Handbook of Compliant Mechanisms E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 236

Veröffentlichungsjahr: 2013

Beschreibung

A fully illustrated reference book giving an easy-to-understand introduction to compliant mechanisms

A broad compilation of compliant mechanisms to give inspiration and guidance to those interested in using compliant mechanisms in their designs, the Handbook of Compliant Mechanisms includes graphics and descriptions of many compliant mechanisms. It comprises an extensive categorization of devices that can be used to help readers identify compliant mechanisms related to their application. It also provides chapters on the basic background in compliant mechanisms, the categories of compliant mechanisms, and an example of how the Compendium can be used to facilitate compliant mechanism design.

• Fully illustrated throughout to be easily understood and accessible at introductory levels
• Covers all aspects pertaining to classification, elements, mechanisms and applications of compliant mechanisms
• Summarizes a vast body of knowledge in easily understood diagrams and explanations
• Helps readers appreciate the advantages that compliant mechanisms have to offer
• Practical approach is ideal for potential practitioners who would like to realize designs with compliant mechanisms, members and elements
• Breadth of topics covered also makes the book a useful reference for more advanced readers

Intended as an introduction to the area, the Handbook avoids technical jargon to assist non engineers involved in product design, inventors and engineers in finding clever solutions to problems of design and function.

Leseprobe

Contents

Cover

Title Page

Copyright

List of Contributors

Chapter Contributors

Managing Library Contributors

Library Contributors

Student Contributors

Acknowledgments

Preface

Part One: Introduction to Compliant Mechanisms

Chapter 1: Introduction to Compliant Mechanisms

1.1 What are Compliant Mechanisms?

1.2 What are the Advantages of Compliant Mechanisms?

1.3 What Challenges do Compliant Mechanisms Introduce?

1.4 Why are Compliant Mechanisms Becoming More Common?

1.5 What are the Fundamental Concepts that Help Us Understand Compliance?

1.6 Conclusion

References

Chapter 2: Using the Handbook to Design Devices

2.1 Handbook Outline

2.2 Considerations in Designing Compliant Mechanisms

2.3 Locating Ideas and Concepts in the Library

2.4 Modeling Compliant Mechanisms

2.5 Synthesizing Your Own Compliant Mechanisms

2.6 Summary of Design Approaches for Compliant Mechanisms

Further Reading

Part Two: Modeling of Compliant Mechanisms

Chapter 3: Analysis of Flexure Mechanisms in the Intermediate Displacement Range

3.1 Introduction

3.2 Modeling Geometric Nonlinearities in Beam Flexures

3.3 Beam Constraint Model

3.4 Case Study: Parallelogram Flexure Mechanism

3.5 Conclusions

Further Reading

Chapter 4: Modeling of Large Deflection Members

4.1 Introduction

4.2 Equations of Bending for Large Deflections

4.3 Solving the Nonlinear Equations of Bending

4.4 Examples

4.5 Conclusions

Further Reading

References

Chapter 5: Using Pseudo-Rigid Body Models

5.1 Introduction

5.2 Pseudo-Rigid-Body Models for Planar Beams

5.3 Using Pseudo-Rigid-Body Models: A Switch Mechanism Case-Study

5.4 Conclusions

Acknowledgments

References

Appendix: Pseudo-Rigid-Body Examples (by Larry L. Howell)

