l’ecole centrale de lyon ecole doctorale de mecanique de lyon

L’ECOLE CENTRALE DE LYONECOLE DOCTORALE DE MECANIQUE DE LYON

Numero d’ordre: 2004-05 Annee 2004

THESE

Pour obtenir le grade de

DOCTEUR

SPECIALITE: ACOUSTIQUE

Par

Fusheng SUI

Titre

THEORETICAL STUDY ON TIME-VARYINGVIBROACOUSTIC ENERGY AT HIGH FREQUENCIES

Soutenue publiquement le 27 janvier 2004 devant le jury

MM C. Soize, Professeur, Universite de Marne-la-Vallee RapporteurJ.-C. Pascal, Professeur, Universite du Maine RapporteurJ. L. Guyader, Professeur, INSA-Lyon President du juryB. Mace, Reader, ISVR, University of Southampton ExaminateurL. Jezequel, Professeur, Ecole Centrale de Lyon Directeur de theseM. N. Ichchou, Maıtre de Conferences, ECL Examinateur

Abstract

A new energy approach, called as Transient Simplified Energy Method (MEST), is proposed in this thesisin order to describe the behavior of time-varying vibroacoustic energy in medium and high frequencies.Two kinds of theoretical demonstrations of this method are presented respectively and the recent mathe-matical results are adopted and generalized.

First, the transient energy equation can be derived directly from generalized wave equation in theframe of linear elasto-acoustics. For hyperbolic waves, the scaled Wigner transform is employed torealize the high frequency limit of vibroacoustic energy. Thus the high frequency energy in dissipativerandom media can be described by the transport equation (Liouville equation). For dispersive waves,the main idea in derivation of MEST equation is to make use of the characteristics of the wave group inphase space. Both of them yield a second-order temporal energy equation. Secondly, MEST equationcan also be derived from fundamental power balance equation where linear assumption and superpositionprinciple are applied for forward and backward energy components in bounded systems.

The properties of MEST equation are discussed in detail. As a hyperbolic partial differential equa-tion, MEST equation demonstrates that vibroacoustic energy propagates in form of wavefront with afinite velocity. On the contrary, time-varying vibrational conductivity approach and Transient StatisticalEnergy Analysis (TSEA) display the diffusion behavior for vibroacoustic energy with an infinite veloc-ity. Especially, it is showed that Fourier law with a delay time can be used to form the proposed energyequation. The derivation of MEST equation and the discussion of its properties form the theoretical coreof the thesis.

The local MEST equation is then discretized by finite element technique to describe the total energybehavior of subsystems. It is compared with TSEA equation, the latter can be actually viewed as the weakform of energy diffusion equation. The validations of MEST are carried out by numerical comparativestudies which are applied to both discrete subsystems and distributed structures. The analytical exami-nation of two-oscillator system shows that the time-varying energy is consistent with the description byMEST equation. A number of examples over a wide range of conditions are investigated. Comparingwith exact energy results and solutions of TSEA, MEST presents rather precise predictions of the peakenergy value, the rise time and the damping rate.

In addition, MEST equation are tried to generalize to multi-dimensional systems. The selected tech-nique is to synthesize two different fields, direct field especially characterized by symmetrical waves andreverberant field by plane waves. Some numerical analysis are given to demonstrate this technique.

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ii Abstract

Resume

On propose dans cette these une nouvelle approche a variables quadratiques pour decrire le comporte-ment vibroacoustique transitoire pour les hautes et moyennes frequences: la Methode Energetique Sim-plifiee Transitoire (MEST). L’approche decoule de deux demonstrations relativement differentes. Ongeneralise ainsi des travaux precedents et on clarifie le cadre qui leur est sous-jacent.

L’equation energetique instationnaire est d’abord obtenue a partir des equations d’ondes generaliseedans le cadre de l’elasto-acoustique lineaire. Pour les ondes hyperboliques, la transformee de Wignerest utilisee afin de realiser la limite hautes frequences de l’energie vibroacoustique. L’energie a hautefrequence dans des milieux heterogenes aleatoires dissipatifs peut etre decrite par une equation de trans-port de type Liouville. Pour les ondes dispersives, l’idee principale dans la demonstration repose sur lescaracteristiques d’un paquet d’ondes dans l’espace des phases. L’equation MEST est ensuite obtenuea partir des bilans de puissances locaux et par application d’un hypothese de superposition lineaire desvariables quadratiques specifiques dans les systemes bornes.

Les proprietes de l’equation obtenue sont explorees. En tant qu’equation aux derivees partielleshyperbolique, l’equation demontre que l’energie vibroacoustique se propage sous la forme de frontsd’ondes avec une vitesse finie. L’approximation d’une telle equation par une equation parabolique detype diffusion est egalement aborde. On etablit alors rigoureusement le cadre dans lequel cette approxi-mation est justifiee. En particulier, des notions de decalage temporelle et de relaxation energetique sontintroduit permetant de justifier certains resultats “empiriques” de la litterature.

L’equation locale MEST est ensuite traite afin de decrire le comportement de l’energie total par sous-systemes (approche globale). Les equations obtenues sont comparees a la SEA transitoire (TSEA). Enparticulier on etablit que cette derniere decoule directement d’une equation de diffusion. Les validationsde ces resultats sont effectuees par des etudes comparatives numeriques. Dans ce cadre, aussi bienles systemes discrets que les systemes distribuees sont analyses. Les examens analytiques d’un systemediscret a deux oscillateurs demontre que l’evolution temporelle de l’energie est conforme a la descriptionpar l’equation local discretisee (MEST). Une etude parametrique faisant intervenir la force du couplageentre les oscillateurs et leurs dissipations est effectuee. Comparee avec les solutions de references et lessolutions de la SEA transitoire, on demontre que les equations proposee dans ce memoire presente desprevisions plus precises des pics energetique et des temps de montee. Les validations dans le cas destructures distribuees confirment egalement ces previsions.

Finalement, la generalisation aux systemes multi-dimensionnels est abordee. La technique choisiese base sur la synthese du champ direct par des ondes symetriques et du champ reverbere par des ondesplanes. La mise en œuvre numerique est ensuite proposee.

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iv Resume

Acknowledgments

I would first like to express my gratitudes to my thesis advisor, Prof. Louis JEZEQUEL for the chanceand financial support provided to realize this research. His guidance and encouragement have been veryimportant in helping me complete this study.

I am very grateful to my supervisors, Dr. ICHCHOU, for his patience, dedication and insight into thedirection of this thesis. His brilliant technical expertise and teaching ability have brought energy andinspiration into every one of our discussions.

This work has been accomplished in the Laboratoire de Tribologie et Dynamique des Systemes (UMRCNRS 5513) and Departement de Mecanique des Solides, Genie Mecanique et Genie Civil, Ecole Cen-trale de Lyon, I am thus thankful for those who kindly helped me in three-years work there. I had thepleasure of meeting so many wonderful people whose friendships were invaluable to make me live andwork better here. The experience of studying in France will be a treasure in my life.

I would like to thank the members of my committee for their input and interest. My acknowledgmentsshould be addressed to the President of committee, Mr. GUYADER and my examiners Professor C.SOIZE, Professor J.-C. PASCAL, and Mr. MACE, whose comments and questions helped to clarify thethesis.

Finally, I express my special gratitude to my parents and my family for their loves and supports in all myyears.

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vi Acknowledgments

Remerciements

Je voudrais en tout premier lieu exprimer ma gratitude a mon directeur de these, Monsieur le ProfesseurL. JEZEQUEL pour le soutien et l’aide financiere qu’il a apporte pour realiser cette recherche. Sesconseils et encouragement ont ete tres importants en m’aidant a achever cette etude.

Je suis tres reconnaissant a mon encadrants, Dr. ICHCHOU, pour sa patience, son devouements et sa per-spicacite dans la direction de cette these. Son expertise technique brillante et ses capacites d’enseignementont apporte l’energie et l’inspiration dans chacune de nos discussions.

Ce travail a ete accompli au Laboratoire de Tribologie et Dynamique des Systemes (UMR CNRS 5513)et Departement de Mecanique de Solides, Genie Mecanique et Genie Civil de l’Ecole Centrale de Lyon,je suis donc reconnaissant a ceux qui m’ont aide durant ces trois annees de travail. J’ai eu le plaisir derencontrer tant de personnes merveilleuses dont l’amities etaient de valeur inestimable qui m’ont aide amieux vivre et travailler. L’experience d’etudier en France aura ete d’un grand enrichissement dans mavie.

Je voudrais remercier les membres de jury pour leur l’interet qu’ils ont manifeste. Mes remerciements de-vraient egalement etre adresses au president de jury, Monsieur GUYADER et mes rapporteurs ProfesseurC. SOIZE, Professeur J.-C. PASCAL, et M. MACE, dont les commentaires et les questions m’ont aide aclarifier la these.

Enfin, j’exprime ma gratitude speciale a mes parents et ma famille pour leur amour et leur appui duranttoutes ces annees.

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viii Remerciements

Table of Contents

Abstract i

Acknowledgements iii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

Chapter 1 General review: high frequency dynamics problem and the methodologies 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Deterministic approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Structural dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Limitations of Element-based Method . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Modal Synthesis Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.4 Enhanced deterministic techniques . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Statistical approaches: based on quadratic variables . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Statistical Energy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2 Wave Intensity analysis: enhancements of SEA . . . . . . . . . . . . . . . . . . 11

1.3.3 A case study: energy density of flexural vibration of thin plates . . . . . . . . . . 13

1.3.4 Vibrational conductivity approach . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.5 Others local energy methods: synthesis of fields . . . . . . . . . . . . . . . . . . 19

1.3.6 Comparative studies on some local energy methods . . . . . . . . . . . . . . . . 23

ix

x TABLE OF CONTENTS

1.4 Transient vibroacoustics in high frequency . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4.1 Transient statistical energy analysis . . . . . . . . . . . . . . . . . . . . . . . . 25

1.4.2 Virtual mode synthesis and simulation . . . . . . . . . . . . . . . . . . . . . . . 28

1.4.3 Time-varying energy conductivity approach . . . . . . . . . . . . . . . . . . . . 30

1.5 Concluding remarks and the aim of the thesis . . . . . . . . . . . . . . . . . . . . . . . 31

Chapter 2 High frequency wave energy: transport theory and wavefront solutions 33

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Waves transport in random media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 High frequency limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.1 Classic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.2 Wigner transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 Transport equation for acoustic wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5 Transport equation for hyperbolic system . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6 Diffusion approximation for high frequency waves in random media . . . . . . . . . . . 44

2.7 The time-delayed diffusion system and wavefront solution . . . . . . . . . . . . . . . . 46

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter 3 Derivation of time-varying energy equation 49

3.1 Time-varying energy described by waves characteristics . . . . . . . . . . . . . . . . . . 49

3.1.1 Derivation by transport theory for hyperbolic waves . . . . . . . . . . . . . . . . 50

3.1.2 Discussions on transport equation . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1.3 Derivation of energy equation for dispersive waves . . . . . . . . . . . . . . . . 57

3.2 Derivation by employing energy law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.1 Energy in undamped systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.2 Energy in damped systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.3 Time-varying energy equation for plane waves . . . . . . . . . . . . . . . . . . 65

3.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

TABLE OF CONTENTS xi

3.4 Properties of MEST equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4.1 Property of MEST equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4.2 Property of diffusion energy equation . . . . . . . . . . . . . . . . . . . . . . . 70

3.4.3 Comparison of properties two kinds of energy equations . . . . . . . . . . . . . 71

3.4.4 Relation between MEST equation and diffusion approximation . . . . . . . . . . 77

3.5 Analysis of the solution of MEST equation . . . . . . . . . . . . . . . . . . . . . . . . 79

3.6 Summary and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Chapter 4 Time-varying vibration energy in discrete systems 83

4.1 The case of two oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.1 Energy equation of two undamped oscillators system . . . . . . . . . . . . . . . 84

4.1.2 Numerical analysis on undamped two oscillators . . . . . . . . . . . . . . . . . 86

4.1.3 Time-varying energy equation for discrete structures . . . . . . . . . . . . . . . 87

4.2 Comparison with reference values from other methods . . . . . . . . . . . . . . . . . . 91

4.2.1 The exact energy expression of two oscillators systems . . . . . . . . . . . . . . 91

4.2.2 Transient statistical energy analysis . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3 Coupling loss factors and the rise time . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4 Numerical study on two oscillators system . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4.1 Two oscillators with identical frequencies . . . . . . . . . . . . . . . . . . . . . 95

4.4.2 Two different oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.3 Energy flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5 The case of two coupled subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.5.1 Two coupled rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.5.2 Two coupled beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.6 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Chapter 5 Time-varying vibrational energy in one-dimensional distributed structures 119

5.1 Longitudinal wave: time-varying energy in a rod . . . . . . . . . . . . . . . . . . . . . 120

xii TABLE OF CONTENTS

5.1.1 Exact result for longitudinal waves in rod . . . . . . . . . . . . . . . . . . . . . 120

5.1.2 Energy density of the rod by MEST equation . . . . . . . . . . . . . . . . . . . 123

5.1.3 Energy density of the rod by Nefske’s equation . . . . . . . . . . . . . . . . . . 125

5.1.4 Numerical simulation and analysis . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2 Flexural wave: time-varying energy in a beam . . . . . . . . . . . . . . . . . . . . . . . 135

5.2.1 Time-varying energy expression by displacement response . . . . . . . . . . . . 135

5.2.2 Solutions of the energy equations . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2.3 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.2.4 Energy shock response spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Chapter 6 Generalization of the energy equation to multi-dimensional systems 147

6.1 MEST equation for symmetric waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.2 Energy propagation in direct field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.3 Boundary condition for reverberant field . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.3.1 The case of isolated system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.3.2 The case of coupled subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.4 Finite element method for Local Energy Approach . . . . . . . . . . . . . . . . . . . . 160

6.4.1 FEM formulations derived from variational method for isolated system . . . . . 160

6.4.2 FEM formulations derived from variational method for coupled subsystem . . . 162

6.5 Applications of energy FEM: synthesis of two fields . . . . . . . . . . . . . . . . . . . . 164

6.5.1 FEM model: mesh and stiffness matrix . . . . . . . . . . . . . . . . . . . . . . 164

6.5.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.6 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Chapter 7 General conclusions and perspectives 173

7.1 Significance of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.2 Summary of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

TABLE OF CONTENTS xiii

7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.4 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Bibliography 177

Appendix 183

A Analytical energy results for two undamped oscillators 183

B Energy expression of two equal oscillators system 187

C MEST solution of two coupled subsystems 189

D MEST equation for one-dimensional longitudinal wave 191

E List of Publications 193

xiv TABLE OF CONTENTS

List of Figures

Figure 1.1 Three-dimensional vibrating body . . . . . . . . . . . . . . . . . . . . . . . . . 3

Figure 1.2 An example of four coupled SEA subsystems . . . . . . . . . . . . . . . . . . . 9

Figure 1.3 Wave incidence and transmission at a boundary . . . . . . . . . . . . . . . . . . 12

Figure 1.4 The thin plate subjected to a harmonic point excitation . . . . . . . . . . . . . . . 14

Figure 1.5 The model of two coupled rectangle plates . . . . . . . . . . . . . . . . . . . . . 15

Figure 1.6 Comparison of energy ratio, case 1 . . . . . . . . . . . . . . . . . . . . . . . . . 16



Figure 1.9 Exact energy distribution in a square thin plate by equation (1.56) for η = 0.05 . . 19

Figure 1.10 Energy distribution in a square thin plate by VCA) for η = 0.05 . . . . . . . . . . 20

Figure 1.11 The reverberant field forms in a sound enclosure . . . . . . . . . . . . . . . . . . 21

Figure 1.12 Single plate model with excitation at (0.532, 0.482) . . . . . . . . . . . . . . . . 24

Figure 1.13 Comparison of energy results along the line Ly = 0.5m . . . . . . . . . . . . . . 24

Figure 1.14 Comparison of energy results along the line Ly = 0.75m . . . . . . . . . . . . . 24

Figure 1.15 Comparison of energy results along the line Ly = 0.1m . . . . . . . . . . . . . . 24

Figure 1.16 Difference between TSEA result and exact solution . . . . . . . . . . . . . . . . 27

Figure 1.17 Two coupled oscillators used to re-examine TSEA . . . . . . . . . . . . . . . . . 27

Figure 1.18 Time response with respect to input frequency Ω . . . . . . . . . . . . . . . . . . 30

Figure 2.1 Sketch of phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

xv

xvi LIST OF FIGURES

Figure 2.2 Wave transports in a random medium . . . . . . . . . . . . . . . . . . . . . . . . 36

Figure 3.1 Conception of wave packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Figure 3.2 Wave designation into a simple waveguide . . . . . . . . . . . . . . . . . . . . . 62

Figure 3.3 Illustration of plane wave in two dimensions . . . . . . . . . . . . . . . . . . . . 65

Figure 3.4 Bounddary condition illustrated by a simple waveguide . . . . . . . . . . . . . . 67

Figure 3.5 Characteristic line of MEST equation . . . . . . . . . . . . . . . . . . . . . . . . 69

Figure 3.6 The initial transient energy density input employed by diffusion energy equation . 71

Figure 3.7 Waveguide under a unit energy source at x=0 and t=0 . . . . . . . . . . . . . . . 72

Figure 3.8 Green’s function versus s-axis for the wave equation at various times values. . . . 73

Figure 3.9 Green’s function versus s-axis for the MEST equation at various times values . . 73

Figure 3.10 Green’s function versus s-axis for the diffusion equation at various times values . 74

Figure 3.11 Green function for the case ∆ > 0 . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 3.12 Green function for the case ∆ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 3.13 Green function for the case ∆ < 0 . . . . . . . . . . . . . . . . . . . . . . . . . 77

Figure 4.1 Two oscillators model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Figure 4.2 Energy motion and partition, case A in Table 4.1 . . . . . . . . . . . . . . . . . . 88

Figure 4.3 Energy motion and partition, case B in Table 4.1 . . . . . . . . . . . . . . . . . . 88

Figure 4.4 Energy motion and partition, case C in Table 4.1 . . . . . . . . . . . . . . . . . . 88

Figure 4.5 Energy motion and partition, case D in Table 4.1 . . . . . . . . . . . . . . . . . . 88

Figure 4.6 Energy motion and partition, case E in Table 4.1 . . . . . . . . . . . . . . . . . . 89

Figure 4.7 Comparison of energy flows. – – –, case A; – · – ·, case B; ——, case E. . . . . . 89

Figure 4.8 Discrete model of FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Figure 4.9 Boundary condition of two coupled elements . . . . . . . . . . . . . . . . . . . . 90

Figure 4.10 The difference of the exact energy expressions. . . . . . . . . . . . . . . . . . . . 93

Figure 4.11 Statistical energy analysis model for two systems. . . . . . . . . . . . . . . . . . 93

Figure 4.12 Kinematic two-degree-of-freedom model . . . . . . . . . . . . . . . . . . . . . . 95

LIST OF FIGURES xvii

Figure 4.13 Comparison of three results for case A in Table 4.2(1). . . . . . . . . . . . . . . . 96

Figure 4.14 Comparison of three results for case A in Table 4.2(2). . . . . . . . . . . . . . . . 97

Figure 4.15 Comparison of three results for case B in Table 4.2. . . . . . . . . . . . . . . . . 98

Figure 4.16 Comparison of three results for case C in Table 4.2. . . . . . . . . . . . . . . . . 98

Figure 4.17 Comparison of three results for case D in Table 4.2. . . . . . . . . . . . . . . . . 99

Figure 4.18 Comparison of three results for case E in Table 4.2. . . . . . . . . . . . . . . . . 99

Figure 4.19 Comparison of three results for case F in Table 4.2. . . . . . . . . . . . . . . . . 100

Figure 4.20 Comparison of three results for case E in Table 4.2(1). . . . . . . . . . . . . . . . 100

Figure 4.21 Comparison of three results for case F in Table 4.2(1). . . . . . . . . . . . . . . . 100

Figure 4.22 The influence of damping, case A and G in Table 4.2. . . . . . . . . . . . . . . . 101

Figure 4.23 The influence of damping, case B and I in Table 4.2. . . . . . . . . . . . . . . . . 101

Figure 4.24 Transmitted energy in test A (Table 4.3) . . . . . . . . . . . . . . . . . . . . . . 103

Figure 4.25 Transmitted energy in test B (Table 4.3) . . . . . . . . . . . . . . . . . . . . . . 103

Figure 4.26 Transmitted energy in test C (Table 4.3) . . . . . . . . . . . . . . . . . . . . . . 103

Figure 4.27 Transmitted energy in test D (Table 4.3) . . . . . . . . . . . . . . . . . . . . . . 103

Figure 4.28 Energy flow in case A (Table 4.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 4.29 Energy flow in case F (Table 4.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 4.30 Energy flow in test A (Table 4.3) . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 4.31 Energy flow in test B (Table 4.3) . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 4.32 Model of two coupled rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Figure 4.33 Velocities density V (x, t) of two coupled rods, D1 = D2 = 50 . . . . . . . . . . 108

Figure 4.34 Velocities density V (x, t) of two coupled rods, D1 = D2 = 200 . . . . . . . . . 109

Figure 4.35 Velocities density V (t) at specific positions in two coupled rods . . . . . . . . . . 109

Figure 4.36 Energy comparison of two coupled rods with a damping coefficient D = 50. . . . 110

Figure 4.37 Energy comparison of two coupled rods with a damping coefficient D = 200. . . 111

Figure 4.38 Model of two coupled beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

xviii LIST OF FIGURES

Figure 4.39 The mobility relation of uncoupled beams . . . . . . . . . . . . . . . . . . . . . 112

Figure 4.40 Frequency response function Vr2/Fs1 . . . . . . . . . . . . . . . . . . . . . . . 113

Figure 4.41 Impulse response function Vr2(t) by inverse FFT; 0.5s . . . . . . . . . . . . . . . 114

Figure 4.42 Time history of total energy in receive beam in frequency band 64 − 800 Hz. . . . 115

Figure 4.43 Time history of total energy in receive beam in frequency band 256 − 1024 Hz. . 115

Figure 4.44 Illustration of the applications and validations of MEST for discretized systems . 116

Figure 5.1 Rod subjected to an impulse force . . . . . . . . . . . . . . . . . . . . . . . . . 120

Figure 5.2 Displacement of the rod subjected to an impulse excitation . . . . . . . . . . . . 121

Figure 5.3 Displacement of the rod subjected to an impulse excitation (x = l/2) . . . . . . . 122

Figure 5.4 Displacement of the rod subjected to a step excitation . . . . . . . . . . . . . . . 122

Figure 5.5 Displacement of the rod subjected to a step excitation (x = l/2) . . . . . . . . . . 123

Figure 5.6 Exact energy density from displacement response . . . . . . . . . . . . . . . . . 126

Figure 5.7 MEST prediction for the rod subjected an impulse excitation . . . . . . . . . . . 126

Figure 5.8 Solution of Nefske’s first order equation . . . . . . . . . . . . . . . . . . . . . . 126

Figure 5.9 Comparison of energy density in the rod: x = 0.4m . . . . . . . . . . . . . . . . 127

Figure 5.10 Comparison of energy density in the rod: x = 0.8m . . . . . . . . . . . . . . . . 127

Figure 5.11 Exact energy density in space domain . . . . . . . . . . . . . . . . . . . . . . . 128

Figure 5.12 MEST prediction of energy distribution in the rod . . . . . . . . . . . . . . . . . 128

Figure 5.13 Energy distribution predicted by the first order diffusion equation . . . . . . . . . 129

Figure 5.14 Distribution of energy at different time points by exact result . . . . . . . . . . . 130

Figure 5.15 Distribution of energy at different time points by MEST solution . . . . . . . . . 130

Figure 5.16 Distribution of energy at different time points by diffusion equation . . . . . . . . 130

Figure 5.17 Energy density in undamped rod by displacement response . . . . . . . . . . . . 131

Figure 5.18 MEST solution for undamped rod . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Figure 5.19 Energy density in damped rod by displacement response . . . . . . . . . . . . . . 132

Figure 5.20 MEST solution for damped rod . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

LIST OF FIGURES xix

Figure 5.21 Comparison of energy results between MEST and exact value. (x = 0.3L). . . . . 133

Figure 5.22 Energy results of the rod subjected to the step impact . . . . . . . . . . . . . . . 134

Figure 5.23 Right- and left-traveling energy densities (x = 0.3L). . . . . . . . . . . . . . . . 134

Figure 5.24 Beam subjected to a transverse unit impulse . . . . . . . . . . . . . . . . . . . . 135

Figure 5.25 Frequency response of the displacement at x = 0.6L . . . . . . . . . . . . . . . 137

Figure 5.26 Prediction of time history of energy density at l = 0.6L, fc = 1000Hz. . . . . . . 138

Figure 5.27 Prediction of time history of energy density at l = 0.6L, fc = 2000Hz. . . . . . . 139

Figure 5.28 Prediction of peak time by MEST, fc = 2000 Hz, l = 1.2 m . . . . . . . . . . . . 140

Figure 5.29 Prediction of peak time by MEST, fc = 1500 Hz, l = 1.6 m . . . . . . . . . . . . 140

Figure 5.30 Concept of SRS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Figure 5.31 Program diagram for energy SRS plotting . . . . . . . . . . . . . . . . . . . . . 142

Figure 5.32 Energy SRS comparison, η = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . 143

Figure 5.33 Energy SRS comparison, η = 0.15. . . . . . . . . . . . . . . . . . . . . . . . . . 144

Figure 5.34 Wave group propagates like an impulse . . . . . . . . . . . . . . . . . . . . . . . 144

Figure 6.1 Illustration of hybrid field: from direct field to reverberant field . . . . . . . . . . 150

Figure 6.2 The square plate used to demonstrate the direct-field energy . . . . . . . . . . . . 152

Figure 6.3 Energy density in direct field plotted in three-dimensions . . . . . . . . . . . . . 153

Figure 6.4 Energy density in direct field plotted in contour curves . . . . . . . . . . . . . . . 153

Figure 6.5 Energy density in direct field plotted in three-dimensional contour curves . . . . . 154

Figure 6.6 Comparison between two energy methods. . . . . . . . . . . . . . . . . . . . . . 155

Figure 6.7 Illustration of boundary condition of an isolated two dimensional structure . . . . 156

Figure 6.8 Illustration of boundary conditions of two dimensional coupled structures . . . . 158

Figure 6.9 Rectangular element used in the FEM model . . . . . . . . . . . . . . . . . . . . 165

Figure 6.10 FEM model of a single plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Figure 6.11 Energy density in the single plate by FEM model . . . . . . . . . . . . . . . . . 167

Figure 6.12 Energy density in the single plate calculated by plate vibration equation. . . . . . 168

xx LIST OF FIGURES

Figure 6.13 Comaprison of energy density by three methods along y = 0.6 m. . . . . . . . . 168

Figure 6.14 Comparison of energy density by three methods along y = 0.15 m. . . . . . . . . 169

Figure 6.15 Reflectivity coefficient influenced by damping factor: η = 0.01. . . . . . . . . . . 170

Figure 6.16 Reflectivity coefficient influenced by damping factor: η = 0.1. . . . . . . . . . . 170

Figure 6.17 FEM model of two coupled plates. . . . . . . . . . . . . . . . . . . . . . . . . . 171

Figure 6.18 Energy distribution in coupled plates for λ1 = λ2 = 0.95 and β1 = β2 = 0.05. . . 172

Figure 6.19 Energy distribution in coupled plates for λ1 = λ2 = 1 and β1 = β2 = 0. . . . . . 172

Figure A.1 Two oscillators model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

List of Tables

Table 4.1 Parameters and data for study of the coupling strength and energy partition . . . . 87

Table 4.2 Parameters and data for two equal oscillators . . . . . . . . . . . . . . . . . . . . 96

Table 4.3 Parameters and data for two different oscillators . . . . . . . . . . . . . . . . . . . 102

xxi

xxii LIST OF TABLES

List of Symbols

A Generalized mass matrix

a Amplitude

a Angularly resolved energy (section 2.2 and 2.4)

B Generalized damping matrix

C Coefficient matrix

C Damping coefficient

c Wave velocity

cg Velocity of wave group

D Generalized state matrix

D Bending stiffness

d Damping coefficient

E Young’s modulus

E Global energy

e energy of oscillator (section 1.4.1)

Ep Peak energy level

F force

f Force vector

f Frequency

G(.) Green function

H(.) Heaviside function

h Thickness

I Identity matrix

I Energy flow; intensity

I0 Zero order modified Bessel function of the first kind

xxiii

xxiv LIST OF SYMBOLS

i Imaginary unit i =√−1

K Reflection matrix (section 1.3.5)

k Wave number

K0 Zero order modified Bessel function of the second kind

K Generalized stiffness matrix

k Stiffness

L Coefficient matrix in SEA equation (section 1.3.1)

L Dispersion matrix

L Differential operator

L() Laplace transform

L Length

l Typical correlation length of the inhomogeneities (section 2.2)

M Mobility

m Mass

n Unit vector

n Modal density

P Input power vector

Pij Exchange power

P Input power

p Pressure

Q Function operator (section 3.1.1)

r Coupling ratio

S Cross-section area

s Eigenvector

T Kinetic energy density

t Time

tr Rise time

U Potential energy density

u Displacement

u Vector of state variables

v Velocity

v Velocity vector

LIST OF SYMBOLS xxv

V General velocity in phase space

W Wigner transform (matrix)

W Energy density

w Scalar Wigner transform

x Space coordinates vector

X General coordinates in phase space

α Absorption coefficient

β Coupling coefficient

∆ Determinant of a matrix

δ Dirac function

ε scale measure

Φ Transform matrix

ϕ wave function (section 2.3.1)

ηi Damping loss factor

ηij Coupling loss factor

κ Compressibility of acoustics media

λ Wave length

λ Energy partition ratio (section 4.1.1)

λτ Eigenvalues

µ Mutual scattering ratio

ν Poisson ratio

Π Variational function

πdiss Dissipated power density

θ Angle of incidence

ρ Density

σ Differential scattering cross-section

τ Mean free time

τ0 Time delay

τij Transmission coefficient

Ω Boundary domain

ω Circular frequency

ω Diagonal frequency matrix

xxvi LIST OF SYMBOLS

Ξ Collision operator

ζ Damping coefficient

∇ Gradient operator

∇· Divergence operator

∇2 Laplace operator

E[ ] Expectation operator

( )T Transpose of vector

[ ]T Transpose of matrix

Be defined by

ψ∗ The conjugate transpose of ψ

diag(.) Diagonal matrix

div Divergence

grad Gradient

Chapter 1

General review: high frequency dynamicsproblem and the methodologies

1.1 Introduction

In the field of acoustics and mechanical vibration engineering, the problems of high frequency dynamicshave always been rather attractive subject to the researchers. Especially in recent years, as technicaladvances has developed in the design of high speed machinery, some special requirements of structuresand more strict design criteria on acoustic properties of products have generated increasing interest ofstudying high frequency dynamics. In addition, an engineer is often faced with the problem of analyzingthe vibrational behavior of a structure. The high frequency vibroacoustics behavior is of special con-cern to engineers mainly because of the potential for excessive sound transmission and radiation (forexample, in architectural acoustics), and in some cases because of the potential for fatigue damage ofstructures in the aircraft and automotive industries. These problems arise in either steady-state or tran-sient cases: the former are associated with stationary high frequency sources while the latter involves thefact that transient excitations always cause high frequency responses. To deal with these problems, it isimperative to establish a set of effective methods to predict high frequency dynamic instead of the verytime-consuming and expensive prototype testing approaches.

However, it is well known that the high frequency dynamics and the related subjects pose specialproblems and difficulties which results from the special characteristics of high frequency range. Then,what is the quantitative meaning of high frequency? It requires qualification according to the actualproblems. Generally, so-called high frequency is viewed as the frequency many times the fundamentalnatural frequency of a system, where the structural wavelengths are much smaller than the typical systemdimension. This is a non-rigorous definition, because the exact description should be accompanied bythe description of specific structures and the research objective. In this thesis, the high frequency band isnot beyond the audible frequency limitation.

In principle, the difficulties of high frequency problems can be ascribed to its complexity and un-certainty. As frequency increases, the resonance peaks increasingly overlap each other. The responseat a specific high frequency comprises substantial contributions from a number of modes, while thesemodes cannot be well identified. Moreover, the high frequency responses are increasingly sensitive tosmall perturbations of the physical parameters of structures and their boundary conditions. These highfrequency characteristics result in some technical difficulties to deal with the problems. To cope with

1

2 General review: high frequency dynamics problem and the methodologies

these challenges, many efforts have been devoted to the development of methodologies about this sub-ject. This chapter will serve both as a review of these methods and also an extension of these viewpointsin preparation to propose of a new technique in the followed chapters.

It is appropriate to attempt a classification of the methods. In general, there are two major categoriesof the methods to deal with high frequency problems, which are sometimes called deterministic andstatistical approaches in the light of study means, or called modal and energy approaches according tothe different variable.

1.2 Deterministic approaches

Conventional vibration analysis is based on a mathematical deterministic model which represents thegeometric forms, boundaries, properties of structure and material. Roughly, the process of predicting thedynamic responses typically involves a series of analytical steps as follows

1. Structural dynamic models

2. A finite element method (FEM) model is created to describe the stiffness and distributed massproperties.

3. The degrees of freedom (DOF) in the FEM model are reduced to create the dynamic model to bevalid within the frequency range of concern.

4. Modal synthesis and analysis.

1.2.1 Structural dynamic model

In a deterministic model, the problems need to seek the solution for the governing partial differentialequation at each position in temporal and spatial domain. These models are in the form of mass andstiffness matrices, usually constructed from FEM models. The solution represent its vibrational responsein terms of displacements or velocities at any position of the structures over a range of frequency ofinterest. Usually, the dependent variables are functions not only of time t, but also of location withinan multi-dimensional spatial domain D, with coordinates x = (x, y, z)T . For example, Figure 1.1shows a three-dimensional solid object of arbitrary shape subjected to external forces. The applicationof appropriate physical laws and certain types of assumptions (for example, damping is suppose to benegligible) leads to Navier equation [1]

ρ

(∂2u(1)

∂t2+∂2u(2)

∂t2

)− (λ+ µ)∇

(∇ · u(2)

)− µ∇2u(1) − µ∇2u(2) = f (1.1)

where u = (u, v, w) is the displacement, and u(1) is a solenoidal field represents the displacementresponse of a shear wave, while u(2) is an irrotational field represents the displacement of a compressivewave. The material of vibrating body is assumed to be linear, isotropic, and homogeneous, so that

µ =E

2(1 + ν)λ =

νE

(1 + ν)(1 − 2ν)(1.2)

1.2. Deterministic approaches 3

dz

dy

Z

Y

X

F1(t)

w

vu

dx

F2(t)

Figure 1.1: Three-dimensional vibrating body

where ν is Poisson ratio and E is the Young’s modulus. Unless otherwise stated, these assumptionswill run through this thesis. Taken the divergence or the curl of both sides of equation (1.1), it can bedecomposed into two equations

ρ∂2u(1)

∂t2− µ∇2u(1) = 0 (1.3)

ρ∂2u(2)

∂t2− (λ+ 2µ)∇2u(2) = 0 (1.4)

In general, the governing partial differential equation of the dynamics can be written in a hyperbolicform (

A(x)∂t +n∑

i=1

Di∂xi + B

)u = f , (1.5)

This equation is the general wave equation, where u is a n−element column vector defined over coor-dinates x ∈ D ⊂ Rn and t ≥ 0. The mass matrix A and state matrix D

i, (i = 1, · · · , n), are realsymmetric n × n matrices; B is a real n × n matrix whose symmetric part models damping (or otherkinds of) loss; f is a the q−element real column vector representing forces or excitations. Initially asimple case of homogeneous equation without damping B = 0 is supposed.

For example, let u = (v, p)T for the velocity v and pressure disturbances p, equation (1.5) repre-sents the acoustics equation

ρ∂v∂t

+ ∇p = 0, κ∂p

∂t+ ∇ · v = 0 (1.6)

where ρ is the density of acoustic media and κ is the compressibility.

If u = (∂u∂t ,

∂u∂xi )T , u is the displacement response, equation (1.5) is equivalent to the conventional

wave equation∂2u

∂t2− c2∇2u = 0 (1.7)


In the case of time-harmonic acoustic/structural problems, that is

u(t,x) = e−iωtu(x) (1.8)

then the steady-state domain is governed by the Helmholtz equation.

Lu = f , (1.9)

where the operator L = −∇2 − k2, k is wave number.

The boundary conditions in equation (1.5) are summarized to three types by specifying a combinationof u and its normal derivatives on the boundary:

• Dirichlet: u = d on the boundary ∂Ωd.

• Neumann: ∇u = g on the boundary ∂Ωn.

• Robin: a∇u + bu = c on the boundary ∂Ωr.

For the complex structures, the numerical solutions of above partial differential equations are usuallysought by some approximation methods. These methods fall into two kinds: element-based approachessuch as Finite Element Method (FEM) or Boundary Element Method (BEM), and modal-based approachlike Normal Modes Analysis (NMA).

1.2.2 Limitations of Element-based Method

The governing equations together with their boundary conditions provide the usual starting point for theanalytical dynamic model of the structures. One can apply the well-established computational technique,Finite Element Method (FEM) to estimate the linear response of local points at specific frequency.

FEM can be summarized in one sentence: Project the weak, or variational, form of the differentialequation onto a finite-dimensional function space. The response is described in terms of a large numberof trial functions defined over various local elements of the domain. The local trial functions are generallyreferred to as basis or shape functions and are chosen to interpolate between various nodal values ofthe response. By building a mesh in term of shape functions, the partial differential equation and theboundary conditions are discretized to obtain a linear system

K u = f (1.10)

where K is the stiffness matrix and the unknown vector u contains the values of the approximatesolution at the mesh points.

In low frequency range, FEM has its advantage to obtain a rather precise solution. However, in thehigh frequency domain, the deterministic methods, for example, FEM [2] or Boundary Element Method(BEM) [3] are found impracticable. The reasons of impediments consist mainly in two aspects :

Technical impossibility. In these element-based methods, the continuum domains or their boundarysurfaces are discretized into small elements. The field variables within each element are described interms of simple, approximate shape functions, a substantial amount of elements must be used in orderto keep the approximation error within acceptable levels. In other words, there is the requirement that

1.2. Deterministic approaches 5

the size of the finite elements used to represent any component must be considerably smaller than theminimum structural wavelength in that component at any frequency. For BEM, it was pointed out in [4]that at least five constant elements should be used for a wavelength to ensure the error in the Euclideannorm is less than five percent, while the number of constant elements per wavelength should be at leasteight when the maximum norm is employed for the error description. Consequently, the required numberof elements (the model size) increases with frequency to a power of between 1.5 and 3, depending on thespecific structures. The effort required to define an FE model increases with size and with the geometricand material complexity of a model. The number of modes involved may run into millions for largestructures such as satellite launch vehicles, aircraft, etc., then the amount of computing execution timewould be so prohibitively large that it is impracticable to run a large number of cases in the search fordesign improvements. The solution of this problem relies seriously on the speed and memory of theadvanced computers in the future.

Uncontrollable uncertainty. For high frequency dynamics, the problems of conventional elementbased methods lie not only in the prohibitive demand on the capability of computer, but also in the er-rors. In addition, the strengths of reflection and scattering (redistribution of incident wave energy intomany directions) of vibrational waves by structural non-uniformities, inhomogeneities, or discontinuitiesof geometry or material properties, generally increase as the wavelength decreases (or as the frequencyincreases). Due to the numerical dispersion coming from the element based methods (FEA or BEM),the wave number of such numerical solution differs from the wave number of the exact solution. Thereexists irreducible uncertainty concerning the high order natural modes and frequencies, which makes thefrequency response at any individual point unpredictable. The frequency response at any point on a com-plex, multi-mode structure is increasingly sensitive to small perturbations of a structure and its boundaryconditions as frequency increases. Even small, variations in geometrical component dimensions, mate-rial properties and assembly tolerances imply large variations in the mid- and high-frequency response,so that one single deterministic prediction model with nominal system property values is no longer rep-resentative for the mid- and high-frequency dynamic behavior of all possible physical realization of thesystem under study.

1.2.3 Modal Synthesis Approach

The dynamic behavior of a discretized, multiple degree-of-freedom (DOF) mechanical system, can bedescribed by the following matrix-form governing equation:

M u + D u + K u = F (t) (1.11)

whereu(t) =

u1(t), u2(t), · · · , unq(t)

T(1.12)

is the nq-dimensional vector which represents the generalized displacement at specific location, and

F (t) =F1(t), F2(t), · · · , Fnq(t)

T(1.13)

is the nq-dimensional vector generalized force. M, D, K are the matrices of mass, damping, and stiffness,respectively. By matrix transform, equation (1.11) can be rewritten as

M

ξ

+ D

ξ

+ K ξ = ΦT F (t) ξ (1.14)

whereξ(t) = ξ1(t), ξ2(t), · · · , ξnm(t)T (1.15)


is modal vector. And M, D, K are the diagonalized mass, damping and stiffness matrices,

M = ΦTMΦ, D = ΦT

DΦ, K = ΦTKΦ. (1.16)

The relation between modal space and physical displacement is expressed by

u(t) = Φ ξ(t) (1.17)

where Φ is a nq × nm matrix whose elements φim are called mode shape coefficients. Then equation(1.17) can also be expressed by

ui(t) =nm∑

m=1

φimξm(t) (1.18)

The decoupled equation (1.14) consist of nm independent linear equation, their Fourier transform reads

ξm(iω) =nq∑

j=1

φjmFj(iω)(ω2

m − ω2) + i2ζmωmω(1.19)

where

ωm =√km

mm, ζm =

dm

2√kmmm

(1.20)

where mm, dm, km are the diagonal terms of M, D and K, respectively. Substitute equation (1.18) into(1.19), it yields

ui(iω) =nq∑

j=1

nm∑m=1

φimφjmFj(iω)(ω2

m − ω2) + i2ζmωmω(1.21)

which is called frequency response function (FRF). It is customary to orthonormalize the mode shapes,namely

ΦTnMΦn = I (1.22)

where I is the identity matrix. It then yields

ΦTnKΦn = ω2

n (1.23)

Assuming that the eigenvalue problem has been solved completely, the modal matrix Φn contains asmany eigenvectors as there are degrees of freedom in the physical dynamic model. The transient re-sponse of each mode can be calculated from closed-form analytical formulas. The classical normalmode analysis provides highly accurate results as a function of both frequency and spatial location, how-ever, this technique requires an accurate knowledge of the normal modes of the structure (i.e., both modeshapes and frequencies) to the highest frequency of interest in the vibration prediction. The determi-nation of normal modes for complex structures by either experimental or analytical (FEM) proceduresbecomes increasing inaccurate at frequencies above about the 50th normal mode. Thus the techniqueis generally limited to the lower end of the frequency range for high frequency vibration predictions inrelatively small assemblies.

1.2.4 Enhanced deterministic techniques

Since the conventional element-based methods is practically restricted to low frequency applications,they need to be improved to meet the challenge of high frequency problems. The subject on the enhanceddeterministic techniques has been well reviewed in reference [5], although this paper’s main goal is todiscuss the various methodologies of mid-frequency problems. Some attempts to look for enhanceddeterministic techniques involved in this paper are concisely cited here:

1.3. Statistical approaches: based on quadratic variables 7

• Seeking efficient numerical solvers to avoid prohibitively large computing time. one of importantwork is to realize the large scale Monte-Carlo simulations of very large finite element models;

• Using of adaptive strategies to optimize the mesh density in view of uniform error distribution;

• Decomposing the whole FE model into many smaller sub-systems, which is perfectly suited forimplementation in a parallel computational environment;

• Using meshless techniques to reduce the efforts involved with mesh generation;

• Developing Trefftz approach, in which the solution expansions exactly satisfy the governing dy-namic equations.

As described in [5], these techniques up to now are not capable of satisfying simultaneously some im-portant properties, e.g., an enhanced computational efficiency and a general applicability to geometricalcomplexity. On the other hand, there are other deterministic approaches, for example, asymptotic modalanalysis (AMA) and classic modal analysis (CMA). In the reference [6], the authors stressed that theresults of statistical energy analysis (SEA) may be obtained as an asymptotic limit of classic modal anal-ysis of a linear structural model under random or sinusoidal forces. Its main conclusion is that whenthe modes number (of plate) in the frequency bandwidth is sufficiently large, the AMA result is quiteaccurate. It agrees to SEA under the same assumption, that is, high modal density in the frequency rangeof interest. However, the successful applications of AMA to coupled structural systems have not beenreported.

Especially in mid-frequency domain, a hybrid approach, in which deterministic and probabilistictechniques had been proposed. At this frequency range, an FE analysis for complex structures is notpossible due to the computational burden. On the other hand, statistical energy analysis (SEA) doesnot apply for strongly coupled and stiff structures with few resonances. Consequently, the mid- andhigh-frequency prediction of noise and vibration in built-up structures such as vehicles requires a hybridapproach. One such hybrid approach is the “fuzzy structure theory” presented in Soize’s works [7, 8, 9].In this theory, the master structure is modeled deterministically. The influence of fuzzy attachments onthe master structure is, upon a stochastic assumption, modeled as an increase of inertia and damping.Langley and Bremner [10] developed a hybrid approach based on a wavenumber partitioning scheme.In an attempt to provide information on the spatial energy distribution within the various subsystems,research efforts are being devoted to the Energy Flow Analysis (EFA) and its numerical implementationthrough the Energy Finite Element Method (EFEM) [11, 12, 13]. The method is based on the extensionof the basic SEA equations for finite modal subsystems into differential terms and enables the descriptionof the dissipation and conduction of energy within each subsystem. Since this description is formallyequivalent to the steady-state equation of heat flow in solids, these methods are also called thermalmethods. A major advantage of this similarity with thermal problems is that the energy distribution andenergy flow within the basic components can be easily computed with readily available finite elementcodes for thermal computations.

