Table Of ContentLena Zentner, Sebastian Linß
Compliant systems
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Lena Zentner, Sebastian Linß
Compliant systems
Mechanics of elastically deformable mechanisms,
actuators and sensors
Authors
Univ.-Prof. Dr.-Ing. habil. Lena Zentner
Technische Universität Ilmenau
Department of Mechanical Engineering
Compliant Systems Group
Max-Planck-Ring 12
98693 Ilmenau
[email protected]
Dr.-Ing. Sebastian Linß
Technische Universität Ilmenau
Department of Mechanical Engineering
Compliant Systems Group
Max-Planck-Ring 12
98693 Ilmenau
[email protected]
ISBN 978-3-11-047731-3
e-ISBN (PDF) 978-3-11-047974-4
e-ISBN (EPUB) 978-3-11-047740-5
Library of Congress Control Number: 2019931431
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available on the Internet at http://dnb.dnb.de.
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Cover image: Feng Yu/iStock/thinkstock
Printing and binding: CPI books GmbH, Leck
www.degruyter.com
Preface
Compliant systems are characterized by complex deformation and motion behavior, which
complicates both their modeling and ultimately their application. This book presents methods
both of classifying and of modeling compliant mechanisms, actuators, and sensors according
to diverse criteria, enabling or simplifying their selection, modeling and design. The content
of this book is based on many years of experience gathered at the Compliant Systems Group
at Technische Universität Ilmenau, and is the subject of a number of classes offered to stu-
dents of the Department of Mechanical Engineering.
This book is intended both for students of engineering, as well as those who have already
qualified. Further, it is written for anyone interested in compliant systems, or who wishes to
implement them. Even without special scientific education, the second chapter of this book,
which describes the classification of compliant systems, will provide an understandable
overview of compliant mechanisms, actuators and sensors, as well as possibilities for their
application. The third chapter, regarding the modeling of compliant systems as rigid-body
systems, is based on the linear theory of material strength science. The content of the fourth
chapter, regarding the modeling of large deflections in compliant mechanisms and actuators,
offers an introduction to non-linear theory and includes detailed derivations, while ensuring
that the calculations and results remain intelligible. This is due in part to mathematical trans-
formations that are purposefully designed to be uncomplicated. The examples given in the
fifth chapter have been taken from a variety of applications, and their results, often taking the
form of mathematical relationships between different model parameters, can be reused for
similar cases. The final chapter suggests methods of synthesizing compliant mechanisms
with concentrated compliance, as well as for the design of flexure hinges with special con-
sideration to optimized notch contours.
Chapters 3 to 5 and partially Chapter 2 of this book are translations of the German book
“Nachgiebige Mechanismen” (“Compliant Mechanisms”) by Lena Zentner [114] with sever-
al improvements and extensions. Chapter 6 and partially Chapter 2 contain knowledge based
on several research papers by Sebastian Linß [58]-[66].
Thanks are due to all members of the Compliant Systems Group at Technische Universi-
tät Ilmenau, who have helped increase the competence and experiences of the field of com-
pliant mechanisms, actuators and sensors through their own efforts, the supervisions of stu-
dent theses and joint discussion at the Group. Special thanks go to Prof. Valter Böhm, his
cooperation was fundamental to the classification of compliant systems. Last but not least,
thanks go to Matthew Partridge for supporting the English translation of this book.