Part Three: Synthesis of Compliant Mechanisms

Chapter 6: Synthesis through Freedom and Constraint Topologies

6.1 Introduction

6.2 Fundamental Principles

6.3 FACT Synthesis Process and Case Studies

6.4 Current and Future Extensions of FACT's Capabilities

Acknowledgments

References

Chapter 7: Synthesis through Topology Optimization

7.1 What is Topology Optimization?

7.2 Topology Optimization of Compliant Mechanisms

7.3 Ground Structure Approach

7.4 Continuum Approach

7.5 Discussion

7.6 Optimization Solution Algorithms

Acknowledgment

References

Chapter 8: Synthesis through Rigid-Body Replacement

8.1 Definitions, Motivation, and Limitations

8.2 Procedures for Rigid-Body Replacement

8.3 Simple Bicycle Derailleur Example

References

Chapter 9: Synthesis through Use of Building Blocks

9.1 Introduction

9.2 General Building-Block Synthesis Approach

9.3 Fundamental Building Blocks

9.4 Elastokinematic Representations to Model Functional Behavior

9.5 Decomposition Methods and Design Examples

9.6 Conclusions

Further Reading

References

Part Four: Library of Compliant Mechanisms

Chapter 10: Library Organization

10.1 Introduction

10.2 Library of Compliant Designs

10.3 Conclusion

References

Chapter 11: Elements of Mechanisms

11.1 Flexible Elements

11.2 Rigid-Link Joints

References

Chapter 12: Mechanisms

12.1 Basic Mechanisms

12.2 Kinematics

12.3 Kinetics

References

Chapter 13: Example Application

13.1 Elements of Mechanisms: Flexible Elements

13.2 Mechanisms: Kinematic

13.3 Mechanisms: Kinetic

References

Index

This edition first published 2013 © 2013 John Wiley & Sons Ltd.

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Library of Congress Cataloging-in-Publication Data

Handbook of compliant mechanisms / edited by Professor Larry Howell, Dr. Spencer P. Magleby, Dr. Brian M. Olsen. pages cm Includes bibliographical references and index. ISBN 978-1-119-95345-6 (cloth) 1. Mechanical movements. 2. Machinery, Kinematics of. 3. Engineering design. I. Howell, Larry L., editor. II. Magleby, Spencer P., editor. III. Olsen, Brian M. (Brian Mark), 1983– editor. TJ181.H25 2013 621.8–dc23 2012037077

List of Contributors

Chapter Contributors

Shorya Awtar – Assistant Professor, University of Michigan, Ann Arbor, MI, USA

Mary Frecker – Professor, The Pennsylvania State University, University Park, PA, USA

Jonathan Hopkins – Assistant Professor, University of California, Los Angeles, CA, USA

Larry Howell – Professor, Brigham Young University, Provo, UT, USA

Brian Jensen – Associate Professor, Brigham Young University, Provo, UT, USA

Charles Kim – Assistant Professor, Bucknell University, Lewisburg, PA, USA

Girish Krishnan – Post-Doctoral Associate, University of Michigan, Ann Arbor, MI, USA

Craig Lusk – Associate Professor, University of South Florida, Tampa, FL, USA

Spencer Magleby – Associate Dean, Brigham Young University, Provo, UT, USA

Chris Mattson – Associate Professor, Brigham Young University, Provo, UT, USA

Brian Olsen – Research Engineer, Los Alamos National Lab, Los Alamos, NM, USA

Managing Library Contributors

G. K. Ananthasuresh – Indian Institute of Science, Bangalore, Bangalore, India

Guimin Chen – Xidian University, Xi'an, P.R. China

Martin Culpepper – Massachusetts Institute of Technology, Cambridge, MA, USA

Mohammad Dado – University of Jordan, Amman, Jordan

Haijun Su – Ohio State University, Columbus, OH, USA

Simon Henein – CSEM Centre Suisse d'Electronique et de Microtechnique SA, Neuchâtel, Switzerland

