Advanced Fixture Design Method and ItsApplication

Guohua Qin

Advanced Fixture DesignMethod and Its Application

Guohua QinNanchang Hangkong UniversityNanchang, Jiangxi, China

ISBN 978-981-33-4492-1 ISBN 978-981-33-4493-8 (eBook)https://doi.org/10.1007/978-981-33-4493-8

Jointly published with Shanghai Jiao Tong University Press.The print edition is not for sale in China Mainland. Customers from China Mainland please order theprint book from: Shanghai Jiao Tong University Press.

© Shanghai Jiao Tong University Press 2021, corrected publication 2021This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whetherthe whole or part of the material is concerned, specifically the rights of translation, reprinting, reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, andtransmission or information storage and retrieval, electronic adaptation, computer software, or by similaror dissimilar methodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publishers, the authors, and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publishers nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publishers remain neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

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Preface

Manufacturing technology is the foundation and key to the development of all indus-tries. But in the manufacturing technology, fixtures are one kind of indispensabletechnical equipment that can directly influence the machining accuracy, productionefficiency and manufacturing cost. Therefore, fixture design plays an important rolein production technology preparation as well as product design and manufacturing.

Fixture design is an important and complex technical work. In the traditionalfixture design method, the determination of fixturing layout, the selection of fixtureelements and the assembly of fixture elements are all completed by fixture designers.The traditional design method not only requires more manpower and a longer designcycle, but also relies on the rich experience of designers. With the wide applicationof computer technology in the manufacturing field, a new fixture design methodis formed by using the integration of the computer with advanced manufacturingtechnologies including the feature technology, group technology and artificial intel-ligence. This is the so-called computer-aided fixture design technology. In the past,the key to develop the computer-aided fixture design system is to collect and expressthe knowledge from the experience of the fixture designers. However, it is eitherimpossible or unrealistic to fully express all fixture design knowledge.

Combined with kinematics, contact mechanics, elastic mechanics, mathemat-ical modeling technology and optimization technology, the advanced fixture designmethod is systematically proposed. The essence of the proposed design method isto iteratively analyze the fixturing performance for the new fixturing layout untilit is satisfied. The established analysis model of fixturing performance includes thelocating determination, the workpiece stability, the clamping reasonability, the work-piece attachment/detachment, and the locating accuracy. By discretizing the valuerange of design variables, some planning algorithms are suggested including theselection algorithm of locating datum, the planning algorithm of clamping force,and so forth.

In the proposed advanced fixture design method, the continuous fixture designproblem is transformed into a discrete problem so that it can be easily realized byprogramming. Accordingly, it can enrich and develop the basic theory of computer-aided fixture design and change the empirical method of fixture design. The combi-nation of theoretical analysis and mathematical modeling technology can resolve

v

vi Preface

the key problems in the process of fixture design, which will play a certain role inpromoting the progress of manufacturing technology, improving the precision andlevel of product manufacturing, and meeting the higher and higher requirements ofthe mechanical manufacturing industry.

The contents covered in this book include major research outcomes of numerousresearch projects sponsored by the National Natural Science Foundation of China(51765047; 51465045; 51165039), Major Discipline Academic and TechnicalLeader Training Plan Project of Jiangxi Province (20172BCB22013), AeronauticalScience Foundation of China (2006ZE56006; 2010ZE56014), Natural Science Foun-dation of Jiangxi Province (2009GZC0104), Key Project of Science and TechnologySupport Plan of Jiangxi Province (2010BGB00300), and China Postdoctoral ScienceFoundation (20070411142). Moreover, several postgraduate students participatedin relevant research work, including Haichao Ye, Huaping Huang, Meidan Zhou,XiyuanGuo,YueCui, HuaminWang, ZikunWang, Shuo Sun,XuiangZhao,YuanjunHou, Zhe Huang, Jiamei Li, Xiuping Dai, Weida Lou, Feng Lin, Jianpeng Qiu, etc.I hereby express my sincere gratitude to them.

