CURRENT ADVANCES IN M E C H A N I C A L DESIGN AND P R O D U C T I O N VII
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CURRENT ADXv~ANCES IN MECHANICAL DESIGN AND P R O D U C T I O N VII Proceedings of the Seventh Cairo University International MDP Conference
Cairo-Egypt February 15-17, 2000
Edited by
Mohamed E HASSAN C~m[-ctcnct, Chairman, Pro]~'.s.sor~.tndHead, Mcchanicttl Dcsign ttn~! Producti{;n Department, Faculty o[ Enginccring, Cairo Univcrsitx; Giza 12316-Egypt and
Cotl lt'lt'~lcc
Said M. MEGAHED Gctlt'~tl[ 5ccFt'ltilV, Editor-in-Chiq[and Prq/cssop;
Mechanical Dcsi,qn ctnd Production Department, Fticttlt\' t!l Erlgint't'ring, Cairo University, Giza 12316-Egypt
Pergamon AMSTERDAM - LAUSANNE - NEW YORK - OXFORD - SHANNON - SINGAPORE - TOKYO
ELSEVIER SCIENCE Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 IGB, UK
9 2000 Elsevier Science Ltd. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use:
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First edition 2000
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PREFACE The Mechanical Design and Production Department, Faculty of Engineering, Cairo University has established its series of international conferences on the Current Advances in Mechanical Desilm and production since 1979. This conference is the 7 ih in the series (MDP-7), held in Cairo during the period February 15-17, 2000. The conference brings together engineers and scientists from allover the world with a view of exchanging experience and highlighting the state of the art in the fields of Mechanical Design and Production as well as Technology Transfer. Several distinguished researchers were invited to address the conference with keynote papers to enrich the sessions and to highlight the recent advances on the various fields of mechanical design and production. A total of 160 papers were submitted to MDP-7 conference from more than 21 countries from the 6 continents. All papers were thoroughly refereed by the conference scientific committee. Following the reviewing process, a total of 104 papers in addition to 15 industrial applications and case studies were accepted for presentation. These MDP-7 proceedings contain 10 invited papers together with 54 selected papers. Further papers are published independently by the Mechanical Design and Production Department, Faculty of Engineering, Cairo University. The conference proceedings include basic and applied research papers in Mechanical Design and Production classified into six main categories: System Dynamics, Solid Mechanics, Material Science, Manufacturing Processes, Design and Tribology, and Industrial Engineering and its Applications. We would like to express our gratitude to our colleagues in the Mechanical Design and Production Department, Faculty of Engineering, Cairo University who spend their time and effort in assisting us to have this fruitful work. We also acknowledge the cooperation of the conference scientific committee members, for their immediate response in reviewing all the papers submitted to the conference and for their valuable suggestions. We would like to take this opportunity to express our gratitude to the Academic Institutions and Industrial Organizations for their financial support. This support made it possible to meet part of the conference expenses. Finally, we dedicate our thanks and respects to all authors whose contributions made it possible to maintain the international reputation of the MDP Conferences. We hope that the MDP-7 proceedings present a useful contribution, reflecting the C u r r e n t Advances in Mechanical Design and Production at the onset of the 21 st century.
Said M. Megahed
Mohamed F. Hassan
General Secretary and Editor.in.Chief
Conference Chairman
vi
Current Advances in Mechanical Design and Production, MDP-7
SCIE .NT.IFIC COMMITTEE AND INTERNATIONAL ADvIsoRY BOARD Abbas, A.T. Abdelaal, R. Abdelhakim, M. Abdel-Kader, S. Abdelmaksoud, H. Abdelraouf, H. Abdo, M.A. Abo lsmail, A. Aboelfetooh, M.N. Adli, A. AI-Ashram, A. All, G.M. Allam, M.N.A. Anis, H.I. Arafa, H.A. Ashour, S. Attalla, W. Badran, F. Bahei Eldin, Y.A. Bahgat, A. Barakat, M.A. Bayoumi, A.M.E. Bayoumi, S.E.A. Bazaraa, A.S. Bedewy, M.K. Choi, B.K. Dardiri, M.A. Elarabi, M.E. Elbestawi, M.A. EI-Demerdash, M.F. Eleiche, A.M. El-Hakim, M.A. EI-Hebeary, M.R. EI-Kharboutly, A. EI-Kousy, M.R.
Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt USA Egypt Egypt Egypt S. Korea Egypt Egypt Canada Egypt Egypt Egypt Egypt Egypt Egypt
EI-Mahalawy,N. EI-Raghy,S. EI-Sabbagh,A. Elsayed,E.A. Elsebai,M.G. El-Sheikh,A. EI-SherbinyM.G. Elwani,M.H. EI-Zoghby,A.A. Ezzat,A. Farag,M. Fat-Halla,N. Gommaa,A.H. Hammouda,M.I. Hanafi,A.A. Hassan,G.A. Hassan,M.A. Hassan,M.F. Hassan,M.F. Hassan,S.D. Hassan,Y.K. Hassanien,A. Hegazi,A. Hosni,Y.A. lbrahim,I.A. Jamshidi,M. Kassem,M.E. Kassem,S.A. Khattab,A.A. Khorshid,S.Y. Koura,M. Krempl,E. Mahmoud,F.F. Mansour,A.M.A. Megahed,M.M.
Egypt Egypt Egypt USA Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt USA Egypt USA Egypt Egypt Egypt Egypt Egypt USA Egypt Egypt Egypt
Megahed,S.M. Meguid,S.A. Mokhtar,M.O.A. Mostafa,A.A-F. Moustafa,M.A. Nassar,M. Rabei,G. Radwan,A.A. Ragab,A.R. Ragab,M.S. Renaud,M. Riad, M.S.M. Riad, S.M. Rizk, M. Sabry,Sh. Sadek,E.A. Said, M.E. Salama,A.E. Salama,A.S. Saleh,I. Salem,H. Sallam,M. Shalaby,M.A. Shash,Y.M.S. Sherif,A.O. Soubeah,M.E. Taha,M. Wahdan,A. Wifi, A.S. Wustof,P. Yehia,N.A. Younan,M. Youssef,M.R.
Egypt Canada Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt France Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt Egypt S. Arabia Germany Egypt Egypt Egypt
STEERING AND ORGANIZING COMMITTEE Arafa, H.A. Bayoumi, S.E. Elaraby, M.E. Eleiche, A.M. EI-Sherbiny, M.G." Fawzy, I. Hassan, M.F. +
Hassan,Y.K. Kassem, S.A. Khattab, A.A Megahed, M.M. Megahed, S.M." Metwalli, S.M. Mokhtar, M.O.A.
* Faculty Dean + conference Chairman ** Conference General Secretary
Radwan, A.A. Ragab, A.R. Riad, M.S. Said, M.E. Salama, A.S. Shalaby, M.A.
vii
Current Advances in Mechanical Design and Production, MDP-7
INVITED AND KEYNOTE LECTURES Bayoumi, A.M.E. Choi, B.K. EIbestawi, M.A. EIsayed, E.A.
USA S. Korea Canada USA
Hosni, Y.A. Jamshidi, M. Krempl, E.
USA USA USA
Meguid, S.A. Renaud, M. Wustof, P.
USA France Germany
EDITORIAL COMMITTEE
ASSISTANTS TO SERETARY
Atria, M.S. EI-Zoghby, A.A. Gadalla, M.A. Hassan, M.F. § Megahed, M.M. Megahed, S.M." Nassef, A.M.O. Omar, M.A. Zeyada, Y.F.
Gommaa, A.H. Shendi, M.
TREASURERS Bedewy, M.K. Riad, S.M.
EXEXUTIVE COMMITTEE Staff Members of the Mechanical Design and Production Department Faculty of Engineering, Cairo University
Emeri.tus Professors Professors Bayoumi, S.EA.. Elarabi, M.E. Hassan,Y.K. Kandeel, S.E. Kouta, F.H. Mostafa, A.A-F. Riad, M.S.M. Riad, S.M.
Arafa, H.A. Bahgat, B.M. Basily, B.B. Bedewy, M.K. EI-Dalil, S.A. Eleiche, A.M. EI-Hebeary, M.R. EI-Sherbiny M.G.' Fawzy, I.
Hasan, M.F.§ Hassan, G.A. Kassem, S.A. Khattab, A.A. Megahed, M.M Megahed, S.M.'" Metawalli, S.M. Mokhtar, M.O.A. Radwan, A.A.
Associate Professors
Assistant Professors
EI-Habak, A.M. EI-Zoghby, A.A. Fouad, A.E.A. Khorshid, S.Y. Mansour, A.M.A. Mawsouf, N.M. Megahed, H.A. Mohamed, M.A.A. Othman, T.A. Salama, M.S.~ Shalaby, M.A.
AbdeI-Aal, O.M. Abdo, M.Z. Abdrabu, M.M.A. Abou-Hamda, M.M. Adly, M.A.~ Anany, A.A. Azzam, B.Sh.N. Bayoumi, L.S.E. Behnam, M.M. EI-Danaf, E.A. EI-Gamil, M.A.
9 ,
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i
,
Radwan, M.A. Ragab, A.R. Ragab, M.S. Said, M.E. Salama, A.S. Salim, F.B. Shash, Y.M.S. Soliman, F.A. Wifi, A.S. ++
ii
EI-Geddawi, M.E. EI-Hossiny, T.M. EI-Shazly, M.H.Y. Gadalla, M.A. Galal, G.M.A. ~ Hassan, M.E. Kamal, B.A. Nassef, A.M.O. Rashad, R.M. Saleh, Ch.A. Zeyada, Y.F.
" Faculty Dean + Conference Chairmanand Departmen'iHead ~* (2onferencr GeneralSecret~ ~ o n leave
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TABLE OF CONTENTS
PREFACE TABLE OF CONTENTS
ix
Section I: S Y S T E M D Y N A M I C S
I.l"
Autonomous Control of Complex Systems: Robotic Applications Jamshidi, M. (USA)
1.2'
A Simplified Inverse Kinematic Model Calculation Method for All 6R Type Manipulators Renaud, M. (France)
15
1.3
A Systematic Algorithm for Flexible Manipulators Simulation Megahed, S.M. and Hamza, K.T. 0ggyp0
27
1.4
Non-Linear Trajectory Control of Flexible Joint Manipulators Bravo, R.R. and Dokainish, M.A. (Canada)
37
1.5
Theoretical and Experimental Investigation of Integrated Structure/Control Design of High Speed Flexible Robot Arm Fanni, M. and EI-Keran, A.A. (Egypt)
45
1.6
Generalized Path Generation for a Mobile Manipulator Bayle, B., Fourquet, J.-Y. and Renaud, M. (France)
57
1.7
Fuzzy Guidance Control for a Mobile Robot Gharieb, W. and Nagib, G. (Egypt)
67
1.8
Fuzzy Logic Sliding Mode Controller for DC Drive Ibraheem, A.A., Bahgat, A. and Abdel Motelb, M.S. (Egypt)
75
1.9
Driver Modeling using Fuzzy Logic Controls for Human-in-the-Loop Vehicle Simulations Zeyada, Y., EI-Beheiry, E., EI-Arabi, M. and Karnopp, D. (Egypt, USA)
85
1.10
Optimal Active Suspension with Preview for a Quarter-Car Model Incorporating Integral Constraint and Vibration Absorber Abduljabbar, Z.S. and EIMadany, M.M. (S. Arabia)
95
1.11
Dynamic Ride Properties of a Roll-Connected Vehicle Suspension Rakheja, S., Ahmed, A.K.W., Liu, P. and Richard, M.J. (Canada, USA)
105
1.12
Bilinear Control Theory of Smart Damping Systems EI-Beheiry, E. (EgYp0
113
1.13
A Neural Adaptive Approach for Relative Guidance of Aircraft Shahzad, M., Slama, J.G. and Mora-Camino, F. (France, Brazil)
123
1.14
Simulation of Turbulence-Induced Vibration of Loosely Supported Heat Exchanger Tubes Hassan, M., Dokainish, M. and Weaver, D. (Canada)
131
1.15
Effect of Port Plate Silencing Grooves on Performance of Swash Plate Axial Piston Pumps Kassem, S.A. and Bahr, M.K. 0ggyP0
139
Section II: S O L I D M E C H A N I C S II.1"
The Modeling of Inelastic Compressibility and Incompressibility using the Viscoplasticity Theory Based on Overstress (VBO) Krempl, E. and Ho, K. (USA)
151
11.2"
On the Mechanical Integrity of Aeroengine Compressor Disc Assemblies Meguid, S.A. (Canada)
161
11.3
Effect of Void Growth on the Plastic Instability of Uniaxially Loaded Sheets Saleh, Ch.A.R. and Ragab, A.R. (Egypt)
173
11.4
Shakedown Analysis of an Infinite Plate with a Central Hole under Biaxial Tension Attia, M.S., Abdel-Karim, M. and Megahed, M.M. (Egypt)
185
11.5
Experimental and Analytical Investigations of Residual Stresses Induced by 195 Autofrettage EI-Shaer, Y.I., Aref, N.A., AbdeI-Kader, M.S., EI-Maddah, M. M. and Megahed, M.M. (Egypt)
11.6
A Mathematical Model for the Study of the Dynamic-Visco-Elastic Contact 205 Problems Ali-Eldin, S.S., Aly, M. F. and Mahmoud, F.F. (Egypt)
11.7
Elastic-Plastic Analysis of Notch Root Stress-Strain and Deformation Fields under Cyclic Loading Hammouda, M.M.I. and Seleem, M.H. 0ggyp0
215
11.8
Elasto-Plastic Thermal Stresses in Functionally Graded Materials Considering Microstrueture Effects Shabana, Y., Noda, N. and Tohgo, K. (Japan)
223
11.9
Fracture Modeling of a Cruciform Welded Joint During Weld Cooling Behnam, W.M., Alkhoja, J., Recho, N. and Zhang, X.B. (Egypt, France)
233
II.10
Application of Elastic-Plastic Fracture Mechanics Criteria to Specimens Cut From Plastic Pipes EI-Zoghby, A.A. and AI-Bastaki, N.M. (Egypt, Bahrain)
243
II.11
Acoustic Emission Detection of Micro-Cracks Initiation and Growth in Polymeric Materials Abo-EI-Ezz, A.E. 0ggyp0
253
11.12
Evaluating the Stress Concentration due to Elongated Defects in Welded Area Abd EI-Ghany, K.M., EI-Mahallawi, I. And EI-Koussy, M.R. (Egypt)
261
11.13
Mechanics of Fracture in Fibrous Metal Matrix Composites Bahei-EI-Din, Y.A. and EIrafei, A.M. (Egypt)
271
11.14
On Simultaneous Failure of Cross-Ply and Angle-Ply Composite Laminates 281 Khalil, M., Bakhiet, E. and EI-Zoghby, A. (Egypt)
11.15
Modelling of Shape Rolling using Three-Dimensional Finite Element Technique Abo-Eikhier, M. (Egypt)
293
II.16
Shape Optimization of Metal Backing for Cemented Acetabular Cup Hedia, H.S., Abdel-Shafi, A.A.A. and Fouda, N. (Egypt)
303
11.17
Properties of Cementitious Composites Containing Non-Reeyelable Glass as a Fine Aggregate Shehata, I.H., Elsawy, A.H., Varzavand, S. and Fahmy, M.F. (USA)
313
Section III: M A T E R I A L S C I E N C E III.1
A Shape Memory Behavior Newly Revealed in Cu-Be Alloy Masoud, M.I., Naito, K., Era, H. and Kishitake, K. (Japan)
323
111.2
Poisoning of Grain Refinement of Some Aluminium Alloys AbdeI-Hamid, A.A. and Zaid, A.I.O. (Jordan)
331
111.3
Effect of Shot Peening on the Fatigue Strength of 2024-T3 Aluminum Alloy in the Unwelded and Welded Conditions Zaid, A.I.O., Ababneh, M.A. and AI-Haddid, T.N. (Jordan)
339
111.4
Creep Behavior of Solid Solution Alloys: Role of Dynamic-Strain Aging Soliman, M.S. and Almakhdoub, S.A. (S. Arabia)
347
xii 111.5
Influence of Intense Plastic Straining on Room Temperature Mechanical Properties of AI-Cu-Li Base Alloys Salem, H.A. and Goforth, R.E. (Egypt, USA)
357
111.6
Structure and Engineering Properties of Some Ductile Irons Refaey, A., H a f ~ M. and Fatahalla, N. (Egypt)
369
111.7
Conventional Versus Thin Slab Casting: A Numerical Simulation Approach for the Comparison of Microstructural Properties Youssef, Y.M., Megahed, G.M. and Lee, P.D. (Egypt, UK)
381
111.8
Simulation and Control of the Cooling of Hot Rolled Steel Wire Rod Labib, H.F., Megahed, G.M., EI-Mahallwi, I., Dashwood, R.J. and Lee, P.D. (Egypt, UK)
389
111.9
Effect of Alumina Additions on the Mechanical Behavior of PM MMC with Low Strength Matrix Mazen, A.A. and Ahmed, A.Y. (Egypt)
397
III.10
Evaluation of Damping Behavior of Spray Deposited SiC Particulates Reinforced Al Composites Abo EI-Naser, A.A. (Egypt)
407
III.11
Bonding and Properties of Explosively Compacted Copper Powder and Polypropylene Granules Hegazy, A. Abosree. (Egypt)
415
Section IV: M A N U F A C T U R I N G P R O C E S S E S IV.l*
New Trends in CIM: Virtual Manufacturing Systems for Next Generation Manufacturing Choi, B.IC and Kim, B.H. (South Korea)
425
IV.2"
Intelligent Machining Systems: Challenges and Opportunities Teltz, R. and Elbestawi, M.A. (Canada)
437
IV.3
CAM System for Efficient Generation of Part-Programs for Wire-EDM EI-Midany, T.T., EI-Keran, A.A. and Radwan, H.T. (Egypt)
447
IV.4
On the Prediction of Surface Roughness in Turning using Artificial Neural Networks EI-Sonbaty, I. and Megahed, A.A. (Egypt)
455
IV.5
On the Adjustment and Validation of Finite Element Models for Hemispherical Cup Forming Wifi, A.S., Ragab, M.S., Hussein, A.A. and Abdel-Hamid, A. (S. Arabia, Egypt)
467
xiii IV.6
Development of Spot-Weld Bonded Low Carbon Steel Damping Sheets Darwish, S.M.H and Ghanya, A. (S. Arabia, Egyp0
477
Section V: D E S I G N AND T R I B O L O G Y V.I"
Contribution of CAD-CAM and Reverse Engineering Technology to the Biomedical Field Hosni, Y.A. (USA)
491
V.2~
Design for Manufacture and Assembly (DFMA): Concepts, Benefits and Applications Bayoumi, A.M.E. (USA)
501
V.3
Inversion of Frusta as Impact Energy Absorbers Aljawi, A.A.N. and Alghamdi, A.A. (S. Arabia)
511
V.4
Effect of Laser Surface Treatment and Work Hardening on the Fretting Wear Resistance of Zr-2.5Nb Alloy at High Temperature Atria, M.H. (Canada)
521
V.5
In-Vitro Model to Evaluate the Effect of Attachment Designs on Stresses Transferred to Surroundings in Implant-Retained Overdentures EI-Wakad, M.T. (Egypt)
531
V.6
Assessment of Reliability Parameters for Maintenance Based Units using Linearized Weibull Model Mostafa, A.A-F. and Khattab, A.A. (Egypt)
539
Section Vl: I N D U S T R I A L E N G I N E E R I N G VI.I"
Optimal Replacement of Components Subject to Degradation Elsayed, E.A. (USA)
553
VI.2
Feature Recognition Algorithm for Process Selection McCormaek, A.D. and Ibrahim, R.N. (Australia)
563
VI.3
Development of a Genetic Algorithm Based on Fuzzy Logic Sets for Solving Facility Layout Problems Ramadan, M.Z. and Abou EI-Ez, S.R.S. (Egypt)
571
VI.4
Manufacturing Cell Formation Problem: A Graph Partitioning Approach Selim, H.M. (UAE)
579
VI.5
An Investigation of the Group Scheduling Heuristics in a Flow-Line Cell Hozayyin, A.S., Badr, M.A. and Helal, M.E. 0ggyp0
591
xiv VI.6
Scheduling Approach for Mixed Networked Batch Processes Soltan, H.A.M. (Egypt)
603
VI.7
A Fuzzy Reactive Approach for the Crew Rostering Problem E! Moudani, W., Brochado, M.R., Handou, M. and Mora-Camino, F. (France, Brazil, Niger)
611
Vl.8
Effects of Static and Dynamic Mean Variation on the Process Capability Mohammed, H.H. (Egypt)
621
VI.9
Short Term Management of Water Resource Systems Faye, R.M., Sawadogo, S., Gonzalez-Rojo, S. and Mora-Camino, F. (France, Senegal, Mexico)
631
AUTHOR INDEX
641
SUBJECT INDEX
643
LIST OF PARTRICIPANTS
645
KeynotePaper
Section I
SYSTEM DYNAMICS
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Current Advances hi Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, Februa O' 15-17, 2000
AUTONOMOUS CONTROL OF COMPLEX SYSTEMS: ROBOTIC APPLICATIONS
Jamshidi, M. Department of Electrical and Computer Engineering and Autonomous Control Engineering - ACE Center, The University of New Mexico
ABSTRACT One of the biggest challenges of any control paradigm is being able to handle large complex systems under unforeseen situations. A system may be called complex here if its dimension (order) is too high and its model (if available) is nonlinear, interconnected, and information on the system is uncertain such that classical techniques can not easily handle the problem. Soft computing, a collection of fuzzy logic, neuro-computing, genetic algorithms and genetic programming, has proven to be a powerful tool for adding autonomy to many complex systems. For such systems the rule base size of soft computing control architecture will be nearly infinite. Examples of complex systems are power networks, national air traffic control system, an integrated manufacturing plant, etc. In this paper a new rule base reduction approach is suggested to manage large inference engines. Notions of rule hierarchy and sensor data fusion are introduced and combined to achieve desirable goals. New paradigms using soft computing approaches are utilized to design autonomous controllers for a number of robotic applications here. I. INTRODUCTION Since the launching of Sputnik in the former Soviet Union, extensive progress has been achieved in our understanding of how to model, identify, represent, measure, control, and implement digital controllers for complex large-scale systems. However, to design systems having high MIQ| (Machine Intelligence Quotient, registered trademark by Lotfi A. Zadeh), a profound change in the orientation of control theory may be required. Currently, one of the more active areas of soft computing is fuzzy logic, and one of the more popular applications of fuzzy logic is fi~zzy control. Fuzzy controllers are expert control systems that smoothly interpolate between hard-boundary crisp rules. Rules fire simultaneously to continuous degrees or strengths and the multiple resultant actions are combined into an interpolated result. Processing of uncertain information and saving of energy using common sense rules and natural language statements are the basis for fuzzy control. The use of sensor data in practical control systems involves several tasks that are usually done by a human in the * This work was supported, in parts, by NASA Grant number NCCW-0087 This paper represents a set of applications of soft computing approaches to complex systems such as mobile robots, flexible arm, etc. The structure of the paper is as follows: Section 2 gives a brief introduction into autonomy through soft computing. Section 3 introduces two notions ot sensory fusion and rule hierarchy. Section 4 constitutes a few applications of autonomous control for complex systems through soft computing approaches. Conclusions are described in Section 5.