Just L. Herder – Delft University of Technology, Delft, The Netherlands

Jonathan B. Hopkins – University of California, Los Angeles, CA USA

Nilesh D. Mankame – General Motors Research & Development, Warren, MI, USA

Ashok Midha – Missouri University of Science and Technology, Rolla, MO, USA

Anupam Saxena – Indian Institute of Technology, Kanpur, Kanpur, India

Umit Sonmez – American University of Sharjah, Sharjah, UAE

Jingjun Yu – Beihang University, Beijing, China

Library Contributors

Imad F. Bazzi – General Motors Research & Development, Warren, MI, USA

Shusheng Bi – Beihang University, Beijing, China

Ozgur Erdener – Istanbul Technical University, Istanbul, Turkey

Bilin Aksun Güvenç – Okan University, Istanbul, Turkey

Huseyin Kızıl – Istanbul Technical University, Istanbul, Turkey

Xu Pei – Beihang University, Beijing, China

Ahmet Ekrem Sarı – Altinay Robot Technologies, Istanbul, Turkey

Nima Tolou – Delft University of Technology, Delft, The Netherlands

Levent Trabzon – Istanbul Technical University, Istanbul, Turkey

Cem Celal Tutum – Technical University of Denmark, Lyngby, Denmark

Hongzhe Zhao – Beihang University, Beijing, China

Guanghua Zong – Beihang University, Beijing, China

Yörükoğlu, Ahmet – R&D Engineer at Arcelik

Student Contributors

Bapat, Sushrut – Missouri University of Science and Technology

Barg, Matt – Brigham Young University

Berg, Fred van den – Delft University of Technology

Black, Justin – Brigham Young University

Bowen, Landen – Brigham Young University

Bradshaw, Rachel – Brigham Young University

Campbell, Robert – Brigham Young University

Chinta, Vivekananda – Missouri University of Science and Technology

Dario, P. – Scuola Superiore Sant'Anna

Davis, Mark – Brigham Young University

Demirel, Burak – KTH Royal Institute of Technology

Duffield, Luke – Brigham Young University

Dunning, A.G. – Delft University of Technology

Emirler, Mümin Tolga – Istanbul Technical University

Foth, Morgan – Brigham Young University

George, Ryan – Brigham Young University

Güldoğan, Bekir Berk – Istanbul Technical University

Greenberg, Holly – Brigham Young University

Hardy, Garrett – Brigham Young University

Harris, Jeff – Brigham Young University

Howard, Marcel J. – Delft University of Technology

Ivey, Brad – Brigham Young University

Jones, Andrea – Brigham Young University

Jones, Kris – Brigham Young University

Kluit, Lodewijk – Delft University of Technology

Koecher, Michael – Brigham Young University

Koli, Ashish – Missouri University of Science and Technology

Kosa, Ergin – Istanbul Technical University

Kragten, Gert A. – Delft University of Technology

Kuber, Raghvendra – Missouri University of Science and Technology

Lassooij, Jos – Delft University of Technology

McCort, Ashby – Brigham Young University

Morsch, Femke – Delft University of Technology

Morrise, Jacob – Brigham Young University

Pate, Jenny – Brigham Young University

Peterson, Danielle Margaret – Brigham Young University

Ratlamwala, Tahir Abdul Husain – University of Ontario Institute of Technology

Reece, David – Brigham Young University

Samuels, Marina – Brigham Young University

Sanders, Michael – Brigham Young University

Shafiq, Mohammed Taha – American University of Sharjah

Shelley, Dan – Brigham Young University

Shurtz, Tim – Brigham Young University

Simi, Massimiliano – Scuola Superiore Sant'Anna

Skousen, Darrell – Brigham Young University

Solomon, Brad – Brigham Young University

Steutel, Peter – Delft University of Technology

Stubbs, Kevin – Brigham Young University

Tanner, Daniel – Brigham Young University

Tekeş, Ayşe – Istanbul Technical University

Telford, Cody – Brigham Young University

Toone, Nathan – Brigham Young University

Wasley, Nick – Brigham Young University

Wengel, Curt – Brigham Young University

Wilding, Sam – Brigham Young University

Williams, David – Brigham Young University

Wright, Doug – Brigham Young University

Yu, Zhiwei – Beihang University

Zhao, Shanshan – Beihang University

Zirbel, Shannon – Brigham Young University

Acknowledgments

The Handbook of Compliant Mechanisms is a result of work by contributors from around the world. Compliant mechanisms experts have authored the chapters in Parts II and III, and many more have contributed entries to the Library of Compliant Mechanisms (Part IV of the Handbook). The contributions of these individuals are gratefully acknowledged.

