Adaptronics – Smart Structures and Materials [1st ed.] 9783662613986, 9783662613993

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Johannes Michael Sinapius

Adaptronics – Smart Structures and Materials

Adaptronics – Smart Structures and Materials

Johannes Michael Sinapius

Adaptronics – Smart Structures and Materials

123

Johannes Michael Sinapius Technische Universität Braunschweig Braunschweig, Germany

ISBN 978-3-662-61398-6 ISBN 978-3-662-61399-3 https://doi.org/10.1007/978-3-662-61399-3

(eBook)

The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content. © Springer-Verlag GmbH Germany, part of Springer Nature 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Vieweg imprint is published by the registered company Springer-Verlag GmbH, DE part of Springer Nature. The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

To Gabriele

Preface

Die Wahrheit hat tausend Hindernisse zu überwinden, um unbeschädigt zu Papier zu kommen, und vom Papier wieder zu Kopf. (The truth has a thousand obstacles to overcome, to get to paper undamaged, and from paper back to head.) Georg Christoph Lichtenberg (1742–1799)

Adaptronics is a comparatively young discipline in the engineering sciences, which is characterized by a pronounced interdisciplinarity, in the sense of the principle of integrative research even by transdisciplinarity. It deals with the research and development of components, structures, whole constructions, but also with manufacturing processes, which are characterized by the ability of self-adaptation of elasto-mechanical properties. Materials science contributes to this with research into multifunctional materials, structural mechanics with methods of modeling for the integration of these material systems, control engineering with research into suitable control concepts, microtechnology and measurement technology with sensors and actuators, design theory with systematics of the development of adaptive designs, structural dynamics with new methods of influencing vibrations and acoustics with concepts of actively influencing sound field characteristics. This makes adaptronics particularly attractive for a modern engineering education in which interdisciplinarity has a high, growing significance. Writing a textbook for an interdisciplinary subject is challenging and associated with the risk of not satisfying the individual disciplines. For example, this textbook may not be sufficient for the control engineer. The materials scientist will miss the depth that is important to him. The mechanic will look for many more advanced methods. I request all of you for your understanding and hope to compensate you by highlighting the relevance of your discipline within the framework of an interdisciplinary field. The book is primarily written for students of engineering sciences of advanced study courses and is the result of my courses at the Technische Universität Braunschweig Carolo Wilhelmina in the master courses in aerospace engineering, mechanical engineering, automotive engineering, and metrology and analytics. The book was published in German in 2018 and thereafter translated into English with the support of the translation program DeepL.

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Preface

I prefix each chapter of the book with an aphorism by Georg Christoph Lichtenberg, the well-known German physicist and master of aphorism from my hometown Göttingen. Braunschweig, Germany January 2020

Johannes Michael Sinapius

Acknowledgement

Over the past 15 years, many students have encouraged me to write a textbook on adaptronics that accompanies my lectures. A sabbatical semester has made this possible. My first thanks therefore go to the Presidium of the Technische Universität Carolo Wilhelmina zu Braunschweig, who approved the research semester and appointed a replacement for my chair for this period. Prof. Dr.-Ing. Christian Hühne has taken on the task of representing the chair with commitment and great care, thereby giving me the necessary freedom. I would like to thank him in particular for this. With gratitude, I think of my doctoral supervisor, who died far too early, and the pioneer of adaptronics in Germany, Prof. Dr.-Ing. Elmar Breitbach, an outstanding expert in the field of structural dynamics, aeroelasticity, and adaptronics. My academic teacher Prof. Dr.-Ing. Horst Irretier laid the foundation for my enthusiasm for structural dynamics. My grateful and honoring remembrance also goes to him. I would like to thank my colleagues at the Institute of Composite Structures and Adaptronics of the German Aerospace Center (Deutsches Zentrum für Luft- und Raumfahrt e.V.), who built up the lecture contents with me during my teaching activities at the Otto-von-Guericke University Magdeburg, especially Prof. Dr.-Ing. Hans Peter Monner, Prof. Dr.-Ing. Peter Wierach, and Prof. Dr.-Ing. Dipl.-Phys. Jörg Melcher. I have learned an immeasurable amount from many years of supervising doctorates in the field of adaptive structures and have constantly broadened my horizons in discussions with doctoral students. I would like to express my sincere thanks to all those who have extended the knowledge of adaptronics and fiber composite technology within the framework of their doctoral theses. The fruitful discussions and the findings are reflected in this book. To be mentioned are the Doctors of Engineering Sciences Johannes Riemenschneider, Tobias Wille, Simon Pansart, Christine Arlt, Erik Kappel, Imad Aldin Khattab, Marco Straubel, Hardy Köke, Fabio da Cunha, Malte Misol, Daniel Schmidt, Alexandra Kühn, Steffen Opitz, Philipp Hilmer, Oliver Unruh, Artur Szewieczek, Daniel Krause, Julian Kuntz, Florian Raddatz, Martin Schulz, Christian Hesse, Paul Lorsch, Andreas Henneberg, Björn Timo Kletz, Thomas Löbel, Julian Kuntz, Daniel Stefaniak, Thomas Haase, Veatriki Papantoni, Daniel Stefaniak, Thomas Weser, Steffen Niemann, Christoph Heinze, Martin Hillebrandt, Dirk Wilckens, Sebastian Geier,

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Acknowledgement

Christian Bülow, Matthias Endres, Hossein Sadri, Falk Heinecke, Christoph Dienel, Christian Pommer, and Christian Behr, as well as all those who are currently still on their way to their doctorate. In addition to the authors mentioned by name in the sections Research Projects, other members of my two institutes were involved in the preparation of this book. I would like to especially emphasize with thanks Mr. Alexander Kyriazis, Mr. M.Sc. Bernd-Christian Hölscher, Dr.-Ing. Anke Lütkepohl, Dr.-Ing. Christian Hesse, Dr.-Ing. Naser Al Natsheh, and Prof. Jörg Melcher, who have carefully proofread and given many further suggestions and support. The English version was carefully proofread by Mr. M.Sc. Jomson Joy. Many of the research projects reported on in the book were or are funded by the German Research Foundation (DFG), especially those in Sections 3..5.1, 5.5.1, 5.5.3, 6.6.2, 7.3.1, and 9.4.1. The support is gratefully acknowledged. Writing a book also involves an editor. I would like to thank Springerverlag for their interest and willingness to publish this book, especially Mrs. Birgit Kollmar-Thoni and Mr. Michael Kottusch for their careful support of the book project. Braunschweig, Germany January 2020

Johannes Michael Sinapius

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johannes Michael Sinapius References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Principles of Adaptronics . . . . . . . . . . . . . . . . . . Johannes Michael Sinapius 2.1 Introductory Examples of Adaptronic Systems 2.2 Elements of Adaptronics . . . . . . . . . . . . . . . . 2.3 Variants of Adaptive Systems . . . . . . . . . . . . 2.4 Components of Adaptive Systems . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Functional Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . Johannes Michael Sinapius and Sebastian Geier 3.1 Classification of Functional Materials . . . . . . . . . . . 3.2 Features of Functional Materials . . . . . . . . . . . . . . . 3.3 Electromechanical Converters . . . . . . . . . . . . . . . . . 3.3.1 Piezoelectric Ceramics . . . . . . . . . . . . . . . . . 3.3.2 Electroactive Polymers . . . . . . . . . . . . . . . . . 3.4 Thermomechanical Converters . . . . . . . . . . . . . . . . . 3.4.1 Shape Memory Alloys . . . . . . . . . . . . . . . . . 3.4.2 Shape Memory Polymers . . . . . . . . . . . . . . . 3.5 Research Project . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Actuators Based on Nanoscale Carbon Tubes 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Adaptronic Functional Elements . . . . . . . . . Johannes Michael Sinapius and Thomas Gries 4.1 Piezo Composites . . . . . . . . . . . . . . . . . 4.1.1 0-3 Piezo Composites . . . . . . . . 4.1.2 1-3 Piezo Composites . . . . . . . . 4.1.3 1-2 Piezo Composite . . . . . . . . .

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4.1.4 2-2 Piezo Composites . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Use of Residual Stresses in Piezo Composites . . . . . 4.1.6 Comparative Consideration of the Designs of Piezo Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stroke Amplifying Actuators . . . . . . . . . . . . . . . . . . . . . . . 4.3 Actuators of Dielectric Elastomers . . . . . . . . . . . . . . . . . . . 4.4 Research Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Radial-Sensory PVDF Fibers . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Active Shape Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johannes Michael Sinapius, Christian Hühne, Hossein Sadri and Johannes Riemenschneider 5.1 Objectives of Active Shape Control . . . . . . . . . . . . . . . . 5.2 Integration of Functional Elements into Components . . . 5.3 Activatable Fiber Composites . . . . . . . . . . . . . . . . . . . . 5.3.1 Use of Deformation Coupling . . . . . . . . . . . . . . . 5.3.2 Effect of Integration on Stiffness and Strength . . . 5.4 Selectively Deformable Lightweight Structures . . . . . . . . 5.5 Research Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Cellular Structures Actuated by Internal Pressure and Variable in Shape . . . . . . . . . . . . . . . . . . . . 5.5.2 Morphing Leading Edge . . . . . . . . . . . . . . . . . . 5.5.3 Adaptive Foil Bearings . . . . . . . . . . . . . . . . . . . 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Active Vibration Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johannes Michael Sinapius, Björn Timo Kletz and Steffen Opitz 6.1 Objectives of Active Vibration Control . . . . . . . . . . . . . . . . . . 6.2 Basics of Active Vibration Control . . . . . . . . . . . . . . . . . . . . . 6.2.1 Wave Propagation in Solids . . . . . . . . . . . . . . . . . . . . . 6.2.2 Structure-Borne Sound Interference . . . . . . . . . . . . . . . . 6.2.3 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Influence of Integrated Functional Elements . . . . . . . . . 6.2.6 Response Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.7 Local and Locally Distributed Destructive Interference . 6.2.8 Discretised Description of the Vibrations of Continuous Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.3 Structural Dynamic Identification of Activatable Structures . . . . 6.3.1 Single-Degree of Freedom Methods . . . . . . . . . . . . . . . 6.3.2 Multiple-Degree of Freedom Methods . . . . . . . . . . . . . . 6.4 Variants of Active Vibration Control . . . . . . . . . . . . . . . . . . . . 6.4.1 Local Vibration Suppression . . . . . . . . . . . . . . . . . . . . . 6.4.2 Global Vibration Suppression . . . . . . . . . . . . . . . . . . . . 6.4.3 Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Active Skyhook-Isolation . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Adaptive Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.6 Electromechanical Absorber Networks . . . . . . . . . . . . . 6.5 Research Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Active Vibration Reduction in Force- and Base-Excited Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Active Vibration Reduction in Parallel Robotics . . . . . . 6.5.3 Active Twist Rotor Blade . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Control of Adaptive Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . Johannes Michael Sinapius and Stephan Algermissen 7.1 Previews and Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Basic Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Control Signals from Digital Filtering . . . . . . . . . . . . . . 7.2.4 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Hybrid Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6 Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Research Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Active Vibration Reduction in Structures with Changing Vibration Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Active Sound Control . . . . . . . . . . . . . . . . . . . . . . . . . . . Johannes Michael Sinapius and Malte Misol 8.1 Objectives of Active Acoustic Control . . . . . . . . . . . . 8.2 Basics of Active Sound Control . . . . . . . . . . . . . . . . 8.2.1 Awareness of Sound . . . . . . . . . . . . . . . . . . . 8.2.2 Sound Field Description . . . . . . . . . . . . . . . . . 8.2.3 Sound Interference . . . . . . . . . . . . . . . . . . . . . 8.2.4 Sound Radiation from Elementary Sources . . . 8.2.5 Sound Radiation of Disks . . . . . . . . . . . . . . . 8.2.6 Sound Propagation in Closed Spaces (cavities)

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8.3 Variants of Active Sound Control . . . . . . . . . . . . . . . . . . . . 8.3.1 Active Reduction of Sound Pressure—Active Noise Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Active Vibration Control . . . . . . . . . . . . . . . . . . . . . 8.3.3 Active Structural Acoustic Control . . . . . . . . . . . . . . 8.4 Research Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Influencing the Sound Transmission Through Aircraft Fuselage Panels with Active Control . . . . . . . . . . . . . 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Integrated Structural Health Monitoring . . . . . . . . . . . . . . . . . . Johannes Michael Sinapius and Florian Raddatz 9.1 Objectives of an Integrated Structural Health Monitoring . . . . 9.2 Modal Method of Structural Health Monitoring . . . . . . . . . . . 9.3 Integrated Structural Health Monitoring Through Guided Ultrasonic Wave Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Research Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Localization of Damages in Complex Fiber Composite Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

Nomenclature1

Symbols @ @t ¼ _ @2 @t2 ¼ € @ 0 @x ¼ 2 @ 00 @x2 ¼ 4 @ IV @x4 ¼

first time derivative (velocity) second time derivative (acceleration) first local derivative (strain) second local derivative (curvature) fourth local derivative

Matrices and Vectors fg hi ½ ½c ½cm  ½k ½km  ½m ½A; ½B ½M ½C ½H hIi ½K ½T ½Tr  ½Te 

vector diagonal matrix matrix damping matrix mass modified damping matrix stiffness matrix mass modified stiffness matrix mass matrix state space matrices modal mass matrix modal damping matrix matrix of transfer functions unit matrix modal stiffness matrix transformation matrix transformation matrix for stresses transformation matrix for distortions

In such a broad and interdisciplinary field as adaptronics, a uniform nomenclature is a particular challenge. Unfortunately, a double use of symbols cannot always be avoided, but it is limited to such an extent that no misunderstandings should arise in the respective narrower context.