4
Current Advances in Mechanical Design and Production, MDP- 7
decision loop,e.g., an astronaut adjusting the position of a satellite or putting it in the proper orbit, a driver adjusting a vehicle's air-conditioning unit, etc. All such tasks must be performed based on the evaluation of data according to a set of rules in which the human expert has learned from experience or training. Often, if not all the time, these rules are not crisp, i.e., some decisions are based on common sense or personal judgment. Such problems can be addressed by a set of fuzzy variables and rules which, if properly constructed, can make decisions as well as an expert.
I
$ u
No. Rules -390.625 No. Rules = "/5
Fig. 1. The sensory fusion - rule hierarchy structure for fuzzy control systems
2. AUTONOMY THROUGH SOFT COMPUTING Soft Computing is an umbrella terminology used to refer to a collection of intelligent approaches such as neural networks (NN), fuzzy logic (FL), genetic algorithm (GA), genetic programming (GP), neuro computing, etc. Soft computing techniques can allow one to design an a u t o n o m o u s controller through learning (NN), optimization (GA) or reasoning (FL). Neural networks, genetic algorithms and genetic programming are augmented with fuzzy logic-based schemes to enhance artificial intelligence of automated systems. Such hybrid combinations exhibit added reasoning, adaptation, and learning ability. In this article, three dominant hybrid approaches to intelligent control are experimentally applied to address various robotic control issues, which are currently under investigation. The hybrid controllers consist of a hierarchical NN-fuzzy controller applied to a direct-drive motor, a GA-fuzzy hierarchical controller applied to position control of a flexible robot link, and a GP-fuzzy behavior based controller applied to a mobile robot navigation task. Various characteristics of each of these hybrid combinations are discussed and utilized in these control architectures. The NN-fuzzy architecture takes advantage of NN for handling complex data patterns, the GA-fuzzy architecture utilizes the ability of GA to optimize parameters of membership functions for improved system response, and the GP-fuzzy architecture utilizes the symbolic manipulation capability of GP to evolve fuzzy rule-sets. 3. SENSORY FUSION AND RULE HIERARCHY In many real-life problems the number of sensory data is way too many for any reasonable sized rule base. For example for a 4-variable system with 5 linguistic labels per variable, 625 rules are nominally needed. For a 10-variable process, the size of the rule base would be over 9.7 million. In other words, the size of the rule base would quickly approach infinity as the number of variables increase. In an effort to reduce the size of the rule base, many approaches are possible. Two of these approaches are sensory fusion and rule hierarchy which have been detailed in the book by the author [5]. Sensory fusion employs a linear combination of the two variables x and y to form a fused variable z= a. x + b. y, where a and b are arbitrary
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parameters. In such a way the number of sensory inputs to the inference engine would be reduced in half. Another approach is to classify the rule of the rule base in groups in according to the roles they play in the performance of the system, e.g. rules for stability, rules for tracking, rules for optimization, etc. In this paradigm, the two most critical variable, say for stability, could first be input to the first sub-set of rules, then the output of the first sub-group would be joining second most important set of sensory variables to be inputted to the second hierarchy of rules, etc. In practice, it can be assessed that neither method by themselves can reduce the rule set substantial when the number of variables is a large number, say 6 or higher.
|
of FIs
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Fig. 2. GP- Fuzzy Mobile Robot Control Architecture A third possibility is to combine the rule hierarchy and sensory fusion jointly. Figure 1 shows this structure for rule base reduction. Jamshidi [5,6] has applied this approach for the balancing of an inverted pendulum with a glass of wine on top of it.
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Fig. 3. Hierarchical decomposition of mobile robot behavior 4. AUTONOMOUS CONTROL IN ROBOTICS In this section many applications of soft computing towards rendering various degrees of autonomy in control systems will be presented. There are too much details in each of these
case studies that we can cover in this short paper. Relevant references will help the readers in all case. Since 1993, University of New Mexico's CAD Laboratory and later on the ACE Center have been active in designing the autonomous behavior of many configurations of robots.
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F_uzze-GP Control In this application a real-time fuzzy controller was designed to guide a
mobile robot through unstructured environments. The paradigm proposed in [ I 0] is based on a hierarchical fuzzy control architecture with respect to primitive and advance behaviors of the robot. The fuzzy rule bases for various behaviors are optimized using genetic programming. Figure 2 presents the architecture of GP-Fuzzy autonomous controller architecture. It is well known [6] that as the number of sensory variables increase the number of fuzzy rules will reach explosive levels. Therefore, for most complex applications of fuzzy control, such as mobile robot control, a rule reduction approach is a very necessary step of the design process. In an attempt to reduce the rules, here the functional operation of the robot is divided into primitive and advance (composite) behaviors, thereby leading to specialized sets of fuzzy rules to be used on needed basis at different levels. Figure 3 shows one possible hierarchical decomposition of a mobile robot behavior. As shown, goal-directed navigation of a mobile robot can be decomposed as a behavioral function of goal-seek (collision-free navigation) and route-follow (a path-planning type direction). These behaviors can be further decomposed into primitive behaviors such as avoid collision, wall-following, etc. linguistic rule-base. Valid behaviors should conform with the syntactic rules of construction. The above theory has been put into both simulation as well as real-time studies. Figure 4 shows the path of the mobile robot under an evolved steady-state genetic programming (SSGP) for coordination and behavior modulation. As compared with a hand-derived coordination and behavior, this approach resulted in a more direct path to the goal due to higher motivation applied to go-to-xy. The resulting path here is executed about 20% faster than the path taken via hand-derived coordination. It is also noted that the behavior modulation under evolved behavior is more complex. Near uniform bouts of competition and cooperation throughout the task evident in the decision-making, thus leading to similar amounts of behavioral influence for each primary behavior. In this section one of many applications of fuzzy logic and its hybridization with genetic programming is represented. The author's research colleagues at the ACE Center have many theoretical and experimental results which can not possibly be covered here. Interested readers can consult some recent references [5,10]. 12
12
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Fig. 4. GP Performance and Resulting Goal-Seeking Task Trajectory Ne.uro-Fuz~ Control another autonomous control approach stemmed from soft computing is
the hybridization of fuzzy logic and neural networks for real-time learning (system identification) and control. The proposed architecture for autonomous control is based on combining the learning capabilities of NNs and reasoning properties of fuzzy logic paradigms. A neural network learns about the behavior of the plant and uses that knowledge
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to modify the parameters of an adaptive fuzzy logic controller. The adaptability of the fuzzy controller is derived from a rule generation mechanism and modification of the scaling factor or the shape of the membership functions. The rule generation mechanism monitors the system response over a period of time to evaluate new fuzzy rules. Non-redundant rules are appended to the existing rule base during tuning cycles. The membership functions of the input variables are adjusted by a scaling mechanism. A multi-layer perceptron neural network classifies the temporal response of the system into various patterns according to oscillatory behavior, response overshoot, steady state error, etc. This information is used by the decision mechanism which determines the scaling factor of the input membership functions. Another neural network identifies the dynamic system, hence acting as a reference model. This model can be used to determine the stability of the new rules generated before applying to the real system. In order to implement this hybrid controller in real time, it is necessary to have substantial computing power. The TMS320C30 digital signal processor from Texas Instruments, with its powerful instruction set, high-speed number crunching capability, and its innovative architecture is ideally suited for such an application (Akbarzadeh, et al., 2000). There are commercially available boards based on TMS320C30 chips, which can be installed on a personal computer (PC). A board from DSP Research has been utilized for this purpose. The software for the control algorithm is developed in C-language and is compiled and downloaded to the DSP board. Collectively, these computing resources are used to implement the neuro-fuzzy controller architecture in real-time to control a direct drive motor used as a robot actuator. Figure 5 presents the neuro-fuzzy autonomous control system which was applied to a directdrive robot. Figure 6 shows the stabilized response of a typical direct drive motor of a scalar robot. The fuzzy logic controller has completely learned to control the direct drive motor after 300 sampling instances.
Fuzzy-GA Contro.!. Genetic algorithms are robust optimization routines modeled after the mechanics of Darwinian theory of natural evolution [3]. GAs do not require gradient evaluation, hence they are applicable to solving a great range of optimization problems including determination of optimal parameters of a fuzzy logic rule-set. Genetic algorithms have demonstrated the coding ability to represent parameters of fuzzy knowledge domains such as fuzzy rule sets and membership functions [4] in a genetic structure, and hence are applicable to optimization of fuzzy rule-sets. Here, several issues pertaining to such integration of the two paradigms are discussed and illustrated through an application on realtime hierarchical fuzzy-GA control of a single-link flexible robotic arm. To understand the actual mechanism of GAs, one may begin with its three most commonly used operators, namely: reproduction, crossover, and mutation. A member of a given population which has a higher fitness is given a higher chance to reproduce identical replicas of itself in an intermediate population. In this fashion, the optimization routine facilitates reproduction of higher fit individuals and hampers the reproduction of lower fit individuals. After reproduction, crossover randomly mates two individuals from an intermediate population and creates offspring which are made up of a random combination of their parent's genetic code. For each generation, the process of crossover is repeated for all individuals in the population. The population size is often a constant equal to the number of individuals in the initial population. The operations of reproduction and crossover create an environment where every generation benefits from the best genetic codes of the previous generations. However, if the building blocks for the optimal genetic structure is not in the initial
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population, these two genetic operators will be unable to find it. The last genetic operator, mutation, randomly mutates one or more of the values in the individual's genetic code in order to create diversity. The mutation operator allows for exploring n e w structures (directions of search) hence allowing the genetic optimization routine to invent new solution and finally locate the optimal solution even though the individuals in the initial population may not have contained the building blocks for the optimal solution. Delm~ TaT O Tempmm~ AdaptiveFizzy
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Fig. 6. Neuro-Fuzzy C o n t r o l S t a b i l i z i n g R e s p o n s e s o f a Direct Drive M o t o r When applying GA to optimization of fuzzy rule-sets, several questions arise. First is the design of the transformation function (the interpretation function) between the fuzzy knowledge domain (phenotype) and the GA coded domain (genotype). This is perhaps the most crucial stage of GA design and can significantly degrade the algorithm's performance if a poor or redundant set of parameters are chosen for a given optimization problem. Two important general categories of fuzzy expert knowledge consist of domain knowledge and meta-knowledge. Meta knowledge is the knowledge used in evaluating rules such as fuzzification (Scaling or Z, cut), rule evaluation (such as Min/Max) and defuzzification (such as Max Membership, Centroid, or Weighted Average) methods. Relatively little research has been performed to study the effect of optimizing the meta-knowledge [2]. Most of the current
Current Advances in Mechanical Design and Production, MDP-7
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research, as in this work, concentrate on optimizing parameters of the domain knowledge. The domain knowledge consists of the following two categories, 9 Membership Function: General Shape (Triangular, Trapezoidal, Sigmoidal, Gaussian, etc.), Defining points (Center, Max Right, Min Left, etc.) : 9 Rule-Base: Fuzzy Associative Memory, Disjunctive (OR) and Conjunctive (AND) operations among antecedents in the rule base. Even though various methods exist to encode both rule-base and membership functions in one GA representation, such coding can have several potential difficulties. In such situation, in addition to the level of complexity and large number of optimization parameter, the problem of competing conventions may arise and the landscape may unnecessarily become multi-modal. This is an important problem since there are often several (or many) fuzzy rulesets which can represent a given nonlinear function. This means that there are more than one optimal solution to a given optimization problem which raises the issue of multi-modality for fuzzy logic systems, or more specifically competing conventions where different chromosomes in the representation space have the same interpretation in the evaluation space. When designing the interpretation function, therefore, the coding needs to contain fewest possible parameters to avoid the problem of dual representation, and yet the coding needs to have enough complexity to contain all possible optimal or near optimal solutions. Evolutionary approaches such as Niched GA [2] are designed to search in complex multimodal landscapes. As a result, in the present approach, this problem is attended to by limiting the optimization parameter space to membership function parameters only. This was a design decision which was made considering the following two considerations. 9 The problem of multi-modality is introduced when the GA string contains both parameters of membership functions and rules. 9 In control of most physical systems, rules can often be derived either intuitively or through operator experience. The ambiguous and fuzzy portion of a knowledge base is often the membership functions. For simplicity in coding the simulation and the real-time algorithms, only triangular membership functions are coded for optimization here. Figure 7 illustrates a triangular membership function whose three determining parameters (a,b, c)are shown. Assuming a normalized membership function, the three parameters are real numbers between-1 and 1. The coding in GA is performed as follows, the real parameter an is first mapped to an n-bit signed binary string where the highest bit represents the sign. This way, the parameter a can take on 2 n different values. Then the binary number is aggregated with other n-bit binary numbers to construct the phenotype representation. l
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Fig. 7. A Triangular membership functions with its three parameters
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Current Advances in Mechanical Design and Production, MDP- 7
The second issue which arises is how to utilize initial expert knowledge for a better and faster convergence. In other search routines such as hill-climbing, it is clear that starting from a "good" point can significantly improve computation time needed for convergence to an optimal solution. However, the conventional GA applications generate a random initial population without using any a-priori expert knowledge. This, in general, will provide a more diverse population while sacrificing convergence time. This convention can indeed be adequate if there is no a-priori knowledge as to where a "good" solution may exist. However, in fuzzy logic applications, there is usually access to some expert knowledge which, even though it may not be the optimal solution, is often a reasonably good solution. Sehultz and Grefenstette [9] addressed the problem of incorporating a priori knowledge by introducing two types of populations, homogeneous and heterogeneous. The homogeneous population consists of individuals created randomly while their string is augmented with the same priori rules. In this sense, they are all identical and hence homogeneous. He concluded that a trade-off exists between manual knowledge and machine learning. The heterogeneous population consists of members which are not identical. This is also referred to as the "seeding" technique. In [7], the process of seeding the initial population with one or more experts' knowledge is proposed. The few seeded chromosomes have the chance of reproducing through mutation and crossover with other randomly generated chromosomes in the population. This method improves the performance of GA by providing the genetic population with a set of highly fit building blocks, as compared with GAs starting with random initial populations. However, such population still requires a large number of iterations before convergence since the "useful" schemata exist in only one or few seeded members and can only be reproduced as fast as the rate of reproduction. Akbarzadeh [1] proposed the grand-parentingscheme where the initial population is comprised of mutations of the "knowledgeable" grandparent. This scheme takes advantage of expert knowledge while maintaining diversity necessary for an effective search. Through grandparenting, an expert's a priori knowledge can be utilized to improve fitness of GA's initial population, thereby increasing the speed and performance of the search routine. In the present approach, the method of grand-parenting is used to improve the convergence rate of the GA optimization process. The third issue is defining a fitness function. A fitness function is a very important aspect of GA design since it determines the direction of the search. Fitness functions come in as many different forms as the systems which they are optimizing. In general, for a lumped parameter system (such as a flexible robot arm), parameters such as control effort u(t), rise-time tr, overshoot y and steady-state error ess are usually incorporated in a quadratic fitness function. Often, constant multipliers define the relative degree of importance which is given to a certain parameter compared to others. The above concepts were applied to controller optimization of a flexible link robotic system as shown in Figure 8. The flexible link can be represented by a distributed-parameter system with spatial as well as temporal parameters. In other words, the states of a flexible robotic system are functions of both space and time. This complicates the modeling of the system and, consequently, the process of designing the controller. Due to the complexity of a mathematical representation for such systems, fuzzy logic is considered an attractive alternative to their control. One of the issues in development of fuzzy controllers is determining faithful expert knowledge. Expert knowledge, however, is difficult to produce since there is often no human expert to consult and training a human expert may not be a
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feasible alternative due to cost and other practical considerations. Furthermore, human psychological issues may prohibit a faithful reproduction of a rule-base from an expert. In addition, the unstructured operating environments associated with space and waste handling projects require the robot controller to also adapt to changing conditions. In the process of designing fuzzy rule sets, membership functions are often chosen through an ad hoe process of random selection and evaluation. As a viable alternative, good results have been achieved by employing genetic algorithms to tune membership parameters within a fuzzy controller's knowledge base [8]. Genetic algorithms equip the fuzzy controller with some evolutionary means by which it can improve its rule-base when faced with inadequate a-priori expert knowledge or varying circumstances in its operating environment.