Nanchang, ChinaSeptember 2020

Guohua Qin

The original version of the book was inadvertently published with incorrect figures in Chapters 4,5, 6 and 10, which have now been corrected. The book and the chapters have been updated with thechanges. The correction to this book is available at https://doi.org/10.1007/978-981-33-4493-8_13

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Fixturing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Alignment Fixturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Fixture Fixturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Fixture Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Types of Fixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Configuration of Fixtures . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Fixture Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Analysis of Locating Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Model of Theoretical DOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Relationship Between Machining Requirementsand DOFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.2 Establishment of Theoretical DOF Model . . . . . . . . . . . . 192.1.3 Undetermined Coefficient Method . . . . . . . . . . . . . . . . . . . 232.1.4 Solving Rank Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 DOF Level Model of Position Sizes . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.1 Position Size of Plane Relative to Plane . . . . . . . . . . . . . . 282.2.2 Position Size of Line Relative to Plane . . . . . . . . . . . . . . . 302.2.3 Position Size of Plane Relative to Line . . . . . . . . . . . . . . . 312.2.4 Position Size of Line Relative to Line . . . . . . . . . . . . . . . . 33

2.3 DOF Level Model of Orientations . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.1 Relation Between Parallelism Tolerance and DOFs . . . . 362.3.2 Relation Between Perpendicularity/Inclination

Tolerance and DOFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4 DOF Level Model of Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.4.1 Relation Between Coaxiality Tolerance and DOFs . . . . . 522.4.2 Relation Between Symmetry Tolerance and DOFs . . . . . 532.4.3 Relation Between Position Tolerance and DOFs . . . . . . . 572.4.4 Relation Between Runout Tolerance and DOFs . . . . . . . . 61

2.5 Model of Locating Point Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

vii

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2.5.1 Establishment of Locating Point Layout Model . . . . . . . 642.5.2 Solution of Locating Point Layout Model . . . . . . . . . . . . 67

2.6 Judgment Criteria of Locating Determination . . . . . . . . . . . . . . . . . 702.6.1 Theoretical Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.6.2 Process Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.6.3 Corollary and Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.7 Application and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.7.1 Verification of Locating Point’s Number . . . . . . . . . . . . . 752.7.2 Verification of the Layout of Locating Points . . . . . . . . . 77

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Analysis of Workpiece Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1 Modeling of Workpiece Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.1.1 Static Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . 853.1.2 Friction Cone Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 903.1.3 Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2 Solution Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2.1 Linear Programming Techniques . . . . . . . . . . . . . . . . . . . . 933.2.2 Solution to the Model of Form-Closure . . . . . . . . . . . . . . 943.2.3 Solution to the Model of Force-Closure . . . . . . . . . . . . . . 100

3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.3.1 Analysis of Force Existence . . . . . . . . . . . . . . . . . . . . . . . . 1023.3.2 Analysis of Force Feasibility Without Friction . . . . . . . . 1043.3.3 Analysis of Force Feasibility with Friction . . . . . . . . . . . 106

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4 Analysis of Clamping Reasonability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.1 Local Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.1.1 Contact Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.1.2 Fixel Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.1.3 Local Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.2 Workpiece Position Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.2.1 Static Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . 1214.2.2 Friction Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.2.3 Relationship Between Local Deformation

and Contact Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.2.4 Relationship Between Workpiece Position Error

and Local Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.2.5 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.3 Workpiece Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.4 Modeling of Clamping Reasonability . . . . . . . . . . . . . . . . . . . . . . . . 1284.5 Application and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.5.1 Analysis of the Clamping Error Due to the WeakStiffness Workpiece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.5.2 Evaluation of the Clamping Error of HighStiffness Workpiece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Contents ix

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

5 Analysis of Workpiece Attachment/Detachment . . . . . . . . . . . . . . . . . . 1395.1 Attachment and Detachment Model . . . . . . . . . . . . . . . . . . . . . . . . . 1395.2 Judgment Method of the Attachment and Detachment . . . . . . . . . 1425.3 Analysis Algorithm of the Attachment and Detachment

Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.3.1 Calculation Algorithm of the Generator Matrix

of the Non-Positive Dual Matrix . . . . . . . . . . . . . . . . . . . . 1465.3.2 Classification Method of Coefficient Matrix . . . . . . . . . . 1485.3.3 Application of Pivot Algorithm . . . . . . . . . . . . . . . . . . . . . 149

5.4 Analysis and Application of Attachment and Detachment . . . . . . 1595.4.1 Three Dimensional Workpiece . . . . . . . . . . . . . . . . . . . . . . 1595.4.2 Two Dimensional Workpiece . . . . . . . . . . . . . . . . . . . . . . . 166

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6 Analysis of Locating Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1756.1 Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.1.1 Locating Source Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.1.2 Workpiece Position Error . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.1.3 Locating Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.2 Position Error Model of Contact Point . . . . . . . . . . . . . . . . . . . . . . . 1836.2.1 Directional Dimension Path . . . . . . . . . . . . . . . . . . . . . . . . 1836.2.2 Directional Dimension Chain . . . . . . . . . . . . . . . . . . . . . . . 185

6.3 Examples and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.3.1 One Vee Bloke-One Supporting Pin Locating

Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.3.2 Two Cylindrical Pins Locating Scheme . . . . . . . . . . . . . . 1926.3.3 One Plane-Two Holes Locating Scheme . . . . . . . . . . . . . . 1976.3.4 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7 Selection Algorithm of Locating Datum . . . . . . . . . . . . . . . . . . . . . . . . . 2097.1 Hierarchical Structure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