We express appreciation to Brian Winder and Jonathan Hopkins for their work on early drafts of the Library of Compliant Mechanisms. The graphic design assistance of Jung-Ah Ahn (Jade) and Stephen Jensen are acknowledged, as is the administrative assistance of Danielle Peterson. The format of the Library of Compliant Mechanisms was inspired by Ivan I. Artobolevskii's seven-volume work Mechanisms in Modern Engineering Design: A Handbook for Engineers, Designers, and Inventors. We also wish to honor the memory of Dr. Umit Sonmez, who passed away unexpectedly during the time that he was contributing to the Handbook.

Preface

Compliant mechanisms are seeing expanded use because they offer advantages such as increased performance (e.g. high precision, low weight, low friction), lower cost (e.g. simplified manufacture, low part count), and ability to miniaturize (e.g. makes possible micro- and nanomechanical devices). However, because compliant mechanisms are relatively new compared to more traditional devices, it is difficult for designers to find examples and resources to guide them in their work. Many people are beginning to understand the advantages of compliant mechanisms but there is still a general lack of knowledge of how to implement them. Although many journal articles and some texts are available to aid in the in-depth engineering of compliant mechanisms, a more concise and visual resource is needed to provide inspiration and guidance in the conceptual stages of compliant mechanism design.

The Handbook of Compliant Mechanisms is intended to provide a summary of compliant mechanism modeling and design methods and a broad compilation of compliant mechanisms that will provide inspiration and guidance to those interested in exploiting the advantages of compliant mechanisms in their designs. Early Handbook chapters provide basic background in compliant mechanisms, summaries of some of the major methods for designing compliant mechanisms, categories of compliant mechanisms, and an example of how the Handbook can be used to facilitate compliant mechanism design. Graphics and brief descriptions of many compliant mechanisms are provided to give inspiration in preliminary design.

The Handbook of Compliant Mechanisms is designed to be a resource for engineers, designers, and others involved in product design. We hope that it is found to be useful by many in the development of compliant mechanisms.

The Handbook is divided into the following Parts:

Part I provides an introduction to compliant mechanisms and describes how to use the Handbook to design compliant mechanisms.

Part II focuses on modeling of compliant mechanisms.

Part III describes methods for the synthesis of compliant mechanisms.

Part IV is a visual library of compliant mechanisms.

We wish to express our sincere thanks to all the contributors that worked to make this handbook possible. We hope that it is found to be useful in creating new compliant mechanism designs.

This material is based upon work supported by the National Science Foundation under Grant No. CMMI-0800606. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Brian M Olsen is an employee of Los Alamos National Security, LLC, the operator of Los Alamos National Laboratory for the US Department of Energy. The views expressed in this book are solely those of Brian and the other authors and do not necessarily reflect the views, positions and opinions of the US Department of Energy or the US Government.

Larry L. Howell Brigham Young University, USA

Spencer P. Magleby Brigham Young University, USA

Brian M. Olsen Los Alamos National Laboratory, USA

November 2012

Part One

Introduction to Compliant Mechanisms

1

Introduction to Compliant Mechanisms

Larry L. Howell

Brigham Young University, USA

1.1 What are Compliant Mechanisms?

If something bends to do what it is meant to do, then it is compliant. If the flexibility that allows it to bend also helps it to accomplish something useful, then it is a compliant mechanism [1]. The idea of using compliant mechanisms in products is catching on, but traditionally when designers need a machine that moves, they commonly use very stiff or rigid parts that are connected with hinges (like a door on its hinge or a wheel on an axle) or sliding joints. But when we look at nature we see an entirely different idea from rigid parts connected at joints – most moving things in nature are very flexible instead of stiff, and the motion comes from bending the flexible parts [2]. For example, consider your heart – it is an amazing compliant mechanism that started working before you were born and will work all day every day for your entire life. Think of bee wings, elephant trunks, eels, sea weed, spines, and the blooming of flowers (Figure 1.1) – all of which are compliant. Even the natural motions that seem to be exceptions to this bending behavior, like your knee or elbow, use cartilage, tendons, and muscles to do their work. We see in nature the possibility of making machines that are very compact – a mosquito (Figure 1.1) is able to fly while carrying its own on-board navigation, control, energy harvesting, and reproduction systems. Would it be possible for us to improve human-designed products if we applied the lessons learned from nature and looked to flexibility to achieve movement?