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½Z ½Zw  ½U ½Pa  ½H fCh g fe1 g; fe2 g; fe3 g ff g frg f/g; f/gi fug; fvg; fwg fux g; fuy g; fuz g

Nomenclature

impedance matrix sound radiation resistance matrix modal matrix, matrix of eigenmodes modal force distributions sound radiation modes chiral vector unit vectors excitation force vector series development coefficients for modal force vectors mode shape (eigenform), ith eigenform distortions, shifts in x; y; z displacements, displacements in x; y; z

Scalars, Vector Elements aikl b; bi c c0 cB cG cT cm ; ck cv cL cB dkk dk e e0 ef er f fa fd fs fA ; fa fB gkk gn h; hs h; ha i; i2

modal constant width damping speed of sound bending wave propagation velocity group speed reduce damping proportionality factors for proportional damping proportionality factor in Weber–Fechner law longitudinal wave propagation velocity bending wave propagation velocity piezoelectric distortion coefficient, piezoelectric charge coefficient piston diameter Euler's number eccentricity in foil bearing error signal controller input force excitation force, adaptive force damping force disturbing force actuator force blocking force voltage coefficient weight factor for element n height, thickness actuator thickness inertia radius

Nomenclature

kikl j kB k kT ks kA kik l l0 ld lL lp lc m mm mu mxx ; mxy ; myy mA mE m mT na nd nh ng np ns p pa pe pr ps ps p0 ps q qxx ; qyy rðsÞ r ry r0

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pulse response of the ith eigenform at location k by excitation at location l imaginary number (j2 ¼ 1) Boltzmann constant stiffness absorber stiffness connection stiffness, structural stiffness actuator stiffness coupling factor length, bar length electrode distance, output length charge spacing coil length polymer dimension ceramic dimension mass machine mass unbalance mass bending moments on the plate actuator mass total mass area related mass absorber mass number of actuators number of domains cell row number number of generalized degrees of freedom, number of eigenmodes of vibration number of measuring points number of sample points sound pressure ambient pressure, absorbed pressure incident pressure reflected pressure pressure in lubricating gap disturbing pressure international reference value for airborne sound transmitted pressure modal coordinate lateral forces on the plate referential variable, setpoint radius inertia radius radius of the spherical emitter

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s skl t ui ; vi ; wi u0 x; y; z vs vx;y;z vA vE vw wðsÞ wa ws w_ 0 w0 A Af As Aa Ab Bp Ci Cp Cs D; Di E ^ E Ec Emax Econv Epot Ekin Erad ED EH EQ EU Er Fi G Gp

Nomenclature

Laplace parameter compliance time displacement, displacement in x; y; z in point i free displacement Cartesian coordinates velocity oscillation velocity in x; y; or z directions draw speed extrusion speed winding speed control parameter secondary path response of the primary path starting speed initial deflection surface, cross-sectional area temperature for end of austenite formation temperature for start of austenite formation Area below the a-peak in the diffraction diagram Area below the b-peak in the diffraction diagram plate bending stiffness modal attenuation equivalent capacitance from electromechanical coupling capacitance of a piezoceramic charge displacement electric field strength, dependent on time and location amplitude of electric field strength coercive field strength Electric field at maximum operating voltage Waste heat by convection potential energy kinetic energy radiated energy energy density of the sound field transition enthalpy heat energy inner energy sensitivity to physical stimuli generalized excitation shear modulus shear modulus of a polymer

Nomenclature

Gf GðsÞ Gwa Gws GðuÞ HðXÞ; Hikl ðXÞ I Iy Im Iact Ireact Iy Iy Ki Kfb Kff KH L LG LL Lp LI LN LW LðAÞ Lr MA Mi Mmol Mf Ms My ; Mz N; Nx ; M1 MDR O Q; Qi Qm QZ R R0V Rm RG Rik

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shear modulus of a fiber transfer function of the controlled system secondary path primary path radiation function transfer function, for the ith eigenform at location k by excitation in location l current moment of area inertia medium sound intensity active portion of sound intensity blind (reactive) portion of sound intensity surface moment of inertia around y length-related moment of inertia around y modal stiffness of i-th eigenmodes feedback controller feedforward controller Hadamard’s condition criterion inductance radiation function in logarithmic form radiation function in logarithmic form with directivity sound pressure level sound intensity level volume level sound power level A-weighted sound pressure level logarithmic sound radiation level actuator moment modal mass, generalized mass of the ith eigenmodes molar mass temperature for end of martensite formation temperature for start of martensite formation bending moment at y normal force (in x direction) spin distortion intersection charge quality factor transverse force in z electrical resistance foil bearing radius before actuation centre distance of the sound field point general gas constant correlation function

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Sik SðsÞ SA Si SE SDR Ti TA T Tg Tm T0 Tc U VðXÞ V Wm We WA WE Ws Ws;k Wact Xi Xo Zec Ze Zm Zn Z0 ZF ZR FL;R R1;2;3;4 I R ap ac ag ad ah ak am al

Nomenclature

spectral power density function sensitivity function centre of area static moment of the ith layer entropy fixed extension time period time center of gravity under the decay envelope temperature glass transition temperature melting temperature reference temperature (288 K) Curie temperature electric voltage magnification function volume mechanical work electrical work actuator work energy density sound power sound power of the spherical emitter active sound power radiated into the far field input size output capacitive reactance electrical impedance mechanical impedance network impedance sound impedance sound impedance in solid bodies impedance left and right traveling waves Rayleigh functions imaginary part real part coefficient of thermal expansion of a polymer coefficient of thermal expansion of a ceramic absorption level ratio of 2 thicknesses ratio of 2 thicknesses stiffness ratio mass ratio ratio of two lengths

Nomenclature

aA a aq bi brik c cv dik dg ex ; ey ei eN e0r eik f fk fr g; gi gu ; go gm gm h h; hN hi hf hv # j jl ; jb ; jxl ; jxb , k kL l l0 m mkk ma md mv n; ni p q

xxi

ratio of 2 cross-sections ratio of 2 moduli of elasticity density ratio factors in Prony’s method impermittivity (electrical permeability) coherence function displacement angle Kronecker delta degree of dissipation dimensionless eccentricity components in foil bearing strain longitudinal strain free strain of an active functional material dielectric constant at constant mechanical stress objective functional Wiener filter convergence factor for adaptive filters radial bearing clearance frequency ratio between excitation frequency and ith natural frequency frequency ratio between excitation frequency in half-width points and natural frequency viscose damping module loss module angle coordinate in circumferential direction angle with respect to the normal at the center of sound modal controllability index for the ith eigenmodes ratio of blocking forces at travel magnification stroke magnification factor reflection factor wave number material-geometric parameters adiabatic coefficient complex eigenvalue wavelength fourth power of bending wave propagation velocity magnetic field constant transverse contraction number electrostriction coefficient proportionality factor to acceleration proportionality factor to displacement proportionality factor to vibration velocity modal damping circle number density

xxii

qA qk qp qpc q0 qg 1 1mn ri rB sij s sA sa ss sg ss t ti uc ui uA;i uFVG x0 ; xi xl;i xb;i xT xg K Cr D H; Hi Hk N P Pr   Eik s a f p L  jjj

Nomenclature

actuator density ceramic density polymer density piezocomposite density rest density degree of reflection sound radiation level modal sound radiation level stress blocking stress shear stresses time difference runtime primary runtime secondary runtime transmission level transmission factor frequency ith natural frequency coincidence angle phase angle element approach functions fiber volume content eigenfrequency, ith eigenfrequency ith longitudinal eigenfrequency ith bending eigenfrequency absorber eigenfrequency coincidence frequency compressibility number shape recovery rate difference (modal) controllability index modal electromechanical coupling probability polarization remanent polarization modulus of elasticity directional modulus of elasticity at constant electric field elastic modulus of embedding material elastic modulus of actuator elastic modulus of fiber elastic modulus of polymer longitudinal modulus of elasticity fiber-parallel modulus of elasticity of the composite

Nomenclature

? ð IÞtot /i /l;i /b;i ^ / ^ /R ^ / I v v0 vM vk vb wk fðvm Þ X Xc

xxiii

modulus of elasticity of the composite transverse to the fiber orientation total bending stiffness of a laminate ith eigenform ith lengthwise eigenform ith bending eigenmode eigenform amplitude real part of complex eigenform amplitude imaginary part of complex eigenform amplitude outgoing stimulus irritation threshold structure fraction of martensite ceramic portion b-phase fraction angle of bending line microstructure-dependent transformation constant excitation frequency coincidence frequency

Abbreviations 1M AFM Ag/AgCl ANC ANVC ASAC AVC BNT CFK CNT corr. CP CRC CVD DEAP DLR DSEC EAP EMIm Tf2 N EPDM FEM FIR

One Molar Atomic Force Microscope Silver/silver chloride reference electrode Active Noise Control Active Noise Vibration Control Active Structural Acoustic Control Active Vibration Control Bismuth Sodium Niobate Carbon Fiber Reinforced Plastic Carbon Nanotube Corrected Conjugated Polymers Collaborative Research Center Chemical Vapor Deposition Dielectric Electroactive Polymer German Aerospace Center e.V Deformable Supportive End Caps Electroactive Polymer 1-Ethyl-2-methylimidazolium bis(trifluorosulfonyl)imide Ethylene-Propylene-Diene Rubber Finite Element Method Finite Impulse Response

xxiv

FRP GFRP IC2 IPMC ITA KCl KNN MgCl2 MIMO MWCNT NaCl NaNO3 Na2SO4 NHE PACS PAN PVDF PZT SCE SHM SISO SLS SMA SMP SWCNT TBAHFP UAV

Nomenclature

Fiber Reinforced Plastics Glass Fiber Reinforced Plastic Interface Conformal Control Ionic Polymer-Metal Composites Institute of Textile Technology, RWTH Aachen University Potassium Chloride Potassium Sodium Niobate Magnesium Chloride Multiple Input/Multiple Output Multi-Walled Carbon Nanotube Sodium Chloride Sodium Nitrate Sodium Sulfate Natural Hydrogen Electrode Pressurized Actuated Cellular Structures Polyacrylonitrile Polyvinylidene Fluoride Lead Zirconate Titanate Saturated Calomel Electrode Structural Health Monitoring Single Input/Single Output Selective Laser Sintering Shape Memory Alloy Shape Memory Polymer Single-Walled Carbon Nanotube Tetrabutylammonium Hexafluorophosphate Unmanned Aerial Vehicle

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Introduction Johannes Michael Sinapius

„Man sollte sich nicht schlafen legen, ohne sagen zu können, daß man an dem Tage etwas gelernt hätte.“ (“You shouldn’t go to sleep without being able to say that you learned something that day.”) Georg Christoph Lichtenberg (1742–1799)

Since the mid-1980s, scientists have been conducting international research into adaptive structures. Especially in America, Japan and Germany, renowned working groups are dedicated to the question of self-adaptive structures in mechanical engineering. The following terms were introduced for this purpose • Smart materials, smart structures, or smart systems • Intelligent materials/structures/systems • Adaptive materials/structures/systems In Germany, the term “Adaptronics” was introduced in 1991 by an expert working group within the VDI (Association of German Engineers). “Adaptronics” is an artificial word that is composed of “adaptive” and “electronics”. According to the working group, the term describes a system in which at least one element of a control loop is multifunctional, on the one hand, and, on the other, the process of generating “intelligent” structures on the basis of multifunctional elements [24]. This condition

J. M. Sinapius (B) Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected] © Springer-Verlag GmbH Germany, part of Springer Nature 2021 J. M. Sinapius, Adaptronics – Smart Structures and Materials, https://doi.org/10.1007/978-3-662-61399-3_1

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also distinguishes adaptronics fundamentally from mechatronic systems in which the multifunctionality of a system element is not assumed. Worldwide research works and results in materials research in the field of energy-converting materials, so-called functional materials, have enabled a new class of technical systems that can adapt to external environmental conditions in terms of their elasto-mechanical behaviour. This opened up the possibility that sensory or actuator properties can be combined with load-bearing properties through the use of active functional materials. This leads to the following definition of the term adaptronics: Definition 1.1 Adaptronics creates a new class of technical, elasto-mechanical systems that can automatically adapt to a wide range of environmental conditions by using activatable materials and real-time digital controllers. A structure or system is adaptronic if all functional elements of a control loop are implemented and a multifunctional element is present at the same time. In this sense, adaptronics is also a multi- and interdisciplinary process in which adaptable structures are generated and integrated into technical systems. This process includes planning and manufacturing from material to system integration. Adaptronics therefore ranges from the search, development, and selection of multifunctional materials to system integration. Adaptronics is based on a scientific and a technical paradigm. The scientific paradigm of adaptronics is the investigation of load-bearing technical products, from material systems to machine elements, devices to plants with properties of biological adaptable systems for the purpose of mass and energy saving and the production of adaptability to different environmental conditions. The technical paradigm of adaptronics is characterized by the investigation of all questions of integration of actuators, sensors and control into a material or a structurally load-bearing component. The resulting multifunctionality leads to a high degree of functional integration. This, in turn, leads inherently to the challenge of harmonizing the requirements of function integration. The requirements can enter into an instrumental relationship, i.e. support each other, but can also have a competitive relationship. The harmonization of all functions of an adaptive structure during the design phase can best be described by the concept of creating functional conformity. Definition 1.2 Functional conformity refers to the optimum of the competing and non-competing objectives for all functions in an adaptive structure. The integration of an electrical conductor into a load-bearing structure may require maximum non-conformity, i.e. electrical isolation, while from a mechanical point of view it may require maximum possible adaptation in terms of stiffness and strength in order to avoid excessive stress concentration or premature failure. Adaptronics is a scientific discipline comprising several technical disciplines, which makes it particularly suitable for the training of young engineers.