1.3 Statistical approaches: based on quadratic variables

In order to cope with the limitations of deterministic techniques, major efforts have been devoted to thedevelopment of alternative prediction techniques in the statistical point of view. In this context, it islogical to apply quadratic variables such as energy or intensity (energy flow) under some asymptotic and


statistical assumptions [9]. According to the different demands and descriptions of structural dynamics,the first level of classification separates the energetic methods into two groups:

• The macroscopic viewpoint ⇒ global measures: Statistical Energy Analysis [14] (SEA) and WaveIntensity Analysis [15] (WIA).

• The microscopic viewpoint ⇒ local measures: Vibrational Conductivity Approach (VCA), En-ergy Flow Analysis (EFA); and Simplified Energy Method (MES) 1, Ray Tracing Method (RTM).

In what follows, these methods are concerned by the relevant review and remarks. A model of twocoupled plates are studied as an example to illustrate their applications.

1.3.1 Statistical Energy Analysis

Statistical Energy Analysis (SEA) [14] is a widely accepted, industrially applied high-frequency mod-eling method, which analyzes energy and power flow in built up systems such as those that occur instructural vibrations, acoustics, and coupled structural-acoustic problems. Originating around 1960,early SEA resulted from a collaboration of two independent efforts by Lyon and Smith. SEA draws onmany of the fundamental concepts from statistical mechanics, room acoustics, wave propagation, andmodal analysis. Significant theoretical refinements and the development of complementary methods hasoccurred. However, the basic SEA theory has changed little since its initial formulation. SEA models avibroacoustic system as an assembly of subsystems, where a subsystem is defined as a group of modeswith similar energetic properties. In contrast to classic deterministic methods, SEA applies some type ofaveraging:

• spatial average energy prediction

• time and frequency band average energyThe vibrational energy state of a subsystem, denoted by E, is defined in terms of the sum ofits time-averaged kinetic and potential energies, and the analysis is made in the average of finitefrequency band.

• modal average parametersfor example, the modal-average “coupling loss factor” (CLF) relates the time-average energystored in any one subsystem to the wave energy flux passing across an interface to another subsys-tem; the modal-average “dissipation loss factor” (DLF) is the arithmetic mean of the dissipationloss factors (twice the damping ratios) of subsystem modes having natural frequencies within aselected range of frequency

Nowadays, SEA is widely applied in high frequency vibration problems although it does not yet meet ageneral agreement for several reasons that are recognized [16, 17].

Overview of SEA

In SEA, the system being analyzed is divided into a set of coupled subsystems. Each subsystem repre-sents a group of modes with similar characteristics. SEA seeks to calculate the average response of each

1The acronym MES was got from its French expression, Methode Energetique Simplifiee


subsystem 4subsystem 3

subsystem 1 subsystem 2

E1 E2

E3 E4

Pin1Pin2

Pin3Pin4

Pdis1Pdis2

Pdis3Pdis4

P12

P21

P13

P14P23

P24P31

P32

P42

P41

P34

P43

Figure 1.2: An example of four coupled SEA subsystems

component of the structure by considering the equilibrium of power flows such as input power, dissipatepower and exchange power. The fundamental assumptions for SEA is the relationship between timeaveraged energy flow and the difference of the modal energy levels in the subsystems

Pij = gijω

(Ei

ni− Ej

nj

)(1.24)

= ω (ηijEi − ηjiEj) (1.25)

where Ni is the number of modes in subsystem i in the given frequency band and ηij is a proportionalityfactor, namely, coupling loss factor. For two dimensional line-coupled structures, the coupling lossfactors are given by [18]

ηij =cgiL

πωS

∫ π2

0τij(θ) cos(θ)dθ (1.26)

where cgi is the group velocity, L and S are the boundary length and area of substructure, and τij is thetransmission coefficient from substructure i to j. This hypothesis in (1.25) was originally derived forbroadband force excitation of two, conservatively coupled single oscillators.

It should be pointed out that the assumption (1.25) is not true in general, since the coupling powerbetween subsystems i and j is not necessarily proportional to the difference in the energy densities,and may also depend on the energy in other subsystems. It may be extended to coupled multi-modesubsystem under an assumption of weak coupling [19]. The coupling was defined to be weak if theGreen function for each uncoupled subsystem is not significantly affected by coupling the subsystemstogether. The implications of assuming (1.25) for a general system have been discussed in [20].

The basic SEA equation can be obtained by expressing the energy balance of each individual sub-system i:

P inputi = ηiωEi +

n∑i=j

ω(ηijEi − ηjiEj) (1.27)

where P inputi is the input power in subsystem i. A complex structure is usually represented by an network

or subsystems shown in Figure 1.2 For all the subsystems, the expression can be combined into a matrixequation:

LE = P (1.28)


where P is a vector with the input powers Pi to subsystem i, E is the vector with the lumped totalenergies Ei of subsystem i, and L is the coefficient matrix which is constituted by the frequency ω, theinternal loss factors ηi, and the coupling loss factors ηij .

For the simple example of four mutually connected subsystems shown in Figure 1.2, the correspond-ing SEA equation is

ω

η1 +4∑

i=1

η1i −η21 −η31 −η41

−η12 η2 +4∑

i=2

η2i −η32 −η42

−η13 −η23 η3 +4∑

i=3

η3i −η43

−η14 −η24 −η34 η4 +4∑

i=4

η4i

E1

E2

E3

E4

=

Pin1

Pin2

Pin3

Pin4

(1.29)

It should be clear that the coupling loss factors used in SEA are generally not reciprocal, that isηij = ηji. If it is assumed, however, that the energies of the modes in a given subsystem are equal,at least within the concept of an ensemble average, and that the responses of the different modes areuncorrelated, a reciprocity relationship can be developed. SEA reciprocity requires that the coupling lossfactors between two subsystems be related by the modal densities,

ni(ω)ηij = nj(ω)ηji (1.30)

This relation allows the energy balance expression (1.29) to be written in a symmetric form. Additional,successful applications of SEA usually require following conditions or hypotheses:

• Mutually uncorrelated input powers. The driving forces must be statistically independent to obtainthe linearity of the energy levels and energy flow in SEA equations.

• The dissipation of power. It is modeled by proportional relation with energy lever, namely, P dissi =

ηωEi.

• The conservative and weak coupling condition. It is supposed that there is no energy loss at thecoupling boundary. Moreover, the coupling should be “weak”, it roughly means each subsystemdissipates much more power than it transfer to other coupled subsystems, that is, the ratio of thecoupling loss factor to the internal loss factor of a subsystem is substantially smaller than unity.Some researchers have proposed or discussed the definitions of “weak coupling” in references[21, 22]

• Each subsystem is assumed to support a reverberant vibrational (or acoustic) wave field, in otherwords, vibrational/acoustic waves are rather strongly reflected at their mutual interfaces.

• the modes in a subsystem should be similar in terms of energetic properties.

Limitations of SEA

Although engineers have greatly benefited from the contribution of SEA, and a lot of successful applica-tions have been presented, research on a established theoretical basis of SEA has still required, because


researchers have found some limitations of SEA. As its global consideration, SEA should be viewed asa useful indicator of the influence of system parameters on vibrational/acoustic response rather than anaccurate predictor of the spatial distribution of vibrational energy. Additional, in SEA, the assumption ofthermal-like exchange of vibrational/acoustic energy among the subsystems is doubtful, so the definitionof subsystems and the corresponding coupling is uncertain. On the contrary of the necessary conditionsfor SEA, the limitations of a good performance of SEA may be summarize as [16, 23]:

• low modal overlap or not many resonances in lower frequency band;

• stronger coupling, for example, stiff and strongly connected elements with small internal dampingloss;

• lower modal density, subsystem has not sufficient number of resonant modes in the frequencyrange of interest.

• the build-in structures are not distinctly different to be modeled as suitable “elements”;

• the difficulty of determining the coupling loss factor is still open in practice, furthermore, sometheoretical estimates of coupling loss factor are not always feasible because of other additionalassumptions.

SEA has its particular merits and some limitations simultaneously. At first, the procedure appears to bea very simple method of analysis. However, because of the diversity of concepts used in formulatingthe basic SEA equations, the method quickly becomes very complex. In contrast with the element basedmethods, the size and subsequent computational effort of an SEA model are very small. It is quite fairto assume that, due to the inherent complexity in practical systems, the frequency- and space-averagedresults (for complex vibro-acoustic systems) give adequate information on the ensemble averaged results.however, the quality and reliability of an SEA model heavily depends on the experience and insight of theanalyst. SEA seems to be simple, but sometimes it is not very clear and straightforward. So in author’sopinion, SEA is worthy to be a semi-empirical method. A rather detailed review and instruction of SEAcan be found in reference [23]

1.3.2 Wave Intensity analysis: enhancements of SEA

Although the basic SEA theory is formulated in terms of modes, it is usually used in practice in an trav-eling wave formulation. The modal SEA model is more useful for the purpose of basic research thanfor practical applications. If many resonant modes contribute to the vibration field in two- or three-dimensional subsystems in a given frequency band, many traveling wave directions are present. SEAadopts essentially an energy diffusion model which admits all directions are present with equal probabil-ity, however, this diffusion model is not always suitable for multi-mode wave field. Langley proposed awave intensity technique [15] which leads to some significant improvements over conventional SEA. Inthis approach, the wave fields are not assumed to be diffuse, but rather the directional dependency of thewave intensity in each component is represented by a finite Fourier series. Quoted from his paper [15],Langley explained the advantages of WIA over SEA: “From a modal rather than a wave point of view,the present method relaxes the standard SEA assumption that there is equipartition of energy betweenthe modes of a particular component: with the present approach the energy of a mode depends upon thedirection of the wave components which constitute the associated mode shape”.


I(θ,ω)iI

BOU

ND

ARY

BOU

ND

ARY

θ θψ

i (θ,ω)

n

SUBSTRUCTURE jSUBSTRUCTURE j

SUBSTRUCTURE i

ψ

(θ,ω)i

COPφ

nCi (θ,ω)

iP

(φ,ω)jI

SUBSTRUCTURE i

Figure 1.3: Wave incidence and transmission at a boundary

In fact, wave intensity analysis (WIA) uses the spectral representation of an homogeneous randomwave field in the high frequency range:

u(x, t) =

∞∫∫−∞

a(k, ω) exp(ik · x − iωt) dk dω. (1.31)

where the random Fourier amplitude a(k, ω) satisfies the condition

E[a(k, ω)a∗T (k′, ω′)] = δ(k − k′)δ(ω − ω′)S(k, ω) (1.32)

whereE[ ] represents the expectation operator, and S(k, ω) is the wavenumber-frequency spectral matrixof the response u(x, t).

In WIA, the directional dependencies of the the wave intensity and energy density in each subsystemare represented by Fourier series. If a single term of the series is used, WIA reduces to conventionalSEA. Under the assumption that the dynamic response of each component is represented in terms of arandom wave field, the total energy density may be expressed as:

E = 2T =∑

j

∫Θ

∫ ∞

−∞ρω2Sj(θ, ω)dθ dω. (1.33)

For the energy flow The power flow vector associated with a particular wave type may be written in theform

Ij(θ, ω) = ej(θ, ω)cgj(θ, ω)n(θ) (1.34)

where n(θ) is the unit vector in the direction of θ. The energy flow equilibrium equation for the wavetype j in direction θ can be written in the form

P inj (θ, ω) = P diss

j (θ, ω) + P coj (θ, ω) − P ci

j (θ, ω) (1.35)

As shown in Figure 1.3, for a two-dimensional component, such as a plate, the power input P cij (θ, ω)


and power output P coj (θ, ω) take the following form

P coj (θ, ω) =

ωL

2πnjcjEj(θ, ω) cos(θ − ψ + π/2) (1.36)

P cij (θ, ω) =

ωL

2πnjcjEj(φ, ω) cos(θ − ψ + π/2)τij(φ− ψ + π/2) (1.37)

where cj is the phase velocity and nj is the modal density which may be expressed as

nj =ωAj

2πcjcgj(1.38)

If assuming that the directional dependence of each wave type may be expressed in terms of a set ofshape functions in the form

Ej(θ, ω) =∑P

EjP (ω)N jP (θ), (1.39)

It should be stated that the WIA approach can be reduced to conventional SEA if it is assumed that theenergy associated with a particular wave type is independent of θ. As shown in the reference [15], WIAis believed to give a better prediction of high frequency energy than SEA. The major difference betweenthem is that SEA cannot deal with the directional wave energy under the assumption of diffusive wavefield, however, WIA employs a non-uniform, directional wave energy with respect to the heading direc-tion. They both use the space-integrated technique so that they can only offer a global energy description.Although the WIA have some theoretical progresses, it need some additional effort to complete the cal-culations. On the other hand, for the very complex structures with many interconnecting substructures,or the energy transmission coefficient has a weak dependency on the wave heading, SEA can be expectedto play a better role because the diffuse wave field approximation is more reasonable in this case.

Some comparisons between WIA and SEA are given below. The energy results from vibrationalmotion equation of two coupled plates are taken as the reference value. These exact results are also usedto make comparisons with energy distribution obtained from some local energy methods which will bementioned in followed sections.

1.3.3 A case study: energy density of flexural vibration of thin plates

The flexural displacement of a thin, transversely vibrating plate (shown in Figure 1.3.3) excited by timeharmonic point force F ejωt, is governed by

D∇4u(x, y) − ρhω2u(x, y) = Fδ(x− xs, y − ys), (1.40)

where D = D0(1 + iη) is the the complex bending stiffness, h is the plate thickness, and (xs, ys) is thelocation of the point source. For stationary time harmonic response, the time-averaged energy density isthe sum of kinetic energy density T and potential energy density U ,

W = T + U, (1.41)

where

T =14ρhω2uu∗, (1.42)


F

y

x

Figure 1.4: The thin plate subjected to a harmonic point excitation

and

U =D0

4

[∂2u

∂x2

(∂2u

∂x2

)∗+∂2u

∂y2

(∂2u

∂y2

)∗+ 2ν

∂2u

∂x2

(∂2u

∂y2

)∗+ 2(1 − ν)

∂2u

∂x∂u

(∂2u

∂x∂y

)∗]. (1.43)

Here ∗ denotes the complex conjugate and ν is the Poisson ratio. The time-averaged flexural waveintensity in the x and y direction can be expressed as [18]

Ix =D0

2[(

∂3u

∂x3+

∂3u

∂x∂y2

)(iωu)∗ −

(∂2u

∂x2+ ν

∂2u

∂y2

)(iω∂u

∂x

)∗− (1 − ν)

(∂2u

∂x∂y

)(iω∂u

∂y

)∗].

(1.44)and

Iy =D0

2[(

∂3u

∂y3+

∂3u

∂x2∂y

)(iωu)∗ −

(∂2u

∂y2+ ν

∂2u

∂x2

)(iω∂u

∂y

)∗− (1 − ν)

(∂2u

∂x∂y

)(iω∂u

∂x

)∗].

(1.45)Because the near field effect is only limited in the neighborhood of the excitation position or discontinu-ities such as boundaries, the approximate expression of a wave field can be written in the form of planewaves [15]:

u(x, y;ω) =∫ 2π

0u(θ)e−ik(x cos θ+y sin θ)dθ (1.46)

The wavenumber k = k0 − i(

ηω2cg

)where k0 is undamping wavenumber, η is the damping coefficient,

and cg is the group velocity of the plate which is given by

cg = 2 4

√ω2D0

ρh. (1.47)

Substituting equation (1.46) into equation (1.42), we get the expression of kinetic energy

T (x, y;ω) =ρhω2

4

∫ 2π

0

∫ 2π

0u(θ1)u∗(θ2)e−ik(x cos θ1+y sin θ1)e−ik(x cos θ2+y sin θ2)dθ1dθ2 (1.48)

If we consider a random wave field, that is, if all wave components were regarded to be statisticallyindependent and the interference between the waves propagating in different direction is neglected, thenequation (1.48) can be approximately written as

T (x, y;ω) =ρhω2

4

∫ 2π

0|u(θ)|2 e−α(x cos θ+y sin θ)dθ. (1.49)

Averaging energy density in the period leads to T = U , thus the total energy density can be approximatedas

W (x, y;ω) =ρhω2

2

∫ 2π

0|u(θ)|2 e−α(x cos θ+y sin θ)dθ =

∫ 2π

0W (x, y;ω, θ)dθ. (1.50)


x

y

Plate 1 Plate 2

lx1 lx2

lyPin1 Pin2

Figure 1.5: The model of two coupled rectangle plates where energy level was examined by SEA andWIA

It is the same as the energy density expression derived in the reference [15].

In practice, the energy density is usually defined in a specific frequency band, that is

W (x, y) =∫ ω2

ω1

∫ 2π

0W (x, y;ω, θ)dθdω ≈ (ω2 − ω1)

∫ 2π

0W (x, y;ωc, θ)dθ (1.51)

where ωc is center frequency, ωc =√ω1ω2.

For the sake of simplicity, we choose a model of two coupled plates to justify the reliability of thesetwo global energy methods, SEA and WIA. This model has appeared in reference [15]. Shown in Figure1.5, two steel plates are simply supported along their boundary, the lengths are lx1 = 1.2m, lx2 = 0.7mwhile the width is ly = 1m, Young’s modulus E = 2 × 1011N/m2, density ρ = 7800kg/m3, Poissonratio ν = 0.3.

In principle, a flat plate will exhibit three type of waves: out-of-plane bending (flexural) wave; in-plane shear wave; and in-plane longitudinal wave. Here, only the pure flexural waves and their couplingbetween two plates are considered. The transverse displacement of each rectangle plate is expressed inthe form

ui(x, y, ω) =∞∑

n=1

uin(x, y, ω) sin(nπy/ly) (1.52)

The exact energy results expressed above are the reference values which will be compared with thesolution of the mentioned energy methods. Some numerical results have been simulated and shown inFigure 1.6–1.8

Figures 1.6-1.8 concern the ratio of the mean energy stored in the second panel to the mean energystored in the first (excited) panel, and the influences caused by different damping loss factor have beenexhibited. Because the exact analysis cannot give directly the space-averaged results, eight random loadpoints are chosen, like the simulation run in the reference [15], to average the responses which take overthe following locations (x, y):

(0.595, 0.540) (0.595, 0.300) (0.400, 0.650) (0.400, 0.500)(0.850, 0.750) (0.850, 0.700) (0.230, 0.450) (0.230, 0.900)


102

103

104

−12

−11

−10

−9

−8

−7

−6

−5

−4

−3

−2

Frequency (Hz)

10*l

og(E

2/E

1)

Figure 1.6: Ratio of the energy of second panel to the energy in first panel for damping loss factorη = 0.01. ——, WIA; – · – ·, exact reference result; – o – o, SEA

102

103

104

−20

−15

−10

−5

Frequency (Hz)

10*l

og(E

2/E

1)

Figure 1.7: Ratio of the energy of second panel tothe energy in first panel for damping loss factorη = 0.05. ——, WIA; – · – ·, exact referenceresult; – o – o, SEA

102

103

104

−22

−20

−18

−16

−14

−12

−10

−8

Frequency (Hz)

10*l

og(E

2/E

1)

Figure 1.8: Ratio of the energy of second panelto the energy in first panel. damping loss factorfor η = 0.1. ——, WIA; – · – ·, exact referenceresult; – o – o, SEA


The almost same result has been obtained shown in Figure 1.6 by comparing with the result in thereference [15] (see Figure 6). In this case, the loss factor taken as η = 0.01, is a slight damping case. Itis found that WIA result agrees with the one-third octave band exact result better than SEA, while SEAresult seem to overestimate the energy ratio level. In Figure 1.7 and 1.8 with respect to the loss factorsη = 0.05 and η = 0.1, the WIA’s and SEA’s results are closer to each other, especially in relative highfrequency band (≥ 1000Hz), however, both of WIA and SEA show the difference from the referencevalues at this frequency band. It illustrates that in the cases of very high frequency and heavy damping,the diffuse wave field approximation is practical, and the advantage of WIA over SEA is not muchobvious.

1.3.4 Vibrational conductivity approach

In recent years it has been founded that the high frequency vibrational/acoustics problems, may be mod-eled by using a partial differential equation to obtain the local spatial response while SEA is not com-petent for it. This partial differential equation is the basis of so-called vibrational conductivity approach(VCA). With this approach, the high frequency vibrational/acoustic energy flow or the structural intensityin a plane wave fields, is supposed to be proportional to the gradient of the energy density, e.g.

〈I〉 = − c2gηω

∇〈W 〉, (1.53)

where 〈. . .〉 denotes a time averaged quantity. It is actually an extension of the SEA law to a differentialform, although differential volume in the differential equation does not agree with the weak couplingcondition in SEA. The motivation of this approach is to overcome some limitations of SEA, especiallythe lack of spatial information of the responses.

In fact, this model was based on the original work [24] in which the authors derived the vibrationaltransport equation for energy flow in a plate. Lately this method had been further developed by other re-searchers like D. J. Nefske and S. H. Sung [12], Wohlever, Bouthier, and Bernhard [13, 25, 26]; Jezequeland his team [27, 28, 29, 30]. Although this method has many different names appeared in literature,such as Energy Flow Analysis (EFA); Simplified Energy Method (MES), etc., they can all be groupedunder the name vibrational conductivity approach, because the basic equations are formally equivalentto steady state heat conduction equation. Normally, the hypotheses for VCA are summarized as

• Linear, elastic, dissipative and isotropic system;

• Small hysteretic damping loss factors;

• Steady state conditions with harmonic excitations;

• Far from singularities, evanescent waves are neglected;

Power balance principle then leads to an equation for the energy density which is fully analogous to thesteady state heat conduction equation,

∇ · 〈I〉 + ηω〈W 〉 = 〈Pin〉 (1.54)

Substituting equation (1.53) into equation (1.54) leads to the basic VCA equation

− c2gηω

∇2〈W 〉 + ηω〈W 〉 = 〈Pin〉 (1.55)


Initially, equation (1.55) had been applied with considerable success to one-dimensional structures [12],lately it has been generalized to two-dimensional components such as plates by Bouthier and Bernhard[26], moreover, they developed the approach to energy finite element method (EFEM)[11, 13].

In the meantime, the research group Modeles vibroacoustiques pour les moyennes et hautes frequencesin our Laboratory (LTDS, UMR CNRS 5513), published successively their investigations on the highfrequency energy models [28, 29, 30, 31] and get the similar results in contrast with VCA, and the de-veloped equations are referred as General Energy Method (GEM) whose energy model accounts for bothnear field and far field effects, both the active and the reactive part of the intensity. This approach madean attempt to solve vibrational problems without approximations. However, due to the complexity andthe higher order equations, this method has not been fully applied in practice. The Simplified EnergyMethod (MES) can be derived from GEM by applying some further assumptions:

• Interferences between propagative waves are neglected.

With this assumption, the superposition principle for the energy variables holds. Moreover, MES doesnot need to perform appropriate space averages, thus it can be viewed as the “modified heat conductivitymethod” [32].

For one-dimensional structures, the results obtained from these energy methods (VCA, EFA, MES)are verified to be reliable, however, for two-dimensional or multi-dimensional problems, the justificationof the VCA became to be more difficult and the validity is no more satisfactory. Xing and Price [33, 34]verified that the solutions of the thermal-like VCA transport equation are inconsistent with the “exact”energy flow results in a semi-infinite case. Langley [35] has reported that the thermal analogy for two-dimensional systems is strictly not correct, that is, the VCA solution is not compatible with known exactresults. In particular for a point load, the vibrational conductivity equation predict a far field energydensity is proportional to 1/

√r, while the exact result is proportional to 1/r. These justifications show

that VCA cannot be verified in direct field arising from a point load while it may be applicable to a re-verberant system in which there are many wave reflections and the correlation between various waves isnegligible. Besides, the determination of boundary conditions for coupled structures is another question,because the boundary conditions in VCA involve the assumption that the wave field is diffuse in thevicinity of a boundary, so that directional wave effect cannot be captured. Although these shortcomingsseriously restrict the use of the method, yet VCA still may give a rough prediction of vibrational energylevel. As shown in reference [35], under some specific condition, the vibrational conductivity approachreduces to SEA by assuming a homogeneous field in each component, say, the energy equation of vibra-tional conductivity approach is the “strong form” of the problem while the energy formalism of SEA isthe “weak form”, in other words, SEA and vibrational conductivity approach express the same physicalnature of vibroacoustics energy motion.

Some numerical simulation are made to compare the predictions of the vibrational conductivity ap-proach with the exact results. The analysis model is a square plate, width (or length) is 1 m, the point loadis located at (0.532, 0.481), the excitation frequency is chosen as 3000 Hz. The solution of vibrationalconductivity equation (1.54) may be expressed by

W =∞∑

m=0

∞∑n=0

Amn cos(mπy/ly) cos(nπy/lx) (1.56)

where Am can be determined by using the orthogonality property of eigenfunctions [18]. The energyspatial distribution predicted by vibrational conductivity approach and exact results were compared in


0

0.5

1

0

0.2

0.4

0.6

0.8

10

2

4

6

x 10−5

Lx (m)Ly (m)

Ene

rgy

dens

ity (

J)

(a) The Surface plot of the exact energy

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Lx (m)

Ly (

m)

(b) Contour lines plot of the exact en-ergy

Figure 1.9: Exact energy distribution in a square thin plate by equation (1.56) for η = 0.05

Figures 1.9 and 1.10. The detailed comparisons will be made in the following section with other energymethods. The outcome of VCA differs from the exact result: the peak energy value at load point islower than that of the latter, while energy level at other region are overestimated in contrast with exactresult. Under some assumptions, VCA could give an approximate prediction, and it still has receivedconsiderable attention and has been regarded as the basis of so-called energy finite element method.

1.3.5 Others local energy methods: synthesis of fields

After the foregoing comments on the vibrational conductivity approach, an interesting question arises:why the vibrational conductivity approach can be successfully applied to one-dimensional structureswhile faces some difficulties when applied to two-dimensional structures? In author’s opinion, the reasonis manifold, but it can be reduced to one point: what is a suitable and reasonable mathematical physicalmodel for two dimensional structures with a point source? Many researchers have paid attention to thisproblem and found some way to treat it. These studies cover the most of important contributions, suchas the energy formulation in rooms [36, 37], the ray tracing method applied to high frequency vibration[18], and the vibroacoustic model for high frequency [31, 38]. The main idea in these studies is thesynthesis of fields, that is, differentiating the direct field from the reverberant field. The prerequisite isthat the different waves pattens exist in two fields, in other words, the direct field governed by a point loadshould be treated as a cylindrical (or spherical waves in three dimensional system) waves field whereasthe reverberant fields then is fraught with the plane waves coming from the reflection and scattering ofsymmetrical (cylindrical or spherical) waves by complex or irregular boundaries.


0

0.5

1

0

0.5

10

0.5

1

1.5

2

x 10−6

Lx (m)Ly (m)

Ene

rgy

dens

ity (

J)

(a) The Surface plot of energy prediction by VCA

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Contour lines plot of the energyprediction by VCA

Figure 1.10: Energy distribution in a square thin plate by VCA) for η = 0.05

Sound energy in rooms

The prediction of sound field at high frequency in an enclosure is usually a subject of geometrical acous-tics (room acoustics). It can be regarded as a high frequency problems in three dimensional systems. Sofar, there are two different algorithms for the calculation of transient and steady state sound propagationin rooms. The first one is image-source model by using second sound source, whereas the other one isthe ray tracing method by making use of sound particle assumption. Michael Vorlnder [39] proposeda new method which combines ray-tracing and image-source together. Universally the high frequencysound field is often simplified to be diffusive, e. g. the sound energy density is the same everywhere inthe concerned volume because the wavelength is much smaller than the enclosure’s dimension. However,this diffuse assumption has been questioned in many practical applications. Relaxing this assumption, R.N. Miles presented an improved (ray-tracing) method based on an integral equation to model the soundenergy in reverberant field [36]. In a sound enclosure shown in Figure 1.11, supposed that the surface isdiffusely reflective, then the reflection matrix follows Lamber’s law:

K(x, x′) =cos θ cos θ′

πr2. (1.57)

The sound intensity in reverberant field, which is incident on x due to the reflected radiation of all otherpoints x′ is

Ir(x, t) =∫

SK(x, x′)I(x′, t− r

c)(1 − α(x′))dS′, (1.58)

then the total intensity at x is the sum of the reverberant portion Ir(x, t) and the direct portion Id(x, t),e. g.

I(x, t) =∫

SK(x, x′)I(x′, t− r

c)(1 − α(x′))dS′ + Id(x, t). (1.59)


dSdS1

θ

θ

θ’θ

r

r r

n n1

x

x

x x′

Figure 1.11: The reverberant field forms in a sound enclosure

Once the surface intensity is known the sound energy at an interior point, (x), can be calculated by asolid angle integral over the surface,

W (x, t) =1πc

∫SK(x, x′)I(x′, t− r

c)(1 − α(x′))dΩ +Wd(x, t), (1.60)

where dΩ = cos θ dS′/r2

K.-S. Chae and J.-G. Ih [18] recently presented their studies on the applications of ray-tracing methodto a thin plate, where the response function was written as

v(x,xs;ω) = vdir(x,xs;ω) +∞∑

n=1

vref,n(x,xs;ω), (1.61)

where xs is the excitation point (source) and vref,n denotes the wave field formed by nth reflection.The energy response was also approximately calculated as the sum of two energy fields: W (x) =Wdir(x) + Wref (x), where Wdir(x) and Wref (x) represent direct energy field and reflected energyfield, respectively. The direct energy portion can be expressed as

Wdir(x) =ρh

2|iω · vdir(x)|2, (1.62)

As to the reverberant energy portion, it depends on the geometrical relation and the Snell’s law sinφi/ci =sinφj/cj , where ci, cj denote the phase velocity of two coupled plates. If the power transmission coef-ficient τij(φi) known, the vibration energy density of reflected and transmitted ray tubes at the couplingboundary can be written as [18]

W ri = (1 − τij(φi))W i

i , (1.63)

W tj =

τij(φi)cg,i cosφi

cg,j cosφjW i

i . (1.64)

In the reference [18], the authors obtained some relatively accurate results compared with SEA, VCA,and WIA techniques, but the major idea in ray tracing method is similar to that in the hybrid energymodel which will be involved in next paragraph.


Hybrid energy model

Although some techniques, for instance, ray-tracing method, are found to be effective to solve problemsin room acoustics, the equivalent approaches for structural problems are not yet sufficient. The ap-proaches mentioned previously such as VCA or energy flow analysis, have been reported to successfullyapply to the one dimensional structures such as rods and beams, but for multi-dimensional systems, theproblem is not completely accomplished, at least, it has been proved to be deficient to generalize simplythe thermal analogy approach from one dimensional structures to multi dimensional systems. In viewof the fact, Lase, et al [27, 31, 40] developed the General Energy Method (MEG) 2. By applying the farfield hypothesis and spatially averaging over a wavelength, this method can be modified to be so-calledSimplified Energy Method (MES) [41, 42]. In addition, M. J. Smith [38] proposed his schemes to dealwith this problem. Although these methods were proposed respectively by several researcher, the mainideas and techniques are the same — they both handle the direct and reverberant fields separately, andthe total energy density is the sum of the two portions. Hereafter, the methods of synthesis of two distin-guished fields is labeled by hybrid energy methods at the author’s disposal. We state that “hybrid” refersspecifically to the combination of two different fields: direct field and reverberant field. As a contrast,in reference [10], “hybrid” means the the combination of FEM and SEA as a technique to deal withmid-frequency problems.

In the references [38, 41, 42], the direct field component is approximated by the far-field response ofan infinite structure, the energy density is governed by

1rn−1

ddr

(rn−1W ) +ηω

cgW = 0, (1.65)

where n = 1, 2, 3 is the dimension of the studied structure. By solving this equation, energy density Wand energy flow I are found to be proportional to the Green functions

W (S,M) ∝ G(S,M); G(S,M) =e− ηωr

cg

rn−1(1.66)

I(S,M) ∝ H(S,M); H(S,M) =cge

− ηωrcg

rn−1nSM (1.67)

where S, M are the source and observation points. nSM =−−→SM/|−−→SM | is unit directional vector, r =

|−−→SM |.The reverberant field results from boundary reflection of the direct field. The boundary condition for

the reverberant field in the reference [38] was expressed as

c2

ηω∇〈Wr〉 · n = ρ〈qd〉 · n (1.68)

where qd is energy flow of direct field incident to boundaries, and ρ represents the reflectivity of theboundary. However, this expression neglects the nth reflection in reverberant field. The suitable boundarycondition which considers the reflections of reverberant field can be written in the form

c2

ηω∇〈Wr〉 · n = βW + λId · nM . (1.69)

2The acronym MEG was from the French expression: Methode Energetique Generale

1.4. Transient vibroacoustics in high frequency 23

where β and α are the generalized reflectivity coefficients. This expression will be discussed and ex-plained in Chapter 4.

As to the coupling loss factors (CLF) of coupled structures, it should be consider the reflection fromand transmission to each coupled sub-structure in the common boundary. A general coupling relationhas been proposes by Ichchou [42] and Cho [43]. For example, in the case of one dimensional coupledsystem, Ichchou [42], Cho [44], and Djimadoum [45] proposed a similar relation,

− c21ηω

∇〈W1〉 =τ12c1

ς12 + ς21〈W1〉 − τ21c2

ς12 + ς21〈W2〉. (1.70)

It is not the same coupling condition as that appeared in the reference [12] which is presented in the formof

− c21ηω

∇〈W1〉 = τ12c1〈W1〉 − τ21c2〈W2〉. (1.71)

This difference of CLF expression has been referred in the reference [46].

1.3.6 Comparative studies on some local energy methods

Based on the introductions about the local energy methods in the last section, a numerical model is usedto simulate and compare their results. The results from VCA and MES, are selected to compare with theexact (reference) and SEA results.

Consider a square plate shown in Figure 1.12, its width is 1m, thickness t = 0.002m, the geometryand material parameters L,E, ρ and ν are the same as the model shown in Figure 1.5. The point source islocalized at x = 0.532, y = 0.482 with center frequency 3000 Hz. The loss factor is taken as η = 0.01.The input power can be calculated by

Pin =F 2

16√Dρh

(1.72)

We choose three parallel lines in direction of X axis in the plate (see Figure 1.12) to examine the energydistributions along them. Figure 1.13 shows the energy results along Ly = 0.5m, which is very closeto point-loaded position. The peak energy values are clearly observed at this position. The input energygot by hybrid method agrees well to the exact value, while VCA’s result is much lower than the former.The outcome of SEA is constant in the plate. Energy behavior in other two lines (Ly = 0.75, 0.1 m)were displayed in Figure 1.14 and 1.15, It was found that energy density predicted by VCA was moreeven-distributed than hybrid method, while at boundaries, the former produces a higher energy valuethan the latter. It is because VCA doesn’t distinguish direct field and reverberant field, thus a total diffusefield occurs. In reference [35], the flaw of vibrational conductivity approach was investigated in detail.The hybrid methods shows a better prediction, however, a successful application of this method need theaccurate description of boundaries, namely, transmission and reflection coefficients.

1.4 Transient vibroacoustics in high frequency

All the works and references mentioned previously only concerne the steady state conditions. In fact, thetransient dynamics study in an energetic point of view at high frequency range is the main goal in thisthesis, and the motivation of this subject will be stressed in the following chapters. Transient phenomena


Y

X

F

S

Ly = 0.1m

Ly = 0.5m

Ly = 0.75m

Figure 1.12: Single plate model with excitation at (0.532, 0.482)

0 0.2 0.4 0.6 0.8 13

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

Lx (m)

Ene

rgy

(dB

)

ExactHybridVCASEA

Figure 1.13: Comparison of energy results: along the line Ly = 0.5m. . . . . . . , exact energy result; ——,MES; – · – ·, SEA; –––, VCA.

0 0.2 0.4 0.6 0.8 14

4.5

5

5.5

6

6.5

7

Lx (m)

Ene

rgy

(dB

)

ExactHybridVCASEA

Figure 1.14: Comparison of energy results alongthe line Ly = 0.75m. . . . . . . , exact energy result;——, MES; – · – ·, SEA; –––, VCA.

0 0.2 0.4 0.6 0.8 14

4.5

5

5.5

6

6.5

7

Lx (m)

Ene

rgy

(dB

)

ExactHybridVCASEA

Figure 1.15: Comparison of energy results alongthe line Ly = 0.1m. . . . . . . , exact energy result;——, MES; – · – ·, SEA; –––, VCA.


are often encountered in practical application. Many engineering structures are subjected to the transientimpact and shock. These transient cases may cause local damage or failure in structural framework. Inthe field of aeronautics, the high frequency transient response of space vehicle structures is due primarilyto the activation of pyrotechnic (explosive) devices that produce structural responses at frequencies up to1 MHz. Pyroshock was defined by NASA TECHNICAL STANDARD 3.

Pyrotechnic shock or pyroshock is the transient response of structural elements, com-ponents, assemblies, subsystems and/or systems to loading induced by the activation ofpyrotechnic (explosive- or propellant-activated) devices incorporated into or attached to thestructure. In certain cases, the pyrotechnic loading may be accompanied by the release ofstored energy due to structural preload, or by impact between structural elements as a resultof the explosive or propellant activation.

Because of the very high frequency character of pyrotechnic loads, classical normal mode analysis andfinite element method (FEM) models are not effective for predicting structural responses to pyrotechnicloads. For example, taking into account only a few modes of a structure, one can use the shock spectra toexpress the transient response in the frequency domain and get the time history by Fourier transform pair.For the ideal mechanical shock and simple structures, it is a straightforward approach. However, for thecomplex structures have several important modes around the frequency of interest, there arise a questionof how to express and how to combine the transient response of these modes of oscillation into a singlemaximum response. Besides, the mechanical shocks (excitation in a very short time) usually cause highfrequency responses, often hundreds of modes in structures are excited. Difficulties in analyzing high-frequency dynamics are well known. In spite of not being a fashionable subject nowadays, the researcheson the transient dynamics can be traced back to the very beginning of the study field of vibration andacoustics, and the classical methods usually were used in the earlier stage. Except empirical models,the prediction of structural responses produced by pyroshocks (or other kinds of transient excitations) isusually accomplished by using two methods:

1. Transient statistical energy analysis (SEA) procedures,

2. Virtual mode synthesis and simulation.

1.4.1 Transient statistical energy analysis

Although statistical energy analysis (SEA) is usually applied to the prediction of steady-state vibrationresponses, similar methods can also be used to predict transient responses. An earlier investigation onthe energy-flow method of transient analysis was explored by Manning and Lee [47] and Lyon [48, 14].For each subsystem, a simple power balance results in the following equations,

dEi

dt= P in

i − P transij − P diss

i . (1.73)

where Ei is the global energy in the subsystem i. Under steady-state conditions, dE(t)/dt = 0. Thesteady-state SEA matrix equation, as given in equation (1.29) is now replaced by the corresponding

3PYROSHOCK TEST CRITERIA: NASA TECHNICAL STANDARD, NASA-STD-7003


transient SEA matrix equation

η1 +n1∑i=1

η1i −η21 . . . −ηn1

−η12 η2 +n2∑i=2

η2i . . . −η42

......

. . ....

−η1n . . . −ηn−1n ηn +nn∑i=n

ηni

E1

E2

...

En

+

dE1

dtdE2

dt...

dEn

dt

=

P in1

P in2

...

P inn

(1.74)

In [47], the authors took the same analytical procedure as SEA by dividing a complex structure into a setof coupled subsystems. They thought that the use of steady-state parameters to study shock transmissionseems valid for quasi-steady vibrations. Thus the damping and coupling loss factors retain their steadystate values. For the steady-state, the power flow is assumed to be proportional to the difference inmodal energy of the two subsystems (see equation 1.25), for transient cases, an assumption that transientresponse is a result of modal resonance was made in the reference [47], which implies that the modalenergy in one band does not contribute to the response of modes with resonances in other bands, so thatthe power flow between subsystems for transient cases was expressed

P transij (ω, t) = ωηijni

[Ei(ω, t)ni

− Ej(ω, t)nj

], (1.75)

where ω is the center frequency of the band, ni,j is the modal density of the subsystem, and ηij is thesteady state coupling loss factor.

Some successful applications of these formulations have been reported (for example, see [49]). Re-cently Pinnington and Lednik [50, 51] published their studies results of TSEA. In their articles [50],they studied the transient energies of a two-oscillator system by TSEA in detail and compare them withthe exact solutions. The numerical tests show that the TSEA results are rather different from the exactsolutions on either rise time or peak energy transferred. An example is referred here with the definitionsof rise time and peak energy transferred. The numerical model is similar to that shown in Figure 1.17,M1 = M2 = 2 kg, K1 = K2 = 16 × 105 N/m, Kc = 4 × 105 N/m, η1 = η2 = 0.1. In Figure 1.16, Ep1

and Ep2 are peak energies of the second oscillator for TSEA and exact solution, respectively, and T1 andT2 are their rise time.

The advantage of the transient SEA to the prediction of high frequency transient responses is be-lieved, if applied with skill, to be able to provide reasonably accurate results without the need for adetailed description of the structural elements involved. The main disadvantages of the technique are asfollows:

• The procedure applies only to the prediction of far-field transient responses, because SEA assumesthat the response is at least quasi-stationary. It means, for example, that the decay time of theresponse should be longer than the period of the oscillations at the frequency of interest.

• The procedure requires estimates for various structural and coupling loss factors, which often canbe only crudely approximated.

• The procedure is not applicable at the lower frequencies where there are less than two or threeresonant modes in the specified bandwidths, usually 1/3 octave bands.


0 0.005 0.01 0.015 0.02 0.025 0.030

0.02

0.04

0.06

0.08

0.1

0.12

Time (s)

En

erg

y (J

)

T2

Ep2

T1

Ep1

Figure 1.16: Difference between TSEA result and exact solution. ——, TSEA solution; – – –, exactprediction

M1

M2

F2

Kc

G

Mc

K1 C1 K2 C2

x1 x2

F1

Figure 1.17: Two coupled oscillators used to re-examine TSEA

In view of the uncertainty of using steady-state parameters in TSEA, Lai and Soom [52, 53] re-examinedthe Manning and Lee’s approach. They found that the use of constant steady-state coupling loss factorto predict time-dependent vibration is not always appropriate. On the contrary, they believed that thecoupling loss factors does not remain constant during transient oscillations, further, they proposed anapparent time-varying coupling loss factor to describe transient power flow, which is simply explainedas followed. Consider two-coupled oscillators shown in Figure 1.17. The equations of motion can bewritten as:

(M1 +

Mc

4

)x1 + C1x1 + (K1 +Kc)x1 = f1 +Kcx2 +Gx2 − Mc

4x2, (1.76)(

M2 +Mc

4

)x2 + C2x2 + (K2 +Kc)x2 = f2 +Kcx1 −Gx1 − Mc

4x1. (1.77)


Power flow relations are got by multiplying equations (1.76) and (1.77) by x1 and x2, respectively

ddte1 = πin

1 − πdiss1 + πtr

21, (1.78)

ddte2 = πin

2 − πdiss2 + πtr

12. (1.79)

where

ei =12(Mi +

Mc

4)(xi)2 +

12(Ki +Kc)(xi)2, i = 1, 2. (1.80)

The energy balance between the two oscillators up to time t can be obtained by integrating both sides ofequations (1.78) and (1.79)

ddtε1(t) = ein1 (t) − ediss

1 (t) + etr21(t), (1.81)

ddtε2(t) = ein2 (t) − ediss

2 (t) + etr12(t). (1.82)

where

εi(t) =∫ t

−∞ei(τ)dτ, eini (t) =

∫ t

−∞πin

i (τ)dτ, (1.83)

edissi (t) =

∫ t

−∞πdiss

i (τ)dτ, etrij (t) =∫ t

−∞πtr

ij (τ)dτ. (1.84)

then the apparent time-varying coupling coefficient Cij(t) is defined by

Cij(t) =etrij (t)

εi(t) − εj(t)(1.85)

It is a creative effort to make amends for TSEA. The concept of apparent time-varying coupling lossfactor is easily to understand, but it is not a practical way to improve the applicability of TSEA. As wewill point out in this thesis, the perplexity of TSEA is not only caused by coupling loss factor.

1.4.2 Virtual mode synthesis and simulation

Another method for high frequency transient response analysis is Virtual Mode Synthesis and Simulation(VMSS). The VMSS process begins with an estimate of the steady-state frequency response magnitudeenvelope, i.e., the peak sinusoidal response at a selected location due to a sinusoidal force at anotherselected location, mapped over the range of selected frequencies. At high frequencies, it is assumed thatthe frequency response envelope can be represented as the peak response from a collection of localizedvibration modes with frequencies spaced according to the estimated modal density of the local structure.

The governing equations of this virtual mode dynamical system come from the classical normal modeanalysis (see subsection 1.2.3). The number of modes nm can be very large at high frequencies, thenapproximations of the actual physical modes is taken. The virtual modes are not pure physical entities,bur rather mathematical coefficients of the same form that produce a frequency response function (FRF)mapping to the measured or predicted FRF. To simulate the time response, the dynamical system is


numerically convolved with a measurement or simulated transient excitation force. The normal modeequations used by VMSS are of the form

Iξ + 2ζωξ + ω2ξ = [φ]T F (1.86)

where

ω = diag(ω1, ω2, · · · , ωm, · · · , ωN )

I the identity matrix

ξ the modal coordinates that represent the virtual modal response

[φ] the virtual mode shape matrix in which each column represents the mode shape for each frequency

F the applied loads

The FRF for the ith response DOF and the kth loaded DOF may be approximated by

|Hik(Ω)| =N∑

m=1

φim · φkm√(ω2

m − Ω2)2 + (2ζmωmΩ)2(1.87)

whereN is the number of vibration modes in the substructure. For convenience, this FRF magnitude canbe rewritten as the product of two vectors, namely,

|Hik(Ω)| = ΦT Λ (1.88)

where

Φ =

φi1 · φk1

φi2 · φk2

...

