Mechanism design : Visual and programmable approaches : Visual and programmable approaches

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Mechanism

Design

Visual and Programmable

Approaches

Kevin Russell

Qiong Shen

Raj S. Sodhi

Russell

•

Shen

•

Mechanism Sodhi

Design

Visual and Programmable

Approaches

Mechanism Design

In the field of mechanism design, kinematic synthesis is a creative means to produce mechanism

solutions. Combined with the emergence of powerful personal computers, mathematical analysis

software and the development of quantitative methods for kinematic synthesis, there is an endless

variety of possible mechanism solutions that users are free to explore, realize, and evaluate for

any given problem in an efficient and practical manner.

Mechanism Design: Visual and Programmable Approaches provides a broad introduction to

kinematic synthesis, presenting and applying motion, path, and function generation methodologies

for some of the most basic planar and spatial single- and multiloop linkage systems. This work

provides numerous in-chapter synthesis examples and end-of-chapter synthesis problems. Users

can also invent their own specialized synthesis problems according to their particular interests.

The commercial mathematical software package MATLAB® and its mechanical system modeling

and simulation module SimMechanics® are thoroughly integrated in this textbook for mechanism

synthesis and analysis. The reader is therefore enabled to readily apply the design approaches

presented in this textbook to synthesize mechanism systems and visualize their results. With this

knowledge of both kinematic synthesis theory and computer-based application, readers will be

well-equipped to invent novel mechanical system designs for a wide range of applications.

Mechanism

Design

Visual and Programmable

Approaches

Boca Raton London New York

CRC Press is an imprint of the

Taylor & Francis Group, an informa business

Mechanism

Design

Visual and Programmable

Approaches

Kevin Russell

Qiong Shen

Raj S. Sodhi

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accu￾racy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not

constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the

MATLAB® software.

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Version Date: 20131021

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To Willie J. Russell and in memory of Ella Russell

Kevin Russell

To my parents, Youlin Shen and Mantian Wang, without

whom none of my success would be possible

Qiong Shen

To my parents, Rachpal S. and Pritpal K. Sodhi

Raj S. Sodhi

vii

Contents

Preface.......................................................................................................................xi

Authors.....................................................................................................................xv

Chapter 1 Introduction to Kinematics...................................................................1

1.1 Kinematics.................................................................................1

1.2 Kinematic Chains and Mechanisms..........................................2

1.3 Mobility .....................................................................................3

1.4 Summary ...................................................................................5

References ............................................................................................6

Additional Reading ..............................................................................6

Chapter 2 Mobility of Mechanisms......................................................................7

2.1 Planar Mechanism Types ..........................................................7

2.2 Links, Joints, and Mechanism Mobility....................................9

2.3 Number Synthesis.................................................................... 12

2.4 Grashof’s Criteria and Transmission Angle ............................ 14

2.5 Summary ................................................................................. 17

Problems............................................................................................. 17

References ..........................................................................................20

Chapter 3 Kinematics of Planar Mechanisms..................................................... 21

3.1 Kinematic Analysis of Planar Mechanisms............................ 21

3.2 Four-Bar Mechanism Analysis................................................ 21

3.2.1 Four-Bar Displacement Equation............................... 21

3.2.2 Four-Bar Velocity Equation........................................22

3.2.3 Four-Bar Acceleration Equation.................................23

3.3 Slider-Crank Mechanism Analysis..........................................25

3.3.1 Slider-Crank Displacement Equation.........................25

3.3.2 Slider-Crank Velocity Equation .................................26

3.3.3 Slider-Crank Acceleration Equation ..........................27

3.4 Multiloop Mechanism Analysis ..............................................29

3.5 Kinematics of Mechanism Locations of Interest .................... 31

3.6 Solution Method for Vector Loop Kinematic Equations......... 33

3.7 Planar Kinematic Modeling in MATLAB®

and SimMechanics® ................................................................34

3.8 Summary .................................................................................38

Problems.............................................................................................39

References ..........................................................................................44

Additional Reading ............................................................................44

viii Contents

Chapter 4 Kinematic Synthesis and Planar Four-Bar Motion Generation.......... 45

4.1 Introduction to Kinematic Synthesis....................................... 45

4.2 Branch and Order Defects....................................................... 47

4.3 Motion Generation: Three, Four, and Five Precision

Positions...................................................................................49

4.4 Branch and Order Defect Elimination:

Three, Four, and Five Precision Positions............................... 62

4.5 Summary .................................................................................66

Problems............................................................................................. 67

References .......................................................................................... 75