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Introduction

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Fig. 1.1 Target fields of adaptronics

An adaptronic system is characterized by the interaction of mechanical and electronic components with the aim of adapting the elasto-mechanical structural properties. The focus is on influencing the external shape and reducing vibration and sound as well as monitoring structural health. Figure 1.1 represents these four target fields of adaptronics, which are also reflected in international priorities of smart structures research. The first publications on adaptronics date back to the middle of the last century. In 1933 Lueg published a first patent [22] for the active influencing of one-dimensional sound fields (see Fig. 8.2 in Sect. 8.1). In 1953 Olson and May [25] took up this idea and combined the proposal with a control. The work documents the first efforts to achieve an active reduction of the sound pressure in the one-dimensional wave field by means of sound interference. Only further progress and knowledge in signal processing opened the way to practical implementation. The transfer of the findings of the interference of waves in gaseous media to solids was also initially limited [32]. In the USA, the first approaches to adaptronic systems were published in the mid-seventies, which were followed up about 10 years later in several working groups, e.g. [35]. At the same time, intensive research work on the technology of adaptive structures also took place in Japan [31]. In Europe, scientists of the German Aerospace Center (DLR) launched the topic of adaptronics in 1989 with the project ARES (Aktiv Reagierende Elastische Strukturen) [3,4]. The first book on “Adaptronics” was published in 1995 by Neumann [24]. From the point of view of a project manager, the monograph gives a brief overview of active functional materials on 150 pages, and thus reflects a widespread view at the time that adaptronics is characterized solely by the use of “energy-converting functional materials”. The book by Elspass and Flemming [7], published in 1998, went a clear step further by taking up the interaction between structure and active elements. However, it is limited to purely static considerations. Aspects of functional conformity are not addressed. The authors tried to coin the term “Struktronik” with the book, but it has not become established in the German language. The first book entitled “Adaptronics” [17] was published in 1999. The anthology of 27 international authors, who highlights various aspects of adaptronics,

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was edited by Janocha. The addressed aspects include the system components actuators, sensors and controllers, but also design aspects and modelling. In the chapter by Baier and Döngi, the inherent optimization task in the development of adaptive structures is already pointed out, which ultimately leads to the production of maximum functional conformity. However, the volume is not a coherent technical textbook because the individual chapters stand on their own. At the same time (1998) the first scientific monograph was published under the title “Theory of Adaptive Structures” [30]. Utku developed approaches for the model-like description of adaptive structures without explicitly addressing active functional materials or addressing the problem of functional conformity. Ten years ago, Wadhawan published his textbook on “Smart Structures: Blurring the Distinction Between the Living and the Nonliving” [36]. With this book, Wadhawan contributes to the whole range of adaptive structural concepts and emphasizes the inherent question of nonlinear material behavior of active functional materials. The book by Chopra and Sirohi entitled “Smart structures theory” [6] dates from 2014. This volume also presents detailed information on the known active functional materials. The last chapter deals with applications of active materials in integrated systems. Thus, this book maintains the view that adaptronics is characterized solely by the application of multifunctional, energy-changing materials. In 2008 Preumont and Seto published a monograph entitled “Active Control of Structures” [27]. The book is an introduction to active vibration reduction and addresses active functional materials, adaptive structural concepts and semi-active control methods. In his book “Vibration Control of Active Structures” [26] Preumont discusses the different control approaches for the purpose of vibration damping in more detail. Gaudenzi wrote a smaller monograph in 2009 titled “Smart Structures” [10]. The book shows the behavior and effect of integrated piezoelectric functional materials up to composites with active functional materials from analytical perspectives. At about the same time as Janocha published “Adaptronics” [17], Ruschmeyer [28] published his volume on piezoceramics, which was limited to the basic and comprehensive presentation of piezoceramic materials, and thus does not belong to the literature on adaptronics in the narrower sense. Also the more recent works of Choi [5], Janocha [18], Jendritza [19] and Uchino [29] are characterized by the detailed description of active functional materials. This includes the standard work on electroactive polymers by Bar-Cohen [1]. All these reference books are valuable sources for adaptronics, but only consider one important aspect. Questions of function-compliant integration are not addressed. In 2010 contributions to the “1st Japanese-Austrian Workshop ‘Mechanics and model-based control of smart materials and structures”’ were published as an anthology [15]. The editors present contributions on the fundamentals of modeling in laminates, functionally graded materials—these are materials with specifically adjusted locally variable functional properties—as well as thermal and piezoelectric actuators. Also topics of passive damping, wave propagation and oscillations in adaptive structures are dealt with. More recent is the book of Fang [8], which is an anthology of several authors who provide contributions to methods of analysis of piezoelectric structures. They also deal with questions of wave propagation, high-frequency oscillations and structure optimization.

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The first books on “Active Vibration Reduction” and “Active Noise Reduction” were published in the early 90’s in England by Fuller, Elliot and Nelson [9,23]. Both works are characterized by the control engineering aspects. Design questions and structural aspects such as the integration of sensors and actuators into the load paths take a back seat, and thus also questions of the production of functional conformity. Books on active vibration reduction have been published in recent years by Gawronski [11], Inman [14], Krysinski and Malburt [20], Jalili [16], Vepa [34] and Landau et al. [21], who all deal with the partial aspect of active vibration control from different perspectives. In his book, Gawronski attaches particular importance to state-space formulation and contributes his many years of experience in space applications. Krysinski and Malburt focus on rotors and their active and passive vibration damping. In addition to piezoceramics, Jalili also considers piezoelectric nanomaterials and their use, especially for micro- and nanosensors. Vepa’s monograph includes chapters on active functional materials, sensors, structural dynamics and control engineering of adaptive structures. A more recent anthology by Hagedorn and Spelsberg-Korspeter “Active and Passive Vibration Control of Structures” was published [13]. The editors present a collection of contributions from the 2013 CISM course “Active and Passive Vibration Control of Structures”. The book explores active and passive vibration damping from different perspectives, such as modelling, electromagnetic and piezoelectric transducers or active magnetic bearings. The book by Valasek “Morphing Aerospace Vehicles and Structures” [33] was published in 2012 on the question of active shape control. Focused on training, the book provides an insight into all aspects of aircraft shape variability with a special focus on the requirements of aerodynamics, flight control, materials, performance and the use of smart actuators. In addition to the encyclopedia on structural health monitoring [2], the research area of integrated component monitoring includes an anthology of numerous authors, in particular the standard work by Giurgiutiu [12], which deals with methods and research results for integrated structural health monitoring, especially with structurally integrated active functional materials. In 2016, Yuan published a new compendium [37] on Structural Health Monitoring, which provides an up-to-date overview of integrated structural health monitoring in four parts. The individual parts include active functional materials and technologies for damage diagnosis and prognosis. This book is intended as a textbook for master students of mechanical engineering, aerospace engineering, mechatronics, automotive engineering and related studies and is also intended to give the professional engineer an insight into the exciting field of adaptronics.

References 1. Bar-Cohen, Y. (ed.): Electroactive Polymers. SPIE-Press, Bellingham (2004) 2. Boller, C., Chang, F.K., Fijino, Y. (eds.): Encyclopedia of Structural Health Monitoring. Wiley, Hoboken (2009) 3. Breitbach, E.: Research status on adaptive structures in Europe. In: Proceedings of the 2nd Joint Japan-USA Conference on Adaptive Structures (1991)

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4. Breitbach, E., Melcher, J.: New approaches for actively controlling large flexible space structures. In: Proceedings of the European Forum on Aeroelasticity and Structural Dynamics (1989) 5. Choi, S., Kim, J. (eds.): Smart Materials Actuators: Recent Advances in Characterization and Applications. Nova Science Publishers Inc, Hauppauge (2015) 6. Chopra, I., Sirohi, J.: Smart Structures Theory. Cambridge University Press, Cambridge (2014) 7. Elspass, W., Flemming, M.: Aktive Funktionsbauweisen. Springer, Berlin (1998) 8. Fang, D., Wang, J., Chen, W. (eds.): Analysis of Piezoelectric Structures and Devices. Higher Education Press and Walter de Gruyter GmbH, Berlin/Boston (2013) 9. Fuller, C., Elliot, S., Nelson, P.: Active Control of Vibration. Academic, Cambridge (1996) 10. Gaudenzi, P.: Smart Structures. Wiley, Hoboken (2009) 11. Gawronski, W.: Advanced Structural Dynamics and Active Control of Structures. Springer, Berlin (2004) 12. Giurgiutiu, V.: Structural Health Monitoring. Elsevier Inc., Amsterdam (2008) 13. Hagedorn, P., Spelsberg-Korspeter, G. (eds.): Active and Passive Vibration Control of Structures. Springer, Berlin (2014) 14. Inman, D.: Vibration with Control. Wiley, Hoboken (2006) 15. Irschik, H., Krommer, M., Watanabe, K., Furukawa, T.: Mechanics and Model-Based Control of Smart Materials and Structures. Springer, Berlin (2010) 16. Jalili, N.: Piezoelectric-Based Vibration Control. Springer, Berlin (2010) 17. Janocha, H. (ed.): Adaptronics and Smart Structures. Springer, Berlin (1999) 18. Janocha, H.: Unkonventionelle Aktoren. Oldenbourg Verlag, Munich (2013) 19. Jendritza, D. (ed.): Technischer Einsatz neuer Aktoren. Expert Verlag, Renningen-Malmsheim (2013) 20. Krysinski, T., Malburet, F.: Mechanical Vibrations Active and Passive Control. ISTE Ltd., London (2007) 21. Landau, J.D., Airimitoaie, T.B., Castellanos-Silva, A., Constantinescu, A.: Adaptive and Robust Active Vibration Control - Methodology and Tests. Springer, Berlin (2017) 22. Lueg, P.: Process of silencing sound oscillations. US Patent 2.043.416 (1933) 23. Nelson, P., Elliot, S.: Active Control of Sound. Academic, Cambridge (1992) 24. Neumann, D.: Bausteine “intelligenter” Technik von morgen - Funktionswerkstoffe in der Adaptronik. Wissenschaftliche Buchgesellschaft, Darmstadt (1995) 25. Olson, H., May, E.: Electronic sound absorber. J. Acoust. Soc. Am. 25, 1130–1136 (1953) 26. Preumont, A.: Vibration Control of Active Structures an Introduction. Kluwer Academic Publishers, Dordrecht (2002) 27. Preumont, A., Seto, K.: Active Control of Structures. Wiley, Hoboken (2008) 28. Ruschmeyer, K. (ed.): Piezokeramik. Expert-Verlag, Renningen-Malmsheim (1995) 29. Uchino, K.: Ferroelectric Devices & Piezoelectric Actuators. DEStech Publications, Lancaster (2016) 30. Utku, S.: Theory of Adaptive Structures. CRC Press, Boca Raton (1998) 31. Utku, S., Wada, B.: Adaptive structures in Japan. J. Intell. Mater. Syst. Struct. 4, 437–451 (1993) 32. Vaicaitis, R., Mixson, J.: Review of Research on Structureborne Noise. In: Proceedings of the 26th AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Paper Nr. 85-0786 (1985) 33. Valasek, J. (ed.): Morphing Aerospace Vehicles and Structures. Wiley, Hoboken (2012) 34. Vepa, R.: Dynamics of Smart Structures. Wiley, Hoboken (2010) 35. Wada, B., Fanson, J., Crawley, E.: Adaptive structures. J. Intell. Mater. Syst. Struct. 1, 157–174 (1990) 36. Wadhawan, V.: Smart Structures: Blurring the Distinction Between the Living and the Nonliving (2007) 37. Yuan, F.G. (ed.): Structural Health Monitoring (SHM) in Aerospace Structures. Woodhead Publishing, Sawston (2016)

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Principles of Adaptronics Johannes Michael Sinapius

„Zweifle an allem wenigstens einmal, und wäre es auch der Satz: zweimal 2 ist 4.“ (“Doubt everything at least once, and would it also be the sentence: twice 2 is 4.”) Georg Christoph Lichtenberg (1742–1799)

Abstract

This chapter first shows the basic principles of adaptronic systems using introductory examples. The elements of the integral development process of adaptronic systems are explained on the basis of system theoretical considerations. Furthermore, the chapter presents the basic variants of adaptronic systems. The most important building blocks of adaptronic systems and the integration variants of sensoric and actuatoric elements into the load path show the red thread through the book.

2.1

Introductory Examples of Adaptronic Systems

Research and development work in adaptronics ranges from material development to multifunctional elements (e.g. sensors and actuators that can be integrated into the structure, machine elements) and adaptive assemblies or components to complete systems. The engineering scientific disciplines are • materials engineering, • structural mechanics,

J. M. Sinapius (B) Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected] © Springer-Verlag GmbH Germany, part of Springer Nature 2021 J. M. Sinapius, Adaptronics – Smart Structures and Materials, https://doi.org/10.1007/978-3-662-61399-3_2

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Fig. 2.1 Shape control of a satellite reflector

• • • • • •

design theory, manufacturing technology, control engineering, systems engineering, information technology, and electrical engineering

and of course the basic disciplines of physics, chemistry and mathematics are involved in the processes. Adaptronics is used in the development of new products as well as new production processes. Three examples illustrate this in the following. Figure 2.1 sketches a satellite reflector which is exposed to the high temperature of solar radiation on one side in space, while the side facing away from the sun experiences low temperatures. The temperature gradient between the top and bottom side leads to a strong thermal expansion gradient in the reflector with the result that the reflector is bent. The primary function, the transfer of information by means of electromagnetic radiation, is impaired. According to [2, p. 463] the deviations from the target contour of a reflector surface are noticeable by a reduction of the antenna gain. The antenna gain summarizes the directivity and efficiency of an antenna. The principle of adaptronics that is applied here is to actively counteract curvature. For this purpose, the curvature must be fed to a controller, e.g. by measuring the relative surface strain, which then generates a control signal, which in turn is transmitted via a power amplifier to a structurally integrated actuator. The actuator acts directly on the elasto-mechanical behavior of the reflector and returns it to the target contour. The multifunctional elements in this case are the structure-integrated sensors and actuators. The second example is a parallel robot. Parallel robots are characterized by closed kinematic chains, which are particularly suitable for very fast movements and the associated high accelerations due to their high rigidity. Figure 2.2 shows an arrangement of a parallel kinematics [5], as it was researched in the Collaborative Research Center SFB562 of the Technische Universität Braunschweig. The goal of the SFB562

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Fig. 2.2 Active vibration control of a parallel robot

was the development of method- and component-related fundamentals for the development of robot systems based on closed kinematic chains (parallel robots). The focus was on high achievable speeds and accelerations, as they are required in the field of application of handling and assembly with simultaneously high accuracy requirements. The end-effector, in which robotics is the last element of a kinematic chain, moves at high speed, which significantly influences productivity, to the desired working position. When reaching the working position, the end-effector must be braked quickly. The resulting strong deceleration induces vibrations in the end-effector, which must decay before the work task can be started, especially in high-precision work steps. This rapid deceleration is indicated by a large arrow in Fig. 2.2. The principle of adaptronics is to calm down induced vibrations as quickly as possible. For this purpose, a sensor at the end-effector must detect the acceleration and feed it to a controller, which generates a control signal from it. This in turn is transmitted via a power amplifier to an actuator, which generates a counter-oscillation and thus causes a damping of the vibration by structure-borne noise interference. The principle of structure-borne sound interference is explained in detail in Sect. 6.2.2. Figure 2.3 shows a section of the research robot Triglide and two activatable rods in detail [5, p. 159ff.]. Chapter 4 deals in detail with the patch actuators used. Together with the rod, they form a multifunctional element. Section 6.5.2 explains the design of the two activatable rods. The rods must transfer the loads and at the same time serves as a vibration exciter that generates the necessary counter-vibrations to calm the end-effector. The third example may initially surprise as an example of the principles of adaptronics. Figure 2.4 schematically shows a pultrusion line. Figure 2.5 depicts the pultrusion research facility of the Institute of Adaptronics and Function Integration of the Technische Universität Braunschweig. In the pultrusion process of fibre composite profiles, fibres impregnated with resin are continuously pulled through a heated

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Fig. 2.3 Parallel robot with active bar structures

J. M. Sinapius

© DLR e.V.