φiN · φkN

; Λ =

[(ω2

1 − Ω2)2 + (2ζ1ω1Ω)2

]− 12[(

ω22 − Ω2

)2 + (2ζ2ω2Ω)2]− 1

2

...[(ω2

N − Ω2)2 + (2ζNωNΩ)2

]− 12

(1.89)

By packing the FRF magnitudes available from steady-state methods into a column vector H, whereeach element of the vector represents the FRF magnitude at the assumed virtual mode frequencies, , onecan obtain the synthesis operation for the virtual modes. It results in

|Hik| = [Λ]T Φik (1.90)

where

[Λ] =[Λ(Ω1)

... Λ(Ω2)... · · · ... Λ(ΩN )

](1.91)

By selecting the evaluation frequencies to be the same as the virtual mode frequencies considered overthe frequency range, one can obtain the virtual mode coefficients as

Φik =([Λ]T

)−1 |H|ik (1.92)

It implies that the input known FRF is averaged to be synthesized FRF. With these coefficients synthe-sized, the governing equations given by equation (1.86) can be solved for the time response.


tt1

Response

Vm

ax

Figure 1.18: Time response with respect to input frequency Ω

In addition, the modal frequency response can be written in time domain to a input frequency Ω

i(Ω, t) =nm∑

m=1

Rm(Ω, t) (1.93)

whereRm(Ω, t) is the analytical time response ofmth mode with respect to a input frequency Ω. Further,one can find the peak value of time response with respect to a input frequency Ω, as shown in Figure 1.18Evaluating all the peak values of time response corresponding to each input frequency roughly producesa new function relation which is called Shock Response Spectra (SRS). Note that SRS is a plot of themaximum response (velocity, acceleration, etc.) of a hypothetical single-degree-freedom oscillator notincluded in the model, as a function of the oscillator’s natural frequency. The detailed description andapplication of SRS is seen in chapter 5.

This method (VMSS) has several advantages, for example, it can estimate near-field response andlower frequency band since the governing equations are in the normal mode form. However at high fre-quency range, it is difficult to identify modes, and it is not realistic to suppose that modes are distributedover frequency according to the modal density estimated. As a consequence, it cannot ensure to recoverof the related modes at high frequency. The concept of phase reconstruction maybe a helpful amend toimprove this method [54, 55].

1.4.3 Time-varying energy conductivity approach

As to the local energy methods in the time domain, it is necessary to refer the energy equation proposedby Nefske and Sung. In their study [12], the authors introduced a term of ∂W

∂t to derive an energy equationfor one-dimensional structure

∂W (x, t)∂t

− c2

ηω

∂2W (x, t)∂x2

+ ηωW (x, t) = 0 (1.94)

where W is the total energy density, c is the group velocity, η is the loss coefficient, and ω is the cir-cular frequency. Equation (1.94) is clearly a parabolic differential equation which is similar to the heatconduction equation. It can be simply obtained by combining followed equation pair:

∂W (x, t)∂t

+ ∇ · I = 0

I = − c2

ηω∇W (x, t)

(1.95)

1.5. Concluding remarks and the aim of the thesis 31

In contrast to equation (1.53), one finds that the same expression of energy flow I is employed in bothsteady-state and time-varying condition. Since VCA can be generalized from one-dimensional struc-ture to multi-dimensional system, it is believed that equation (1.94) is not just limited itself in one-dimensional structures. One can use variational principle to get the weak form of equation (1.94). For anexample of one-dimensional structure, denote Ei = LiWi represents global energy variable, where Li isthe length of one-dimensional element. Then the variational function for equation (1.94) is given by

Π = −∫

Li

1Li

dEi

dtEi +

(c2i

2Liηiω

)∇Ei · ∇Ei +

(ηiω

2Li

)E2

i

dLi − ηijω(Ei − Ej)2 (1.96)

Suppose that energy in each component is constant, thus ∇Ei = 0. Thus the variational principle maybe written as

∂Π/∂Ei = 0 ⇒ dEi

dt+ ηiωEi + ηijω(Ei − Ej) = 0 (1.97)

This result shows that Nefske’s equation (1.94) is equivalent to the standard Transient SEA. From thispoint of view, the time-varying VCA can be considered to be a generalization of the SEA formula. Thedetailed examination on this method will be discussed in following chapter with some examples.

1.5 Concluding remarks and the aim of the thesis

On the basis of above reviews on the methodologies for high frequency dynamics, we found that thestatistical methods in term of energy/intensity variables are privileged techniques. For steady state con-ditions, in spite of some problems being open yet, these energy approaches can claim many notableand well-documented triumphs when being applied to simple or complex structures. For unsteady statecondition however, at least up to now, the energy approaches such as TSEA (for discrete subsystems)and time-varying VCA (for continuous structures, see equation (1.94) ) have not offered satisfactorysolutions.

In fact, all steady state processes must always be started up and occasionally shut down, and some-times the unsteady state cases may represent critical condition, therefore researchers and engineers wouldbe vitally concerned with the transient dynamics. In contrast to so many studies results on energy meth-ods for steady state condition, the researches on unsteady state condition are not thoroughly explored.Motivated by the task to predict transient (unsteady state condition) dynamics in terms of energy vari-ables, we are going to carry out an analytical study on this subject. Moreover, in the physical pointof view, a kind of mature approach for dynamics should be complete. It means that the expected ap-proach may be expressed by a time-space-varying continuous partial differential equation. It should be,of course, capable of describing precisely the space state evolution in time and spatial domain. It isnaturally suited to solve transient dynamic problems, and in the meanwhile, the steady state conditioncan just be regarded as a special application case by averaging the time-varying terms in the equation.For this reason, the focus of this thesis is aimed at developing a time-varying energy description in highfrequency.

This thesis is organized into three major parts: the first part includes a fairly detailed reviews on themethodologies for high frequency dynamics; the second part includes chapter 2 and chapter 3, which aredevoted to proposing a new time-varying energy equation. Chapter 2 generalizes the recent mathematicalresults on the high frequency waves in random media and serves as preparatory stage. Chapter 3 focuson the derivation of the time-varying energy equation by two different techniques, and the properties ofpropose equation will be discussed at full length. The validation and applicability of proposed method are


investigated in last part, where chapters 4 and 5 involve the discrete subsystems and distributed structure,respectively. Chapter 6 makes an attempt to generalize the proposed energy equation to multi-dimensiondomain. The conclusion and perspectives are given at last chapter 7. We hope that our work in the thesiscan provide not only an approach to deal with transient dynamic problems but also a solid foundation forthe further studies on energy methods in high frequency domain.

Chapter 2

High frequency wave energy: transporttheory and wavefront solutions

2.1 Introduction

The high frequency waves in structural vibration and acoustics, which is concerned in this thesis, havesome common features that are shared by other classic waves. Here, “classic waves” are defined by thewave equation or more general symmetric hyperbolic system (see equation (1.5, 1.8) in chapter 1, andflexural waves in beams or plates are dispersive.), although the definition like “a wave is a solution ofwave equation” seems to be an argue in a circle. Electronic wave is thus not included in this scope becauseit obeys the Schrodinger equation. In addition, classic waves are often classified as scalar wave (acousticwave), vector wave (electromagnetic wave) and tensor wave (elastic wave). Although electromagneticwave is also governed by wave equation (Maxwell’s equation), it is not taken into account either inthe thesis, because propagation of electromagnetic wave is not dependent on media. For example, theimportant characteristic parameter such as frequency ω and velocity c of light are not eventually decidedby the media; they just can be influenced by the media where it go through, light can even transportin vacuo. So, the wave of light is said to be media-independent. However, the elastic (or mechanical)waves are completely media-dependent, that is, the elastic waves propagate definitely in media, all theparameters such as frequencies ω, propagation velocity v, etc, are decided by the properties of media.Acoustic wave and structural (elastic) waves are the main subject to be discussed in what follows.

In fact, waves propagating in media is a wide-ranging and difficult problem. The whole subject canbe categorized into diverse fields in accordance with different waves types and different media properties.Although the substantial content of this subject is of course beyond the coverage of this thesis, we stilltry to begin with a more general situation, then draw out our conclusions after some reasonable simplifi-cation. To some extent, this chapter is devoted to making preparations for the further derivation of energyequation in following chapters. The behaviors of waves propagating, for example, velocity, frequency,decay, linear or nonlinear, etc., are definitely influenced or decided by the characteristic of media. Me-dia where waves propagate are generally classified by the description of uniformity from point to point.Uniform media are called homogeneous; nonuniform inhomogeneous. A medium can be homogeneouswith respect to one aspect of behavior (say, density) and inhomogeneous for another (say, modulus ofelasticity, E). The inhomogeneities can be strong, weak, or random. The study of waves propagatingin inhomogeneous media has been for many years (and continues to be) the subject of intensive investi-

33

34 High frequency wave energy: transport theory and wavefront solutions

z

x

y

θ

φr

n

X

Y

Z

Figure 2.1: Sketch of phase space

gation and it has produced many important achievement. An important application is geometrical opticapproximation [56]. On the condition that the wavelength is small compared to the scale, the wave func-tion behaves in the limit regime can be approximated as a physical point.Another mathematical subjectinvolved is the homogenization, which characterized by a small parameter, for example, density ρ(x)oscillates in small scale ε. Thus the approximations for inhomogeneous media are fundamental. Thus,we can take advantage of the methods and results of study on waves in inhomogeneous media althoughthe subject of our study is aimed at waves in homogeneous media or structures. The former can be ofcourse regarded as the special case of the latter.

In fact, the study of waves propagating in inhomogeneous media is a subject with long and richhistory. At the first stage, the interest of the physics community was focused on quantum 1. The motionof a collection of particles such as electrons and photons is described by the Boltzmann equation (or,transport equation) in phase space. One can specify both the position r and velocity v of a particle atgiven time t in phase space, or in other words, a point in phase space corresponds to both a position and avelocity. The concept of phase space is illustrated in a sketchy Figure 2.1. Define X as a vector of phasespace coordinates, i.e.,

X ≡ (r,v) = (x, y, z, x, y, z), or, X ≡ (r,n, E), (2.1)

where n is a unit three-dimensional vector representing the direction of particle flow, andE is the particleenergy.

As our main interest isn’t the derivation of Boltzmann equation, a rather simply introduction is pre-sented here. A statistical description of a system can be made in terms of the distribution function,for example, particles density, f(r,v, t) defined in phase space. Then f(r,v, t)drdv is the number ofparticles at time t positioned between r and r + dr which have velocities in the range v → dv.

Consider an external force F acts on these particles and assume initially that no collisions take placebetween these particles. In time t the velocity v of any particle will change to v +Fdt and its position r

1For example, electronic wave described by the Schrodinger equation as Planck constant goes to zero. see [57] for a moredetailed introduction.

2.2. Waves transport in random media 35

will change to r + vdt. Thus the number of particles f(r,v, t)drdv is equal to the number of particlesf(r + vdt,v + Fdt, t+ dt)drdv, that is

f(r + vdt,v + Fdt, t+ dt)drdv − f(r,v, t)drdv = 0 (2.2)

Consider the collisions occurred between the particles, there will be a net difference between the numberof particles f(r+vdt,v+Fdt,t+dt)drdv and the number particles f(r,v,t)drdv. Denote collision operator asΞ(f), then the above equation is rewritten as

f(r + vdt,v + Fdt, t+ dt)drdv − f(r,v, t)drdv = Ξ(f)drdvdt (2.3)

Dividing equation (2.3) by drdvdt and letting dt→ 0 gives the Boltzmann equation

∂f(t, P )∂t

+drdt

· ∇rf(t, P ) +dvdt

· ∇vf(t, P ) = Ξ(f) (2.4)

While we are not concerned here with the explicit form of the collision term, note that it should satisfyconservation laws. If there is no scattering (collision), equation (2.4) is reduced to Liouville equation inclassical mechanics.

Lately it was found that the analogy between quantum and classical waves sometimes works well,this theory thus was transplanted to study the classical waves scattering phenomenon in random media,and it has been the subject of intense research either in the applied mathematics or the engineeringmechanics literature. At the first of 20th century, the radiative transport theory was originally developedto interpret some observed phenomena like light propagation through a turbulent atmosphere [58]. Fromthe seventies of last century, considerable attention has been focused on this theory and it has been muchdeveloped and applied in various physical fields from radar, optics to underwater sound propagation,geophysics, and seismology. Now it has became mainstream theory in the field of wave propagation in acomplex medium.

2.2 Waves transport in random media

The subject of waves propagating in random media is actually a mathematical-physical problem with faircomplexity. Of course, anyone familiar with modern physics will realize that understanding the behaviorof an apparently physical problem, if rigorously pursued, also encounters boundless complexity. Thus theassumptions and restricted conditions may be necessary to reduce the complexity and make the problemsbe modeled in a relatively practical and simple fashion. For the wave propagation in the random media,it is necessary to introduce three basic length scales

• The typical wavelength λ.

• The typical correlation length of the inhomogeneities l.

• The typical propagation distance L

Transport equation is a good way to describe the propagation of wave energy if the following assumptionshold [58]

• high frequency approximation: typical wavelengths are much short compared to the typical prop-agation distance, i.e., λ L;


k’

k

k’

Figure 2.2: Wave transports in a random medium

• wavelengths are comparable to the correlation lengths of the inhomogeneities, λ ∼ l;

• weak localization: the fluctuation of the inhomogeneities are weak, that means the heterogeneitiesare not too strong to induce interference.

The general transport equation was proposed to deal with the high frequency wave propagating withscattering in a randomly inhomogeneous medium. The conception was shown schematically in Figure2.2 as a means of introducing basic viewpoints of scattering. For this moment we just introduce thetransport equation in a simple way. The detailed investigation will be explored in later. When a wave withwave vector k′ propagates through the inhomogeneous medium, it can be scattered into any direction kwith wave vector k, where k = k/|k|. A scalar function a(t,x,k) denotes the angularly resolved energydensity defined for all wave vector k at each point x and time t. Similar to Boltzmann equation (2.4),Energy balance leads to the transport equation

∂a(t,x,k)∂t

+ ∇kω(x,k) · ∇xa(t,x,k) −∇xω(x,k) · ∇ka(t,x,k)

=∫

Rn

σ(x,k,k′)a(t,x,k′)dk′ −∑

(x,k)a(t,x,k) (2.5)

Here n is the dimension of space, ω(x,k) is the local frequency at x of the wave with wave vectork, σ(x,k,k′) is the differential scattering cross-section, expressing the rate at which energy with wavevector k′ is converted to wave energy with wave vector k at position x, and∑

(x,k) =∫σ(x,k,k′)dk′ (2.6)

is the total scattering cross-section. The function σ(x,k,k′) is non-negative and usually symmetric in kand k′.

The left hand side of equation (2.5) is the total time derivative of a(t,x,k) at a point moving alonga trajectory in phase space (x,k) and may be written as a Liouville equation

∂a(t,x,k)∂t

= ω, a(t,x,k), (2.7)

where

f, g =n∑

i=1

(∂f

∂xi

∂g

∂ki− ∂f

∂ki

∂g

∂xi

)(2.8)

2.3. High frequency limit 37

is the Poisson bracket. The right side of equation (2.5) represents the effects of scattering. Clearly, if theenergy dissipation is not considered, the total energy is independent of time∫ ∫

a(t,x,k,k′)dxdk = constant (2.9)

In fact, radiative transport equation (2.5) has been derived by many authors (see [59] for reference) indivers physical fields. These contributions made transport theory and the underlying transport equationsplay more and more important role in those fields.

2.3 High frequency limit

The difficulty arising in the studies of so-called high frequency wave problems is how to justify the localanalysis in a small scale comparable to wavelength. Mathematically speaking, it requires tools to expandof the solution in powers of the small parameter. We first introduce briefly the traditional technique, then,a rigorous method, Wigner distribution, will be presented as followed.

2.3.1 Classic approximation

In cases in which the inhomogeneous of the medium varies gradually relative to the wavelength of wavespropagating in it, we can use approximation solution to the wave equation consisting of plane waves withgradually varying amplitude and wavelength. For example, a traditional approximation technique is theWKB approximation2. This approximation can be used for the classic waves and Schrodinger equationand is a very useful guide for estimating solutions to these equations [60, 61].

Usually, wave equations can be written in the form

Lϕ = ϕtt (2.10)

where L is some linear operator independent of t, periodic solution may be written

ϕ(x, t) = ϕ(x)e−iωt (2.11)

it leads to

Lϕ+ω2

c2ϕ = 0 (2.12)

For large value of ω/c (high frequency cases), a standard method of finding the asymptotic solution is totake

ϕ ∼ eiωσ(x)∞∑

n=0

ϕn(x)(−iω)−n (2.13)

where the function σ(x) and ϕn(x) are to be determined. In term of ϕ(x, t) this is

ϕ(x, t) ∼ eiω(t−σ(x))∞∑

n=0

ϕn(x)(−iω)−n (2.14)

2Three researchers, G. Wentzel, H. A. Kramers, and L. Brillouim, who published this approximation technique applied toSchrodinger equation in 1926. Their initials give the term WKB approximation.


It may be rewritten as

ϕ ∼∞∑

n=0

ϕn(x)fn(S) (2.15)

with

S = t− σ(x), fn(S) =e−iωS

(−iω)−n(2.16)

Acoustics waves can serve as an example of application. To reach the high frequency limit, we introducethe scaled wave function , uε(x, t) = u(x/ε, t/ε). The solution of wave equation (1.6) then can bewritten as

u(x, t) = AeiS(x,t)/ε (2.17)

Insert (2.17) into (1.6) and equating the power of ε, we get evolution equation for the phase and amplitude

∂S

∂t− v2(∇S)2 = 0 (2.18)

and∂ |A|2∂t

+ ∇ ·(|A|2 v ∇S

|∇S|)

= 0 (2.19)

Equation (2.18) is called the eiconal equation and equation (2.19) is a kind of transport equation. Notethat (2.19) should not be confused with the radiative transport equation defined in phase space.

2.3.2 Wigner transform

In high frequency domain, the objective functions (vibrational energy, displacement, or velocity) areoften statistically described by using the concept of average quantities, namely, ensemble average, thatis

E[u(x1)] limN→∞

∑Ni=1 ui(x1)N

(2.20)

where the brackets E[· · · ] denote an ensemble average. Beginning with the fundamental definition ofthe nonstationary autocorrelation for time variable t,

Rx(t1, t2) = E[u(t1)u(t2)] (2.21)

Similarly for space variables. Here we assume that the wave field is homogeneous (on the scale of thewave length, at least), i.e. the two point correlation only depends on the distance x1 − x2.

Definex =

x1 + x2

2, s = x1 − x2 (2.22)

thenx1 = x +

s2, x2 = x − s

2(2.23)

The main idea here is to recast the variables into a center-of-mass coordinate x, taking care of thesmooth variations in the problem, and the difference coordinate s, dealing with the oscillatory one. Thecoordinate x will play the role of the classical coordinate, while s is the Fourier transformed into thewavenumber k. So, if we define a two-point correlation function

φ(s) = E[u(x1)u(x2)] = E[u(x +

s2)u(x − s

2)]

(2.24)

2.3. High frequency limit 39

We now impose a small parameter ε to scale function u, then equation above is rewritten as

φε(s) = E[uε(x1)uε(x2)] = E[u(x +

εs2

)u(x − εs2

)]

(2.25)

If ε → 0, it leads to the measure scale in the wavelength approaching to zero, thus high frequency limitis taken.

In fact, the technique for high frequency limit has been found to be Wigner transform [62, 63], whichis closely related to Fourier transform. Given a function ψ(x) in S(Rn), the Schwartz space of testfunctions, its Wigner transform W (x,k) is defined by

W(x,k) =1

(2π)n

∫Rn

eik·yψ(x − y

2

)ψ∗

(x +

y2

)dy. (2.26)

Its scalar Wigner transform isw(x,k) = trace(W(x,k)). (2.27)

The function ψ(x) can be scalar or vector-valued (ψ∗ denotes the transposed conjugated of the vectorψ); in the latter case W is an hermitian matrix.

Denote the Fourier transform pair

ψ(k) =∫ψ(x)e−ik·xdx

ψ(x) =1

(2π)n

∫ψ(k)eik·xdk

(2.28)

One can regard Wigner transform as a generalized Fourier transform

W(x,k) =∫

Ψ(x,y)e−ik·ydy (2.29)

whereΨ(x,y) = ψ

(x − y

2

)ψ∗

(x +

y2

)(2.30)

as x is fixed.

Using the property of Fourier transform and equation (2.29), one get the integral of W(x,k) over allwave vector k is ∫

W(x,k)dk = ψ (x)ψ∗ (x) . (2.31)

Obviously, Wigner transform W can be viewed as energy density in phase space, and this is one ofimportant properties. Further, in term of the Fourier transform ψ(k) of ψ(x), Wigner transform can alsobe read 3

W(x,k) =1

(2π)d

∫eip·xψ

(k +

p2

)ψ∗

(k − p

2

)dp. (2.32)

3The duality in x and k can be simply expressed by

W(x,k) =1

(2π)n

∫Rn

eik·yψ(x − y

2

)ψ∗

(x +

y

2

)dy

=1

(2π)n

∫Rn

eik·yψ(x − y

2

) [∫Rn

eix·pψ∗(k +

p

2

)dp

]dy

=1

(2π)n

∫Rn

eix·pψ∗(k +

p

2

) [∫Rn

ψ(x − y

2

)eik·ydy

]dp

=1

(2π)n

∫Rn

eix·pψ∗(k +

p

2

)ψ(k − p

2

)dp


We can use the Wigner distribution to realize the high frequency limit of classic wave. It is carried out byscaling ψ(x) to be ψε(x) which are oscillating on a scale ε as ε→ 0. Thus W is rescaled in a nontriviallimit form:

Wε(x,k) =1

(2π)d

∫eik·yψε

(x − εy

2

)ψ∗

ε

(x +

εy2

)dy. (2.33)

and

Wε(x,k) =1

(2πε)d

∫eip·xψε

(kε

+p2

)ψ∗

ε

(kε− p

2

)dp. (2.34)

At this stage, one can find another property of Wigner transform:∫W(x,k)dx =

1(2πε)d

ψε

(kε

)ψ∗

ε

(kε

). (2.35)

Other important properties of Wigner transform can be found in [63, 64]. In general, one can define ann× n “Wigner matrix”

Wε(x,k) =1

(2π)d

∫eik·yψε

(x − εy

2

)⊗ ψ∗

ε

(x +

εy2

)dy. (2.36)

where ⊗ denotes the tensor product of vectors. In the term of “Wigner matrix”, we denote the elementin n× n Wigner matrix Wε[ψε]

wijε [ψε] = wε[ψi

ε, ψjε ], wε[ψε] = traceWε[ψε]. (2.37)

One can get to limits of Wigner transform by expanding it in powers of ε,

Wε = W(0) + εW(1) +O(ε2) (2.38)

High frequency limit, i.e. wave function ψε(x) is oscillating on a scale ε as ε → 0. Thus W0 can bea good and rigorous description. Due to that Wε[ψε] is hermitian, W0 is a nonnegative matrix-valuedmeasure which is sometimes called Wigner measure [65].

The followed section is used to explain how to apply Wigner transform to acoustic wave and generalhyperbolic waves.

2.4 Transport equation for acoustic wave

Acoustic wave is a kind of simple and standard scalar wave. We begin with acoustic wave to illustratehow to use Wigner transform to study the high frequency behavior. The acoustic equation for velocity vand pressure p are

ρ∂v∂t

+ ∇p = 0,

κ∂p

∂t+ ∇ · v = 0

(2.39)

These equations may be reformed to a symmetric hyperbolic system

A(x)∂u∂t

+3∑

i=1

Di ∂u∂xi

= 0 (2.40)

2.4. Transport equation for acoustic wave 41

where u = (v, p)T . The matrix A(x) = diag(ρ, ρ, ρ, κ) is symmetric and positive definite and thematrices Di are symmetric and independent of x and t,

D =3∑

i=1

Di =

0 0 0 10 0 0 10 0 0 11 1 1 0

(2.41)

It should be noted that system (2.40) is just limited itself to the non-dissipation case, the more generalone will be discussed in the next chapter.

The spatial energy density for this system (2.40) is given by

W (x, t) =ρv2

2+κp2

2=

12〈A(x)u(x, t) · u(x, t)〉 =

12Aij(x)ui(x, t)u∗j (x, t) (2.42)

where Aij are elements of matrix A and 〈· · · 〉 denotes inner product. The energy flow I(x, t) by

Ii(x, t) =12⟨D

iu(x, t) · u(x, t)⟩. (2.43)

Here, The energy conservation law holds if the damping or others dissipation effects are not considered

∂W

∂t+ ∇ · I = 0. (2.44)

The dispersion matrix of the system (2.40) is defined by

L(x,k) = A−1(x)kiD

i =

0 0 0 k1/ρ0 0 0 k2/ρ0 0 0 k3/ρ

k1/κ k2/κ k3/κ 0

(2.45)

It has one double eigenvalue ω1 = ω2 = 0 and two simple eigenvalues

ωf = −ωb = v(x) |k| (2.46)

where |k| =√k2

1 + k22 + k2

3 and v is the sound velocity v = 1√κρ .

If we denote k = k|k| , then the corresponding basis of eigenvectors are

s1 =1√ρ(α, 0)t, s2 =

1√ρ(β, 0)t

sf =

(k√2ρ,

1√2κ

)t

, sb =

(k√2ρ,− 1√

2κ

)t

where the vectors α, β and k form an orthonormal triplet. The eigenvectors s1 and s2 correspond totransverse advection modes, orthogonal to the direction of propagation, k. These modes are not prop-agative due to ω1,2 = 0. The eigenvectors sf and sb represent forward and backward acoustic waveswhich propagate in the directions k.


By applying the Wigner distribution, the angularly resolved energy densities for forward and back-ward waves can be expressed as

af,b(k,x, t) =1

(2π)3

∫R3

eik·y〈u(t,x − y/2) · sf,b〉〈u(t,x + y/2) · sf,b〉∗dy (2.47)

For non dissipation cases, the scalar function af,b are related by af (t,x,k) = ab(t,x,−k), and af (t,x,k)satisfies the Liouville equation [59]

∂af

∂t+ ∇kω · ∇xaf −∇xω · ∇kaf = 0

∂ab

∂t−∇kω · ∇xab + ∇xω · ∇kab = 0

(2.48)

The energy density and the energy flow are the integration of the angularly resolved energy densitiesover all the directions, which have been shown in equation (2.31). These two energy or energy flowcomponents are associated with orthogonormal eigenvectors, thus the superposition principle can beused here,

W (x, t) = Wf (x, t) +Wb(x, t)

=∫af (t,x,k)dk +

∫ab(t,x,k)dk, (2.49)

I(x, t) = v

∫k [af (t,x,k) − ab(t,x,k)] dk

= vk(Wf (x, t) −Wb(x, t)) (2.50)

Specifically, if acoustic waves propagate in a homogeneous medium, acoustic velocity is a constant, say,c, then

∇kω = ck, ∇xω = 0 (2.51)

The acoustic energy associated with forward and backward wave are

∂Wf (x, t)∂t

+ ck∇xWf = 0 (2.52)

∂Wb(x, t)∂t

− ck∇xWb = 0 (2.53)

Substitute equation (2.52, 2.53) in (2.49, 2.50) we obtain a simple relation of acoustic energy and inten-sity:

∂W (x, t)∂t

+ ∇xI(x, t) = 0 (2.54)

∂I(x, t)∂t

+ c2∇xW (x, t) = 0 (2.55)

Equation (2.54) is the same as (2.44) which reflects the energy conservation law, while (2.55) revealsthe energy motion behavior. This equation pair (2.54, 2.55) may be employed to describe dynamiccharacteristics of the acoustic wave.

2.5. Transport equation for hyperbolic system 43

2.5 Transport equation for hyperbolic system

Acoustic equation (2.40) is actually belonged to hyperbolic systems. A more general case can be realizedon the basis of the above discussion. Here we borrow some ideas from [63]. Let D

1,D2, · · ·Dm be real,symmetric, n× n matrices. Consider a hyperbolic system with a scaled function uε,

A(x)∂uε

∂t+

m∑i=1

Di∂uε

∂xi= 0, x ∈ Rm

x , t ∈ R (2.56)

uε(t = 0) = uε(0) (2.57)

By calculating

L(k) = A−1(x)

m∑i=1

Diki, k = (k1, · · · , km)T , (2.58)

we seeks the eigenvalues from dispersion matrix of the system. Because of the symmetrical hyperbolicsystem (2.57), we can assume that the eigenvalues ω1(k), · · · , ωn(k) of the matrix L(k) are simple andsmooth. They can be ordered as follows

ω1(k) < · · · < ωn(k) (2.59)

Suppose the medium is homogeneous, one have

ωi(αk) = αωi(k), ∀α ∈ R (2.60)

and in particular

ωi(k) = |k|ωi(k) = |k| vi. (2.61)

where k = k|k| . Let b1(k), · · · , bn(k) be eigenvectors of matrix L(k) corresponding to ω1(k), · · · , ωn(k),

respectively, and they are normalized by |ωi(k)| = 1.

Denote the Wigner transform of function uε is W(uε) and W0 is its high frequency limit. Similar tothe above discussion in previous subsection, transport equation can be expressed by [63]

∂w0i

∂t+ |k|ωi(k) · ∇xw

0i = 0 (2.62)

where

w0 =m∑

i=1

w0i and, w0 = traceW0 (2.63)

It was found that equation (2.62) is equivalent to equation (2.48). If we consider the inhomogeneity inmedia, the mathematical manipulation will be a little complex, this subject can be found in [59].

As viewed in above-discussed examples, Wigner transform play an important role in derivation ofhigh frequency limit of acoustic or hyperbolic wave equation in terms of energy variable. It has beenalso used to describe the high frequency limit of Helmholtz equations [66]. Our further derivation oftime-varying energy equation will be based on these conclusions.


2.6 Diffusion approximation for high frequency waves in random media

In general, the propagation and scattering of classic waves in a random medium can be described by aradiative transfer equation. It is well known that strong scattering leads to diffusive behavior rather thanwave propagation. For the random inhomogeneous media, the diffuse approximation can be obtained byconsidering the scattering effects. When the density ρ and compressibility κ are random and oscillatingon the scale of the wavelength, they are assumed mathematically [58] in the form,

ρ→ ρ(1 +

√ερ1(

xε)), κ→ κ

(1 +

√εκ1(

xε)). (2.64)

where ε is the ratio of wavelength to propagation distance, ρ1 and κ1 are mean zero space homogeneouswith power spectral densities Rρρ, Rκκ, and cross spectral density Rκρ. The limit ε → 0 results in thescaled radiative theory equation (2.5) [58, 59].

For acoustic waves, equation (2.5) can be simplified as

∂a

∂t+ vk · ∇xa− |k|∇xv · ∇ka+

∑(k,x)a =

∫R3

σ(x,k,k′)a(t,x,k′)dk′ (2.65)

where

σ(x,k,k′) =πv2 |k|2

2

[(k · k′)2Rρρ(k − k′) + 2(k · k′)Rρκ(k − k′) + Rκκ

]. (2.66)

Let τ denotes the mean free time4, which is given by

τ−1 =∑

(k,x) =∫σ(x,k,k′)dk′, (2.67)

To avoid excessive mathematical derivation, we propose a straightforward way to analyze the dif-fusion approximation of transport equation in what follows. Define the energy exchange between twowave vector, for example

Πk′→k energy density a(t,x,k′) scattering to wave vector k

Πk→k′ energy density a(t,x,k) scattering to wave vectors k′

we introduce a coefficient µ(x,k) to represent the ratio of the mutual scattering effect:

µ(x,k) =Πk′→k

Πk→k′

=

∫R3 σ(x,k,k′)a(t,x,k′)dk′∫R3 σ(x,k,k′)a(t,x,k)dk′ (2.68)

=

∫R3 σ(x,k,k′)a(t,x,k′)dk′∑

(k,x)a(t,x,k)4mean free time τ is related with mean free path lm, τ = lm/v, v is velocity. The mean free path was originally defined

as the average distance that a particle will travel between interactions. For sound waves in an enclosure, it is defined as theaverage distance traveled between successive reflections. An approximation value for the mean free path in room acousticsproposed by Sabine, is lm = 4V

S, where V is the total volume of the room, and S is total surface area enclosing the room. This

approximation requires the diffuse field assumption, i.e., the plane waves are traveling randomly in all direction [67, 68].

2.6. Diffusion approximation for high frequency waves in random media 45

Suppose the scattering effect is the same for forward and backward waves, then the transport equation(2.65) for homogeneous media can be rewritten as

∂af

∂t+ vk · ∇xaf +

1 − µ

τaf = 0 (2.69)

∂ab

∂t− vk · ∇xab +

1 − µ

τab = 0 (2.70)

Applying the superposition principle shown in (2.49) and (2.50), the radiative transport equation can bewritten in the form

∂tW (x, t) + ∇ · I(x, t) +(1 − µ)W (x, t)

τ= 0 (2.71)

∂tI(x, t) + v2∇W (x, t) +(1 − µ)I(x, t)

τ= 0 (2.72)

The further discussion about equation pair (2.71, 2.72) need to examine the value of µ. One can readilyfind its possible value by the definition:

µ > 1, if Πk′→k > Πk→k′ : net energy gain (2.73)

µ = 1, if Πk′→k = Πk→k′ : energy dynamic equilibrium (2.74)

µ < 1, if Πk′→k < Πk→k′ : net energy loss (2.75)

If µ = 1, equation pair (2.71, 2.72) represent a hyperbolic system, and the diffusion approximation canbe reach. As to two other cases where µ = 1, we introduced an assumption, that is, if we ignored theterm ∂tI(x, t) in equation (2.72), a generalized Fick law was found

I(x, t) = − τv2

1 − µ∇W (x, t) (2.76)

It should be paid more attention to the reasons of ignoring the term ∂tI(x, t). If the scattering is enoughstrong, the scattering cross section

∑(k,x) must be huge, then according to the definition (2.67), τ will

be rather small. If τ → 0, the term ∂tI(x, t) will vanish.

Upon substituting ∇ · I(x, t) in equation (2.71) by the divergence of equation (2.76) leads to thediffusion equation

∂tW (x, t) − τv2

1 − µ∇2W (x, t) +

(1 − µ)W (x, t)τ

= 0 (2.77)

However, for the cases of fairly week scattering, the term ∂tI(x, t) in equation (2.72) could not beneglected arbitrarily, the combination of equation (2.71) and (2.72) leads to

∂2W (x, t)∂t2

− v2∇2W (x, t) +2(1 − µ)

τ

∂W (x, t)∂t

+(1 − µ)2

τ2W (x, t) = 0. (2.78)

It is a hyperbolic equation rather than a parabolic one.

In fact, some references like [59, 67, 69] were found involved with the diffusion approximation in thecontext, and some important conclusions are drawn on basis of it. As pointed out in [70], the differentdynamic characteristics of waves in a random medium depends on the its elastic (scattering) mean freepath lm: over a typical distance of the order of lm, an acoustic wave propagates roughly ballistically, theequation governing the acoustic wave propagation is the hyperbolic wave equation

∂2A

∂t2= c2∇2A, (2.79)


where c is the sound speed and the wave amplitude A is related to wave energy density E by E = |A|2.We can proof that this equation also holds if the variable A is taken place by E (this proof is found innext Chapter); beyond the distance lm, the nature of the transport of the wave intensity crosses over fromballistic to diffusive. This means that the mathematical model for wave propagation changes from thehyperbolic wave equation to the parabolic one. In summary, the scatters are the major factors whichdestroy the wave propagating and ultimately lead to diffusive behavior.

2.7 The time-delayed diffusion system and wavefront solution

The diffusion approximation of transport equation (2.77) was used to describe the evolution of wavesenergy in random media. However, from a physical point of view, this equation give rise to the infinitespeed of waves energy, which must lead to an overestimation of waves energy speed in time domain. Theinfinite velocity cause physically unsatisfactory result. On the other hand, from a philosophy point ofview, there should exist a time interval between cause and effect. In equation (2.76), the cause is a waveenergy density W , and the corresponding effect is the energy flow I: the cause disappears (∇W = 0)result in an instantaneous vanishing effect (I = 0), therefore it yields an unrealistic description. Thisproblem have attracted considerable interest for a long time (see the references in [71, 72]), it even canbe tracked to the times of Maxwell, who actually had his doubts about the diffusion equation whichgoverns heat conduction. To deal with this problem several researchers have introduced the hyperboliccorrection to the diffusion approximation, taking into account the finite speed of the transport process[69, 71].

Here we will follow the idea (see [69] and its references therein) that Fourier’s law with allowancefor a time delay τ0 is applied to take into account the finite speed of the transport process. We rewrite theequation (2.76) in such form:

I(x, t+ τ0) = −D∇W (x, t) (2.80)

Where D =τv2

1 − µrepresents the diffusion coefficient. By expanding I(x, t+ τ0) up to the first order in

τ0, we have

τ0∂I(x, t)∂t

+ I(x, t) = −D∇W (x, t) (2.81)

which is the same form as equation (2.72).

If τ0 = 0, equation (2.81) reduces to the classical Fick law (2.76). Combining equation (2.81) and(2.71) leads to a hyperbolic energy equation like (2.78).

Another technique used by Sergei Fedotov [71] is to assume the flux, which refer to wave energyflow I(x, t)) here, is determined as an integral over time of the energy density gradient multiplied by theflux kernel R(t− s),

I(x, t) = −D∫ t

0R(t− s)∇W (x, s)ds (2.82)

We also choose an exponential kernel with a single relaxation time τ0,

R(t− s) =1τ0

et−sτ0 (2.83)

2.8. Summary 47

Derivative of I(x, t) with respect to t reads

∂

∂t

[I(x, t) = −D

∫ t

0

1τ0

et−sτ0 ∇W (x, s)ds

]

⇒ ∂I(x, t)∂t

= − 1τ0

I(x, t) − D

τ0∇W (x, s) (2.84)

Surprisingly, it is found that equation (2.84) is identical to equation (2.81). Mathematically speaking,the so-called time-delayed diffusion equation is actually equivalent to a hyperbolic equation rather thana parabolic one. The hyperbolic equations yield wavefront solutions such as W (x, s) = ψ(t− cx) (Thedetailed investigations of the solution format will be discussed in the next Chapter). Wavefronts areusually defined as solutions such that they travel with constant wave shape.

2.8 Summary

Although in many physical phenomena there is no clear distinction between wave-like and diffusive be-havior, the hyperbolic (exactly, hyperbolic reaction-diffusion) equations and their wavefront solution arefound to be more suitable and realistic to describe a space-time evolution process, because the unphys-ical feather of an infinite velocity can be avoided. Except being employed in structural dynamics, thehyperbolic systems have recently got widely applications ranged from biology, chemical systems, crys-tal growth to forest fire models, propagation of epidemics, etc. Even heat conduction, which is basicallyregarded as a typical diffusion case, has been found wave-like behavior under some condition. An ele-gant instructive paper [73] referred to this discovery. In the matter, lattices are composed of atoms andmolecules. Lattice vibrations5 are responsible for the transport of heat. If the lattice is cooled to nearabsolute zero, lattice vibrations no longer behave diffusively but are actually wave-like. In this case, it isno surprise that heat moves like wave propagating!

In summary, transport theory is an effective tool to deal with waves propagation in random media.Many problems involved with scattering in waves can be explained by transport equation. Specially, inhigh frequency range, it is scattering phenomenon that results in the diffusive behavior of waves prop-agation. In this thesis, the main study objective is restricted to homogeneous structures/systems whichare widely utilized in engineering practice. Only if wavelengths are short compared to the heterogeneityfeatures of the medium, scattering by inhomogeneities should be considered. However, some aspectsshould be taken into account. In classical transport theory or Boltzmann equation, the energy loss termcaused by media is neglected mainly because that the transport loss of media-independent waves is notaffected by media. But for elastic mechanical waves, however, the loss term can not be ignored be-cause damping and energy emission from structural(media-related) wave is the important subject whichshould be concerned, especially in the case of high frequency. The assumption of waves without damp-ing loss is not realistic. These aspects can illustrate us how to break some limits and develop a suitabledescription for high frequency elastic mechanic waves. We will take advantage of the contribution andrecent progresses in transport theory and seek a standard time-varying energy equation to predict thehigh frequency acoustic/structural dynamic in next Chapter.

5The atoms in a crystal are not locked into a rigid pattern but can oscillate around their average position. When a wholegroup of them does this in a synchronized way we call it a lattice vibration.

Chapter 3

New time-varying energy equation and itsproperties

As reviewed in Chapter 1, energy methods to deal with high frequency vibroacoustic problems in tran-sient condition are underdeveloped up to now. The typical ones, such as transient statistical energyanalysis (TSEA) and the time-varying vibrational conductivity approach (see equation (1.94)) have beenproposed and discussed. Some researchers [47, 49] claimed to realize the successful applications ofTSEA comparing with experimental results. However, other studies [50, 52, 53, 74] showed that TSEAdoes not always agree with the exact energy results. In addition, one can find that TSEA is just thediscretized form of time-varying vibrational conductivity approach. In the followed chapters, we willdemonstrate the flaw and limitation of these methods. Besides, radiative transport equation, which isintroduced at some length in Chapter 2 offers an alternative attempt to describe the waves energy evolu-tion in inhomogeneous domain, thus it can not be directly applied to the practical engineering structuresunder the assumption of homogeneity. In this chapter, we dedicate our major effort to proposing a newtime-varying energy equation based on the work mentioned above. It is hoped that the method based onthe new equation can be an effective tool to predict the time-varying energy in high frequencies.

In general, there are two ways to understand and describe energy of high frequency waves. On theone hand, we can extract the concept of vibrational energy by investigating the behavior of diverse wavestypes; on the other hand it is possible to study this subject by directly employing the universal energylaws, thus some characteristics of waves become minor important at this stage. we will explore thederivation of time-varying energy equation along these two guidelines. Of special importance is at thelate of this chapter, where the properties of new energy equation will be discussed and compared withthe diffusion energy equation (1.94).

3.1 Time-varying energy described by waves characteristics

Waves are ubiquitous in nature. Generally, a wave is a propagating imbalance. The imbalance conceptis present in a simple oscillator where kinetic energy and potential energy are interchanged during theoscillation. It tells that a wave propagating is equivalent to the energy propagating. In the structuralvibration and acoustics field, linear waves can be generally divided into two main classes [75]: hyperbolicwaves and dispersive waves. The former is formulated mathematically in terms of hyperbolic partial

49

50 Derivation of time-varying energy equation

differential equation; the latter is often encountered as the cases of flexural vibrations in beams, plates,etc. Simple speaking, hyperbolic waves are characterized by the form:

u(x, t) = f(x− ct) + g(x+ ct) (3.1)

with a constant wave speed c, while dispersive system is any system which admits solution of the form

u(x, t) = a cos(kx− ωt) (3.2)

where the frequency ω is a definite real function of the wave number k. The phase speed is then ω(k)/kand the wave are usually said to be dispersive if this phase speed is not a constant but depends on k. Itmeans the modes with different k will propagate at different speed; they will disperse.

3.1.1 Derivation by transport theory for hyperbolic waves

Many physical problems involved with dynamics in time-space domain can be summarized mathemat-ically to be a hyperbolic system. This mathematical model plays an important roles in miscellaneousphysical topics ranging from electromagnetic waves, the voltages and currents in circuits, classic fluidmechanics, continuum mechanics, modern quantum mechanic, to seismic dynamics, acoustics and me-chanical vibration in structures. The study on hyperbolic system itself has a long and rich history, and thissubject has still been an attractive field both in “industry oriented aspects” (e.g. semiconductors, plas-mas, fluid mechanics, · · · ) and in “society oriented aspects” (vehicular traffic, environmental sciences,· · · ) [76, 77].

In the mathematical-physical point of view, a hyperbolic model marks a time-dependent wavefrontphenomenon. The most simple hyperbolic example may be acoustic wave equation in R3:

ρ∂v∂t

+ ∇p = 0

κ∂p

∂t+ ∇ · v = 0

(3.3)

where ρ = ρ(x) is the density and κ = κ(x) is the compressibility. Equation (3.3) can be put in the formof a symmetric hyperbolic system

A(x)∂u∂t

+3∑

i=1

Di ∂u∂xi

= 0 (3.4)

where u =(vp

), and the matrix

A =

ρ(x) 0 0 0

0 ρ(x) 0 00 0 ρ(x) 00 0 0 κ(x)

D =3∑

i=1

Di =

0 0 0 10 0 0 10 0 0 11 1 1 0

(3.5)

It should be clear that the hyperbolic system (3.4) does not consider the unavoidable energy lossby acoustic damping and, examination of references [58, 59, 62, 64] shows that researchers (especiallymathematician) had not mentioned the energy loss condition either. The highlight of our work in thissection is to derive a general transport equation taking dissipative term into account. Based on the generaltransport equation, some conclusions suited to several special cases will be reported.

3.1. Time-varying energy described by waves characteristics 51

In fact, general wave equations can be written as symmetric hyperbolic systems accounting for damp-ing loss,

A(x)∂u∂t

+n∑

i=1

Di ∂u∂xi

+ Bu + f = 0 (3.6)

Here, u, the state, is a q-element column vector defined over coordinates x ⊂ Rn and t ≥ 0. A andD

i, i = 1, ..., n, are real symmetric q × q matrices; in particular, A is assumed to be positive definite.B is a real symmetric matrix models energy loss factors, and the q-element real column vector f is aforcing function or excitation. For all the systems to be discussed in this thesis, the matrices D

i and B

are assumed to be constant, while A is allowed to depend on x.

