDL
In order to further verify the models that were either built in MATLAB according to their description in the
literature, a model of a simply supported beam is developed in ANSYS Mechanical APDL (see [40]).
With this, besides aiming for a verification of already produced models, one also desires to give
an introduction to how one can work around the ANSYS APDL software in order to implement a V-A
interaction model and extract data from it, with the purpose of verifying V-A transmissibility further ahead
in this work.
Figure 3.2: BEAM3 element (source [41])
The beam element that will be used in this new model is the BEAM3 element (a legacy element avail-
able via APDL, used for EB theory based implementation [40]). Besides having nodal vertical displace-
ment (Oy) and rotation about Oz, the BEAM3 element also considers nodal horizontal displacement
36
(Ox), which ultimately gives out, in a certain more widely spread range of frequencies, natural frequen-
cies that do not appear in more simplified 2 nodes beam element, as the one that will be reproduced as
announced back in section 3.1.2, which was based in [29].
In PREP7 (see [42]), the APDL routine used for this section is as follows:
• Firstly, the type of beam element to be used is chosen (BEAM3, see [41]), along with the proper-
ties of the material (Young’s Modulus, Poisson’s Ratio, mass density), and Section Real constant
(section area, width, height second moment of area). For illustration, a portion of the APDL script
is presented:
et ,1 ,3 ! sets element type 1 to BEAM3, the 2d beam element i n ANSYS
mp, ex ,1 ,208e9 ! sets Young ' s modulus to 208e9 Pa
mp, prxy ,1 ,0 .4 ! sets Poisson ' s r a t i o to 0.4
mp, dens ,1 ,7840 ! sets dens i t y to 7840 kg /mˆ3
r ,1 ,1e−4 ,2.0833e−10 ,0.005 ! de f ines area , second moment o f area and he igh t
• Then, the nodes/elements are generated with functions such as NGEN, in order to have nodes
sequentially numbered from 1 to N, and EGEN (see [42]), for element construction from the nodes
generated;
• Boundary conditions are imposed at nodes (simply supported beam), to allow remesh:
u0y = 0 ; uLy = 0 (3.5)
This is done through (inside APDL environment):
d , node 0 , uy ,0 ! cons t ra ins displacements i n y a t x=0 , to zero
d , node L , uy ,0 ! cons t ra ins displacements i n y a t x=L , to zero
d , a l l , ux ,0 ! cons t ra ins the a x i a l x displacement i n a l l nodes ,
! so t h a t the displacements / r o t a t i o n s i n both cases are the same
• Vertical loads are imposed to allow remesh;
In the solution phase (/SOL, see [42]):
• The Modal analysis type is chosen (type 2 in APDL). The analysis options (number of modes to
extract and frequency range for the case of a modal analysis) are established and the solution
is generated. The global system is assembled and the matrices are written into a file, through a
hbmat function. The following code sample does what is being mentioned:
/ f i lname , ebeam ! creates new d i r e c t o r y i f needed
/SOL ! en ter the s o l u t i o n phase
ANTYPE,2 ! ana l ys i s type
EQSLV,SPAR ! so l ve r
MXPAND,20 , , ,0 ! modes to expand
LUMPM,0 ! lumped mass mat r i x or not
37
PSTRES,0
MODOPT,LANB,20 ,0 ,300 , ,OFF ! s o l u t i o n type , Block Lanczos
! w i th 20 modes (0−300 Hz)
/STATUS,SOLU
SOLVE
/AUX2 ! b inary f i l e dumping processor
FILE , ebeam , f u l l ,
hbmat , HBMstiffbeamEB , t x t , , a s c i i , s t i f f , yes , yes
hbmat ,HBMmassbeamEB, t x t , , a s c i i , mass , yes , yes
SAVE
FINISH
• After assembled and written onto a file (dumped in Harwell-Boeing (HB) format [43]), the file is read
through a MATLAB routine which converts HB format sparse matrices into MATLAB ones, using
[44]. Essentially, the routine reads the header of the .txt file, where the type of matrix and total
number of DOFs is stored. Then, coordinates inside the matrix to be filled with actual non-zero
values are displayed vertically, just like these same values, following a certain order (determined
intrinsically by ANSYS as it is writing the matrices). The other matrix coordinates that do not appear
in the file are assumed to be zero and are written as so (sparse). The storage of the matrix for
further processing is done as shown in:
K_ansys=hb_to_msm('HBMstiffbeamEB.txt');
for a=1:1:34
K_ansys(a,:)=K_ansys(:,a);
end
M_ansys=hb_to_msm('HBMmassbeamEB.txt');
for b=1:1:34
M_ansys(b,:)=M_ansys(:,b);
end
• Then, a MATLAB script [44] reads and processes these matrices, after being converted, and stores
them. Once these are stored, these are manipulated in a way similar to section 3.1.2;
• Through the Receptance method (sections 2.2.3.1 and 2.2.3.2), for either the cases of load or
displacement, the Transmissibility matrices are assembled accordingly to the DOFs desired and,
again, following the principles in [23].
• Finally, the rest of the processing and post-processing is done in MATLAB environment:
– Input data for modelling in MATLAB for comparison with implementation using BEAM3, based
on the EB 2D beam theory (see [29]):
∗ Material Properties and Section Constants;
38
∗ Number of elements and nodes in the model;
∗ Boundary Conditions;
∗ Step of each element;
∗ Definition of element matrices for assembling;
– Computation of Z and H for the excitation frequencies ω, followed by the force and displace-
ment transmissibility matrices T (f)UK and T (d)
UK ;
– At last, plot superimposition of Transmissibility (from receptance) values between chosen
nodes along the same range of frequencies, for each model (APDL and MATLAB), to verify
the APDL routine.
3.2 Acoustic Pressure Transmissibility
In this section is presented the methodology followed to obtain pressure transmissibility inside a 1D tube,
as well as for a 3D one (for SDOF and MDOF), with either purely reflective or anechoic top extremities.
This is done following two different paths. Primarily, for a single imposed pressure in a certain point and
compared with the literature, as in [3], both for the 1D and 3D cases. Since there is only one point (in a
1D or a 3D section) from which there is either a source or an imposed pressure, the transmissibility in
this case will be solely scalar, as dictated by equation (2.74). Then, sets of coordinates (U ,K and C) will
be considered (fig. 3.3) and MDOF Transmissibility will be computed through the receptance matrices.
In [2], the author tries to reproduce what was produced by Devriendt in [3], where an acoustic source
(or pressure imposition) is put in one of the considered tube’s ends. In this section one wants also
to reproduce the results already obtained in the literature, and will try and implement that through the
MATLAB and APDL environments.
There is relevance in adding that the implementation with APDL is considered similar to the one used
for the beam. This presents itself to be the case because the main idea behind this work is to develop a
methodology which is able to utilize the global matrices from the ANSYS code used to calculate the FRFs
(receptance matrix) , which were extracted with an APDL macro, in order to produce V-A Transmissibility.
So, essentially, the methodology to be followed for the case of acoustics is the verification of results
for transmissibility inside an acoustic tube with both reflective and anechoic tops, either in MATLAB and
ANSYS Mechanical APDL. The main proposed method (in the specific case of APDL) being 3D model
creation (appropriate modelling and meshing), analysis (modal and harmonic), data-acquisition (writing
K, M and C matrices data onto ANSYS output files with extensions .full and .emat) and finally, through
MATLAB, post-processing, as in reading those files and converting the matrices to proper format (sparse
HB [43] to sparse MATLAB) to be further processed, with the help of [44], as it was previously mentioned
for the beam.
39
Set K
Set U Set C
FU
FK
FC
Figure 3.3: Sets of Coordinates U , K and C for an Acoustic Enclosed Domain
Fig. 3.3 illustrates how the regions for harmonic acoustic load application/pressure imposition are
organized inside an enclosed volume of fluid. This figure serves as support for the theoretical projec-
tions made back in equations (2.77) to (2.79), (i.e. to establish the acoustic transmissibilities), which
essentially relate the FRFs for dynamic stiffness, and Receptance with transmissibility matrices wher-
ever, whilst inside the acoustic domain (MDOF transmissibility determination). With this, having a set of
known pressures, while knowing the set of matrices that dictate how pressure is transmitted from one
region to another will enable pressure determination basically anywhere.
3.2.1 Transmissibility Using a Code in MATLAB
The implementation of all the Acoustic transmissibility in 1D is done with resource to MATLAB, and is
based on the theoretical principles announced in chapter 2. The code to be used in this step is described
as follows [39]:
• Defining material constants (air speed, density) to the acoustic environment, as well properties of
the 1D tube, like width and length;
• Defining the number of elements per wavelength, based on the frequency of the travelling wave,
just like it will be done for the 3D case;
• Defining hat functions and K, M , C assemblage, based on section 2.3.9.;
• Obtaining Natural frequencies and corresponding vectors of the system with the help of eigs, a
MATLAB function which receives the mass and stiffness global matrices as arguments, and outputs
said frequencies and vectors [39];
• Scalar Transmissiblity computation, from a single reference excitation force in the first node of the
tube, or from pressure imposition in the same node, based in [3], for a reflective and anechoic tube.
This difference exists only if the system has damping or not. This damping will only be introduced
in the last node of the system;
• Post-processing of the obtained transmissibilities, as in drawing and comparing them, taking into
account the node in the tube to which, acoustic pressure is being transmitted.