i
....
I
Fig. 8. Autonomous Control Architecture Through a Fuzzy-GA Approach The GA-optimized architecture proposed here needs several parameters defined. Spatial variables are fuzzified for use in a rule base at the higher level of hierarchy. Other parameters in the knowledge base are not allowed to vary. The fitness criterion used to evaluate various individuals within a population of potential solutions was based on the error e(t), effort u(t) and 3, as an inverse square function. Consequently, a fitter individual is an individual with a lower overshoot and a lower overall error (shorter rise time) in its time response. Here, results from previous simulations of the architecture are applied experimentally. The method of grand-parenting [ 1] was used to create the initial population. Members of the initial population are created through mutation of the knowledgeable grandparent(s). As a result, a higher fit initial population results in a faster rate of convergence as is exhibited in Figure 9. Figure 9a shows the time response of the GA-optimized controller when compared to previously obtained results through the non-GA fuzzy controller. The rise time is improved by 0.34 seconds (an 11% improvement), and the overshoot is reduced by 0.07 radians (a 54% improvement). The average fitness of each generation is shown in Figure 9. A total of 10 generations were simulated. The mutation rate for creating the initial population was set at 0.1. This value was chosen to increase diversity among members of initial population. GA depends on this diversity to exploit a large number of differing path of solutions in parallel. The mutation rate throughout the rest of the simulation, however, was set to 0.01. Since a high mutation rate delays convergence. The probability of crossover was set to 0.6. Initial experimental results demonstrate that the GA-learned controller is able to control the actual experimental system as in Figure 9.
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Current Advances in Mechanical Design and Production, MDP- 7
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Fig. 9. Computer Simulation and Experimental Results of GA-Fuzzy Autonomous Control System. (a) Simulation, (b) Fitness Function Behavior and (c) Experimental Results. The hardware used to implement the above algorithms is the same as was explained in the previous section on neuro-fuzzy control with a few modifications pertaining to flexible robot control such as a tip end position sensor and several strain gauges distributed evenly across the length of the flexible beam. Control update was performed at 250 Hz. 7. CONCLUSIONS The basic theme of this paper was autonomous control and autonomy through several architectures of soft computing. A number of robotic applications were used to illustrate these architectures. These autonomous controllers are simple to implement in a laboratory environment on either a PC or on a chip-level board. Soon, autonomous control through intelligent paradigms technology will be a matter of economy and not controversy. It features applications in a wide variety of fields such as control, pattern recognition, medicine, finance, marketing, etc. should be given serious considerations as additional tools for the solution of problems which are most suitable for this technology, i.e. problems where a mathematical model is neither available nor feasible. Applications to complex systems require careful considerations of the system's model, its structure, the behavior and means of sensory data, and rules specialization and hierarchy. Finally, new avenues should be opened for new software design and analysis of control systems utilizing the power and efficiency of all tools such as fuzzy logic, neural networks and genetic algorithms. ACKNOWLEDGEMENTS The author wishes to thank his peers and associates; Dr. M. Akbarzadeh, Dr. E. Tunstel, Dr. K. Kumbla, Mr. A. EI-Osery and Ms. Sandi Avrit for their help in preparation of the manuscript. REFERENCES Akbarzadeh, M. R., Fuzzy Control and Evolutionary Optimization of Complex Systems, PhD Dissertation, University of New Mexico, (I 998).
Current Advances in Mechanical Design and Production, MDP- 7
2. 3. 4. 5. 6. 7. 8. 9. 10.
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Akbarzadeh, M., Kumbla, K., Tunstel, E. Jr., and Jamshidi, M. "Soft Computing for Autonomous Robotic Systems," International Journal on Computers in Electrical Engineering", Vol. 27, (2000), (to appear). Goldberg, D. E., "Genetic Algorithms in Search, Optimization and Machine Learning," Addison-Wesley, (1989). Homaifar, A. and McCormick, E., "Simultaneous Design of Membership Functions and Rule Sets for Fuzzy Controllers Using Genetic Algorithms," IEEE Transactions on Fuzzy Systems, Vol. 3, No. 2, pp. 129, (1995). Jamshidi, M., "Large-Scale Systems - Modeling, Control, and Fuzzy Logic", Prentice Hall Series on Environmental and Intelligent Manufacturing Systems (M. Jamshidi, Ed.),Vol. 8., (1996). Jamshidi, M., "Fuzzy Control of Complex Systems," Soft Computing, Vol. 1, No. 1, pp. 42-56, (1997). Lee, M. A. and Takagi, H., "Embedding Apriori Knowledge into an Integrated Fuzzy System Design Method Based on Genetic Algorithms," Proceedings of the 5th IFSA World Congress, (1993). Lee, M. A. and Takagi, H., "Integrating Design Stages of Fuzzy Systems Using Genetic Algorithms," Proceedings of the 1993 IEEE International Conference on Fuzzy Systems, San Francisco, CA, pp. 612-617, (1993). Schultz, C. and Grefenstette, J.J., "Improving Tactical Plans with Genetic Algorithms," Proceedings of the 2nd International Conference on Tools for AI, Herndon, (1990). Tunstel, E. W. and Jamshidi, M., "Intelligent control and evolution of mobile robot behavior," in Applications of Fuzzy Logic - Towards High MIQ(TM) Systems, (Jamshidi, M., Titli, A., Zadeh, L. A., and Boverei, S. - eds.), Prentice Hall Series on Environmental and Intelligent Manufacturing Systems (M. Jamshidi, Ed.), Vol. 9, Chapter 1.
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
15
A SIMPLIFIED INVERSE KINEMATIC MODEL CALCULATION METHOD FOR ALL 6R TYPE MANIPULATORS
Renaud, M. LAAS-CNRS, 7, Avenue du Colonel Roche, 31077 Toulouse, Cedex 4, France
ABSTRACT An inverse kinematic model of a manipulator - equipped with an end effector- is a function which allows to calculate a manipulator configuration corresponding to a given end effector location (position and orientation). In this paper, we consider the case of 6R type manipulators having a single open kinematic chain structure and six revolute joints R. The first method, which brought back the determination of this model to the computation of the roots of a polynomial equation whose degree - equal to sixteen - is minimal, was proposed in 1988 by Lee and Liang. Nevertheless, Lee and Liang's technique was extremely complicated and necessitated divisions which prevent it from being used in any case. Here, we present a simplified method for this calculation, which does not involve any division; as a consequence it works in any case. We present also an example of a particular, but non academic, manipulator for which we actually obtain sixteen different real configurations corresponding to a particular given end effector location; this result demonstrates that the degree of the previous polynomial equation cannot be reduced. KEYWORDS Inverse Kinematic Model, Configuration, Location, Generalized Coordinates, Operational Coordinates. I. INTRODUCTION In this paper, we consider manipulators having a single open kinematic chain structure equipped with an end effector (gripper or tool). An Inverse Kinematic Model (IKM)is a function which allows to calculate one configuration of this manipulator corresponding to a given end effector location (position and orientation). The configuration is defined by a set of generalized coordinates and the location by a set of operational ones. In fact, as we will see later in this paper, each IKM associates to one real location (i.e. defined by real operational coordinates) one complex configuration (i.e. defined by complex generalized coordinates). We study the case - well known to be the most difficult among all the six joints manipulators of the 6R type manipulators having six revolute joints R, and for which several IKM exist. Nevertheless each IKM is defined on a particular subspace of the location space; as a consequence the number of complex configurations, corresponding to one given location, is function of this particular location. The number of real configurations - which is, in any case, less than the previous one - is also function of this particular location. This last number-
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Current Advances in Mechanical Design and Production, MDP- 7
which is a very complicated function of the given location - and the number of IKM - which is independent of this location - must not be confused. Here, we are looking for the IKM and, more particularly, for their number. One of the earliest attempts to calculate the IKM has been made by Pieper [1 ] who, in 1968, demonstrated, by naive elimination, that their number was at most 524,288. Several years after, in 1976, Roth [2] conjectured that this number was at the most of thirty two. In 1980, Duffy and Crane [3] proved this conjecture by combining spherical geometry and Sylvester dialytic elimination [4,5]. Afterwards, in 1985, Tsai and Morgan [6] used homotopy continuation and produced several examples of 6R type manipulators for which the number of IKM was equal to sixteen and supposed this result general for all the 6R type manipulators. The demonstration of this result was established for the first time in 1988 by Lee and Liang [7,8] who have, moreover, proposed a computational method of these sixteen IKM. This technique used, extremely complicated, involves divisions and as a consequence fails in some cases. Several solutions were proposed to simplify this technique, but they all share the same drawback, namely the use of divisions. See for example the works developed by Raghavan and Roth, in 1989, [9], Manocha and Canny, in 1992 and 1994, [10,11], Kohli and Osvatic in 1992, [12], or Mavroidis, Ouezdou and Bidaud, in 1994, [13] . . . In this paper, we propose a new method, much more simpler than the previous ones, and which presents no singularities at all; therefore, it applies without restrictions to any kind of 6R type manipulators. Nevertheless our method relies on the original method of these authors and shares the following original ideas: it transforms the 6R open single kinematic chain into an hypothetic 7R closed single kinematic chain, 9
it uses a vector, specified in the sequel, already considered by these authors, it uses the subtle artifice which consists of writing in an affne way cosine and sine (here of the angle q6 ) as a function of the half angle tangent (here of the angle ~-),
9
it uses the dialytic calculation.
Our method consists of bringing back the calculation of all the IKM to the computation of the roots of a sixteen degree polynomial equation whose unknown is a generalized coor- dinate ql
half angle tangent (here of y ). Because of the impossibility to express these roots literally our method will be only numeric, although it allows the literal writing of this equation coeffcients. Nevertheless this literal writing is generally too complicated to be of any use. For a particular 6R type manipulator, the algorithm we have developed allows to compute numerically all the complex configurations corresponding to a particular given end effector location. Among these configurations, we extract the real ones. Finally, we give an example of a very simple, but non academic, particular 6R type manipulator- the RMS of the American shuttle - for which we obtain sixteen real configurations corresponding to a particular given end effector location. This demonstrates that the previous polynomial equation degree cannot be reduced and, as in Lee and Liang's calculation, allows us to prove that our method is minimal.
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2. D E F I N I T I O N OF THE M A N I P U L A T O R T O P O L O G Y
The considered manipulator consists of six rigid links jointed in a single open kinematic chain with revolute joints R. The first link in the chain is connected, via the first joint, to the base and the last one supports the end effector. Links are designated by Ci and joints by L,, i being an index increasing from one to six starting from the base; thus (:6 designates the end effector. The Fig. 1 represents an example of a particular 6R type manipulator: The RMS of the American shuttle.
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Fig. 1: The R~IS 6R type manipulator
3. D E F I N I T I O N OF T H E M A N I P U L A T O R G E O M E T R Y
The orthogonal affine frame R0 = (O0,_Xo, Y-o'z0) is embedded in the base, named link Co and an orthogonal affine frame Ri = (Oi, x i, Yi' zi) is embedded in each link C,i (see. Fig. 2) using the modified Denavit Hartenberg (mDH)iterative procedure [14] [15] [16] (i=l, 2 , . . . 6): 9 0i-1 is the foot of the perpendicular common to the axes of Li-1 and Li, located on the axis of Li-1, while f~i is the foot of this perpendicular located on the axis of Li, 9 xi_ 1 is the unit vector along this perpendicular, oriented from the axis of Li-1 to the one of Li, 9 zi_ ~ is the unit vector (of arbitrary orientation) along the axis of Li-1, 9 -Y~-I is a unit vector such that Ri be direct. To initialize this procedure we consider that L0 is a fictitious joint whose axis is identical to the one of L0 and we fix O0 = O1..XIoreover we assign z__0 = _z1. To end this procedure, we choose 06 = f~6 and we assign z__6 as the unit vector of the axis of L6, with an arbitrary orientation. The location of Ci with respect to its antecedent C~_~ is defined by the four mDH parameters of the Fig. 2: 9 ai-1 algebraic angle between zz_ 1 and z i measured around xi_l, 9 ai-l length of the perpendicular common to the axes of Li-1 and Li (ai-1 3> 0), 9 Oi = q~ algebraic angle between x,_~ and x i measured around z__i, 9 ri = fliOi, algebraic distance measured along z i. Note that a0 = a0 = r~ and r6 = 0, and let Ci = cos 0i = cos qi and & = sin Oi= sin qi. The homogeneous transform matrix T~_I,, [14] [15], from R,-1 to R~, is then defined by:
18
Current Advances in Mechanical Design and Production, MDP- 7 Li.i Z-i~i.i I i xi Oi Oi ai- I
Oi.!
ri
Fig. 2: Modified Denavit Hartenberg parameters Ci
-Si
0
cosc~i_ 1 S/
COSCei-I Ci sin C~i_ 1 Ci
-sincei_l cos O~i- 1
T i - 1,i =
ai-1 - r i - 1 sinai_l
sin ai-1 Si r i - 1 sin a i - 1 " 0 0 0 1 R e m a r k : these mDH parameters must not be confused with the classical Denavit Hartenberg parameters (cDH) used by numerous authors [17] [18] [13]...; modified parameters are much more general than classical ones as, contrary to the classical ones, they can be generalized to manipulators having an open complex kinematic chain structure. Among the previous four mDH parameters three are constants (ai-1, a i - 1 , ri) and one is variable (0i = qi). The geometry of a particular manipulator is then defined by the table of the twenty four mDH parameters among whose eighteen are constants and six variables. For example the table corresponding to the RMS 6R type manipulator is indicated hereafter:
i
1
2
3
4
ai-1
o 0
0
o as
a3
Oi ri
ql 0
q2 0
q3 0
q4 0
.
5
.
.
.
.
6
a4
0
q5 0
q6 0
, ,
4. DEFINITION OF THE MANIPULATOR CONFIGURATION AND OF THE END E F F E C T O R LOCATION The manipulator configuration q is defined by six generalized coordinates: q = (ql q2 . . . q6) t and its end effector location x by six operational coordinates: x = ( x l x~ . . . x6) t. Numerous choices of operational coordinates can be made. Our proper choice is specified in the sequel.
4.1 Transformation of the open 6R type manipulator into an hypothetic closed 7R type one
Let L7 be an hypothetic joint whose axis is defined by point 06 and unit vector z 7 - Y_Y-6" Then a6 = - r / 2 and a6 = 0. Moreover 06 is located in the right place according to the mDH iterative procedure. Furthermore f~7 - 06. Now let us consider the perpendicular
Current Advances in Mechanical Design and Production, MDP-7
19
common to the axes of LT and Lo let O7 and l~0 be, respectively, the feet of this perpendicular on the axes of L7 and L0. and J:7 the unit vector of this perpendicular oriented from the axis of L7 to the one of L0. The location of R6, with respect to R0, (i.e. the location of the en(l effector with respect to the base), is defined by the six independent parameters" 07, /'7. 07. a7. 00, r0,
which correspond to the mDH parameters of the homogeneous transform matrix from /~6 to R7 (for 07 and rT) and from RT to R0 (for aT, aT, 00 and ro). We choose these parameters as operational coordinates. Remark: let us particularize R0. for each end effector location, by choosing x_.0 = x_T. This leads to impose 0o = 0 and the previous location is now only defined by five independent parameters.