Symmetric hyperbolic systems are important because they form a subclass of strongly hyperbolicsystems, for which the initial-value problem is well-posed, i.e., its solution is bounded in a well-definedway. In addition, the unsymmetric systems, which means the matrix A and D

i are not symmetric, can betransformed to a symmetric one [76].

At first, let’s assume that the problem is defined over an unbounded spatial domain Rn so that theboundary conditions is out of consideration, and also that the forcing function f = 0. Take the innerproduct of uT (the transpose of u) with equation (3.6) to get

uTA∂u∂t

+n∑

i=1

uTD

i ∂u∂xi

+ uTBu = 0 (3.7)

Due to the symmetry of A and Di, equation (3.7) can be written as

12∂

∂t

(uT

Au)

+12

n∑i=1

∂

∂xi

(uT

Diu

)+ uT

Bu = 0 (3.8)

The spatial energy density and the energy flow for the solutions of equation (3.6) are given by

W (t, x) =12(uT · Au), Ii(t, x) =

12(uT · D

iu) (3.9)

then the energy balance law reads

∂W

∂t+ ∇ · I +Wdiss = 0 (3.10)

where

Wdiss = uTBu (3.11)

which denotes the energy loss.

In what follows, the transport equation with energy loss term were derived. Introduce Wigner func-tion [59, 62] with a wave field u, which is defined by the formula

Wε(t,x,k) =1

(2π)3

∫R3

ejk·yu(t,x − εy

2

)u∗

(t,x +

εy2

)dy (3.12)


The following identities are easily checked1

12

∫R3

trace(WεA)dk =12uT

Au, Energy density (3.13)

12

∫R3

trace(WεDi)dk =

12uT

Diu, Energy flow (3.14)∫

R3

trace(WεB)dk = uTBu, Energy dissipation. (3.15)

The time derivative of equation (3.12) was given by

∂Wε

∂t=

1(2π)3

∫R3

ejk·y[∂u

(t,x − εy

2

)∂t

u∗(t,x +

εy2

)+ u

(t,x − εy

2

) ∂u∗ (t,x + εy2

)∂t

]dy

(3.16)

Substituting hyperbolic waves equation (3.6)2 in (3.16), it yields

∂Wε

∂t= − 1

(2π)3

∫R3

ejk·y

A−1(x − εy

2)Di ∂

∂xi

[u(t,x − εy

2

)u∗

(t,x +

εy2

)]+

∂

∂xi

[u(t,x − εy

2

)u∗

(t,x +

εy2

)](A−1)∗(x +

εy2

)Di

dy

+1

(2π)3ε

∫R3

ejk·y[A−1(x − εy

2)Diu

(t,x − εy

2

) ∂u∗ (t,x + εy2

)∂yi

− ∂u(t,x − εy

2

)∂yi

u∗(t,x +

εy2

)(A−1)∗(x +

εy2

)Di

]dy

− 1(2π)3

∫R3

ejk·y[A−1(x − εy

2)Bu

(t,x − εy

2

)u∗

(t,x +

εy2

)+ u

(t,x − εy

2

)u∗

(t,x +

εy2

)(A−1)∗(x +

εy2

)B]dy (3.17)

By applying the properties of Wigner transform (2.31–2.35) and doing the second integral by parts inequation (3.17), thus equation (3.17) can also be rewritten as

∂Wε

∂t= Q1

εWε +1εQ2

εWε + Q3εWε (3.18)

1The identities (3.13) is fully expanded as

1

2

∫R3

trace(Wε(t,x,k)A(x))dk =1

2

∑m=n=1

Amn(x)

∫R3Wmn(t,x,k)dk

=1

2

∑m=n=1

Amn(x)um(t,x)un(t,x) =1

2(uT · Au)

The identities (3.14, 3.15) are got by the similar fashion.2The term of forcing function or excitation f is omitted for this moment because it is nothing to do with the generalized

wave function u.


where Q1ε , Q2

ε and Q3ε are operators, which are given by

Q1εWε =

12

∫ejp·x

[A−1(p)Di∂Wε

(t,x,k + εp

2

)∂xi

+∂Wε

(t,x,k − εp

2

)∂xi

DiA−1(p)

+jA−1(p)piDiWε

(t,x,k +

εp2

)+ jWε

(t,x,k − εp

2

)piD

iA−1(p)

]dp (3.19)

and

Q2εWε =

∫ejp·x

[jA−1(p)kiD

iWε

(t,x,k +

εp2

)− jWε

(t,x,k − εp

2

)kiD

iA−1(p)

]dp (3.20)

and

Q3εWε =

∫ejp·x

[A−1(p)BWε

(t,x,k +

εp2

)+ Wε

(t,x,k− εp

2

)BA

−1(p)]dp (3.21)

It was found that expressions (3.19, 3.20) were the same as that in [59] while (3.21) is the originaldevelopment in this thesis.

The next step is to expand the above operators in the terms of order ε. For Q1ε and Q3

ε , the limit weregiven

if ε→ 0, then Q1ε → Q1 Q3

ε → Q3 (3.22)

while Q2ε should be expanded to the second-order term because it is multiplied by the coefficient 1

ε in(3.18)

Q2ε = Q2 + εQ21 +O(ε2) (3.23)

After expansion of above operators, equation (3.18) can be written as

∂Wε

∂t+

12

[A−1

Di∂Wε

∂xi+∂Wε

∂xiD

iA−1

]− 1

2

[∂A

−1

∂xiD

iWε + WεDi∂A

−1

∂xi

]︸︷︷︸

Q1Wε

+1ε

[jA−1kiD

iWε − jWεkiDiA−1

]︸︷︷︸Q2Wε

−12

[∂A

−1

∂xikiD

i∂Wε

∂ki+∂Wε

∂kikiD

i∂A−1

∂xi

]︸︷︷︸

Q21Wε

+ A−1

BWε + WεBA−1︸︷︷︸

Q3Wε

= 0 (3.24)

Note that there is no other term to balance 1εQ2Wε. This means that the limiting Wigner transform

W (t,x,k) (Wε →W as ε→ 0) must belong to the null space of the limit operator Q2, where Q2ε → Q2

as ε→ 0 [59]. As discussed in last chapter, the high frequency energy limit can be reached by expandingWε, that is

Wε = W(0) + εW(1) +O(ε2) (3.25)

This leads to

Q2W(0) = 0 (3.26)


and

∂W(0)

∂t+

12

[A−1

Di∂W

(0)

∂xi+∂W(0)

∂xiD

iA−1

]− 1

2

[∂A

−1

∂xikiD

i∂W(0)

∂ki+∂W(0)

∂kikiD

i∂A−1

∂xi

]

− 12

[∂A

−1

∂xiD

iW(0) + W(0)D

i∂A−1

∂xi

]+ A

−1BW(0) + W(0)

BA−1

= jε(A−1kiD

iW(1) − W(1)kiDiA−1

)(3.27)

Introduce the dispersion matrix L(x,k), defined by

L(x,k) = A−1kiD

i (3.28)

All its eigenvalues λτ and the corresponding eigenvectors sτ satisfy the eigenvalue equation

L(x,k)sτ (x,k) = λτsτ (x,k), or A−1kiD

isτ = λτsτ (3.29)

Thus from equation (3.26) we get the equation

LW(0) − W(0)L

T = 0 (3.30)

Suppose the eigenvalues decided by (3.29) is N dimensions, then a general solution of equation (3.30) is

W(0) =N∑

τ=1

wτsτ (sτ )∗ (3.31)

Substituting (3.31) in equation (3.27) and multiplying by (sβ)∗A on the left and by Asβ on the right of(3.27), one finds that the right-hand of equation (3.27) must be zero because eigenvectors are orthonormalabout A, that is

(Asβ , sτ ) = δβτ (3.32)

where ( , ) denotes inner product. From equation (3.29), we get the identities

∂λτ

∂ki= (sτ )∗A−1

Disτ

∂λτ

∂xi= (sτ )∗ki

∂A−1

∂xiD

isτ

Applying these identities, we finally get the transport equation from (3.27)

∂wτ

∂t+∂λτ

∂ki

∂wτ

∂xi− ∂λτ

∂xi

∂wτ

∂ki+ bτwτ = 0 (3.33)

where

bτ = (sτ )∗(

A−1

B − 12∂A

−1

∂xiD

i

)sτ (3.34)

We now have obtained the transport equation (3.33) which includes the energy loss term. However,equation (3.33) is not involved with scattering effect, therefore, it can be regarded as a modified Liouvilleequation in phase space.


3.1.2 Discussions on transport equation

Equation (3.33) can be transformed or simplified according to some special cases.

1. For homogeneous media.

In this case, A(x) is no longer the function of x, consequently we have

∂λτ

∂xi= (sτ )∗ki

∂A−1

∂xiD

isτ = 0

∂λτ

∂ki= (sτ )∗A−1

Disτ = vki

and equation (3.34) is changed to

bτ = (sτ )∗(

A−1

B − 12∂A

−1

∂xiD

i

)sτ = (sτ )∗A−1

Bsτ (3.35)

thus equation (3.33) is reduced to

∂wτ

∂t+ vki

∂wτ

∂xi+ bτwτ = 0 (3.36)

Equation (3.36) is equivalent to (3.10).

2. Simple and distinct eigenvalues.

It means the eigenvalues are and only are zero (λβ = 0) or positive-negative dual values, namely,±λτ . The physical meaning for the zero eigenvalues corresponds to transverse advection modes,orthogonal to the direction of propagation, while positive-negative dual eigenvalues represent theforward and backward propagation direction. This hypothesis is satisfied in the case of acousticand elastic waves.

For general structures, it is possible that several different wave types coexist. For example in[65], there are three different wave types in a Mindlin plate: shear wave, compressional wave,and flexural (bending) wave. Except non-propagative modes, the eigenvalues are λ±S = ±cS |k|,λ±P = ±cP |k|, and λ±T = ±cT |k|, where cS , cP , and cT are respective wave velocity. Theseeigenvalues and their eigenvectors s±S , s±P , and s±T are also governed by equation (3.29).

Some assumptions are proposed here to avoid an excessive expenditure of mathematical effort:

• Interferences between all waves are neglected.

• A general elastic displacement is the incoherent sum of each individual wave.

• Absence of polarization: simple eigenvalues are present by the dispersion matrix.

• The non-propagative amplitudes (near field effect) is ignored.

• The damping matrix B in equation (3.6) is a constant diagonal matrices with identical ele-ments b.

Thereby, the angularly resolved energy density for specific wave type j can be expressed as

w±j =

1(2π)d

∫eik·yf±j (t,x − y/2,k)f±j

∗(t,x + y/2,k)dy (3.37)


where ± represent the forward (+)and backward (−) propagation directions for wave type j. f±jis the inner product of A, u, and eigenvectors b±

j )

f±j = (Au,b±j ) (3.38)

Energy density w±j , as shown in last subsection, satisfy the generalized Liouville equation in the

high frequency condition, as derived

∂w+j

∂t+ cj(x)k · ∇xw

+j − |k|∇xcj(x) · ∇kw

+j + bw+

j = 0 (3.39)

∂w−j

∂t− cj(x)k · ∇xw

−j + |k|∇xcj(x) · ∇kw

−j + bw−

j = 0 (3.40)

According to the assumptions list above, the incoherence between waves and initial non-propagationterm are neglected, then energy density are an integral over wave vector k of the angularly resolvedenergy density

W±j (t,x) =

∫w±

j (t,x,k)dk (3.41)

We can regard W+j and W−

j as the forwards and backwards energy densities corresponding to

±c(x)k. Not like the state in reference [59], W+j is not equal to W−

j because of the effect ofdamping loss.

3. Synthesis of two cases above. For this moment the subscrit j in wj denoting the wave type isomitted. Specially in bounded system, the energy component W+ is not equal to W− whendissipative effect is taken into account (It differs from the discussion shown in reference [59]where W+ = W− was employed for a conservative system). Thus equations (3.39) and (3.40) arefurther modified to be

∂W+

∂t+ ck · ∇xW

+ + bW+ = 0 (3.42)

∂W−

∂t− ck · ∇xW

− + bW− = 0 (3.43)

By adding equations (3.42) and (3.43), one obtain

∂W

∂t+ ∇x · I + bW = 0 (3.44)

where

W =W+ +W−

2, I =

W+ −W−

2ck (3.45)

subtracting equations (3.42) and (3.43), it yields

∂I∂t

+ c2k · ∇xW + bI = 0 (3.46)

Combination of equations (3.44) and (3.46) results a new second-order energy equation

∂2W (x, t)∂t2

− c2∇2W (x, t) + 2b∂W (x, t)

∂t+ b2W (x, t) = 0 (3.47)


Eventually, we obtain a new time-varying energy model described by the simultaneous set of equations(3.44) and (3.46) or a second order temporal equation (3.47). Based on the equation (3.44) or (3.44), wepropose a new method to predict the time-varying high frequency dynamics, which is named MethodeEnergetique Simplifiee Transitoire (MEST), and for the sake of being referred easily, equation (3.44,3.46) or (3.47) will be called MEST equation hereafter in this thesis.

While we make use of the transport theory to develop the derivation above, we stress again that theconclusions and deduction in this subsection are of major importance. We are aware of the limitation thatthe our time-varying description can be applied only in situations where inhomogeneous waves play aminor role. In the meantime, there has been other efforts from some recent mathematical results appliedto develop energy approaches. Although transport theory are normally employed to describe geomet-rical optics or quantum mechanic, the propagation and scattering of elastic waves in a random mediacan be described by a radiative transfer equation [59, 78], or say, it can be used to interpret the highfrequency phase space energy of general elastic wave field. This result can also be obtained in the frameof the mathematical theory of microlocal asymptotic analysis developed by some mathematicians [62].These studies express the wave scattering phenomenon in extreme high frequency range and randominhomogeneous media where the wavelength is short compared to macroscopic features of the medium.For acoustic or high frequency mechanical waves propagating in homogeneous structures, the radiativetransfer equation can be much simplified but the energy loss due to structural inner friction should betaken into account. Moreover, the mechanical or elastic wave is much different from light or electromag-netic waves, properties of the former are completely decided by the elastic or compressible characteristicof structures, while the latter is only affected but not decided by the medium (Light even propagatesin vacuum). So, the description of elastic wave should not only be radiative, this will be stressed andexplained in the derivation of new time space energy equation in followed section.

It is well known that the complexity of radiative transfer equation can be greatly reduced by the useof the diffusion approximation [59, 78]. This diffusion approximation is only valid after the waves havescattered many times, or it can be obtained in a bounded homogeneous medium for multiply reflectedwaves. The diffusion approximation of transport equation has been mentioned in chapter 2, and theproperties of equation (3.47) will be discussed lately.

3.1.3 Derivation of energy equation for dispersive waves

The discussion in section 3.1.1 was concerned primarily with hyperbolic system which describe by equa-tion (3.6). However, most wave motion are not described by hyperbolic equation in the first instance.These nonhyperbolic wave motions can be grouped largely into a main class called dispersive. In gen-eral, the classification is less precise than that for hyperbolic waves, that is, hyperbolic waves can bedescribed by hyperbolic equation (3.6) while dispersive waves can not represented by a unique equation,but, the latter may be characterised by their solutions.

For linear dispersive waves, they are usually recognized by the existence of elementary solution inthe form of sinusoidal wavetrains

u(x, t) = Aei(kx−ωt) (3.48)

where k is the wave number, ω is the frequency, and A is the amplitude. For example, if φ satisfies thetransverse vibration equation of beam,

∂2u(x, t)∂t2

+ a2∂4u(x, t)∂x4

= 0, a2 =EI

ρS(3.49)


Figure 3.1: Wave packet: the amplitude and phase of high frequency wave are slowly varying function.

it yieldsω2 − a2k4 = 0. (3.50)

The relation between ω and k is called the dispersion relation. Further, a single linear equation withconstant coefficient may be written

P

(∂

∂t,∂

∂x1,∂

∂x2,∂

∂x3

)φ = 0 (3.51)

where P is a polynomial. Substituting the elementary solution (3.48) into (3.51), yields the dispersionrelation

P (−iω, ik1, ik2, ik3)φ = 0 (3.52)

where the dispersion relation consists of the correspondence

∂

∂t↔ −iω ∂

∂xj↔ ikj (3.53)

For dispersive waves it is known that they come in groups (wave packets) 3.1. If the wave groupsare sufficiently long, a reasonably accurate description by using a plane wave solution can be accepted,however, with slowly varying phase and amplitude [79].

u(x, t) = A±(x, t)e±iθ(x,t) (3.54)

= A±(x, t)e±i(kx−ωt) (3.55)

where both amplitude A and phase θ are slowly varying functions of space and time. Here, slow has arelative meaning; it refers to the basic length (and time) scale imposed by high frequency wave lengthand period. Since the phase also is slowly varying, one can define the oscillatory solutions of lineardispersive waves with a local wave number k(x, t) and a local frequency ω(x, t).

ω =∂θ

∂t, k = ∇θ (3.56)

it yields∂k∂t

+∂ω

∂x= 0 (3.57)

This equation shows that, if the frequency of the wave depends on x (because of slowly varying depthand or currents), the wave number will change in time. Equation (3.57) can be interpreted in other way.That is, k(x, t) is the density of the waves—the number of wave crests per unit length, while ω(x, t) isthe flux—number of wave crests crossing the position x per unit time. If we expect that wave crests willbe conserved in the propagation, the conservation equation in differential form (3.57) should hold.


In addition, for a continuum of wave groups, assuming there are two propagation direction (a positivefrequency and a negative frequency mode), we write

u(x, t) =∫ ∞

−∞u+ei(k·x−ω+t)dk +

∫ ∞

−∞u−ei(k·x−ω−t)dk (3.58)

Note thatω = ω(k) (3.59)

we can define the the group velocity as

c =∂x∂t

=∂ω

∂k(3.60)

It is the important velocity for a group of waves with a distribution of wave number. It should bepointed out that the energy of the dispersive waves propagates with the group velocity [80]. (In the1880s Lord Rayleigh introduced the concept of group velocity in order to characterize the speed ofpropagation of non-monochromatic signals in purely dispersive media. Its rigorous derivation for wavepackets was possible when Lord Kelvin developed the stationary phase method. The derivation dependson the absence of frequency-dependent attenuation. Consequently it applies to dispersive modes but itdoes not extend to dispersion associated with attenuation.)

In the phase space of (x,k), x and k represent a canonical vector pair. In fact, equation (3.57) and(3.60) describe simultaneously the propagation of a wave group (also known as Hamilton’s equations ofmotion) read:

∂x∂t

= +∂ω

∂k⇒ xi = +

∂ω

∂ki(3.61)

∂k∂t

= −∂ω∂x

⇒ ki = − ∂ω

∂xi(3.62)

Dispersive waves group is characterized by equation pair (3.61, 3.62), which will be used frequently inthis section.

In addition, the homogeneous and stationary statistical theory of random waves is introduced fordispersive waves. In such a theory, wave components are necessarily independent (random phases). Asa consequence, the probability distribution of high frequency plane waves is approximately Gaussian.The waves are approximately independent because they have propagated into a given area from differentdistant regions. And, even if initially one starts with a highly correlated state then, because of thedispersion, waves become separated, thereby decreasing the correlation (in fact, for dispersive waves theloss of correlation is exponentially fast).

Moreover, just to start from a general case,we also define the dynamics problem of high frequencydispersive wave in phase space. The bases of phase space consist of spatial coordinates x = (x1, x2, x3)and wave coordinates k = (k1, k2, k3). Also, the phase space can be regards as an expansion of spatialspace.

Denote the general coordinate X in phase space

X = (x,k) = (x1, x2, x3, k1, k2, k3), (3.63)

the corresponding general velocity is given by

V = (x, k) = (x1, x2, x3, k1, k2, k3) (3.64)


Thus energy flow in phase space is expressed as

I(x,k, t) = VW (x,k, t) (3.65)

We start from the basic energy balance equation in phase space. Similar, it is reasonably supposedthat the total energy is the sum of forward and backward energy components corresponding to forwardand backward waves. Here, we take the forward energy component as an example, where sign + isomitted.

∂W

∂t+ ∇(x,k) · I + bW = 0 (3.66)

or∂W

∂t+∂(VW )∂X + bW = 0 (3.67)

where b characterizes the dissipative effect. Equation balance equation (3.66, 3.67) were defined in thesame way but extended the space from spatial space to phase space, and they were the most fundamentalexpression of vibrational energy.

Note that x and k are mutual orthogonal and independent, then equation (3.67) was rewritten by

∂W

∂t+∂(xW )∂x

+∂(kW )∂k

+ bW = 0 (3.68)

The chain derivatives rule was applied to the second and third term, it yields

∂W

∂t+ x

∂W

∂x+W

∂x∂x

+ k∂W

∂k+W

∂k∂k

+ bW = 0

⇒ ∂W

∂t+ x

∂W

∂x+ k

∂W

∂k+W

(∂x∂x

+∂k∂k

)+ bW = 0 (3.69)

Substituting dispersive relation (3.61, 3.62) in equation (3.69), one found that

∂x∂x

+∂k∂k

=∂

∂x∂ω

∂k+

∂

∂k

(−∂ω∂x

)= 0. (3.70)

Thus equation (3.69) was reduced to

∂W

∂t+ x

∂W

∂x+ k

∂W

∂k+ bW = 0. (3.71)

We use the dispersive relation (3.61, 3.62) once again to replace x and k in equation (3.71),

∂W

∂t+∂ω

∂k∂W

∂x− ∂ω

∂x∂W

∂k+ bW = 0

⇒ ∂W

∂t+ ∇kω · ∇xW −∇xω · ∇kW + bW = 0

(3.72)

It was found that the high frequency energy equation (3.72) for dispersive waves can also be representedby Liouville equation. Especially, frequency ω of wave group play the same roles as particle with aHamiltonian H . As a contrast, for particles motion, the ordinarily Hamilton’s equations are written asfollows

qi =∂H

∂pi, pi = −∂H

∂qii = 1, 2, · · · (3.73)

3.2. Derivation by employing energy law 61

where qi is general coordinate and pi the kinetic momentum. Denote

∇ =

(∑i

∂

∂qi,∑

i

∂

∂pi

)(3.74)

thus

∇ · v =∑

i

(∂

∂qi

∂H

∂pi, − ∂

∂pi

∂H

∂qi

)(3.75)

If the equation of continuity∂ρ

∂t+ ∇ (ρv) + κρ = 0 (3.76)

holds , it can be written as∂ρ

∂t+ ρ,H + κρ = 0 (3.77)

where · · · , · · · is Poisson bracket, and κ represents the dissipative coefficient. Note that equation(3.77) is equivalent to (3.72), while Hamiltonian in the latter is frequency ω.

Moreover, it was found that energy evolution of dispersive waves, if the wave group concept em-ployed, can also be described by a hyperbolic equation (3.72). In the following discussions we will seethat this hyperbolic-type energy equation allow for an analytical solutions and yield a realistic descriptionof some relevant phenomena, for example, energy velocity.

3.2 Derivation by employing energy law

It has a long history to describe dynamics of particles, rigid body, and structures by energy variable,either in continuum mechanics or thermodynamics. So is for waves: energy is the actual kernel whichimplicitly embodied in the wave phenomenon although wave propagation is in many situations describedby a linear differential equation in terms of non-energy variables such as displacement, velocity, orpressure. While the fundamental energy law is universal, it can be a start point to derive a new energyequation to describe high frequency waves.

There are two energy variables usually concerned: energy density and energy flow. Energy densityis defined as the sum of the potential and the kinetic energy density,

W = T + U, and T = U = W (3.78)

where T and U are their envelop. Note that the latter in (3.78) requires an fundamental assumption

• H1: The energy density is defined as the quantity time-averaged over the oscillatory period.

Energy flow defined as the energy per unit time and unit section. Our study objective in this thesis is highfrequency structural/acoustic waves propagating in homogeneous media, and both energy density andenergy flow are generally defined in R

3 with space variable x = (x1 x2 x3). For the sake of analyticalsimplicity, we have to make other hypotheses:

• H2: The wave field is supposed to be linear in a bounded system.


x

forward wave W+backward W−

Figure 3.2: Wave designation into a simple waveguide

• H3: Interferences between waves are neglected.

These hypotheses imply that the different energy types are independent, which allow us to considereach wave mode (flexural wave, shear wave, and longitudinal wave, and etc.), respectively. Further, thesuperposition principle also becomes applicable.

Wave propagation is dependent on the elastic restore forces in media. The restore characteris-tics allows waves definitely consist of both forward and backward wave motion. An example of one-dimensional aveguide can be used to illustrate this characteristics, which is shown in Figure 3.2. How-ever, it should be noted this characteristics is not limited in one-dimensional waveguide.

Corresponding to the waves components, energy density includes both forward and backward energycomponent. Mathematically, the total energy density and energy flow are given by the superpositionprinciple:

W (x, t) = W+(x, t) +W−(x, t)

I(x, t) = I+(x, t) + I−(x, t)(3.79)

where + and − are used to represent the components of variables corresponding to forward and backwardwave.

Similar to the energy conservation law in thermodynamics, the mechanical energy also obey theuniversal conservation law [81]

∂W

∂t+ ∇ · I = Pin (3.80)

Note that eqaution (3.80) is only valid for isolated conservative system. A realistic system is usuallycharacterized by dissipation.

The conservation condition is broken in this case, however, one can still find an energy balancerelation to replace the conservative law, that is,

∂W

∂t+ ∇ · I + πdiss = Pin (3.81)

where πdiss represents the dissipated power density.

For energy propagating in infinite systems, the relation between energy density W and energy flowI is simple

I = cW (3.82)

Thus equation (3.80) and (3.81) can be easily solved. In bounded system, however, the reflection ofwaves result in an implicit relation between energy densityW and energy flow I. Therefore, the problemis now turned to determine this relation in bounded system.


3.2.1 Energy in undamped systems

Let us first consider an undamped wave field. As discussed above, the energy conservation law holdsrespectively for forward and backward wave,

∂W

∂t+ ∇ · I = 0 ⇒

∂W+

∂t+ ∇ · I+ = 0

∂W−

∂t+ ∇ · I− = 0

(3.83)

It is known [82, 83] that for a pure propagating wave, a constitutive relation exists between the energyflow and the energy density:

I+(x, t) = +cW+(x, t)I−(x, t) = −cW−(x, t)

(3.84)

where c is the group velocity of waves. Introducing (3.84) into the power balance (3.83) leads to partialdifferential equations in term of the partial energy densities.

∂W+

∂t+ c∇W+ = 0

∂W−

∂t− c∇W− = 0

(3.85)

This is exactly the energy equation obtained by Xing and Price for longitudinal wave in simple semi-infinite rods which is derived by other way. In their papers [33, 34] the authors started from the governingequations of small disturbances in a continuum elastic medium and derive equation (3.85) without anysimplification.

Inserting equations (3.84) into (3.85), we find that:

∂I+

∂t+ c2∇W+ = 0 (3.86)

∂I−

∂t+ c2∇W− = 0 (3.87)

By applying the linear superposition relationships (3.79), equation (3.86) plus (3.87) yields:

∂I(x, t)∂t

+ c2∇W (x, t) = 0 (3.88)

At this stage, one can clearly confirm that the relationship relating the energy flux and the energy densityis of different kind when comparing it to the classical Fourier’s law (3.141), even for very simple vibratingsystems.

Finally, by combining the relationship (3.88) into the energy balance equation (3.83), it yields:

∂2W (x, t)∂t2

− c2∇2W (x, t) = 0 (3.89)

Equation (3.89) describes the energy propagating in non damping system. It is a wave type equation


3.2.2 Energy in damped systems

The similar procedures of derivation of the energy equation for damped systems was carried out asfollows. The dissipative factor was taken into account. Once again from the local energy balance (3.81)for damped systems. For a particular traveling wave solution, the power balance (3.81) may still beapplied,

∂W+(x, t)∂t

+ ∇ · I+(x, t) + π+diss(x, t) = 0

∂W−(x, t)∂t

+ ∇ · I−(x, t) + π−diss(x, t) = 0

(3.90)

The damping model adopted here is the same as that in SEA: dissipated power density is proportional tothe energy density, that is

π±diss(x, t) = ηωW±(x, t) (3.91)

where η is the damping loss factor and ω the center frequency of a frequency band. The validity of thisrelationship has been discussed in the literatures [52, 50, 82, 14].

The constitutive relation between energy flow I and energy density W for a pure propagating waveare identical to (3.84),

I±(x, t) = ±cW±(x, t) (3.92)

Inserting (3.91, 3.92) into the power balance equations (3.90), it yields:

∂I+(x, t)∂t

(x, t) + c2∇W+(x, t) + ηωI+(x, t) = 0

∂I−(x, t)∂t

(x, t) + c2∇W−(x, t) + ηωI−(x, t) = 0

(3.93)

Adding the forward and backward components in equations (3.93) we obtain,

∂I(x, t)∂t

+ c2∇W (x, t) + ηωI(x, t) = 0 (3.94)

⇒ I(x, t) = − c2

ηω∇W (x, t) − 1

ηω

∂I(x, t)∂t

(3.95)

At this stage, we introduce again the thermal-like constitutive relation employed in Nefske’s equation(2.7)

I = − c2

ηω∇W (x, t) (3.96)

By comparing (3.96) with (3.2.2) , we found that the former lacks a time-derivative term of I. In fact,the relationship (3.2.2) generalizes the Fourier’s law (3.141) by adding the time-derivative term revealedby the relationship (3.88).

Deriving the local power balance equation (3.81) respect to time leads to the expression:

∂2W (x, t)∂t2

+ ηω∂W (x, t)

∂t− c2∇2W (x, t) − ηω∇ · I(x, t) = 0 (3.97)


wave fronts

wave vector

λ|k|

Figure 3.3: Illustration of plane wave in two dimensions

where ∂I/∂t has been evaluated from (3.2.2).

Finally the energy equation is obtained from (3.97) by substituting the expression of ∂I/∂x from theenergy balance (3.81)

∂2W (x, t)∂t2

− c2∇2W (x, t) + 2ηω∂W (x, t)

∂t+ (ηω)2W (x, t) = 0 (3.98)

Equation (3.98) is the energy equation representing the time and space evolution, which is identical toMEST equation (3.47) proposed in section 3.1.2. This equation is a telegraph type equation and it gen-eralizes the wave-like energy equation (3.89) obtained for undamped systems.

3.2.3 Time-varying energy equation for plane waves

In this section we consider time-varying energy of plane waves which is fairly important in structuralvibration and acoustics. Plane waves are characterized by their wavefronts being everywhere parallelplanes normal to the direction of propagation. The sketch of plane wave in two dimensions is shown inFigure 3.3, where the wave fronts are constant phase surfaces separated by one wavelength and the wavevector is normal to the wave fronts, its length is the wavenumber. Suppose a plane wave propagates in thestructure, in which there exists relation between energy flow I and energy densityW in wave propagatingphenomenon

I = cWn (3.99)

where c is the energy velocity, the same as the group velocity of the plane wave in the slight dampingstructure. n is a normalized vector indicating the wave direction (in Figure 3.3, n = k/|k|).

The gradient of the energy density can be got as

gradW =∂W

∂xn (3.100)


The local power balance obeying conservation principle can be expressed as

∂W (x, t)∂t

+ πdiss + div I(x, t) = 0. (3.101)

where x is the spatial variable in R3. The dissipated energy is still described by a heristic damping

model,πdiss = ηωW (x, t) (3.102)

where η is the damping loss coefficient and ω is the center frequency of a narrow frequency band.

By inserting equation (3.99, 3.102) into the power balance equation (3.101), it yields:

∂W (x, t)∂t

+ cn · gradW (x, t) + ηωW (x, t) = 0 (3.103)

Then substituting the equation (3.100) into above yields

∂W (x, t)∂t

+ c∂W (x, t)

∂x+ ηωW (x, t) = 0 (3.104)

Substituting the energy density W by energy flow I according to(3.99) gives

∂I(x, t)∂t

+ c2 gradW (x, t) + ηωI(x, t) = 0 (3.105)

Equation (3.101) and (3.105) consist of simultaneous equations which are of the same form as the basicacoustics equation. Differentiating equation (3.105) and equation (3.101) with respect to spatial and withrespect to time, respectively, then adding them, leads to the expression:

∂2W (x, t)∂t2

− c2∇2W (x, t) + 2ηω∂W (x, t)

∂t+ (ηω)2W (x, t) = 0 (3.106)

Equations (3.106) can be served as a theoretical basis of transient simplified energy method (MEST).Thus from now on, we temporarily call (3.106) MEST equation for a plane wave. Obviously equation(3.106) can be reduced in case of the damping loss effect being neglected,

∂2W (x, t)∂t2

− c2∇2W (x, t) = 0. (3.107)

Note that equation (3.107) and equation (3.106) can be transformed respectively to discrete format whichare discussed lately.

3.3 Boundary conditions

A closed form of the partial differential equation (3.106) needs both initial and boundary conditions. Theinitial condition

W (x, t)|t=0 = W0(x), and∂W

∂t|t=0 = I0(x) (3.108)

physically depends on external excitation, while the boundary condition are associated with the structureitself. Here only the boundary condition for a one-dimensional structure is given. The two-dimensionand three-dimensional complex boundary conditions will be treated in considerable detail in chapter 6.

3.3. Boundary conditions 67

xx = 0 x = L

I+(x, t) I−(x, t)

Figure 3.4: Bounddary condition illustrated by a simple waveguide

The simplest boundary conditions, if the extremities are non dissipative, for a waveguide with lengthL are:

I(0, t) = 0 (3.109)

I(L, t) = 0 (3.110)

Applying equation (3.79) and (3.84), one can manipulate the boundary conditions shown in figure 3.4into a form in term of energy density:

At boundary x = 0 : W+(0, t) = W−(0, t),At boundary x = L : W+(L, t) = W−(L, t);

(3.111)

or,∂W

∂x|x=0 = 0,

∂W

∂x|x=L = 0. (3.112)

In the presence of absorbing boundaries shown in Figure 3.4, the absorption coefficient α (0 ≤ α ≤ 1) isintroduced. Another related parameter is reflection coefficient, R = 1 − α, which is defined as the ratioof reflected energy flow to incident energy flow. Thus, for the left extremity located at x = 0

I+(0, t) = −(1 − α)I−(0, t) (3.113)

The net or total energy flow is then:

I(0, t) = I+ + I− = αI−(0, t) (3.114)

Consequently, the boundary condition (3.84) in term of energy densities can be rewritten accordingto (3.113)

W+(0, t) = (1 − α)W−(0, t) (3.115)

and for total energy density one obtains

W (0, t) = W+ +W− = (2 − α)W−(0, t) (3.116)

By comparing (3.114) and (3.116), a local relationship involving total energy quantities solely is derived,

I(0, t) = − c α

(2 − α)W (0, t) (3.117)

Proceeding in similar fashion for the right boundary x = L with a reverse sign, we obtain now twoboundary conditions for absorbing extremities.

I(L, t) =c α

(2 − α)W (L, t) (3.118)


Indeed, the case α = 0 matches the conditions (3.109, 3.110). Of special interest is the case α = 1yielding

I(0, t) = −cW (0, t) I(L, t) = cW (L, t) (3.119)

In fact, equation (3.119) are the description of special boundaries, from which there is no reflection atall. They make the system with such boundaries seem to be an infinite one, as shown in (3.84).

3.4 Properties of MEST equation and comparison with the diffusion en-ergy equation

To better understand the physical meaning of MEST equation, we have to examine carefully its proper-ties. While the derivation itself in last section shows some characteristics of MEST equation, we do wishto explore the properties embodied in this mathematical model, because the MEST equation (3.98) is anew time-varying description of vibroacoustic energy rather than the diffusion energy equation (1.94).

3.4.1 Property of MEST equation

MEST equation (3.98) is rearranged in the followed form:

−2ηω∂W (x, t)

∂t− (ηω)2W (x, t) =

∂2W (x, t)∂t2

− c2∂2W (x, t)

∂x2(3.120)

Denote

B ∂W (x, t)∂x

, C ∂W (x, t)∂t

,

D ∂2W (x, t)∂x2

, G ∂2W (x, t)∂x∂t

, H ∂2W (x, t)∂t2

then equation (3.120) can be written as

−2ηωC − (ηω)2W (x, t) = H − c2D + 0 ·G (3.121)

and the total differential of B and C are

dB =∂B

∂xdx +

∂B

∂tdt = 0 ·H +Gdt+Ddx (3.122a)

dC =∂C

∂xdx +

∂C

∂tdt = Hdt+Gdx + 0 ·D (3.122b)

B and C are variables known as the initial and boundary conditions. As for the unknown variablesD, G,andH , they can be got by solve the simultaneous set of equations (3.121) and (3.122a, 3.122b), providedthat

∆ =

∣∣∣∣∣∣1 0 −c20 dt dxdt dx 0

∣∣∣∣∣∣ = 0 (3.123)

The requirement ∆ = 0 yields (dxdt

)2 = c2 (3.124)

3.4. Properties of MEST equation 69

M

A B

t

s

Figure 3.5: Characteristic line of MEST equation

Equation (3.124) has two real roots so that it is hyperbolic, and the two real roots corresponds to the tworeal characteristics lines, x + ct and x − ct are constant.

Denoting T = x + ct and S = x − ct and applying the chain rule for partial derivatives leads to:

∂W

∂t= − c

∂W

∂x+ c

∂W

∂T,

∂2W

∂t2=c2

∂2W

∂x2− 2c2

∂2W

∂x∂T+ c2

∂2W

∂T 2(3.125a)

∂W

∂x=∂W

∂x+∂W

∂T,

∂2W

∂x2=∂2W

∂x2+ 2c2

∂2W

∂x∂T+∂2W

∂T 2(3.125b)

Inserting these derivatives into MEST equation (3.120) gives

∂2W

∂x∂T= (

ηω

2c)∂W

∂T− (

ηω

2c)∂W

∂x+ (

ηω

2c)2W (3.126)

Equation (3.126) suggests one to search a general solution in form of

W (x, t) = F (x, t)ea(S−T ) (3.127)

Applying the chain rule for partial derivatives again and comparing with the equation (3.126), one findsthat a = ηω

2c .

Thus the general solution of MEST equation is

W (x, t) = F (x, t)ea(S−T ) = F (x, t)e−ηωt (3.128)

Its equivalent form isW (x, t) = F (x − ct)e−ηωt +G(x + ct)e−ηωt (3.129)

Its characteristics is shown in Figure 3.5.

From the discussions of the general solution presented above, one can deduces that the MEST equa-tion provides a reasonable description of the energy propagation within systems. In fact, solution (3.129)proves that the energy propagates with a finite velocity, namely, c. The justification of such property iswell known in physics [81].


Further, for a given initial energy density and its first temporal derivative given asW0(x) and W0(x),one can find the general expression of energy density in the system:

W (x, t) = e−ηωtW0(x − ct) +W0(x + ct)2

+e−ηωt

2c

∫ s+ct

s−ct(W0(ξ) + ηωW0(ξ))dξ (3.130)

Examination of equation (3.130) with the help of Figure (3.5) demonstrates that vibrational energyspreads in a finite velocity. According to (3.130), one readily finds that the energy density in a pointM(x, t) only depends on initial values at points A and B. We just try to express the properties of MESTequation in relatively simple mathematics, the oscillatory characteristics of telegraph equation can befound in [84].

3.4.2 Property of diffusion energy equation

A time-vary vibrational conductivity equation has been proposed by Nefske and Sung [12],

∂W

∂t− c2

ηω

∂2W

∂x2+ ηωW = 0 (3.131)

When applied to discrete structures like the subsystems in SEA models, equation (3.131 can be dis-cretized to ordinary differential equations by variational principle. The energy level in subsystem isviewed as a constant, the only remaining unknown variation is then with respect to time. The discretiza-tion of energy equations will be discussed in a later chapter devoted completely to comparing the differentenergy methods. Here we just followed the discretization process shown in [35].

Supposing that the energy density in each component Wi(x, t) is constant, one can define a newenergy variable Ei such that

Ei(t) =ΨWi(x, t)

νi(3.132)

where νi is the modal density of the ith component and Ψ is a geometric quantity which is denoted as

Ψ

L the length of one-dimensional structures,

S the area of two-dimensional structures,

V the volum of three-dimensional structures.

(3.133)

Keeping the term with respect to time in equation (3.131), namely, ∂W∂t unchanged, then the Reyleigh-

Ritz procedure yieldsdEi

dt+ ηωEi +

∑j

ωηijνi(Ei − Ej) = 0 (3.134)

Equation (3.134) is actually the energy equation employed by TSEA. Physically speaking, equation(3.134) has the same properties as equation (3.131).

To discuss the properties of diffusion energy equation (3.131), the method of Green’s function isemployed. Supposing that an infinite model subjected to an impulsive energy load δ(x − ξ)δ(t − τ),diffusion energy equation takes the form

− c2

ηω

∂2G(x, t)∂x2

+∂G(x, t)∂t

+ ηωG(x, t) = δ(x− ξ)δ(t− τ) (3.135)


1 W∆

−ε 0 +εx

W

Figure 3.6: The initial transient energy density input employed by diffusion energy equation

A double Laplace transform approach t→ s, x→ k leads to

˜G(k, s) = −ηωc2

· e−ξke−τs

k2 − ηωc2

(s+ ηω). (3.136)

Then inverse procedures (3.136) in two steps corresponding to the double-transform, yields

G(x/ξ, t/τ) =√ηω√

4πc2(t− τ)· e−[

ηω(x−ξ)2

4c2(t−τ)+ηωt]

H(t− τ). (3.137)

Specifically when τ = 0, Green’s function is got by

G(x/ξ, t) =√ηω√

4πc2t· e−[

ηω(x−ξ)2

4c2t+ηωt] (3.138)

Suppose that the initial transient energy density shown in Figure 3.6, is input at the spatial origin. Theinitial energy is a rectangular impulse with the amplitude ∆W and an interval 2ε, the energy density wasobtained by the diffusion equation

W (x, t) =∫ −ε

−∞G(x/ξ, t)W1 dξ +

∫ +ε

−εG(x/ξ, t)∆W dξ +

∫ +∞

+εG(x/ξ, t)W1 dξ (3.139)

It reduces asymptotically to the following expression:

W (x, t) = W1e−ηωt +

ε√ηω∆Wc√πt

e−( ηωx2

4c2t+ηωt) (3.140)

This expression shows an instantaneous effect of energy density varying ∀x and ∀t > 0. That meansdiffuse vibrational energy spreads in an infinite speed due to a localized initial perturbation at the origin.This result corresponds to the phenomenon of thermal motion governed, but it is not an appropriate inter-pretation for vibroacoustic energy propagated in structures. As a contrast, MEST equation demonstratesthat the vibrational energy propagates with a finite velocity c whose value is decided by the mediumitself.

3.4.3 Comparison of properties two kinds of energy equations

Mathematically speaking, MEST equation (3.98) is a hyperbolic partial differential equation while dif-fusion energy equation (3.131) is a parabolic one. The properties of them and the relations betweenthem are very interesting mathematical topics. In the book of Volpert [85], traveling wave solutions areeven sought by parabolic systems. The remarks that the problem of the repartition of vibrational energywithin absorbing systems is similar to the heat conduction problem can be found in [24]. In [12], this


x = 0

δ(x− 0, t− 0)

−∞ +∞

Figure 3.7: Waveguide under a unit energy source at x=0 and t=0

analogy is invoked to postulate that Fourier’s law (or Fick’s law) should be valid for vibrational energyflow, namely, the energy flow is proportional to the gradient of the energy density,

I(x, t) = − c2

ηωgradW (x, t) (3.141)

which predicts that the instantaneous appearance of a energy flow I stems from an energy gradient ∇W ,consequently, energy flow reaches a infinite transfer velocity. This physically unpleasant property hasbeen stressed repeatedly. Of course homogeneous media are concerned here, the diffusion phenomenacaused by strong scattering in inhomogeneous media is another subject (seen last chaper).

In the section 3.1-3.2, we have put forward an alternative energy flow equation,

∂I(x, t)∂t

+ c2 gradW (x, t) + ηωI(x, t) = 0 (3.142)

which is a Maxwell-Cattaneo equation. We will demonstrate in next subsection that equation (3.142) canbe asymptotically obtained by expand the time-delayed equation (3.141).

The properties presented in previous subsections 3.4.1-3.4.2 allow us now to usefully augment thediscussion of this subject by means of Green function. Green’s functions of equations (3.131, 3.98, and3.89) corresponding to the one-dimensional system represented in Figure 3.7 are summarized as follows:

for the wave-like energy equation (3.89),

G(x, t) =12cH(c2t2 − x2) (3.143)

. for the telegraph-type MEST equation (3.98),

G(x, t) =12ce−ηωtH(c2t2 − x2) (3.144)

for the diffusion energy equation (3.131)

G(x, t) =√ηωt

2c√πte

−ηω

4c2tx2

e−ηωt (3.145)

In these expressions, H denotes the Heaviside function.

The Green’s function (3.143) for the wave-like equation can be, as expected, obtained from theGreen’s function (3.144) of the telegraph equation when the damping loss factor η goes to zero.

In Figures 3.8, 3.9 and 3.10, Green’s functions are plotted versus x-axis for various time values. Infact, let

φ(t, x) =1

2c√πt

e−x2

4c2t (3.146)


−10 −5 0 5 100

0.5

1

1.5

s−axis

G(s

,t) a

mpl

itude

Wave equation (30)

Figure 3.8: Green’s function versus s-axis for the wave equation at various times values: solid line ct=1,dashed line ct=4 and dashdot line ct=9.

−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s−axis

G(s

,t) a

mpl

itude

Figure 3.9: Green’s function versus s-axis for the MEST equation at various times values: solid line ct=1,dashed line ct=4 and dashdot line ct=9.