Fig. 2.4 Pultrusion with adaptronic tool

mould, in which the resin cures, by a puller unit. The material is created with the product. The exact position of the fibres in the load direction (geometric challenge to the process) is decisive for the high mechanical properties. During the process, this geometric fibre position must remain fixed until the resin has cured (a time challenge for the process). The speed at which the impregnated fibers can be pulled through the tool is determined by the speed of curing. For this reason, the position of the gelling point in the tool, i.e. the point at which the liquid state of the resin changes into a gel-like state, defines the pulling speed. The dimensional stability of the drawn profile depends on the degree of curing. If it is pulled out of the mould too quickly, it can still deform. However, if it is drawn too slowly, it may stick to the tool. If now a further functionality is added to the tool, namely a metrological function to the shape-giving and fiber orientation-giving function, a multifunctional tool is created, which influences the elastomechanical behavior of the pultrudate completely in the sense of adaptronics. The sensors that can be integrated must therefore be able to detect the curing state without influencing the fiber orientation and the temperature distribution in the tooling. In his dissertation [4], Pommer emphasizes the advantages of in-situ curing monitoring in pultrusion proceses. This is a typical task of adaptronics, namely the production of functional conformity. The three examples come from different areas: product, means of production and production process. However, together they show the basic principles of an adaptive system, namely state observation, control and the active influence of actuators and the use of elasto-mechanical behaviour. A multitude of further examples of

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Fig. 2.5 Pultrusion research plant

the research and development of adaptronic systems are contained in the volume “Adaptive, Tolerant and Efficient Composite Structures” [6].

2.2

Elements of Adaptronics

The development of adaptronic systems takes place as an integral process in four areas. The first realization of multifunctionality is already taking place within the framework of material development. Even though the use of multifunctional, energyconverting materials is not a mandatory feature of an adaptronic system, this class of materials is nevertheless of great interest in adaptronics since the integration of functions towards multifunctionality is already takes place at this level. In the second area, multifunctional elements are developed, which already produce part of the functional conformity, e.g. the harmonization of the sensory properties with the stiffness and strength requirements, but also with electrical functionalities. In the third area, the specific design task takes place, which is usually associated with an optimization of the design zones in order to harmonize the most varied requirements on the adaptive structure. The last major area is characterized by system integration, which includes questions of energy supply and control. Figure 2.6 represents the four areas of the development and gives exemplarily the elements of an adaptive reflector antenna, whose operating principle is illustrated in Fig. 2.1. An essential difference to a widely held opinion on adaptronics cited in the introduction (Chap. 1) is that although the use of active functional materials is often the basis of adaptronic systems, it is not the sole and mandatory prerequisite. Due to the limits of active functional materials, e.g. with regard to achievable actuator strokes, multifunctionality can only be achieved at the next level.

2.3

Variants of Adaptive Systems

Adaptronic systems, like all technical systems, perform a task specified in the requirements. They “serve a technical process in which energies, substances and signals are

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Fig. 2.6 Elements of adaptronic systems

Fig. 2.7 Adaptronic Systems

conducted and/or modified” [3, p. 42]. Figure 2.7 (left) illustrates this connection according to the theory of design [3]. It considers technical entities as systems characterized by the conversion of input variables into output variables, such as Fig. 2.7 (right) represents [1]. However, this state is a state that cannot be reached completely, rather only approximately, in the philosophical sense thus an ideal state. In the physical sense, a system is ideal if the transformation is loss-free. In the three examples illustrated with Figs. 2.1, 2.2 and 2.4 the energy, material and signal flows can be identified. The satellite reflector transmits signals, the parallel robot converts electrical energy into kinetic energy, and the pultrusion system converts the fibers and resin into a profile. In an ideal system, the technical system translates the input variable(s) without losses into the desired output variable(s), which is (are) the target variable(s) at the same time. At the same time, however, the three examples also show that disturbance variables acting on the system can lead not only to the desired effect (target variable) but also to malfunctions or deviations from the desired output variable (target variable), as shown in Fig. 2.8. This is where adaptronics comes in by enabling the technical system to react to the disturbance variable and achieve the desired effect (target variable). As Fig. 2.9 shows, two ways are possible. On the one hand—if possible—the disturbance vari-

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Fig. 2.8 Technical system under the influence of disturbances

Fig. 2.9 Adaptive, malfunction suppressing technical system

Fig. 2.10 Adaptive technical system with changing characteristics

able can be detected and converted into an adaptation variable. On the other hand the output variable can be measured and from a comparison with the target variable, an adaptation variable can be determined, which acts on the adaptronic system. Adaptability can also mean that a system can switch between different target functions in response to external conditions. Figure 2.10 illustrates this variant of adaptronic systems. The variable wing leading edge of a commercial aircraft is an example of this from research. The variable wing leading edge should be switchable between the gap-free condition of the cruise flight with the desired maximum laminar flow of the wing and the maximum possible high lift for take-off and landing. It is presented in Sect. 5.5.2 as an example of shape variability.

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Components of Adaptive Systems

From the preceding considerations, the essential, characteristic building blocks of adaptronics emerge: • Construction material: composite materials are particularly suitable for function integration in the manufacture of multifunctionality. • Multifunctional materials: the focus here is on functional materials because their energy-converting properties can be exploited. • Design rules aiming in particular at harmonising the different functions. • Control: fast to real-time capable and adaptive control algorithms are developed and applied in adaptronics. Ultimately, the degree of integration determines an adaptronic system and distinguishes it from purely mechatronic systems. The integration stages of sensory and actuator properties currently in research and development are visualized in Fig. 2.11. The interaction of mechanical structure, discrete actuators and sensors, control engineering and electrical engineering is typical for mechatronic systems. According to the German VDI Guideline 2206, mechatronic systems describe the synergetic interaction of mechanical systems, electronic systems and associated information processing.

Fig. 2.11 Integration steps of sensory and actuator properties

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Principles of Adaptronics

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In the first integration steps of adaptronic systems, sensors and actuators are brought close to the load path, usually over a wide area by application. In the second integration stage, they are integrated into the load path and perform load-bearing functions. Further integration stages include signal processing and control.

2.5

Summary

The chapter shows the principle characteristics of adaptronic systems. The basic variants are adaptive systems that suppress malfunctions and adaptive technical systems with changing properties. The building blocks of such systems are developed from basic system considerations and the engineering challenges in development are shown. The next chapter will describe the most important characteristics of functional materials that can be integrated into structures and present the most important functional materials currently used in adaptronics.

References 1. 2. 3. 4.

Hubka, V. (ed.): Theorie Technischer Systeme. Springer, Berlin (1984) Kark, K. (ed.): Antennen und Strahlungsfelder. Springer, Berlin (2011) Pahl, G., Beitz, W., Feldhusen, J., K.-H., G.: Konstruktionslehre, 7th edn. Springer, Berlin (2007) Pommer, C.: Geregelter Pultrusionsprozess mit In-situ-Aushärtungsüberwachung. Ph.D. thesis, Technische Universität Braunschweig (2019) 5. Schütz, D., Wahl, F. (eds.): Robotic Systems for Handling and Assembly. Springer, Berlin (2010) 6. Wiedemann, M., Sinapius, M. (eds.): Adaptive. Tolerant and Efficient Composite Structures. Research Topics in Aerospace. Springer, Berlin (2013)

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Functional Materials Johannes Michael Sinapius and Sebastian Geier

„Durch das Einfache geht der Eingang zur Wahrheit.“ (“Through simplicity goes the entrance to truth.”) Georg Christoph Lichtenberg (1742–1799)

Abstract

This chapter first shows the basic principles of functional materials that have energy-converting properties. It derives the basic parameters of functional materials. The idea of two now widely used classes of functional materials, electromechanical transducers and thermomechanical transducers, concretizes these basic properties. The chapter presents in detail the phenomenological properties of energy conversion and presents simple models for the complex conversion processes. A concluding overview provides information on other functional materials, some of which are currently being researched. An example from current research on functional materials completes this overview.

3.1

Classification of Functional Materials

In the development of adaptronic systems (Fig. 2.6) functional materials play an essential role as a basis for the realization of multifunctionality. Multifunctional materials are characterised by the fact that, in addition to the features of a construc-

J. Michael Sinapius (B) Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected] S. Geier German Aerospace Center (DLR e.V.), Braunschweig, Germany e-mail: [email protected] © Springer-Verlag GmbH Germany, part of Springer Nature 2021 J. M. Sinapius, Adaptronics – Smart Structures and Materials, https://doi.org/10.1007/978-3-662-61399-3_3

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tion material, i.e. primarily load-bearing properties, they also have other features, e.g. actuator or sensory properties. Construction materials are materials from which components are designed and manufactured. They cannot be activated, i.e. they are passive and primarily characterized by their mechanical properties of stiffness and strength. Functional materials, on the other hand, fulfil certain functions when they are excited or controlled by ambient conditions. In contrast to discrete functional devices, multifunctional materials retain their functional properties, i.e. their physical effect when divided. For example, a motor loses its ability to generate motion when divided. A functional actuator material, however, basically retains its motiongenerating physical property after splitting. In the literature, functional materials are divided into two groups, passive and active functional materials. Passive functional materials are characterized by an outstanding physical property that can be used functionally because it changes over a wide range. With crystalline materials, this is often due to a phase transition of the crystals in a limited temperature range, e.g.: • glass fibers used for sensory purposes, or • superconductors whose electrical resistance becomes minimal at certain temperatures. In terms of adaptronics, however, functional materials are energy converters which are occasionally referred to in the literature as active functional materials. The basic forms of energy are mechanical, thermal, electrical and magnetic energies. Functional materials can convert e.g. • mechanical energy into electrical energy, • thermal energy into electrical energy, or • magnetic energy into mechanical energy. Fig. 3.1 Classification of multifunctional materials

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Typical examples are • piezoelectric materials that convert pressure into charge displacement, • shape memory alloys that convert heat into deformation, or • magnetostrictives that convert a magnetic field into deformation. Figure 3.1 classifies the currently best known functional materials of adaptronics.

3.2

Features of Functional Materials

Most of the functional materials mentioned can be used for sensory and actuator purposes. Sensorically acting and used material ideally converts a physical quantity, independent of other external influences, loss-free into a signal, usually an electrical signal. For example, functional materials with sensory properties can be used to convert • motion quantities into electrical signals, or • thermal changes into electrical signals. They thus transform state variables of a system into input variables of the adaptive system. Since the focus of adaptronics is on influencing the elasto-mechanical behavior, mechano-electric transducer materials are of particular interest, i.e. functional materials that convert a mechanical structure size, such as material elongation, into an electrical signal. An essential requirement for functional materials in adaptronics is the uniqueness of the conversion. An input variable X i is thus to be converted into the output variable X o via a unique functional relationship. In this sense, the transformation of the mechanical measurement information by the active functional material is also a mathematical representation: X o = F (X i )

(3.1)

Mathematically speaking, a unique function means that exactly one output variable is assigned to each input variable. A further requirement of an active functional material used for sensory purposes is that the functional relationship between the input variable X i and the output variable X o must be continuous. Preferably, active functional materials with a linear or almost linear relationship between input X i and output X o are developed and used. A linear relationship is characterized by a constant sensitivity defined by the differential quotient d X o/d X i . Figure 3.2 shows in the left part the ideal linear characteristic curve of a functional material. The sensory properties of functional materials are mostly based on crystalline processes, such as stability limits or phase transitions. Therefore, they usually behave nonlinearly and are characterized by hysteresis behavior, as Fig. 3.2 in the bottom part shows.

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Fig. 3.2 Characteristics of sensory functional materials, top ideal, bottom real

The output variable X o not only depend on the input variable X i , but also on its history. The uniqueness and linearity requirements are therefore usually violated in the case of functional materials. Due to the losses which are reflected in the hysteresis behaviour, they are not ideal converters. One goal in the development of functional materials for adaptronics is to keep the deviation of uniqueness and linearity as well as losses as low as possible. Other significant interfering influences for sensory material are • cross sensitivity, • drift of sensitivity, and • noise. Cross-sensitivity is a major interference factor in sensor technology for adaptronic systems because vector quantities are to be measured with the mechanical quantities. If mechanical structural changes that are orthogonal to the intended measuring direction (e.g. transverse contraction) influence the measurement result, this is a disturbance that influences the unambiguous assignment of the measured variable to the cause. In addition to the non-linearity of the functional relationship between input X i and output X o , the sensitivity may be time-dependent due to temporal changes of the material itself. This can also include the electrical fatigue of the active functional materials. Noise refers to randomly distributed disturbances. Actuating material converts a control signal, usually an electrical signal, into a physical quantity independent of other external influences. Thus, mainly electrical signals are converted into motion quantities by functional actuator materials. They thus define the response of the adaptive system. In adaptronics, electro-mechanical and thermo-mechanical transducers are mainly used. Just like functional materials used for sensory purposes, an ideal actuatorically used energy-converting material should establish a linear and unambiguous functional relationship between input variable X i and output variable X o , as Fig. 3.3 on the bottom shows. The same causes as for functional materials used in sensors lead to a non-linear, hysteresis-prone transducer behaviour in real functional materials used in actuators. The following are also considered significant interferences for actuator material

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Fig. 3.3 Characteristic curves of actuator functional materials, top ideal and linear, bottom real

• • • •

nonlinear conversion, cross-sensitivity, drift of sensitivity, and noise.

