Electrically Interconnected Periodic Arrangements of Piezoelectric Elements on Mechanical ...

142 Pages·2014·20.89 MB·English

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Electrically Interconnected
Periodic Arrangements of
Piezoelectric Elements on
Mechanical Substrates
Master Thesis
Author(s):
Zündel, Manuel
Publication date:
2014
Permanent link:
https://doi.org/10.3929/ethz-a-010583736
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In Copyright - Non-Commercial Use Permitted
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Master Thesis
Ref. Nr.: 14-014
Electrically Interconnected Periodic
Arrangements of Piezoelectric Elements on
Mechanical Substrates
Author:
Zündel
Manuel
Advisor:
Bergamini
Andrea
Prof. Dr. Paolo Ermanni
Lab of Composite Materials and Adaptive Structures
Eidgenössische Technische Hochschule Zürich
Prof. Dr. Massimo Ruzzene
Vibration and Wave Propagation Lab
Georgia Institute of Technology
30. September 2014

Abstract
This thesis deals with a phononic metamaterial consisting of periodically
structured mechanical and electrical waveguides, which are coupled by piezo-
electricelements. Thegoalisthebroadbandattenuationofmechanicalwaves
for the reduction of vibrations in mechanical structures.
The investigations have shown, that the interaction between the electrical
and the mechanical wave mode is expressed in form of wave mode veering
and is characterized by a strong energy exchange between the two energy
domains. Both, numerical and experimental results have shown the ability
of this waveguide to create wide attenuation bands around the frequency of
interaction of the wave modes, whose position can be influenced by an appo-
site tuning of the electric waveguide.
To benchmark this coupled waveguide, its attenuation properties have been
compared to the ones of an equivalent locally resonant phononic metamate-
rial. The comparison of the numerical and experimental results of the two
configurations has demonstrated that the investigated waveguide features
weaker, but significantly wider attenuation bands.
Zusammenfassung
Im Rahmen dieser Arbeit wurde ein phononisches Metamaterial untersucht,
das aus periodisch strukturierten mechanischen und elektrischen Wellenleiter
besteht, wobei diese Wellenleiter durch piezoelektrische Elemente gekoppelt
sind. Das Ziel ist die Erzeugung breiter Frequenzbänder in denen mechanis-
che Wellen stark gedämpft werden, was zur Reduzierung von Schwingungen
in mechanischen Strukturen führt.
Es konnte gezeigt werden, dass sich die Interaktion der mechanischen und
elektrischen Wellenmoden durch mode veering und durch einen starken En-
ergieaustausch zwischen den zwei Systemen kennzeichnet. Die numerischen
und experimentellen Resultate haben gezeigt, dass in diesem Wellenleiter im
Bereich der Interaktion ein breites Frequenzband entsteht, in dem mecha-
nische Wellen gedämpft werden und dessen Lage durch das Design des elek-
trischen Wellenleiters beeinflusst werden kann.
Um das Potential dieses gekoppelten Wellenleiters zu ermitteln, wurden die
Dämpfungseigenschaften mit denjenigen eines äquivalenten Metamaterials
verglichen, dass durch lokale Resonanzen charakterisiert ist. Der Vergleich
der numerischen und experimentellen Ergebnissen dieser zwei Strukturen hat
gezeigt, dass der untersuchte Wellenleiter Frequenzbänder erzeugt, die zwar
eine schwächere Dämpfung aufweisen, aber dafür wesentlich breiter sind.
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Acknowledgment
I would like to thank my advisor Dr. Andrea Bergamini for all the support
and advices he gave to me during the work on this thesis.
I would also like to express my gratitude to Prof. Dr. Paolo Ermanni and
Prof. Dr. Massimo Ruzzene for making this thesis possible and creating the
opportunity to carry out the experimental part at the Vibration and Wave
Propagation Lab. The stay in Atlanta was a profitable experience for both
my scientific knowledge and my personal growth.
Finally I want to thank all the people I had the pleasure to have discussions
with during this work. These were very helpful, sometimes off-topic and
always interesting.
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Contents
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1 Introduction 11
1.1 Motivation and Goals . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 State of Research in Phononic Crystals and Coupled Waveguides 13
1.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Theoretical Background 23
2.1 Piezoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Physical Explanation . . . . . . . . . . . . . . . . . . . 24
2.1.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Basics Concepts . . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 Useful Dispersion Relations . . . . . . . . . . . . . . . 30