−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s−axis

G(s

,t) a

mpl

itude

Figure 3.10: Green’s function versus s-axis for the diffusion equation at various times values: solid linect=1, dashed line ct=4 and dashdot line ct=9.

then

limt→0+

φ(t, x) → δ(x) (3.147)

It means that the energy expressed by Nefske’s (diffusion) equation rapidly decrease after t > t → 0+,it will be verified in the following chapters.

Under steady state conditions, no difference exists between the model of the diffusion equation(3.131) and the model (3.98) presented here. In this way, the equation (3.98) proposed in this thesisgeneralizes effectively the vibrational conductivity equation under steady state conditions widely studiedin the literature [35, 13, 25, 26]. However, when time and space domain are simultaneously involved,many fundamental differences appear between equations (3.131) and MEST equation (3.98).

From a pure mathematical point of view, it should be addressed that the energy equation (3.131) isof different nature than the two others (3.89, 3.98). In fact, the energy equation (3.131) is a parabolicequation whilst the equations (3.89, 3.98) are hyperbolic (if the time t is considered as being the secondcoordinate). More precisely, equations (3.89, 3.98) admits two families of real characteristics. Thed’Alembert’s solutions for these two equations are of the following form:

W (x, t) = F (x − ct) +G(x + ct) (3.148)

where F (x − ct) and G(x + ct) are functions representing the right- and left-traveling energy densitiesW+(x − ct) and W−(x − ct). Finally, the parabolic equation (3.131) has only one family of character-istics which leads to a drastically different solution behavior. In particular, the oscillatory properties oftheir fundamental solutions are different.

In addition, as stated in reference [81] for the heat flow analysis, diffusion equation results in infinitetransfer velocity. However, it is physically unsatisfactory. Thus in [81], a correction of the diffusionequation was proposed. That is, an additional term 1/c2(∂2./∂t2) is introduced. Of course, it is necessaryto examine whether a similar procedure can be introduced to deal with energy behavior. Without loss thegenerality, we start from a more general case, that is a three-dimensional version of telegraph equation,


which is written as∂2W

∂t2+ a

∂W

∂t+ bW − c2∇2W = δ(x)δ(t) (3.149)

Note that a, b, and c are the algebra coefficients, and they have not explicit physical meaning for thismoment. We will also derive an explicit Green’s function for (3.149). Suppose the initial condition are

W (0) = 0Wt(0) = 0

Thus Laplace transform of (3.149) is given by

(s2 + as+ b)W − c2∇2W = 1 (3.150)

The spatial Fourier transform U(s,k) of W (s,x) satisfies the equation

(s2 + as+ b+ c2k2)U = 1 (3.151)

The inverse Fourier transform gives

W (s,k) =1

irc2(2π)2

∫ ∞

∞keikr

β(s)/c2 + k2dk (3.152)

wherek = |k|, r = |x|, β(s) = s2 + as+ b (3.153)

The inverse Laplace transform yields

G(t,x) =e−

at2

4πrc2×

δ (t− r

c

)− r

√∆

c

√t2 − r2

c2

J1

(√∆

(t2 − r2

c2

)) if ∆ > 0

δ(t− r

c

)if ∆ = 0δ (t− r

c

)+

r√|∆|

c

√t2 − r2

c2

I1

(√|∆|

(t2 − r2

c2

)) if ∆ < 0

(3.154)

where J1 is first order Bessel function and I1 the first order modified Bessel functions of the first kind.And ∆ is denoted by

∆ b− a2

4(3.155)

It was found that the parameter ∆ actually controls the behavior of Green function. The three kindof Green function were depicted in Figure 3.11–3.13. In these figures, let t = 10, c = 10, then thewavefront should be occurs at r = 100. In fact, ∆ > 0 dominates the dispersive behavior, an oscillatorytail can be found in Figure 3.11, the oscillations are concentrated at the wavefront; while ∆ > 0, shownin Figure 3.13, represents predominantly dissipative cases and a monotone tail decaying from the sourcetoward the wavefront. The case ∆ = 0 is indeed the Green function of MEST equation, where

∆ b− a2

4= (ηω)2 − (2ηω)2

4= 0 (3.156)


0 20 40 60 80 100 120−1

−0.5

0

0.5

1

r

Gre

en fu

nctio

n G

(r,t)

Figure 3.11: Green function for the case ∆ > 0

0 20 40 60 80 100 120−1

−0.5

0

0.5

1

r

Gre

en fu

nctio

n G

(r,t)

Figure 3.12: Green function for the case ∆ = 0


0 20 40 60 80 100 120

0

0.2

0.4

0.6

0.8

1

r

Gre

en fu

nctio

n G

(r,t)

Figure 3.13: Green function for the case ∆ < 0

Figure 3.12 shows that impulse function δ(t− r

c

)propagates in the form of wavefront. Therefore,

∆ = 0, as the characteristics of MEST equation, is an equilibrium point.

Although equation (3.149) of the cases ∆ > 0 or ∆ > 0 is also of telegraph type, their evolutionbehavior have been changed, as shown in Figure 3.11 and 3.13.

In particular, the first derivative of the energy density with respect to time, namely the term ∂W/∂tappears in both the diffusion equation (3.131) and the telegraph equation (3.98). This term breaks thesymmetry with respect to time. This means that a different kind of solution is obtained, if the sign of thetime t is reversed. This non conservation of energy is accompanied by a continuous increasing changein entropy as time goes on.

While equation (3.131) is a “perfect” diffusion equation with respect to both time and space, itpredicts that the energy flow propagates instantaneously from a source to an observation point on thevibrating system. The infinite velocity of energy flow is obviously physically impossible. In anotherhand, the wave or the telegraph equation predicts a finite velocity of propagation of mechanical energyflow. Consequently, there are some lacks in the analogy between the vibrational energy flow and thethermal flow. This systematic analogy leads to misunderstanding of physical phenomenon.

3.4.4 Relation between MEST equation and diffusion approximation

We have discussed in detail the remarkable difference between Nefske’s (diffusion) equation and MESTequation in last subsection. Our present interesting now is turned to search the mathematical relationsbetween these two types description. We found that if, say, a time scale/delay technique is adopted, thesetwo equation can approximately approach to each other under specific conditions.

In the derivation of MEST equation (3.157), two energy components, the forward energy density and


the backward were employed in bounded system.

∂2W (x, t)∂t2

− c2∇2W (x, t) + 2ηω∂W (x, t)

∂t+ (ηω)2W (x, t) = 0 (3.157)

It means that MEST equation has a natural property to represent the total energy level including two kindsenergy components. If the damping is not very heavy, or say, with an assumption of slight damping, it isexpected that energy flow propagates in specific duration before complete dissipated.

We use a scaled time t/ε to represent the sufficient time taken by plenty of reflections. ε is sufficientsmall quantity. Then energy density is defined as

W ∗ = W

(t

ε,x

)(3.158)

Substituting W in equation (3.157) by W ∗, one obtains

ε2∂2W ∗

∂t2− c2∇2W ∗ + 2ηωε

∂W ∗

∂t+ (ηω)2W ∗ = 0 (3.159)

or,

ε∂2W ∗

∂t2− c2

ε∇2W ∗ + 2ηω

∂W ∗

∂t+

(ηω)2

εW ∗ = 0 (3.160)

Since ε is sufficiently small, the term ε∂2W ∗∂t2

can be approximately neglected. Thus the hyperbolic energyequation reduces to a parabolic diffusion equation

∂W ∗

∂t− c2

2ηωε∇2W ∗ +

ηω

2εW ∗ = 0 (3.161)

It was found that equation (3.161) is equivalent to Nefske’s (diffusion) equation (3.131). At this stage,diffusion energy equation (3.161) can be viewed as a special case of hyperbolic energy equation (3.98).

Moreover, equation (3.161) was rewritten in the form

2ε∂W ∗

∂t− c2

ηω∇2W ∗ + ηωW ∗ = 0. (3.162)

Clearly, if ε → 0, then in equation (3.161), 2ε∂W ∗∂t → 0. Equation (3.162) reduces again to an elliptic

equation,

− c2

ηω∇2W ∗ + ηωW ∗ = 0 (3.163)

which happened to be the local energy equation to describe vibroacoustic energy in steady state field.

In addition, from equation (3.159), one found that ηω plays a same role as ε. If damping coefficientη is fixed, the higher frequency ω will make the MEST solution to be closer to the result of diffusionequation.

On the other hand, motivated by the unphysical feather of the diffusion equation (causing an infiniteenergy velocity), one have to revise the expression of energy flow formula (3.96) to satisfy the physicalinterpretation. As discussed early in section 2.7 of chapter 2, one can use a delay time to modify equation(3.96) or employ a flux kernel (2.82) to precisely define the relation between energy flow I and energygradient ∇W .

3.5. Analysis of the solution of MEST equation 79

Of special interest is in equation (2.82), which can be rewritten as

I(x, t) = −c2eηωt

∫ t

0

∇W (x, s)eηωs

ds (3.164)

DefineI(x, t) I(x, t)e−ηωt, W (x, t) W (x, t)e−ηωt (3.165)

then equation (3.164) is replaced by

I(x, t) = −c2∫ t

0∇W (x, s)ds (3.166)

Derivative of equation (3.166) with respect to time t, one finds another equivalent expression

∂I(x, t)∂t

= −c2∇W (x, t) (3.167)

The equation (3.167) is the same as (3.88). By the similar technique, one can transform Nefske’s (diffu-sion) equation to MEST equation.

The discussion in this subsection just gives some ideas to understand the difference and relationbetween two kind of energy equation. The results got here will be observed explicitly in followingchapters.

3.5 Analysis of the solution of MEST equation

The MEST equation defined in R3 is in the form of hyperbolic system

∂W

∂t+ ∇ · I + ηωW = 0 (3.168)

∂I∂t

+ c2∇W + ηωI = 0 (3.169)

If we assume that the energy velocity is same in isotropic media in the direction of three coordinatexi = x1, x2, x3, then equation (3.168, 3.169) can be written as

1 0 0 00 1 0 00 0 1 00 0 0 1

∂

∂t

WIx1

Ix2

Ix3

+

0 1 0 0c2 0 0 00 0 0 00 0 0 0

∂

∂x1

WIx1

Ix2

Ix3

+

0 0 1 00 0 0 0c2 0 0 00 0 0 0

∂

∂x2

WIx1

Ix2

Ix3

+

0 0 0 10 0 0 00 0 0 0c2 0 0 0

∂

∂x3

WIx1

Ix2

Ix3

+ ηω

1 0 0 00 1 0 00 0 1 00 0 0 1

WIx1

Ix2

Ix3

= 0

(3.170)

If we denote

W =

WIx1

Ix2

Ix3

, B = ηω

1 0 0 00 1 0 00 0 1 00 0 0 1

(3.171)


then a concise form of equation (3.170) is shown

A∂W∂t

+n∑

i=1

Di∂I∂xi

+ BW = 0 (3.172)

Note that the matrix Di in equation (3.170) is non-symmetric, however, it is possible to transform it intoa symmetric one.

The characteristic equation of equation (3.172) is

det|Di − λI| = 0 (3.173)

where I is the special unit matrix corresponding to the Di. It is easy to find that, all the solutions ofequation (3.173) corresponding to each Di are identical, i.e. they are

λ = ±c (3.174)

In equations (3.168, 3.169), we make a linear transform as

W∗(1,2) = I ± cW, that is

W∗

1 = I + k+cW

W∗2 = I − k−cW

(3.175)

where ±k correspond to the directions of velocity c. Hence equations (3.168, 3.169) can be rewritten by

∂W∗1

∂t+ c∇ · W∗

1 + ηωW∗1 = 0 (3.176)

∂W∗2

∂t− c∇ · W∗

2 + ηωW∗2 = 0 (3.177)

or in the combined form

A∗∂W∗

∂t+ D

∗∂W∗

∂x+ B

∗W∗ = 0 (3.178)

where

A∗ =

[1 00 1

], D

∗ =[c 00 −c

], B

∗ = ηω

[1 00 1

](3.179)

Matrices A∗ and D

∗ are symmetric and diagonal, and equations (3.176, 3.177) are uncoupled, W∗1 and

W∗2 can be resolved independently by characteristic curves [86].

3.6 Summary and comments

In this chapter, a new energy method and the underlying energy equation defined in space temporal do-main has been derived. The derivations were carried out from two different view point: general waveequations and fundamental energy relation. The former were classified by two types of wave patten:hyperbolic waves and dispersive waves. For hyperbolic system, high frequency limit of vibroacousticenergy was obtained by Wigner (matrix) measure and the high frequency waves propagating in inhomo-geneous media can be described by transport equation. Although the subject of transport equation forelastic waves had been involved in some references, the originality in this thesis is to append the dissipa-tive term to the conventional Liouville equation. This improvement is important for vibroacoustic fieldbecause the damping effect is the key characteristics. For the dispersive waves, the wave group concept

3.6. Summary and comments 81

was stressed. The main idea of derivation of MEST equation for dispersive waves is define the wavegroup in phase space. The same energy equation were found for both hyperbolic and dispersive waves.On the other hand, we can also make use of the fundamental energy balance equation to derive the newtime-varying energy equation. The linear assumption and superposition principle were applied.

It was found that MEST equation can be extract from transport equation. However, for vibroacousticfield, transport theory could not be applied directly. There are some shortcomings in transport equationworthy to be notified:

1. It just expresses the one-way (single direction) energy propagation, the difference between for-wards and backward wave is not considered. In convensional transport theory, the energy associ-ated with ±ω are treated as the same amplitudes.

2. It does not consider energy loss in propagation. In fact, it does not distinguish the energy dampingdissipation from energy exchangeed by scattering.

3. It just describes the energy propagation in plane wave patten, not involving with symmetricalwaves.

The propagation of high frequency vibroacoustic energy can always be expressed by a hyperbolic system,for either hyperbolic waves or dispersive waves. MEST does not limit itself in plane waves, it coversso-called symmetrical waves which will be discussed in next chapter.

The properties of MEST equation have been discussed in detail by comparing with Nefske’s (diffu-sion) energy equation. Green function was employed to carry out these discussion and comparison. Theproperties of MEST equation provide a reasonable interpretation of energy velocity.

Chapter 4

Time-varying vibration energy in discretesystems

We have completed the theoretical preparation of Time-varying Simplified Energy Method (MEST) anddiscussed its properties in last chapter. A mathematical model has been developed. The following workis to verify its accuracy by numerical analysis and comparison with other existed methods, especiallywith “exact” results. Such validation procedures will be held by applying MEST to several kinds ofstructures and systems which are simulated to be subjected to shock excitations. The numerical studieson time-varying vibrational energy consist of two parts. The first involves the discrete systems and it isthe objectives of our study in this chapter. We frequently need to discretize the physically distributedstructures/systems, not only because the partial differential equation that describe such situations areoften unsolvable, but also because the practical configurations can often be viewed as the assemble ofsubsystems at the macroscopic level. It suggests us to break large problems into pieces of manageablesize. It is somewhat similar to the idea in SEA: dividing whole system into subsystems. The second partaims to distributed structures which will be discussed in next chapter.

4.1 The case of two oscillators

To illustrate and describe the vibrational energy motion in the structure, it may prove useful and instruc-tive to first consider the most elementary dynamic system. The two oscillators system is a typical oneof discrete structures, which had been also dealt with in SEA as its beginning stage [14] to explore theenergy coupling relation between two oscillators, and the basis of statistical energy analysis is the cor-respondence in steady response between two coupled energy elements with the exact response of twocoupled single-degree-freedom systems. Physically, more complex discrete systems can be modeled bysuperposing or combining with the simplest discrete units.

We are going to develop a technique to simplify the energy analysis. To a great extent, the vibra-tional energy transferring motion consists of two divisions: energy is dissipated by damping (mechanicalenergy is converted to heat energy) and energy is exchanged between subsystems by coupling element.While the mechanisms of two kinds of energy motion are certainly different in time domain, thus onecan treat these two energy motions as mutually independent process. As a consequence, one can firststudy the case without damping to search the clue for building up the energy equation with accurate

83

84 Time-varying vibration energy in discrete systems

k1 k2k

C1C2

m1m2

F1 = δ(t)

x1(t) x2(t)

Figure 4.1: Two oscillators model

coupling relation, then the more complex case with damping will be studied on the basis of the formerresult. In one word, we are allowed to first analyze the partial energy motion then synthesize them. Suchan idea is reasonable from the point of view of trying to separate a complicated dependence into easilyunderstandable components.

In order to find the exact energy description, it is imperative to get an analytical solution of motionequation. While the absolutely analytical solution without any assumption and simplification has notbeen found by the author, in what followings we devote to proposing a method to find the closed-formanalytical expression and it is expected to find something new from the exact solutions.

4.1.1 Energy equation of two undamped oscillators system

Consider two oscillators coupled by a spring with an arbitrary stiffness which is shown in Figure 4.1. Atthe first step, let C1 = 0 and C2 = 0 as discussed above. Other parameters such as m1, m2, k1, k2 canbe input any values. As shown in Figure 4.1, an initial unit impulse excitation is applied to mass m1, theinitial energy is got from the initial velocity response

v1(0) =1m1

, then the initial energy is E1(0) =1

2m1(4.1)

The analytical displacement and velocity responses are sought by Laplace transform method (see theAppendix A).

Introduce

sin θ =2Rω1ω2

ω21 + ω2

2

, θ ∈ (0,π

2) (4.2)

where R2 = 1 − k2

(k1+k)(k2+k) .

4.1. The case of two oscillators 85

Thus the velocities

x1(t) =I1

[(ω2

1 − ω22) + (ω2

2 + ω21) cos θ

](ω2

1 + ω22) cos θ

×

cos

(cos( θ

2) − sin( θ2)

2

√ω2

1 + ω22 t

)cos

(cos( θ

2) + sin( θ2)

2

√ω2

1 + ω22 t

)(4.3)

x2(t) =2I1r22ω

22

(ω21 + ω2

2) cos θ×

sin

(cos( θ

2) − sin( θ2)

2

√ω2

1 + ω22 t

)sin

(cos( θ

2) + sin( θ2)

2

√ω2

1 + ω22 t

)(4.4)

where I1 = 1/m1. For two oscillators system, we apply the assumption shown in 3.78, i.e. the kineticenergy envelope is viewed as total energy of a single oscillator, thus

Ei(t) =12mi(ˆxi)2 (i = 1, 2) (4.5)

where ˆx1 and ˆx2 are the envelopes of velocity responses. While cos( θ2) + sin( θ

2) > cos( θ2) − sin( θ

2) in(4.3, 4.4), the envelops of velocities ˆx1 and ˆx2 may be written

ˆx1(t) =I1

[(ω2

1 − ω22) + (ω2

2 + ω21) cos θ

](ω2

1 + ω22) cos θ

cos

(cos( θ

2) − sin( θ2)

2

√ω2

1 + ω22 t

)(4.6)

ˆx2(t) =2I1r22ω

22

(ω21 + ω2

2) cos θsin

(cos( θ

2) − sin( θ2)

2

√ω2

1 + ω22 t

)(4.7)

Moreover, the energies expressions above based on the analytical terms of ˆx1 and ˆx2 can be rearrangedin the concise formalism: E1(t) = M +N +Neiωt

E2(t) = N −Neiωt(4.8)

where i =√−1, and

M =12m1I

21 (ω2

1 − ω22)

2

(ω21 − ω2

2)2 + 4k2

m1m2

, N =I21

k2

m2

(ω21 − ω2

2)2 + 4k2

m1m2

, ω =(cos( θ

2) − sin( θ2))√

ω21 + ω2

2,

Note that in equation (4.8), M + 2N stands for the steady-state energy level of oscillator m1 and 2Nfor oscillator m2. ω is not equal to the definition of center frequency ωc =

√ω1ω2. In additional, the

time-varying energy is conservative

E1(t) + E2(t) =12m1I

21 = E1(0) (4.9)

We now examine the first and second derivatives of energies with respect to time variable

dE1

dt= −dE2

dt= iNωeiωt,

d2E1

dt2= −d2E2

dt2= −Nω2eiωt (4.10)


By observing the expressions of energies and their derivatives listed in equation (4.8, 4.11), one caneasily obtain two sets of different equations:

dE1

dt+ α1E1 − α2E2 = 0

dE2

dt+ α2E2 − α1E1 = 0

(4.11)

where

α1 = −12iω(1 − M

M + 2N), α2 = −1

2iω(1 +

M

M + 2N)

or d2E1

dt2+ β1E1 − β2E2 = 0

d2E2

dt2+ β2E2 − β1E1 = 0

(4.12)

Where the coupling coefficients β1 and β2 are

β1 =12ω2(1 − M

M + 2N) =

2k2ω2

m1m2(ω21 − ω2

2)2 + 4k2, (4.13)

β2 =12ω2(1 +

M

M + 2N) = ω2 − β1. (4.14)

Moreover, the energy partition can be described by partition ratio λ

λ1 =β1

ω2, λ2 =

β2

ω2= 1 − λ1 (4.15)

They have definite physical meanings, for example, λ1 indicates the proportion of transmitted energy E2

in the total energy level.

At first glance, the first-order equations (4.12) seem to be transient SEA equations without dampingloss effect, however, α1 and α2 are unexpectedly imaginary values which are physically inconsistent. Itmeans that one cannot use a first-order system to describe the energy motion in a two oscillators systemunless the non-real coupling loss factors are used. On the other hand, a second-order system (4.14) isfortunately found suitable to establish the energy relation between two oscillators.

4.1.2 Numerical analysis on undamped two oscillators

Physically, α1 and α2 in (4.12) or β1 and β2 in (4.14), which are the functions of structural parametersin terms of the mass, the blocked frequency and the stiffness, play the same role as coupling loss factorsin SEA. The coupling strength and energy partition were investigated in a numerical study, for a range ofparameters shown in Table 4.1: varying masses (m1, m2) and springs stiffness (k1, k2, and k). Input andtransmitted energies of the two oscillators with parameters in Table 4.1 were calculated by using equation(4.8). Both Case A and B are two identical oscillators, while case A has a strong coupling strength(β1 = 200.48) and case B has a weak one (β1 = 42.37). The both partition ratio read 0.5, which meansvibrational energy exchange completely between two oscillators. Figure 4.2 and 4.3 correspond to case Aand case B, respectively. One can also see, however, the “speed” of energy exchange are different–energyexchange in case A much more quickly than in case B. In fact, the “speed” of energy exchange only


Mass (kg) Spring (N/m) Coupling Frequency Oscillatory Partition ratioTest m1 k1 stiffness ω1 (rad/s) frequency λ1

m2 k2 k (N/m) ω2 (rad/s) ω(rad/s) λ2

A 4 12 × 105 12 × 105 774.6 400.96 0.54 12 × 105 774.6 0.5

B 4 12 × 105 2 × 105 591.6 84.73 0.54 12 × 105 591.6 0.5

C 4 10 × 105 11 × 105 724.57 574.14 0.302 12 × 105 1072.4 0.69

D 2 12 × 105 6 × 105 948.68 421.31 0.214 10 × 105 632.46 0.79

E 2 12 × 105 2 × 105 836.66 307.34 0.054 10 × 105 547.72 0.95

Table 4.1: Parameters and data for study of the coupling strength and energy partition

depends on the different energy oscillatory frequency ω. Similarly, the rise time of receiver oscillator(m2), which is defined as the time taken to reach the (first) peak energy value of receiver oscillator, canalso be expressed by ω:

tr =π

ω(4.16)

Thus, the energy oscillatory frequency ω is higher, the rise time is shorter. Figure 4.4 and 4.5 correspondto case C and case D. The two oscillators are different, m1 = m2, ω1 = ω2. The energy partitionindicators, λ1,2, are actually decided by coupling strength and the difference of two block frequencies∆ω = |ω1 − ω2|. Qualitatively speaking, a bigger ∆ω and a smaller coupling stiffness k cause lesspartition ratio λ1. Hence in case E, a rather small energy partition ratio λ1 = 0.5 occurs. It means thatthe transmitted energy E2 is 5% of total energy E1(0). The energy flow from oscillator m1 to m2 in allthese cases also represent energy transmission which is shown in Figure 4.7.

4.1.3 Time-varying energy equation for discrete structures

While the equations of two undamped oscillators have been analytically developed, the next task is seek ageneral form taking damping effect into account. Unfortunately, it is nearly impossible to find an explicitenergy relation for two damped oscillators system by means of a similar analytical procedure applied inlast subsection. However, MEST equation (3.98) derived in chapter 3 is a partial differential one, onemay apply it to a discrete system by FEM technique. Before involved within this topic, we at first try tosynthesize both the energy dissipation and energy transmission between two damped oscillators to searcha general equation.

For two oscillators, the equivalent global (overall) loss factor [50] ηeq can be written as,

ηeq =η1E1 + η2E2

Etotal(4.17)

then the energies of subsystem may be approximaterly represented in followed form with exponentialdecaying

E1d(t) = E1(t)e−ηeqt, E2d(t) = E2(t)e−ηeqt (4.18)


0 0.02 0.04 0.06 0.080

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (s)

Ene

rgy

(J)

Figure 4.2: Energy motion and partition, caseA. ——, input energy E1, – – – –, transmittedenergy.

0 0.02 0.04 0.06 0.080

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (s)

Ene

rgy

(J)

Figure 4.3: Energy motion and partition, caseB. ——, input energy E1, – – – –, transmittedenergy.

0 0.02 0.04 0.06 0.080

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (s)

Ene

rgy

(J)

Figure 4.4: Energy motion and partition, caseC. ——, input energy E1, – – – –, transmittedenergy.

0 0.02 0.04 0.06 0.080

0.05

0.1

0.15

0.2

0.25

Time (s)

Ene

rgy

(J)

Figure 4.5: Energy motion and partition, caseD. ——, input energy E1, – – – –, transmittedenergy.


0 0.02 0.04 0.06 0.080

0.05

0.1

0.15

0.2

0.25

Time (s)

Ene

rgy

(J)

Figure 4.6: Energy motion and partition, caseE. ——, input energy E1, – – – –, transmittedenergy.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08−30

−20

−10

0

10

20

30

Time (s)

Ene

rgy

flow

(W

)

Figure 4.7: Comparison of energy flows. – – –,case A; – · – ·, case B; ——, case E.

One can get the first- and second-order derivatives of E1d and E2d. By substituting equation (4.20) intothese derivatives, it leads to

d2E1d

dt2+ 2ηeq

dE1d

dt+ η2

eqE1d + β1E1d − β2E2d = 0

d2E2d

dt2+ 2ηeq

dE2d

dt+ η2

eqE2d + β2E2d − β1E1d = 0

(4.19)

Equations (4.21) are the standard second-order ordinary differential equations, and obviously, they differcompletely from the first-order TSEA equations (1.97 or 3.134).

Rigorously speaking, it is just a special example to find the time-varying energy equation for twooscillators system by utilizing the analytical expression of responses. In fact, for almost all the discretestructures/systems subjected to transient excitations, the governing energy equation should be sought bymeans of discretizing MEST differential equation. Rather than a general FEM technique used in lastchapter (chapter 6), a zero-order finite element technique is selected to apply the MEST equation to thediscrete structures/systems. Figure 4.8 shows an example of discrete model. For the sake of brevity,only one-dimensional substructures are considered here. In Figure 4.8, the coupled substructures areequivalent to two coupled oscillators if we use zero-order elements, which means the energy in eachcomponent is constant so that each node stands for an element. Thus the concept of total energy ratherthan energy density is employed in MEST differential equation (3.106),

∫V

[1ηω

∂2W (x, t)∂t2

− c2

ηω∇2W (x, t) + 2

∂W (x, t)∂t

+ ηωW (x, t)]

dV = Pinput (4.20)

the total energy flow term in equation (4.22) for a one-dimensional structure shown in Figure 4.9


Figure 4.8: Discrete model of FEM

∆ij Aij

Wi Wj

Li Lj

Figure 4.9: Boundary condition of two coupled elements

can be expressed as∫V

[− c2

ηω∇2W (x, t)

]dV = − c2

ηω

∫Aij

(n · ∇W )dA =c2

ηωAij

W1 −W2

∆12(4.21)

where Aij is the cross-sectional area connecting two element and V is the volume of element. Substitut-ing (4.23) into equation (4.22) and using the total energy in the element, Ei = AiLiWi, the zero-ordertransient elemental energy for element 1 will be

1ηiω

d2Ei

dt2+ 2

dEi

dt+ ηiωEi +

c2

ηiω∆ijL1Ei − c2

ηiω∆ijLjEj = Pinput (4.22)

The simplified form may be

1ηiω

d2Ei

dt2+ 2

dEi

dt+ ηiωEi + ηeijωEi − ηejiωEj = Pinput (4.23)

where coupling coefficients are defined as

ηeij =c2

ηiω2∆ijLi, ηeji =

c2

ηiω2∆ijLj(4.24)

Comparing with TSEA equation

dEi

dt+ ηiωEi + ηijω(Ei − Ej) = Pinput (4.25)

4.2. Comparison with reference values from other methods 91

one find that equation (4.25) differs from the TSEA equation (4.27), the former is a second-order tempo-ral equation while the latter is a first-order one. The numerical method to resolve equation (4.25) can befound in Appendix C.

4.2 Comparison with reference values from other methods

The preceding solution (4.24) is applied to a two oscillators system (Figure 4.1 with damping loss) andcompared with TSEA and exact result. Rather than the numerical analysis on undamped two oscillatorsin section 4.1.2, the two oscillators system includes damping loss effect in this section so that it is arelatively general model. Before making numerical comparison among these three methods, we shouldintroduce the exact energy result and the basic theory of transient statistical energy analysis (TSEA).

4.2.1 The exact energy expression of two oscillators systems

The equations of motion for two coupled oscillators subject to an impulse shown in Figure 4.1 are:

m1x1 + C1x1 + (k1 + k)x1 − kx2 = δ(t), (4.26)

m2x2 + C2x2 + (k2 + k)x2 − kx1 = 0 (4.27)

They can be rearranged as

x1 = − ω21x1 + p1

2x2 − ∆1x1 (4.28)

x2 = − ω22x2 + p2

2x1 − ∆2x2 (4.29)

where ω21 = k1+k

m1, ω2

2 = k2+km2

, ∆1 = η1ω1 = C1m1

, ∆2 = η2ω2 = C2m2

, p12 = k

m1, and p2

2 = km2

.

Let X = x1 x2 x1 x2T and the state space method is employed, then the equations can bewritten in the form of matrix

dX

dt= AX (4.30)

where

A =

0 0 1 00 0 0 1

−ω21 p1

2 −∆1 0p2

2 −ω22 0 −∆2

(4.31)

The initial values of matrix equation (4.32) is

Xt=0 = 0 0 1m1

0T . (4.32)

One can deal with the eigenvalue problem associated with matrix A. Supposed D = d1 d2 d3 d4T

and V = V1 V2 V3 V4T are the eigenvalues and eigenvectors of matrix A, thus the solution ofequation (4.32) can be expressed as

X =4∑

i=1

ciViedit, i = 1, . . . 4. (4.33)


where the coefficient ci is got by

C = c1 c2 c3 c4T = V−1

Xt=0. (4.34)

In addition, one can also apply Laplace transform to seek the exact responses of two oscillators (seeAppendix B). Once xi and xi are known, the exact energy results of each oscillator is conventionallyobtained as the sum of kinetic and potential energy,

Ei(t) =12mi(xi)2 +

12(ki + k)(xi)2, (i = 1, 2). (4.35)

However, the definition (4.37) is just an approximation because the exact potential energy of each oscilla-tor is impossible to expressed clearly due to the coupling stiffness. Obviously, the potential energy storedin coupling stiffness is calculated twice in (4.37) — the exact total potential energy for two oscillatorsare

Ep(t) =12k1(x1)2 +

12k2(x2)2 +

12k(x1 − x2)2, (4.36)

while in (4.37) the total potential energy is

Ep(t) =12k1(x1)2 +

12k2(x2)2 +

12k[(x1)2 + (x2)2]. (4.37)

It means that the energy expression (4.37) always overestimate the exact value. In the case of weakcoupling, this error is not remarkable, however, as the coupling strength becomes stronger and stronger,the “blocked” potential energy concept is no longer appropriate because the difference between (4.38)and (4.39) is more and more notable. Thus, the reliable exact energy expression may be the envelope ofkinetic energy

Ei(t) =12m1(ˆxi(t))2 (4.38)

Figure 4.10 shows the difference between energy expression (4.37) and (4.40). Hereafter, the envelopeof kinetic energy will represent the exact energy as a reference value.

4.2.2 Transient statistical energy analysis

Transient statistical energy analysis (TSEA) is the time varying version of SEA. A model of two-coupledsubsystem is shown in Figure 4.11. In TSEA, the power balance is

P i =dEi

dt+ ηiωEi + ηijωEi − ηjiωEj (i, j = 1, 2) (4.39)

where the damping loss factors are ηi = cimiωi

(i = 1, 2).

The set of linear equations (4.41) yields the solution [50, 51]

E1(t) =E1(0)

2be−aωt

[(D1

ω+ ηb)ebωt − (

D2

ω+ ηb)e−bωt

](4.40)

E2(t) =E1(0)

2bη12e

−aωt(ebωt − e−bωt) (4.41)

where

ηa = η1 + η12, ηb = η2 + η21, D1, D2 = −ω(a∓ b),

a = (ηa + ηb)/2, b =12

√(ηa − ηb)2 + 4η12η21.

4.2. Comparison with reference values from other methods 93

0 10 20 30 400

0.05

0.1

0.15

0.2

0.25E

nerg

y (J

)

0 10 20 30 400

0.05

0.1

0.15

0.2

Ene

rgy

(J)

Time (ms)

(a)

(b)

(a) Strong coupling

0 10 20 30 400

0.05

0.1

0.15

0.2

0.25

Ene

rgy

(J)

0 10 20 30 40−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Ene

rgy

(J)

Time (ms)

(a)

(b)

(b) Weak coupling

Figure 4.10: The difference of the exact energy expressions. Figures marked with (a): Input energy;figures marked with (b): transmitted energy. ——, envelope of kinetic energy; - - - - -, “blocked” energy

P1 P2

Ei Ej

ηijωEi

ηjiωEj

ηiωEi ηjωEj

Figure 4.11: Statistical energy analysis model for two systems.


4.3 Coupling loss factors and the rise time

The determining of coupling loss factors is one of major difficulties in SEA applications. For two-oscillators system, the coupling loss factor (CLF) η12 and η21 employed in TSEA (actually the same asCLF used in SEA for steady state condition) can be obtained by

η12ω = η21ω = β (4.42)

where ω is the average frequency in an excitation band, and

β =k2(∆1 + ∆2)

m1m2

[(ω2

1 − ω22)2 + (∆1 + ∆2)(∆1ω2

2 + ∆2ω21)] (4.43)

where ∆1, ∆2 are equal to c1 and c2, respectively. This expression (4.45) had been deduced in [14, 82].For MEST, we still follow the idea: the damping loss factor (DLF) and coupling loss factors (CLF) aremutual independent. Then substituting ξ with ηω in equation (4.21)

d2E1d

dt2+ 2ηω

dE1d

dt+ (ηω)2E1d + β1E1d − β2E2d = 0 (4.44)

In order to compare the expression of coupling loss factors (CLF) with TSEA, one rewrites 4.46 as

1ηω

d2E1d

dt2+ 2

dE1d

dt+ ηωE1d +

β1

ηωE1d − β2

ηωE2d = 0 (4.45)

thus nominal CLFs applied in MEST can be represented by

η12ω =β1

ηω(4.46)

Moreover, the expression of transmitted energy (energy of an indirectly excited subsystem) may befound by integrating (4.8) and (4.20),

E2(t) =(N −Neiωt

)e−ηωt (4.47)

then it is easy calculate the rise time tr and the peak value of transmitted time-varying energy Emax. LetdE2dt = 0 in (4.49) , thus

tr =π

ω=

π(cos( θ

2) − sin( θ2))√

ω21 + ω2

2

. (4.48)

It is found that the rise time is the same as (4.18). Subsequently the peak energy value is

Emax = 2Ne−ηωπ

ω (4.49)

Fahy, James and Ruivo [87, 88, 89, 90] found that the time delay to the peak kinetic energy levelof an indirectly excited SEA subsystem can be used as an indicator of coupling strength. Here, the risetime tr takes the same concept as “time delay” in [87]. They have investigated several different one- andtwo-dimensional coupled subsystems, however, they did not derive an explicit expression which indicateapparently the relation between time delay and CLFs–the time delay is just a simple, empirical indicator.With the help of explicit formula (4.50), we are able to draw some conclusion. Substituting (4.50) into(4.15), it yields

β1 =2k2π2

t2r[m1m2(ω2

1 − ω22)2 + 4k2

] (4.50)

4.4. Numerical study on two oscillators system 95

k1

C1

m1

F1

x1(t)

k

F2

x2(t)

m2

k2

C2

Figure 4.12: Kinematic two-degree-of-freedom model

It shows that the the coupling strength is inversely proportional to the square of rise time tr. Similarly,one can also denote it in the form of CLFs used in SEA:

η12ω =2k2π2

ηωt2r[m1m2(ω2

1 − ω22)2 + 4k2

] (4.51)

Specially for two oscillators with equal frequency, ω1 = ω2, then (4.53) was reduced to be

η12ω =π2

2ηωt2r. (4.52)

Although the relation (4.53, 4.54) were restricted to the two-oscillators system, it is promising to drawout a more general expression for other coupling structures (rods, beam, plates, and etc.).

4.4 Numerical study on two oscillators system

A wide range examples of two oscillators system are analyzed numerically in this section based on theabove-discussed theory. The major purpose consist of two points: validation of MEST and comparisonwith TSEA. While R. J. Pinnington and D. Lednik [50] have compared the transient energy responseof a two-degree-of-freedom system with that for a transient SEA two oscillators system, we are at firstgoing to realize their numerical test and join MEST results in the comparison. As the second step, thesimulation will be extended to more general cases where the block frequencies and damping factors don’tmatch.

4.4.1 Two oscillators with identical frequencies

In their study [50], R. J. Pinnington and D. Lednik chose a relative simple model, two equal oscillatorswhich have identical blocked natural frequencies but different masses. This simplification is justified bythe view that the main interest is resonant energy exchange between the modes in a moderately narrowbandwidth. In the two-DOF model employed in [50] shown in Figure 4.12, two coupled oscillatorshave the same blocked natural frequencies and damping, the coupling stiffness, k and damping factor(η1 = η2) of each oscillator are variables. m1 = m2 = 2 kg, η1 = η2 = 0.1 and the bandwidth wasmaintained constant at 100 rad/s.

For this case, one find that η12 got from (4.44) is identical to η12 from (4.53). Followed the idea in[50], we set the “coupling ratio” as a variable to investigate the relation between time-varying energyexchange and coupling strength. The “coupling ratio” is defined as

r =η12

η(4.53)


Mass Spring Coupling Frequency Damping Coupling ratioCase (kg) (N/m) stiffness (rad/s) loss factor r

m1 = m2 k1 = k2 k (N/m) ω1 = ω2 η1 = η2 r = η12/η1

A 2 16 × 105 16 × 105 1265 0.1 20B 2 16 × 105 4 × 105 1000 0.1 2C 2 17.17 × 105 2.8 × 105 1000 0.1 1D 2 18 × 105 2 × 105 1000 0.1 0.5E 2 19.1 × 105 0.9 × 105 1000 0.1 0.1F 2 19.8 × 105 0.2 × 105 1000 0.1 0.005G 2 16 × 105 16 × 105 1265 0.316 1.25I 2 16 × 105 4 × 105 1000 0.4 0.125

Table 4.2: Parameters and data for two equal oscillators (cases B - F used in [50])

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

Ene

rgy

(J)

Time (ms)

Input energy

Transmitted energy

Figure 4.13: Comparison of three results for case A with the coupling ratio r = 20. ——, MESTsolutions; - - - - -, exact results; - -•- -, TSEA solutions.

The parameters of all the tests are found in Table 4.2. The first example (case A) is the case of verystrong coupling (r = 20), where the coupling stiffness k is equal to the blocked stiffness k1,2. Thiscase can be used to simulate the coupling of two continuous substructures, like two coupled rods. Bothof the two kinds energies: input and transmitted energy, are shown together in Figure 4.13. For strongcoupling the energy is exchanged much more rapidly between the two oscillators than it is dissipated,and the two oscillators share the energy as it decays with time. Hence, both the input energy (energy ofmass 1) and transmitted energy (energy of mass 2) exhibit an oscillatory character by sharing the totalenergy. MEST solutions are found to be an ideal prediction in contrast with the exact energy results.TSEA solutions, however, do not show the oscillatory character due to the its first order system, and itpredicts a gradual identical decay for both oscillators. The MEST solutions and exact results show theenergy exchange about this mean value. In addition, from Figure 4.13 one can find the partition ratioof input and transmitted energy predicted by exact result and MEST solution keep stable. It means thatdamping behaviour does not affect the partition ratio.

To observe the rise time tr and peak values of transmitted energy clearly, input and transmitted


0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

Ene

rgy

(J)

Time (ms)

(a)

(a) Input energy

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

Ene

rgy

(J)

Time (ms)

(b)

(b) Transmitted energy

Figure 4.14: Comparison of three results for case A with the coupling ratio r = 20. ——, MESTsolutions; - - - - -, exact results; - -•- -, TSEA solutions.

energies are respectively displayed in Figure 4.14. In Figure 4.14(b), TSEA predicts a rather quicklyincreasing transmitted energy, its peak energy value is 0.105J and the rise time is about 1 ms, while theexact result and MEST solution show that the peak energy value and rise time should be 0.1505J and5ms.

The comparison of time-varying energy for case B is shown in Figure 4.15. It can be seen that thesolution of MEST is a precise and smooth curve along the exact energy data. The rather good agreementbetween MEST solution and exact results is also found in following figures.

Figure 4.16 and 4.17 show the same similarity of MEST solutions and the exact results, however,the oscillatory trend of energy is less obvious due to slightly weaker coupling (r = 1 and r = 0.5).The transient SEA solutions, however, predicts the energies in different decay rate compared with MESTsolutions and exact results. In addition, one always finds that the rise time predicted by TSEA is much“shorter” that that predicted by MEST and exact result, the reason will be explained lately. The energylevel for rather weak coupling conditions, case E and F, are shown in Figure 4.18 and 4.19. Althoughthe difference of transmitted energy predicted by three methods is visible in Figure 4.18(b) and 4.19(b),however, in a large scale they seem to be in a good agreement that can be seen in Figure 4.20 and 4.21.In fact, when the coupling is very weak, the energy exchanged between two oscillators is a very smallquantity. The oscillator 1 behave like a single-DOF one, consequently the oscillator of mass 2 is at a verysmall energy level.

In the cases above studied, the stiffness values are variated while keeping damping loss factor asη1 = 0.1. Other examples (case G and I) are presented to check the influence of damping on energytransmission. In fact, case G and I are respectively equivalent to case A and B except for different damp-ing factors. The comparison between case G and case A is shown in Figure 4.22, and the comparison forcase I and case B is shown in Figure 4.23. Case G has the same strong coupling strength as that of caseA, but damping loss factor in the former (η1 = 0.316) is bigger than the latter (η1 = 0.1). Compared withcase A shown in Figure 4.22(a), transmitted energy in Figure 4.22(b) loses its oscillatory character due to


0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

Ene

rgy

(J)

Time (ms)

(a) Input energy

0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Ene

rgy

(J)

Time (ms)


Figure 4.15: Comparison of three results for case B with the coupling ratio r = 2. ——, MEST solutions;- - - - -, exact results; - -•- -, TSEA solutions.

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Ene

rgy

(J)

Time (ms)

(a) Input energy

0 5 10 15 20 25 30−0.01

0

0.01

0.02

0.03

0.04

0.05

Ene

rgy

(J)

Time (ms)


Figure 4.16: Comparison of three results for case C with the coupling ratio r = 1. ——, MEST solutions;- - - - -, exact results; - -•- -, TSEA solutions.


0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25E

nerg

y (J

)

Time (ms)

(a) Input energy

0 5 10 15 20 25 30−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Ene

rgy

(J)

Time (ms)


Figure 4.17: Comparison of three results for case D with the coupling ratio r = 0.5. ——, MESTsolutions; - - - - -, exact results; - -•- -, TSEA solutions.

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

Ene

rgy

(J)

Time (ms)

(a) Input energy

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10x 10

−3

Ene

rgy

(J)

Time (ms)


Figure 4.18: Comparison of three results for case E with the coupling ratio r = 0.1. ——, MESTsolutions; - - - - -, exact results; - -•- -, TSEA solutions.


0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Ene

rgy

(J)

Time (ms)

(a) Input energy

0 5 10 15 20 25 30

0

2

4

6

8

10x 10

−4

Ene

rgy

(J)

Time (ms)


Figure 4.19: Comparison of three results for case F with the coupling ratio r = 0.005. ——, MESTsolutions; - - - - -, exact results; - -•- -, TSEA solutions.

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

Ene

rgy

(J)

Time (ms)

Input energy

Transmitted energy

Figure 4.20: Comparison of three results forcase E with the coupling ratio r = 0.01. ——,MEST solutions; - - - - -, exact results; - -•- -,TSEA solutions.

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

Ene

rgy

(J)

Time (ms)

Input energy

Transmitted energy

Figure 4.21: Comparison of three results forcase F with the coupling ratio r = 0.005. ——, MEST solutions; - - - - -, exact results; - -•- -,TSEA solutions.


0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

Ene

rgy

(J)

Time (ms)

Input energy

Transmitted energy

(a) Case A, η1 = 0.1

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Ene

rgy

(J)

Time (ms)

(b) Case G, η1 = 0.316

Figure 4.22: The influence of damping. ——, MEST solutions; - - - - -, exact results; - -•- -, TSEAsolutions.

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

Ene

rgy

(J)

Time (ms)

(a) Case B, η1 = 0.1

0 5 10 15 20 25 30−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Ene

rgy

(J)

Time (ms)

(b) Case I, η1 = 0.4

Figure 4.23: The influence of damping. ——, MEST solutions; - - - - -, exact results; - -•- -, TSEAsolutions.