The suitability of a material as a functional material in adaptronics is determined by its “mobility” and reactivity to energy supply. Reactivity is often related to crystalline stability limits or structural phase transitions. Phase transitions are thermodynamically induced crystalline instabilities to which a critical temperature Tc (Curie temperature) can be assigned. These relationships are illustrated in Sect. 3.3 using the example of piezoceramics. The supply of energy to functional actuator materials used to influence the elastomechanical behaviour causes a change in length, provided there is no external constraint. In a broader sense, it is referred to as the free stroke. In the narrower sense used in this book, the free stroke u 0 describes the maximum unhindered change in length of a functional material, which of course still depends on the actuator geometry. Figure 3.4 illustrates the general case of a typical transducer with non-linear characteristic. The supplied thermal, magnetic or electrical energy is converted into potential energy in the form of a change in length. The change in length is due to material strain:

Fig. 3.4 Free stroke and free expansion of functional actuator materials

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du (3.2) dz The maximum unhindered strain is called free strain ε0 and can be linearized by εz =

ε0 =

u0 . l0

(3.3)

l0 denotes the original length of the actuator according to Fig. 3.4 u 0 the extension due to the energy input. In addition, Fig. 3.4 specifies the energy conversion for an electromechanical transducer. The electric field E generated by an electric voltage leads to a strain εz of the functional material. To prevent a functional material from its free strain, i.e. to block it, the blocking force f B is required. The stress σz is generated in the material, which is related for linear-elastic material behavior to the elongation by σz = ϒ A εz .

(3.4)

ϒ A is the modulus of elasticity of the actuator material. From this follows the blocking stress σ B = ϒ A ε0 ,

(3.5)

which is related to the blocking force f B for an actuator cross-section A by f B = σB A .

(3.6)

The blocking force f B of an actuator indicates the force that pushes an actuator deflected by energy back into its original shape. Together with Eqs. (3.5) and (3.3) the blocking force is determined by f B = k Au0

(3.7)

This contains the stiffness k A ϒA A (3.8) l0 of the actuator in longitudinal direction. It determines the gradient of the straight line in the working diagram of the actuator shown in Fig. 3.5. In the stress-strain diagram, there is a relationship between an external load and the strain due to energy input: kA =

ε z = ε0 +

σz . ϒA

(3.9)

The normal stress σz results from the compressive force f A , i.e. σz = − f A /A. With Eq. (3.8), u = u0 −

fA kA

(3.10)

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Fig. 3.5 Blocking force and working diagram of actuator functional materials

describes the straight line in the working diagram of Fig. 3.5. A central task of the design and development of adaptronic structures is the integration of actuators into the load path of a load-bearing material. When integrating a functional material into a construction material, the stiffness ratio between the actuator and the load-bearing structure plays an important role in the usable work. This shall be illustrated by a simple, model-like thought experiment with Fig. 3.6. The figure shows the construction material as a spring which acts on the functional material. The force-dependent deflection of the actuator is as follows: f A (u) kA

u = u0 −

(3.11)

f A (u) denotes the interface force between the actuator material and the construction material. It depends on the deflection u and spring stiffness k S of the structural material: f A (u) = k S u

(3.12)

With the stiffness ratio αk =

kS kA

(3.13)

it follows for the actuator deflection by inserting Eq. (3.11) into Eq. (3.12): u = u0

1 1 + αk

Fig. 3.6 Integration of functional actuator materials

(3.14)

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Fig. 3.7 Working capacity of the integrated functional material

Fig. 3.8 On the structural conformity of integrated functional materials

The actuator deflected by energy supply is located in the point O (Fig. 3.6, right). The area under the elasticity line of the construction material characterizes the work performed: 1 (3.15) kS u2 2 The work done W A can be expressed with the actuator characteristic values free stroke u 0 and actuator stiffness k A : WA =

WA =

αk

1 k A u 20 (1 + αk ) 2 2

(3.16)

The factor αk/(1+αk )2 expresses the influence of the stiffness ratio between actuator and construction material on the actuator effect and is shown in Fig. 3.7. Maximum working capacity exists for αk = 1, i.e. if structural stiffness and stiffness of the actuator are equal. With a stiffness ratio of αk = 1 maximum structural conformity is reached, as Fig. 3.8 depicts. The figure illustrates an embedding in a very flexible structural material on the left and a very rigid one on the right. In addition to material properties, structural conformity of course also depends on geometric conditions. In the simplest case shown, the consideration refers to longitudinal stiffness. Definition 3.1 Structural conformity is an adaptronic characteristic, a degree of adaptability. Structure-compliant actuators conform to the structure, especially with regard to their stiffness. The concept of structural conformity can be extended to

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function conformity if further properties of sensors and actuators are harmonized with properties of the construction (see Definition 1.2). The process of establishing structural conformity is also referred to as impedance matching. The term impedance comes etymologically from the Latin, in which “impedire” means “inhibit”. In adaptronics, the question of obstacles to adaptability must play a central role. In active shape control (Chap. 5) the mechanical impedance matching, in active vibration control (Chap. 6) the mechanical and electrical impedance matching, and in active sound control (Chap. 8) the acoustic sound impedance matching plays a central role. These three physical quantities are visualized in Fig. 3.9 • mechanical impedance in Eq. (6.162) as ratio of stimulating force f a (t) and vibration velocity vs (t). • electrical impedance in Eq. (6.275) as ratio of electrical voltage U (t) and current I (t). • acoustic sound characteristic impedance in Eq. (8.31) as ratio of sound pressure p(t) and sound velocity v(t). If the maximum possible working capacity, i.e. for αk = 1 or αk/(1+αk )2 = 14 is related to the available volume of the active functional material, which forms the actuator, the relationship energy per volume results. It is called the volumetric energy Fig. 3.9 To the impedance concept

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Table 3.1 Overview of central common parameters of actuatorically used functional materials u0 l0

Free strain

ε0 =

Blocking stress Blocking force Maximum working capacity

σ B = ϒ A ε0 f B = ϒ A Aε0 W A,max = 18 k A u 20

Energy density Specific energy density

(–)

W E = 18 ϒ A ε02 W Es =

(N/mm2 ) (N) (J) (J/m3 )

2 1 ϒ A ε0 8 ρA

(J/kg)

density: WA V WA = Al0 1 k A u 20 = 8 Al0

WE =

WE =

1 Υ A ε02 8

(3.17)

The energy density W E is a parameter of the performance of a functional material and depends, besides the modulus of elasticity Υ A , only on the square of the maximum achievable active strain ε0 , as Eq. (3.17) for linear-elastic material behavior shows. Especially for lightweight applications the specific energy density W E s is an additional parameter: W Es =

1 Υ A ε02 8 ρA

(3.18)

The specific energy density refers the energy density of the functional material to its density ρ A . Table 3.1 gives a summary overview of the previously derived parameters of functional materials used in actuators. Further parameters are explained in the following sections and summarized in the Tables 3.3, 3.4 and 3.5 for the presented functional materials. A comprehensive overview of the properties of functional materials is contained in Newnham’s monograph [57].

3.3

Electromechanical Converters

The most common transducer materials currently used in adaptronics are electromechanical transducers. According to Fig. 3.1 they can essentially be divided into two categories:

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Fig. 3.10 On the dipole moment of electromechanical functional materials in the electric field

1. piezoelectric ceramics 2. electroactive polymers Electrorheological fluids are also listed in Fig. 3.1 because they are often mentioned in the literature in connection with adaptive structures. However, they play a rather minor role in the research and development work on adaptronics and are therefore not dealt with in this book. Essential prerequisites for electromechanical conversion within an active functional material are a material with electrical dipoles on an elementary level and the presence of an electric field. A dipole is characterized by the physical asymmetric arrangement of positive and negative charges of the same size, i.e. the center of the charges does not coincide. However, they compensate each other in terms of amount. The dipole itself is characterized by the amount of the opposite charges Q and their distance ld . This assumes an asymmetry on the elementary level. The product of charge Q and spatial distance ld forms the existing dipole moment. Such dipole moments can be found in crystalline materials as well as in polymers with near orders of the polymer chains (see Fig. 5.9). If an electric dipole, as shown in Fig. 3.10, is located in an electric field formed between two electrodes to which an electric voltage U is applied, a moment M y acts on the dipole: M y = f a ld sin ϕ = Q Eld sin ϕ U = Q ld sin ϕ l0

(3.19)

In general spatial notation Eq. (3.19): {M} = Q · {l} × {E}

(3.20)

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If the electric dipole is perpendicular to the electric field lines, the moment is maximum. If the direction of the dipole is in the direction of the electric field lines, the moment is zero. The acting moment causes a mechanical distortion of the active functional material, which is used actuatorically. So there is an interaction between electrical and mechanical quantities. The following sections introduce two of the most important piezoelectric materials, one based on ceramic ion crystals, the second based on semi-crystalline thermoplastics.

3.3.1

Piezoelectric Ceramics

Piezoelectric ceramics belong to the class of crystalline materials. Their crystalline structure is characterized by an ion lattice, i.e. by a regular spatial arrangement of anions and cations of a homogeneous substance in the solid-state. The cohesion of the lattice formation is effected by the ionic bond. Ceramics are non-metallic, inorganic materials. They consist of fine-grained raw materials which are sintered after a mixing process in an aqueous medium and subsequent dewatering. This process, called sintering, produces a typical crystalline structure. Single crystals are only formed under special production conditions, otherwise a polycrystalline material is formed. The prerequisite for piezoelectricity is the absence of a center of symmetry in the crystal structure, which results in electrical dipole moments. If these dipole moments can be changed by applying an external electric field and distorting the grid, the ceramics are piezoelectric. They are therefore ferroelectric materials. Ferroelectric substances are always piezoelectric and pyroelectric. The prefix “ferro-” is an analogy to the property of ferromagnetism, in which magnetism disappears at high temperatures. Heckmann [25] represents the energy conversion in a triangle between mechanics, electrical engineering and thermodynamics, as can be seen in Fig. 3.11. The piezoelectric effect was discovered in 1880 by the brothers Pierre and Jacques Curie.1 They observed that for certain crystals mechanical stress leads to an electrical charge displacement. The inverse piezoelectric effect was predicted and theoretically justified in 1881 by Gabriel Lippmann2 and later experimentally confirmed by the Curie brothers. Due to the inverse piezoelectric effect, piezoelectric crystals expand when an electrical voltage is applied. Barium titanate (BaTiO3 ) and lead zirconate lead titanate (PZT) were the first representatives of this class of materials to be researched. Today, ceramics made of lead zirconate lead titanate, which was discovered in 1971 as a piezoelectrically functional material, are mainly used. Due to the heavy metal content in the PZT, however, alternatives in the form

1 Jacques

Curie (1855–1941) received in 1903 together with Marie Curie and Henri Becquerel the Nobel Prize for Physics for their “Joint work on the radiation phenomena discovered by H. Becquerel”. The Curie brothers were awarded the Prix Gaston Planté of the Académie des sciences in 1895 for their discovery of piezoelectricity. 2 Gabriel Lippmann (1845–1921), French physicist, who received the Nobel Prize for Physics in 1908 for his “method of photographically reproducing colors based on the interference phenomenon”.

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Fig. 3.11 Relationships between mechanics, electrical engineering and thermodynamics of ferroelectrics, according to Heckmann [25]

of lead-free ceramics with the same performance are being sought intensively, see Sect. 3.3.1.7.

3.3.1.1 Lead Zirconate Lead Titanate (PZT) Lead zirconate lead titanate (PZT) consists of the basic substance lead (Pb), oxygen (O) and titanium (Ti) or zirconium (Zr) in the stoichiometric ratio PbZr1−x Tix O3 . From the crystal structure PZT belongs to the class of perovskite structures, an orthorhombic crystal system. Perovskite is a calcium titanate (CaTiO3 ), a mineral

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Fig. 3.12 Perovskite, mineral and crystal structure

from nature. The general grid structure and the corresponding structural model are shown in Fig. 3.12. In the lattice structure shown, the atom with the designation A for PZT corresponds to zirconium or titanium, B to lead and O to oxygen atoms. PZT has different lattice structures depending on temperature and composition. This can be represented in the diagram for multiphase materials, the so-called phase diagram. The phase diagram for PZT shows the temperature-dependent lattice structures as a function of the proportion of lead zirconate (PZ) in relation to the lead titanate (PT). The actual position of the phase boundaries between the different lattice structures depends, among other things, on additives to the PZT and can thus be influenced. The so-called additives (donor), which also influence the electromechanical properties, are referred to on, are explained below. Above the temperature limit Tc shown in Fig. 3.13, which is called the Curie temperature (about 200–350◦ C), PZT exists in a cubic lattice. Here all edge lengths of the grid are the same. Because of the point symmetry then present, the charge centre of gravity and geometric centre coincide. Thus, the material is not piezoelectric. When cooling below the Curie temperature Tc a spontaneous lattice transformation takes place. The grid folds from the cubic grid, depending on the Zr/Ti ratio, in asymmetric grids. The phase diagram of the PZT (Fig. 3.13) shows a further phase boundary at temperatures below the Curie temperature, which is slightly above 50% lead titanate. If the titanate content is higher, there is a tetragonal lattice where one lattice parameter changes compared to the cubic lattice, the edge lengths are no longer equal. The grid distorts constant volume in one of the orthogonal grid directions. The angles of the tetragonal grid remain 90◦ . At lower titanate proportions there is a rhombohedral lattice, where all edge lengths are equal, but the angles to the cubic lattice are distorted, i.e. = 90◦ . The boundary between tetragonal and rhombohedral lattice is nearly temperature independent and is called morphotropic phase boundary. The morphotropic phase boundary lies between 40 and 45% titanate, i.e. between 55 and 60% zirconate, respectively. It is blurred because it consists of a mixture of the ferroelectric, tetragonal and ferroelectric, rhombohedral phases. Figure 3.14 shows the ferroelectric domains in morphotropic PZT ceramics in a in a scanning electron microscope image. The structure of the domains suggests that rhombohedral (Rh) and tetragonal (T) phase regions occur within a grain. Particularly favourable values of the electromechanical properties can be found for ceramic compositions near this phase boundary. The setting of the piezoelec-

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Fig. 3.13 Schematic phase diagram of the lead zirconate lead titanate Fig. 3.14 Ferroelectric domains in morphotropic PZT ceramics ©picture Fraunhofer IKTS (2002), production of ceramics KIT (IAM-KWT)

tric characteristic values (piezoelectric constant d33 and coupling factor k33 , see Sects. 3.3.1.2 and Sect. 3.3.1.3) is primarily done by the relationship between lead zirconate and lead titanate. During the transition at the Curie temperature from the cubic lattice to the ferroelectric, tetragonal phase, the titanium or zirconium ions and the oxygen ions shift axially towards each other. The grid expands in a constant volume in the direction of the displacement. There are six equal directions of ion displacement, as Fig. 3.15 shows. The axial displacement of the ions to each other and the associated lattice distortion lead to the formation of dipole moments. This process is called spontaneous polarization. From it, six possible areas with uniform polarization direction result in

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Fig. 3.15 Spontaneous polarization

each grain, so-called domain. Domains are thus areas of grids with homogeneous orientation of spontaneous polarization. If only a homogeneous polarization orientation prevails in a crystal, a single domain state is present. However, multi-domain crystals are common because the phase transformation process starts at many points simultaneously. The areas of homogeneous polarization are separated by domain walls. If there is no external electric field in the process of spontaneous polarization, a multi-axial ferroelectric is formed, i.e. are usually 6 energetically stable polarization directions. In general, entropy plays a decisive role in phase transitions. According to the second law of thermodynamics, the equilibrium of isolated thermodynamic systems is characterized by a maximum of entropy. According to the statistical formulation of the second main theorem, the entropy of a closed system increases in spontaneous processes. This can be explained by a statistical analysis. The most probable arrangement of the polarization direction in the numerous starting points of the phase transformation in the many grains is the one with the most realization possibilities of the polarization states of the domains. The thermodynamic probability is equal to the number of realization possibilities of these microstates. In spontaneous polarization, the axial displacement in the material is accompanied by a state of greatest probability, which corresponds to a state of equilibrium. The maximum probability for the polarization direction of n D domains can be specified with nD! =  nD 6

!