Chapter 5 Planar Four-Bar and Multiloop Path and Motion Generation............77

5.1 Path Generation versus Motion Generation.............................77

5.2 Coupler Curves and Dwell Motion.......................................... 78

5.3 Approximate Four-Bar Path and Motion Generation..............80

5.4 Alternate Four-Bar Kinematic Equations................................86

5.5 Alternate Approximate Four-Bar Motion and

Path Generation Equations......................................................88

5.6 Constructing Cognates ............................................................92

5.7 Analytical and Approximate Multiloop Path

and Motion Generation..........................................................100

5.8 Summary ............................................................................... 104

Problems........................................................................................... 105

References ........................................................................................ 116

Chapter 6 Planar Four-Bar Function Generation.............................................. 117

6.1 Introduction to Function Generation ..................................... 117

6.2 Function Generation: Three, Four, and

Five Precision Points............................................................. 119

6.3 Approximate Function Generation........................................ 122

6.4 Velocity and Acceleration Constraints

for Function Generation......................................................... 125

6.5 Function Generation with Finite and Multiply

Separated Positions................................................................ 127

6.6 Approximate Function Generation with Finite

and Multiply Separated Positions.......................................... 129

6.7 Summary ............................................................................... 130

Problems........................................................................................... 131

References ........................................................................................ 136

Additional Reading .......................................................................... 136

Chapter 7 Spatial Mechanism Kinematics and Synthesis ................................ 137

7.1 Introduction to Spatial Four-Bar Mechanisms...................... 137

7.2 RRSS and 4R Spherical Mechanism Analysis...................... 137

Contents ix

7.3 RSSR and 4R Spherical Mechanism Analysis...................... 141

7.4 Approximate RRSS and 4R Spherical Motion and

Path Generation ..................................................................... 143

7.5 Approximate RSSR and 4R Spherical

Function Generation ............................................................. 160

7.6 RSSR–SS Mechanism Analysis............................................ 166

7.7 R–S Dyad and Approximate RSSR–SS

Motion Generation................................................................ 170

7.8 Spatial Kinematic Modeling in MATLAB®

and SimMechanics® .............................................................. 176

7.9 Summary ............................................................................... 177

Problems........................................................................................... 178

References ........................................................................................ 186

Chapter 8 Adjustable Planar and Spherical Four-Bar Mechanism Synthesis..... 187

8.1 Introduction to Adjustable Mechanism Synthesis................. 187

8.2 Approximate Adjustable Planar Four-Bar Motion

Generation ............................................................................. 190

8.3 Approximate Adjustable Planar Four-Bar Motion

and Path Generation: Alternate Equations............................ 194

8.4 Approximate Adjustable Planar Four-Bar Function

Generation ............................................................................. 199

8.5 Approximate Adjustable 4R Spherical Motion Generation.....202

8.6 Approximate Adjustable 4R Spherical Motion

and Path Generation: Alternate Equations............................207

8.7 Summary ............................................................................... 211

Problems........................................................................................... 212

References ........................................................................................223

Appendix A: Solution Algorithm for Analytical Four-Position

Synthesis and MATLAB® File User Instructions......................225

Appendix B: Solution Algorithm for Analytical Five-Position

Synthesis and MATLAB® File User Instructions....................... 231

Appendix C: User Instructions for Chapter 5 MATLAB® Files..................... 239

Appendix D: User Instructions for Chapter 6 MATLAB® Files..................... 255

Appendix E: User Instructions for Chapter 7 MATLAB® Files.....................263

Appendix F: User Instructions for Chapter 8 MATLAB® Files......................299

Appendix G: User Instructions for MATLAB® and SimMechanics® Files...... 327

xi

Preface

In kinematic synthesis, mechanism designs are produced to fulfill given design func￾tions. Current textbooks on mechanism design deal mostly with analyzing existing

mechanisms and devote only a chapter or two to synthesis, which is not enough

to provide a comprehensive introduction. Most textbooks present mainly qualita￾tive methods for kinematic synthesis, which are based on drafting techniques.

Although qualitative methods are easy to implement, they are typically limited to

two-dimensional, small-scale problems. In recent times, personal computers have

facilitated the design of complex systems. Computation-based or quantitative meth￾ods are far more robust in terms of the type and scale of problems that one can

consider. In addition, the computing technologies available today, both hardware and

software, make quantitative methods very practical.

This book, which is dedicated entirely to kinematic synthesis, is based on our

teaching, research, and industry experience in mechanism design. The purpose of

this textbook is to give the reader an introduction to kinematic synthesis that is both

broad and practical. In terms of providing a broad introduction, this textbook pres￾ents applications of planar and spatial motion, path, and function generation meth￾odologies (both classical and modern). Additional topics include number synthesis,

synthesis using finite and multiply separated positions, and the synthesis of adjust￾able mechanisms (a more modern area of focus in kinematic synthesis). In terms of

providing a practical introduction, this textbook considers some of the most basic

planar and spatial single- and multiloop linkage systems.