40
3.2.2 Transmissibility Using a Code in ANSYS From a 3D Model
Initially, with the purpose of validating by superposition, the results obtained in 1D through the MATLAB
Software, the same method for scalar transmissibility is used to calculate/estimate this same transmis-
sibility with the help of equation (2.74) which, as already mentioned, is based upon [3]. But, in this
particular situation, this will be done through a FE commercial software, ANSYS MECHANICAL APDL,
followed by further processing in MATLAB, just like before. The implementation was developed as fol-
lows:
• First, the properties of the 3D acoustic fluid element (FLUID30, see [40]) will be established (air
speed, density, absorption) and assigned to a specific material type;
Figure 3.4: Base element with 3 nodes per width
• Then, the model, a tube, is to be developed. Based upon the number of elements per wavelength,
the model will have it’s nodes orderly generated from a single one, which will be copied along the
x and y−axis (and z−axis, just once for now) of the graphical users interface (GUI), while aiming
to produce a square base with square elements. This is done by:
n ,1 , width ,0 ,0 ! generate f i r s t node
! face nodes generat ion
ngen , nodes per width ,1 ,1 ,1 ,1 ,0 , width / ( nodes per width −1) ! copy along +y
ngen , nodes per width , nodes per width ,1 , nodes per width ,1 , ! copy along −x
−width / ( nodes per width −1) ,0 ,0
ngen ,2 , nodes per width * nodes per width ,1 , nodes per width * nodes per width ,
1 ,0 ,0 ,−1/ nd iv ! copy the pa t t e rn once along −z ( double node pa t t e rn to be
! connected and form elements )
! c reate f i r s t se t o f elements
41
e ,1 , nodes per width * nodes per width +1 ,
nodes per width * nodes per width+nodes per width +1 , nodes per width +1 ,2 ,
nodes per width * nodes per width +2 ,
nodes per width * nodes per width+nodes per width +2 ,
nodes per width+2
egen , nodes per width −1, nodes per width , 1 , , , , , , , ,
−width / ( nodes per width −1) , , ,
egen , nodes per width −1 ,1 ,1 , nodes per width −1 , , , , , , ,0 ,
width / ( nodes per width −1) ,0
Where ndiv is the division element mesh parameter along the length (its meaning is further clarified in
section 4.3). If the mesh is produced by using commands like the standard vmesh, the nodes are ordered
randomly, so, because the position and number of the node matters in the methodologies presented, the
method above was chosen.
Ultimately, this pattern will be reproduced along the z− axis and the elements will be generated and
the mesh assembled, through interconnection of the nodes created, by means of the APDL function
EGEN (see [42]):
! generate the r e s t o f the mesh
egen , leng th * ndiv , nodes per width * nodes per width ,1 ,
( nodes per width −1)*( nodes per width −1) ,1 , , , , , ,0 ,0 ,−1/ nd iv
The divisions between elements in the x0y plane will depend on the width of the tube, and in the z0y
one, on the length of the same. The number of elements per wavelength also weighs heavily on the
number of elements per width and length. The order of the coordinates of each node in the element was
chosen according to figure 3.5;
Figure 3.5: fluid30 - 3D Acoustic Fluid Element (source [40])
• Once the model is assembled and the mesh is generated, the top boundary of the tube which is
naturally reflective, will stay as so or converted to an anechoic one, by applying a unit impedance
to the surface:
nsel , s , loc , z,− l eng th ! Se lec t downstream end nodes
enode ! Corresponding elements
sfe , a l l , 1 , impd , , 1 ! Un i t Impedance
42
emodif , a l l , mat ,2 ! Changes new mate r i a l to type 2 ( absorbing )
Then, a modal analysis will be done to generate the global matrices K, M and C (which will only
exist if a surface enforces damping for instance an anechoic one) onto a .full binary file, resorting
to a hbmat function which writes under the ascii language, on HB sparse matrix format:
/AUX2
FILE , Tese , f u l l ,
hbmat , HBMst i f f tube , t x t , , a s c i i , s t i f f , no , no ! s t i f f n e s s
hbmat , HBMmasstube , t x t , , a s c i i , mass , no , no ! mass
hbmat , HBMdamptube , t x t , , a s c i i , damp, no , no ! damping
save
FINISH
• The matrices are then converted (again) to MATLAB format, through [44];
K=hb_to_msm('HBMstiff.txt');
• Using (2.75) these matrices will originate the global dynamic stiffness matrix Z(ω) and conse-
quently, by inverting, the FRF receptance matrix H(ω), like in equation (2.15), since this study is in
a spectrum of excitation frequencies, as well;
• The reference nodes will therefore be chosen as well as the nodes, with which transmissibility
wants to be established/obtained, while considering a straight line along the z− axis that connects
the two nodes (to make an, as best as possible, approximation to the uni-dimensional case). After
chosen, their respective position will be represented in equation (2.74), through:
T kir(ω) =pi(ω)
pr(ω)=
Hz=i,x=zref (ω)
Hz=zref ,z=zref (ω)(3.6)
With zref being the position of the reference nodes, about which T kir(ω) will be calculated, and i
being the point where it is actually computed.
• Finally, the values obtained are then compared with the equivalent 1D case, taking into account
that the number of elements per length is essentially the same in either cases (not accounting for
the elements distributed along the width of the 3D tube).
When the objective is a transmissibility calculated in regions of the fluid (from 2 points onward),
matrices that specify grouping of DOFs inside a region have to be used, much like it already was in
Dynamic Structures.
The basis for this particular part of the methodology is explained in section 2.3.8.2. The FR matrices
HKU will be nKxnU , HUU will be nUxnU and T aKU , also nKxnU . So basically, the global stiffness, mass
and damping (if there is any) matrices were written onto .full files with HB format, and converted.
In this case, a pressure excitation of 1 Pa is applied on all the nodes of one of the tubes’ ends and
”measured” at the centre of the same tube, and saved onto a results file.
43
p1=NODE( width /2 , width / 2 , 0 ) ! ( x , y , z ) end z=0
p2=NODE( width /2 , width /2 ,− l eng th ) ! mid po in t o f the tube
freqmin=0 ! Range of frequency sample
freqmax=500 !
NSTEPS=500 !
! So lu t i on
/SOLU
ematwri te , yes
ANTYPE,3 ! harmonic ana lys i s
HROPT, FULL
HROUT,ON
LUMPM,0
EQSLV,FRONT,1e−008,
PSTRES,0
HARFRQ, freqmin , freqmax ,
NSUBST,NSTEPS,
KBC,1
SAVE
SOLVE
SAVE
/POST26
FILE , ' f i lename ' , ' r s t ' , ' . ' ! r e s u l t s f i l e
! *nsol , 2 , p1 , pres , , p1 ! so lves nodal pressure
nsol , 3 , p2 , pres , , p2
! w r i t t i n g sec t ion
*DIM , UP PLATE,TABLE,NSTEPS,2
VGET, UP PLATE(1 ,0 ) ,1
VGET, UP PLATE(1 ,1 ) ,2 , , 0
VGET, UP PLATE(1 ,2 ) ,2 , , 1
/OUTPUT, ' p1 ' , ' t x t ' , ' . '
*VWRITE, 'FREQ' , ' p1 ' , ' '
%14C %14C %14C
*VWRITE, ' ' , 'REAL ' , ' IMAGINARY '
%14C %14C %14C
*VWRITE, UP PLATE( 1 , 0 ) ,UP PLATE( 1 , 1 ) ,UP PLATE(1 ,2 )
%14.5G %14.5G %14.5G
! *
44
*DIM , UP PLATE,TABLE,NSTEPS,2
VGET, UP PLATE(1 ,0 ) ,1
VGET, UP PLATE(1 ,1 ) ,3 , , 0
VGET, UP PLATE(1 ,2 ) ,3 , , 1
/OUTPUT, ' p2 ' , ' t x t ' , ' . '
*VWRITE, 'FREQ' , ' p2 ' , ' '
%14C %14C %14C
*VWRITE, ' ' , 'REAL ' , ' IMAGINARY '
%14C %14C %14C
*VWRITE, UP PLATE( 1 , 0 ) ,UP PLATE( 1 , 1 ) ,UP PLATE(1 ,2 )
%14.5G %14.5G %14.5G
! *
The computed transmissiblity will convert the applied pressure set (U ) into, what is saved as the pressure
set (K) in the middle of the tube. Finally (in MATLAB), the pressures obtained from both extraction and
transmissibility will be compared.
%Matrices conversion
K=hb_to_msm('HBMstifftube.txt');
dim=size(K,2);
for k=1:1:dim
K(k,:)=K(:,k);
end
M=hb_to_msm('HBMmasstube.txt');
for l=1:1:dim
M(l,:)=M(:,l);
end
%Pressure measure in ANSYS
importfileANSYS('p2.txt') %Since the wave is plane only one value needs to
p2=data(:,2); %be measure
freq=data(:,1); %frequency
%nodes to measure pressure at K region (4 nodes)
npk1=168;
npk2=169;
npk3=170;
npk4=171;
%nodes to apply pressure at U region (4 nodes)
npu1=1;
npu2=2;
npu3=3;
45
npu4=4;
P=[1 1 1 1]; %applied pressure
for i=1:1:ciclos
Z_asy=K-w(i)^2*M+1j*w(i)*C; %Dynamic Stiffness Matrix
H_asy=inv(Z_asy); %FRF
Hku=; %Desired H_asy sub-matrices from npk (4x4)
Huu=; %Desired H_asy sub-matrices from npu (4x4)
Tku=Hku*inv(Huu); %Transmissibility Matrices 4x4
p_vec(i,:)=Tku*P'; %pressure vector from transmissibility
end
The acoustic analysis (coupled as well) will be done, based upon [40] and [45].
If an anechoic end is regarded, instead of the reflective one (default), it is done as follows:
NSEL,S,LOC, Z,− l eng th ! Se lec t downstream end nodes
ENODE ! Corresponding elements
SFE, ALL ,1 , IMPD, ,1 ! Un i t Impedance
EMODIF, ALL ,MAT,2 ! anechoic
3.3 Vibro-Acoustic Transmissiblity
In acoustics, the procedure used to obtain pressure transmissibility through the FRFs method, as it
was already stated before in subsection 2.3.8.2, is based on pressure imposition (with acoustic load
cancellation) in regions (U , K and C) inside the enclosed volume, without the existence of an interface
(Fluid-Structure). In this case, the pressure set imposed in region U will be converted through a matrix
T aKU , into a new set of known pressure coordinates, as in:
PK = T aKU PU (3.7)
Assuming this logic as a basis for further implementation, an analogy will be established for the V-A
interaction.