Figure 1.1 A few examples of compliance in nature: a spine, bee wings, elephant trunks, blooming flowers, a mosquito, sea weed, and eels

It is interesting that some early man-made machines were compliant mechanisms. Is that because we were closer to nature then? An example of a compliant mechanism with a multi-millennia history is the bow (Figure 1.2). Ancient bows were made using a composite of bone, wood, and tendon, and they used the flexibility of their limbs to store energy that would be released into propelling the arrow. It is interesting to see the sketches of Leonardo da Vinci [3] and see many compliant mechanisms (see Figure 1.2 for an example). Even one of the great achievements of engineering – sustained human flight – began with a compliant mechanism when the Wright brothers (Figure 1.3) used wing warping to achieve control of their early aircraft [4].

Figure 1.2 Early compliant mechanism designs include the ancient bow and many compliant mechanism designs by Leonardo da Vinci

Figure 1.3 The Wright brothers used wing warping to achieve control of their aircraft for sustained human flight

This may all sound good, but it turns out that compliant mechanisms can be difficult to design. Nature has done it, but nature employed very different design methods from those we mortals use. Great strides were made in the design of machines when compliance was left to nature and we moved to the much easier-to-design realm of rigid parts connected at hinges. For example, the too-sophisticated-for-its-time wing warping of the Wright Flier was eventually replaced by the much-easier-to-work-with control surfaces provided by an aileron pivoting on a hinge.

However, over the past few decades our knowledge has advanced. We have developed new materials, increased our computational capabilities and expanded the ability to design more sophisticated devices. At the same time, society has developed new needs that cannot be easily addressed using traditional mechanisms. This means that there is an increased ability to create compliant mechanisms, and an increased motivation for doing so. As an example, reconsider the example of aircraft control. The Wright Flier started out with wing warping for its control surfaces, but other aircraft quickly moved to approaches using traditional mechanisms. But with the increased computational power available and improved materials that have been developed, researchers are returning to the idea of wing warping to get the advantages, such as reduced weight, that would come from the approach.

One of the things that make traditional design of mechanical components compelling is that designers can separate different functions to be done by different parts, and each part is assigned to do that one function. The blessing and curse of compliant mechanisms is that they integrate different functions into fewer parts. Compliant mechanisms may be able to accomplish complex tasks with very few parts, but they can be much more difficult to design.

1.2 What are the Advantages of Compliant Mechanisms?

The integration of functions into fewer parts leads to compelling advantages for compliant mechanisms. For one, there is a potential for significantly lower costs. This comes from reduced assembly, fewer components to stock, and the possibility of simplified manufacturing (such as fabricating a mechanism from a single mold).

Another advantage is the potential for increased performance. This includes high precision [5,6] due to reduced wear and reduced or eliminated backlash. The low weight of compliant mechanisms can be useful for shipping and for weight-sensitive applications such as spacecraft. Eliminating the need for lubrication at joints is also a useful performance improvement that is helpful in many applications and environments.

Another category of advantages lies in the ability to miniaturize compliant mechanisms. Microelectromechanical systems (MEMS) for example, are often fabricated from planar layers and compliant mechanisms offer a way to achieve motion with the extreme constraints caused by the resulting geometry (Figure 1.4) [7,8]. Compliant mechanisms will likely be central to the creation of nanoscale machines.