Mass (kg) Spring (N/m) Coupling Frequency Damping factor Coupling ratioTest m1 k1 stiffness ω1 (rad/s) η1 r1

m2 k2 k (N/m) ω2 (rad/s) η2 r2A 4 16 × 105 12 × 105 836.66 0.06 40.8

2 12 × 105 1095.4 0.09 17.25B 4 12 × 105 2 × 105 591.08 0.08 1.69

2 16 × 105 947.36 0.1 13.16C 2 16 × 105 12 × 105 1182.2 0.08 9.92

4 12 × 105 774.19 0.06 110.1D 2 12 × 105 2 × 105 836.66 0.12 0.44

4 16 × 105 670.35 0.07 12.67

Table 4.3: Parameters and data for two different oscillators

the relatively strong damping, although the differences of rise time and peak values between TSEA andMEST are easily distinguished. Similarly, case B and I have the same coupling strength (k = 4 × 105

N/m) and different damping loss factors: η1 = 0.1 in case B while η1 = 0.4 in case I. Not like Figure4.23(a) where the difference between MEST and TSEA is apparently observed, one finds that, in Fig-ure 4.23(b) the energies predicted by three methods approximate in a same trend. The heavy dampingplays the same role as weak coupling that makes the MEST solution to approach the TSEA solution. Onan extremity condition, when the coupling loss factor η12 and η21 in MEST equation (4.25) and TSEAequation (4.41) are set to be zero, they both yield the same energy expressions for single-DOF oscillator,

EMEST = ETSEA = E(0)e−η1ωt. (4.54)

In general, it is found that TSEA solution cannot accurately predict the time-varying energy level al-though it is roughly applicable in the very heavy damping and weak coupling condition. Surprisingly,the same limitation for SEA is “weak coupling”.

4.4.2 Two different oscillators

It should be state again that, the foregoing numerical simulations is just to compare three energy resultsunder the same condition as that used in [50]. However, the two identical oscillators system in previoussubsection, cannot completely represent many actual structures in practice. The two completely differentoscillators system is studied here to illustrate a more general case, the numerical parameters are list inTable 4.3. Since the transmission indicators, such as rise time and peak energy value of indirect-excitedsubsystem, are very important to evaluate the dynamic characteristics, the transmitted time-varying en-ergy (energy of oscillatorm2) is worthy to pay more attention. Thus, only the comparisons of transmittedenergy in each case listed in Table 4.3 are presented. The first example (test case A)

There are two different coupling loss factors (η12, η21) and inherent loss factors (η1, η2) in the dif-ferent oscillators, the coupling rates are defined as

r1 = η12/η1, r2 = η21/η2 (4.55)

Figure 4.24, 4.25 gives the comparison of energy results from three methods for two different oscil-lators associated with the strong coupling, and Figure 4.26, 4.27 displays the comparison in the case of


0 5 10 15 20 25 30−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Ene

rgy

(J)

Time (ms)

Figure 4.24: Transmitted energy in test A: ——, MEST solutions; - - - - -, exact results; - -•--, TSEA solutions.

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−3

Ene

rgy

(J)

Time (ms)

Figure 4.25: Transmitted energy in test B: ——, MEST solutions; - - - - -, exact results; - -•--, TSEA solutions.

0 5 10 15 20 25 30−0.02

0

0.02

0.04

0.06

0.08

0.1

Ene

rgy

(J)

Time (ms)

Figure 4.26: Transmitted energy in test C: ——, MEST solutions; - - - - -, exact results; - -•--, TSEA solutions.

0 5 10 15 20 25 30−0.005

0

0.005

0.01

0.015

0.02

0.025

Ene

rgy

(J)

Time (ms)

Figure 4.27: Transmitted energy in test D: ——, MEST solutions; - - - - -, exact results; - -•--, TSEA solutions.


very weak coupling. It is similar to the figures shown in the previous section, although there are a fewdisagreements occurred between MEST solutions and the exact energy results.

4.4.3 Energy flow analysis

The above-discussed comparisons show the great difference between MEST and TSEA. While the energyexchanged between two coupled subsystem is clearly of major importance, the study on energy flowamong different methods may be necessary. For both TSEA and MEST, the energy flow transmittedfrom oscillator m1 to m2 can always be written as

P12 =dW2

dt+ η2ωW2 =

η12W1 − η21W2 in TSEA,

− 1ηω

d2W2

dt2− dW2

dt+ η12W1 − η21W2 in MEST.

(4.56)

The energy flow in TSEA has been found to be the same as that in SEA where the energy flow betweenthe oscillators is assumed to be proportional to the difference of their time-averaged energies.

In addition, we can get the direct energy flow expression from motion equation. Multiply equatoneqn:ceni1 by x1, then the energy flow can be denoted by

P12 = kx1x2 (4.57)

The numerical analysis of energy flow from oscillator 1(m1, input) to oscillator 2 (m2, transmitted) isinvestigated and comparisons among MEST, TSEA, and exact result are shown in Figure 4.28 – 4.31. Atfirst for two identical oscillators, case A and F in Table 4.2 are chosen to examine their energy flow bythree methods, because they have completely different coupling strength: case A has a strong couplingwhile case F weak. The comparisons among them are shown in Figure 4.28 and 4.29 which correspondto case A and case F. In Figure 4.28, one finds that the energy flow p1→2 predicted by TSEA keepspositive while MEST and exact energy flow represented by oscillatory curves. The positive value ofTSEA energy flow forces the input energy level to be always higher than transmitted energy level (thisphenomenon have been observed in previous subsections.), while the oscillatory energy flow in MESTand exact results yields oscillatory energy exchange.

As for the examples of two different oscillators, the simulation of energy flow analysis were madeon test A and test B in Table 4.3. The comparisons are shown in Figure 4.30 and 4.31.

It is visible that, from Figure 4.28 – 4.31, the energy flow term (from element 1 to 2) predictedby MEST can be viewed as the mean value of the exact ones, while the energy flow terms in TSEAbehave the remarkably different decay rate and different initial value. For example, at the initial time, thetransmitted energy flow level obtained from both MEST and exact results are zero, however, TSEA givesout a very large quantity. At least, it illustrates that the assumed proportional relation between energyand energy flow is no longer valid in the time domain. The numerical analysis and comparisons aboutenergy flow may be useful in consolidating our thinking about the energy density behavior and it showsthat the different energy flow behaviour heavily affects the energy results. Of course, the principal andfundamental reason lies in that vibrational energy is differently described by a first-order system (TSEA)or a second-order system (MEST).


0 5 10 15 20 25 30−100

0

100

200

300

400

500

600

Ene

rgy

flow

(W

)

Time (ms)

Figure 4.28: Energy flow in case A (Table 4.2):——, MEST solutions; – – – – , exact results;– · – ·, TSEA solutions.

0 10 20 30 40 50−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Ene

rgy

flow

(W

)

Time (ms)

Figure 4.29: Energy flow in case F (Table 4.2):——, MEST solutions; – – – –, exact results; –· – ·, TSEA solutions.

0 5 10 15 20 25 30−40

−30

−20

−10

0

10

20

30

40

50

Ene

rgy

flow

(W

)

Time (ms)

Figure 4.30: Energy flow in test A (Table 4.3):——, MEST solutions; – – – –, exact results; –· – ·, TSEA solutions.

0 5 10 15 20 25 30−3

−2

−1

0

1

2

3

Ene

rgy

flow

(W

)

Time (ms)

Figure 4.31: Energy flow in test B (Table 4.3):——, MEST solutions; – – – –, exact results; –· – ·, TSEA solutions.


4.5 The case of two coupled subsystems

Although the analytical discussions on two-oscillators system in earlier sections have provided a soundframework for understanding the general character of MEST and its application, it does not explore allthe subtleties of the situations and structures. In the two coupled oscillator system, the energy in eachsubsystem is stored at a point and energy transfer between subsystems commences at the moment ofthe initial impulse. Such a model is not always capable of being representative of coupled distributedsystems. In fact, rather than mass-spring oscillators, the practical engineering structures are often mod-eled by one-dimensional and two-dimensional coupled subsystems. We restrict ourself to the numericalsimulation of rather simple structures as contrasted with the actual applications. The former is of moreinterest to validation of MEST as a new energy approach in time domain than to applying the method toan actual structures concerned with complexe geometrical form or boundaries.

4.5.1 Two coupled rods

A one-dimensional system represented by quasi-longitudinal waves travelling in two coupled rods isanalyzed, because of the existence of analytical Green function in the frequency domain. Consider thetwo coupled rods in Figure 4.32, the similar model has been studied by Fahy and James [87] to investigatean indicator of the strength of coupling.

X

L1 L2

x1 x2

Fδ(t)

Figure 4.32: Model of two coupled rods

Each rod is considered as a uniform elastic medium. A viscous damping model has been preferredto the hysteretic damping model [87]. The properties of the driven rod are as follows: length L1 = 7m;Young’s modulus E1 = 2.1 × 1011N/m2; density ρ1 = 7800kg/m3; cross-section S1 = 0.005m2. Thesame parameters are for the received rod except length and L2 = 5m and cross-section, S2 = 0.005m2.

Suppose rod 1 is driven by an axial impulse force at left end x = 0. The resulting displacements u1

and u2 are given by

ρS1

∂2u1

∂t2− ES1

∂2u1

∂x21

+ d1∂u1

∂t= F1δ(x1)δ(t)

ρS2∂2u2

∂t2− ES2

∂2u2

∂x22

+ d2∂u2

∂t= 0

(4.58)

4.5. The case of two coupled subsystems 107

where d1 and d2 is viscous damping coefficient. The boundary conditions are

ES1∂u1(0, t)∂x1

= 0

ES2∂u2(L2, t)

∂x2= 0

u1(L1, t) = u2(0, t)

E1S1∂u1(L1, t)

∂x1= E2S2

∂u2(0, t)∂x2

(4.59)

Thus the Laplace transform of displacements of two rods are

U1(s) = L[u1(t)] =F1c

ES1· A1(s)B1(s)

(4.60)

U2(s) = L[u2(t)] = −2F1c

E· A2(s)B2(s)

(4.61)

where

A1(s) =e(L1−x1)

√s2+D1s

c

[S2

√s2 +D2s

(1 − e

2L2

√s2+D2s

c

)− S1

√s2 +D1s

(1 + e

2L2

√s2+D2s

c

)]

−e(x1−L1)

√s2+D1s

c

[S2

√s2 +D2s

(1 − e

2L2

√s2+D2s

c

)+ S1

√s2 +D1s

(1 + e

2L2

√s2+D2s

c

)]

B1(s) =√s2 +D1s

[S1

√s2 +D1s

(e

L1

√s2+D1s

c − e−L1

√s2+D1s

c

)(1 + e

2L2

√s2+D2s

c

)]

−√s2 +D1s

[S2

√s2 +D2s

(e

L1

√s2+D1s

c + e−L1

√s2+D1s

c

)(1 − e

2L2

√s2+D2s

c

)]

and

A2(s) =e(x2−L2)

√s2+D2s

c + e(L2−x2)

√s2+D2s

c

B2(s) =S1

√s2 +D1s

(e

L1

√s2+D1s

c − e−L1

√s2+D1s

c

)(e

−L2

√s2+D2s

c + eL2

√s2+D2s

c

)

−S2

√s2 +D2s

(e

L1

√s2+D1s

c + e−L1

√s2+D1s

c

)(e

−L2

√s2+D2s

c − eL2

√s2+D2s

c

)

where

c =

√E

ρ, Di =

di

ρSii = (1, 2) (4.62)


0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

Time (ms)

Vel

ocity

(m

/s)

(a) Velocity of rod 1

0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

Time (ms)

Vel

ocity

(m

/s)

(b) Velocity of rod 2

Figure 4.33: Velocities density V (x, t) of two coupled rods with a damping coefficient D1 = D2 = 50

It was found impossible to analytically express u1(t) and u2(t) by inversing Laplace transform of equa-tion (4.62) and (4.63) with all algebra parameters. A numerical methods with parameters values listedabove is adopted. Suppose the viscous damping coefficient is same in two rods. Two kinds of case:different cross-section S1 = S2 or different rod length L1 = L2 were investigated and the displacementsu1(t) and u2(t) with respect to these two cases are listed as fllows.

For the case of same cross-section S and different rod length L1 and L2:

u1(t, x1) = −2F1c

πESe−

Dt2

∞∑n=1

(−1)n

ncos

nπ(L1 + L2 − x1)L1 + L2

sinnπct

L1 + L2; (4.63)

u2(t, x2) = −2F1c

πESe−

Dt2

∞∑n=1

(−1)n

ncos

nπ(L2 − x2)L1 + L2

sinnπct

L1 + L2. (4.64)

For the case of same rod length L and different rod length S1 and S2:

u1(t, x1) = − 2F1c

πES1e−

Dt2

∞∑n=1

(−1)n

n

[cos

nπ(2L− x1)2L

− S2 − S1

S1 + S2cos

nπx1

2L

]sin

nπct

2L;

u2(t, x2) = − 4F1c

πE(S1 + S2)e−

Dt2

∞∑n=1

(−1)n

ncos

nπ(L− x2)2L

sinnπct

2L. (4.65)

Another studied parameter is damping coefficient Di. The velocities distributions of each rods with adifferent damping coefficient are shown in Figures 4.33–4.34. In fact, they are kinds of velocity densityrepresenting the velocity evolution in spatial and time domain. The velocities of a position in coupledrods, for example, x1 = L1

2 and x2 = L22 , are given in Figure 4.35, where the reflection of velocity by

the junction and the each ends of two rods is cleared observed.

The spatial distribution of energy density can be evaluated from the obtained velocity distributiondisplayed above. The total energies stored in each rod are the spatial integration of the distribution of


0 5 10 15 20−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time (ms)

Vel

ocity

(m

/s)

(a) Velocity of rod 1

0 5 10 15 20−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Time (ms)

Vel

ocity

(m

/s)

(b) Velocity of rod 2


0 5 10 15 200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Time (ms)

Vel

ocity

(m

/s)

reflected from the junction

reflected from the end of rod 2

(a) Velocity at x1 = L12

in rod 1

0 5 10 15 200

0.005

0.01

0.015

0.02

0.025

0.03

Time (ms)

Vel

ocity

(m

/s)

(b) Velocity at x2 = L22

in rod 2



0 5 10 15 200

1

2

3

4

5

6x 10

−3

Time (ms)

Ene

rgy

(J)

(a) Time-varying energy in rod 1,

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−3

Time (ms)

Ene

rgy

(J)

(b) Time-varying energy in rod 2,

Figure 4.36: Energy comparison of two coupled rods with a damping coefficient D1 = D2 = 50. ——,MEST solutions; – – – –, exact results; – · – ·, TSEA solutions.

energy density. In order to reveal the energy evolution in the time domain, we adopted the technique of“the temporal moving average” which was presented in [87]. Thus the moving average of time-varyingenergy E(t) over an interval T is defined by

< E(t) >T =1T

∫ t+T/2

t−T/2E(τ)dτ (4.66)

The interval T is normally evaluated as the period related to the lower frequency of the band-pass filteremployed above.

The energy calculated in equation (4.68) is comparied with the solution of MEST and TSEA. Thecoupling loss factors used in MEST and TSEA is determinded by (see Appendix 2 in [51])

ηsr = [τsr/ω(1 − τ/2)](csg/2ls) (4.67)

The comparison of total energy from three methods: MEST, TSEA, and exact result are shown in Figure4.36- 4.37

As viewed in these figures, MEST solutions show a very good agreement with exact results and theyexhibit the identical oscillatory characteristics even if the technique of “the temporal moving average”was applied to reference energy values. That the two rods are rigidly coupled certainly brings about avery strong coupling relation, thus the energy is exchanged much more rapidly than it is dissipated. Thisprediction has been verified by MEST solutions and the exact results. The TSEA solutions, however,have a sharp change at initial time and then attenuate in a slightly different decay rate. It is found thatthe TSEA solutions are not capable of predicting the time history of time-varying vibrational energy,in neither the source rod (rod 1) nor the received rod (rod 2). Moreover, the transmitted energy givenby TSEA shown in Figure 4.36(b) reaches its peak value instantaneously, it obviously underestimatedthe rise time and peak value of transmitted energy, and it violates the physical fact that time-varyingvibrational energy propagates in a finite velocity.


0 5 10 15 200

1

2

3

4

5x 10

−3

Time (ms)

Ene

rgy

(J)

(a) Time-varying energy in rod 1,

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−3

Time (ms)

Ene

rgy

(J)

(b) Time-varying energy in rod 2,

Figure 4.37: Energy comparison of two coupled rods with a damping coefficientD1 = D2 = 200. ——,MEST solutions; – – – –, exact results; – · – ·, TSEA solutions.

In addition, the TSEA solutions in 4.37 appear to be rather closer to MEST solution and exact results.It suggests that TSEA solution may likely be an approximation only in some special cases, for example,a heavy damping system. This conjecture actually accords with the remarks and conclusions discussedin previous sections.

4.5.2 Two coupled beams

In fact, the study on transient energy flow between two coupled beams had been investigated by Pinning-ton and Lednik in [51]. In their model, two coupled beams were excited with an impulse at one end ofthe source beam, and the response monitored at the remote end of the receiver beam. Three kinds of nu-merical results, the exact wave solution, TSEA, and WPA (Wave Propagation Analysis), were presentedsimultaneously to make some comparisons. The applicability of TSEA and WPA was studied in termsof modal overlap. They found that the appropriate applicability of TSEA or WPA significantly dependson the range and value of modal overlap. This conclusion suggested the limitations of TSEA (or WPA).In order to validate the applicability of MEST for dispersive waves, we try to supplement the numericalanalysis of MEST and compare its simulation with TSEA results presented in [51].

Consider two coupled beams in Figure 4.38. They are two ideal Euler-Bernoulli beams whose prop-erties are as follows: Young’s modulus E1 = E2 = 2.1 × 1011 N/m2; density ρ1 = ρ2 = 7800 kg/m3;cross-section S1 = 0.01 m2, S2 = 0.02 m2; length L1 = 3 m, L2 = 3.5 m.

Unlike the longitudinal waves in last section, the flexual waves are dispersive, so the group velocitycg is associated with frequency.

cgi = 2 4

√ω2

cEiIiρiSi

, i = 1, 2. (4.68)

It is twice the phase velocity and the speed of propagation of energy. Introduce the hysteretic loss factor


Fδ(t)

X

L1 L2x0

x1 x2

Figure 4.38: Model of two coupled beams

Fs1

Vs1

Fs2 Fr1 Fr2

Vs2 Vr1 Vr2source beam receiver beam

Figure 4.39: The mobility relation of uncoupled beams

ηi, the bending wavenumber ki is given by

ki = 4

√ω2

cρiSi

EiIi(1 − j · ηi

4), i = 1, 2 and j =

√−1. (4.69)

The exact expression of velocities (or energies) in time domain can be obtained by using a fourier trans-form. Following the idea in [51], we also evaluate the time-varying transmitted energy by employing theend velocity in the receiver beam. Shown in Figure 4.39, the mobility relation can be written as (takingthe source beam as an example denoted by subscript s)

Vs1

Vs2

=

[Ms11 Ms12

Ms21 Ms22

]Fs1

Fs2

(4.70)

and likewise for the receiver beam (beam 2) with s replaced by r,Vr1

Vr2

=

[Mr11 Mr12

Mr21 Mr22

]Fr1

Fr2

(4.71)

Thus the velocity response at the remote free end of the receive beam is defined by

Vr2 =Ms12Mr12

Ms22 +Mr11Fs1 (4.72)

Let M be the characteristic mobility of an infinite beam,

M =1

ρScg(4.73)

then the input mobility and transfer moblity can be written as [51, 91]

Minput = 2M(

1 − αi

1 + αi− j

), Mtr =

4M√αi

1 + αi(4.74)

where αi is the attenuation and phase change in a wave travelling 2Li,

αi = e−2jkiLi , j =√−1. (4.75)


101

102

103

10−5

10−4

10−3

10−2

10−1

Frequency (Hz)

Mod

ulus

(m

/s /N

)

Figure 4.40: Frequency response function Vr2/Fs1

For the purpose of comparing the existed results, the same parameters are adopted from [51](see testC in Table 1). The properties of two beams (subscripted as 1 and 2) are as follows: length L1 = 2m,L2 = 2.2m; Young’s modulus E1 = E2 = 2.1 × 1011N/m2; density ρ1 = ρ2 = 7800kg/m3; cross-section S1 = 5 × 10−4m2, S2 = 2 × 10−3m2; loss factor is 0.01 for two beams.

The transfer mobility Vr2/Fs1 from equation (4.74) was reexamined and shown in Figure 4.40, whichwas found to be identical with the figure in [51]. A bandwidth 64 − 800 Hz was chosen to filter thefrequency response in (4.74). Thus the centre frequency is 219 Hz which will be applied in TSEA andMEST. The velocity, vr2(t), in time domain is found by using an inverse Fourier transform:

vr2(t) =∫ ∞

−∞Vr2 ej2πftdt (4.76)

The exact time response with the certain bandwidth defined in equation (4.78) was calculated by theinverse FFT program and filter program in MATLAB c©. It was shown in Figure 4.41.

Similarly, the instantaneous total energy of the receiver beam can be estimated by using the endvelocity, vr2(t). Given a position x, the instantaneous kinetic energy density is

Wk(x, t) =12ρSv2(x, t). (4.77)

Because the strain energy density is equal to the kinetic energy density (it holds under the assumptionH1, see 3.2), thus the energy density , W (x, t) is

W (x, t) = ρSv2(x, t). (4.78)

The total instantaneous energy of the receive beam, Er at an instant, t = t0, is the integral of equation(4.80) at t = t0 over twice the length of receive beam [51],

Er(x, t0) = ρSrcrg

∫ t0+ τ2

t0− τ2

(v2r (x, t))

2

8dt, (4.79)


0 0.1 0.2 0.3 0.4 0.5−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time (s)

Vel

ocity

(m

/s)

Figure 4.41: Impulse response function Vr2(t) by inverse FFT; 0.5s

where τ is the time for velocity vr to run through (incident and reflection) the length of receive beam,

τ =2L2

crg. (4.80)

It is interested to note that the whole energy expression (4.81) is similar to so-called temporal movingaverage energy in (4.68).

The time history of vibrational energy in the receive beam is displayed in Figure 4.42. The resultsfrom three methods, TSEA, MEST, and exact solution, are compared. The solution of TSEA and MESTare always found in equation (4.43) and Appendix (C). The coupling loss factors for flexural waveshave the same expression with that for longitudinal waves in last section (see equation (4.69)). Thecomparison in Figure 4.42 showed again that MEST gave a good energy estimate. Even the rise timefor peak energy predicted by MEST appears acceptable although there was a minor error. This occursbecause the coupling loss factors used here cannot accurately describe the time-varying energy transferbetween subsystems. This problem arised several times in previous discussions. And one also finds thatthe time-varying energy in dispersive wave form has the oscillatory character, it cannot be smeared orswept off by averaging and integral technique. In the TSEA, time-varying energy transferfing betweensubsystem is supposed to begins the instant the excitation is applied, so it always arrives its peak energyvalue much earlier than the exact one, which can be viewed in Figure 4.42. The peak value of transmittedenergy predicted by TSEA happens to be close to the exact and MEST solution, but it was just thoughtas an accidental phenomena.

The second example was associated with another bandwidth: 256 − 1024 Hz with the centre fre-quency 512 Hz. The comparison of three methods for the transmitted energy in receiver beam wasshown in Figure 4.43. The peak value of transmitted energy for this band is about 0.022 J which isrelatively lower than that (0.037 J) in band 64 − 800 Hz, because the frequency is higher, the stored en-ergy component is lower. Moreover, one finds that TSEA solution underestimates the transmitted energylevel.


0 0.05 0.1 0.15 0.2−0.01

0

0.01

0.02

0.03

0.04

Time (s)

Ene

rgy

(J)

Figure 4.42: Time history of total energy in receive beam in frequency band 64 − 800 Hz. ——, exactresult; – – – –, MEST solution; – · – ·, TSEA solutions.

0 0.01 0.02 0.03 0.04 0.05 0.060

0.005

0.01

0.015

0.02

0.025

Time (s)

Ene

rgy

(J)

Figure 4.43: Time history of total energy in receive beam in frequency band 256− 1024 Hz. ——, exactresult; – · – · –, MEST solution; · · · · · · , TSEA solutions.


Analytical energy expression Ei(t)

Forming the energy equation dealing with discrete systems

1ηiω

d2Ei

dt2+ 2

dEi

dt+ ηiωEi + ηeijωEi − ηejiωEj = Pin

Validation of MEST equation discretized by FEM technique

1ηω

∂2W (x, t)∂t2

− c2

ηω∇2W (x, t) + 2

∂W (x, t)∂t

+ ηωW (x, t) = pin

Figure 4.44: Illustration of the applications and validations of MEST for discretized systems

4.6 Brief summary

The principal aim of this chapter has been to verify the applicability of MEST to the discretized orcoupled subsystem/substructures. The investigation were caught out by the application of MEST totwo-oscillators systems, coupled rods and beams and by the comparisons with exact results and TSEAsolutions.

First for two-oscillators system, an analytical temporal energy expression (4.8) has been proposedindependently. The important idea is to separate the energy exchange from energy dissipation. It wasdefinitely found that the time-varying vibrational energy is governed by a second-order temporal equationwhich is potentially consistent with MEST equation. Thus MEST equation has been proved out onceagain by equation (4.14) and (4.21) for two-oscillators system – the most fundamental discretized system.On the other hand, MEST differential equation (3.106) can also be discretized into the form of (4.21)by a basic FEM technique. It can be interpreted by Figure 4.44. With the help of exact expression (4.8and (4.14), it was facilitated to build up the relation between energy partition and exchange by couplingfactors. It suggested that the coupling loss factors describe the energy partition to some extent. Inaddition, it was found that the coupling loss factors for MEST are not the same as those for TSEA unlessthe two oscillators have the identical nature frequency.

Secondly, the theoretical study on MEST is helpful to understand the link between the couplingstrength and the time delay defined in [87] (or, rise time labeled in [51]). We try to propose an explicitexpression (2.7) to determine the relation between coupling loss factor ηij and the rise time tr. In fact,the rise time tr and peak value of transmitted energy Emax are two important indicators to quantify thetime-varying energy and to evaluate damage potential in structures subjected to shock excitations. Due tothe different equation, MEST and TSEA predict the different tr and Emax. For two coupled subsystems,

4.6. Brief summary 117

MEST equations write

Pin1 =1

η1ω1

d2E1

dt2+ 2

dE1

dt+ η1ωE1 + η12ωE1 − η21ωE2 (4.81)

Pin2 =1

η2ω1

d2E2

dt2+ 2

dE2

dt+ η2ωE2 + η21ωE2 − η12ωE1 (4.82)

To solve MEST equations, the initial energies E1(0), E2(0) and the initial energy derivatives dE1(0)dt ,

dE2(0)dt are required. Supposed an impulse excitation is applied on the system, the input powers, Πin1,

Πin2 are zero, and E2(0), dE2(0)dt are also zero. Here is the same case as adopted by Pinnington and

Lednik [50], where, η1 = η2. By applying the Laplace transform method, the two second order differen-tial equations (4.83) and (4.84) are converted into two linear algebraic equations, then the solutions areobtained,

E1(t) =E1(0) e−η1ωt

1 + r1[r + cos (

√η1 η12 (1 + r1)ωt)] (4.83)

E2(t) =E1(0) e−η1ωt

1 + r1[1 − cos (

√η1 η12 (1 + r1)ωt)] (4.84)

where r1 = η21/η12, r = η12/η1 and E1(0) can be found in formula (4.1). The coupling loss factor usedin MEST and TSEA is the same definition in steady state condition. For the purpose of comparing thesolutions of MEST and TSEA with the exact results, the coupling loss factor described in the model oftwo equal oscillators system is used here [14]

η12 =k2

2η1(k + k1)2(4.85)

Furthermore, the coupling ratio is defined

r =η12

η1=

k2

2η21(k + k1)2

(4.86)

Thus, the rise time predicted by MEST is

tr1 =√

2Φη1ω

√r

(4.87)

where

Φ = arcsin(√η12 + η21√

η1 + η12 + η21), r =

η12

η1(4.88)

In addition, TSEA solution listed in (4.43) yields a rise time

tr2 =1

2bωln

(a+ b

a− b

)(4.89)

One can always find that tr1 > tr2. It means TSEA will predict an early arrive of transmitted energy,this conclusion have been verified by all the examples investigated in this chapter.

Thirdly, as pointed in previous discussions, TSEA can be applied in some special cases withoutproducing remarkable error. The limitation may be located in very weak coupled structures and veryhigh frequency cases. Anyway, we observe that such a modus operandi is fraught with the danger thatthe TSEA does not maintain an adequate physical understanding of the basic phenomena and thus over-simplifies reality. One must be willing to risk the possibility of some inaccuracy in detail in order to gainthe benefits of overall energy trend by a relative simple methods like TSEA.

Chapter 5

Time-varying vibrational energy inone-dimensional distributed structures

In last chapter the focus was concentrated on discrete systems like coupled oscillators or substructures.Some examples were used to validate the applicability of MEST as a new method to deal with transientdynamics. The validity of MEST was examined in these models because almost all the actual structurescan be taken apart into several substructures using judgment, past experience, and available rules-of-thumb. To some extent, these investigation are mainly considered from the point of view of engineering.We now wish to extend the applicability analysis of MEST to distributed-parameter models. Maybe it ismore logical to discuss distributed-parameter models earlier than the discrete models, because the studyof energy motion in space-time domain facilitate to understand better the physical implications of wavespropagating in structures.

Since real-world physical systems are, at the microscope level, of a spatially distributed nature, andsome problems related to the distributed-parameter structures become very important in practical engi-neering, researches on applications of MEST appears indispensable. While MEST is a new-borne one,it is of course not possible to develop the engineering background needed for a thorough and generalapplications. Such a background is necessary to relate material properties, structures dimensions andcomplex configurations. At the initial stage, one need not deal at this level of detail, simply acceptingthe existence of a physical effect. For the purpose of validity and demonstration of MEST, we accept thisviewpoint and now proceed to analyze those typically simple but important structures.

It is necessary to refer another method which had been used to describe vibrational energy in spacetime domain. This is the equation proposed by D.J. Nefske and S.H. Sung in [12]. It has been introducedin chapter 1 and discussed its properties in chapter 3. We now call it Nefske’s equation,

∂W (x, t)∂t

− c2

ηω

∂2W (x, t)∂x2

+ ηωW (x, t) = 0 (5.1)

It is limited in one-dimensional structures with a spatial variable x (see in [12]). Equation (5.1) isevidently a parabolic differential equation which is usually used to describe thermal or thermal-liketransfer. MEST solutions will be compared with the solution of Nefske’s equation (5.1) and with exactenergy values.

119

120 Time-varying vibrational energy in one-dimensional distributed structures

5.1 Longitudinal wave: time-varying energy in a rod

5.1.1 Exact result for longitudinal waves in rod

Consider a free-clamped rod subjected to a transient excitation, which is shown in Figure 5.1.

x=0 x=l

u(x, t)F0

Figure 5.1: Rod subjected to an impulse force

The partial differential equation for longitudinal vibration of the slender rod is written as

∂2u(x, t)∂t2

+ d∂u(x, t)∂t

− E

ρ

∂2u(x, t)∂x2

= F0δ(x)δ(t)

u(x, t)|t=0 = 0,∂u(x, t)∂t

|t=0 = 0

ES∂u(x, t)∂x

|x=0 = F0δ(t), u(l, t) = 0

(5.2)

where u(x, t) is the axial displacement of any plane in the rod, d is the damping coefficient, . If d = 0,equation (5.2) reduce to the standard wave equation

∂2u(x, t)∂t2

− E

ρ

∂2u(x, t)∂x2

= F0δ(x)δ(t) (5.3)

In general, there are two methods to get the exact solution of the rod subjected to a transient excitation.One is seeking the synthesis of normal modes, another is Laplace transform. The former is used forequation (5.2) to avoid the very complex inverse procedure of Laplace transform, while we use the latterfor equation (5.3) to get more explicit solution. Two kinds excitation are applied: a unit impulse (δ(t),Dirac function) and a unit step excitation (H(t), Heaviside function). The latter is defined by

H(t− τ) =

1 t ≥ τ,

0 t < τ.or H(t) =

∫ t

−∞δ(τ)dτ (5.4)

The solution for an impulse excitation are:

u(x, t) =c

E

∞∑n=0

(−1)n

[H

(t− x

c− 2nl

c) −H(t+

x

c− 2(n+ 1)l

c

)](5.5)

ud(x, t) =2cρl

∞∑n=1,3,5,···

e−ξt√ω2

n − ξ2sin

(√ω2

n − ξ2 t)

cosβnx (5.6)

5.1. Longitudinal wave: time-varying energy in a rod 121

0 1 2 3 4 5 6

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x 10−8

Time (ms)

Dis

plac

emen

t (m

)

(a) Damping coefficient d = 0

0 1 2 3 4 5 6−4

−3

−2

−1

0

1

2

3x 10

−8

Time (ms)

Dis

plac

emen

t (m

)

(b) Damping coefficient d = 400

Figure 5.2: Displacement of the rod subjected to an impulse excitation

For step excitation:

u(x, t) =c

E

∞∑n=0

(−1)n

[H

(t− x

c− 2nl

c

)(t− x

c− 2nl

c

)

−H

(t+

x

c− 2(n+ 1)l

c

)(t+

x

c− 2(n+ 1)l

c

)](5.7)

ud(x, t) =2cρl

∞∑n=1,3,5,···

cosβnx

ω2n

[1 − e−ξt

(cos

(√ω2

n − ξ2 t)

+ξ√

ω2n − ξ2

sin(√

ω2n − ξ2 t

))](5.8)

where

c =

√E

ρ, ξ =

d

2,

βn =nπ

2l, ωn = cβn, n = 1, 3, 5, · · · (5.9)

The study parameters are as follows: length of rod L = 1 m; cross-section A = 0.01 m2; materialdensity 7800 kg/m3; Young’s Modulus 2.1 × 1011 N/m2; viscous damping coefficient d = 400 Ns/m.Longitudinal waves are not dispersive, the wave velocity is 5.19 × 103m/s.

The displacement about the two excitation cases are shown in Figure 5.2-5.5. Note that thedisplacements of six positions x = 0, l/6, l/3, l/2, 2l/3, 5l/6, have been plotted together in Figure5.2 and 5.4, while Figure 5.3 and 5.5 display the displacement of the position at l

2 . In Figure 5.2(a), ifthe rod is subjected an impulse excitation, displacements of each position have the same amplitude withdifferent time shift, and the period of the placement is 2l

c , however, if the rod is excited by a step function,


0 1 2 3 4 5 6−4

−3

−2

−1

0

1

2

3x 10

−8

Time (ms)

Dis

plac

emen

t (m

)


0 1 2 3 4 5 6−4

−3

−2

−1

0

1

2

3x 10

−8

Time (ms)

Dis

plac

emen

t (m

)(b) Damping coefficient d = 400

Figure 5.3: Displacement of the rod subjected to an impulse excitation (x = l/2)

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−11

Time (ms)

Dis

plac

emen

t (m

)


0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−11

Time (ms)

Dis

plac

emen

t (m

)


Figure 5.4: Displacement of the rod subjected to a step excitation


0 1 2 3 4 5 6−1

0

1

2

3

4

5x 10

−12

Time (ms)

Dis

plac

emen

t (m

)


0 1 2 3 4 5 60

1

2

3

4

5x 10

−12

Time (ms)

Dis

plac

emen

t (m

)


Figure 5.5: Displacement of the rod subjected to a step excitation (x = l/2)

the displacements amplitude and periods of each position are different, as shown in Figure 5.4(a). Whendamping exists, behavior of displacement corresponding to those two kinds excitation are also different.Shown in Figure 5.2(b), 5.3(b), 5.4(b), 5.5(b), the displacement caused by an impulse excitation decaysand reduces to zero with time, while the displacement invoked by a step excitation decay to a specificvalue.

One can derive the velocity ∂u(x, t)/∂t and strain ∂u(x, t)/∂x from the expressions (5.5)-(5.8).Thus energy density can be defined as the sum of the potential energy density and the kinetic energydensity by the displacement response u(x, t),

W (x, t) =12ρS

[(∂u(x, t)∂t

)2

+ c2(∂u(x, t)∂x

)2]

(5.10)

5.1.2 Energy density of the rod by MEST equation

Energy density in the rod is solved by MEST equation

∂2W (x, t)∂t2

− c2∇2W (x, t) + 2d∂W (x, t)

∂t+ d2W (x, t) = 0. (5.11)

Note that equation (5.11) includes a viscous damping model (Detailed derivation can be found in Ap-pendix D).The boundary conditions for the free-clamped rod at ends x = 0 and x = l are

I(0, t) = 0, I(l, t) = 0 (5.12)

It is easy to understand the boundary condition (3.113), it means the right travelling energy wave isreflected to be the left travelling one at the free boundary, where the slope is zero. It is much different


from the boundary condition of wave equation. Or, they can be rewritten as:

∂W (x, t)∂x

|(x=0)= 0 where x = 0 (5.13a)

∂W (x, t)∂x

|(x=l)= 0 where x = l (5.13b)

In addition, the initial condition are imposed by

W (x, 0) = w0δ(x),∂W (x, 0)

∂t= −dW (x, 0) (5.14)

where w0 denotes the energy input, w0 =∫∞0 F0(t)v0(t)dt

The equations of MEST with their boundary conditions can be solved by Laplace Transform method.Let Ψ(s) is the Laplace transform function of ψ(t), namely, Ψ(s) = L[ψ(t)]. The general solution ofMEST is

W (x, t) = L−1

Ψ(s+ d) cosh

[(s+ d)(l − x)

c

](s+ d) cosh(

(s+ d)lc

)

= e−κt · L−1

Ψ(s) cosh

[s(l − x)

c

]s cosh(

sl

c)

(5.15)

If d = 0, W (x, t) is the undamped energy density obtained by (D.10).

The time function of energy input have two cases

ψ(t) =

δ′(t) rod subjected to an impulse excitation

δ(t) rod subjected to a step excitation(5.16)

It is found that the time functions of energy input are not the same as time function of excitation. Thesolutions of MEST list as follow:

Wi(x, t) =A

E

∞∑n=0

(−1)n

[δ

(t− x

c− 2nl

c

)+ δ

(t+

x

c− 2(n+ 1)l

c

)]

Ws(x, t) =A

E

∞∑n=0

(−1)n

[H

(t− x

c− 2nl

c

)+H

(t+

x

c− 2(n+ 1)l

c

)]

Wid(x, t) = e−kt · AE

∞∑n=0

(−1)n

[δ

(t− x

c− 2nl

c

)+ δ

(t+

x

c− 2(n+ 1)l

c

)]

Wsd(x, t) = e−kt · AE

∞∑n=0

(−1)n

[H

(t− x

c− 2nl

c

)+H

(t+

x

c− 2(n+ 1)l

c

)](5.17)

where, the subscripts i and s is corresponding to the impuse input and step input about undampedcases,the subscripts id and sd is corresponding to the impuse input and step input about damped cases;δ(t) denotes Dirac impulse function, H(t) is Heaviside function. The solutions of MEST above showthat the energy density of longitudinal wave is also a function of t − x

c , and energy velocity is identicalto the wave velocity, namely, c.


5.1.3 Energy density of the rod by Nefske’s equation

A first-order temporal partial differential equation had been employed by Nefske and Sung [12] to de-scribe the time-varying vibrational energy in one-dimensional structure. It reads

∂W (x, t)∂t

− c2

ηω

∂2W (x, t)∂x2

+ ηωW (x, t) = 0 (5.18)

The properties of equation (5.18) (called Nefske’s equation) have been discussed in chapter 3. It canviewed as the differential form of TSEA equation (4.41). In this chapter, the solution of Nefske’s equationis another reference value to compare with exact energy result and MEST solution.The boundary condition for Nefske’s equation are listed here

∂W (0, t)∂x

=∂W (L, t)

∂x= 0 boundary condition

W (x, 0) = w0δ(0) initial condition

(5.19)

where w0 is the initial energy input at x = 0. By the method of separating variables, the solution of Eq.(5.18)-(5.19) is expressed as

W (x, t) =w0

L+

2w0

Le−κt

∞∑n=1

e−( nπc

L√

κ)2t cos

nπx

L(5.20)

Comparing with the solutions of MEST equation shown in 5.17, one find that the solution (5.20) doesn’tdispaly the oscillatory character with respect to time.

5.1.4 Numerical simulation and analysis

Energy density expressed in (5.10, 5.17, 5.20) by three methods are applied in this section. The firstexample is the rod subjected to an impulse excitation. The numerical simulation is expressed in non-dimensional form: Wn = W/w0. Three results of non-dimensional energy density versus time variableare shown in Figure 5.6, 5.7 and 5.8. In these figures, the energy time history at six locations x =0, l/6, l/3, l/2, 2l/3, 5l/6, are plotted together. It is seen that the solutions of MEST equation shownin Figure 5.7 is very similar to the exact energy in Figure 5.6. Energy density at each position is a seriesof pulse function with a period of T = 2l

c , and they display a same decay rate. However, the solutionof first order diffusion equation (5.18) in Figure 5.8 is much different from the former: energy valuesof each position, decrease very rapidly and then keep in a rather lower energy level, and it has not anexplicit concept of energy velocity.

In order to distinguish the difference among them, we select two positions (x1 = 0.4 m and x2 = 0.8m) to observe the time history of energy density got from three methods. The comparisons are madeamong MEST, exact energy results, and the solution of first-order diffusion equation. The numericalresults is shown in the Figure 5.9 and 5.10. It is found that the shapes and magnitude of energy densitypredicted by MEST are almost the same as the exact results, and energy form are periodic pulses, themaximum non-dimensional energy value is 0.5 which corresponds the travel time t1 = x1

c . At the timebefore t1, the transient longitudinal wave has not yet arrived at the position x1 = 0.4, so the energylevel is zero before t1. It demonstrates that the energy behavior predicted by MEST agrees well to thephysical explanation. Energy level predicted by diffusion (Nefske’s) equation, however, is much lower


0 0.5 1 1.5 2 2.5 3

x 10−3

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

ener

gy d

ensi

ty

Figure 5.6: Exact energy density from dis-placement response

0 0.5 1 1.5 2 2.5 3

x 10−3

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

ener

gy d

ensi

ty

Figure 5.7: MEST prediction for the rod sub-jected an impulse excitation

0 0.5 1 1.5 2 2.5 3

x 10−3

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

ener

gy d

ensi

ty

Figure 5.8: Solution of Nefske’s first order equation


0 0.5 1 1.5 2 2.5 3

x 10−3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Ene

rgy

dens

ity

Figure 5.9: Comparison of energy density in the rod: x = 0.4m. ——, MEST; – – – –, exact results; – ·– ·, solution of Nefske’s equation.

0 0.5 1 1.5 2 2.5 3

x 10−3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Ene

rgy

dens

ity

Figure 5.10: Comparison of energy density in the rod: x = 0.8m. ——, MEST; – – – –, exact results; –· – ·, solution of Nefske’s equation.


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Length (m)

Nor

mal

ized

ene

rgy

dens

ity

Figure 5.11: Exact energy density in space do-main

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Length (m)

Nor

mal

ized

ene

rgy

dens

ity

Figure 5.12: MEST prediction of energy distri-bution in the rod

than MEST energy results. Its peak value is about one tenth of that obtained from MEST, and energyvelocity is so high that it arrive at the position instantaneously. In fact, this result has been discussed bythe relevant analysis of the Green function for Nefske’s equation in section 3.4.

In addition to observing energy density in time domain, the behavior of energy density in spatialdomain are also available according to the expressions of energy density listed in last section. Theenergy results by three methods are shown in Figures 5.11-5.13, respectively.

Not like Figures 5.6-5.8 which display the time history of energy density at several locations, Figures5.11-5.13 show the spatial distribution of energy density at many time point. Note that only six locations(0, 0.2, 0.4, 0.6, 0.8, 1) are taken as spatial coordinates. It is found again that MEST solution in Figure5.12 is very similar to exact energy results in figure 5.11. One find that, at each time point, the values ofenergy density at the ends (x = 0 and x = 1) are twice of energy level in other positions. It is becausethe energy densities at ends are augmented by the returning energy wave.

Similarly, the comparisons of three kinds of energy results at given time points are made and shownin Figures 5.14-5.16. The energy distributions in the rod were observed at five time points, t0 = 0ms,t1 = 0.048ms, t2 = 0.096ms, t3 = 0.144ms, and t4 = 0.192ms. Exact energy result and MEST solutionshow the energy pulse moves from the excited end to another end, its velocity is c = 5.19 × 103 m/s.The duration of time taken by transient wave propagates from excited end to another end is so small,0.192ms, that the energy decay was not apparently viewed. At the first time point, the Nefske’s equationgive a same prediction as two others, however, the energy distribution at followed time points (t1, t2, t3,t4) were completely different. Energy does not behave like a moving pulse but spread instantaneouslyalong the rod. The sub-figures about these time points (t1, t2, t3, t4) in Figure 5.16 were zoomed in toobserve the energy level, and it was found energy density in these positions is rather small.