Just like previously, as observed in chapter 2, from the fluid-structure coupling equation, one can
make the statement that, just like in pure structural mechanics and acoustics, the matrices appearing in
this equation can be grouped into a general dynamic stiffness matrix [14]:
Z
ui
Pi
=
Fsi
Fai
⇐⇒
ui
Pi
= H
Fsi
Fai
(3.8)
Now, adopting the same perspective as when Eq.(3.7) was generated, an analogy can established
between the loads of acoustic and structure interaction (Fai and Fsi), and the loads applied in regions
46
U and K inside an enclosed volume where an additional fluid-structure interface is regarded (see also
[17]). The same is done for the displacement and pressure in the first set:
Fsi ⇐⇒ FK , Fai ⇐⇒ FU , ui ⇐⇒ uK , Pi ⇐⇒ PU (3.9)
resulting in: uK
PU
=
HKK HKU
HUK HUU
FK
FU
(3.10)
If the second equation of the system above is solved with respect to FU and FK=0, the first one solved
with respect to uK yields:
uK = HKUH−1UUPU ,where HKUH−1
UU = TFSKU
If an imposed dynamic displacement on the plate is considered instead of pressure in the fluid (now uU
and PK , and applying the same switch to the interaction loads), one obtains, through equation (2.106):
TSFUK = HKUH−1UU
K U
FSI FAI C
z = 0 z = −L
Structure Int. Acoustic Fluid
u P←
Figure 3.6: Vibro-Acoustic Interaction depicted in Sets of Coordinates U , K and others C, with imposedpressure
The figure above illustrates the proposed interaction (equation (2.101)). A pressure is imposed in
set U near the interface (while in the acoustic medium). Then, this pressure is converted into dynamic
displacements (in K), after it reaches the interface.
The algorithm that is going to be employed to calculate the FRFs and estimate the resulting Trans-
missibility matrix that relates displacements in a structure with a pressure imposed in a fluid within the
vicinity will be the following:
• Definition of material type and properties for the acoustic cavities and structural environments. For
47
the fluid is the same 3D fluid element FLUID30 (see [40],[45]), whereas for the structure, which will
be a plate, the SHELL181, which designates the Mindlin-Reissner (see [46]) 4 node shell element,
in the APDL environment (all the values for the presented parameters were defined apriori):
! f l u i d and r e l a t e d p r o p e r t i e s ( n o f s i )
ET,1 , FLUID30 , , 1 ! f l u i d i n s i d e the tube
MP,SONC,1 , a i r speed
MP,DENS,1 , dens i t y
MP,MU,1 ,0 .00 ! damping i n s i d e o f the c a v i t y
R,1 ,20e−6 ! re ference pressure 20uPa
! f l u i d and r e l a t e d p r o p e r t i e s ( w i th f s i )
ET,2 , FLUID30 , , 0 ! f l u i d i n s i d e the tube
MP,SONC,2 , a i r speed
MP,DENS,2 , dens i t y
MP,MU,2 ,1 ! damping on i n s i d e o f the c a v i t y
R,2 ,20e−6 ! re ference pressure 20uPa
! s t r u c t u r e ( M ind l i n P la te )
R,3 , p l a t e t h i c k n e s s
ET,3 ,SHELL181
SECTYPE, ,SHEL
SECDATA, p l a t e t h i c k n e s s
MP,EX,3 , p la te E
MP,DENS,3 , p l a t e d e n s i t y
MP,NUXY,3 , p l a t e p o i s s o n r a t i o
• Generating the model and meshing, with the mesh parameter depending on the same variables as
in subsection 3.2.2. It is of major importance to refer that in this methodology both the plate and
the fluid need to have corresponding nodes at the interface, to ensure that the following analysis
are properly done [47];
• After the structural and fluid mesh is produced, the FSI flag is switched on, in the manners of [47].
! s e l e c t the elements touch ing the s t r u c t u r e ( p l a t e ) to be changed to FSI
ESEL,S,TYPE, ,3
NELEM
ENODE
ESEL,R,TYPE, ,1
! t u rn a l l the new elements i n t o FSI type elements
TYPE,2
REAL,2
MAT,2
EMODIF, ALL
48
SF, ALL , FSI ,1
Then a modal analysis will be executed primarily, so that the global mass, stiffness and damping
sparse matrices are extracted from the simulation. This extraction will be done just like before,
to a .full file in the corresponding directory where the software is being run, with the help of the
APDL function hbmat, which extracts, again, HB sparse matrices. Once this analysis is done, there
will be a slight difference in terms of nodal association between the dimension of these matrices
(ndofxndof ) and the number of total DOFs, compared to when the medium was solely acoustic (and
there were only pressure DOFs). For this case, the total number of DOFs will be a bit different from
the total number of nodes. This difference comes from the insertion of the plate inside the model,
which inserts DOFs of rotation and translation (three of each) into the system, at the interface,
point which will be further clarified ahead in chapter 4.
! S ing le Layer P la te I n s e r t i o n
NSEL,S,LOC, Z,− l eng th ! a t the downstream end
TYPE,3
REAL,3
MAT,3
ESURF, ALL
From this analysis, the natural frequencies for the whole will also be extracted;
• Pressure is imposed in the U region,in a set of chosen nodes (could be a boundary as well). At
the same time, the nodes that will represent the U region corresponding to plate displacement will
be chosen, also within a set. It is also considered applying a force F at the centre of the plate and
evaluating the P/u ratio along the frequency range.
NSEL,S,LOC, Z,0 ! s e l e c t nodes at z=0
D, a l l ,PRES,1 ! apply a pressure e x c i t a t i o n o f 1 Pa on
! a l l the se lec ted nodes
! or
F , node at the midd le , fz ,−1 ! f o r ins tance
• Once the sets are chosen, a harmonic analysis is ran inside a excitation frequency range
! cons t ra i n the displacement DOFs on the FSI element t h a t
! are not touch ing the p la te
ESEL,S,TYPE, ,2
NELEM
nsel , s , loc , z ,0 ,− l eng th ! s e l e c t a l l the nodes along the tube
nsel , u , loc , z,− l eng th ! unse lec t the nodes of the p la te
d , a l l , ux
d , a l l , uy
49
d , a l l , uz
! change the boundary cond i t i ons on the p la te to f i x e d
nsel , s , type , , 3
nsel , s ,1
nsel , r , ex t
d , a l l , a l l
/SOLU
ematwri te , yes
ANTYPE,3 ! Harmonic Ana lys is
HROPT, FULL
HROUT,ON
LUMPM,0
EQSLV,FRONT,1e−008,
PSTRES,0
HARFRQ, freqmin , freqmax ,
NSUBST,NSTEPS,
KBC,1
SAVE
SOLVE
ALLSEL
FINISH
And either the sets of displacements and pressures will be written onto a .txt file, for each step of
frequency, to be later imported into MATLAB:
/POST26 ! postprocess the r e s u l t s w i th the time−h i s t o r y processor
! *FILE , ' f i lename ' , ' r s t ' , ' . ' ! choose r e s u l t s f i l e ( . r s t ) f o r w r i t t i n g
u 1=NODE( width /2 , width /2 ,− l eng th ) ! node d e f i n i t i o n f o r s t r u c t u r a l d isp .
p u1=NODE( width /2 , width /2 ,− l eng th / 2 ) ! node gen . f o r pressure
! Ret r ieve the z displacement a t the d r i ve po in t
! on the p la te
NSOL,2 , u 1 ,UZ, , u 1
! Ret r ieve the pressure i n a sa id po in t
! i n s i d e the tube
NSOL,4 , p u1 ,PRES, , p u1
! Reading and W r i t t i n g the u 1 values f o r NSTEPS i n frequency range
! based on FORTRAN
*DIM , UP PLATE,TABLE,NSTEPS,2
VGET, UP PLATE(1 ,0 ) ,1
50
VGET, UP PLATE(1 ,1 ) ,2 , , 0 ! 2 represents u 1 and 4 p u1 from nsol ,
VGET, UP PLATE(1 ,2 ) ,2 , , 1 ! f o r ins tance
/OUTPUT, ' u 1 ' , ' t x t ' , ' . '
*VWRITE, 'FREQ' , ' u 1 ' , ' '
%14C %14C %14C
*VWRITE, ' ' , 'REAL ' , ' IMAGINARY '
%14C %14C %14C
*VWRITE, UP PLATE( 1 , 0 ) ,UP PLATE( 1 , 1 ) ,UP PLATE(1 ,2 )
%14.5G %14.5G %14.5G
• After every needed analysis is done, the rest of the process will be done in MATLAB. The set
of pressures/displacements will be imported through a function designated importfileANSY S,
which receives as an argument the produced file where all the sets were written.
function importfile1(fileToRead1)
%IMPORTFILE1(FILETOREAD1)
% Imports data from the specified file
% FILETOREAD1: file to read
% Auto-generated by MATLAB on 12-Jul-2017 11:42:17
% Import the file
newData1 = importdata(fileToRead1);
% Create new variables in the base workspace from those fields.
vars = fieldnames(newData1);
for i = 1:length(vars)
assignin('base', varsi, newData1.(varsi));
end
Then, the imported information, not just the sets but also the range of frequencies, will be stored
into different vectors. The matrices previously generated will also need to be converted into
MATLAB format, in order to be processed [44]. Afterwards, within a cycle covering the entire
frequency range, the following operations will be done:
– Assemblage of dynamic stiffness matrix Z and its inversion to obtain H;
– Proper choosing of H’s sub-matrices HKU and HUU (and HUK and HKK as well, for P/u
comparison) according to the specified regions;
– Calculation of TFSKU (and TSFKU ) for each frequency ω;
– Point-by-Point pressure/displacement uK
PU(and displacement/pressure) ratio computation;
51
– Post-Processing (comparison of Ratios versus obtained Transmissibility values, within TFSKU ).
importfileANSYS('u1.txt'); % Import all the data regarded in the .txt
% namely freqeuncy range and disp. values
freq=data(:,1); % Frequency [Hz]
u1=data(:,2); % Real
importfileANSYS('pk3.txt');
pk3=data(:,2); % Real
for i=1:1:ciclos
Z_asy=K-w(i)^2*M+1j*w(i)*C; %Dynamic Stiffness Matrix
H_asy=inv(Z_asy); %FRF
Hku=; %Desired H_asy sub-matrices or scalar
Huu=; %Desired H_asy sub-matrices or scalar
T_dev=; %Point by point FRF ratio (Hku/Huu) for corresponding DOF inside the model
%Compare with pressure/displacement ratio
Tku=Hku*inv(Huu); %Transmissibility Matrices
T(i)=Tku(1,1); %Choosing a specific transmissiblity between MDOF sets
TdB=20*log10(abs(real(T))); %Value from matrix to be printed in dB
pu(i)=u1(i)\pk3(i); %and vice-versa
end
52
Chapter 4
Results and Discussion
Primarily, following the methodology described in the previous chapter, one wants to present the ob-
tained results and put them to test against those present in adequate literary sources. In a primary
instance this will be done for verification of solutions regarding the field of Vibrations, namely the topic
of Vibration Transmissibility (either Force or Displacement).