(3.21)

The important relationship between the thermodynamic probability and the entropy S E of a state is described by the Boltzmann equation S E = k B ln

(3.22)

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In Eq. (3.22) k B = 1, 381 · 10−23 J/K is the Boltzmann3 constant. From this statistical observation it follows that after spontaneous polarization by cooling, a state is most likely to occur in which the dipoles of all domains, which form at the same time, cancel each other out. The material is therefore macroscopically nonpolar. After cooling, the ceramic consists of grains with a large number of areas of random crystal orientation. Figure 3.16 shows a fracture pattern of a PZT ceramic in which the individual grains are clearly visible. Within the grains, areas of equal polarization are formed, so-called Weiss4 domains (or simply domains). The term Weiss domain comes from the field of ferromagnetic materials, where it describes microscopically small magnetized domains in the crystals. If two Weiss domains with different polarization directions collide with each other, the direction of the polarization changes fluently. The boundary between the Weiss domains in ferroelectric materials below the Curie temperature is called Bloch5 wall. The polarization change in the Bloch wall has a helical structure, as Fig. 3.17 illustrates. Both 90◦ and 180◦ domain boundaries exist within a grain. The domains are statistically distributed after cooling from the sintering process. The dipole moments cancel each other out, i.e. they have no macroscopic effect. The dissertation of Guyonnet [23] provides a comprehensive study on ferroelectric domain boundaries. Starting from the macroscopically nonpolar state (zero point of the dielectric hysteresis shown), the polarization curve runs on the so-called maiden curve. At the beginning, the areas with a dipole moment, which is nearly opposite to the field direction, are reoriented to about 90◦ . This initially leads to stagnation of the strain curve with increasing polarization. If the field strength E reaches the electric coercivity E c , the change of domains starts for all areas. The domain structure changes permanently during polarization. It reaches a saturation level at a maximum field strength of E max at which the maximum posFig. 3.16 Grains of a PZT ceramic in microscopic fracture

3 Introduced by Max Planck(1858–1947), German physicist, named after Ludwig Boltzmann (1844–

1906), Austrian physicist and philosopher. after Pierre-Ernest Weiss (1865–1940), French physicist. 5 After Felix Bloch (1905–1983), Swiss-American physicist, was awarded the Nobel Prize for Physics in 1952. 4 Named

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Fig. 3.17 Bloch walls (inside the grains the domains are recognizable by light-dark contrast)

Fig. 3.18 Polarization process

sible domain orientation is reached. Thus the maximum polarization is reached at the same time. If the applied electric voltage U is removed again, the electric field goes back to zero and the domain structure is reoriented while the newly formed domains are retained. This results in a remanent polarization Πr . If now a voltage is applied, which is opposite to the remanent polarization, the material is repolarized when the negative coercivity −E c is reached. The titanium and zirconium ions centrally located in the lattice change their position, as do the oxygen atoms. When the maximum possible domain orientation is reached, i.e. the corresponding electric field −E max , the material is completely repolarized. In the cycle between −E max and +E max the hysteresis curve is formed, which encloses the area caused by losses due to internal friction processes. The area of the hysteresis curve is a measure for the lost work converted into heat in a cycle. Domain wall movements, domain shifts and domain formation during the polarization process are shown schematically in Fig. 3.19. The polarization leads to a volume-constant elongation of the material in the direction of the electric field. In the polarized state with the remanent polarization Πr also a remanent strain remains compared to the unpolarized state. Before polarization, piezoelectric ceramics are electrically and mechanically isotropic. After polarization, piezoelectric ceramics show an electrical and mechanical anisotropy, i.e. directional material behavior. Additionally, in Fig. 3.19 the dipole orientations are sketched under the different states. If this representation is used, the dipole orientations can be displayed along the polarization hysteresis, see Fig. 3.20.

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Fig. 3.19 Polarization process on crystalline level

Fig. 3.20 Dipol orientations along the polarization hysterse

Voids are created in the grid by additives (doping), i.e. specific grid errors. They thus increase the mobility of the domain wall, and thus the inducible elongations. In the perovskite lattice ABO3 (Fig. 3.12) ions of lower or higher valence are introduced to the A and B sites. The valence denotes the electric charge relative to the charge of the ion normally present there. In a more general notation, the chemical sum formula for a PZT with dopants is (Pb1−y−ε B y )(Ti1−x Zrx )1−z Az O3−δ ). A denotes a doping on an A-place (Zr or Ti), B a doping on a B-place (Pb), see Fig. 3.12. ε denotes the resulting proportion of lead spots, δ the resulting proportion of oxygen vacancies. Doping influence the property profile of piezoceramics and are therefore called “ferroelectric soft” or “ferroelectric hard” doping with so-called donors, i.e. ions of higher value such as lanthanum (La), neodymium (Nd), antimony (Sb) or bismuth (Bi), cause lead spots. They result in a so-called “soft” ceramic. Doping with so-called acceptors, i.e. ions of low valence such as potassium (K) or silver (Ag) on the A-place, magnesium (Mg) or aluminium (Al) on the B-place, lead to oxygen vacancies. They result in a so-called “hard” ceramic. Figure 3.21 shows the effect of doping on polarization hysteresis. While “hard” PZT ceramics generate an increased coercivity and a higher Q-factor (see Sect. 3.3.1.5), which is a measure of losses due to structure damping, and a smaller piezoelectric distortion coefficient

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Fig. 3.21 Polarization hysteresis “soft” and “hard” piezo ceramics

(see Sect. 3.3.1.2), “soft” PZT ceramics have a comparatively smaller coercivity, a relatively larger piezoelectric distortion coefficient and a smaller Q-factor (see Sect. 3.3.1.5). A polarized PZT ceramic is sensorial. The piezoelectric effect of the PZT is based, as described above, on the asymmetry of the crystal structure, i.e. charge center and geometric center do not coincide. An external mechanical load leads to lattice distortion, which in turn leads to a displacement of charge D, as Fig. 3.22 illustrates. This is the direct piezoelectric effect. If the acting force is further increased, this can lead to mechanically caused depolarisation. The inverse piezoelectric effect denotes the mechanical crystal distortion caused by the application of an electric field. The field can lie in the direction of the polarization, which leads to a longitudinal strain, or it can be applied orthogonally to the polarization direction, which leads to a shear distortion of the crystals. Figure 3.23 represents in the left part a measurement of the deflection u of a PZT actuator depending on the applied electric field strength E. The measurement starts at the thermally depolarized state and reaches at the maximum electric field strength the maximum possible deflection of the actuator corresponding to point B of the polarization hysteresis from Fig. 3.20. The point P1 of the new curve corresponds to the coercive field strength. After the electric field is brought back to zero, the deflection goes back to the remanent value, which is marked P2 in the maiden curve. In the right part of the figure, the behavior of a polarized actuator is shown. The zero point in this representation corresponds to the remanent strain of the actuator, which is drawn with the remanent deflection u r in the measurement curve (left) in relation to the unpolarized state. In the schematic relationship between the electric field and the strain, the strain runs from zero along the dotted line and then goes back to the negative coercive field strength as the electric field decreases on the curve called butterfly hysteresis. The ceramic is repolarized here. As the negative field strength continues to increase, the ceramic expands again because the dipole moments are again aligned with the electric field. The dipole orientations are shown at characteristic points of butterfly hysteresis of a polarized ceramic in Fig. 3.24. The butterfly hysteresis is supplemented by the maiden curve of A to ). B Even if the technically usable work area is the polarization process (from  always between −E c /3 and E max , the area up to E max /2 is usually used in order to avoid flashover. If the coercive field strength falls below −E c , depolarization occurs and can lead to mechanical destruction of the ceramic. The inverse piezoelectric behavior shows a pronounced hysteresis. Hysteresis characterizes a behavior in which

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Fig. 3.22 Direct piezoelectric effect

Fig. 3.23 Butterfly hysteresis of a PZT actuator

Fig. 3.24 Dipol orientations along the butterfly hysteresis of a PZT actuator

the elongation depends not only on the independently variable input quantity, i.e. the electric field, but also on the previous state of the output quantity, the elongation. The piezoceramic can exhibit several strain states with the same electric field, as can be seen in Fig. 3.25. The hysteresis behaviour becomes clear with different field time curves. The piezoceramic hysteresis is caused by the domain wall movements. These movements cause friction and, in turn, heat in the ceramic. The losses are generally proportional to the area within the hysteresis curve.

3.3.1.2 Small Signal Behavior and Piezoelectric Effects Figure 3.25 indicates that the electromechanical behavior is amplitude-dependent. Furthermore, the figure also shows that the hysteresis area becomes very small for the control with small electric fields. Here the hysteresis effects and heat losses can be neglected and the behaviour can be regarded as linear. This is called smallsignal behavior in contrast to large-signal behavior with its pronounced non-linear converter characteristic. In the small-signal range, the relationship between strain

38

J. Michael Sinapius and S. Geier

Fig. 3.25 Hysteresis behavior of a PZT actuator Fig. 3.26 small-signal behaviour of a PZT-Actuator

ε and electric field E can be linearized in a very good approximation. From this, a phenomenological definition of the small-signal behaviour can be derived: The characteristic values can be described approximately amplitude-independently. At high field strengths, so-called domain switching processes occur in which the polarity of domain regions is changed. This leads to a physically based Definition 3.2 Definition of small-signal behavior: If almost reversible polarization and strain changes occur, a typical small-signal behavior is present. If ελ = f (E k ) describes the functional relationship of the expansion of the ceramic in the direction of λ due to an electric field in the direction of k, the smallsignal behaviour can be linearly determined with the gradient at zero (see Fig. 3.26) dkλ =

 dελ  d E k  E k =0

(3.23)

which in turn yields a linear relation between the electrical field and the strain: ελ = dkλ E k Uk = dkλ l0

(3.24)

dkλ is the piezoelectric charge coefficient of the piezoelectric material, also referred to in the literature as the piezoelectric distortion coefficient to distinguish it from the

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Fig. 3.27 Convention of the coordinate system for piezoceramics

piezoelectric stress coefficient according to Eq. (3.31). The piezoelectric distortion coefficient contains two indices that indicate the directions of the quantities it links. The first index therefore indicates the direction of the cause and the second the direction of the effect generated. For example, the first index at the piezoelectric distortion coefficient dkλ indicates the direction of the applied electric field, the second the direction of the generated strain. For the directions of the following equations (e.g. Eq. (3.25)) the coordinate system shown in Fig. 3.27 is agreed, which follows the right-hand rule. The z-direction or 3-direction is defined as the direction of the polarization. Besides the small-signal coefficient, which is indicated with Eq. (3.23) for a unipolar control, i.e. for a control with an electric field exclusively in the direction of the dipoles, there are further formulations of the piezoelectric distortion coefficient. Analogous to Eq. (3.23), a differential large-signal coefficient is defined which corresponds to the derivative of the distortion function ε(E) for an input variable Eˆ for large-signal control. Also common is the large-signal secant coefficient, which results from the quotient of the difference values ΔE and the corresponding Δε. The reference value must of course be specified. The value for ΔE = Eˆ − 0 has special significance. It is important to note the following when determining the secant coefficients, that for its formation the strain difference of the reversal point on the ascending branch is selected, so that due to the hysteresis effects the characteristic value is unique. For both large-signal coefficients, the specification of the reference value Eˆ k is important. The possible piezoelectric distortions, and thus the complete matrix of piezoelectric distortion coefficients, are as follows: ⎤ ⎡ 0 0 d31 ⎢ 0 0 d32 ⎥ ⎥ ⎢ ⎢ 0 0 d33 ⎥ ⎥ (3.25) [d] = ⎢ ⎢ 0 d24 0 ⎥ ⎥ ⎢ ⎣ d15 0 0 ⎦ 0 0 0

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Table 3.2 Overview of the different definitions of the piezoelectric distortion coefficient Description

Equation

Small-signal coefficient

dkλ =

dελ d Ek

   

Differential large-signal coefficient

d¯kλ =

dελ d Ek

   

Large-signal secant coefficient

E= Eˆ = d¯kλ

Defn

E k =0

E k = Eˆ

Δελ ΔE

Sketch

Derivation of the strain function ε(E) for E = 0; this coefficient is also called piezoelectric constant. Derivation of the distortion function ε(E) for a certain value Eˆ in the large-signal range

Quotient of difference values of strain and electric field with reference value Eˆ

This equation uses the coordinate convention given with Fig. 3.27. Because of transverse material isotropy, the piezoelectric distortion coefficients d24 and d15 are equal, and d31 is equal to d32 . Table 3.2 gives an overview of the different parameter definitions. The dissertation of Pertsch [62] gives a very detailed overview of the large-signal behavior of piezoceramics. In addition, piezoceramics is an elasto-mechanically compliant material whose linear-elastic behavior is characterized by the Hooke’s6 Law E σμ . ελ = sμλ

(3.26)

E is measured The superscript index E indicates that the compliance coefficient sμλ in the tensile test at constant electric field (E = 0). This ensures that the mechanically induced elongation is considered independently of the electrically induced elongation. Then the superposition principle for linear problems can be applied for Eqs. (3.24) and (3.26): E σμ + dkλ E k . ελ = sμλ

6 Named

after Robert Hooke (1635–1702), English universal scholar.