5. INVERSE KINEMATIC MODEL CALCULATION 5.1 Loop Equation To each set of the five previous independent parameters correspond several sets of generalized coordinates ql, q2. q3, q4, qs, q6 that verify the loop equation: T61(ql)T12(q2)... T56(q6) -- E where E represents the four order unit matrix. Let us remark that: T6~(q~) = T67TT~(q~) with" c07 -sO7 0 0 C1 - $1 0 a7 COz7S1 c0~7C1 - s o l 7 - r l so~ T 1 r7 and Tzl(ql) = 0 0 -sOTS1 sa~ S1 cat 7"lCaT 0 1 0 0 0 1 the loop equation using the "preferential" index p [18] such that' and as a consequence p 3" T61 (ql )T12(q2)T23(q3) = T65(q6)T54(qs)T43(q4). The first member of the previous equation corresponds to the sub chain A and depends only on q~, q2 and q3. while the second one is related to the sub chain B and is only function of q6. q5 and q4. We are going to eliminate the q3 variable by taking into account only the first two columns of the T 0 matrices. Let To be the 4 x 2 matrix constituted by these first two columns. Then the matrix T23 is constant and we can write successively"
T67 =
0 0 -sO7 -c07 0 0 It is possible to write p - E(~). with n - 6
T6, (q, )T12(q2)T'23 = T65(q6)T54(qs)T43(q4), T6~ (q~)T13(q2) = T65(q6)T53(q5, q4), T63(q,. q2)= T6a(q6, qs, q4). The last matrix equation defines a system of six equations, in five unknowns q~, q2, q6, qs, q4, related by one constraint determined hereafter. Let us write" o~ a b
o
The six equations to be solved are then:
1
.
20
Current Advances in Mechanical Design and Production, MDP- 7
c~(ql,q2)
= =
a(q,, q2)
c~(q6,qs, q4)
/3(q,,q2)
a(q6, qs, q4)
b(ql, q2)
= =
/3(q6, qs, q4)
7(q,,q2)
b(q6, q5, q4)
c(q,, q2)
= =
7(q5, q4)
c(qs, q4),
while the constraint relation is given by: (c~? + ~2 + 72)(q,, q2) = (c~2 + f12 + 72)(q6, qs, q4) = 1.
5.2 Resolution of The Previous Six Equations System Let P-63 be a vector whose origin is Os and extremity 03. In order to solve the previous six equations system we propose to solve an augmented system in which appear eight other equations. To this aim, let us add eight other scalars (u, v, w, l, m, n, s and a) to the six already introduced (c~,/3, 7, a, b and c). The first three represent the vector z__3 x P-63 1 2 components, with respect to Rs, the next three the v e c t o r ~P_63z_3(z3.P63)P63 components, with respect to the same frame, and the last two are: s = z__3.P_s3 = c~a + fib + 7c and 1 2 = ~1 (a2 + b2 + c2)" = ~/2_63
R e m a r k : the ~p_~3z3 ~ 2 -- (Z_3.P_63)P_63vector was already introduced by Lee and Liang [7] [8] and used by Manocha and Canny [10]; nevertheless, as these authors, we were unable to explain its introduction. Then, the a u g m e n t e d system to be solved is given as follows: c~(q,,q2)
a(q,, q2) u(qi, q2) l(ql,q2)
= =
c~(q6,qs, q4)
/3(q,,q2)
=
/3(q6, qs, q4)
b(q6, qs, q4) v(qs, qs, q4)
=
l(q6, qs, q4)
b(q,, q2) v(q,, q2) m(q,,q2)
=
=
a(qs, qs, q4) u(q6, qs, q4)
= =
7(ql,q2)
=
7(qs, q4)
c(q,, q2) = w(ql, q2) = m(q6, qs, q4) n(ql,q2) =
c(qs, q4) w(qa, q4) n(qs, q4)
s(ql, q2) = s(q5, q4) a(q,, q2) = a(q5, q4). R e m a r k : it seems judicious to consider the following six equations in four unknowns sub system: 3'(q1,q2)
=
n(ql,q2)
= n(q5,q4)
7(qs,q4)
c(ql,q2) s(ql,q2)
= c(q5,q4) = s(qs,q4)
w(ql,q2) a(ql,q2)
= w(qs, q4) = a(q5,q4)
Although it is possible to calculate all the IKM starting with this sub system the calculation leads to solve a polynomial equation whose degree is thirty two and not sixteen. That is why we have only worked on the augmented system.
5.2.1 Inspection of the left terms The scalar (~(ql, q2) can be written: ~-
( C2
$2
1 )
c~s ~1
with
c~s Ctl
= [~]
1
and [c~l =
c~
~.
O/lc
C~ls a l l
c~1
.
The [c~] m a t r i x is independent of q2 and ql; it depends on the considered manipulator and on the end effector given location. This matrix can be calculated in a literal way very easily. It is possible to do the same thing with the other thirteen scalar terms and therefore to obtain the other thirteen matrices: [~], ["/], [a], [b], [c], [u], [v], [w], [l], [m],
N,
[ol.
Current Advances in Mechanical Design and Production, MDP-7
5.2.2 Inspection
21
of the right terms
After various a t t e m p t s and tedious calculations it a p p e a r s t h a t it is p a r t i c u l a r l y interesting to introduce the two following vectors: q______. (13 8003 8004 C005 ( ~p6a~3 1 2 . , - (za.p63)P63) + aa r5 saa set4 z 6 x (z 3 x P6a) --r4 (a3 cct3 cot5 -F a5 set3 s o 5 ) sct3 s o 4 P63
+~ (~ a~ ~o~ ~o~ - ~1 (ag - a] + ~,~2 + ,'~ - ~) ~~
CCt5) 80~4 _Z3
--a3 a4 s003 c04 c 0 5 "z3 x P-63 -t- a3 a4 c a 3 8ct3 z 6 x P-63 --t- a3 a4 (r4 + r5 cc~4) s o 3 _z6 x z 3
-?
1,.,2 Z -_ r__ -- 03 8 0 4 (~ff_63_ 3 (Z_3.P_63)P63) + a3 r5 8a4 c0~5 2:6 x (2:3 x /)63 ) a3 r4 col3 8 a 4 P63 2 (aZ3 - a ] - 0 5 + r24 - r 2) s004 z__3 - aa a4 ca,, z_a x P63 + a4 (a3co~3cc~5 + assaasc~5) z6 x P6a
+a3 a,4 (r4 + r5 ca4) ca5 z_6 x z a
Indeed, these two vectors lead to the following f u n d a m e n d a l properties: 9 the first two c o m p o n e n t s of q and L', with respect to R6, can be w r i t t e n as a linear function of some of the fourteen previous quantities, 9 the first c o m p o n e n t s of z_6 x q, q, _.r and _z6 x r_, with respect to Rs, can be written as an affine function of some of the fourteen previous quantities. In order to calculate the first two c o m p o n e n t s of q and r_, with respect to R6, let us use the following table:
m 37(6)
l
--V
a
o'
-Y(6)
m
u
b
/3
z__6x p~a b
z6xza
a
a
v
q
-/3
B
A
r
A' B'
This table lead us, w i t h o u t an3" difficulty, to the c o m p o n e n t s we are looking for: -B
= a3 so3 so4 ca5 1 - a3 r5 saa so4 v - r4 (a3 cc~3 ca5 + a5 sa3 SOs) saa so4 a 1
'
'
2
+ a 3 (a3 a5 c003 s005 -- ~ (a~ -- a~ + a 5 + r~ -- r~) 8 a 3 c a 5 ) 8t~4 o' -- a3 a4 8 a 3 c a 4 c a 5 u --a3 a4 c003 set3 b -- a 3 a4 (r4 + r5 cot4) 8 a 3 j~
.4
- " a 3 8 ( l 3 8 ( 1 4 c o 5 l l l -[- a 3 r5 8 0 3 800'4 u - - r 4
(a3 c~3 c~5 + a5 s~3 s~5) s~a s~4 b
+a3 a4 co'3 s003 a + a3 a4 (r4 + r5 cct4) set3 c~
A' = a3 ~0~ t-~.~ ,.~ ~
co~ ~ , - ~
~ ~0~ .~,~ a - ~ ( ~ - a ] - a g
+~-~)
~4 ~-a~4
c~4
- a 4 (a3 ca3 cc~ + as sa3 sos) b - a3 a4 (r4 + r5 cc~4) cc~5/~ B ' = a 3 8004 Ill +(/3 1"5 8004 C(15 t l - - a 3 r 4 c003 8(I 4 b - ?
((7,2--(/, 2 --a~ + r 4 2 - r 2) 8 ~ 4 ~ - - a 3 a 4 COl4 V
+a4 (a3 c003 c05 + a5 s003 s00s) a + a3 a4 (r4 + r5 cc~4) coe5 oz T h e z 6 x q. q. s and Z 6 X _F first c o m p o n e n t s , with respect to Rs, are d e s i g n a t e d by: _xs.r _ , X = ~ . ( z 6 x q_) 9 l ' = s . .y ' . . and Y' = -xs.(z 6 • r__). Then, after tedious calculations, we obtain"
Current Advances in Mechanical Design and Production, MDP- 7
22
X -- - - a 3 8 a 3 Sa4 S a 5 n ~- a 3 a4 s ~ 3 c a 4 s a 5 w 4- r4 (a3 c a 3 s a 5 -- a5 s a 3 c a s ) s a 3 s a 4 c 1 +a3 (a3 as ca3 cas + 5(a3 2 - a 2 + a 2 + r42 - r 2) sa3 sas) sa4 3' - a ~ (~] c ~ + r~ (r~ c ~ +
~) ~ )
~
Y -- sc~4 (a3 a5 r5 80/3 7 - a3 (a3 c a 3 s ~ 5 - a5 80/3 c a 5 ) s A- r4 82a3 sol5 a -4- a3 r4
x' = ~ ~ ~.~ ~.~ ~ + ~ (~ c~ ~
- ~ ~
as ca3 s~3 ca5
c . ~ ) ~ + a. ~ ( ~ + ~ r
~
Y ' = - a 3 a5 s ~ 4 w - a3 a4 a5 c0~4 3' + a4 so~3 sc~5 t7 -4- a3 a4 a5 c0~3 c0~5 - a4 T4 r5 8~3 c0~4 80~5
+~ (~
- ~ + ~ - ~ - r~)~
~
The interest of the previous components appears when one uses the relation: = c 6 ~ - $6 y__6,
indeed:
X = C6 (x_6, z_s, q) - $6 ( ~ , ~ , q) = - A C6 + B Ss Y = C6x_~.q- S6y_s.q = - B C 6 _ + $6 ~ . r_ = X ' = - C 6 x__s.r
A'
- AS6 C6 + B' $6
Y' = C6 ( 2 , n-s, r) - $6 ( ~ , _z6, r__)= - B ' C6 - A' $6 Let us replace the unknown angle q6 by its half angle tangent ~, according to Lee and Liang's proposal. Then, let be: x6 = tan ~ 9 Taking into account that: x6 - _3a_ - 1-ce 1+C686 ' the previous four equations system can be written as follows: Jx6+K Lx6+I j' x6+K' L' x6 + I'
with: I = K'=Y'
A+X,
J=
+ B ' , L' = Y ' -
A-X,
K = Y+B,
= =
0 0 0 0
L =Y-B,
I' - A ' + X ' ,
J' - A ' - X ' ,
B' .
5.2.3 Inspection of the augmented system equations The quantities I, J, K, L, f , J', K', L' are given by affine functions of the fourteen basic scalar terms. As a consequence it is possible to calculate the matrices [I], [J], [g], [L], [I'], [J'], [K'], [L'] which are given by the same relations as functions of the matrices [c~], [~], [7], [a], [b], [el, [u], [v], [w], [/], [m], [n], [s], [a]. These matrices are independent of q2 and ql; they depend only on the considered manipulator and on the end effector given location. Then it is possible to write, successively:
I~ It
= [1]
S', and I = ( (72 $2 1 ) Is and to do the same thing with: J, 1 I1 K , L, I', J', K ' and L'. If one introduces the q2 half angle tangent" x2 = tan ~ and take into account that C2 = !l+x~ _ ~ and $2 = ~l+x 2 it is possible to write the previous equations as follows:
Current Advances in Mechanical Design and Production, MDP-7
[(J~ [(L, [(J', [(n]
./~).r.~ + 2.1~.r.2 + - Lc).r.~ + 2Lsx2 g:).r.~ + 2Ji.r2 + - L;).v~9 + 2L:x2
(Jr + J~)]x6 + [(K~ - K~)x~ + 2K, x2 + (L, + Lc)]x6 + [(I, - Ic)x~ + 2Isx2 (J; + J~)lx6 + [(K'I - K : ) x 2 + 2Ki.'r2 22 + 2I~x2 + (L,' + L;)]x6 + [ ( I ' 1 - I~)x ' '
23
+ (K, + Kc)] = O, + (11 3t- I~)] = 0,
+ (K', + K:)] = 0, + ( I x' + I'~)] = 0,
In order to be able to use the dialytic calculation let us multiply these equations by x2" [(.S~ - Jc)x 3 + [(L,[(J,' - go) -r3' + [(n] - L:)x~
2J, x.~ + (J, + J~)z2]x6 + [(hq - Kc)x~ + 2K~x~ + (K, + K~)x2I = o, = O, + 2L x ~ + (L, + n~)x2]x6 + [(I, - Ic)x 3 + 2Isx~ + (I, + Ir t
!
t
9g'x~-~ 2 + (J, + g~)x2]x6 + [(K'l - K:)x 3 + 2t(;x~ + (K' 1 + K~)x2] = O, + 2Lix~ + (L', + L;)x2lx6 + [(I; - I:)x 3 + 2Iix~ + (I', + I'~)x~] = O.
Then. we obtain an eight linear equations in eight unknowns system whose second member is zero (the unknowns are: x~.r6, x22x6, x2xe,, xe,, x 3, x~, x2, 1). This system is not a Cramer one and therefore its determinant must be zero. Each determinant element being an affine function of C1 and $1 this leads to a polvnomial~ equation, whose unknown is Xl = tan q~2 if 1-x 2
one uses the relations: C1 = ~ equation is sixteen, and its roots software (which was our choice). values offers no difficulty, and. as
and S1 = ~l+~," The maximal degree of this polynomial can be obtained very easily, for instance using Maple The computation of the others generalized coordinates a consequence, is not presented here.
6. E X A M P L E
We have calculated all the IK~I for the RMS manipulator (see Fig. 1) with the following numerical values: a2 = 1, a3 = 0.1, a4 = 1 and for the given location defined by the following operational coordinates: 07 = 7r. r7 = 0.75, a7 = arcsin 0.8, ar = 0.8, Oo = 0 (obligatory!), ro = 0.45, We have obtained the following sixteen real configurations (in radians)"
So,. 1I a b c d e f g h i j k / m n o p
qi
....
6'.0031 0.0235 0.3411 0.4605 1.4286 1.5433 2.0131 2.0553 -3.138~ .3.1181 .2.800~ .2.6811 .1.713( .1.5982 .1.1284 .1.0862
q3' [ q4 2.3749 -0.9159 2.4950 2.1294 2.2901 0.6670 3.0395-0.1247 2.1570 3.1387 3.1392 1.1040 0.3763 -2.8325 1.0695 0.2953 0.3597 2.1868 -1.0934 2.2177 0.4849 -0.8661 -1.0286 2.6988 0.7667 0.9159 -2.49513 1.0122 -2.2901 -0.667C 0.1021 0.1247 -2.15713 0.0029 -3.1392 .1.104G 2.7653 2.8325 -1.0695 2.8463 -0.3597 .2.1868 .2.0482 -2.2177 .0.4849 -2.2755 1.0286 .2.6988
qs' 1.4795 1.8177 2.3955 0.6458 3.0445 0.0202 -0.583~ -2.430c~ 1.6621 1.3239 0.7461 2.4957 0.0971 3.1214 -2.5581 -0.710(
qs~ -0.0648 2.9808 1.2532 -1.5786 -2.7523 0.5941 -0.7943 2.3144 3.0768 -0.1608 -1.8884 1.5630 0.3893 -2.5475 2.3473 -0.8272
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Current Advances in Mechanical Design and Production, MDP-7
7 CONCLUSION In this paper, we have presented a simplified inverse kinematic model calculation method for 6R type manipulators. Our method simplifies significantly the original one due to Lee and Liang, and contrary to this one, can be used, for all 6R type manipulator without any restriction as it does not involve any division. The inverse kinematic model calculation is brought back to the computation of the roots of a polynomial equation whose degree, equal to sixteen, is minimal and a non academic example shows that every sixteen roots must be real. Nevertheless we think that our method can still be simplified and that the previous sub system of six equations in four unknowns suffices to obtain also a sixteen degree polynomial equation. In spite of numerous attempts we have not succeeded in obtaining this minimal degree polynomial equation using this sub system. ACKNOWLEDGEMENTS We acknowledge V. Cadenat and J.-Y. Fourquet for their help concerning the preparation of this paper. REFERENCES 1. Pieper, D.L., "The Kinematics of Manipulator under Computer Control", PhD. Thesis. Stanford University. Stanford. California. USA. October (1968). 2. Roth, B., "Performance Evaluation of Manipulators from a Kinematic Viewpoint", Cours de Robotique, IRIA, Le Chesnay, France, August (1976). 3. Duffy, J. and Crane, C., "A Displacement Analysis of the General Spatial 7-Link, 7R Mechanism", Mechanism and Machine Theory, Vol. 15, pp. 153-169, (1980). 4. Cayley, A., "On the Theory of Elimination", Cambridge and Dublin Mathematical Journal, 3, pp. 116-120, (1848). 5. Salmon, G., "Lessons Introductory to the Modem Higher Algebra", Hodges and Foster, Dublin, Ireland, (1876). 6. Tsai, L.W. and Morgan, A.P., "Solving the Kinematics of the Most General Six and Five-Degree-of-Freedom Manipulators by Continuation Methods", Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 107, pp. 189-200, June (1985). 7. Lee, H.Y. and Liang, C.G., "A New Vector Theory for the Analysis of Spatial Mechanisms", Mechanism and Machine Theory, Vol. 23, No.3, pp. 209-217, (1988). 8. Lee, H.Y. and Liang, C.G., "Displacement Analysis of the General Spatial 7-Link 7R Mechanisms", Mechanism and Machine Theory, Vol. 23, No.3, pp. 219,226, (1988). 9. Raghavan, M. and Roth, B., "Kinematic Analysis of the 6R Manipulator of General Geometry", The fifht International Symposium of Robotics Research, Tokyo, Japan, (1989). 10. Manocha, D. and Canny, J.F., "Real Time Inverse Kinematics for the General 6R Manipulators", IEEE International Conference on Robotics and Automation, Nice, France, (1992). 11. Manocha, D. and Canny, J.F., "Efficient Inverse Kinematics for General 6R Manipulators", IEEE Transactions on Robotics and Automation, Vol. 10, No.5, pp. 648-657, October (1994).