Figure 1.4 A multi-layer compliant microelectomechanical system (MEMS). A scanning electron micrograph of the device (top left) with a close up of compliant segments (top right), and the device shown in two stable equilibrium positions (bottom)

1.3 What Challenges do Compliant Mechanisms Introduce?

While the advantages of compliant mechanisms are amazing, they also have some challenges that have to be carefully considered in their design. For example, the integration of different functions into fewer parts offers advantages, but it also requires the simultaneous design for motion and force behavior. This difficulty is increased further by the fact that the deflections are often well into the nonlinear range and simplified linear equations are not adequate to define their motion.

Fatigue life needs to be addressed for most compliant mechanisms. Because their motion comes from bending of flexible parts, compliant mechanisms experience stress at those locations. When that motion is repeated during its life, fatigue loads are present and the fatigue life must exceed the expected life of the mechanism. Fortunately, methods for analyzing and testing fatigue life are available to help design compliant mechanisms for their needed fatigue life (see Chapter 2), but it requires special attention and effort to ensure that the mechanism has the life required.

Although properly designed and tested compliant mechanisms can achieve needed fatigue life, there can still be a consumer perception that flexible components are flimsy or weak. This can be a particular concern where the flexible component is visible to the consumer and it may require special care in the design for adequate life and for its appearance.

The motion of compliant mechanisms is often more limited than for traditional rigid-link mechanisms. For example, a shaft connected to bearings has the ability to undergo continuous revolution, whereas the motion of a flexible component will be limited by the deflection it can undergo before failure.

The fact that strain energy is stored in a deflected beam can be either an advantage or a disadvantage. Advantages include that a compliant element integrates both a spring and hinge function into a single component providing a “home” position where the device will go when unloaded. This integration also allows certain behaviors, such as bistability (the characteristic of having two distinct preferred positions, such as the on-off positions of a light switch) [9]. However, there are times when these qualities are not desired, and the properties become a disadvantage in the device design.

If certain materials are held under stress for long periods of time or at elevated temperatures, they can take on a new shape associated with the stressed position. This is called “stress relaxation.” Some compliant mechanisms have functions where they must maintain positions where they are under stress, and so are subject to stress relaxation conditions. This requires careful design and thoughtful choice of material.

1.4 Why are Compliant Mechanisms Becoming More Common?

Advances in our understanding of compliant mechanisms, combined with general technological developments, have resulted in a rapid growth in compliant mechanism applications (the library portion of this handbook is a testament to that growth). These applications range from high-end, high-precision devices to ultralow-cost packaging; from nanoscale featured components to large-scale machines; from weapons to healthcare products.

We mentioned that many early devices were compliant mechanisms, but then rigid-link devices connected at hinges gained favor because of the simplicity offered for analysis and design. So what is different now and why are there so many more compliant mechanisms than before? The answer lies at least partly in technological advances that have been made over recent decades. For example, new materials are available that are well suited for compliant mechanisms. There have been dramatic improvements in computational hardware and software available to analyze compliant mechanism motion and stresses. Developers and researchers have also increased our ability to design and analyze compliant mechanisms. Considerable effort has gone into creating methods to facilitate compliant mechanism design (some of the resulting methods are summarized in this handbook). There is also an increased awareness of the advantages of compliant mechanisms. As some commercial applications have been successful, they provide examples and inspiration for other applications to follow. Finally, as society and technology have advanced, new needs have risen, and some of these needs are best addressed by compliant mechanisms. This includes devices at very small size scales, devices with relatively complex motion but must be made at extremely low cost, compact medical implants, and high-precision machines.

1.5 What are the Fundamental Concepts that Help Us Understand Compliance?

There are a few straightforward but counterintuitive concepts that can help us understand the fundamentals of compliant mechanisms.