0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Length (m)

Nor

mal

ized

ene

rgy

dens

ity

Figure 5.13: Energy distribution predicted by the first order diffusion equation

130T

ime-varying

vibrationalenergyin

one-dimensionaldistributed

structures

0 0.5 10

0.5

1

Ene

rgy

dens

ity

t=t00 0.5 1

0

0.2

0.4

0.6

0.8

t=t10 0.5 1

0

0.2

0.4

0.6

0.8

t=t20 0.5 1

0

0.2

0.4

0.6

0.8

t=t30 0.5 1

0

0.5

1

t=t4

Figure 5.14: Distribution of energy at different time points by exact result

0 0.5 1−0.5

0

0.5

1

Ene

rgy

dens

ity

t=t00 0.5 1

−0.2

0

0.2

0.4

0.6

t=t10 0.5 1

−0.2

0

0.2

0.4

0.6

t=t20 0.5 1

−0.2

0

0.2

0.4

0.6

t=t30 0.5 1

−0.5

0

0.5

1

t=t4

Figure 5.15: Distribution of energy at different time points by MEST solution

0 0.5 1−0.5

0

0.5

1

Ene

rgy

dens

ity

t=t00 0.5 1

−0.05

0

0.05

t=t10 0.5 1

−0.05

0

0.05

t=t20 0.5 1

−0.05

0

0.05

t=t30 0.5 1

−0.05

0

0.05

t=t4

Figure 5.16: Distribution of energy at different time points by diffusion equation


0 0.5 1 1.5 2 2.5

x 10−3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−12

Time(s)

Ene

rgy(

J)

Figure 5.17: Energy density in undamped rod by displacement response

Since the solution of Nefske’s equation (5.18) has been found untenable to predict time-varyingenergy, we turned our attention to the comparison of MEST solution and exact results. In followingexample, the rod was subjected to a step excitation. For undamped rod, exact energy result and MESTsolution were shown in Figures 5.17-5.18 while Figures 5.19-5.20 displayed the time history of energydensity in damped rod.

0 0.5 1 1.5 2 2.5

x 10−3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−12

Times(s)

Ene

rgy(

J)

Figure 5.18: MEST solution for undamped rod

A very good agreement was found again for undamped case by comparing two figures. However,


0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

8

9x 10

−13

Time (ms)

Ene

rgy

dens

ity (

J)

Figure 5.19: Energy density in damped rod by displacement response

if damping effect was considered in second case, the comparison of energy results from two methods(MEST and exact result) in Figures 5.19 and 5.20 showed some differences. It is partly because thedifferent numerical methods: MEST solution was sought by Laplace transform, while exact energy camefrom the displacement response which calculated by normal mode method (see expression (5.8)). In thecalculation of the latter, the more normal modes were synthesized, the result better. On the other hand,the assumption for MEST requires that kinetic energy density (Wk) is equal to potential energy density(Wp) which have been averaged in the period. This assumption results in some minor error in contrastwith exact results. A specific position, say x = 0.7l, was taken to observe the comparison of two resultssimultaneously. It was shown in Figure 5.21. To some extent, predication by MEST seems to be morestraightforward to find MEST solution while so-called exact energy results was indirectly acquired bydisplacement response as demonstrated by (5.10).

A bird’s-eye view of energy evolution in space-time domain, of the damped rod subjected to a stepexcitation, was expressed in Figure 5.22. Further, the solutions of MEST expressed in (5.17) showthat total energy density can also be viewed as the sum of the forward and backward traveling energydensities. In other words, it means vibrational energy transmitting accompanies waves propagating. Inthe case of the rod subjected to a impulse excitation, two energy components were decomposed and theirbehavior were exhibited in Figure 5.23.

The above discussions with examples verify the applicability of MEST to describe the energy motionby longitudinal waves. Note that the energy analysis of longitudinal wave in a free-clamped rod is justa simple example. Since our major purpose just aims to validating the method, thus an application to arather complicated structures has not been involved at this stage.


0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

8

9x 10

−13

Time (ms)

Ene

rgy

dens

ity (

J)

Figure 5.20: MEST solution for damped rod

0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

8

9x 10

−13

Time (ms)

Ene

rgy

dens

ity (

J)

Figure 5.21: Comparison of energy results between MEST and exact value. (x = 0.3L). ——, solutionof MEST equation; – – – –: energy results from displacement


0

0.2

0.4

0.6

0.8

10 0.5 1 1.5 2 2.5

x 10−3

0

0.5

1

x 10−12

Time (s)

Rod length (m

)

Ene

rgy

dens

ity (

J)

Figure 5.22: Energy results of the rod subjected to the step impact

0 0.5 1 1.5 2 2.5 3

x 10−3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−13

Time(s)

Ene

rgy(

J)

Figure 5.23: Right- and left-traveling energy densities (x = 0.3L). ——, W+; – – – –, W−.

5.2. Flexural wave: time-varying energy in a beam 135

F

X

l=0.6L

x=Lx=0

Figure 5.24: Beam subjected to a transverse unit impulse

5.2 Flexural wave: time-varying energy in a beam

This section was devoted to an application of MEST to a beam characterized by traverse vibration andflexural waves. Of all the various wave types, flexural waves (which also called bending waves) are themost important for sound radiation. As discussed at some length in sections 3.1.3 and 4.5.2, in contrastwith longitudinal waves, the characteristics of flexural waves and the underlying differential equationsdiffer much greatly – they must be represented by four field variables instead of two, and therefor alsothe boundary conditions are more complex.

5.2.1 Time-varying energy expression by displacement response

The derivation of the governing equation for flexural waves in a beam was not mentioned again, as it canbe found in every textbook of vibration mechanics. We just introduce the governing equation for flexuralwaves in an Euler-Bernoulli beam with viscous damping coefficient κ,

EI∂4u

∂x4+ ρS

∂2u

∂t2+ ρSd

∂u

∂t= F0δ(x− 0)δ(t− 0) (5.21)

where S is the section area,E is Young’s modulus, and section inertia moment I = bh3/12, d is dampingcoefficient. Note that the damping effect can be taken into account by complex Young’s modulus E∗ dueto the characteristics of flexural waves. In frequency domain, equation (5.21) becomes

E∗I∂4u

∂x4− ρSω2u = F0δ(x− 0) (5.22)

where complex Young’s modulus E∗ = E(1 + iη), and η = dω . Thus the viscous damping coefficient κ

is replaced by hysteretic damping coefficient η.

Let the general solution in the form of

u(x, t) = u0ei(kx−ωt) = u(x)e−iωt (5.23)

Substituting (5.23) into (5.21), it yields

k = 4

√ω2ρS

E∗I(5.24)

For the cantilever beam (free-clamped) shown in Figure 5.24, the boundary conditions are(d2u

dx2

)x=0

= 0,(

d3u

dx3

)x=0

= 0, (u)x=L = 0,(

dudx

)x=L

= 0 (5.25)


by imposing these boundary conditions, one can get the following frequency equation:

cos kL cosh kL = −1 (5.26)

It yields the approximate roots which may be calculated with the formula

knL ≈(n− 1

2

)π, n = 1, 2, 3, · · · (5.27)

and the normal modes were found as

ϕn(x) = cos knx+ cosh knx+An (sin knx+ sinh knx) (5.28)

where

An =sin knl − sinh knL

cos knL+ cosh knL(5.29)

Several methods can used to solve equation (5.21, 5.22), for example, Fourier transform, Laplace trans-form, and normal modes methods, etc. [92]. Equation (5.22) with initial conditions u(x, t)t=0 = 0 and(

dudt

)t=0

= 0 was solved by a Laplace transform — normal mode expansion. In frequency domain, thesolution was given

u(x, ω) =2F0

ρSL

∞∑n=1

ϕn(x)ϕn(0)ω2

n − ω2

=4F0

ρSL

∞∑n=1

ϕn(x)ω2

n − ω2

(5.30)

where nature frequencies are

ω2n = k4

n

E∗IρS

(5.31)

The frequency response of the beam at position x = 0.6L is shown in Figure 5.25 where peak values ofanti-resonant were found at ω = 1457.7, 4699.8, 9877.2 rad/sec, as probably because that the positionx = 0.6L is the node for the modes associated with these frequency components.

In addition, the solution of equation (5.21) in time domain is

u(x, t) =4F0e−ζt

ρSL

∞∑n=1

ϕn(x) sin(√

a2k4n − ζ2 t

)√a2k4

n − ζ2(5.32)

where

a2 =EI

ρS, ζ =

κ

2(5.33)

Since the flexural waves are dispersive, the group velocity is thus frequency dependent. The responses ineither time domain or frequency domain are the superposition of successive modes in a specific frequencyband. So is the energy density in the beam. The energy density is defined as the sum of the potentialenergy density and the kinetic energy density in terms of the displacement frequency response u,

W (x, ω) =ρSω2

4|u(x, ω)|2 +

Re(E∗I)4

∣∣∣∣∂2u(x, ω)∂x2

∣∣∣∣2 (5.34)


101

102

103

104

−100

−50

0

50

Frequency (rad/sec)

Fre

quen

cy r

espo

nse

(dB

)

Figure 5.25: Frequency response of the displacement at x = 0.6L

Or, the velocity response in time domain corresponding to a specific frequency band, v(t), can be ob-tained by a band-pass filter designed in Signal Processing Toolbox of Matlab c©. The time history ofkinetic energy density thus was expressed

Wk(t) =ρS

2v2(t) (5.35)

We will use this expression as the exact energy result to compare the prediction given by energy ap-proaches.

5.2.2 Solutions of the energy equations

Similarly, the solutions of MEST equation and Nefske’s equation were given to compare the exact energyresult obtained above. In MEST equation,

∂2W (x, t)∂t2

− c2∂2W (x, t)

∂x2+ 2ηωc

∂W (x, t)∂t

+ (ηωc)2W (x, t) = 0, (5.36)

note that ωc is the center frequency of a frequency band, and group velocity is associated with the centerfrequency ωc,

c = 2(E∗Iω2

c

ρA

) 14. (5.37)

These definition are also adopted in Nefske’s equation

∂W (x, t)∂t

− c2

ηωc

∂2W (x, t)∂x2

+ ηωcW (x, t) = 0 (5.38)

The boundary conditions for them are ∂W∂x |(x=0)= ∂W

∂x |(x=L)= 0, while the initial values for

MEST equation is W (x, 0) = w0δ(0) and ∂W (0,t)∂t = −ηωW (x, 0); for diffusion equation it is only


0 0.002 0.004 0.006 0.008 0.01 0.0120

0.02

0.04

0.06

0.08

0.1

0.12

Time (s)

Ene

rgy

dens

ity (

J)

Figure 5.26: Prediction of time history of energy density at l = 0.6L, fc = 1000Hz. . . . . . . , exact energyresult; ——, MEST; – · – ·, diffusion result

W (x, 0) = w0δ(0), where the input energy, w0, to the impulse input, is given by [51]

w0 = 2 |F |2M∆f (5.39)

where M = (ρSc)−1, is the input mobility for a semi-infinite beam, ∆f is frequency bandwidth.Then the solutions can be expressed as

WMEST =w0

L+

2w0

Le−ηωt

∞∑n=1

cosnπct

Lcos

nπx

L(5.40)

Wdiffusion =w0

L+

2w0

Le−ηωt

∞∑n=1

e−( nπc

L√

ηω)2t cos

nπx

L(5.41)

5.2.3 Numerical analysis

The study parameters are as follows: the length of beam, L = 2m, the height of section h = 0.04m,width b = 0.06m, material density, ρ = 7800Kg/m3, Young’s Modulus, E = 2.1 × 1011N/m2, inputforce F is a unit impulse, δ(t). The out-of-plane force F would excite flexural waves in the beam.

The first case is a 1/3 octave band with center frequency 1000 Hz and the observation position wherethe energies are calculated is located at l = 0.6L = 1.2 m. For this case, group velocity c = 1227.1 m/s.The simulation comparison of energy density in the time domain is shown in Figure 5.26. Rather than anenvelop of kinetic energy density to describe the exact energy result, the kinetic energy density withoutbeing averaged by an integral technique was directly to use, and the oscillatory character was viewedobviously in this manner. MEST solution predicted impulse-like energy behavior, as its energy levelwas concentrated at the center frequency ωc, while the exact result can be thought as an energy groupassociated with the frequency band. The solution of Nefske’s equation, however, displayed a quickly-spread energy trend in function with time. A series of peak time (the time point when energy arrives the


0 2 4 6 8

x 10−3

0.02

0.04

0.06

0.08

0.1

0.12

Time (s)

Ene

rgy

dens

ity (

J)

Figure 5.27: Prediction of time history of energy density at l = 0.6L. . . . . . . , exact energy result; ——,MEST; – · – ·, diffusion result

observation position) predicted by MEST and exact result were also found very close. The first energyimpulse arrives after 0.001 s, it is equal to the value of l/c. Another case is a 1/3 octave band with centerfrequency 2000 Hz. The comparison of three methods was shown in Figure 5.27 where similar outcomeof MEST and results result were found except the peak times.

In addition, we may verify again the peak time. Here peak time is defined as the time point when thetime-varying energy response propagates through the observation position every time. For an impulseexcitation at the left end of beam x = 0, the input energy propagates in form of a wave group, thus thepeak time can be easily calculated by the following formula,

Tn =l + (n− 1)L

c+

(1 − (−1)n−1)(L− 2l)2c

, n = 1, 2, 3, · · · (5.42)

where c is the group velocity, L = 2 m is the beam length, and l is the length from the left end (energyinput) to the observation position. For the second case, l = 0.6L = 1.2 m, center frequency was 2000Hz, then energy velocity was 1.7354 × 103 m/s by equation (5.37). In this case the first seven peak timewill be

T1−7 = 0.0007, 0.0016, 0.0030, 0.0039, 0.0053, 0.0062, 0.0076 (s) (5.43)

MEST solution about the this case was re-plotted in Figure 5.28. It was readily found that MEST gaveout accurate prediction which is completely identical to the data above. Another example is a 1/3 octaveband with center frequency 1500 Hz. The observation position was chosen at x = 0.8L = 1.6 m. Thegroup velocity was 1502.9 m/s which got by equation (5.37). The peak time were obtained directly byformula (5.42)

T1−7 = 0.0011, 0.0016, 0.0037, 0.0043, 0.0064, 0.0069, 0.0090 (s) (5.44)

In this case, MEST solution also agrees with the theoretical prediction.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 90

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (ms)

Ene

rgy

dens

ity

Figure 5.28: Prediction of peak time by MEST, fc = 2000 Hz, l = 1.2 m

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.50

0.02

0.04

0.06

0.08

0.1

0.12

Time (ms)

Ene

rgy

dens

ity

Figure 5.29: Prediction of peak time by MEST, fc = 1500 Hz, l = 1.6 m


5.2.4 Energy shock response spectrum

As pointed out previously, the characteristics of dispersive waves are highly dependent on the frequency,or a specific frequency band. It is found complicate to represent the transient responses in time domaincorresponding to each frequency or frequency band, however, the peak values of transient response cor-responding to each frequency band are much desired. To meet the needs, we adopt the concept of shockresponse spectrum (SRS). The shock response spectrum has been traditionally used as a measure of thedamaging potential of a given transient environment, either deterministic or random. For deterministicenvironments, the shock response spectrum concept has also been extended to calculate bound loads fordesign purposes.

Simple speaking, SRS analysis is a tool to provide an estimate of the response of an item to inputshock, with the information presented as a frequency spectrum. The concept of the SRS involves ahypothetical single degree-of-freedom system (oscillator) consisting of a mass supported by a springand a dash pot, both attached to a rigid base. Suppose these single DOF oscillators characterized byundamped natural frequency fn and damping ratio η, SRS, denoted by S(fn, η), then come from thelargest acceleration response of the oscillators when the transient acceleration a(t) is applied at thebase of the oscillators. Although defined above in terms of acceleration for both the input and theoscillator response, the SRS is applicable to any combination of input and response parameters, includingdisplacement, velocity, or acceleration. An implicit assumption is made in the SRS definition that themass of the oscillator is so small compared to the mass of the base that its presence has no effect on thebase input, i.e., the impedance between the oscillator mass and its base is zero. SRS is a useful tool forestimating the damage potential of a shock pulse, and this method can be traced back to the pioneeringwork by Lyon [48] . Note that SRS is essentially different from Power Spectral Density (PSD). Toconstruct the relation between the time domain and the frequency domain, the key was Fourier theoryfor PSD while for SRS, the key lies in the behavior of spring-mass system.

Normally SRS is a calculated function based on the acceleration time history which is a plot ofthe maximum acceleration of a single-degree-freedom oscillator, as a function of the oscillator’s naturalfrequency [93]. When used to estimate design loads of a multi-degree-of- freedom system in responseto deterministic environments, the same definition is used with a modifications — each elastic modeof the system is viewed as a single degree-of-freedom system (oscillator) with a mass represented bythe so-called effective modal mass, leading to a modal SRS. A recent application to pyroshock can beseen in reference [94]. Here, SRS was adjusted itself to express the maximum energy values of everynormal mode of flexural waves in the time domain, thus here it was called energy SFS. The energySRS concept is shown in Figure 5.30. The plot of energy SRS was carried out as follows: we pickedup the maximum energy values from their time history associated with a specific frequency band andcenter frequency, then make these energy peak value correspond to those discrete frequencies. Thisprocedure was depicted in Figure 5.31. SRS results was calculated by following the flowchart 5.31 forthe cantilever beam. As it is just a simple structure, one can easily identify each modes by (5.28) and(5.30) even at high frequencies. Thus the center frequencies in energy SRS calculation were taken bythe nature frequencies corresponding to each modes. For every nature frequency, we can always findthe maximum energy values from their time history predicted by MEST, exact results, and Nefske’sequations. The comparison of them for the slight damping case η = 0.01 was shown in Figure 5.32.SRS of MEST and Nefske’s equation were the smooth curves while the SRS of exact energy resultsdisplayed some oscillatory behavior . It shows MEST agrees well with the exact results, especially in thehigh frequency range. SRS of Nefske’s equation displays much lower value.

Another case was characterized by a relatively higher damping η = 0.15. SRS by three methods


t

r

t

r

t

r

t

r

The eigenvalue problem in

Frequency

Peak

A set of ordinary equations related to each nature frequency

ω3ω1m1, m2, ω2 m3, ωmn n

distributed stucture

Figure 5.30: Concept of SRS model

Group velocity

)( nn fc ω=

Frequency band withcenter frequency

nω

Energy prediction intime domain

)(tEn

Maximum value inabove time history

nS

Energy ShockResponse Spectrum

history

nnS ω~

Next centerfrequency

n=n+1

Figure 5.31: Program diagram for energy SRS plotting

5.3. Summary 143

100

101

102

103

104

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

Energ

y densi

ty (

J)

2nd−order energy Eqnexact resultdiffusion energy Eqn

Figure 5.32: Energy SRS comparison, η = 0.01. – – – –, exact energy result; ——, MEST; – · – ·,diffusion result

were plotted on logarithmic scale in Figure 5.33. In contrast to Figure 5.32, the peak energy level inFigure 5.33 were much lower. Specially at higher nature frequency, three results gradually approach toeach other. It implies that high frequency and high damping would be the condition to narrow the gap ofMEST and Nefske’s equation.

5.3 Summary

This chapter was devoted to validating MEST for distributed structures. The most typical example ofvalidating MEST for distributed system is to examine its Green function for infinite system, which hasbeen discussed in detailed in chapter 3. In fact, MEST equation was derived from continuous system andoriginated in the form of partial differential equation, its applicability to distributed system is logicallyself-adaptive.

Two kinds waves, longitudinal waves and flexural waves are studied by MEST. And the most simplestructures, rod and beams were applied because we can easily find the exact energy results. The numericalanalysis showed that the MEST solutions agree very well with the exact energy results, either in energyamplitudes or rise time; while the diffusion energy equation was not capable of providing an accurateprediction.

It is easily understood that MEST can describe the energy of longitudinal waves. While for dispersivewaves, for example, flexural waves, waves group propagation can be view as an impulse propagates witha velocity c(ω) and energy amplitudeE(ω) (ω is the center frequency of the waves group). It is illustratedby Figure 5.34.

In this case, there is no difference between longitudinal waves and dispersive waves when MEST is


100

101

102

103

104

105

10−12

10−10

10−8

10−6

10−4

10−2

100

Nature frequency (Hz)

Pea

k va

lue

of e

nerg

y de

nsity

Figure 5.33: Energy SRS comparison, η = 0.15. ——-, exact energy result; – – – –, MEST; – · – ·,diffusion result

Figure 5.34: Wave group propagates like an impulse

5.3. Summary 145

applied. Of course, It might not be right when we do it reversely. It means that we can apply MEST topredict the energy level for dispersive wave associated with a certain frequency, but we are unable to tellclearly the motion of each mode. This is the deficiency of energy approach in statistical point of view,however at least, MEST release us from solving fourth-order beam equation.

That’s why Shock response Spectrum (SRS), developed in the early 1960s, continues to grow inpopularity with vibration test lab, even if modern computer and signal processing technology makesit possible to directly compute the response of specific structures resulting from well defined actualtransient forces. In this thesis, SRS has been modified a little to adjust itself to MEST applications.The advantage of SRS is that we don’t need to care about resonance frequencies are represented bythe components mounted on the structure, because we will simply apply the shock transient from thestructure to every conceivable SDOF (Single-Degree-Of-Freedom) structure that could be mounted onthe structure.

Although the analysis and the simulation present in this chapter restricts itself to very simple struc-ture, the MEST equation can be generalized to multi-dimensional application. This subject will bediscussed in next chapter.

Chapter 6

Generalization of the energy equation tomulti-dimensional systems

Although the derivations of MEST do not restrict itself only to one-dimensional structures or only forplane waves, the explicit form of MEST equation in multi-dimension space, especially for symmetricwaves, has not been expressed yet. Here, symmetric waves are specified by cylindrical waves (two-dimensional) and spherical waves (three-dimensional). In fact, many practical vibroacoustic problemsare hitched together with symmetric waves. For example, spherical waves occur in three-dimensionalacoustic field when the sound sources (for example, loudspeaker) are often modeled as a point source; intwo-dimensional structures, the point excitations or point-like loading (very small local load and ignoringof near field effect) also cause cylindrical waves. Of course, the prerequisite for existing of symmetricwaves is isotropic and homogeneous media. Sometimes conditions of an infinite system or direct fieldare necessary. The former means a unbounded system without boundary, thus excited symmetric wavesalways keep their form. The latter arises in the bounded systems, those symmetric waves in directfield are likely broken into a population of plane waves after reflecting, scattering, and diffracting byboundaries, as a result, a complex field including direct field and reverberant field occurs .

As a start point, we will discuss the energy behavior of symmetric waves. However, it is not ouronly motivation. In fact, a complete vibroacoustic problem in a bounded system definitely include twofields: direct field and reverberant field. Moreover, it should be noted that direct field and reverberantfield are characterized by different wave types if the excitations can be modeled as point-like sources.The problem is how to describe this kind of vibroacoustic model: two kinds wave types dominate in twodifferent fields.

In the vibrational conductivity approach, the two different waves and fields seem not be distinguishedclearly, where the fields is uniquely supposed to be diffusive [25, 26] and governed by one equation.Unfortunately, it is found that the treatment of complex field in vibrational conductivity approach cannot provide an accurate estimate of the direct field subjected to point loading [35, 41]. In fact, the physicalnature of two different fields marked by different waves make it logical to consider synthesizing them.Some researchers, like Miles in [36] and Smith in [38] proposed an idea of combination of the directfield and reverberant field, as being reviewed at some length in chapter 1.

In addition, as pointed out in last chapter, MEST can be transformed to Simplified Energy Method(MES) by applying time-averaging technique while the latter is used in steady state problems. In fact, theproblem of complex field mentioned above is of special importance for steady state condition. Therefore,

147

148 Generalization of the energy equation to multi-dimensional systems

the model and numerical analysis are restricted in steady state condition although we will all start fromMEST equation. The validation and numerical simulation of MEST will be held in followed chapters.

Following the ideas in [36, 38], we will first discuss two different fields respectively and demonstratethe difference of LEA and vibrational conductivity approach. Since the two-field problem can be settledby synthesis, then the accurate evaluation of reverberant field become imperative while the direct field isrelatively simple to manage. Thus our second task is to find a complete description of boundary conditionbecause it is a key factor to calculate reverberant field. In addition, we try to build up a FEM model oftwo-dimensional problem with a general boundary condition derived in the second task. Some examplesfollowed the FEM model are given in the later.

6.1 MEST equation for symmetric waves

At first we review the characteristics of symmetric waves. The damping effect and source condition havenot taken into account for this moment.

For waves symmetric about the origin u = u(r, t), where r is the distance from the origin. Thespherical wave equation reads

1c2∂2u

∂t2− ∂2u

∂r2− 2r

∂u

∂r= 0 (6.1)

this may also be written1c2∂2(ru)∂t2

− ∂2(ru)∂r2

= 0 (6.2)

which is the one dimensional wave equation. Thus the general solution takes the simple form

u =1r

[f(r − ct) + g(r + ct)] (6.3)

where f and g are function operators. In the direct field only outgoing waves are considered, thus thesolution is

u =f(r − ct)

r(6.4)

As for cylindrical waves in cylindrical coordinates (r, θ, z), we take the rather simple case, for exam-ple of a thin plate, the effect of thickness is neglected. Then the cylindrical coordinates (r, θ, z) reducesto plane polar coordinates (r, θ). Thus the governing equation for symmetric cylindrical waves may bewritten by

1c2∂2u

∂t2− ∂2u

∂r2− 1r

∂u

∂r= 0 (6.5)

Note that the third term in equation (6.5) differs from that in (6.1), thus (6.5) can not be transformedinto a simple form like (6.2). For a periodic source q(t) = e−iωt, the fundamental solution of (6.5) maybe expressed as

u = C1I0(ωr

c)e−iωt + C2K0(

ωr

c)e−iωt (6.6)

6.1. MEST equation for symmetric waves 149

where I0 is zero order modified Bessel function of the first kind, while M0 is zero order modified Besselfunction of the second kind. The solution (6.6) have different behavior with respect to position.

u ∼ q(t)2π

logr, near field (6.7)

u ∼√

c

2rQ(r − ct), far field (6.8)

where Q if function operator. Note that fundamental solutions of symmetrical waves (6.3, 6.7) representa wavefront behavior. These characteristics of symmetric waves are related to the MEST equation forsymmetric waves.

It may be helpful to have in mind the derivation of time-varying energy equation for plane wavesas discussed in last chapter. For the cylindrical wave and spherical wave, the derivation of basic energyequation is almost the same procedure. Consider a two- or three-dimensional systems Ω (its coordinatex ⊂ R

3) with a point excitation source without near-field effect, the energy balance equation still reads

∂W (x, t)∂t

+ ∇ · I(x, t) + ηωW (x, t) = 0 (6.9)

By considering the symmetric properties of cylindrical and spherical waves, one finds that the diver-gence of energy flow I can be written as

∇ · I =1

rα−1

∂(rα−1I)∂r

(6.10)

andI = cW r (6.11)

where r denotes radius vector, r = r|r| , and α indicates the wave types: α = 2 stands for cylindrical waves

in two-dimensional structures, and α = 3 spherical waves in three-dimensional structures. Surprisingly,it is found that α = 1 can be used to express one-dimensional cases.

Inserting (6.10, 6.11) into (6.9), we get

∂I∂t

+c2

rα−1

∂(rα−1W )∂r

+ ηωI = 0 (6.12)

Similarly, the MEST equation for symmetric waves in multi-dimensional structures can be expressedby the simultaneous set of equation:

∂W

∂t+

1rα−1

∂(rα−1I)∂r

+ ηωW = 0 (6.13)

∂I∂t

+c2

rα−1

∂(rα−1W )∂r

+ ηωI = 0 (6.14)

or, they are transformed to a second-order temporal equation where only energy density W remains,

∂2W

∂t2− c2

rα−1

∂2(rα−1W )∂r2

+ 2ηω∂W

∂t+ (ηω)2W = 0 (6.15)


S

Figure 6.1: Illustration of hybrid field: from direct field to reverberant field

It is interesting that equation (6.15) has a fundamental solution

W (r, t) =1

|r|α−1[f(|r| − ct) + g(|r| + ct)] (6.16)

Comparing with the fundamental symmetrical waves solutions (6.3, 6.7), it is found

W (r, t) ∝ u2(r, t) (6.17)

Recall the discussion of properties of MEST equation for plane waves, we can draw a conclusion: foreither plane waves or symmetric waves, their time-varying energy density all have wavefront character.

6.2 Energy propagation in direct field

Suppose the point sources are mutually independent and uncorrelated, then the total response can besuperposed by the contribution of individual source. Hereby, the discussion below is concentrated on asingle-source condition. From the physics point of view, the response of point-loaded fields consist oftwo part, direct field and reverberant field. If the system is infinite or unbounded, the vibroacoustic energycan be viewed as a only forward propagating wave, then equation (6.13, 6.14) and their fundamentalsolution (6.16) will be reduced as

∂W

∂t+

c

rα−1

∂(rα−1W )∂r

+ ηωW = 0 (6.18)

and

W (r, t) =1

|r|α−1f(|r| − ct) (6.19)

If the system is bounded, energy waves will be reflected by boundary. In fact, near boundaries andinterfaces the waves undergo coherent or partially coherent reflection and transmission, as the interfacesare smooth or randomly rough. Due to the scattering of the symmetrical wave by irregular boundary, thereverberant field then is formed in a diffuse field with plane waves. This concept is shown in Figure 6.1.

6.2. Energy propagation in direct field 151

The distribution of energy density in bounded system can be described as the sum of two field: indirect field with source:

1ηω

∂2Wd

∂t2− c2

ηωrα−1

∂2(rα−1Wd)∂r2

+ 2∂Wd

∂t+ ηωWd = Pin in Ω

0 ≤ r ≤ R(θ)∂Ω

(6.20)

and in reverberant field without source1ηω

∂2Wr

∂t2− c2

ηω∇2Wr + 2

∂Wr

∂t+ ηωWr = 0 in Ω

∀(x, y, z) ∈ ∂Ω : I(r, θ) · n = Φ(r, θ)

(6.21)

where Φ(r, θ) describes the undetermined boundary condition. In the steady state condition, the termsinvolved with time in the above equations will vanish, then in direct field:

− c2

ηωrα−1

d2(rα−1Wd)dr2

+ ηωWd = Pin in Ω

0 ≤ r ≤ R∂Ω

(6.22)

By applying the appropriate boundary condition at r = 0, the solution is

Wd =Pin

2πcrα−1e−ηωr/c (6.23)

and in reverberant field− c2

ηω∇2Wr + ηωWr = 0 in Ω

∀(x, y, z) ∈ ∂Ω : boundary condition undetermined

(6.24)

These energy equations for steady state condition have been investigated by some researchers, like LeBot [41], Ichchou [42]. In their studies, this method was also called Methode Energetique Simplifiee(MES).

As demonstrated above, it is reasonable to separate the problem into two kind of field, becausethe two field are governed by two different equation: equation (6.22) is for symmetric waves whileequation (6.24) for plane waves. For the purpose of comparison, we intentionally deal with the vibrationconductivity equation by the same manner.

Recall the diffusion vibrational conductivity equation− c2

ηω∇2Wr + ηωWr = Pin in Ω

∀(x, y, z) ∈ ∂Ω : I · n = −cW (1 −R)1 +R

(6.25)

where R is the reflectivity coefficient. Applying the polar coordinate, the direct field described by equa-tion (6.25) is

− c2

ηωrα−1

ddr

(rα−1 dWd

dr) + ηωWd = Pin in Ω

0 ≤ r ≤ R∂Ω

(6.26)


F

1 m

1 m

Figure 6.2: The square plate used to demonstrate the direct-field energy

For vibrational conductivity approach, the energy govern equations in direct field and reverberant fieldare the same. However, equation (6.26) is not the same as (6.22), the former yields different fundamentalsolutions,

Wd =Pin

4πcK0(ηωr/c) for 2D cylindrical wave (6.27)

Wd =Pin

4πcre−ηωr/c for 3D spherical wave. (6.28)

In reverberant field, vibrational conductivity approach also reads− c2

ηω∇2Wr + ηωWr = 0 in Ω

∀(x, y, z) ∈ ∂Ω : I · n = −cW (1 −R)1 +R

(6.29)

By comparing equation (6.29) with (6.24), One finds that the two energy approaches fall into the sameformat in the reverberant field: both of them are plane wave solution.

Since there is no difference in reverberant fields between two approaches, we may compare theenergy behavior in the direct field. In fact, the expressions (6.23) and (6.28) are the fundamental solutionsor called Green function corresponding to the two kind energy equation (6.22) and (6.26), respectively.We take a two-dimensional thin plate as an example to illustrate the different solutions from two methods:local energy approach and vibrational conductivity approach. Consider a simple square plate shown inFigure 6.2, the parameters are as follows: length L = 1m; thickness h = 0.001m; material density 7800kg/m3; Poisson ratio ν = 0.3, Young’s Modulus 2.1 × 1011 N/m2; damping loss factor η = 0.001. Thepoint load is located at xo = 0.532, y0 = 0.481; center frequency is 3000 Hz. The input power is get by

Pin =F 2

16√Dρh

(6.30)

where D id the plate stiffness.

The comparison of the direct field solution are shown in following Figures 6.3-6.6. Figure 6.3 isthe three-dimensional plots for solutions of (6.23) and (6.28). It is found that the distribution of energydensity are clearly different between them. The decrease slope from peak energy (at excitation point)predicted by local energy approach differs from that calculated by vibrational conductivity approach.This can be seen in the Figure 6.4 by contour plot and Figure 6.5 is the three-dimensional version.

6.2. Energy propagation in direct field 153

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

165

70

75

80

85

90

X (m)Y (m)

Ene

rgy

dens

ity (

dB)

(a) local energy approach

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

160

65

70

75

80

85

90

X (m)Y (m)

Ene

rgy

dens

ity (

dB)

(b) vibrational conductivity approach

Figure 6.3: Energy density in direct field plotted in three-dimensions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Figure 6.4: Energy density in direct field plotted in contour curves


00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1−5.5

−5

−4.5

−4

−3.5

−3

X (m)y (m)

Ene

rgy

dens

ity


00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1−6

−5.5

−5

−4.5

−4

−3.5


Figure 6.5: Energy density in direct field plotted in three-dimensional contour curves

The more explicit comparison of decrease slope from peak energy was demonstrated in Figure 6.6(a)where the energy density is displayed along y = 0.5m. The peak amplitude of energy density by vibra-tional conductivity is found 8dB lower than the solution of local energy approach; energy distributionof the former seems more flat off the near field. At the boundary, however, vibrational conductivity ap-proach offer a higher energy level than the other. Figure 6.6(b) shows the energy density near boundary(along y = 0.15m), where the prediction of vibrational conductivity approach is rather higher than thatof local energy approach. The comparison between these two energy methods and exact energy resultsare given in the later of this chapter. Of course the energy distribution in direct field is neither the com-plete energy value nor the major topics discussed in this chapter. The following content is concentratedon reverberant field.

6.3 Boundary condition for reverberant field

While the problems of reverberant field in the multi-dimensional structures discussed above are clearly ofmajor difficulty, the analysis on boundary conditions should be treated carefully because they are the keyfactors to solve the problem. Here we take two-dimensional structures (a single plate and two coupledplates) as examples to demonstrate the problem of boundary condition.

A wave decomposition technique is adopted here. The components of I and W , I+(x, t), I−(x, t),W+(x, t) andW−(x, t) will represent respectively the energy flows and the energy densities correspond-ing to the forward traveling and the backward traveling waves, s is the space variable. First consider apure progressive plane wave propagating in the specific direction. The superposition principle is appliedby

W (x, t) = W+(x, t) +W−(x, t) (6.31)

I(x, t) = I+(x, t) + I−(x, t) (6.32)

6.3. Boundary condition for reverberant field 155

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

15

20

25

30

35

40

45

50

Length (m)

Ene

rgy

dens

ity (

dB)

(a) Energy distribution along y = 0.5m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 113

14

15

16

17

18

19

20

21

Length (m)

Ene

rgy

dens

ity (

dB)

(b) Energy distribution along y = 0.15m

Figure 6.6: Comparison between two energy methods. ——, local energy approach; - - - -, vibrationalconductivity approach

The constitutive law between partial energy density and energy flow can be expressed as

I+(x, t) = c ·W+(x, t) · n, I−(x, t) = −c ·W−(x, t) · n (6.33)

where n is unit normal vector.

The relation between components and total of energy flow can also obtain from (6.32) and (6.33)

I+ =12(I + cW · n) (6.34)

I− =12(I − cW · n) (6.35)

The problem of boundary condition for transport equation can be found in reference [95]. In this thesis,we take an engineer’s (rather than a mathematician’s, see in [95]) point of view and restrict the scopeand detail so as to include the boundary conditions regularly used in practical applications in multi-dimensional system dynamics. Thus the discussion follows some assumptions:

• incident waves and reflected waves are incoherent.

• waves are diffusely scattered at boundary.

• height and length of roughness smaller than or comparable to the wavelength.

In addition, the different treatment on boundary condition and coupling condition leads to the discussionon isolated system and coupled system, respectively.


6.3.1 The case of isolated system

First we consider the boundary condition of an isolated system. An two-dimensional isolated structureis depicted in Figure 6.7.

Source

nM

Mds

I−(θ,M)dθ

I+(θ,M)dθ

Id

Ω

α

θ

Figure 6.7: Illustration of boundary condition of an isolated two dimensional structure

Suppose ds is the differential element on the given point M in boundary, and the energy flowsinvolved here are defined in ds. Thus the incident energy flow in reverberant field along θ within dθ∫

θI+(θ,M)dθ · nM (6.36)

where θ ∈ [−π2 ,

π2 ]. The reflected energy flow consists of two parts: one comes from the reflection of

incident energy flow, r(θ)I+(θ,M)dθ · nM ; the other is the reflection of direct-ield energy flow Id, thatis ∫

θI−(θ,M)dθ · nM = −

∫θr(θ)I+(θ,M)dθ · nM + r(α)Id(α,M) · nM (6.37)

where r(θ) is the reflectivity coefficient at a given direction θ, while r(α) is the reflectivity coefficient atM along a fixed angle α from source point, Id represents the the angularly resolved energy flow in directfield. Then imposing the constitutive relation (6.34, 6.35) into equation (6.37), we get∫

θ(1+r(θ))I(θ,M) · nMdθ −

∫θ(1−r(θ))cW (θ,M)dθ = −r(α)Id(α,M) · nM (6.38)

The assumption of diffusely reflecting boundary make the energy flow and energy density to be angularlyresolved, that are

I(M) =∫ π

2

−π2

I(θ,M)dθ = πI(θ,M) (6.39)

W (M) =∫ π

2

−π2

W (θ,M)dθ = πW (θ,M) (6.40)


Applying assumptions (6.39, 6.40) to equation (6.38), it yields

I(M) · nM =

∫θ(1 − r(θ))dθ∫θ(1 + r(θ))dθ

cW − 2π r(α)∫θ(1 + r(θ))dθ

Id · nM (6.41)

Introduce a general reflecting coefficient

R =∫

θr(θ)dθ (6.42)

then equation (6.41) is written as

I(M) · nM =π −R

π +RcW − 2π r(α)

π +RId · nM (6.43)

More generally, it can be rewritten as

I(M) · nM = βW + λId · nM (6.44)

Note that 0 ≤ R ≤ π and 0 ≤ r(α) ≤ 1, thus 0 ≤ β ≤ 1, 0 ≤ λ ≤ 1 and they can be viewed asgeneralized reflection coefficient. The diffusely reflecting boundary condition described by (6.44), isactually so-called generalized Neumann (or Robin) boundary condition. There are two extremities casesshould be investigated:

• If it is perfect reflecting boundary, then r(θ) = 1, r(α) = 1, and R = π, the boundary condition(6.43) will be

I(M) · nM = −Id · nM (6.45)

• If it is non-reflecting boundary, namely, boundary tends to infinite, then r(θ) = 0, r(α) = 0, andR = 0, the boundary condition is reduced as

I(M) · nM = cW (6.46)

The boundary condition expressed by (6.44) covers completely the reflections from direct field andreverberant field itself. As a contrast, the boundary condition proposed in reference [38] only considerthe reflection from direct field, it writes

I = −qId (6.47)

where q is the reflection coefficient. In fact, the energy flows in reverberant field would be, if the dampingloss is not extremely heavy, reflected numberous times, thus the reflection of energy flow in reverberantfield should be taken into account.

6.3.2 The case of coupled subsystems

The derivation of coupling boundary conditions is similar to the subsection above. Suppose M is thepoint located on the coupling boundary which is shown in Figure 6.8.


nM

M

I−1 (θ1,M)dθ1

I+1 (θ1,M)dθ1

I−2 (θ2,M)dθ2

I+2 (θ2,M)dθ2

I1d

I2d

Ω1

Ω2

α1

θ1

α2

θ2

S1S2

Figure 6.8: Illustration of boundary conditions of two dimensional coupled structures

For subsystem 1, the reflected energy flow in reverberant field may be∫θ1

I−1 (θ1,M)dθ1 · nM = −∫

θ1

r1(θ1)I+1 (θ1,M)dθ1 · nM − r1(α1)I1d(α1,M) · nM

−∫

θ2

t2(θ2)I+2 (θ2,M)dθ2 · nM − t2(α2)I2d(α2,M) · nM (6.48)

where r1, r2 are the reflectivity coefficient along common boundary in the two coupled plates, respec-tively; t1, t2 are the transmitting coefficients, and r1 + t1 = 1; r2 + t2 = 1.

For subsystem 2, the reflected energy flow in reverberant field can be written in the similar form,∫θ2

I−2 (θ2,M)dθ2 · nM = −∫

θ2

r2(θ2)I+2 (θ2,M)dθ2 · nM − r2(α2)2I2d(α2,M)dθ2 · nM

−∫

θ1

t1(θ1)I+1 (θ1,M)dθ1 · nM − t1(α1)I1d(α1,M)dθ1 · nM (6.49)

First consider the subsystem 1. We transform equation (6.48) to[∫θ1

r1(θ1)I+1 (θ1,M)dθ1 +

∫θ1

I−1 (θ1,M)dθ1

]· nM = −r1(α1)I1d(α1,M) · nM

−∫

θ2

t2(θ2)I+2 (θ2,M)dθ2 · nM

−t2(α2)I2d(α2,M) · nM (6.50)

Substituting the relations (6.34, 6.35) into equation (6.50), it produces[∫θ1

(1 + r1(θ1))I1(θ1,M)dθ1 +∫

θ2

t2(θ2)I2(θ2,M)dθ2

]· nM

=∫

θ1

(1 − r1(θ1)) c1W1(θ1,M)dθ1 −∫

θ2

t2(θ2) c2W2(θ2,M)dθ2

− 2 [r1(α1)I1d(α1,M) + t2(α2)I2d(α2,M)] · nM

(6.51)


Integrating (6.51) and applying the diffusive reflecting assumption ( 6.39, 6.40), it yields[I1(M)

∫θ1

(1 + r1(θ1))dθ1 + I2(M)∫

θ2

t2(θ2)dθ2

]· nM

=c1W1(M)∫

θ1

(1 − r1(θ1))dθ1 − c2W2(M)∫

θ2

t2(θ2)dθ2

− 2π [r1(α1)I1d(M) + t2(α2)I2d(M)] · nM

(6.52)

Similarly for subsystem 2, the power balance at the point M for plate 2 follows the same pattern[I2(M)

∫θ2

(1 + r2(θ2))dθ2 + I1(M)∫

θ1

t1(θ1)dθ1

]· nM

=c2W2(M)∫

θ2

t2(θ2)dθ2 − c1W1(M)∫

θ1

(1 − r1(θ1))dθ1

− 2π [r2(α2)I2d(M) + t1(α1)I1d(M)] · nM

(6.53)

Denote

R1 =∫

θ1

r1(θ1)dθ1, R2 =∫

θ2

r2(θ2)dθ2, (6.54)

T1 =∫

θ1

t1(θ1)dθ1, T2 =∫

θ2

t2(θ2)dθ2, (6.55)

then R1 + T1 = π, R2 + T2 = π.

Thus coupled boundary condition (6.52, 6.53) can be written in matrix format:I1(M)I2(M)

· nM = A

−1B

c1W1(M)

c2W2(M)

− A

−1C

I1d(M)I2d(M)

· nM

= D1

c1W1(M)c2W2(M)

+ D2

I1d(M)

I2d(M)

· nM (6.56)

where

A =[π +R1 T2

T1 π +R2

], B =

[T1 −T2

−T1 T2

],

C = 2π[r1(α1) t2(α2)t1(α1) r2(α2)

], D1 =

1R1 +R2

[T1 −T2

−T1 T2

],

D2 =1

R1 +R2

[T2 − 2πr1(α1) T2 − 2πt2(α2)T1 − 2πt1(α1) T1 − 2πr2(α2)

]Or, they may be expressed separately

I1(M) · nM = β11W1 − β22W2 + (λ11I1d + λ12I2d) · nM (6.57)

I2(M) · nM = β22W2 − β11W1 + (λ21I1d + λ22I2d) · nM (6.58)


where

β11 =T1c1

R1 +R2, β22 =

T2c2R1 +R2

,

λ11 =T2 − 2πr1(α1)R1 +R2

, λ12 =T2 − 2πt2(α2)R1 +R2

,

λ21 =T1 − 2πt1(α1)R1 +R2

, λ22 =T1 − 2πr2(α2)R1 +R2

The analogous formulas have been obtained in [95]. We state again that the boundary loss is not consid-ered here, which means r + t = 1. Thus, by adding (6.57) and (6.58), one finds

I1(M) · nM + I2(M) · nM = I2d + I1d (6.59)

It suggests that all the energy flows in reverberant fields come from the reflection of direct fields. More-over, if t2(α2) and t2(θ2) be zero in equation (6.52), or if t1(α1) and t1(θ1) be zero in equation (6.53), itmeans two plates are isolated separately, the boundary conditions of two plates are reduced to equation(6.43).

6.4 Finite element method for Local Energy Approach

Generally, it is almost impossible to solve previous equation (6.21) analytically if structures have arbi-trary, complex or irregular boundary condition. Other approximate numerical methods, like FEM, BEM,etc, serve to deal with this kind of problems. This section try to use variational principle to build up afinite-element model.