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12. Kohli, D. and Osvatic M., "Inverse Kinematic of General 6R and 5R,P Serial Manipulators", Journal of Flexible Mechanisms, Dynamics and Analysis, DE-Vol. 47, pp. 619-627, (1992). 13. Mavroidis, C., Ouezdou, F. B. and Bidaud, P., "Inverse Kinematics of a Six-Degree-ofFreedom General and Special Manipulators Using Symbolic Computation", Robotica, Vol. 12, pp. 421-430, (1994). 14. Khalil, W. and Kleinfinger, J.F., "A New Geometric Notation for Open and Closed Loop Robots", IEEE International Conference on Robotics and Automation. San Francisco, California, USA, (1986). 15. Craig, J.J., "Introduction to Robotics. Mechanisms and Control", Addison Wesley, Reading, Massachussets, USA, (1986). 16. Smith, D.R., "Design of Solvable 6R Manipulators", PhD. Thesis, Georgia Institute of Technology, Georgia, USA, May (1990). 17. Paul, R.P.C., "Robot Manipulators. Mathematics, Programming and Control", MIT Press, Cambridge, USA, (1981 ). 18. Gorla, B. and Renaud, M., "Modeles des Robots Manipulateurs. Application a leur Commande", CEPADUES, Toulouse, France, (1984).
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
27
A SYSTEMATIC ALGORITHM FOR FLEXIBLE MANIPULATORS SIMULATION
Megahed S.M.'and Hamza K.T.*" * Professor, ** Graduate Student, Mechanical Design and Production Department Faculty of Engineering, Cairo University, Giza 12316- Egypt. E-mail
[email protected] and
[email protected] Abstract This paper presents a systematic simulation algorithm for flexible manipulators with revolute joints. This algorithm takes into consideration the flexibility of links and joints, structural damping and joints' internal clearance. Flexible multi-body dynamics formulations are employed to obtain the equations of motion of such manipulators by considering their links as a lumped and as a consistent mass system. Simulation of one and two link planar manipulators is provided and comparison between the techniques of lumped mass and consistent mass is performed. In both cases, the system oscillates about the trend of rigid body motion, but better performance is achieved in the case of consistent mass. This algorithm can be readily adopted to simulate the performance of a control system of flexible manipulators.
Keywords Flexible Manipulators, Multi-body Dynamics, Finite Segment, Structural Damping.
1. Introduction Recently, there has been an increasing interest in studying flexible manipulators because of their advantage of high load to weight ratio. Since there is no such thing physically as a perfect joint or a rigid body, control schemes of manipulators that are based on rigid body mechanics often resulted in degraded control performance especially at high speeds. Higher speeds being a constant demand for higher production rates, extensive research work is directed towards developing good performance controllers for flexible manipulators [1-4]. Accordingly, much research work is also directed towards modeling and simulation of flexibility effects on the dynamic behavior of manipulators. The dynamic equations of motion of flexible manipulators are highly nonlinear and are often coupled. Therefore, the problem of simulating the dynamic behavior of a general flexible manipulator is difficult. Moreover, there is no exact analytical solution for such systems except for some limited special cases. Flexible manipulators, being a special class of flexible multi-body systems, have been treated as such [5]. However, there are many techniques and methods for studying flexible multibody systems, and there is a disagreement about accuracy and efficiency of such methods [6,7]. Most popular techniques are the Floating Frame of Reference [6-8], incremental Finite Element [6], Finite Segment [6,9] and Absolute Nodal Coordinate [6,7,10].
28
Current Advances in Mechanical Design and Production, MDP-7
In this paper, a variation of the Finite Segment approach is employed to formulate the dynamic equations. The global position and orientation angle of each node are used as generalized coordinates. The use of finite rotation as nodal coordinates, as demonstrated by Shabana [7] does not lead to an exact representation of flexible body inertia. Shabana proposed the use of slopes as nodal coordinates to define orientation in the Absolute Nodal Coordinate method [7,10]. The formulation utilized in this algorithm, however makes computing stiffness and damping forces easier while providing acceptable accuracy for cases of small deformation per element which is applicable in most manipulators. A complete formulation of the equations of motion of such manipulators is presented along with the algorithm for their implementation. One and two link manipulators are simulated to compare the lumped mass formulation and the consistent mass formulation and to demonstrate the use of the algorithm for simulating controller performance. Both techniques have been incorporated in the algorithm. 2. Equations of Motion
A variation of the Finite Segment approach is used to formulate the dynamic equations of motion. The Finite Segment approach assumes a flexible member to be composed of a number of discrete rigid bodies that are connected by springs. This makes the treatment of a flexible link similar to the treatment of a set of several rigid bodies. Each link in the manipulator is broken into several elements connected at nodes (Fig. 1). Equivalent masses are computed at these nodes, then by applying the principles of rigid body dynamics on those masses, the manipulator equations of motion are: Mi=F
(I)
where M is the manipulator mass matrix, i is the nodal acceleration vector referenced to the global axes and F is the total force vector representing the effect of stiffness, damping and actuator forces, referenced to global axes. The generalized coordinates (Fig. 2) are the global position (xi, Yi) and the angular orientation 0~ of the i~ node in the system. Since the manipulator is a single open chain, it is convenient to assemble the stiffness and damping forces directly rather than first assembling the stiffness and damping matrices and then multiply them by the deflections and velocities respectively. Assembling the stiffness and damping forces directly also facilitates the use of nonlinear expressions for computing the forces. 2.1 Element Stiffness Forces
An element in a general configuration is shown in Fig. 2. The stiffness forces due to this element at nodes i and j are first computed in the direction of element axis ui and vi then transformed to the direction of the global axes. These forces are given by: Fu i
--
Fvi
=
Mzi
-Fuj -F~j Mzj
= = = =
Ka du KbjdV - Kb2d0 -Kb2dv + Kb3 dO -Mzi - FviLo
(2)
Current Advances in Mechanical Design and Production, MDP-7
29
with: EA
Ka =
'
6EI K b 2 = - -L2o
12EI Kbl = ' - L To '
du : c , ( x , - x,)+ s , ( y j - y , ) - L o
&
,
4EI Kb3 = - - ~ ,
dO = Oj -0 i
-x,)+ c,(yj-y,) - - o
d v : -si(x i
where Ka is the axial stiffness constant, Kbl, Kb2, Kb3 are bending stiffness constants, du, dv, dO are the axial, transverse and angular deflection of the element respectively, E is the modulus of elasticity, A is the link cross sectional area and I is its second moment of area. Transformation to the direction of the global axes gives:
LFy,
]I
I
s,
c, JLF,,
_
[ c'
LF,j
s,
c,
jLF~j]
ci=cos(Oi) & s i=sin(oi) 2.2 Element Damping Forces
Several methods are customarily used to express the damping effect of structural members [11,12]. However, in most cases, the exact damping properties of a structural member are uncertain and have to be determined experimentally [ 11 ]. In many cases, it is acceptable to use an equivalent viscous damping, on the other hand, solid (or structural) damping is mainly believed to be proportional to the stress (or deformation), but in phase with the velocity [ 11 ]. An advantage to assembling the damping force directly instead of the damping matrix first is the ease of using multiple formulae that are either linear or nonlinear to express the damping forces. Damping forces of an element are first computed in the direction of the element local axes then transformed to the direction of the global axes. The computation of damping forces is very similar to that of the stiffness forces. Equivalent Viscous Damping
Fui Fvi Mzi
=
=
-Fuj -Fvj Mzj
=
Ca dO
= = =
Cbl d';' - Cb2d0 -Cb2 d~' + Cb3 dO -Mzi - FvjLo
(3.a)
Solid Damping
Fui Fvi
= =
Mzi
=
-Fuj -Fvj Mzj
--- Ca du sign(dti ) = Cbidvsign(d~') - Cb2 dO sign(dO ) = -Cb2dv sign(de,) + Cb3 dO sign(dO ) -Mzi - Fvi Lo
(3.b)
where Ca, Cbl, Cb2, Cb3 are the damping constants. These damping constants are best identified experimentally for the members to be simulated. However, it is sometimes acceptable to estimate approximate values for them such that the simulated response is closely matching similar systems. The time derivatives of the element deflections are given by:
30
Current Advances in Mechanical Design and Production, MDP-7
dE)-'Oj-E)i, du=ci(:~j-:~i)+si(~'j-~'i)
&
d~,=-si(:~j-:~i)+ci(~'j-~'i)
2.3 Joint and Actuator Forces Revolute joints can be modeled as perfect joints (Fig. 3) or elastic joints (Fig. 4). The actuators driving the joint can also be perfect (Figs. 3,4) or elastic (Fig. 5). Perfect actuator joints join the last node of a previous link (Node 1) with the first node of the next link (Node 2). In the case of joints with elastic actuators, an extra node is added in between so as to simulate the internal dynamics of the actuator. Generally, all values of bearing stiffness, damping coefficients, internal lumped mass and bearing clearance should be determined separately for each joint prior to simulation, either experimentally or based on an acceptable model. The following cases are studied: Perfect Joint: The effect of adding a perfect joint is incorporated in the dynamic equations of the system as algebric constraint equations: xt = x2
&
yt = y2
(4)
Elastic Joint: The radial deflection between Nodes 1 and 2 is permitted. Joint stiffness and damping forces are computed as function of this deflection, then transformed to the direction of the global axis and added to the global force vector. A possible formula for computing radial forces is:
Frt = " F,z = {
KsoftSr + CrS, Ksoa 8c + Kstiff (Sr " 8c ) + Cr~r
For 5c-> 8r For 5c < 8r
(5)
where 5c is the joint radial clearance, 5r is the radial deflection between nodes 1 and 2, ~ir is the radial relative velocity, Kson and Kstiffare respectively a small and a large stiffness value used to simulate the effect of internal beating clearance. Perfect Actuator: Actuator torque Ta is added to the global force vector at Node 2 and with the same magnitude but in opposite direction at Node 1 (Figs. 3,4).
Elastic Actuator: To simulate the internal actuator inertia, an extra node with a lumped mass is added between the last node of the previous link and the first node of the next link. The actuator torque is added at nodes 1 and 2 (Fig. 5) in equal and opposite directions. However, the torque is transmitted from Node 2 to Node 3 through a torsional spring and damper whose characteristics can be set to simulate the effect of the actuator transmission system. 2.4 Mass Matrix The system mass matrix is obtained by assembling the element mass matrices of all the elements in the system. Optionally, additional lumped masses can be added at specific nodes to simulate special effects like the presence of a payload, or the mass of an actuator. Equivalent Lumped Masses can be computed at nodes i and j (Fig. 2 ) o f an element by dividing the total element mass (m) and inertia by two (since the element has a uniform cross section). Lumping results in a diagonal element mass matrix given by:
Current Advances in Mechanical Design and Production, MDP-7
I
0
0
0
0
0
0
l_ 2
0
0
0
0
0
0
L', 24
0
0
0
0
0
0
~ 2
0
0
0
0
0
2
M
=m
0
o
0
0
~ 2
0
o
31
0
L',
(6.a)
Lumped mass formulation leads to a fully diagonal system mass matrix so the dynamic equations are uncoupled. Efficient computation of global nodal acceleration vector can be performed. Integration of the acceleration vector gives the system state at the next time step. Lack of dynamic coupling in lumped mass formulation generally results in bad accuracy of simulation results and requires the use of a large number of elements per member in order to achieve acceptable accuracy. Alternatively, Consistent Mass formulation takes into account the dynamic coupling between the nodes of the element. Ignoring the coriollis, centripetal and tangential acceleration terms resulting from element deformation [ 11 ] (usually valid for small deformation per element) can simplify an expression for element consistent mass matrix. The element mass matrix will be [11 ]: m
1
0
0
3
0 o
M=mR T 1
6 0
m
I
6
--"--'
!!L,
35
210
~!L.
e.
210
105
0
0
"~ 70 -13L. 420
13Lo 420 - L~ 140
o o I
3 0 0
0
0
9_~ 70 13L. 420
-13L. 420 - L~ 140
0
0
13 35 -ILL. 210
-ILL. 210 L20 105
R,
Where" R =
c,
s,
0
0
0
0
- s,
c,
0
0
0
0
0
0
l
0
0
0
0
0
0
c,
s,
0
0
0
0
- s~
c~
0
0
0
0
0
0
I
(6.b)
Since the dynamic equations of motion become coupled when the consistent mass formulation is used, a set of linear algebric equations will have to be solved at every time step in order to determine the global nodal acceleration vector. This is however a trade-off of computational speed for better accuracy of solution. 3. Computation algorithm
.
A computation algorithm is developed for simulation of flexible manipulators either as a lumped mass system or a consistent mass system. The flow chart of the computational algorithm is shown in Fig. 6. The required pre-processing includes: input of the problem data; namely the number of links and joints, links' geometry and material properties, joints' stiffness and damping properties, number of elements per link and parameters for the numerical time step integration technique. The next step is to compute the mass, stiffness and damping constants for each element (Sec. 2) and establish element-node connectivity. Next, the initial state (Positions & Velocities) of the system has to be determined. Then, the simulation procedure starts, the current system state is used to compute stiffness and damping forces due to elements and joints and are added to form a global force vector. Then, the actuator forces (torques)are computed according to the control law whose performance is
32
Current Advances in Mechanical Design and Production, MDP-7
Elements
Sta.
Nodes ~
|
yJ/
~
/
Input No. of links, joints, / their properties meshing and time step parameters.
-., '
~.~"
..f-'e,f~a'/ :l ~y". y z/,,'~//,'//,'~oun d
]
I
~._Revolute
Joint
-
I
' Fie. 1. A Flexible Manipulator
. . . .
Compute Mass, Stiffness & Damping Constants of Elements. 'I:
Establish Element Connectivity
vj V
Compute Lumped
du
[
Y~
,
FiR. 2. An Element in a General ConfiRuration
,.
F TM
.........
Compute Initial State Vector. ,,L! v1
I-m~ ~ {
Next L i n k ~
,,
,!
.
Use current state to compute stiffness, damping and actuating forces; Assemble global force vector.
,,_?,0
Previous Link
Fig. 3. Perfect Joint with Perfect Actuator
Next Link---~
T.
N.._ PreviousLink Fig. 4. Elastic Joint with Perfect Actuator
NextLink--~
/.
"rI - ' ' ' Divide - ' ' ' ' ~ - by Compute element [ nodal masses mass matrices, [ to obtain assemble global mass ] global matrix and multiply I accelerations force vector by its [ inverse to obtain global accelerations "l~ ["q .
!
,,
Integrate numerically to get the System State at next time step; Record new State. Time = Time + Step Ye
_ ( E n d ) Fig. 5. Elastic Joint with Elastic Actuator
Fig. 6. Computation Algorithm Flow Chart
Current Advances in Mechanical Design and Production, MDP-7
33
being simulated and added to the global force vector. The global force vector is multiplied by the inverse of the system mass matrix to obtain the global accelerations, which are integrated numerically to obtain the system state at the next time step. The new system state is recorded and the process is repeated with the new state being the new current state until the desired simulation time is reached. It should be noted that in lumped mass formulation, the system mass matrix is constant and diagonal so it is evaluated only once during pre-processing, while in case of consistent mass formulation, it has to be re-evaluated at every time step. A tailored computer program is developed using MATLAB. Fourth order Runge-Kutta [13] is used for time step integration. 4. Example 1: One-Link Manipulator A single link (Fig. 7) of length L = 1.0 m, Mass = 1.0 kg, modulus of elasticity multiplied by second moment of area EI = 100 N.m 2, modulus of elasticity multiplied by cross sectional area EA = 1000 N, is subjected to a constant torque T = 1.0 N.m, (Fig. 8), no damping is considered and the revolute joint is assumed to be perfect. Initial state is 0(0) - 0.0 rad and 6(0) - 0.0 rad/s with no initial strain. Simulation is performed for 4 cases: a rigid body system, 4 and 8 element per link lumped mass, and 4-element per link consistent mass. The expected motion of the link is to be vibrating but its gross motion should be following the motion of the rigid body system. Figs. ( 9 . a - d ) show the recorded system state at various locations of the link for the first 0.5 (s) of motion. It can be seen that the consistent mass formulation is rippling very closely about the rigid body motion, while the simulation using lumped mass formulation has the same trend, but its values tends to diverge. This is further seen in Fig. l0 where the difference between angular orientation of the link tip of the flexible system and the rigid one is plotted against time. This divergence can be understood since lumping is an approximation of mass distribution in the system, it results in a slightly different response, while consistent mass formulation only ignores the inertia change due to deformation, and thus will simulate the gross motion (rigid body mode) very accurately.