2.2.2.1 Transversal Waves in Beams . . . . . . . . . . 30
2.2.2.2 Discrete Lossless Transmission Line . . . . . . 31
2.2.3 Analogies Mechanical and Electrical Systems . . . . . . 32
3 Numerical Investigations 35
3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Geometry and Materials . . . . . . . . . . . . . . . . . 35
3.1.2 Eigenfrequency Solution of the Periodic Unit Cell . . . 36
3.1.2.1 Boundary Conditions . . . . . . . . . . . . . . 36
3.1.2.2 Dispersion Diagrams . . . . . . . . . . . . . . 37
3.1.3 Frequency Domain Solution of the Finite Structure . . 37
3.1.3.1 Excitation and Boundary Conditions . . . . . 38
3.1.3.2 Wave Transmittance . . . . . . . . . . . . . . 39
3.2 Open Circuit Configuration . . . . . . . . . . . . . . . . . . . 40
3.2.1 Mechanical Wave Modes . . . . . . . . . . . . . . . . . 41
3.2.2 Wave Propagation Properties . . . . . . . . . . . . . . 42
3.2.3 Effect of the Piezoelectric Element Size . . . . . . . . . 42
3.3 Local Resonators Configuration . . . . . . . . . . . . . . . . . 44
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3.3.1 Electric Boundary Conditions . . . . . . . . . . . . . . 44
3.3.2 Wave Propagation Properties . . . . . . . . . . . . . . 45
3.3.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Electrically Interconnected Configuration . . . . . . . . . . . . 49
3.4.1 Electric Boundary Conditions . . . . . . . . . . . . . . 49
3.4.2 Wave Propagation Properties . . . . . . . . . . . . . . 51
3.4.3 Energy Considerations . . . . . . . . . . . . . . . . . . 52
3.4.3.1 Energy Flow . . . . . . . . . . . . . . . . . . 52
3.4.3.2 Energy Velocities . . . . . . . . . . . . . . . . 57
3.4.4 Parameter Studies . . . . . . . . . . . . . . . . . . . . 59
3.4.4.1 Geometrical Parameters . . . . . . . . . . . . 60
3.4.4.2 Electromechanical Coupling Strength . . . . . 61
3.4.4.3 Dielectric Properties of the Piezoelectric Ele-
ment . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.4.4 Resistance . . . . . . . . . . . . . . . . . . . . 63
3.4.4.5 Number of Interconnected Unit Cells . . . . . 65
3.5 Other Electrically Periodic Configurations . . . . . . . . . . . 66
3.5.1 Diatomic Transmission Line Configuration . . . . . . . 66
3.5.2 Non-Nearest-Neighbour Interaction . . . . . . . . . . . 68
4 Experimental Phononic Cristal 71
4.1 The Experimental Structure . . . . . . . . . . . . . . . . . . . 71
4.1.1 Geometry and Materials . . . . . . . . . . . . . . . . . 71
4.1.2 Numerical Results of the Experimental Configuration . 72
4.1.3 Manufacturing . . . . . . . . . . . . . . . . . . . . . . 74
4.1.3.1 Assembly Template . . . . . . . . . . . . . . . 74
4.1.3.2 Assembling the Experimental Structure . . . 74
4.1.4 Simulated Inductor Array . . . . . . . . . . . . . . . . 76
4.2 Electric Tuning Procedure . . . . . . . . . . . . . . . . . . . . 79
4.3 Characterization of the Electric Components . . . . . . . . . . 81
4.3.1 Measurement of the Capacities . . . . . . . . . . . . . 81
4.3.2 Estimation of the Unit Cell Resistance . . . . . . . . . 82
4.4 Wave Propagation Investigations . . . . . . . . . . . . . . . . 84
4.4.1 Boundary Conditions and Coordinate System . . . . . 84
4.4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.2.1 Measurement Equipment . . . . . . . . . . . . 84
4.4.2.2 Excitation . . . . . . . . . . . . . . . . . . . . 86
4.4.2.2.1 Tone Burst . . . . . . . . . . . . . . 86
4.4.2.2.2 Pulse Signal . . . . . . . . . . . . . . 87
4.4.2.3 Data Postprocessing . . . . . . . . . . . . . . 87
4.4.2.3.1 Concepts and Techniques . . . . . . 88
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4.4.2.3.2 TransformationtotheFrequency-Wavenumber
Domain . . . . . . . . . . . . . . . . 91
4.4.2.3.3 ComputationoftheExperimentalWave
Transmittance . . . . . . . . . . . . 91
4.4.2.3.4 Estimation of the Exact Wavenumber 92
4.4.3 Open Circuit Results . . . . . . . . . . . . . . . . . . . 94
4.4.4 Local Resonator Results . . . . . . . . . . . . . . . . . 96
4.4.4.1 Wave Propagation Properties . . . . . . . . . 96
4.4.4.2 Electrical Response to the Mechanical Wave . 99
4.4.5 Results of the Interconnected Configuration . . . . . . 100
4.4.5.1 Mechanical Wave Propagation Properties . . 100
4.4.5.2 Electrical Response to the Mechanical Wave . 103
4.4.5.3 Electrical Excitation of the Coupled Waveguide105
5 Conclusions 109
5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
References 115
List of Figures 123
List of Tables 129
Appendix 131
A. Thesis Definition . . . . . . . . . . . . . . . . . . . . . . . . 133
B. Declaration of Originality . . . . . . . . . . . . . . . . . . . 137
C. PCB Layout . . . . . . . . . . . . . . . . . . . . . . . . . . 139
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