Ilmenau, January 2019 Lena Zentner and Sebastian Linß
https://doi.org/10.1515/9783110479744-005
Contents
Preface V
1 Introduction (L. Zentner, S. Linß) 1
2 Definition and classification of compliant systems (L. Zentner, S. Linß) 3
2.1 Compliance ................................................................................................................... 4
2.1.1 Classification of compliance ......................................................................................... 4
2.1.2 Variation of compliance ............................................................................................... 5
2.2 Compliant mechanisms ................................................................................................. 8
2.2.1 Classification of compliant mechanisms ...................................................................... 8
2.2.2 Compliant joints ........................................................................................................... 9
2.3 Compliant actuators and sensors ................................................................................. 14
2.3.1 Compliant actuators .................................................................................................... 14
2.3.2 Compliant sensors ....................................................................................................... 15
2.3.3 Multi-functionality ...................................................................................................... 19
2.4 Motion behavior of compliant systems ....................................................................... 20
2.4.1 Stable motion behavior ............................................................................................... 21
2.4.2 Unstable motion behavior ........................................................................................... 23
3 Modeling compliant systems as rigid-body systems (L. Zentner) 29
3.1 Assumptions for modeling .......................................................................................... 29
3.2 Modeling for individual load cases ............................................................................. 33
3.3 Modeling for complex loads ....................................................................................... 34
3.4 Modeling for concentrated compliance ...................................................................... 36
3.5 Comparison of the methods ........................................................................................ 37
3.6 Serial cascading rigid-body joints ............................................................................... 38
3.7 Modeling examples ..................................................................................................... 41
3.7.1 A gripping system with two joints .............................................................................. 41
3.7.2 A gripping system with multiple joints ....................................................................... 43
3.7.3 Parallel cascading compliant elements ....................................................................... 45
VIII Contents
4 Modeling large deflections of curved rod-like structures (L. Zentner) 49
4.1 Assumptions for modeling ..........................................................................................49
4.2 Equilibrium conditions for a rod element ....................................................................52
4.2.1 Equilibrium conditions in vector form ........................................................................52
4.2.2 Derivative of the unit vectors ......................................................................................54
4.2.3 Natural coordinate system ...........................................................................................56
4.2.4 Relationship between the natural coordinate system and attached coordinate
system ..........................................................................................................................58
4.2.5 Further development of the equilibrium equations ......................................................60
4.3 Inclusion of material properties ...................................................................................60
4.4 Transformation of unit vectors ....................................................................................66
4.5 Shape of the rod in coordinate systems .......................................................................70
4.6 Displacement vector ....................................................................................................72
4.7 Summarizing representation of equations for large deflections...................................74
4.7.1 Vector form of model equations in attached coordinate system in a space .................74
4.7.2 Scalar form of model equations in attached coordinate system in a space ..................76
4.7.3 Scalar form of model equations in attached coordinate system in a plane ..................78
4.7.4 Vector form of model equations in Cartesian coordinate system in a space ...............80
4.7.5 Scalar form of model equations in Cartesian coordinate system in a space ................82
4.7.6 Scalar form of model equations in Cartesian coordinate system in a plane.................84
5 Examples of modeling large deflections of curved rod-like structures
(L. Zentner) 87
5.1 Plane problems in the attached coordinate system ......................................................89
5.1.1 A pneumatically actuated gripping finger ...................................................................89
5.1.2 A pipe with flowing liquid ..........................................................................................92
5.1.3 A coated hollow rod ....................................................................................................96
5.1.4 Rods with non-constant cross-sections ........................................................................99
5.2 Spatial problems in the attached coordinate system ..................................................102
5.2.1 A helical rod under internal pressure .........................................................................102
5.2.2 A drill-bit under load due to a moment .....................................................................107
5.3 Plane problems in the Cartesian coordinate system ...................................................110
5.3.1 A sensor for measuring dynamic pressures ...............................................................110
5.3.2 Compliant elements for monitoring angular velocity ................................................112
5.3.3 A gripping device with a compliant body ..................................................................115
5.3.4 Two compliant mechanisms ......................................................................................121
5.4 Spatial problems in the Cartesian coordinate system ................................................124
5.4.1 A compliant valve .....................................................................................................124
5.4.2 A gripping tool with curved compliant fingers ..........................................................127
Contents IX
6 Synthesis of compliant mechanisms and design of flexure hinges (S. Linß) 133
6.1 Rigid-body replacement approach ............................................................................ 133
6.2 Types of flexure hinges ............................................................................................ 135
6.3 Types of notch flexure hinges and suggested contour shapes ................................... 138
6.4 Angle-based synthesis method for individually shaped flexure hinges in a
mechanism ................................................................................................................ 141
6.4.1 Design of polynomial flexure hinges with variable order using design graphs ........ 144
6.4.2 Design of various flexure hinges with variable dimensions using design
equations ................................................................................................................... 146
6.5 Examples of compliant mechanisms with polynomial flexure hinge design ............ 149
6.5.1 Angle-based synthesis of a high-precision and large-stroke straight-line
mechanism with different polynomial hinges ........................................................... 150
6.5.2 Further examples of synthesized mechanisms .......................................................... 153
References 157
Index 165