After the Transmissibility is verified for this section, the results for MDOF presented in [3] and FRF
originated pressure transmissibility will be presented, for Acoustics.
Finally, following the same approach as before, the results for numeric V-A Transmissiblity are pre-
sented.
4.1 Force Transmissibility in a Mass/Spring System
In this section it is aimed to verify the method suggested back in section 3.1, in order to obtain the load
transmissibility matrix in an MDOF arbitrary spring/mass dynamic system. This was achieved with the
help of a small MATLAB Script that will be described at once.
The chosen system is much similar to the one present in Neves and Maia [1], with slight differences
in terms of force vectors and mass display. The announced system is represented in figure (4.1). The
connectivities for the same system are represented in table 4.1.
Table 4.1: Spring System Connectivity Table
Spring Element Corresponding Nodes
1 1 3
2 2 4
3 3 5
4 4 5
53
5
3
1
4
2
K K
KK
Figure 4.1: Mass/Spring System
Since a two node discretized spring element was used (as described in section 2.1.3), the connec-
tivity table, presented bellow, with the rigidity weights from every spring in every node was constructed
and through it, the global stiffness matrix (5x5 since there are 5 nodes) was assembled.
Each spring was assigned with a rigidity of k = 103 [N/m] and mass element with mass m = 5 [kg].
Thus, the final Mass and Rigidity matrices of the system will then be:
[K] =
k 0 −k 0 0
0 k 0 −k 0
−k 0 2k 0 −k
0 −k 0 2k −k
0 0 −k −k 2k
, [M] =
0 0 0 0 0
0 0 0 0 0
0 0 m 0 0
0 0 0 m 0
0 0 0 0 m
(4.1)
A set of constant amplitude known loads FK = 10, 10T were applied in nodes 4 and 5 of the system,
while the reaction loads would be present where the system was constrained, so to say, in nodes 1 and
2.
After the assembling of the final K matrix, as well as the final M one, the receptance and dynamic
stiffness matrix were calculated, with again, equations (2.15) and (2.26), for the of angular frequency ω
range of 0 to 151 rad/s, which essentially corresponds to f from 0 to 25 Hz. The load transmissibility
matrix was calculated as described in section 3.1.1 and was also evaluated through the same frequency
range. Since there are two unknown load coordinates and two other that are known (#U=#K), the final
54
[T] (transmissibility) matrix will have four entries describing transmissibility from external to rection loads
(so from 4 and 5 to 1 and 2). The results are shown in fig. 4.2.
(a) (b)
Figure 4.2: Comparison of T14 (from the receptance method), obtained in this work (a) with the one in[1] (b)
As it can be observed in fig. 4.2, in a range of frequencies that goes from 0 to 25 Hz, the transmissi-
bility between nodes 1 and 4 of the system, obtained from the MATLAB script, is pretty much overlapped
with the one in [1]. This can be stated due to the fact that the behaviour of the curves is identical and
the resonance frequencies are pretty much the same. Having said this, there is but one difference, and
that would be the amplitude of the curves.
In order to verify even further the model used, T obtained from the receptance matrix and from the
dynamic stiffness were compared just like in [32]. The comparison was made for T14 as well.
(a) (b)0 5 10 15 20 25
Frequency (Hz)
-140
-120
-100
-80
-60
-40
-20
FR
F (
dB)
H1-4
Figure 4.3: T14 (receptance and dynamic) obtained through H and Z (a) and receptance between nodes1 and 4 (b)
Figure 4.3 represents a perfect match between both methods. As it was referred in [26], the peaks
for the FRF in (b) represent, as predicted, the zeros of the transmissibility function for (a).
Table 4.2: Natural Frequencies of the Considered Spring-Mass System
N.F. f (Hz) 7.12 12.33
The values in table 4.2 were obtained with the MATLAB function eig (does a modal analysis, see
55
[39]), which solves the eigenvalues problem |K−ω2M | = 0 for the said system and gives out, ultimately,
the natural frequencies of the system. Besides showing the expected similarity of transmissibility values
(between both methods shown), the same is possible with the reactions in node 1. This is shown in the
same array of frequencies as the ones in figure 4.3:
Figure 4.4: Comparison of reactions obtained in node 1, obtained from the Receptance and Dynamicstiffness methods
As it can be observed while comparing figure 4.4 with figure 4.3, the resonance frequencies of the
system are the same (so the transmissibility zeros are in the same places), as expected, and since
the transmissibility is adimensional, it can only stated a change in amplitude of the curves, while the
behaviour stays the same throughout the 25 Hz of frequency range. The reaction was computed from
equation (3.3) and using T14.
4.2 Transmissibility in a Simply Supported Beam
4.2.1 Using a Code in MATLAB
The methodology followed to obtain the results about to be presented, since the theoretical background
for both examples (either the spring/mass system or the supported beam) is practically the same, will
be quite similar to the previous one. So to say, a MATLAB script was developed in order to study load
transmssibility in a simply supported beam (supported at both ends), but with a small difference. In this
case, the displacement transmissibility was also obtained and then compared with the load one. Finally,
the results obtained will be compared with the ones in article [23].
Since only vertical displacement and rotation about the z axis was considered, the beam element
used in the script for the FEM, was the one announced back in section 3.1.2, which has precisely four
DOF. So axial displacements were disregarded for the transmissibility, and it is witnessed further ahead
56
that this approximation is legitimate since no natural frequencies appeared from axial displacement, at
least in the range of frequencies considered. The element Mass and Stiffness matrices are the ones
represented in equation 3.4.
1
7
17
Figure 4.5: 16 Finite Element Beam, generated in ANSYS APDL
The beam assembled in the script has the properties indicated in [23], which are numbered in table
4.3.
Table 4.3: Beam Properties
Young’s Modulus - E 208 GPa
Mass Density - ρ 7840 kg/m3
Length - L 0.8 m
Section Width - b 20x10−3 m
Section Height - h 5x10−3 m
Second Moment of Area - Izz 2.0833x10−10 m4
Element Length - le 0.05 m
A beam with 16 discretized elements (so 17 nodes and 34 DOF, total) was used. These elements
were connected longitudinally with no discontinuities between them, as shown in figure 4.5. The Global
Mass and Stiffness matrices were assembled through a connectivity table where each group of four
DOFs would be defined by:
[2(n− 1) + 1; 2(n− 1) + 2; 2(n− 1) + 3; 2(n− 1) + 4]
where n is the corresponding element number, in the whole beam.
The initial purpose, as in [23] was to apply a dynamic load FK=100 N, in node number 7 (x = 0.3 m),
and study how this load would be transmitted to the nodes where the structure was supported, so nodes
1 (x = 0) and 17 (x = L), which would, as already mentioned would have their transverse DOF fixed
(as the beam is being simply supported exactly in these nodes). With this in mind, the dynamic stiffness
57
matrix Z was then calculated, with resource to equation (2.22), for a range of excitation frequencies
defined from 0 to 300 Hz. During these calculations, the transmissibility matrices were obtained with the
receptance method, through equation (2.20), for the case of load transmissibility.
Afterwards, in order to further verify results, the pseudo inverted transposed displacement transmis-
sibility matrix was obtained (which is the same as just inverting, since the matrix is a square one). For
this to be done, an additional dynamic load was considered in the load vector (of known applied loads)
applied in node 9 (whose value was taken as null later). Under only this condition are we able to in-
vert the load transmissibility matrix, whose conditions are present in section 2.2.3, for the displacement
transmissibility section. As this additional load was only considered so that both transmissibility matri-
ces could be mutually inverted, for the sake of mathematical simplicity, this load was considered null
afterwards.
From (2.31) the displacement transmissibility matrix was obtained, resorting to the H sub-matrices
that relate the sets U and K. Then, from equation (2.32) the load transmissibility matrix was obtained
(once again). From this matrix, only the coordinates (1,1) and (2,1) were considered, since these were
the ones referring to load transmissibility between nodes 1 and 7, and 17 and 7, respectively (as the
others involved the load applied in node 9). Finally the results were plotted in the same frequency range
[0:300] Hz, and compared with the literature [23], as is represented in figures 4.6 and 4.8.
(a)0 50 100 150 200 250 300
Frequency (Hz)
-60
-50
-40
-30
-20
-10
0
10
20
30
40
T1-7
(dB
)
T(f)1,7
((T(d))-1)T1,7
(b)
Figure 4.6: a) Transmissibility T1,7 Obtained in This Work; b) From [23]
The receptance that relates nodes 1 and 7 was plotted to detect frequency peaks and compared with
the transmissibility (See Appendix B). This is shown in figures 4.6 and 4.7.
58
0 50 100 150 200 250 300
Frequency (Hz)
-140
-120
-100
-80
-60
-40
-20
H1-
7(dB
)
Figure 4.7: H1,7 plotted in the frequency spectrum
Just as figure 4.6 suggests, for the range of frequencies taken, the results in Lage et al [23] match
with the ones obtained numerically, for the load transmissibility between nodes 1 and 7, through either
the regular method (previously mentioned), and the transposed pseudo-inverse matrices method. The
only detail that is not an actual complete match would be the amplitude of the curves which might infer
the presence of damping in one case [23], and the absence of it numerically.