5. Example 2: Two-Link Manipulator A two-link manipulator (Fig. 11) is employed to evaluate the performance of a controller based on rigid body dynamics when used with a flexible system. Both the direct and inverse dynamic model of the problem are provided in [14]. Initial state is at rest, with both links aligned on the global X-axis. The first link is to move with constant angular acceleration for a duration of 1.0 (s), constant angular velocity for 1.0 (s), then constant angular deceleration for 1.0 (s), during which, the link will have completed a complete revolution, all the while, orientation of the second link is maintained parallel to the X-axis. Actuator torques for such maneuver of a rigid system are given in Figs. (12.a,b). The actuating torques are employed for flexible systems and their response is simulated. Flexible system data is: lengths Ll - L2 = 1.0 m, Masses MI - M2 - 1.0 kg, modulus of elasticity multiplied by second moment of area EI = 40,100,1000 N.m 2, modulus of elasticity multiplied by cross sectional area EA -- 1000 N, no damping is considered, and the revolute joints are assumed perfect. Simulation using consistent mass formulation is performed with 4-elements per link. Figs. (13.a-d)showthe system state at different locations while Figs. (14.a,b) show the path of the end point of each link. It can be seen that lower link stiffness results in degraded performance of a controller based on rigid body dynamics, which can still give acceptable performance, if the links are stiff enough.
34
Current Advances in Mechanical Design and Production, MDP-7
jo~
@
o.
-
Fig. 7. E x a m p l e 1: O n e - L i n k M a n i p u l a t o r
o.~s o~
o'.= o~+ oi,
o.~
Tim,+ ll )
o s
o.ii
Fig.8. A c t u a t o r T o r q u e
0.4
.....
0.35
,
..
,
.,,
~,
0.3S
,
,
9~
...
.
.
.
,
.
;
:
~ o,
~o. ~
i
o.~1
~
o.1!
0.2
~
o.1
I
ILilid
0.1S
-41-- 4-IEI COlll~lem M u s I
"
"
olol !
0.1
r m o (el
~
- - , -
~._--:
.+ .
.
.
.
.
,
o.s
Time 1-)
s
(b) Link Orientation at Point A
.
0.38
Oils
9
v
.
..
:
.
. .
.
. .
.
v .
.
.
.
.
.
.
+ .
.
03
~oolm
j
"
.+,].,,r//t.
o
, ~.:..
.
o~
(a) Link Orientation at Point G
.~._--_
~
4 - -+ t,,.+ ~ .
o.I
I+o1
1
"-
4.El C m ~ l ~ l
~o~
097
i
0.2
..L__ R ~
:
s o.Is Rillid
--
O.95
.-.-. . .,,..t, . .t.,,. . . .., II
O.94 0.~
0.05
0.1
O. 1S
0.2
\ \ II 0.2S 0.3 T i m * (e)
0 36
0.4
0 45
0 S
(c) X - Position of Point A
-0
O+OS
0.1
0.1S
0.2
0.2S
0.3
0.3S
0.4
0.4S
O.S
(d) Y - Position of Point A
Fig. 9. E x a m p l e I Results [ El--100 N.m 2, E A - 1 0 0 0 N, L = I . 0 m ] O02--
I
.
,
,
,
,
,
,
,
--r--- --
B
0
~ -0.02 11
e
0
0~
o, i
o. t5
o.2
0.2s
Time ( m c )
03
0 3s
o.4
0.4s
0.5
Fig. 10. E x a m p l e 1 Difference in Orientation from Rigid System at point A
Fig. 11. E x a m p l e 2: T w o - L i n k M a n i p u l a t o r
Current Advances in Mechanical Design and Production, MDP-7
i!!.
I-T~
35
J -3
(Sec)
T'~ne (See)
(a) Actuator 1 Torque
(b) Actuator 2 Torque
Fig. 12. Example 2 Actuator Torques
'Fi ,i
///
:L
,
|~
~,~
t~,-,,: Ii l++',',=_.. ! l
/
I,.., 1/ o
[
"~
os
I
i s l~,ne (-)
2
2'S
'
I
.
i
3
3s
(a) Orientation of Linkl at Point A
o
os
t
t
l ' * . e (l)
2
2S
3
3S
(b) Orientation of Link2 at Point B .-!
/y "~
o's
I
is 1 " ~ (i)
~
! 2s
3
o$
0
3s
(c) X-Position of Point B
os
Is
-
-
2
=s
(d) Y-Position of Point B Fig. 13. Example 2 Results
[
4
os oe t.
'
,~ go
os,
i
j ~ .o2i I I
"1t
-oeI
! .1iS-,,
'
.o's
,;
x - POl,llUm - "A" ira)
o's -~T-----~s
(a) Path of Point A
"I .,
o
o$ ! x - PosJoon. " i t (m)
is
(b) Path of Point B
Fig. 14 Example 2: Path of Links' End-Points
3
3s
36
Current Advances in Mechanical Design and Production, MDP-7
6. Conclusion
Simulation of flexible manipulator dynamics is an essential tool in developing and assessing their control in a more accurate manner. This paper presents a systematic algorithm for such a purpose based on the mathematical modeling of flexible manipulators as flexible multi-body systems. The equations of motion of planar single open chain manipulators, with only revolute joints are provided. Minor modifications are required to expand these equations to spatial manipulators. A comparison between lumped mass and consistent mass formulation is performed. Simulation results showed that the lumped mass formulation is only suitable for determining the trend of motion, or qualitative analysis, while the consistent mass formulation provides better accuracy which is more suitable for quantitative analysis. In the second example, consistent mass formulation was employed to simulate the performance of a controller as a demonstration of the basic motive of developing this algorithm. References
1. Hastings, G.G. and Book, W.J., "Experiments in Control of a Flexible Robot Arm", Robots 9, Conf. Proceedings, Vol. 2, Detroit, Michigan, USA, pp. 20.45 - 20.57, 1985. 2. Schmitc, E., "Dynamics and Control of a Planar Manipulator with Elastic Links", Proceedings of 25th conf. on Decision and Control, Athens, Greece, pp. 1135-1139, 1986. 3. Yim, W., Zuang, J. and Singh, S.N., "Experimental Two Axis Vibration Suppression and Control of a Flexible Robot Arm", J. of Robotic Systems, 10 (3), pp. 321-343, 1993. 4. Eisler, G.R., Robinett, R.P., Segalman, D.J. and Feddema, J.D., "Approximate Optimal Trajectories for Flexible-Link Manipulator Slewing using Recursive Quadratic Programming", J. of Dynamic Systems Measurement and Control, Vol. 115, pp. 405-410, 1993. 5. Idler, S.K., "Open Loop Flexiblity Control in Multi-body System Dynamics", Mechanism and Machine Theory, Volume 30, No. 6, pp. 861-869, August 1995. 6. Shabana, A.A., "Flexible Multi-body Dynamics: Review of Past and Recent Developments", Multi-body System Dynamics, Volume 1, pp. 189-222, March 1997. 7. Shabana, A.A., "Dynamics of Multi-body Systems", Cabridge University Press, 1998. 8. Shabana, A.A. and Schwertassek, R., "Equivalence of the Floating Frame of Reference Approach and Finite Element Formulations", Int. J. of Nonlinear Mechanics, Vol. 33, No. 3, pp. 417-432, 1998. 9. Connelly, J.D. and Huston, R.L., "The Dynamics of Flexible Multi-Body Systems: A Finite Segment Approach - I Theoretical Aspects & II Example Problems", J. of Computers and Structures, Vol. 50, No. 2, pp. 255-261, 1994. 10.Shabana, A.A., "An Absolute Nodal Coordinate Formulation for the Large Rotation and Deformation Analysis of Flexible Bodies", Technical Report No. MB96-1-UIC, Department of Mechanical Engineering, University of Illinois at Chicago, March 1996. l l.Sandor, G.N. and Erdman, A.G., "Advanced Mechanism Design: Analysis and Synthesis", Prentice-Hall Inc., New Jersey, USA, 1984. 12.Sun, C.T. and Lu, Y.P., "Vibration Damping of Structural Elements", Prentice Hall PTR, USA, 1995. 13.Woods, R.L. and Lawrence, K.L., "Modeling and Simulation of Dynamic Systems", Prentice Hall Inc., New York, USA, 1997. 14.Megahed, S.M., "Principles of Robot Modeling and Simulation", John Wiley & Sons Ltd., London, UK, 1993.
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
37
NON-LINEAR TRAJECTORY CONTROL OF FLEXIBLE JOINT MANIPULATORS
Bravo, R.R.* and Dokainish, M.A.** *PhD Candidate, ** Professor, Department of Mechanical Engineering McMaster University, Hamilton, Ontario, Canada, LSS-4L7 ABSTRACT It has been known for years that the presence of flexibility in the joints can severely limit the performance of industrial manipulators when performing trajectory tracking tasks. This paper addresses the problem of control of manipulators, taking into account the existence of flexibility in the joints. Dynamic models are developed for the general n-link manipulator and for the particular case of a two-link articulated flexible joint manipulator. The model is used for the simulation and synthesis of the control strategies: Feedback Linearization and Variable Structure (Sliding Mode) control. These techniques are numerically simulated for the tracking of a prescribed trajectory in the wrist space, and the results compared to the case of a manipulator controlled by a PD plus Gravity controller, which does not account explicitly for the presence of flexibility in the joints. The results of the simulations show that the PD plus Gravity controller is not capable of achieving satisfactory tracking, while the Feedback Linearization Controller shows excellent performance for the case of no uncertainties. The Sliding Mode Controller results in excellent performance, and exhibits good robustness in the presence of parametric or modelling uncertainties, showing the advantages of the proposed approach. KEYWORDS Flexible Manipulators, Non-Linear Control, Feedback Linearization, Variable Structure Control. 1. INTRODUCTION Industrial manipulators have been part of our lives for the last 30 years. They are taking charge of activities that are tedious, repetitive or dangerous for human operators. These activities include traditional applications such as material handling, spot welding, etc., which involve only point-topoint control, and more complex tasks such as arc welding, spray painting, mechanical and electronic assembly, etc. which often involve trajectory tracking control and force control. Applications have become more demanding and the manipulators are required to be smaller, lighter, faster and more precise. This evolution results in increased flexibility, in both the structure and the joints of the new generations of manipulators. The induced compliance will degrade the tracking performance of the robots [1 ], since it can cause lightly damped oscillations whose resonant frequencies are low enough to fall inside the bandwidth of the controller. Research has revealed that for most manipulators, the principal source of flexibility resides in the drive system rather than in the links. For this reason, considerable attention is given by the researchers to the problem ofjoint flexibility in manipulators. The appearance, in recent years, of powerful low-cost microprocessors has spurred great advances in the theory and application of nonlinear control.
38
Current Advances In Mechanical Design and Production, MDP-7
This has resulted in rapid development in the areas of feedback linearization, sliding control and adaptive control techniques. Several approaches have been proposed to control flexible joint manipulators: Spong [7] employed a static feedback linearization approach, which was coupled to a variable structure scheme by Sira-Ramirez and Spong [5]. Sira-Ramirez et al [4] used dynamic feedback linearization, while Khorasani [2] and Slotine and Hong [6] made use of a singular perturbation approach that divides the dynamic model into slow and fast subsystems that are controlled separately. 2. MODELLING Figure I shows a schematic diagram of a two link articulated manipulator, as well as the relevant parameters and variables, consistent with the Denavit-Hartenberg conventions. Figure 2 depicts a flexible joint, where the joint compliance is modelled by a linear torsional spring. I Direction of the gravity Vector g ,
~"-----~--~ la, nil i
~..-~~ ! ....... ~.. __~I
-
'P
l-t, m t-:"~ "
Link i Transmission
at
-A, i
) L,~
i
-..~..__ J~__~
Actuator i r,
Fig. 1. Coordinate systems and parameters
Fig. 2. A flexible joint
The symbols shown in Figs. 1 and 2 are defined as follows: qh: qm,:
r,: m~"
I:,: J,,.:
K,: Lc,: ~7,:
angular displacement of gh link (rad) with respect to the x,.! axis. angular displacement of i'h actuator (rad) with respect to the x,.! axis. torque applied by i'h actuator. mass of i'h link moment of inertia of i'h link with respect to its centre of mass moment of inertia of i 'h actuator with respect to its axis of rotation stiffness of i'h joint length of gh link distance from i'h joint to the centre of mass of i'h link Transmission ratio between the i'h link and actuator
The presence of the joint compliance introduces dynamics associated with the link position and the actuator position, thus causing the vector of generalized coordinates of the manipulator to contain both link and actuator positions: q = [qt r, qmr]rwhere qt is the vector representing the angular displacements of the n links and qm is the vector of angular displacement of the n actuators. Defining Tand Uas the kinetic and potential energies of the arm, respectively, then the Lagrangian function of the arm can be defined as"
Current Advances in Mechanical Design and Production, MDP-7
39
L (q,q) = r (q,q) - U (q)
(1)
where q = dq/dtis the vector of link and actuator velocities. The general equations of motion of a robotic arm can be formulated in terms of the Lagrangian function as:
d 0 L(q,q) - 0____ L(q,dl) = F, dt adl e aq ,
i = 1, 2, ..n
(2)
where n is the number of links and/7, is the resultant torque applied to the i 'n joint. The kinetic energy can be expressed as follows
1
r(q, dl) = -~ dl
r
[D(q)] 4
(3)
where the inertia matrix [D(q)] and the coordinates q for links and actuators are given respectively for a two link manipulator by" [D(q~)] [D(q)] =
I._i+ [O(q)]
[0]
2m2atLc2c~
[J.]]
2
2
mlL:!+ /=2+ m2Lc2+ m2a I +
:
to]]
t2)
I:2+ m2L~2§ m2atLc2COs(qt2) '
2
/.2 + m2L~2+ m2aiLceCOS(qt2)
o
1
q={ qt, qt2 qm, qm2 }r
d,,,2
(4)
The potential energy of a flexible joint manipulator contains a gravitational component, as well as a contribution caused by the elasticity of the joints. It is given by"
1
U (q) = -~( qt- [111-Iqm
)T
-!