1.5.1 Stiffness and Strength are NOT the Same Thing

Usually when we want something to be strong (meaning that we don't want it to break), we also want it to be stiff (meaning that we don't want it to bend). For example, the floor in the upper story of a building we want to be both stiff and strong. We obviously don't want it to break, but we also don't want it to move around when people walk on it. So it needs to be stiff and strong. The crank shaft in an engine? Stiff and strong. A bridge? Stiff and strong. A desk? Stiff and strong.

We so often design things that need to be both stiff and strong that it is easy for our intuition to begin to tell us that stiffness and strength are the same. But they are NOT the same. Strength relates to resistance to failure, while stiffness relates to resistance to deflection. These are different and are governed by different properties. Consider a piece of steel with a rectangular cross section as shown in Figure 1.5. The steel will withstand a certain stress until it will fail. But its strength is the same whether it is loaded about its thin or thick axis (assuming it is isotropic), while its stiffness is very different for these two conditions.

Figure 1.5 The rectangular piece of steel may have the same strength in different directions, but it will have very different stiffness for the two orientations shown

1.5.2 It is Possible for Something to be Flexible AND Strong

Consider examples of things that are both flexible and strong. Flexible endoscopes, such as that shown in Figure 1.6, are used to examine the interior parts of the body. The endoscope must be flexible to undergo the required motion and to minimize any trauma from its use within the body. It must also be strong to withstand the loads that it will undergo during its use. As another example, consider the pulleys on the cables of a ski lift (Figure 1.7). They must be strong enough to reliably lift the skiers to their destination but must be flexible enough to go around the pulleys.

Figure 1.6 A flexible endoscope is an example of a device that needs to be both flexible and strong

Figure 1.7 A ski lift cable must be flexible enough to go around the pulley and strong enough to carry the loads

So why is it that many things we want to be stiff and strong, but others we want to be flexible and strong? What is it that determines the difference between these two situations? The answer lies in whether the device needs to hold a force, or if it needs to be deflected (like a cable going around a pulley). A bridge is an example of something that needs to be stiff and strong because we want it to hold the weight of traffic going across them without moving. The endoscope and the pulley cable are both examples of things that need to bend to perform their function. If they were too stiff, they would be overstressed and would break when they were forced to undergo the needed motion. So if something needs to hold a weight or other force, it should be stiff and strong; if it needs to go through a certain deflection, it should be flexible and strong.

1.5.3 The Basics of Creating Flexibility

There are three primary ways that we can influence flexibility. These are

1. material properties (what it is made of);

2. geometry (its shape and size);

3. loading and boundary conditions (how is it held and loaded).

Each of these is described below.

1.5.3.1 Materials Properties

Different materials have different stiffnesses as measured by the material's Young's modulus (or modulus of elasticity). Consider the three rods in Figure 1.8. Each rod has identical size and shape and each has the same size weight hanging from it, but they are each made of a different material: steel, aluminum, and polypropylene. The Young's modulus of steel (207 GPa) is about three times that of aluminum (72 GPa), so for the same geometry and same weight, the aluminum rod will deflect three times as much as the steel. But polypropylene's Young's modulus (1.4 GPa), is about one fiftieth that of aluminum, so it deflects much farther than either aluminum or steel for the same applied load.

Figure 1.8 Material properties contribute to stiffness. These three rods all have the same geometry but they experience different deflections even with the same size weight hanging from them

Because we want something both flexible and strong when we are designing compliant mechanisms, we look for materials with a high strength and a low Young's modulus. One way to compare different materials for use in a compliant mechanism is to compare the ratio of strength to Young's modulus, with a higher ratio being better. A similar approach is to compare the “resilience” of materials, where the modulus of resilience is one-half the yield strength squared divided by the Young's modulus. The modulus of resilience is a measure of how much energy per unit volume the material can withstand without a permanent change. Table 1.1 lists some materials and their ratio of strength to modulus and their resilience.

Table 1.1 Yield strength to Young's modulus ratio and resilience for several materials

1.5.3.2 Geometry

Shape and size have a large effect on flexibility. Let's start with an obvious example. Consider the two parts shown in Figure 1.9