(a) (b)
Figure 4.8: a) Transmissibility T17,7, obtained in this work ; b) From [23]
0 50 100 150 200 250 300
Frequency (Hz)
-160
-140
-120
-100
-80
-60
-40
-20
H17
-7(d
B)
Figure 4.9: H17,7 plotted in the frequency spectrum
59
For the transmissibility between nodes 7 and 17, the results are, as well, a match with, again, the
exception of the curve’s amplitude, where there was some deviation, despite having the same peak
position (see Appendix B). The same was done as before, for the receptance between nodes 7 and 17,
in figure 4.9.
Now there is a need to come back to the 4 DOF beam element considered instead of the 6 DOF
one. One thing that could happen, which is not present in the figures above, could be the existence of
natural frequencies coming from the axial displacement, which is something that does not happen in this
range of frequencies, but could happen if a wider range was to be considered in the literature. If that
were to be the case, a beam element which considers longitudinal/axial displacement would have to be
considered.
4.2.2 Using a Code in ANSYS APDL
The methodology proposed to solve the main problem in this work was firstly applied for the case of a
16 element beam (BEAM3), discretized in APDL, to reproduce what was previously obtained through
MATLAB.
Table 4.4: Natural Frequencies of the Simply Supported Beam
N.F. f (Hz) 18.247 72.984 164.21 291.95
Through the script developed in APDL, one was able to recreate with some proximity, the results
already previously obtained. The natural frequencies for the simply supported beam, obtained with
APDL, are presented in table 4.4.
Even though the results are quite similar, there are still some differences that have to be addressed.
These differences could come from several sources. Since the element mass and stiffness matrices
used in APDL are slightly different from the typical Euler-Bernoulli ones, presented in [29]
(a)0 50 100 150 200 250 300
Frequency (Hz)
-60
-50
-40
-30
-20
-10
0
10
20
30
40
T1
-7(d
B)
T(f)1,7
((T(d))-1)T1,7
(b)0 50 100 150 200 250 300
Frequency (Hz)
-30
-20
-10
0
10
20
30
T1
-7(d
B)
Figure 4.10: a) Transmissibility T1,7 (same as figure 4.6a ) ; b) The one obtained through APDL
The results obtained for the simulation in APDL are presented and compared in figure 4.10 and a
comparison between FRFs and transmissibility, in the ways of, for instance Maia in [26], in figure 4.11.
60
0 50 100 150 200 250 300
Frequency (Hz)
-150
-100
-50
0
50
Mag
nitu
de (
dB)
T(f)1,7
from Matlab
H1-7
from Matlab
H1-7
from APDL
Figure 4.11: FRF and Transmissibility Plots (for odes 1 and 7)
As it can be stated through observation, what immediately stands out is the depth/intensity, which
appear to be much more shallow in the graph obtained with APDL. One of the reasons for this to happen
might be the growing error that might have come from the fact that either the element mass and stiffness
matrix that ANSYS uses intrinsically (embedded into the code of the software) differ in a certain amount
φ (correcting factor, acting as a margin), when compared to the Euler-Bernoulli ones used in MATLAB.
This φ factor is defined as:
φ =12EI
GleAs(4.2)
where the only still unknown factors are the shear modulus G (was not relevant) and As the shear area
As = F s
A , with F s being the shear force applied on the transverse area A of the beam.
The element stiffness matrix in element coordinates is the same one used by Przemieniecki in [48],
and the element mass matrix in element coordinates is the same as Yokoayama in [49], where the φ
factor is regarded more in depth.
The Global matrices of the beam were assembled within the software and written into a file, which
was in HB format (see chapter 3). Just by doing this, there is already some error to be expected, as it is
not done in binary but in .txt format, considering the hypothesis that information might be lost between
conversions. After being assembled, these matrices had to be converted again into a MATLAB format
in order to be further processed.
0 50 100 150 200 250 300
Frequency (Hz)
-60
-40
-20
0
20
40
60
T1-
7(dB
)
0
5
10
15
20
25
30
35
40
45
50
Err
or (
%)
From MatlabFrom AnsysRelative Error
Figure 4.12: Superposition of results obtained with different methods from figure 4.10, and relativedeviation calculation
61
This was done (as mentioned in chapter 3), resorting to an open-source HB-MATLAB sparse matrix
converter in [44], which essentially read the header of the files (which has a specific order and logic to
it [43]), and then proceeded to process each column/row of the matrices into vectors, which would then
give birth to the final sparse matrices already in the intended format.
This successfully concludes the verification for vibration Transmissibility.
4.3 Acoustic Transmissibility
In this segment, the results for a 1D and 3D acoustic medium FE simulation (inside a tube), will be
presented. The results were obtained and processed through a mixed usage of both MATLAB and
APDL. For the 1D case, solely MATLAB was used for establishing the FEA, whereas for the 3D one, the
model was generated in APDL and the post-processing was done in MATLAB.
The FEM used in APDL consisted of FLUID30 parallelepipedic acoustic fluid elements, with and
without absorption (anechoic or not), as it appears in [40]. With this, the analytic solution for 1D plane
wave propagation with both with a reflective and anechoic end will be presented primarily.
(a)0 0.5 1 1.5 2 2.5 3 3.5 4
Tube Length (m)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Pre
ssur
e (P
a)
Reflective End
(b)0 0.5 1 1.5 2 2.5 3 3.5 4
Tube Length (m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Pre
ssur
e (P
a)
Anechoic End
Figure 4.13: Analytic Wave Equation Solution for: a) Reflective and b) Anechoic End
The solution presented in figure 4.13, for a reflective and anechoic boundary, respectively, was ob-
tained from the following equations [8]:
p(x) = cos(kz) + tan(kL)sin(kz) (4.3a)
p(x) = sin(kz) (4.3b)
After having plotted the analytic solutions for a wave propagating along theOz direction in the considered
tube, the results for a plane wave obtained in APDL are presented, as well as the input parameters to
obtain such results.
The number of elements per length of the tube was generated through:
NELE = L ∗NDIV with NDIV = nint
(Nfrefc
)(4.4)
where:
62
• NELE is the number of elements per L;
• L is the length of the tube;
• NDIV is the number of element divisions inside the mesh, along the direction of propagation;
• fref is the reference frequency from which the wavelength is estimated;
• N is the number of elements per wavelength.
The properties of the tube are represented in table 4.5.
Table 4.5: Tube Properties (FLUID30 - Ansys)
Sound Speed - c 344 m/s
Mass Density - ρ 1.21 kg/m3
Reference Pressure - Pref 20x10−6 Pa
Length - L 4 m
Heigth and Width - b 0.1 m
Frequency for # elements per λ (N ) - fref 200 Hz
(a) (b)
Figure 4.14: Acoustic tube simulated in APDL with N = 12 (a) and N = 24 (b), evidencing a greaterdifference between mesh refinements
Since the wave propagating in the acoustic medium is plane, and propagates along Oz, the real rele-
vance in terms of refinement and minimum requirements for elements per wavelength is along the length
of the tube. In other words, the refinement in either ends can be whichever takes less computational
effort, while still maintaining a center-line between nodes, well defined along the model. So, with this
purpose, the ends were modelled with 4 elements each. Figures 4.15 and 4.16 were obtained with a
pressure P of 1 Pa, imposed at z = 0 on the tubes presented in fig.4.14.
63
(a) (b)
Figure 4.15: Pressure along the tube for N=12 (a) and N=24 (b) with a reflective end, extracted fromAPDL
(a) (b)
Figure 4.16: Pressure along the tube for N=12 (a) and N=24 (b) with an anechoic end, extracted fromAPDL
Figures 4.15 and 4.16 show that N = 12 is enough to show the period of the propagating wave and
about N = 24 could be taken, with a small deviation, to plot the amplitude, when compared with figure
4.13.
(a) (b)
Figure 4.17: Pressure plot for reflective top (z = −L).
One last analysis was done for N = 36, and the pressure was plotted (figure 4.17).
4.3.1 Modal Analysis of the tube (APDL)
In order to analyze the order of convergence for the implemented acoustic environment (tube) a simula-
tion was done in APDL for 10, 20, 30 and 40 elements per wavelength (N ), assuming a frequency of 200
64
Hz (since a modal analysis was done, there is no necessity for a travelling wave though). This analysis
was done for the first six eigenfrequencies. A Block-Lanczos mode extraction method was used, as well
as a sparse method solver, as advised in [50].
On the account of the fact that the number of elements per wavelength depends on the frequency of
the wave and travelling speed, by means of equation (2.61), the value of the parameter used for element
length division was decimal, therefore altering the mesh into a heterogeneous tube (non-rectangular).
This detail was corrected in this work, by utilizing a function inside the APDL environment nint [42],
which converts the value of the parameter into the nearest integer. Thanks to this, there was a necessity
to use paralellepipedic elements for the assemblage instead of cubes, to minimize the error introduced
by this approximation.
Table 4.6: Convergence Analysis For the Acoustic Tube, from APDL
Natural Frequencies (Hz) N=10 N=20 N=30 N=40 Analytic Values
1 43.031 43.008 43.004 43.002 43
2 86.246 86.061 86.031 86.017 86
3 129.83 129.21 129.10 129.06 129
4 173.97 172.49 172.24 172.13 172
5 218.86 215.96 215.48 215.26 215
6 264.67 259.66 258.83 258.45 258
The analytic natural frequencies were obtained for the 1D case, from the following equation [51]:
fn =nc
2L,with n = 0, 1, 2, 3, 4, ... (4.5)
Also it is valid only up to a cut-off frequency fc (plane wave), with properties shown in table 4.5.
This has importance when regarding computational power optimization, taking into account that, the
bigger the dimension of N , the harder the solver will work, and the slower the simulations will be.
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Step
0
0.5
1
1.5
2
2.5
3
Err
or (
%)
N=10N=20N=30N=40
Figure 4.18: Error plot between the first six natural frequencies for increasing N and the analytic solution
65
Even if the first six natural frequencies do not diverge too much in terms of value (not beyond 2.5%),
a higher refinement is advised acoustic model meshing, as shown in figures 4.15 and 4.16.