6.4.1 FEM formulations derived from variational method for isolated system

After getting the boundary condition, the governing equation of energy in reverberant field may be ex-pressed as

1ηω

∂2Wr

∂t2− c2

ηω∇2Wr + 2

∂Wr

∂t+ ηωWr = 0

∀M(x, y) ∈ Γ, I(M) · nM = βWr + λId · nM

(6.60)

where Id denotes the energy flow along Γ coming from direct field, which is supposed to be known bysolving equation (6.22).

Suppose the elements of the function space is denoted by Ωe without reference to a specific geometricshape. For the sake of simplicity, we denote D = c2

ηω , G = ηω, Wr = ∂2Wr∂t2

, and Wr = ∂Wr∂t . Multiply

the equation (6.60) with an arbitrary test function variation v, and integrate the product by parts on Ωe :∫∫Ωe

(D∇Wr∇v +GvWr +

1GvWr + 2vWr

)dxdy −

∮∂Ωe

D∇Wrv · ndΓ = 0 (6.61)

The boundary integral can be replaced by the non-essential boundary condition:∫∫Ωe

(D∇Wr∇v +GvWr +

1GvWr + 2vWr

)dxdy −

∮Γv (βWr + λId · nM ) dΓ = 0 (6.62)

6.4. Finite element method for Local Energy Approach 161

An approximate solution of Wr is interpolated as the form

Wr = [N ]1×n

W er

n×1=

n∑i=1

ψei (x, y)(W

er )i (6.63)

where i is the element node, [N ] is the matrix of shape function, and ψei (x, y) is the shape function (If

elements have corner nodes only, then n = 3 for a triangle, n = 4 for a rectangle.).

Substituting equation (6.63) and ψej (x, y) for v into equation (6.62), we obtain the system of equa-

tionsn∑

i=1

(∫∫Ωe

1Gψiψj dxdy

)(W e

r )i +n∑

i=1

(∫∫Ωe

2ψiψj dxdy)

(W er )i

+n∑

i=1

[∫∫Ωe

(D∇ψi∇ψj +Gψiψj) dxdy −∮

Γβψiψj dΓ

](W e

r )i

−∮

ΓλIdψj · ndΓ = 0 (6.64)

Use the following notations:

[mr] =∫∫

Ωe

1Gψiψj dxdy

[cr] =∫∫

Ωe

2ψiψj dxdy

[k1r ] =

∫∫Ωe

D∇ψi∇ψj dxdy

[k2r ] =

∫∫Ωe

Gψiψj dxdy

[k3r ] = −

∮Γβψiψj dΓ

[br] =∮

ΓλIdψj · ndΓ

equation (6.64) is rewritten by

[mr]W er + [cr]W e

r + [kr]W er − [b] = 0 (6.65)

where [kr] = [k1r ] + [k2

r ] + [k3r ].

Upon assembly process of elements, the nodal values W er is replaced by the global vector Wr,

which contains all nodal energy density of the structure. The global equation is therefore(∑[mr]

)Wr +

(∑[cr]

)Wr +

(∑[kr]

)Wr −

∑[b] = 0,

or,[Mr]Wr + [Cr]Wr + [Kr]Wr − [B] = 0 (6.66)

Similarly, we can get the FEM equation of direct energy by same process, it can be expressed as

[Md]Wd + [Cd]Wd + [Kd]Wd − [P ] = 0 (6.67)


where [Kd1], [Kd2], and [Kd] are the analogous meaning, [P ] is the global matrix of input power. Finally,the total local energy is the sum of two different energy field:

([Md] + [Mr]) W + ([Cd + [Cr]) W + ([Kd] + [Kr]) W − [P ] − [B] = 0 (6.68)

When used in steady state condition, it reduces to

([Kd] + [Kr]) W − [P ] − [B] = 0 (6.69)

6.4.2 FEM formulations derived from variational method for coupled subsystem

Now we turn to consider two coupled subsystems which depicted in Figure 6.8. The direct fields in twocoupled subsystem are similarly dealt with as single system mentioned above, while the reverberant fieldof subsystem 1 can be described

1η1ω

∂2Wr1

∂t2− c21η1ω

∇2Wr1 + 2∂Wr1

∂t+ η1ωWr1 = 0

∀M1(x, y) ∈ Γ1,c21η1ω

∇Wr1 · nM1 = β1Wr1 + λId1 · nM1

∀M(x, y) ∈ Γ,c21η1ω

∇Wr1 · nM = β11W1 − β22W2 + (λ11Id1 + λ12Id2) · nM

(6.70)

and subsystem 2

1η2ω

∂2Wr2

∂t2− c22η2ω

∇2Wr2 + 2∂Wr2

∂t+ η2ωWr2 = 0

∀M2(x, y) ∈ Γ2,c22η2ω

∇Wr2 · nM2 = βWr2 + λId2 · nM2

∀M(x, y) ∈ Γ,c22η2ω

∇Wr2 · nM = β22W2 − β11W1 + (λ21Id1 + λ22Id2) · nM

(6.71)

Consider the first system in Ω1. Denote D1 = c21η1ω , G1 = η1ω, and multiply the equation (6.71) with

an arbitrary test function v1 and integrate on Ω1:∫∫Ω1

(−D1v1∇2Wr1 +G1v1Wr +

1G1

v1Wr1 + 2v1Wr1

)dxdy = 0 (6.72)

Integrating by parts, it yields∫∫Ω1

(D1(∇Wr1∇v1) +G1v1Wr1 +

1G1

v1Wr1 + 2vWr1

)dxdy

−∫

Γ1

D1∇Wr1v1 · ndΓ1 −∫

ΓD1∇Wr1v1 · ndΓ = 0 (6.73)

6.4. Finite element method for Local Energy Approach 163

The two boundary integrals can be replaced by the two kinds boundary condition list in equation (6.71):∫∫Ω1

(D1(∇Wr1∇v1) +G1v1Wr1 +

1G1

v1Wr1 + 2v1Wr1

)dxdy

−∫

Γv1(β11W1 − β22W2 + (λ11Id1 + λ12Id2) · nM ) dΓ

−∫

Γ1

v1(β1Wr1 + λId1 · nM1),dΓ1 = 0 (6.74)

Since the differential operator is linear, we seek an approximation of the test function as ψi (Suppose theelement type in system Ω2 is same as the Ω1, then the same test function will be used in Ω2), i.e.,∫∫

Ω1

(D1(∇Wr1∇ψi) +G1ψiWr1 +

1G1

ψiWr1 + 2ψiWr1

)dxdy

−∫

Γψi [β11W1 − β22W2 + (λ11Id1 + λ12Id2) · nM ] dΓ

−∫

Γ1

ψi(β1Wr1 + λId1 · nM1),dΓ1 = 0 (6.75)

Expanding Wr1 in the same basis function

Wr1 =n∑

j=1

ψj(Wr1)j , Wr2 =n∑

j=1

ψj(Wr2)j (6.76)

we obtain the system equations

n∑j=1

(∫∫Ω1

1Gψiψj dxdy

)(Wr1)i +

n∑j=1

(∫∫Ω1

2ψiψj dxdy)

(Wr1)i

+n∑

j=1

[∫∫Ω1

(D∇ψi∇ψj +Gψiψj) dxdy −∫

Γ1

β1ψiψj dΓ1 −∫

Γβ11ψiψj dΓ

](Wr1)i

+n∑

j=1

(∫Γβ22ψiψj dΓ

)(Wr2)i −

∫Γ1

λId1ψi · ndΓ1 −∫

Γ(λ11Id1 + λ12Id2)ψi · ndΓ = 0 (6.77)

Denote the following terms:

[mr1] =∫∫

Ωe

1Gψiψj dxdy, [cr1] =

∫∫Ωe

2ψiψj dxdy,

[k1r1] =

∫∫Ωe

D∇ψi∇ψj dxdy, [k2r1] =

∫∫Ωe

Gψiψj dxdy

[k3r1] = −

∫Γ1

β1ψiψj dΓ1, [kr12] = −∫

Γβ11ψiψj dΓ

[kr21] = −∫

Γβ22ψiψj dΓ, [br1] =

∫Γ1

λIdψi · ndΓ1

[bcr1] =∫

Γ(λ11Id1 + λ12Id2)ψi · ndΓ


Thus the solution in domain Ω1 can be written in finite element model:

[mr1]W er1 + [cr1]W e

r1 + [kr1]W er1 + [kr12]W e

r1 − [kr21]W er2 − [br1] − [bcr1] = 0 (6.78)

By assembly process of elements, the local nodal values is replaced by the global energy density of thestructure,

[Mr1]Wr1 + [Cr1]Wr1 + [Kr1]Wr1+[Kr12]Wr1 − [Kr21]Wr2−[Br1] − [Bcr1] = 0 (6.79)

where [Kr1] = [k1r1] + [k2

r1] + [k3r1].

For steady state conditions, the equation (6.80) reduces to

[Kr1]Wr1+[Kr12]Wr1 − [Kr21]Wr2−[Br1] − [Bcr1] = 0 (6.80)

It is an analogy format of SEA formula.

The direct field solution can be easily obtained since no need of considering the boundary condition,

[Mr1]Wd1 + [Cd1]Wd1 + [Kd1]Wd1 − [P1] = 0 (6.81)

The combination of direct field and reverberant field leads to the final solution:

W1 = Wr1 + Wd1 (6.82)

It should be noted that solving of equation (6.81) Wr1 involves with the value of Wr2, so thecoupled subsystem should be solved simultaneously.

For the another subsystem Ω2, similar procedures may be taken to build up the finite element modelwhich coupled with subsystem Ω1.

6.5 Applications of energy FEM: synthesis of two fields

It is believed that the Energy Finite Energy Method (EFEM) is an alternative to the more establishedstatistical energy analysis (SEA) for modeling of high frequency dynamic behavior of vibro-acousticstructures [96]. The main advantages over SEA are the use of conventional description of the model,similar to classical finite element models, and the modeling of the special distribution of vibrationalenergy throughout the structure. However, EFEM developed in [96] based on vibrational conductivityapproach. It means that the difference of two fields with different waves has not been taken into account.The applications introduced here is an attempt to realize the ideas of synthesis of two fields.

6.5.1 FEM model: mesh and stiffness matrix

The procedures of energy FEM has been discussed in last section. For simplicity, we take a two-dimensional structure, say, a plate or two coupled plates as examples. Suppose the plates are all dividedby rectangular elements shown by Figure 6.9. A linear variation in rec element e can be expressed as

W er (x, y) = c1 + c2x+ c3y + c4xy (6.83)

6.5. Applications of energy FEM: synthesis of two fields 165

1 2

34

a a

b

b

x

y

Figure 6.9: Rectangular element used in the FEM model

We can express the constants c1, c2, c3 and c4 in terms of the nodal energy density ee1, ee2, ee3 and ee4 as

W e1

W e2

W e3

W e4

=

1 x1 y1 x1y1

1 x2 y2 x2y2

1 x3 y3 x3y3

1 x4 y4 x4y4

c1c2c3c4

(6.84)

By some matrix proceeding, we get

W er (x, y) =

4∑i=1

W ei ψ

ei (x, y) (6.85)

where

ψe1 =

(a− x)(b− y)4ab

(6.86)

ψe2 =

(a+ x)(b− y)4ab

(6.87)

ψe3 =

(a+ x)(b+ y)4ab

(6.88)

ψe4 =

(a− x)(b+ y)4ab

(6.89)

where x = x−xeo, y = y− ye

o, x, y are the global coordinates, and xeo, ye

o are global coordinates of localcoordinates origin of the element.

[N ] =1

4ab[(a− x)(b− y) (a+ x)(b− y) (a+ x)(b+ y) (a− x)(b+ y)] (6.90)

[N,x ] =1

4ab[(y − b) (b− y) (b+ y) (−b− y)] (6.91)

[N,y ] =1

4ab[(x− a) (−a− x) (a+ x) (a− x)] (6.92)


and assuming a unit thickness, and it is known that∫∫

s 1dx dy = 4ab, it yields

[mr] =ab

9G

4 2 1 2

4 2 1sym- 4 2

-metric 4

=ab

9G[S] (6.93)

[cr] =2ab9

[S] (6.94)

[k1r ] =

bD

6a

2 −2 −1 1

2 1 −1sym- 2 −2

-metric 2

+aD

6b

2 1 −1 −2

2 −2 −1sym- 2 1

-metric 2

(6.95)

[k2r ] =

Gab

9[S] (6.96)

[br] =∮

Γ

ψ1

ψ2

ψ3

ψ4

λId · ndΓ (6.97)

6.5.2 Numerical analysis

First we apply the FEM formulation to a single plate with the rectangular elements mesh. The plate issquare whose length is 1.2m. The other physical and geometry parameters are the same as that of platemodel shown in Figure 6.2. The plate is equally divided into 24 × 24 elements, and 25 × 25 nodes arenumbered, the excitation point is located at x0 = 0.6m, y0 = 0.6m. Figure 6.10 shows a portion of themeshed plate.

The energy density calculated by above FEM model is shown in Figure 6.11. For the purpose ofcomparison, energy density obtained from governing vibration equation is offered. It is based on theequation (1.50) seen in subsection 1.3.3. The exact energy result got from plate vibration equation isshown in Figure 6.12. These two figures agree with each other. The comparisons among local energyapproach (hybrid) by FEM model, vibrational conductivity approach, and exact energy results are madein Figures 6.13–6.14. The same conclusions can be explored as that mentioned in section 6.2, VCA’sprediction is lower for peak value while higher in the locations near the boundary. In section 6.3.1, wehave derived the boundary condition by

I(M) · nM = βW + λId · nM (6.98)

Suppose an averaged reflectivity coefficient can be expressed by

r =1π

∫ π2

−π2

r(θ)dθ

=1π

∫ π2

−π2

r(α)dθ

then the coefficients β and λ in equation (6.99) may be simplified as

β =c(1 − r)1 + r

, λ =2r

1 + r(6.99)


S

Figure 6.10: FEM model of a single plates.

05

1015

2025

05

1015

2025

5

5.5

6

6.5

7

7.5

8

Figure 6.11: Energy density in the single plate by FEM model


05

1015

2025

05

1015

20254.5

5

5.5

6

6.5

7

7.5

8

8.5

9

Figure 6.12: Energy density in the single plate calculated by plate vibration equation.

0 0.2 0.4 0.6 0.8 1 1.2 1.45.5

6

6.5

7

7.5

8

8.5

9

9.5

10

Length of the plate

Ene

rgy

dens

ity (

dB)

hybridVCA exact

X=0.6 m

Figure 6.13: Comaprison of energy density by three methods along y = 0.6 m. ——, hybrid FEMmodel; · · · · · · · · · , vibrational conductivity approach; – – – –, exact energy result.


0 0.2 0.4 0.6 0.8 1 1.2 1.46

6.5

7

7.5

Length of the plate

Energ

y densi

ty (

dB

)

data1data2data3

X=0.3 m

Figure 6.14: Comparison of energy density by three methods along y = 0.15 m. ——, hybrid FEMmodel; · · · · · · · · · , vibrational conductivity approach; – – – –, exact energy result.

The averaged reflectivity coefficient r is definitely a key factor to effect the energy value. We try to ex-amine how the energy density influenced by reflectivity coefficient. Two examples related with differentdamping factors are taken. First, a slight damping factor, η = 0.01 is assumed, energy with respected totwo different averaged reflectivity coefficient r = 0.8 and r = 1 is displayed in Figure 6.15. The secondexample involves a relative strong damping η = 0.1.

Clearly, the difference of energy density associated with different reflectivity coefficient is found inFigure 6.15, while there is no remarkable difference discovered in Figure 6.16 where damping factorη = 0.1. In strong damping case like the example shown in Figure 6.15, energy level decreases sorapidly with energy dissipation that the reverberant field obtains less energy flow from direct field. Whilereverberant field contributes little to total energy density, total energy level is not easy to be influencedby reflectivity coefficient which is only affects reverberant field. In the slight damping case, however,reverberant field plays a relative important role, thus the influences of reflectivity coefficient on the totalenergy level is very significant, as demonstrated in 6.15. At this stage, we find that the description ofreflected energy flow by the equation (6.47) is actually oversimplified boundary conditions. The abovediscussions show that the accurate boundary condition is rather necessary and important when the methodof energy synthesis is employed.

Now we turn to the application of FEM model to two coupled plates. The model shown in Figure 6.17where two identical square plate are coupled. Each plate has the same physical and geometry parameteras that of single plate employed above. The point load is located at x0 = 0.6m, y0 = 0.6m in plate 1.

Two examples are given to demonstrate the application of FEM model for local energy approach. Inthe first example, the frequency is 1 kHz and damping loss factor η = 0.05. As the coupling boundary


0 0.2 0.4 0.6 0.8 1 1.2 1.460

65

70

75

80

85

90

95

100

Length (m)

Ene

rgy

dens

ity (

dB)

Figure 6.15: Reflectivity coefficient influenced by damping factor: energy density alongy = 0.6 m.η = 0.01. ——, r = 1; – – – –, r = 0.9;

0 0.2 0.4 0.6 0.8 1 1.2 1.435

40

45

50

55

60

65

70

75

80

85

Length (m)

Ene

rgy

dens

ity (

dB)

Figure 6.16: Reflectivity coefficient influenced by damping factor: energy density alongy = 0.6 m.η =0.1. ——, r = 1; – – – –, r = 0.9;

6.6. summary 171

S

S

Figure 6.17: FEM model of two coupled plates.

condition was expressed by equation (6.57, 6.58), the generalized reflectivity coefficients are denoted asλ1 = λ2 = 0.95 and β1 = β2 = 0.05. It is found that the flexural wave energy is discontinuous at theplate joint showing the lower level in plate 2 as seen in Figure 6.18

In the second example, the damping is changed to η = 0.1 and the generalized reflectivity coefficientsare modified to λ1 = λ2 = 1 and β1 = β2 = 0. This boundary condition allows the energy density crossthrough the coupling boundary without reflection and two plate can be viewed as a single plate. Due toa larger damping factor, the energy density in Figure 6.19 is lower than that in first example, and energydistribution is continuous at the plate joint. The similar energy results have been observed in [97].

6.6 summary

In this chapter, we have tried to generalized Time-varying Simplified Energy Method (MEST) to multi-dimensional system. The symmetrical waves were mainly concerned. For two- or three-dimensionalsystems with complex boundary, we followed the idea of combining the two fields: direct and reverberantfield associated with different wave types. In fact, the point-loaded energy in direct field is the solutionof energy equation for infinite system which is easier to obtain.

On the other hand, the energy result in reverberant field need an accurate description of boundarycondition. We then explored the expression of boundary condition at some length which expressedby general formula. On the basis of boundary conditions, we developed a FEM model to apply LEAequation to plates. Plate is always an important subject because plates are indispensable in a varietyof engineering practice. However in this chapter, the numerical analysis carried out by FEM modelare just used to validate the applicability of the expression of boundary condition. Much more detailedinvestigation will be done in further studies.


0 5 10 15 20 25 30 35 40 450

10

20

30

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

Figure 6.18: Energy distribution in coupled plates for λ1 = λ2 = 0.95 and β1 = β2 = 0.05.

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

2

3

4

5

6

7

8

Figure 6.19: Energy distribution in coupled plates for λ1 = λ2 = 1 and β1 = β2 = 0.

Chapter 7

General conclusions and perspectives

7.1 Significance of the study

The transient excitations such as shocks and impacts give rise to medium and high frequency vibrationwhich can cause structural failure and unwanted noise. However, as reviewed in chapter 1, the matureand complete techniques to predict the time-varying response in various engineering structures is stilllacking at present.

Chapter 1 devoted some initial effort to explaining the general scope, purpose, and the methodologieswhich are usually applied in medium and high frequency domain. It was discussed that how each of thesemethods have significant advantage and drawbacks in their approximations. There are certain difficultiesto describe the dynamic features because of structural sensitivity and uncertainty in this frequency range.It was stressed that the statistical methods in term of energy/intensity variables may be practicable andeffective to meet the requirements. Among them, statistical Energy Analysis (SEA) is a typical andwidely accepted one. Of course, with these statistical methods one must be willing to risk the loss ofsome information in details in order to gain the benefits of relative simple calculation procedures.

Examination of references showed that most researches of energy approach have focused on steadystate condition, while the tools to predict the transient dynamics are relatively rare. Two methods whichdeal with the time-varying energy in vibroacoustic domain were mentioned:

1. Transient Statistical Energy Analysis (TSEA).

2. Time-varying vibrational conductivity approach.

The former is the time-varying version of well-known SEA. The coefficients in TSEA, for example, cou-pling loss factors, are borrowed from SEA. The latter is the time-varying version of so-called vibrationalconductivity approach. It can be expressed by Nefske’s equation (1.94) which is an analogy to heat con-duction equation. In the thesis we have proved that this equation could be discretized into TSEA equationby zero order finite element procedure. It means that these two approaches have the same physical na-ture. It is interesting to know that these two time-varying methods are the minor by-products of theirsteady state portion (Only three pages in [14] was taken to address TSEA). Unfortunately, As remarkedin chapter 1, it had been reported that these methods were not always satisfactory to provide an accurateprediction. Thus the question as how to find a effective tool remains open. This thesis is devoted to

173

174 General conclusions and perspectives

proposing a new method to deal with this problem.

7.2 Summary of the study

Before presenting our approach, we spend chapter 2 introducing some recent mathematical results ob-tained for the high frequency asymptotic of hyperbolic partial differential equation. Based on the trans-port theory, an exact transient energy flow equation can be derived to describe the propagation of highfrequency waves in random, heterogeneous media. The theory shows that the angularly-resolved en-ergy density in phase space satisfy Liouville-type transport equation. In fact, these results are obtainedwith the aid of Wigner transform which carries out high frequency asymptotic of classic waves. Someimportant properties and simple applications of Wigner transform was given in chapter 2. The brief in-troduction of transport equation and Wigner transform serves also as an extension of these viewpointsin preparation for the more advanced treatment to follow in later chapters. In addition, we demonstratethat the transport equation has two possible extensions: in inhomogeneous media with strong scatteringit can be approximated to diffusion equation while in homogeneous media it can be expressed by a hy-perbolic equation. It implies that the behavior of high frequency classic waves depends on the propertiesof media.

The main theoretical work was fulfilled in chapter 3 where a new time-varying energy equation wasderived in two directions. From the waves point of view, the behavior of high frequency waves en-ergy in random media can be described by the transport equation. We studied the hyperbolic waves anddispersive waves, respectively. Specially, we have taken the energy dissipation into account in the deriva-tion of generalized Liouville equation. The consideration of dissipation energy is necessary because thedamping plays an important role in vibroacoustic domain. While Wigner transform was applied to hy-perbolic system, the characteristics of the wave group in phase space was imposed into the derivationof TLEA equation for dispersive waves. On the other hand, TLEA equation was also derived from fun-damental energy balance law. In bounded system, the forward (incident) energy wave and backward(reflected) energy wave were treated respectively. Under the linear assumption and superposition princi-ple, a second-order temporal energy equation was discovered. Since TLEA equation is a new descriptionof time-varying energy, the second part of chapter 3 was concentrated to discuss its properties. Theproperties of TSEA and time-varying vibrational conductivity approach were also investigated to makea comparison with TLEA.

The validation of TLEA was held in chapters 4 and 5. We examined carefully the energy relationof two oscillators system. In this case, the principal idea is to separate the damping loss factor fromcoupling loss factor because they represent different energy behavior. The energy relation of two oscilla-tors revealed in terms of analytical responses was proved to be consistent with the description of TLEAequation. TLEA equation was also discretized by FEM technique so that it can be applied to discretesubsystems. In chapter 4, the numerical analysis were concentrated to discrete subsystems: two oscil-lators system, two coupled rods and two coupled beams. A number of examples over a wide range ofconditions were investigated. In chapter 5, TLEA was applied to simple distributed structures to verifyits applicability. The longitudinal waves in a rod and dispersive waves in a beam subjected to an impulse-like excitation were studied by TLEA, and the solution of TLEA was compared with two other referencevalues. Specially, energy Shock Response Spectrum (SRS) has been introduced to evaluate the severityof transient excitations by analyzing each frequency components. It is suitable for study on transientdynamics of a beam because the dispersive behavior depends heavily on frequency values.

7.3. Conclusions 175

In chapter 6 we have made an attempt to generalized MEST to multi-dimensional structures. On thecondition of point loads, the direct field can be governed by symmetrical waves. The important idea forthese cases is separating direct field from reverberant field and treating them respectively, then synthesiz-ing them together to get integral result. The connection between two fields is boundary condition, whichhas been investigated in detail for isolated system and two coupled systems. We also discussed the FEMmodel to settle the problem of synthesizing two fields.

7.3 Conclusions

The derivation of MEST equation from hyperbolic system shows that the behavior of high frequencywaves energy in time-spatial domain can be expressed by the fundamental transport equation. Wignertransform plays an important role to reach the high frequency limit for hyperbolic waves. In homoge-neous media, the so-called angularly resolved energy density becomes angularly-constant energy den-sity, it means energy density is independent of wave vector. At this case, energy density of high frequencywaves can be described by the hyperbolic equation – that was called MEST equation in this thesis. How-ever, it should be pointed out that radiative transport equation cannot be directly applied to vibroacousticsdomain because the assumptions are not suited to the latter.

It has been shown that the hyperbolic description of the high frequency waves energy holds for bothhyperbolic waves and dispersive waves (Here, dispersive waves are treated as wave groups). As a hyper-bolic partial differential equation, MEST equation demonstrates that vibroacoustic energy propagates inform of wavefront with a finite velocity. On the contrary, the Nefske’s equation which is analogy to heatequation, predicts a diffusive behavior of time-varying energy with a infinite energy velocity.

The comparisons made among MEST, exact energy results from displacement responses, and TSEAwere displayed by the examples shown in chapter 4 and 5. The characteristics of transient dynamicssystem may be described by three important parameters: the peak value of transmitted energy, rise timeand energy decay rate. On predicting these quantities, good agreements were found between MESTsolutions and exact results. Nevertheless, TSEA or the diffusion equation exhibited remarkable errors:the rise time calculated by TSEA or time-varying vibrational conductivity approach were always muchless than the exact values, while the decay rates were overestimated.

From the physical point of view, MEST equation indicated that the time-varying vibroacoustic energyare oscillatory. The time-varying energy stored in each subsystem is reciprocal and reversible; on thecontrary, energy exchange described by TSEA (or Nefske’s diffusion equation) is irreversible, so thatthe input energy level in subsystem 1 predicted by TSEA is always higher than the energy transmittedinto subsystem 2. In fact, the dissipated energy is irreversible, it will never come back; the energyexchange among subsystems is reciprocal. For this moment, we understand that energy dissipating andenergy exchanging are two completely different process for time-varying dynamics. Thus, terminology“coupling loss factors” is somewhat of a misnomer since the energy exchange in time domain is reversiblerather than lossy.

It was found that the predictions of TSEA are sometimes close to the exact energy results, for exam-ple, in the cases of very weak coupling cases where the energy exchange is rather weak. Unfortunately,a major drawback of TSEA has been found in above discussion and it makes TSEA face a problem of asomewhat physical nature rather than technical one. Therefore, it is concluded that TSEA is strictly notcorrect, although, under particular condition, it can provide approximate results.

176 General conclusions and perspectives

For multi-dimensional problems, there is no single technique which can be applied with confidenceto various engineering structures for high frequency prediction at present. However, the idea of synthesisof two fields (direct and reverberant field) is a creative motivation, both in transient condition and steadystate condition. We found in chapter 6 that an integral description of boundary condition plays theimportant role to calculate the reverberant field.

In fact, MEST allows the vibroacoustic energy to be analyzed exactly in the time domain by usingthe second order energy equation, which gives an insight into the physical nature of vibroacoustic energypropagating. It shows energy flow is governed mainly by a propagation process with an oscillatorycharacteristic rather a diffusion one. The fact that both the MEST equation and the diffusion equationcan be reduced to the same form if the temporal derivatives vanish by time averaging technique for steadystate condition implies that the MEST can be used as a solid basis of a promising energy approach.

7.4 Perspectives

Our major contribution is to propose a new energy approach to describe time-varying dynamics of acous-tics and engineering structures at high frequency. For the purpose of validation, the applications havebeen just limited to deal with some simple structures/systems, because the analytical reference value(so-called exact energy results) can be easily found for them. The work in this thesis offers a start pointto further study, the investigations in the near future and some possible perspectives may be summarizedas follows.

Based on the theorectical investigation and initial guideline in chapter 6, MEST should be applied tomore complex systems. Moreover, the validation by experimental results will be an important step forMEST to be applicable.

A promising alternative direction is to explore the link between MEST and TSEA or SEA. Thecharacteristics of MEST equation make us reconsider the fundamental relation and description of energyand energy flow. There still exsits a need to determine the dissipation loss factor and coupling loss factorin SEA. The idea of using in-situ measurements of response energy and power input to obtain an estimateof damping and coupling loss factors was considered early in Lyon’s text [14]. However, the inversionof a matrix of measured steady state response may become numerically unstable whenever the couplingloss factors become as large as the damping loss factor. MEST offer another technique to deal with thisproblem, as James and Fahy [87] had proposed that the kinectic energy impulse response could be anindicator of the strength of coupling between SEA subsystems. These work will be helpful to improvethe applicability and reliability of energy approaches. Related to this subject, MEST can also be used toevaluate the reverberant time in acoustic field.

On the basis of accurate prediction of time-varying energy, MEST may be eligible to be integral partof real-time vibration control technique. The prediction of MEST may be integrated with time-frequencyanalysis for unsteady cases. Another interesting subject may be exploring the strategies provided byMEST to control unwanted impact noise and shock response in some specific applications of technicalengineering.

There should be other directions to continue the research and much work left to be done. This thesisshould be an accessible start point or platform where an active area of future study can be exploited.

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Appendix A

Analytical energy results for twoundamped oscillators

Consider two coupled oscillators system shown in Figure A.1. As discussed in Chapter 4, for time-varying cases energy dissipating is a different physical nature from that of energy exchange. It is thuspossible to separate them and the investigation of analytical expression can be simplified. At first damp-ing losses are not taken into account for the moment, then the damping coefficients C1 = 0, and C2 = 0.Without losing the generality, other parameters are taken arbitrary values. The equations of motion fortwo coupled oscillators subjected to a unit impulse shown in Figure A.1 are:

m1x1 + (k1 + k)x1 − kx2 = δ(t), m2x2 + (k2 + k)x2 − kx1 = 0 (A.1)

where δ(t) denotes the Dirac delta function. They can be rewritten as

x1 + ω21x1 − r21ω

21x2 =

δ(t)m1

, x2 + ω22x2 − r22ω

22x1 = 0 (A.2)

where ω1 and ω2 are the blocked natural frequencies of either oscillator, ω1 =√

k+k1m1

, and ω2 =√

k+k2m2

,

the ratio r1 =√

kk1+k , and r2 =

√k

k2+k represent the stiffness coupling relation.

Equations (A.2) can be solved by Laplace transform. Denote Laplace transform pairs X1(s) = Lx1(t)

k1 k2

k

C1C2

m1m2

F1

x1(t) x2(t)

Figure A.1: Two oscillators model

183

184 Appendix A: Analytical energy results for two undamped oscillators

and X2(s) = Lx2(t), then

X1(s) =1m1

s2 + ω22

s4 + (ω21 + ω2

2)s2 +R2ω21ω

22

(A.3)

X2(s) =1m1

r22ω22

s4 + (ω21 + ω2

2)s2 +R2ω21ω

22

(A.4)

where R2 1 − r21r22.

To obtain a close-form analytical solution, we have to introduce a key parameter θ,

sin θ =2Rω1ω2

ω21 + ω2

2

, θ ∈ (0,π

2) (A.5)

The definition (A.5) holds because

0 <2Rω1ω2

ω21 + ω2

2

< 1 (A.6)

With the help of parameter θ, equations (A.3, A.4) may be written as

X1(s) =4m1

s2 + ω22

(2s2 + ω21 + ω2

2)2 − (ω21 + ω2

2)2 cos2 θ(A.7)

X2(s) =4m1

r22ω22

(2s2 + ω21 + ω2

2)2 − (ω21 + ω2

2)2 cos2 θ(A.8)

The inverse transform of (A.7, A.8) yield the expressions of displacement:

x1(t) =I1

[(ω2

2 − ω21) + (ω2

1 + ω22) cos θ

]sin

(sin( θ

2)√ω2

1 + ω22 t

)2(ω2

1 + ω22)

32 sin( θ

2) cos θ+

I1[(ω2

1 − ω22) + (ω2

1 + ω22) cos θ

]sin

(cos( θ

2)√ω2

1 + ω22 t

)2(ω2

1 + ω22)

32 cos( θ

2) cos θ(A.9)

x2(t) =I1r

22ω

22 sin

(sin( θ

2)√ω2

1 + ω22 t

)(ω2

1 + ω22)

32 sin( θ

2) cos θ−I1r

22ω

22 sin

(cos( θ

2)√ω2

1 + ω22 t

)(ω2

1 + ω22)

32 cos( θ

2) cos θ(A.10)

where I1 1m1

. Then the velocities are listed as

x1(t) =I1

[(ω2

2 − ω21) + (ω2

2 + ω21) cos θ

]cos

(sin θ

2

√ω2

1 + ω22 t

)2(ω2

1 + ω22) cos θ

+

I1[(ω2

1 − ω22) + (ω2

2 + ω21) cos θ

]cos

(cos θ

2

√ω2

1 + ω22 t

)2(ω2

1 + ω22) cos θ

=I1

[(ω2

1 − ω22) + (ω2

2 + ω21) cos θ

](ω2

1 + ω22) cos θ

×

cos

(cos( θ

2) − sin( θ2)

2

√ω2

1 + ω22 t

)cos

(cos( θ

2) + sin( θ2)

2

√ω2

1 + ω22 t

)(A.11)

Appendix A: Analytical energy results for two undamped oscillators 185

x2(t) =I1r

22ω

22 cos

(sin( θ

2)√ω2

1 + ω22 t

)(ω2

1 + ω22) cos θ

−I1r

22ω

22 cos

(cos( θ

2)√ω2

1 + ω22 t

)(ω2

1 + ω22) cos θ

=2I1r22ω

22

(ω21 + ω2

2) cos θ×

sin

(cos( θ

2) − sin( θ2)

2

√ω2

1 + ω22 t

)sin

(cos( θ

2) + sin( θ2)

2

√ω2

1 + ω22 t

)(A.12)

Since cos( θ2) + sin( θ

2) > cos( θ2)− sin( θ

2) if θ ∈ (0, π2 ), thus the envelops of velocities ˆx1 and ˆx2 can be

expressed as

ˆx1(t) =I1

[(ω2

1 − ω22) + (ω2

2 + ω21) cos θ

](ω2

1 + ω22) cos θ

cos

(cos( θ

2) − sin( θ2)

2

√ω2

1 + ω22 t

)(A.13)

ˆx2(t) =2I1r22ω

22

(ω21 + ω2

2) cos θsin

(cos( θ

2) − sin( θ2)

2

√ω2

1 + ω22 t

)(A.14)

The total energy of the oscillators can be expressed as the kinetic energy envelope, namely

E1(t) =12m1(ˆx1)2, E2(t) =

12m2(ˆx2)2 (A.15)

Inserting the expressions of R and θ in (A.13, A.14), one find

E1(t) =12m1I

21 (ω2

1 − ω22)

2

(ω21 − ω2

2)2 + 4k2

m1m2

+I21

k2

m2

(ω21 − ω2

2)2 + 4k2

m1m2

[1 + cos

((cos( θ

2) − sin( θ2))√

ω21 + ω2

2 t

)](A.16)

E2(t) =I21

k2

m2

(ω21 − ω2

2)2 + 4k2

m1m2

[1 − cos

((cos( θ

2) − sin( θ2))√

ω21 + ω2

2 t

)](A.17)

Denote

M =12m1I

21 (ω2

1 − ω22)

2

(ω21 − ω2

2)2 + 4k2

m1m2

, N =I21

k2

m2

(ω21 − ω2

2)2 + 4k2

m1m2

, Q =(cos( θ

2) − sin( θ2))√

ω21 + ω2

2,

then time-varying energy E1(t) and E1(t) can be written asE1(t) = M +N +N cosQt = M +N +NeiQt

E2(t) = N −N cosQt = N −NeiQt(A.18)

186 Appendix A: Analytical energy results for two undamped oscillators

Appendix B

Energy expression of two equal oscillatorssystem

The equations of motion for two coupled oscillators subject to an impulse shown in Figure 4.12 are:

m1x1 + c1x1 + (k1 + k)x1 − kx2 = δ(t), m2x2 + c2x2 + (k2 + k)x2 − kx1 = 0 (B.1)

where δ(t) denotes the Dirac delta function.

The Laplace transform method is applied on the simultaneous motion equations, so that

x1(t) = L−1

[ 1m1

(s2 + ∆2 s+ ω22)

(s2 + ∆1 s+ ω21)(s2 + ∆2 s+ ω2

2) − k2

m1m2

], (B.2)

x2(t) = L−1

[ km1m2

(s2 + ∆1 s+ ω21)(s2 + ∆2 s+ ω2

2) − k2

m1m2

](B.3)

where ω21 = k1+k

m1, ω2

2 = k2+km2

, ∆1 = η1ω1 = c1m1

, and ∆2 = η2ω2 = c2m2

.

For the possibility to compare the exact energy results with the solution of MEST and TSEA, somesimplifications adopted by reference [50] are taken here, such as m1 = m2, k1 = k2, c1 = c2.These simplifications result in two equal oscillators so that ω1 = ω2 and η1 = η2. The results of Laplacetransform give the expressions of displacements of two equal oscillators

x1(t) =e−

12η1ωt

m1

[ 1B

sin (12Bt) +

1A

sin (12At)

], (B.4)

x2(t) =e−

12η1ωt

m1

[ 1B

sin (12Bt) −

1A

sin (12At)

](B.5)

where A,B =√

4ω2 ± 4p2 − (η1ω)2 and p2 = km1

.

187

188 Appendix B: Energy expression of two equal oscillators system

Then the velocities are obtained from the expressions of displacements

x1(t) =e−

12η1ωt

m1

[√ω2 − p2

Bcos (1

2Bt+ φ1) +

√ω2 + p2

Acos (1

2At+ φ2)], (B.6)

x2(t) =e−

12η1ωt

m1

[√ω2 − p2

Bcos (1

2Bt+ φ1) −√ω2 + p2

Acos (1

2At+ φ2)]

(B.7)

where φ1 = arcsin( η1ω

2√

ω2−p2), and φ2 = arcsin( η1ω

2√

ω2+p2).

The total energy of the oscillators can be expressed as the kinetic energy envelope, namely

E1(t) =12m1(ˆx1)2, E2(t) =

12m2(ˆx2)2 (B.8)

where the velocities envelopes are

ˆx1(t) =e−

12η1ωt

m1

[ω2 − p2

B2+ω2 + p2

A2+

2√ω4 − p4

ABcos(1

2(B −A)t+ (φ1 − φ2))] 1

2, (B.9)

ˆx2(t) =e−

12η1ωt

m1

[ω2 − p2

B2+ω2 + p2

A2− 2

√ω4 − p4

ABcos(1

2(B −A)t+ (φ1 − φ2))] 1

2. (B.10)

Appendix C

MEST solution of two coupled subsystems

The MEST solution of two different subsystems shown in Figure 5.30 can be got in a similar way. TheMEST equations (4.83, 4.84) can be rearranged as

d2E1

dt2= −2∆1

dE1

dt− (∆1

2 + ∆1η12ω1)E1 + ∆1η21ω2E2 (C.1)

d2E2

dt2= −2∆2

dE2

dt− (∆2

2 + ∆2η21ω2)E2 + ∆2η12ω1E1 (C.2)

Let E = E1 E2dE1dt

dE2dt T , then the equations will be

dEdt

= DE (C.3)

where

D =

0 0 1 00 0 1 0

−(∆12 + ∆1η12ω1) ∆1η21ω2 −2∆1 0∆2η12ω1 −(∆2

2 + ∆2η21ω2 0 −2∆2

(C.4)

The initial values of matrix equation (C.3) is Et=0 = E(0) 0 − ∆1E(0) 0T .

One can find the eigenvalues and eigenvectors associated with matrix B. Suppose λ = λ1 λ2 λ3 λ4T

and S = s1 s2 s3 s4T are the eigenvalues and eigenvectors, respectively. Then the solution ofequation (C.3) is

E =4∑

i=1

fisieλit, i = 1, . . . 4. (C.5)

where the coefficient fi can be got from

F = f1 f2 f3 f4T = S−1Et=0. (C.6)

189

190 Appendix C: MEST solution of two coupled subsystems

Appendix D

MEST equation for one-dimensionallongitudinal wave

Consider the energy density inside a rod subjected to a longitudinal transient excitation. Denote ρ isthe density, A is section area, and c represents the group velocity. The wave equation for undampedlongitudinal wave is ,

∂2u(x, t)∂t2

− c2∂2u(x, t)∂x2

= 0 (D.1)

equation (D.1) is actually equivalent to the equation (D.2) via a change of the independent variables.

∂2u(x, t)∂t∂x

= 0, (D.2)

Furthermore, some useful conditions can be obtained from equation (D.1, D.2),

∂3u(x, t)∂t2∂x

= 0,∂3u(x, t)∂x2∂t

= 0,∂3u(x, t)∂x3

= 0,∂3u(x, t)∂t3

= 0 (D.3)

By differential manipulating on equation (5.10) and substituting equation (D.2) and condition equalities(D.3), we can get some expressions simultaneously as followed

∂W (x, t)∂t

=12ρA

[2∂u(x, t)∂t

· ∂2u(x, t)∂t2

+ 2c2∂u(x, t)∂x

· ∂2u(x, t)∂t∂x

]

= ρA∂u(x, t)∂t

· ∂2u(x, t)∂t2

(D.4)

∂2W (x, t)∂t2

= ρA

[(∂2u(x, t)∂t2

)2 +∂u(x, t)∂t

· ∂3u(x, t)∂t3

]

= ρA(∂2u(x, t)∂t2

)2 (D.5)

(D.6)

191

192 Appendix D: MEST equation for one-dimensional longitudinal wave

∂W (x, t)∂x

=12ρA

[2∂u(x, t)∂t

· ∂2u(x, t)∂t∂x

+ 2c2∂u(x, t)∂x

· ∂2u(x, t)∂x2

]

= c2ρA∂u(x, t)∂x

· ∂2u(x, t)∂x2

(D.7)

∂2W (x, t)∂x2

= c2ρA

[(∂2u(x, t)∂x2

)2 +∂u(x, t)∂x

· ∂3u(x, t)∂x3

]

= c2ρA(∂2u(x, t)∂x2

)2 (D.8)

These expressions yields the following formula,

∂2W (x, t)∂t2

− c2∂2W (x, t)

∂x2= ρA(

∂2u(x, t)∂t2

)2 − c4ρA(∂2u(x, t)∂x2

)2 = 0 (D.9)

Finally, we get the undamped energy equation for transient condition, namely,

∂2W (x, t)∂t2

− c2∂2W (x, t)

∂x2= 0 (D.10)

Furthermore, it is straightforward to apply these processes on damped cases by considering the lossfactor in propagation and get the similar expression. Allow for the damping influence, the dynamicmotion equation can be written as:

ρA∂2u(x, t)∂t2

+ c1∂u(x, t)∂t

− EA∂2u(x, t)∂x2

= 0 (D.11)

By some mathematics manipulation on equation(D.11) and using equalities (D.4-D.8), an equation de-scribing damped case can be got as

∂2W (x, t)∂t2

+ 2k∂W (x, t)

∂t+ k2W (x, t) − c2

∂2W (x, t)∂x2

= 0 (D.12)

where k = c1ρA , a parameter related with damping loss factor. If using the damping model adopted from

SEA, let k = ηω, the damping energy equation can be written in another patten

∂2W (x, t)∂t2

+ 2ηω∂W (x, t)

∂t+ (ηω)2W (x, t) − c2

∂2W (x, t)∂x2

= 0 (D.13)

The equations (D.10, D.13) for rod are the same as the MEST equations (3.89, 3.98).

Appendix E

List of Publications

1. F. Sui, M. N. Ichchou and L. Jezequel 2002. “Prediction of Vibroacoustics Energy Using a Dis-cretized Transient Local Energy Approach And Comparison with TSEA”’, Journal of Sound andVibration, 251(1):163–180.

2. F. Sui and M. N. Ichchou 2004. “Prediction of Time-varying Vibro-Acoustic Energy Using A NewEnergy Approach”, Journal of Vibration and Acoustics, volume 127 April 2004.

3. F. Sui, M. N. Ichchou and L. Jezequel 2004. “Application of Transport Equation to Time-varyingVibroacoustic Energy at High Frequency”, Wave Motion (submitted)

4. F. Sui, M. N. Ichchou and L. Jezequel 2001 “Study of Transient Energy of Coupled Systems Usinga Local Energy Technique”, Proceedings of Internoise 2001.

5. F. Sui, M. N. Ichchou, and L. Jezequel 2002. “A Second-Order Vibroacoustic Energy Equation andIts Applications”, Proceedings of ISMA 2002, International Conference on Noise and VibrationEngineering.

6. F. Sui, M. N. Ichchou and L. Jezequel 2001. “From Local to Global Energy Approaches - Cor-rection of Instationnary SEA”, Proceedings of 17th International Congress on Acoustics, Rome,Italy.

7. F. Sui, M. N. Ichchou and L. Jezequel 2000, “Transient Local Energy Approach and its Applica-tion”, 4th SEANET Meeting, TU Dresden, Dresden, Germany.

8. M. N. Ichchou, F. Sui, and L. Jezequel 2000. “Transient Local energy: Theory and Application”,Proceedings of CAA’2000 Congress, Sherbrooke, Canada.

193

194 Appendix E: List of Publications

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