Chapters 1 and 2 include fundamental concepts in mechanism kinematics

that are also considered in mechanism synthesis. Chapter 2 goes on to intro￾duce number synthesis—a systematic approach where mechanism solutions are

produced according to mechanism mobility. The kinematic analysis of planar

four-bar and multiloop mechanisms is covered in Chapter 3. Analytical planar

four-bar motion generation is covered in Chapter 4. Chapter 5 covers numerical

motion and path generation of planar four-bar and multiloop mechanisms. Both

analytical and numerical planar four-bar function generation are presented in

Chapter 6. Chapter 7 covers the kinematic analysis and numerical synthesis of

spatial four-bar and multiloop mechanisms for motion, path, and function gener￾ation. Chapter 8 introduces adjustable planar and spherical four-bar mechanism

synthesis for motion and path generation. Appendices A through G include user

instructions for each of the MATLAB® and SimMechanics® files that are avail￾able for download with this textbook.

This work considers quantitative methods for kinematic analysis and synthe￾sis exclusively. The commercial mathematical software package MATLAB and its

mechanical simulation module SimMechanics are thoroughly integrated in this text￾book for kinematic analysis and synthesis. Both MATLAB and SimMechanics are well

established (and often the de facto standard for mathematical analysis and simulation)

xii Preface

in colleges and universities. A library of MATLAB and SimMechanics files has

been developed for this textbook. This and additional material can be downloaded

from the CRC website: http://www.crcpress.com/product/isbn/9781466570177. These

files will help the reader apply any of the dozens of equation systems presented in

this textbook for kinematic analysis and synthesis. The combination of theoretical

breadth and depth provided in this work, in addition to the library of MATLAB and

SimMechanics files (for immediate application in analysis and synthesis), make this

textbook a valuable tool for engineering teaching, engineering research, and every￾day engineering practice.

Because this work is dedicated to kinematic synthesis—an advanced area in

kinematics—it is hoped that the reader would have been exposed to the content

covered in a typical undergraduate course in mechanism kinematics. In terms of

MATLAB knowledge, the user should know how to launch MATLAB and open

and run MATLAB files (these skills can be acquired easily using MATLAB help

resources). Of course, the user should also have access to the MATLAB software

package itself.

All of the synthesis problems in this textbook were solved using the 2009 version

of MATLAB (specifically version R2009b, win32). This was because the develop￾ment of this textbook began in 2008, and we preferred to use a single version of

MATLAB throughout this work. It is possible that the user could occasionally pro￾duce solutions that do not precisely match those in the textbook when using more

recent versions of MATLAB. However, because an infinite number of mechanism

solutions exist for a given synthesis problem in general, the user should not be at any

disadvantage.

By definition, kinematic synthesis is a creative means to produce mechanism

solutions. There is an endless variety of possible mechanism solutions that the user

is free to explore, realize, and evaluate for any given problem. Even though this work

provides numerous synthesis examples and end-of-chapter synthesis problems, the

user can invent his or her own specialized synthesis problems according to his or her

interests. The resources in this textbook help make this possible. The textbook can

be used

• As a companion text (for synthesis topics) for an undergraduate course in

kinematics

• As a text for an undergraduate or graduate course in advanced (computer￾based) kinematics

• As a text for an undergraduate mechanical design course (where synthe￾sized mechanisms are analyzed using statics/dynamics, stress analysis,

machine design, computer aided engineering, etc.)

We encourage the reader to explore the topics covered in this book and look for￾ward to receiving valuable feedback, if any. We can be reached through e-mail at

[email protected]. We hope you enjoy utilizing this work as

much as we have enjoyed preparing it.

The authors would like to acknowledge the contributions of those who assisted

in the review and evaluation of this textbook. In particular, we thank J. Angeles,

Preface xiii

McGill University; B. Corves, RWTH Aachen University; A. Ghosal, Indian Institute

of Science; T. Huang, Warwick University; W. Lee, National Pingtung University

of Science and Technology; J. Merlet, INRIA; Y. Nakamura, University of Tokyo;

T. Shoup, Santa Clara University; and K. Ting, Tennessee Technological University.

Professor Sodhi wishes to thank Dr. Terry Shoup and Dr. Ken Waldron who

helped him learn kinematics and mechanism design and introduced him to a lot of

exciting concepts in mechanical engineering.

MATLAB® is a registered trademark of The MathWorks, Inc. For product infor￾mation, please contact:

The MathWorks, Inc.

3 Apple Hill Drive

Natick, MA, 01760-2098 USA

Tel: 508-647-7000

Fax: 508-647-7001

E-mail: [email protected]

Web: www.mathworks.com

Kevin Russell

Qiong Shen

Raj S. Sodhi