4.3.2 Transmissibility Through the Tube Containing the Acoustic Fluid
The models developed and tested either inside the MATLAB environment and the APDL one, respec-
tively for the 1D and for the 3D tube, were compared with regards to how acoustic transmissibility would
be obtained and modelled as in [3], following the principles in section 2.3.8. These models disre-
gard acoustic sources, which means that any travelling wave only exists by pressure imposition, as
the Helmholtz equation (2.69) suggests.
On a first instance the results that are being presented are referring to DOFs positioned at the mid-
section of the tube relative to a pressure DOF at the end upstream from the fluid (considering positive z
axis close to the observer, instead of away)
The simulations were done for 6, 10, 12 and 24 elements per wavelength λ (but only some are
presented), considering as reference from which λ is calculated, a frequency of 200 Hz (being 20 the
ideal breakpoint for convergence, even though it takes a bit computational power for the 3D case since
the total number of elements is very large and the calculations, as seen in chapter 2, take quite a bit
of Algebra and time as well). An analysis made for a larger N (higher but not much different) will be
presented ahead.
4.3.2.1 1D Case Using a Code in MATLAB
In this section, the 1D case (most simple one) is presented using a MATLAB. Following the principles
detailed in [3], 1D pressure transmissibility was computed and plotted accordingly as is shown is figures
4.20 and 4.21, while considering pressure imposition and a single excitation source in x = 0. This was
based on the model in fig. 4.19. The wave propagation is taken along x and is positive away from the
origin of the reference frame (for the 3D case, propagation will be along z and negative away from the
origin).
Source or Impositon "Measured" MidpointReflective/Anechoic
Figure 4.19: 1D acoustic medium for case illustration
66
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency [Hz]
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
T [
dB]
Tirk( ) with source on x=0
Tirk( ) with imposed pressure on x=0
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency [Hz]
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
T [
dB]
Tirk( ) with source on x=0
Tirk( ) with imposed pressure on x=0
Figure 4.20: Transmissibility obtained from [3], using the FRF ratio with imposed pressure in x = 0 andfrom pressure ratio with a source in x = 0, for N=12 (a) and (b) N=24, with an anechoic top. Thetransmission is done to a node in x = 2 m
For the anechoic case, the absolute value of the transmissibility is plotted, since the insertion of
damping generates an imaginary part in the system.
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency [Hz]
-60
-40
-20
0
20
40
60
T[d
B]
Tirk( ) with source on x=0
Tirk( ) with imposed pressure on x=0
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency [Hz]
-60
-40
-20
0
20
40
60
80
100T
[dB
]T
irk( ) with source on x=0
Tirk( ) with imposed pressure on x=0
Figure 4.21: Transmissibility obtained from [3], using the FRF ratio with imposed pressure in x = 0 andfrom pressure ratio with a source in x = 0, for N=12, 25 nodes (a) and (b) N=24, with an reflective top.The transmission in done to a node in x = 2 m
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency [Hz]
-80
-60
-40
-20
0
20
40
60
80
H[d
B]
H(x=0,x=2)
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency [Hz]
-20
-10
0
10
20
30
40
H[d
B]
H(x=0,x=2)
Figure 4.22: FRF plot for the DOF in the middle of the 1D tube (x = 2) and N =12, for a reflective (a)and anechoic end (b), with a source in x = 0
Both figures above (4.20 and 4.21), present what was already predicted. For the different degrees
of refinement, the graphics for pressure and source excitation are quite similar, for either the anechoic
end and the reflective one, which means the methods used are identical. For the case of fig. 4.22,
67
it can be seen that, just like in previous cases, the peaks from the receptances correspond to the flat
Transmissibility spots (for a reflective end, with no absorption).
4.3.2.2 3D Case Using a Code in APDL
In this segment, the results for acoustic transmissibility in APDL will be presented, and ultimately com-
pared with the ones where MATLAB was used, at least for the 1D results. This comparison assumes a
plane wave condition is verified. The verification for MDOF acoustic transmissibility was done by mea-
suring pressure values in certain nodes of the model, divide those into two input and output vectors,
multiplying the input vector by the T aKU that relates both sets and ultimately comparing the measured
outputs with the ones calculated.
The tube was also discretized with basis on the number of elements per wavelength, according to
previous statements. The walls of the tube were designated as reflective (default).
The same logic was used as in section 4.3.1 in order to obtain the global Mass and Acoustic Stiff-
ness matrices, where these were written into a file, and where the dimension of these matrices would
correspond to the number of unsconstrained DOFs within the structure.
For each iteration of the developed APDL script, the number of elements was updated with respect
to the N parameter.
4.3.2.2.1 Scalar Transmissibility
In fig. 4.24, Transmissibility results are presented for a tube discretized with exactly 117 pressure DOFs
for (a), 224 for (b) and 513 for (c) (fluid30 only regards pressure DOF since it is a purely acoustic element,
while reflective ends are regarded, or KEYOPT(2)=1 [45], which enforces wave reflection in the element
used, within APDL language [40]). On the other hand, as comes from before, the 1D tube does not need
as many elements (DOFs) for it’s discretization, having only 15, 24 and 57 nodes/DOFs for the same
number of elements per wavelength, respectively.
In either cases, the FRF matrices were used to compare scalar pressure transmissibility between
the midpoint/midsection of the tube with the imposed pressure on one of the ends, in this case, the one
upstream from the fluid.
MidsectionCenterline
Figure 4.23: Side View of the 3D tube model (z0y plane) with N = 24 elements per λ, 56 elements alongthe length and a reflective end at z = −L
A pressure excitation of 1 Pa was applied in z = 0, the plots in figure 4.24 were obtained following
T kir =Hz=−2,z=0
Hz=0,z=0, where k = r (z = 0) and i corresponding to z = −2.
68
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency [Hz]
-60
-40
-20
0
20
40
60
T[d
B]
Tirk( ) based on Matlab (1D)
Tirk( ) based on APDL (3D)
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency [Hz]
-80
-60
-40
-20
0
20
40
60
T[d
B]
Tirk( ) based on Matlab (1D)
Tirk( ) based on APDL (3D)
(c)0 50 100 150 200 250 300 350 400 450 500
Frequency [Hz]
-80
-60
-40
-20
0
20
40
60T
[dB
]T
irk( ) based on Matlab (1D)
Tirk( ) based on APDL (3D)
Figure 4.24: Results obtained based on [2, 3] compared against the 3D model developed in APDL, fora) 6; b) 10 and c) 24 elements per wavelength, having as reference f=200 Hz, and a reflective end.Pressure ratio in black, for 1D, and FRF ratio in blue for 3D
The point can be made that, since a reference frequency of 200 Hz is being used in order to establish
an element discretization parameter, and since the range of excitation frequencies goes to 500 Hz, the
model may start losing convergence after passing through the fref . In this case, having either a higher
fref or N , according to equation (4.4) would increase the level of refinement, hence quite possibly
increasing the solution proximity between the 1D and 3D cases.
Be that as it may, the pollution that comes from h-FEM methods is always a factor in terms of solution
convergence, and a higher N does not always ensure that the model has converged (see [10, 11]).
Nevertheless, before proceeding any further, it is important to mention that a decent level of refining
for a 3D acoustic model that exports HB format matrices to be processed and manipulated through
MATLAB, demands quite a bit from a computer, which implies somewhat large simulation times, which
is not a problem for 1D (if the wave is plane), where convergence comes much earlier because there
are only elements over one direction.
One could also argue that the slight disparity between models in represented in fig.4.24c might come
from transverse modes originating from perpendicular vibration frequencies in the 3D model (that do not
exist in 1D), which would justify how some areas are a bit off. As the following equation [51] suggests,
based on table 4.5:
fm,n,l =c
2
√(m
b/2
)2
+
(n
b/2
)2
+
(l
L
)2
,with m,n, l = 0, 1, 2, 3, 4, ... (4.6)
However, since the width of the tube is quite small, comparing with the length, as already referred, this
69
detail might not be that much of an issue.
Another source of error that would be worth mentioning would be the condition number for the global
dynamic stiffness matrix Z, which will basically tell how well the matrix is conditioned to be inverted,
without having the solution dispersing. Essentially, a badly conditioned matrix will cause loss of data
between inversions inside the routine, as explained in [52].The dimensions of each of the computed
Z matrices were respectively, 117x117, 224x224, 261x261 and 513x513. The condition number was
calculated along the frequency range, resorting to the MATLAB function condest which receives as
inputs the matrix and a positive integer parameter equal to the number of columns in the underlying
iteration matrix [39].
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
1
2
3
4
5
6
7
8
9
10
Z C
ondi
tion
Num
ber
104
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
1
2
3
4
5
6
7
8
9
10
Z C
ondi
tion
Num
ber
104
(c)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
1
2
3
4
5
6
7
8
9
10
Z C
ondi
tion
Num
ber
104
(d)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
1
2
3
4
5
6
7
8
9
10
Z C
ondi
tion
Num
ber
104
Figure 4.25: Z matrix condition number for (a) N=6 ,(b) N=10, (c) N=12 and (d) N=24, having asreference f=200 Hz, and a reflective end (no damping)
The plots represented in figure 4.25 correspond to as N grows, it does appear to exist somewhat of
a decrease in terms of conditioning for Z, effect that may justify why the calculated transmissibilities may
differ, even if just slightly. The growth of N is inherently connected to the size of the matrices extracted
and inverted, so, as the refinement grows higher, some error will have to be accounted as due to bad
conditioning.
Meanwhile, other simulations were done in APDL, for an harmonic analysis with an imposed pressure
P of 1 Pa, with either a reflective or anechoic end, for N=6, 10, 12, 20, and the transmissibility T kir(ω)
was calculated for both cases, Since these calculations were done assuming pressure ratio between
a reference point (section where pressure is imposed) and the other one along a line parallel to the
main axis of wave propagation (center of the tube), no inversions of Z were done, which means that the
refinement can go much further, weighing much lightly computationally.