7.1.1 Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2107.1.2 Surface Feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.1.3 Valid Locating Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.1.4 Dimension Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

7.2 Judgement Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2127.3 Layer Weight Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2147.4 Combination Weight Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2157.5 Reconstruction of Judgment Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 2177.6 Algorithm and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7.6.1 Algorithm and Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2187.6.2 Practical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

x Contents

8 Planning Algorithm of Locating Point Layout . . . . . . . . . . . . . . . . . . . . 2298.1 Locating Point Layout Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2298.2 Causes of Locating Non-determination . . . . . . . . . . . . . . . . . . . . . . 2328.3 Generative Planning Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2378.4 Application and Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

9 Planning Algorithm of Clamping Forces for Rigid Workpieces . . . . 2459.1 Mechanical Model of Fixturing Scheme . . . . . . . . . . . . . . . . . . . . . 245

9.1.1 Workpiece Static Equilibrium Constraints . . . . . . . . . . . . 2469.1.2 Direction Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

9.2 Analysis of Force Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2479.3 Analysis of Force Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2489.4 Planning Algorithm of 1-clamping Force . . . . . . . . . . . . . . . . . . . . 2509.5 Planning Algorithm of n-clamping Force . . . . . . . . . . . . . . . . . . . . 2539.6 Calculation of Clamping Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

9.6.1 Single Clamping Force of 2D Workpiece . . . . . . . . . . . . . 2579.6.2 Single Clamping Force of 3D Workpiece . . . . . . . . . . . . . 2619.6.3 Double Clamping Forces of 2D Workpiece . . . . . . . . . . . 263

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

10 Determination Method of Clamping Point Layout for RigidWorkpieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27110.1 Workpiece Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

10.1.1 Stability Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27110.1.2 Stability Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

10.2 Planning Algorithm of Application Region . . . . . . . . . . . . . . . . . . . 27610.3 Planning Method of Fixturing Layout . . . . . . . . . . . . . . . . . . . . . . . 28010.4 Numerical Tests of Application Region Planning . . . . . . . . . . . . . . 283

10.4.1 Planning of Application Region for One ClampingForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

10.4.2 Calculation of Stable Region in DynamicMachining Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

10.4.3 Planning of Application Region for MultipleClamping Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

10.5 Examples of Fixturing Layout Optimization . . . . . . . . . . . . . . . . . . 29410.5.1 Fixturing Layout Optimization of 2D Workpiece . . . . . . 29410.5.2 Fixturing Layout Optimization of 3D Workpiece . . . . . . 297

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

11 Fixturing Layout Optimization of Thin-Walled Workpieces . . . . . . . 30311.1 Finite Element Analysis of Fixturing Deformation . . . . . . . . . . . . 303

11.1.1 Static Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . 30311.1.2 Friction Cone Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 30411.1.3 Unilateral Contact Constraints . . . . . . . . . . . . . . . . . . . . . . 305

11.2 Prediction Method for Fixturing Deformation . . . . . . . . . . . . . . . . 305

Contents xi

11.2.1 Structure of Neural Network . . . . . . . . . . . . . . . . . . . . . . . 30511.2.2 Selection of Training Samples . . . . . . . . . . . . . . . . . . . . . . 30611.2.3 Update of Weight Coefficients . . . . . . . . . . . . . . . . . . . . . . 306

11.3 Optimization Technology of Fixturing Layout . . . . . . . . . . . . . . . . 30711.3.1 Optimal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30811.3.2 Solution Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

11.4 Numerical Tests and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 31111.4.1 Prediction of Workpiece Deformation . . . . . . . . . . . . . . . . 31211.4.2 Optimization of Fixturing Layout . . . . . . . . . . . . . . . . . . . 317

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

12 Matching Method of Fixture Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 32512.1 Analytical Hierarchy Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32512.2 Construction of Hierarchical Structure Model . . . . . . . . . . . . . . . . 327

12.2.1 Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32812.2.2 Selection Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

12.3 Determination of Layer Weight Vector for Criterion Layer . . . . . 33112.3.1 Construction of Judgement Matrix . . . . . . . . . . . . . . . . . . 33112.3.2 Check of Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

12.4 Determination of Layer Weight Vector for Scheme Layer . . . . . . 33312.4.1 Construction of Judgment Matrix . . . . . . . . . . . . . . . . . . . 33312.4.2 Check of Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33512.4.3 Decision Index of Locators . . . . . . . . . . . . . . . . . . . . . . . . . 336

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Correction to: Advanced Fixture Design Method and ItsApplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C1