(3.27)

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41

Equation (3.27) reads together with Eq. (3.25) in complete matrix notation: ⎧ ⎫ ⎡ E E E ⎤ ⎤⎧ ⎫ ⎡ s11 s21 s31 0 0 0 0 0 d31 σ1 ⎪ ε1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ E sE sE 0 0 ⎧ ⎫ ⎪ ⎥⎪ 0 s12 ε2 ⎪ σ2 ⎪ 0 0 d31 ⎥ ⎪ ⎪ ⎢ ⎪ ⎢ 22 32 ⎪ ⎪ ⎥ ⎨ E1 ⎬ ⎥⎪ ⎨ ⎪ ⎨ ⎪ ⎬ ⎢ ⎬ ⎢ E E E ⎢ ⎢ ⎥ ⎥ ε3 0 s s s 0 0 ⎥ σ3 + ⎢ 0 0 d33 ⎥ E 2 . = ⎢ 13 23 33 E ⎥ ⎪ σ4 ⎪ ⎢ 0 d15 0 ⎥ ⎩ ⎭ 0 0 0 s44 0 0 ⎪ ε4 ⎪ ⎪ ⎢ ⎢ ⎪ ⎪ ⎥ ⎥⎪ ⎪ ⎪ ⎪ ⎢ E ⎪ ⎪ ⎪ ⎣ ⎣ d15 0 0 ⎦ E 3 ⎦⎪ ε5 ⎪ σ5 ⎪ ⎪ ⎪ ⎪ 0 0 0 0 0 s44 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ E − sE ) ⎩ σ ⎭ ε6 0 0 0 0 0 0 0 0 2(s11 6 12 (3.28) This equation also uses the coordinate convention introduced with Fig. 3.27. It makes use of the symmetrical properties of the transverse piezoelectric distortion coefficients. Just like the indirect piezoelectric effect, the direct piezoelectric effect can also be described linearized in the small-signal range. According to Fig. 3.22 a force in the direction of μ generates a stress σμ . The distortion leads to a charge shift Di for the dipole moments pointing in the i direction: Di = dμi σμ .

(3.29)

At the same time, a piezoceramic is also a capacitor, i.e. a passive electrical component, in which an applied electric field generates charge displacement and which is thus able to store electrical energy, as Fig. 3.28 represents. The charge displacement (displacement density) dQ i dA σ Uk = εki l0 σ = εki E k .

Di =

(3.30)

describes the capacitive behavior of an electroplated ceramic with the electrode surσ is the permittivity, which is face A on which the charges Q i are located. In it, εki a multiple of the electric field constant εc = 8,854 · 10−12 mF (usually referred to in literature as ε0 , which is used in this book for free strain), which characterizes permitσ the first index refers to the direction tivity in vacuum. With the dielectric constant εki of the electric field, the second to the direction of the charge shift caused by it. The σ is measured in the experiment superscript index σ indicates that the permittivity εki at constant mechanical stress (σ = 0). The test at constant mechanical stress ensures that the mechanically induced charge displacement is considered independent of the electrically induced charge shift. Fig. 3.28 PZT-ceramics as capacitor

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J. Michael Sinapius and S. Geier

Equations (3.29) and (3.30) result in the relationship between electric field and mechanical stress: E k = gμk σμ .

(3.31)

gλk denotes the piezoelectric stress coefficient. The product of the piezoelectric stress coefficient and the piezoelectric distortion coefficient is also referred to in the literature as the piezoelectric quality factor, which, however, unlike the quality factor introduced in Eq. (3.90), does not characterize the hysteresis losses. With the superposition principle for linear problems, Eqs. (3.29) and (3.30) can be superimposed: σ Ek . Di = dμi σμ + εki

(3.32)

Equation (3.32) reads together with Eq. (3.25) in complete matrix notation: ⎧ ⎫ ⎪ ⎪ ⎪ σ1 ⎪ ⎪ ⎡ ⎪ ⎫ ⎡ ⎧ ⎤⎧ ⎫ ⎤⎪ σ2 ⎪ ⎪ ⎪ σ 0 0 ⎪ ε11 0 0 0 0 d15 0 ⎨ ⎪ ⎬ ⎨ E1 ⎬ ⎨ D1 ⎬ σ3 σ 0 ⎦ D2 = ⎣ 0 0 0 d15 0 0 ⎦ E 2 . (3.33) + ⎣ 0 ε22 ⎪ σ4 ⎪ ⎩ ⎭ ⎭ ⎩ σ ⎪ D3 E3 d31 d31 d33 0 0 0 ⎪ 0 0 ε ⎪ ⎪ 33 ⎪ σ5 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ σ6 Again, this equation also uses the coordinate convention given in Fig. 3.27. In summary, Eqs. (3.27) and (3.32) yield the following coupled equations for the linearized description of the isothermal electromechanical behavior of a piezoelectric ceramic:      σ εki dμi E k λ, μ = 1...6 Di = (3.34) E dkλ sμλ ελ σμ i, k = 1...3 The relations are given here in the engineering notation. Please note, in the nomenclature of solid state physics or crystallography Sλ describes the strain and Tμ the mechanical stress (e.g. in [31,32]). Equation (3.34) represents the electromechanical coupling via the piezoelectric distortion coefficient. An alternative formulation couples the electrical quantities with the mechanical properties via the piezoelectric stress coefficient (Eq. (3.31)): 

Ek ελ





σ −g βik μk = E giλ sμλ



Di σμ



λ, μ = 1...6 i, k = 1...3

(3.35)

σ denotes the impermittivity (electrical permeability) in the inverse Therein βik relationship between charge displacement (displacement density) Di and electric

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43

Fig. 3.29 Technically useful effects of piezoelectric material behavior σ must field E k . Again, the superscript index σ indicates that the impermitivity βik be determined at constant mechanical stress (σ = 0). It is related via the Maxwell7 equations with εc μ0 c2 = 1 (μ0 is the magnetic field constant, c the speed of light) to the permittivity of the vacuum. Due to the direction dependence of the dielectric σ β σ is not its reciprocal value. constant, εik ik The electromechanical behavior of piezoceramics can be divided into three effects. They are characterized by the direction of the electric field in relation to the main orientation of the dipoles of the ceramic and the deformation used for actuators, respectively the direction of elongation in relation to the polarization direction and the direction of charge displacement for sensors. A distinction is made between:

1. Longitudinal effect: The electric field lies in the direction of polarization, the strain or force is used in the same direction. This effect is also called the d33 -effect. 2. Transversal effect: The electric field lies in the direction of polarization, the strain or force due to transverse contraction is used. This effect is also called the d31 -effect. 3. Shear effect: The electric field is orthogonal to the polarization, the caused shear deformation is used. This effect is also called the d15 -effect. The three different effects are illustrated in Fig. 3.29. Longitudinal and transversal effects are the technically most frequently used forms of operation. Next the free strain and the blocking force for an actuator with longitudinal effect are considered as introduced with Eqs. (3.3) and (3.7). With Eq. (3.27) a free strain of a piezoceramic actuator with d33 -effect is obtained ε0,3 = d33 E 3,max ,

(3.36)

because no external forces are applied, i.e. σ3 = 0. With E 3,max =

7 Named

Umax l0

after James Clerk Maxwell, 1831–1879, Scottish physicist.

(3.37)

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J. Michael Sinapius and S. Geier

Fig. 3.30 Characteristic curve of a preloaded d33 -actuator

the free stroke results to u 0 = d33 Umax .

(3.38)

The blocking stress results from the condition that u 0 = 0 must be, i.e. with Eq. (3.27): σB = −

d33 E . E max s33

(3.39)

With u 0 and the blocking force resulting from the blocking stress f B the working diagram of an actuator according to Fig. 3.5 is also defined. If a constant external force f 3 loads the actuator (with d33 -effect as example), this causes a constant compression of the actuator: E u c = s33

f3 l0 . A

(3.40)

The actuator characteristic curve is shifted by this amount.  u = l0

E f3 d33 E 3 − s33

A

 ,

(3.41)

as can be seen in Fig. 3.30. For an actuator preloaded in this way, σ3 is generated when additional mechanical stress is applied:     f3 E + d33 E 3 . σ3 − u = l0 s33 A

(3.42)

The changed characteristic is shown in Fig. 3.30. If the actuator is elastically embedded in a structure with a connection stiffness of ks , an internal force f s = −ks u

(3.43)

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Fig. 3.31 Characteristic curve of an embedded d33 -actuator

is exerted. With the actuator longitudinal stiffness kA =

A El s33 0

,

(3.44)

the characteristic curve of the d33 actuator changes to u=

E 3 d33l0 ks kA

+1

,

(3.45)

as Fig. 3.31 shows. An extended consideration leads to geometric influences of the structure when the ceramic is embedded. A complete structural embedding can be modelled very simplified by a complementary spring acting in series to the actuator, as Fig. 3.32 illustrates. The acting stiffnesses for the actuator k A (subsystem III of the sketch), the parallel-acting structural stiffness ks, p (subsystem II of the sketch) and the seriesacting structural stiffness ks,r (subsystem I of the sketch) are as follows A , E l s33,A 0 αΥ = kA , αA

kA = ks, p

ks,r = k S, p αl (1 + α A ) = k A

(3.46a) (3.46b) αΥ αl (1 + α A ) . αA

(3.46c)

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J. Michael Sinapius and S. Geier

Fig. 3.32 Fully integrated d33 -actuator

For the sake of clarity of consideration, the following definitions are made here: s33,A , s33,s l0 αl = , l0,s AA αA = . As

αϒ =

(3.47a) (3.47b) (3.47c)

These α-values are ratios of the areas, the compliances and the lengths between the actuator and the structure, which determine the respective stiffnesses. Free stroke u 0 and blocking force determine the working capacity of the fully embedded actuator: 1 (3.48) u 0 fb 2 For the free stroke, only the ratio of the parallel-connected structural stiffness to actuator stiffness is decisive, i.e. with Eq. (3.45): W A,max =

u0 =

E 3 d33l0 k S, p kA + 1

= E 3 d33l0

αA αΥ + α A

(3.49)

The blocking force is the force required to deform the entire spring arrangement in Fig. 3.32 with the total stiffness k ges about its free stroke u 0 : f b = u 0 k ges

(3.50)

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The total stiffness of the parallel and series stiffnesses results from this:  k ges =

1

k S,r ⎛

1 + k S, p + ka

−1

⎞−1 α 1 A  ⎠ =⎝ + k A αΥ αl (1 + α A ) k 1 + αΥ A αA  −1 αA αA = kA + αΥ αl (1 + α A ) α A + αΥ

(3.51)

From Eqs. (3.48)–(3.51) the work capability of the fully embedded actuator is derived: 1 2 u k ges 2 0  2  −1 αA αA 1 αA = ka (E 3 d33l0 )2 (3.52) + 2 αΥ + α A αΥ αl (1 + α A ) α A + αΥ

W A,max =

This relationship is illustrated in Fig. 3.33 for αl = 1, αϒ = 1 and α A = 1. The three characteristic curves illustrate the basic task of adaptronics in the design of adaptive structures, which is the creation of structural conformity. It is achieved by stiffness matching (impedance matching) which is an optimization task.

3.3.1.3 Coupling Factor Functional materials are energy converters. The piezoceramics considered here are electromechanical converters. The consideration of the stored mechanical and available electrical energy provides a measure for the ability of energy conversion that depends on the crystalline composition. The stored mechanical work in an d33 actuator (longitudinal effect) considered as an example results from the deformation u and the normal force N3 from dWm =

1 N3 du, 2

(3.53)

which is proportional to the mechanical stress for linear-elastic material behaviour: 1 σ3 Adu 2 1 E 2 σ3 Adx = s33 2

dWm =

(3.54)

For the actuator, this results in a mechanical work: Wm =

1 E 2 s σ Al0 2 33 3

(3.55)

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J. Michael Sinapius and S. Geier

Fig. 3.33 Characteristic curves of a fully integrated d33 -actuator

For a blocked actuator with an applied electric field strength E it follows from the material equation (3.27) because of ε3 = 0, see Fig. 3.34: σ3 = −

d33 E E 3 s33

(3.56)

The applied electric field strength results from the distance of the electrodes l0 and the applied electric voltage: E3 =

U3 l0

(3.57)

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Fig. 3.34 To the coupling factor

From Eqs. (3.55)–(3.57) follows for the mechanical energy: 2 1 d33 E 2 Al E 3 0 2 s33

Wm =

2 A 1 d33 U2 E 2 s33 l0

=

(3.58)

At the same time, the stored electrical energy of the piezoceramic dielectric is 1 C pU 2, 2

We =

(3.59)

with the capacity C p and the applied voltage U . The capacitance C p depends on the cross-section and distance of the actuator electrodes: σ C p = ε33

A l0

(3.60)

This results in the stored electrical energy: We =

1 σ A 2 U ε 2 33 l0

(3.61)

Piezoceramics convert this energy into mechanical energy. The ratio of converted mechanical energy to available electrical energy defines the degree of coupling and is called the coupling factor, which is in the considered case of an d33 -actuator: 2 k33 =

=

Wm We 2 d33

σ sE ε33 33

(3.62)

or in generalized notation: 2 kiλ =

2 diλ

σ sE εik λμ

(3.63)

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J. Michael Sinapius and S. Geier

Inscribed therein diλ σ εik E sλμ

— — —

piezoelectric distortion coefficient dielectric coefficient compliance coefficient

in m/V, bzw. C/N in F/N in m2 /N

The coupling factor is a direction-dependent, dimensionless parameter and a measure for the energy conversion of the converter. The coupling factor is not an efficiency because it is not a measure of losses in the form of heat dissipation. It results from the scattering of the orientation of the dipoles in the material, as can be seen in Fig. 3.35 in point C, which characterizes the polarized ceramic, comparative for piezoceramics with higher and lower coupling factor. For perfectly aligned dipoles throughout the ceramic, k33 = 1. Only piezoelectric single crystals come close to this optimum. Here coupling factors of k3 = 0.9 are achievable with a piezoelectric distortion coefficient of d33 = 1500 pC/N. They also have minimal hysteresis. Piezoelectric single crystals, however, are more brittle than ceramics because grain boundaries of the polycrystals slow down crack propagation. They are demanding in their production and have a much higher price than polycrystalline piezoceramics. An essential aspect in the use of ceramic piezoelectric materials is their strength. Ceramics are among the brittle materials, i.e. they have a comparatively low elongation at break. Brittle materials such as ceramics fail close to the elastic limit without plastic deformation with a brittle fracture. The data available on the strength characteristics of PZT ceramics is not very comprehensive. While the compressive strength is given at about 600 MPa, the tensile strength with values between 40 and 80 MPa is an order of magnitude lower. Fett et al. [12,13] investigated in 1999 and 2003 the fracture behaviour of soft piezoceramics in the uniaxial tensile test and the 4-point bending test. The tensile strength values depend strongly on the direction of polarization, i.e. position of the dipoles in comparison to the mechanical tensile stress. In comparison to unpolarized ceramics, the fracture stress of polarized ceramics is reduced, with the lowest tensile strength being achieved with a load parallel to the polarity direction. The presence of an electric field also has a strength-reducing influence, whereby there are hardly any differences between positive and negative fields.