70
1 Pa
Midsection
Figure 4.26: 3D tube model with N = 20, 48 elements along the length and a reflective en at z = −L
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-60
-40
-20
0
20
40
60
80
P(d
B)
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-60
-40
-20
0
20
40
60
P(d
B)
(c)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-80
-60
-40
-20
0
20
40
60
80
100
P (
dB)
(d)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-80
-60
-40
-20
0
20
40
P(d
B)
Figure 4.27: Pressure measured in point (0.05;0.05;-2) of the 3D model, with a reference pressureimposed at z=0 of 1 Pa, for (a) N=6 ,(b) N=10, (c) N=12 and (d) N=20, having as reference f=200 Hz,and a reflective end
71
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-70
-60
-50
-40
-30
-20
-10
0
10
P(d
B)
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-50
-40
-30
-20
-10
0
10
P(d
B)
(c)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-30
-25
-20
-15
-10
-5
0
5
P(d
B)
(d)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-50
-40
-30
-20
-10
0
10
P(d
B)
Figure 4.28: Pressure measured in point (0.05;0.05;-2) of the 3D model, with a reference pressureimposed at z=0 of 1 Pa, for (a) N=6 ,(b) N=10, (c) N=12 and (d) N=20, having as reference f=200 Hz,and an anechoic end
Considering both the 1D and 3D cases for an anechoic end, it can be stated that the results are quite
different. This may have to do with the fact that in the first case, wave propagation is is taken along a
single direction where the length is assumed to be infinite, therefore existing no resonance along this
distance, as it can be observed in sub-section 4.3.2.1. In the latter, the wave also propagates in the x
and y directions, which are finite. This indicates that, most likely, there will exist resonance frequencies
along this direction, fact which may explain why, in the 3D case, the pressure plotted with each excitation
frequency exhibits such pattern.
There is one small remark to be added. If only the real values of the pressure are plotted, for a high
refinement, the 1D and 3D models exhibit proximity, but with small deviations, which can still be due to
what was mentioned above.
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-70
-60
-50
-40
-30
-20
-10
0
10
P(d
B)
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency [Hz]
-70
-60
-50
-40
-30
-20
-10
0
T[d
B]
Tirk( ) with source on x=0
Tirk( ) with imposed pressure on x=0
Figure 4.29: a) ”Measured” pressure at the centre of the tube with N = 36, in APDL; b) Transmissibilityfor N = 36, in the manner of [3], with MATLAB, except for the fact that in this case, only the real part ofthe Transmissibility was regarded
72
Figure 4.29 still evidences some differences, but the model is closest for 36 elements per wavelength.
For the case of fig. 4.29b only the real part was considered, and not the absolute transmissibility value.
4.3.2.2.2 MDOF Pressure Transmissibility from FRFs
When analyzing the results obtained for this section, one has to consider some assumptions which may
alter the accuracy of the results, towards a better one.
For the MDOF pressure transmissibility four nodes were considered both for PU and PK , so the
resulting FR matrices were 4x4. If the tube has four nodes to model each extremity (and basically
each section between elements), the level of refinement can go much higher, without increasing the
computational demand too greatly. The base of the tube was discretized in a single element, since
for the fref considered, the propagating wave is still plane, and transverse discretization is barely not
needed. In opposition with section 4.3.2.2.1, where the analysis was done along a center-line, so the
minimum number of elements to define the base had to be four, or nine nodes.
1 Pa
Figure 4.30: Pressure Imposition on z = 0 in a tube model with N = 36, 84 elements along its’ length,and a reflective boundary at z = −L
The model in figure 4.30 has 340 pressure DOFs. Further ahead, for the results regarding V-A
Transmissibility, the discretization of the inserted plate into a more refined mesh.
(a) (b)
Figure 4.31: Plane Pressure wave along the tube at 200 Hz and a reflective end in (a) and anechoic in(b), with P= 1 Pa applied at z = 0
Having already shown results regarding single point Transmissibility, the results that follow were
73
obtained for specific regions K and U (fig. 3.3) inside the volume enclosed by the tube, where the
pressure values were respectively known and unknown.
Fig. 4.32 was obtained by:
• Imposing pressure excitation (set U , so PU ) of 1 Pa in all the pressure nodes positioned in z = 0
with the help of the APDL command D [45]:
• Extraction of K, M and C (for the case of an anechoic boundary), as well as the pressure in the
midsection of the tube (set K), by reading the HB sparse matrices and importing the ”measured”
pressure values into MATLAB;
• Assemblage of the matrices and T aKU computation for each excitation frequency, from the HKU
and HUU sub-matrices. To determine transmissibility, equation (2.85) was used;
• Representation of the measured pressures versus the ones computed from the transmissibility
matrix, for each coordinate, through: PK(ω) = T aKU (ω)PU , with PU = [1, 1, 1, 1]T , as described in
sub-section 2.3.8.2.
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-25
-20
-15
-10
-5
0
5
10
15
20
25
P(d
B)
Extracted from APDL
Obtained with TKU
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-25
-20
-15
-10
-5
0
5
10
15
20
25
P(d
B)
Extracted from APDL
Obtained with TKU
Figure 4.32: Pressure measured at the centre of the tube (black) and calculated with T aKU (green) withan imposed pressure of 1 Pa at z=0, and a reflective top, for N = 12 (a) and N = 36 (b)
0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-25
-20
-15
-10
-5
0
5
10
15
20
25
P(d
B)
Extracted from APDL
Obtained with TKU
Figure 4.33: Pressure measured at the centre of the tube (black) and calculated with T aKU (green) withan imposed pressure of 1 Pa at z=0, and an anechoic top, for N = 36
The coordinates inside the APDL model, through which Fig.4.32 was obtained, were based on the
reference frame in figure 4.30, basically being the four coordinates defining the sets of nodes in each
considered section. Since the wave is plane, any of the resulting nodal pressure chosen would be equal,
so any one of the four (number of nodes) would give out the same results. The results for the reflective
74
case are matched with some deviation, for N = 36. This deviation might be reduced if a higher N is
regarded. Previously mentioned factors like matrix conditioning could also enhance the divergence.
This successfully concludes the verification for acoustic transmissibility.
4.4 Vibro-Acoustic Transmissibility
Finally, the results for V-A Transmissiblity are presented in this part.
The proposed model conisted of an acoustic tube connected to a plate by means of a fluid structure
interface, similar to the one proposed by Howard in [47]. However, the model in this work has as plate
inserted at the end in z = −L, whereas the other model has a plate in the midsection of the tube. This
choice was made to try to have an even simpler model for analysis, since otherwise a double pressure-
displacement conversion had to be taken into account (propagation to either sides of the tube).
It was verified that, along this work, the analysis for waves travelling to anechoic surfaces has some
issues shown between the 1D and 3D cases.
(a) (b)
Figure 4.34: Model for the coupled System, with the plate at the end, in red. Mesh generated with 567DOFs
The plate observed in fig.4.34b and, when it is joined to the acoustic tube inserts DOFs of displace-
ment and rotation (six total), so, instead of the previous 513, for the same model without the plate, the
new mesh will have 567 DOFs, 513 plus the new 9 (nodes in the plate) times 6 DOFs. The properties of
the plate are presented in table 4.7 and the ones for the tube are maintained.
Table 4.7: Plate Properties (SHELL181 - Ansys)
Young’s Modulus - E 210 GPa
Mass Density - ρ 7800 kg/m3
Section Width and Height - b 0.1 m
Thickness - t 0.001 m
Poisson’s Ratio - ν 0.3
The oscillating modes (structural or acoustic) of the coupled system were first studied through a
modal analysis, as in table 4.8.
75
Table 4.8: Modes of the Coupled System Tube+Plate (without constraints), with Corresponding ElementType, referring to Fig. 4.34
N.F. f (Hz) Type of Element N.F. f (Hz) Type of Element N.F. f (Hz) Type of Element
9.7847 A 216.25 A 391.43 A
45.531 A 259.68 A 435.94 A
87.374 A 303.33 A 480.79 A
130.05 A 347.10 A - -
173.04 A 354.35 S - -
4.4.1 Pressure-Displacement Ratio for Scalar Vibro-Acoustic Transmissibility
For this segment, the results for point V-A transmissibility, by Pressure-Displacement ratio, are pre-
sented.
A dynamic load of 1 N was applied at the centre of the plate, as is evidenced in fig.4.35. This was
done through the F command in APDL [42].
(a)
Harmonic Load F(N)
plate
Interface
Measured Pressure P
Reflective Boundary
(b)
plate
Interface Harmonic Load F(N)
Measured Pressure P
Anechoic Boundary
Figure 4.35: Pressure Profile along the tube (plane wave), with displacement excitation applied to theplate, for a reflective (a) and anechoic boundary (b)
76
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-20
0
20
40
60
80
100
120
140
TV
A(d
B)
P/u
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-40
-20
0
20
40
60
80
100
120
140
TV
A(d
B)
P/u
Figure 4.36: Pressure/displacement Ratio results from the model in fig.4.34. Model with a reflective endin z = 0. The results were plotted for an imposed load at the center coordinates of the plate, and a”measured” pressure at the midsection of the tube. N = 12, or 315 DOFs (a), and N = 24 or 567 DOFs(b)
(a)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-20
0
20
40
60
80
100
TV
A(d
B)
P/u
(b)0 50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
-20
0
20
40
60
80
100
TV
A(d
B)
P/u
Figure 4.37: Pressure/displacement Ratio results from the model in fig.4.34. Model with an anechoicend in z = 0. The results were plotted for an imposed load at the center coordinates of the plate, and a”measured” pressure at the midsection of the tube. N = 12, or 315 DOFs (a), and N = 24 or 567 DOFs(b)
The results in figures 4.36 and 4.37 are then computed for both a reflective and anechoic end. For
the latter, the real part of the complex transmissibility was computed, since the anechoic boundary
introduces damping (imaginary) into the system. Between the same figures, it can be stated that for
the anechoic case, partial peaks can be observed. This might be related to the fact that for this specific
case, the absorption induced by the anechoic boundary is not total, but almost total. Case which may
suggest that there is still some reflection. Another argument could be considered, which is the fact that
the tube does not prolong itself to infinity (contribution towards resonance frequencies).