[g] ( ql- [111 qm ) - g
r ~ mk ck(qt)
(5)
k=l
where [rl] is a n x n diagonal matrix of containing the transmission ratios r b and [K] is the n x n diagonal matrix ofjoint stiffness, c*(q) are the vectors of the positions of the centres of gravity of each link k. For a two link manipulator, they are given by:
cl(q) - { Lcl cos(qi) /
c2(q) = / a ! cos(ql) + L c2 cos(ql + q2)1
[
Lct sin(q!) J
(6)
al sin(ql) + L c2 sin(ql + q2)J
The generalized forces acting on the joints are produced by electric, pneumatic or hydraulic actuators and by friction. Friction is a complex, nonlinear effect that is hard to model accurately. The following model is adopted for the links and for the actuators" v
c
v
c
bt,(dl) = Itt, Clt, + Itt, sgn(dtt) , b,,,,(dl) = It,, dim, + It,,,, sgn(dl,,) i = 1, 2, ..n (7) where Itv and ~rr are the viscous and coulomb friction coefficients. The vector of generalized forces is F = [- b t , [, - bruit] r, or, for the two joint manipulator: F-
{-btl
-bt2
,l-brat
~.2_bm2 }T
(8)
Substituting equations (3) and (5) in (I), and equations (I) and (8) in (2), the following matrix equations are obtained, which represent the dynamic model of the flexible joint manipulator:
40
Current Advances in Mechanical Design and Production, MDP-7
[D(qt)]#~+ c(q~, 4,)+ b~(4)+ h(q)+ [K](qt- [vl]-' qm) = 0 [J,,,]q., + b,.(q.,) - [rl]-' [K](q,- [111-' q,n)
(9)
= 1;
where, for a two link manipulator, the velocity coupling, gravity and input torque vectors are given respectively by the following expressions: 2m2 al Lc2 Clll 012 sin(q/2) - m2 al L c2 ql2 sin(qt2
c(qt,4, ) = m2 al Lr q~ sin(q12) miLe / sin(qtt)+ m2a I sin(qu)+ m2Lc2 sin(qt/+qt2) ! h(q~) = -go'
I
m2 L,:2 sin(q/t +qt2)
0o)
It is important to point out that this model is based on the following two assumptions [7]: the kinetic energy of each rotor is only due to its own rotation, and each actuator is symmetric with respect to its associated axis of rotation. The first assumption removes the dynamic coupling of the actuator and link variables in the inertia matrix while it allows for the modelling of the elasticity in the joints; the second assumption is very easy to justify and it makes the potential energy of the manipulator and the velocity of the rotor centre of mass independent from the actuator position. Defining the state space vector by: 9
T
T
x=tx, ,x, ,x,
T
"
,x{]' = [q,~,4,~,[n]
'q.r,[nl-'#.q~
then the dynamic model of the flexible joint robot in state space form is given by: "fl = X2
"~Z = - [D%)]-I{C(Xl.Xz)+
b,(xz)+ h(Xl)+ [Kl(x t - x3) }
(ll)
.i' 3 = x 4
-~4 = - [3.1-~[rll-~{b..(x4)-
[rll-~[Kl(x,
-x3)-
~}
3. FEEDBACK LINEARIZATION CONTROL It has been demonstrated that the dynamic model of a rigid joint manipulator is linearizable using static state feedback allowing for the familiar computed torque control method. However, for the flexible joint case, this is not straightforward, since the presence of the compliance makes the relation between the actuator input and the link position weak. In order to extend the feedback linearization approach to the flexible joint manipulator, it is necessary to introduce a change in the state space variables of the dynamic model, transforming them from actuator and link states to a link-only state. The state space transformation that accomplishes that is given next: y,-r,(x)=x I yz-T2(x)=~, = x,
Y3=T3(x)=-~z= -[D(x, )]-' {c(x, ,xz) +h (x~)+b,(xz) +[K](x, -%)} y 4 = T4(x)= T3(x)= -
(12)
d([D(x,)]- ' {c(x,,xz) +h (x,)+b ,(xz) +[Kl(x,-x3)})
This mapping from the old to the new state space is globally invertible. The new state space representation of the system can now be written in the feedback linearizable transformed form: .p = [A] y + [B] [M(y0]-I[1: -flY)]
(13)
Current Advances in Mechanical Design and Production, MDP-7
41
with [o1 [fl [o] [o1 [A]-
[0] [0] [/] [0] [o1 [o] [o] [rl
[B]-
[o] [o]
(14)
[o]
[o1 [o1 [o] [o] [fl where [/] and [0] are n x n identity and zero matrices, respectively, and [mo, l)l=[M(Tl(x))l=[N(xt)l:[D(xt) ] [K]-I[rl][Jm] J(Y):J(T(x)):[N(xl)]
~[D(xl)]I{c(xt,x~)+h(xt)+bt(xz)+[K](x I -xs)} + dt z
[-~t2{c(xl,x2)+h(Xl)+bt(x2)}
+[K].~ 2 - [ N ( x , ) ] - '
[rl]-'
[K] (x, - x3)
The control law that cancels the nonlinear terms present in the transformed model is given by: 1: = f(y) + [M(yt) ] v
(16)
Applying the control law to the feedback linearizable model (13), assuming no uncertainties, yields the next system of linear, first order, controllable differential equations: .P : [A] y + [B] v
(17)
The vector v is chosen to stabilize the transformed linear system tracking the desired trajectory: I,' = J ) i d - [ K i ] { Y 4 -
Y4d}-[K2]{Y3- Y3d}-[K3]{Y2- Y2d}-[K4]{Yl- Yld}
(18)
with [K,], [K_~],[K3] and [K4] being gain matrices which are selected in such a way as to place the poles of the closed loop system in the left hand side of the complex plane, andy,a being the desired trajectory of the i'h state variable. Selecting the gain matrices as follows: [KI]
= 4~.[/] ,
[K2] = 6~2[/] ,
[K3] = 4~3[/] ,
[K4] = ~4[/]
(19)
where ~. is a positive number, results in the next uncoupled, exponentially stable error dynamics: e (4) + 4)re (s) + 612e ~2) + 413e (t) + ~,4e = 0
(20)
with e = Yl -Yld, etl)= de/dt, e ~2)= d2e/dt :, etc. 4. SLIDING MODE CONTROL The Feedback Linearization control strategy requires exact knowledge of the kinematic and dynamic parameters of the manipulator to achieve best results. However, most of the times, such information is available only in approximate values, and some parameters may change due to normal operation conditions. For this reason, it is desirable to seek a control system which is able to tolerate some degree of parameter uncertainty and still maintain a good dynamic performance. In a Variable Structure Control system, a discontinuous control law is employed to guide the system to a switching surface. Once the system is on the switching surface, the control law guarantees that the dynamics are insensitive to parameter variations. The feedback linearization control law is employed with a discontinuous outer loop, to construct the new "sliding mode" control law. Starting with the transformed feedback linearizable model (13), one can apply a control law which is evaluated using approximate values for the manipulator parameters:
Current Advances in Mechanical Design and Production, MDP-7
42
:]Cv)
[Mfy,)]
+
v
(21)
where ^ means that the term is calculated with approximate parameters, instead of the actual ones. Implementing the control law leads to the following nonlinear perturbed model" ~= [All, + [B][M(yl)]- I[~(y)_ f(y))+ h~/r0,l)v] = [A]y+ [B][~(y)+ ([Ill(y)] + [/])v]
(22)
where the nonlinear perturbations are defined as ~(V) = [M(Y|)]- | { ](.V) -f(Y) }
[ql(V)] = [M0'|)] -| [/I;/(,Vl)] - [ / ]
(23)
The problem at this point consist of selecting the outer loop control law v to stabilize the system in spite of the bounded modelling errors represented by ~(.V) and [r For this purpose, a Lyapunov based approach is used. A switching vector of the form S : [K,I{ y, - Yd, } + [K21{ Y2 - Ydz } +[/(31{ Y3 - YdJ } + { Y4 - Yd~ }
(24)
is selected. The scalar Lyapunov function is defined as: i =
v(s) : ~
/1
i=l
IS, I
(25)
where St are the scalar components of the switching vector S. The function V(S) >0 for S , 0. The control is chosen such that I;" ~ -p < 0 for a given p > 0 whenever S , 0. This assures stability, according to Lyapunov's second method, and it leads to the following conditions, which are known as the sliding conditions:
S t . S~< 0
i : 1, 2, ..n
(26)
which imply that Si > 0
if
Si < 0
or
Si < 0
if
S i> 0
(27)
To satisfy the condition for the existence of the sliding regime (26), the derivative of S is found:
"~ : [KI]0~2 -Yd2)
+
[K2]0~3 -Yd3)
+
[KI](Y4 -fld4)
(28)
+ (f~4 -Yd4)
= [g,]{v= -Yd~) + [r2]~v~ -YdJ) + [/3]0'4 -Yd,) + (0{v) + ([*(Y)] + [q)v - ~ d ,
The outer loop control v is chosen as v - ~
- [r,]
ly2 - ya2}
- [to21 Lv3 - y ~ }
- [r31 {y4 - yo~} - [ r , = ]
sgn{s}
(29)
where sgn {S} is the vector formed by applying the scalar sign function to the switching vector and [K.,m] is specified below. Substituting (29) in (28) results in the following expression:
S=r
-Yd2} - [KI]{Y3-YaJ}- [KI]{Y4-Yd4}}-{II/{Y} +[l]}[Ksm]$gn(s) (30)
Assuming some bounds in the uncertainty, the matrix of switching gains [Ksm] can be specified: II~(.v)*
[r
[gt]{Y2-Yd2}-
[K2]{Y3-Yd3}-[K3]{Y4-Yd4}}ll
(1 - ~)-160,) [/]
(32)
If the switching surface gain matrices are designed as follows [Kl] = 3X [/]
[K2] = 3~.2 [ / ] ,
[K3] = Z3 [/]
(33)
where ~ is a positive real number, then when the system is in the sliding regime, i.e. S=0, the dynamics of the system are given by the following reduced order, stable differential equation:
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Current Advances in Mechanical Design and Production, MDP-7
(34)
e (3) + 3~. e (2) + 33.2 e (t) + ~3 e = 0
which is independent from the manipulator parameters. In practical applications, the presence of the discontinuous switching may result in control chattering. To avoid this the boundary layer concept is used, where instead of trying to keep the system dynamics on S = 0, one tries to keep S close to S = 0, by employing a continuous approximation of the sgn function, the sat function:
: I sgn(S/f3), IS,/13,1 > ] [ s/f~, Is/f~, l
Mobile robot
Recommended Steer angle & Linear velocity
"~ Left
obstacle avoidance Fig. 3. Fuzzy guidance control.
I-- -PLinear velocity
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Each navigator is a task oriented control algorithm and designed as a separate fuzzy system to compute the recommended steering angle and linear velocity independently. In goal seeking, the distance between the robot position and aim point (d)is used to compute the recommended linear velocity (vr). While the deviation angle (~-h) to move towards the aim point is used to determine the recommended steering angle (0~). In obstacle avoidance, the capturing information from sensors determines the recommended steering angle, while the recommended velocity is reduced to half in order to decrease the robot inertia. Finally, the minimum risk criterion detects the applied control to the robot and the final decision will be taken as illustrated in Fig. 3.
In goal seeking, the crisp input to the fuzzy system is the distance (d) to compute the recommended velocity. Input membership functions are triangular in form (VC, CL, SM, ME and LA). They are very close, close, small, medium and large respectively. Output membership functions are singleton (ZE, VS, SM, ME, LA). They are zero, very small, small, medium and large respectively. Fuzzy rules are very simple as: If d VC Then v~ ZE If d CL Then vr VS If d SM Then v~ SM If d ME Then vr ME If d LA Then Vr LA The recommended steering angle Or is computed as function of the deviation angle (~-h). The input membership functions are triangular in form (NL, NS, ZE, PS and PL). They are negative large, negative small, zero, positive small and positive large respectively. The output membership functions are singleton (LH, LF, ZE, RG and RH). They are turn left hard, turn left, turn zero, turn right and turn right hard respectively. The fuzzy rules are: If (~-h) ZE Then Or ZE If (~-h) PS Then Or LF If (~-h) PL Then Or LH If (~-h) NS Then Or RG If (~-h) NL Then Or RH
In obstacle avoidance, the measured distances RS, LS and FS are fuzzified using membership functions CL, NE and FA. They are Close, Near, and Far respectively. The visual zone for front side is a symmetric 30~ Right obstacle avoidance rules: If RS CL Then If RS NE Then If RS FA Then Left obstacle avoidance rules: 9If LS CL Then If LS NE Then If LS FA Then Front obstacle avoidance rules: If FS CL and If FS CL and If FS NE and If FS NE and If FS FA Then
Or Or Or
LH LF ZE
Or Or Or
RH RG ZE
RS LS RS LS Or
FA FA FA FA ZE
Then Or Then Or Then Or Then Or
RH LH RG LF
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Current Advances In Mechanical Design and Production, MDP- 7
When the robot sensors detect the front obstacle only, either right or left steering can be taken. The decision depends on the situation and which is much better to be taken. In decision making, Each fuzzy system produces its output, then the decision making block
has to select the values of recommended variables. The minimum risk criterion aims to avoid the nearest obstacle by computing the minimum distance as follows:
(5)
J = Min(LS, RS, FS, d) The recommended steering angle is selected as" If J=LS Then Left obstacle avoidance If J=RS Then Right obstacle avoidance If J=FS Then Front obstacle avoidance If J=d Then Goal seeking 4. S I M U L A T I O N W O R K
The robot motion is simulated on a PC using MATLAB 5. The sampling time is chosen as 0.1 see., xl=l see., x2 = 0.5 see. and x3 = 1 see. Input membership functions are given in Fig. 4. _.__
0 V
0.5
FA
1 PL
LA d(m)
(,-h) p~
0
0.1
0.5
1
2
-90
-30 0 30
90
Fig. 4. Input membership functions. While the output membership functions are as shown in Fig. 5. Z
VS
SM
ME
LA
RH
RG
E
LF
LH
9 ...............................................................
0 0.01 0.1
1).3
(1.5
. 3 0 ............. i 5
............
Fig. 5. Output membership functions.
..........
.............
~
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Oarrent Advances in Mechanical Design and Production, MDP-7
First results are obtained to show the goal seeking navigation without obstacles as in Fig 6. The robot starts to move from (0,0) toward the aim point (10,10). 12 1(
y-axis in meters
,
/
,/
x-axis in m e t e r s i
-2
I
0
I
2
,
I
4
,
I
6
,
I
8
10
12
Fig. 6. Goal seeking in working plan (no obstacles).
The obtained result shows that the goal seeking is achieved. Next results show the robot navigation to avoid obstacles at (2,2) and (6,4). We observe that the proposed controller has been succeeded to avoid obstacles and reached to the aim point within a finite time. The sharp turn to avoid the second obstacle is due to the high-speed value at this point. 1; 1(
y-axis in meters t"
/
/
/
0
x-axis in m e t e r s !_
!
-2
l,
0
,
I
2
I
4
I
6
,
8
I
12
10
Fig. 7. Robot navigation to avoid obstacles.
Next figures show the velocity and heading angle curves during obstacles avoidance. O
. ~
.
O1
,
150
.............
Robot velocity
............. Heading angle
10
i" 0
5 seconds |
2O 40 60 Fig. 8. Linear velocity.
,
0
20
,rn-e,n secon,,,, 40
Fig.9. Heading angle.
60
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Current Advances in Mechanical Design and Production, MDP- 7
We observe that the robot velocity is reduced to half two times while it was trying to avoid obstacles. Robot reached the aim point and is stopped after 63 seconds. The robot starts to move with 0 heading angle. It turned left towards the aim point then it turned right to avoid the first obstacle and then turned left to avoid the second one. Finally, it turned right and reached with a steady state 37 degrees heading angle. 5. CONCLUSIONS This paper presents a simple fuzzy guidance control to navigate a mobile robot in uncertain environment. The design objectives are broken down to implement each one independently. Then, the final decision is taken based on which one of all navigation problems has the minimum risk to avoid obstacles. Each navigator is designed using a simple fuzzy system (single-input single-output). The obtained results by simulation affirmed that the proposed control algorithm has the potential to achieve the control objectives such as: goal seeking and obstacle avoidance. In the future work, path planning will be considered as well as the dynamic obstacles avoidance. Also, coupling problem between steering and velocity control loops will be treated to improve the system performance. REFERENCES 1. Arrue, B.C., Cuesta, F., Braunstingl R.and Ollero A., "Fuzzy Behaviors Combination to Control a Non-Holonomic Mobile Robot Using Virtual Perception Memory", Proc.of 6th IEEE Int.Conf. on Fuzzy Systems, vo1.(3/3), PP. 1239-1244, Spain, July, 1-5, (1997). Garcia-Gerezo, A., A-Mandow and Lapez-Baldan, M. J., "Fuzzy Modelling Operator Navigation Behaviors", Proc.of 6th IEEE Int.Conf. on Fuzzy Systems, vol. (3/3), PP. 1339-1345, Barcelona, Spain, July, 1-5, (1997). 3. Yoichiro Maeda, "Beahvior Learning and Group Evolution for Autonomous Multi- Agent Robot", Proe. of 6 th IEEE Int. Conf. on Fuzzy Systems, vol. (3/3), PP. 1355-1360, (1997). 4. Urzelai, J., Uribe, J. P. and Ezkerra, M.," Fuzzy Controller for Wall-Following with a Non-Holonomous Mobile Robot", Proc.of 6th IEEE Int.Conf. on Fuzzy Systems, vol. (3/3), PP. 1361-1368, Barcelona, Spain, July, 1-5, (1997). 5. Steven G. Goodridge, Michael G. Kay and Ren C. Luo, "Multi-Layered Fuzzy Behavior Fusion for Reactive Control of an Autonomous Mobile Robot", Proc. of 6th IEEE Int. Conf. on Fuzzy Systems, vol. (1/3), PP. 579-584, Barcelona, Spain, July, 1-5, (1997). 6. Vaneck, T. W.,"Fuzzy Guidance Controller for an Autonomous Boat", IEEE Control Systems, vol. 17, n~ 2, PP. 43-51, (1997). 7. Bing-Yung Chee, Sherman Y. T. Lang and Peter W. T. Tse, "Fuzzy Mobile Robot Navigation and Sensor Integration", Proc. of 5th IEEE Int. Conf. on Fuzzy Systems, vol. (1/3), PP. 7-11, New Orleans, USA, Sept., 8-11, (1996). 8. Sng Hong Lian, "Fuzzy Logic Control of an Obstacle Avoidance Robot", Proc. of 5th IEEE Int. Conf. on Fuzzy Systems, vol. (1/3), PP .26-30, USA, Sept., 8-11, (1996). 9. Debay, P., Eude, V., Said Hayat and Edel, M., "Fuzzy Control for the Future Automatic Guidance Near the Bus Stations", Proe. of 5~ IEEE Int. Conf. on Fuzzy Systems, vol. (1/3), PP. 660-666, New Orleans, USA, Sept., 8-11, (1996). 10. Zadeh, L. A., "Fuzzy Sets", Inform. Contr, Vol.8, pp. 338-353, (1965). 11. Mamdani, E. H., and Assilian, S., "An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller", Int. J. Man-Machine Studies, Vol. 7, pp.l-13, (1975). 12. Lee, C. C., "Fuzzy Logic in Control Systems: Fuzzy Logic Controller-Part I & II", IEEE Trans., Syst. Man, Cybern., vol. SMC-20, n~ 2, pp. 404-435, (1990). .
Current Advances in Mechanical Design and Production Seventh Cairo University htternational MDP Conference Cairo, February 15-17, 2000
75
FUZZY LOGIC SLIDING MODE CONTROLLER FOR DC DRIVE
Ibraheem, A.A.', Bahgat, A.'" and Abdel Motelb, M.S.* "Electronics Research Institute, National Research Center, Dokki, Cairo, Egypt "Faculty of Engineering, Cairo University, Giza, Egypt,
ABSTRACT Sliding mode control is used now in the speed control of electric drive systems. It provides attractive features such as fast dynamic response, insensitivity to variations in plant parameters and external disturbance. However, chattering is one problem, which limits the use of the sliding mode to control DC motors in industry. This paper presents a new fuzzy sliding mode controller for the speed control of DC motor drives. The fuzzy controller is used to adjust the switching gain constant in the sliding mode controller to guarantee the system stability and practically to eliminate the chattering problem. The performance of the proposed system is compared with that of traditional sliding mode under different operating conditions. KEYWORDS Fuzzy Logic Control, Sliding Mode Control, Robust Control, Speed Control of DC Motor 1. INTRODUCTION The DC motors have been commonly accepted in industrial applications, where fine speed adjustment is needed. Some of applications, such as steel rolling mills and paper mills, are characterized by fast and large changes in operating conditions. Electric traction systems require high starting torque. In other applications such as computer numerical control (CNC) precision speed pattern has to be followed with minimum absolute error. Variable structure system theory has gained attention over the past decade as the method may offer robustness in the control of electric drives and power electronics [1-4]. In the variable structure system control (VSC), the system response is forced onto a predetermined sliding surface and towards a pre-specified point in the state-space, using predetermined switching gains and variables. The switching surface is solely defined by parameters that are independent of the plant model, so the sliding mode dynamics are completely insensitive, or invariant, to bounded plant parameter changes. Under certain conditions, the sliding mode dynamics are also invariant to bounded external disturbances. Included in the basic design approach are the specifications of the switching surface and the design of a control input that meets the reaching conditions, which declare that the control always drives the state onto the switching surface [5-8]. A disadvantage of the sliding mode controllers is that the discontinuous signal produces chattering dynamics, so that the control system can be switched from one value to another in very fast way. In practical systems, it is impossible to achieve the high switching control that
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Current Advances in Mechanical Design and Production, MDP- 7
is necessary to realize most of VSC designs. There are several reasons for this. One reason is the presence of finite time delays for control computation. Another reason is the physical limitations of actuators [9]. Different schemes [4, 10, 12] have been proposed in literature to eliminate chattering. However, these do not solve the problem completely. One of these schemes is to define a boundary layer in the vicinity of the sliding surface and then use a continuous approximation of the switching function in the layer. Although chattering can be eliminated by the use of boundary layer, a loss of robustness can result. The main objective of this paper is to present a new VSC with fuzzy logic, which is used to adjust the switching control gain of the sliding mode. The Fuzzy Variable Structure Controller (FVSC) is simulated using the Matlab. The speed control of a DC motor has been tested under different loading and operating conditions using the developed FVSC. 2. SLIDING MODE CONTROL Behavior based sliding mode control (or variable structure system) control (SMC) is a robust one that depends very little upon the model of the plant to be controlled. The system response in the phase-plane is forced onto a predetermined "sliding surface" and towards a prespeeified point in state-space, using predetermined switching gains and variables as shown in Fig. 1, where x is the state and x is its derivative. In position control, x represents the position error and x" represents speed, whereas in speed control, x presents speed and x presents acceleration. And the controlled system response is insensitive to parameter of the motor and depends only on the slope of the sliding surface s ( x , t ) [4].