Fig. 3.35 Ceramics with different coupling factors in the dipole orientations of polarization hysteresis

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3.3.1.4 Dynamic Parameters of Piezoceramic Transducers: Frequency Constant, Quality Factor and Impedance Piezoelectric transducers are also used dynamically, i.e. under an alternating electric field. The deflection then also changes with the frequency of the alternating field, but due to the hysteresis effects (see Fig. 3.25) with a phase shift, as shown in Fig. 3.36. The right part of the figure shows the periodic deflection resulting from the temporal change of the electric field in the characteristic curve diagram of the piezoceramic (Fig. 3.23). The reversal point is magnified to show that due to the hysteresis effects the maximum of the deflection occurs temporally offset to the maximum of the electric field. Two further characteristic values are derived from the dynamic behavior: the frequency constant and the quality factor, which are derived next. For this purpose, Fig. 3.37 considers an actuator to which an AC voltage is applied. It generates an alternating electric field, as shown in Fig. 3.36. For the time-dependent forces acting on a section of the length Δz, the equilibrium of forces applies: ¨ t) N (z + Δz, t) − N (z, t) = ρ A AΔz u(z,

(3.64)

The inertial force caused by the acceleration u(z, ¨ t) of the mass m forms the right side of the Eq. (3.64). The mass m results from the section volume AΔz and the density ρ A . For an infinitesimal thin actuator section (Δz → 0) Eq. (3.64) is converted to: ¨ t) N  (z, t) = ρ A Au(z,

Fig. 3.36 Piezoceramics in the alternating electric field

Fig. 3.37 Internal forces of a longitudinally oscillating piezoceramic

(3.65)

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J. Michael Sinapius and S. Geier

The normal force N (z, t) results from constant stress in the cross-section σ3 by N (z, t) = Aσ3 .

(3.66)

The normal stress in the excited actuator can be specified with the extended material law from Eq. (3.26): E ε3 − σ3 = Υ33

d33 E E 3 s33

(3.67)

This material equation describes linear-elastic behavior. As shown in Fig. 3.25, the piezoceramic shows hysteresis behaviour. This complex relationship can be described phenomenologically simplified by mathematical-physical models. A simple model is the Kelvin/Voigt8 model. It extends the material law according to Hooke (Eq. (3.26)) by a linear-viscous term: E σμ = Υ33 ελ + ηv ε˙ λ

(3.68)

Thus the material behaviour would be frequency-dependent, but this is not observed. The damping properties remain constant with increasing frequency in good approximation, which leads to a modification of Eq. (3.68) and thus to an extended formulation of the material law: ηv ε˙ λ Ω∗ ∗ = Υλ (1 + jηv )ελ

σμ = Υλ ελ +

(3.69)

The expression in brackets can be understood as a complex modulus of elasticity, which maps the linear-elastic behaviour (storage modulus) in the real part and the hysteresis losses as loss modulus in the imaginary part. The disadvantage is that this formulation only applies to harmonic oscillations at a frequency Ω ∗ . Therefore, the more general linear elastic, linear viscous material law from Eq. (3.68) is used in the following. It leads to an extended formulation of the material law for the piezoceramic material: E ελ + ηv ε˙ λ − σ3 = Υ33

d33 E E 3 s33

(3.70)

From Eqs. (3.65), (3.69) and (3.70), the gradient of the normal force is derived: N  (z, t) = ρ A Au(z, ¨ t) 

d E  = A Υ33 u (z, t) + ηv u˙  (z, t) − dz

8 Named



d33 E E 3 s33

 (3.71)

after William Thomson Kelvin, Scottish-Irish physicist, 1824–1907 and Woldemar Voigt, physicist, 1850–1919.

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53

From this, the motion equation of the actuator follows as a homogeneous partial differential equation: u(z, ¨ t) −

E Υ33 ηv  u  (z, t) − u˙ (z, t) = 0 ρA ρA

(3.72)

Equation (3.72) is the descriptive wave equation of the longitudinal wave propagation E and the density of the in the actuator. The quotient of the modulus of elasticity Υ33 actuator ρ A , E Υ33 , (3.73) ρA describes the square of the corresponding wave propagation velocity. As a solution to the wave equation (3.72), it is assumed that reflection of the waves induced by the electric alternating field at the actuator ends produces a standing wave φ, the amplitude of which changes over time with q(t):

c2L =

u(z, t) = φ(z)q(t)

(3.74)

This approach is explained in detail in Sect. 6.2.3. This allows the location- and time-dependent behavior of the oscillating actuator to be separated:  q(t) ¨ 2 φ (z) = c = −ω02 L φ(z) q(t) + ΥηvE q(t) ˙

(3.75)

33

An ordinary differential equation for the temporal behaviour of the oscillating actuator ηv q(t) ¨ + ω02 E q(t) ˙ + ω02 q(t) = 0 (3.76) Υ33 and a location-dependent ordinary differential equation for the local amplitude distribution of the actuator displacements u(z) ω02

φ  (z) +

c2L

φ(z) = 0,

(3.77)

are derived. The solution of Eq. (3.77) is a sum of harmonic functions: φ(z) = φˆ c cos

ω0 ω0 z + φˆ s sin z cL cL

(3.78)

The form factors φˆ c and φˆ s are determined by the boundary conditions at the actuator ends. For a free actuator, no electric field is applied, i.e. with E = 0 at the edges: N (z = 0) = 0 N (z = l0 ) = 0

i. e. i. e.

u  (z = 0) = φ  (z = 0) = 0 u  (z = l0 ) = φ  (z = l0 ) = 0

(3.79a) (3.79b)

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J. Michael Sinapius and S. Geier

Fig. 3.38 First longitudinal eigenform of the longitudinal piezoceramic

Strictly speaking, the strain at the edge u  (0, l) after Eq. (3.70) is proportional to the applied electric field. However, this leads to time-dependent boundary conditions. The boundary conditions in Eq. (3.79) are therefore taken as an approximation, from which follows: ω0 ω0 φˆ c sin l0 = 0 . (3.80) cL cL This is true for the zeros of the sine function, i.e. for multiples of the circle number π : ω0 l0 = iπ . cL

(3.81)

The resonance frequencies of the actuator result from this: iπ ω0i = c L l 0 ΥA π =i ρ A l0

(3.82)

Taking the first longitudinal eigenfrequency for actuators of arbitrary length, ω01 = c L

π , l0

(3.83)

the characteristic quantity frequency constant is defined by relating it to a characteristic length l ∗ : ωk∗ = ω01l ∗ .

(3.84)

The index k at the frequency constant refers to the direction of the resonance oscillation. The distribution of the oscillation amplitudes along the actuator belongs to the lowest longitudinal natural frequency ω01 : φ1 = φˆ cos

π z. l0

(3.85)

It is represented in Fig. 3.38 both as a deformation and as a graph φ(z). There is a vibration node in the center of the actuator, i.e. here is no deflection u.

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Another parameter, the quality factor, is derived from the damping behaviour of the ceramic. The damping determines the temporal behavior of the oscillating actuator, which is described by Eq. (3.76). An exponential approach, q(t) = eλt ,

(3.86)

leads to the temporal decay behaviour of the excited actuator. Applying  λ

2

2 + ω0i

 ηv λ + 1 = 0, i.e. E Υ33

2 λ2 + 2ξ ω0i λ + ω0i =0

(3.87)

yields: λi1,2 = −ξi ω0i ± ω0i 1 − ξi2 .

(3.88)

This contains the damping value that refers to each resonant frequency according to Eq. (3.82): ξi =

ηv 1 ω0i E . 2 Υ33

(3.89)

The damping value ξ1 of the lowest resonance of the actuator defines the so-called quality factor: Qm =

2 ξ1

(3.90)

The quality factor can be determined by measuring the frequency response of the actuator. Transforming the equation of motion (3.72) with the separation approach Eq. (3.74) and the solution found for the actuator with free boundary conditions (Eq. (3.79)), the first eigenform according to 

      π π2 π π2 π E φˆ ρ A A cos z q(t) ¨ + ηv A 2 cos z q(t) ˙ + Υ33 A 2 cos z q(t) = 0 . l0 l0 l0 l l 

0

0

(3.91) After multiplication with the eigenmode (Eq. (3.79)) and integration over the actuator length, the simple equation of oscillation for the first actuator longitudinal eigenfrequency is obtained M1 q¨ + C1 q˙ + K 1 q = 0 with the quantities

(3.92)

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J. Michael Sinapius and S. Geier

• generalized mass M1 = φˆ 2 ρ A A

!

l0

0

π cos2 ( z)dz l0

1 = φˆ 2 ρ A Al0 2 1 = φˆ 2 m A . 2

(3.93)

m A is the mass of the actuator. The generalized mass M1 is an energetic quantity. It is a measure for the kinetic energy of the actuator vibrating in its lowest eigenfrequency. • generalized damping C1 = φˆ 2 ηv A =

π2 l02

!

l0

0

π cos2 ( z)dz l0

π2

1 2 φˆ ηv A . 2 l0

(3.94)

• generalized stiffness E A K 1 = φˆ 2 Υ33

π2 l02

! 0

l0

π cos2 ( z)dz l0

π2

1 2 E φˆ Υ33 A 2 l0 1 2 2 = φˆ π k A 2 2 = ω01 M1 .

=

(3.95)

k A is the actuator stiffness according to Eq. (3.8). The generalized stiffness K 1 is an energetic quantity. It is a measure for the potential energy. The previously derived energetic quantities make use of the property of orthogonality of the modes of natural vibration (Eq. (3.85)), which is explained on Sect. 6.2.4. The work done by the alternating electric field E 3 = Eˆ 3 E 3 (t)

(3.96)

on the elongation of the vibrational characteristic Eq. (3.85) π dφ1 π = −φˆ sin z dz l0 l0

(3.97)

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Fig. 3.39 Single-Degree-of-freedom representation of the first longitudinal waveform of a piezo ceramic actuator and frequency response

results from ! F1 = 0

l0



 d33 ˆ π π −A E 3 φˆ sin zdz . E l l s33 0 0

d33 ˆ E ·φˆ E 3 s33 " #$ %

= 2· A

(3.98)

 1

ˆ The term  1 of the equation represents the force from the piezoelectric effect, φ the amplitude of deflection of the first mode shape (Fig. 3.38). With the energetic quantities, the lowest longitudinal eigenvibration generated by an alternating electric field can be described very simply as a degree of freedom oscillator: M1 q¨ + C1 q˙ + K 1 q = F1 E 3 (t)

(3.99)

The equivalent single-degree of freedom oscillator is shown in Fig. 3.39 on the left. The Fouriertransformation9 of Eq. (3.99) − Ω 2 M1 q + jΩξ1 q + K 1 q = F1 E 3 (Ω)

(3.100)

yields the frequency response behavior that can be measured: |H (Ω)| =

F1 q (Ω) = 2 E3 M1 (−Ω + jΩω1 ξ1 + ω12 )

(3.101)

The magnitude of the function from Eq. (3.101) is shown in √ Fig. 3.39 on the right near the resonant frequency ω01 . It marks the half-width at 2 of the maximum resonance response amplitude from which the quality factor can be determined: Qm =

9 Named

ω01 ωo1 − ωu1

(3.102)

after Jean Baptiste Joseph Fourier (1768–1830), French mathematician and physicist.

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J. Michael Sinapius and S. Geier

ωo1 and ωu1 are the circular frequencies of the two half-width points. The derivation of the half-width method can be found in the Sect. 6.3. The frequency-dependent, electrical input impedance of the actuator U¯ Z¯ e (Ω) = I¯

(3.103)

is essentially determined by its dynamic properties, as Melcher points out in his dissertation [52]. Far below the first mechanical resonance the piezoceramic acts as a capacitor with the impedance Z¯ ec (Ω) =

1 , jΩC p

(3.104)

see Fig. 3.40. The mechanical impedance, which is derived from Eq. (3.101) as ratio of acting force to vibration velocity, 2 ) Z¯ m (Ω) = M1 (−Ω 2 + jΩω01 ξ1 + ω01

(3.105)

dominates the first resonance. In the first mechanical resonance Ω = ω01 , the actuator opposes the excitation voltage with minimal resistance. Both impedances act in parallel, i.e. in the sense of Eq. (3.103) d2 k2 1 1 = + 33 A . Z¯ e Z¯ ec Z¯ m

(3.106)

The actuator stiffness k A is given with Eq. (3.44). The capacity Cs = 2

2 d33 kA π2

(3.107)

combines the product of the square of the piezoelectric distortion coefficient and the actuator stiffness under consideration of Eq. (3.95). This formes in addition to the mechanical resonance ω01 , which leads to a minimum in the impedance, to a second resonance  Cs , (3.108) ω p = ω01 · 1 + Cp which leads to a maximum in the impedance gradient, as Fig. 3.40 shows. Above this parallel resonance the impedance then drops again with 1/Ω . Operation of the actuator above the natural frequency is usually avoided due to the strong phase shifts.

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Fig. 3.40 Impedance curve of an unloaded piezo actuator

3.3.1.5 Property Profile of Lead Zirconate Lead Titanate The piezoelectric effect is temperature-dependent, as Fig. 3.41 shows for three different piezoceramics. The diagram shows the temperature dependence of the piezoelectric distortion coefficient and the coupling factor exemplary for three different piezoceramics. In the normal room temperature range, the temperature dependence of the piezoelectric properties can be neglected. In the range of low temperatures (below 77 K), 20–30% of the stroke at room temperature can be expected. Piezoelectric ceramics can also be used at very low temperatures (