4.4.2 Vibro-Acoustic Transmissibility Estimation Through FRFs
4.4.2.1 Frequency Response Sub-Matrices Ratio For Vibro-Acoustic Transmissibility
Now that the scalar ”measured” P/u is already computed and plotted, V-A transmissibility calculation
through the FRFs follows, in this part of the work. The results presented followed the methodology
regarded in section 3.3 where a method is proposed for computation from the HB matrices extracted
directly from the APDL software. This is somewhat of an uncharted area, so there is still some uncer-
77
tainties toward the approach chosen, namely for data verification. In reality it was not found a description
of DOF indexing in HB matrices obtained from ANSYS APDL.
In this case, the same force of 1 N is applied, as the one shown in figure 4.35a, but instead of
comparing ”measured” pressure and displacement values by ratio, the HB global K and M matrices
were extracted from the software, and the FRFs were obtained. The results obtained were based on
the methodology described in [3], in a way that H sub-matrices were used to establish Fluid-Structure
Transmissiblity, for the applied force set U (corresponding to a dynamic displacement in the centre of
the plate, along the Oz axis) and the known K pressure set, as in (2.106): TSFKU = HKU
HUU=
Hz=−L/2,z=−L
Hz=−L,z=−L,
with x, y = b/2, so, precisely in the middle. Inside the projected model, these coordinates were picked
as DOFs inside the global FRF, as in, the DOFs mapped to the corresponding node. So the model is
ready and prepared to extract results, but so far unsuccessfully.
The plots obtained differed from the P/u ratio ones, in a way, so far unaccounted for. Initially, it
would be expected that the computed FRF ratios would at least be close to the P/u ratios (as it was for
pressure ratios in acoustic), but that was not the case, being verified a clear deviation in terms of results.
So, verification is yet to be achieved, hence, the results obtained won’t be showed in this work. Some
reasons for this not to work could be:
• Unknown mapping/assigning of DOFs when the plate is inserted into the model;
• Little information about the inner workings of APDL (internal processes/routines);
• Low Refinement for a coupled model, since the one of the limitations of coupled FE/FE would be
the hardship of requiring high computational power for decent structural and acoustic refinement.
4.4.2.2 MDOF Vibro-Acoustic Transmissibility
A routine was also developed for estimating V-A Transmissibility between a pressure imposition PU in
a section of the tube U and a ”known” displacement set (K) in the plate. This process is quite similar
theoretically to the one presented in section 4.3.2.2.2.
Commencing with obtaining H just like previously, the sub-matrices are then chosen accordingly to
the DOFs to be related, so to speak, pressure somewhere inside the tube (where it wants to be es-
timated) and nodal displacement along Oz. In this case, K consisted of a set with the DOF number
corresponding to the nodal uz displacement inside the vector of DOFs (pressure, rotations and displace-
ments), and U to the pressure DOFs inside the same vector.
78
Chapter 5
Conclusions
In the work presented along this document, a methodology is studied with the purpose of estimating
transmissibility between points of a structure, inside an acoustic medium (cavity) or an acoustic fluid-
structure medium, in the frequency domain.
By primarily verifying, based on the available literature on SDOF/MDOF transmissibility in dynami-
cally excited structures (steady-state) as well as acoustic fluids, a V-A methodology was then proposed
with the same goal. Provided that the available literature was scarce regarding V-A transmissibility (close
to just some experimental results in [15]), the method developed was based on Vibration and Acoustic
Transmissibility computation, and some theoretical projections in [14] for coupling and [17].
So, before actually suggesting a method for V-A Transmissibility computation in 3D, other models
were developed to predict load and displacement transmissibility in structures (spring/mass system and
beam), by means of a FEM. However, in Acoustics, the level of refining for a model to be proper requires
a rather high number of elements per wavelength, which means that simulation times (hours) would have
to get higher in order to avoid pollution [10, 11], and try to guarantee proper meshing. In some cases,
this was not entirely possible, given the relatively high number of DOFs in the mesh and the low sparsity
of the FR matrix used. Indeed, by using finite elements one limits the analysis, sometimes to a certain
maximum mesh refinement.
One last methodology was proposed to predict V-A Transmissibility based on FR matrices (sub-
matrices), used to relate sets of pressure in acoustic coordinates with displacements in structure co-
ordinates, and vice-versa. The FR matrices were extracted from a commercial FEM software (using a
modal analysis) and the procedure for transmissibility computation was proposed (scalar u/P and P/u
ratio and from the FRF the methodology was proposed and awaits verification), with some deviation.
This deviation reflects the difficulty of interpreting how the mapping of the mixed DOFs (in FSI models)
is done within the software and loaded into the matrices, after the structure is inserted into the model
(Displacements, Rotations and Pressure),topic regarding which there is not much clear and concise
information.
In sum, the established FEM models were in fact satisfactorily verified for both vibrational and acous-
tic Transmissibility. The same did not happen for the case of V-A Transmissibility, but, the model is
79
created and ready to operate, missing only the proper indexing between DOFs and model nodes.
The model proposed in this work to estimate Vibration, Acoustic and V-A response through Trans-
missibility functions used a specific method based on the extraction of globally assembled matrices (K,
M and C) from a commercial software and computation of the final T matrices, with Matlab environment.
There are some considerations regarding future work on the topic that were not developed and tested
due to lack of time. These are:
• Clarify the indexing of entries of matrices from the files in HB format for V-A models. Afterwards, it
is expected that the methodology can be quickly tested and verified;
• Verify the V-A model while considering noise;
• Validation of the V-A model;
• Since the interaction is modelled by the commercial software itself, as well as the DOF mapping
into the matrix, a more user friendly option could be used, for instance developing the whole 3D
model in Matlab, from scratch:
• Try and apply (after verification) the developed model inside an array of other acoustic cavities also
with Fluid-Structure interaction;
• Instead of using the receptance method, one could try to obtain T from the Dynamic stiffness matrix
of the system;
• Study V-A Transmissibility in the domain of time instead of frequency.
80
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86
Appendix A
Finite Element Formulation for EB
Beam
To obtain the FE model of equation (2.6), a FE approximation of the following form is assumed:
W (x) =
4∑j=1
∆ejφej(x) (A.1)
where φej(x) are the Hermite cubic polynomials (hat functions).
The FE model is obtained by:
([Ke]− ω2[Me]
)∆e = Qe (A.2)
where
Keij =
∫ xe+1
xe
EId2φeidx2
d2φejdx2
dx, Meij =
∫ xe+1
xe
(ρAφeiφ
ej + ρI
dφeidx
dφejdx
)dx (A.3)
Also from equation (2.6):
Qe1 =
d
dx
(EI
d2W
dx2
)+ ω2I
dW
dx
∣∣∣∣xe
, Qe2 =
(EI
d2W
dx2
)∣∣∣∣xe
Qe3 = −
d
dx
(EI
d2W
dx2
)+ ω2I
dW
dx
∣∣∣∣xe+1
, Qe4 = −
(EI
d2W
dx2
)∣∣∣∣xe+1
(A.4)
For constant EI and ρA, the resulting matrices are the following:
[Ke] =EI
l3e
12 6le −12 6le
6le 4l2e −6le 2l2e
−12 −6le 12 −6le
6le 2l2e −6le 4l2e
, [Me] =
ρAle420
156 22le 54 −13le
22le 4l2e 3le −3le
54 13le 156 −22le
−13le −3l2e −22le 4l2e
(A.5)
87
In (A.5), the rotary Inertia term ρI is neglected, so the equation for mass matrix is halved, comparing to
[29].
The Hermite cubic polynomials are defined by:
φe1 = 1− 3
(x− xele
)2
+ 2
(x− xele
)3
φe2 = −(x− xe)(
1− x− xele
)2
φe3 = 3
(x− xele
)2
− 2
(x− xele
)3
φe4 = −(x− xe)
[(x− xele
)2
− x− xele
](A.6)
88
Appendix B
Tables for Peak Representation
Regarding the Beam Results
Table B.1: Comparison of Peaks (approx.), in Hz, From Fig. 4.6
From Model 26 32 39 38
From Literature 26 30 21 35
Table B.2: Comparison of Peaks (approx.), in Hz, From Fig. 4.8
From Model 30 36 47 45
From Literature 24 30 25 35
89
90
Appendix C
The Problem of obtaining
Harwell-Boeing Sparse Matrices from
ANSYS APDL and converting to
MATLAB
In this section, since this issue was one of the most present ones throughout the development of this
work, given its’ major importance for calculating the FRFs from the dynamic stiffness matrix, it would be
of importance to refer the problems that might have occurred in the extraction.
In order to obtain the desired transmissibility matrices, the proposed methodology included the ex-
traction of K, M and C from the APDL software, after an analysis was executed.
Initially, it was found that these matrices could be extracted from a .sub file which essentially de-
pended upon running a substructure analysis where there was a master-slave relation between nodes,
which in some way (intrinsic to the software itself) would assemble the mentioned global matrices. This
method ended up not working because there was no master-slave stated within the developed APDL
macro.
Afterwards, it was through literature research [50, 53] that it was uncovered that the matrices could
be written from the .full file by using the mentioned hbmat function, which received as arguments, the
file where these would be written (and would generate them for each iteration of the developed macro),
the file format (.txt) and the format of the writing (ascii or binary). Instead of building and running the
macro, it is also possible to extract the matrices from the pre-processor of the software as described in
[53].
Typically, the nodes and elements of the model developed in APDL generally have an arbitrary order
to them, so if the user does not force order through already mentioned commands like NGEN and
EGEN (see [42]), the model will be assembled as the software ”wishes” (the software re-orders equa-
tions as it sees fit according to graph partitioning [50], when it is executing the solution part). This factor
91
would difficult the establishment of a direct equivalence between node number, DOF and line/column
within the matrices.
For symmetric cases, as it is mentioned in [43], only the lower triangle of the matrix is to be stored
(diagonal included) in the resulting file, in order to optimize memory spending. This would consequently
set the requirement of generating the rest after converting (inside MATLAB), for further processing. This
was not an issue for the damped or unsymmetrical cases, since APDL automatically recognizes that
these matrices are not symmetric.
92