S0
X
S =0
S >0
I
S 0. s(x,t) = 0
(3)
when e(O)= 0
( x = x~ )
The sufficient condition for this behavior is to choose the control value so that [5],[7] 1 d (s2(x,t)) < _qls I "~-~ -
.
'
rI > 0 -
(4)
Considering s 2 (x,t) a Lyapunov function, it follows from (4) that the controlled system is stable. From phase plane, we obtain that the system is controlled in such a way that the state always moves towards the sliding surface. The sign of the control value u must change at the intersection of the state trajectory e(t) and sliding surface. In this way, the trajectory is forced to move always towards the sliding surface, then the existence and convergence condition can thus be re-writing as: Then
s.sO
(10)
such that k is the switching gain constant and the sliding line s is defined as a function of the dco error signal (e = -x~ = co - co,,/) and the error derivative ( e = x 2 = - - ~ ) .
Current Advances hz Mechanical Design and Production, MDP-7
79
(11)
s = 2e+e = (co"~ef - 2 e
- f)
(12)
,JL o
K,,~
Such that f is the compensation term defined as:
(13) f:-
Z - . + 7 o., +
t,oJ
) JK,#
Since the control law (10) is a discontinuous across the sliding surface, it might lead to chattering. Thus the S a t ( s ) function is replaced instead of s i g n ( s ) function for improving the sliding mode controller characteristic [5-7]. (14)
u = fi - k * s a t ( s )
The sliding line slope 2 is taken to be 5 and therefore the sliding line time constant is (0.2 Sec.) and the control switching gain is chosen such that the existence condition in [5-7] given in (5) is always satisfied. The schematic diagram of Simulink program used for simulation of the DC motor, three-phase controlled Thyristor Bridge, and the sliding mode controller is shown in Fig. 4.
I
Torque
v.
~[
Lo~ Tortl~
~
""~1
A.SpeedOol
Nmse
~eT~o,.
[.... i SI I "~i
x
~s)
~...,'mm,d
,..o.o-2, i I
il ~
[
I
~ ~-~~ Sum 1'
Fig. 4. SIMULINK model of Sliding Mode controller 5. DESIGN OF THE FUZZY SLIDING MODE CONTROLLER In the fuzzy sliding mode controller (FSMC) scheme the sliding surface s ( t ) forms the input space of the fuzzy implication of the major switching rule and the switching gain is written in the form of fuzzy rule, given by:
Current Advances in Mechanical Design and Production, MDP-7
80
I F s is Aj and s" is B j T H E N K is C j
where j = 1.... n and n is the number of rules and s(t) is the sliding surface, and the major principal of a FSMC can be represented by the following equation
(15)
u = -KF,,,,(e,e,2).sgn(s )
such that the control gain KF=:z (non-linear, non-continuous and positive function of error e, derivative of error e and 2 ) is driven according to the following rule: "above the switching line a negative control output is generated and a positive one below it (ANBP), where close to the sliding surface the control outputs are smaller than at a larger distance" [13, 14]. The fuzzy input membership functions (on a normalized range of 1) are illustrated in Fig. 5, and the rule base in the fuzzy association matrix (FAM) is shown in Table 1. The output membership functions "N", "Z", and "P" correspond to fuzzy singletons a t - l , 0, and 1, respectively. The composed output is calculated using center of sum approach. NB
I
-1
NM i~ISAZ PS PM
I~B
PB
-0.4-0.2 0 0.2 0.4
1
NM NS
?VV
'1
-0.4 -0.2 0 0.2 0.4
J
Fig. 5. Input membership function of FSMC Table 1. FAM matrix of FSMC
O
rp~
!pB PM PS AZ NS NM NB
NB p N N N N N N
NM P P N N N N N
NS p p p N N N N
S ZE P P ....p Z N N N .,
PS P P P P P N N
PM P P p P P N N
PM P P P P P P N
6. SIMULATION RESULTS STUDY To indicate the effectiveness of the developed FSMC three cases are studied: 1. Transient response in case of motor starting without load 2. Sudden load change 3. System parameter variation
1
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Current Advances in Mechanical Design and Production, MDP-7
In addition a comparison between the FSMC and SMC regarding chattering phenomena is studied. Table 2. Show the parameters of the DC motor for simulation study. Table 2. Parameters of the DC motor
175 Watt rated power 220 V rated armature voltage 1.3 A. Rated armature current 1500 Rated motor speed Armature resistance Armature inductance Voltage constant L Torque constant Moment, , _ of inertia constant Viscous fraction constant
26.6 0.1h 1.47 V/rad/sec. 1.47 N.m./amp 0,0041-kg~.m2 0.00i9 m-/Sec. ~
Figure 6 shows the motor speed in case of starting at no load using each of the two controllers SMC and FSMC. Figure 7 shows a zoom into motor speed at the instant of sudden load change at t =1.5 seconds. The corresponding armature current responses are illustrated in Fig. 8. It is clear from these figures that the speed response is relatively faster to reach a steady state value and has a good disturbance rejection capability for load disturbance in case of FSMC compared to the case of SMC. 1540
12001400 / f
(__
"-"-'-"
;s.c I
.~ IOOO'
I
15.10 1520 1510
Fs.cl
. SMC ]
LoadingIn8tant -: ............r....... l . ~ o ~ . . , ~ "
1480 1470
0
0
--
,!
0'5 Time (See)
Fig. 6. Motor speed transient of the SMC and FSMC
1440 i
"
215. . . . . . ~me ( $ e c )
2
Fig. 7. Zoom in to loading instant of the SMC and FSMC
Figure 9 shows the motor speed and the armature current in case of starting at no load using SMC controller for two different values of motor load inertia. Figure 10 shows the motor speed and armature current in case of starting at no load using FSMC controller for the same study case. It is clear from these two figures that both controllers are robust to system parameter variation, but the peak current drawing in case of SMC is greater than that in case of SMC. Figure 11 shows the motor armature voltage for both controllers SMC and FSMC. It is clear from this figure that the control signal (armature voltage) in case of FSMC is smoother than that of the SMC, because the fuzzy logic controller (FLC) provides better damping and reduces chattering. The oscillations of the armature voltage in case of SMC controller may
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Current Advances in Mechanical Design and Production, MDP-7
cause damage for the motor drive and the control equipment. The amplitude of these oscillations can grow up when increasing the gain in order to reduce the rise time as FSMC. ,-~ 1500 r
2.S
'
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7. CONCLUSIONS This paper presents the design of Fuzzy Sliding Mode Controller for the speed control of DC motor drive. The controller is based on a sliding mode control where its switching control gain can be adjusted by a fuzzy controller. This controller not only improves the system transient response, but also yields better performance in terms of steady state-error, stability, chattering elimination and robustness criteria based on rejecting disturbance. ACKNOWLEDGMENT This research work has been conducted in the frame of FRCU project No. 205. This project is financed through USAID in Egypt and is executed in collaboration with Oakland University and Case Westem and Reserve University in USA.
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REFERENCES 1. Emelyanov, S. V., "Theory of Variable Structure Systems", Moscow, Nauka, (1970). 2. Borojrvic, D., Naitoh, H, , and Lee, F. C, " Soft Variable Structure Based Adaptive PI Control for DC Motor Position Control", in Proc., IEEE, Industrial Applications Society Annual Meeting, pp. 283-288, (1986). 3. Utikin, I.V., "Sliding Mode Control Design Principles and Applications to Electric Drives", IEEE, Trans, Industrial Electronics, Vol. 40, NO. 1, Feb., (1993). 4. Habibi, S.R., and Richards, R.J., "Sliding Mode Control of An Electrically Powered Industrial Robot", lEE, Proc.-D, Vol. 139, No. 2, March. (1992). 5. Hung, J.Y., Hung, J.C., "Variable Structure Control: A Survey," IEEE, Trans, Industrial Electronics, Vol. 40, No. 1, Feb (1993). 6. Nandam, P.K. and Sen, P.C., "Accessible-State-Based Sliding Mode Control of a Variable Speed Drive System", IEEE, Trans, Industrial Appl., Vol. 31, NO. 4, Jun/Agu. (1995). 7. Slotine, J.J.F.," Applied Nonlinear Control", Prentice Hall, 2nd edition, (1991). 8. Buja, G.S., Menis, R. and Valla, I., "Variable Structure Control of An SRM Drive", IEEE, Trans, Industrial Electronics, PP. 56-79, Vol. 40, NO. 1, Feb., (1993). 9. Furuta, K., "VSS Type Self-Tuning Control," IEEE, Trans, Industrial Electronics, PP. 3744,Vol. 40, NO. 1, Feb., 1993. 10. Ackermann, J. and Utkin, V., "Sliding Mode Design", from Authors Publications WWW (e-mail:
[email protected]) 11. Spurgeon, S.K. and Patton, R.J., " Robust Variable Structure Control of Model Reference Systems," lEE, Proc., Vol. 137, Pt. D, No. 6, Nov. (1990). 12. Hung, J.Y., Nelms, N. M. and Steven, P.B., "An Output feedback Sliding Mode Speed Regulator for DC Drives", IEEE, Trans., Industrial Appl., Vol. 30, No. 3, May/June (1994). 13. Lo, J.C. and Kuo, Y.H., "Decoupled Fuzzy Sliding-Mode Control", IEEE, Trans., Fuzz., Syst., Vol. 6, No. 3, Aug. (1998). 14. Glower, J.S. and Munighan, J.," Designing Fuzzy Controllers from a Variable Structure Standpoint," IEEE, Trans, Fuzzy Sys., Vol. 5, No. 1, Feb. (1997).
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Current Advances in Mechanical Design and Production Seventh Cairo UniversiO, International MDP Conference Cairo, February 15-17, 2000
85
DRIVER MODELING USING FUZZY LOGIC CONTROLS FOR HUMAN-IN-THE-LOOP VEHICLE SIMULATIONS Zeyada, Y.*, EI-Beheiry, E. **, EI-Arabi, M. ~ and Karnopp, D. ~176
*Assistant Professor, ~ Mechanical Design and Production Department, Faculty of Engineering, Cairo University, Giza 12316-Egypt. Lecturer, Department of Production Engineering and Mechanical Design, Faculty of Engineering, Menoufia University, Shebin EI-Kom, Egypt. oo Professor, Mechanical, Aeronautical and Materials Engineering Department, University of California at Davis, CA 95616, USA.
ABSTRACT It is a universal trend that automotive engineers tend to use fast digital computers to develop advanced vehicle systems. They can design, analyze, and test their systems using computer simulation before physically manufacturing them. It is still questionable how these advanced systems would react with the human driver. One way to deal with this problem is to develop a computer model that is capable of controlling the vehicle in a way similar to human driver behavior. Fuzzy logic inference systems are known of their great ability to simulate human reasoning process as well as the possibility of being further trained to mimic specific human control process. This paper presents a new driver model using fuzzy logic controls. The model is designed to control the longitudinal as well as the lateral motions of the vehicle by performing simultaneous steering and braking commands. The model is tested on a vehicle model having an integrated active steering and direct yaw control strategy as developed by this paper's authors in [1]. The results show success of this fuzzy model in simulating driver control actions in curve following and collision avoidance maneuvers. KEYWORDS Driver Modeling, Vehicle Dynamics, Fuzzy Logic, Collision Avoidance, Lane Following. I. INTRODUCTION One of the methods to include the human behavior in vehicle dynamics simulation is to design a driver model capable of performing control task close enough to what an average driver would behave. For such a task, the use of fuzzy inference systems is strongly recommended. This is due its ability to create a decision-making process based on the logical analysis of the system inputs, which suits the way of human analytical thinking procedure, i.e., they can map those functions that have no equivalent mathematical model or whose mathematical models are very complicated, s e e Sugeno [2]. Most recent developments in the application of the neural networks (NN) and fuzzy logic (FL) to vehicle systems have been summarized in a review article by Ghazi Zadeh et al. 1997 [3]. Wuertenburger and Isermann [4] have presented
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supervision of lateral vehicle dynamics by using a model-based scheme. The model is able to monitor the lateral motion of the vehicle utilizing an estimation of the road friction coefficient as determined by a driving state observer. Four wheel steering controllers were studied by Nagai et al [5]. The tire and the suspension nonlinearities were taken into account in the identification of the vehicle motion by integrating a NN parallel to a linear controller to improve overall performance of the controller. A driver model has been presented by Kageyama and Pacejka [6] which utilizes the so-called risk level concept for the course decision making process and a FL controller for the course tracking. A speed controller and steering torque controller are also integrated within the system. A feedforward neurocontroller with one hidden layer was employed to emulate driver's behavior by Nuesser et al. [7]. It has the vehicle speed, yaw angle, road curvature, road width, and lateral deviation of the vehicle from the path as inputs. The model returns the steering angle as an output. A FL driver utilizing yaw velocity and lateral displacement signals to produce the steering angle has been Studied by Xi and Qun [8]. This article is mainly devoted for developing FL model of human driver behavior for humanin-the loop simulations of advanced vehicle systems. The FL model not only performs driver steering inputs, as was the case in the majority of publications, but also provides human driver braking events. The model is trained to react to driving situations in both collision avoidance and curve following. 2. VEHICLE MODEL Two axes systems namely called OXYZ fixed in space axes-system and oxyz moving axissystem are used to describe a comprehensive nonlinear 14-DOF vehicle model as shown in Fig. 1. A condensed model of 4-DOF has been extracted which includes tire nonlinearities. These nonlinearities are essential to the study vehicle dynamics in emergency situations.
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Fig. 1. Schematics of vehicle body in pitch and roll modes The 4-DOF nonlinear model includes the longitudinal, lateral, yaw and roll motions. No pitch, heave, or tire deflections are considered. Moreover, the wheel spin rates are determined by implying a targeted slip ratio for each wheel separately. Those slip ratios are inputs to the model and are controlled by an appropriate control system such as the (ABS) anti-lock brake
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system or the (DYC) direct yaw control system. Details of both the comprehensive and the simplified models are in Zeyada [9]. 3. FUZZY LOGIC DRIVER MODEL
3.1. Sensigg System For the frizzy driver model, a sensing system is basically a group of variables that provide simple and enough information about the vehicle position w.r.t, roadway. Consider Fig 2., where there are five rays goes out from the vehicle center in left, right, straight-ahead, straight-left, and straight-right directions. The respective lengths of those rays measured from the vehicle center to the roadway borders are indicative quantities to determine the position of the vehicle. These quantities are referred to as DL, DR, Ds, DsL, and Dsn respectively. These distances can be measured in reality by many techniques such as laser ranging or electromagnetic sensing as implemented in intelligent vehicle highway systems. The angle between Ds and DSL is equal to the angle between Ds and DSR, and is referred to as 19.
DSL
Vehicle
Direction of motion
Fig.2. Sensing road geometry for fuzzy driver 3.2. Decision-Making System The decision-making system is a part of the driver model responsible for determining the amount of steering and braking necessary to keep the vehicle in the right course. It is mainly consisting of two modules, the steering module and the braking module. Both modules will be designed using fuzzy logic systems. The system will utilize the sensing variables described above as inputs and produces the appropriate steering and braking commands.
3.2.1. The steering module Generally, the amount of steering necessary to guide the vehicle in a roadway would be generated by two mechanisms. The first is called the preview mechanism that resulted from the anticipation of incoming curve on the roadway. The second is called the deviation mechanism which resulted from the deviation of the vehicle center from the center the roadway. The steering portion contributed by the first mechanism can be summarized by the following rules:
lf DsL > DSR then left steering is required, If Dsn > DsL then right steering is required. For the deviation mechanism the following rules can be written,
If DL > DR then right steering is required, If DR > DL then left steering is required.
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Current Advances In Mechanical Design and Production, MDP-7
The fuzzy steering module is designed using the above rules. It consists of two main parts, the left steering part and the right steering part. The left steering part is focused on the sensing variables that causes a left steering command. While the fight steering part is concerned with the sensing variables that causes a right steering command. The left steering part is mainly a fuzzy inference system that utilizes two normalized inputs and produces one normalized output that represents the left steering angle required. The normalized inputs are:
[)R
=
DW ~2-DR D w 12
DSR DSR max
,andbsR=
(1)
,
where Dw is the road width and Ds~,~ is the maximum allowable value for DsR. The normalized output is, ~L = 6/./Smax, where ~,~ is the maximum allowable steering angle. The above-normalized variables are only allowed to vary in the range from 0 to 1. Figure 3 shows the membership functions of the inputs and output of the left steering fuzzy system. Table 1 shows the inference rules. Similarly the right steering fuzzy system can be handled. For a vehicle runs on the road center with no incoming curves ahead the left steering system opposes the right steering one, and consequently no net steering output is generated. Now consider the vehicle gets closer to the right shoulder (deviation case), this will cause/)R to increase while /)L is held zero. The result is a left steering command. Also, for an incoming
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Current Advances in Mechanical Design and Production, MDP-7
left turn (preview case) DSR will get shorter causing the system to react by producing a left steering command. The above behavior can be justified by inspecting the inputs/output surface for the left steering fuzzy system shown in Fig 4. The figure also shows how the system performs for simultaneous preview and deviation inputs. The above discussion is also applicable for the right steering fuzzy system. Notice the flat zero area (i.e. no steering output is assigned) associated with low values of/gR, it is introduced to simulate the human driver behavior of tolerating a small shift from the center of the lane. Table 1. Fuzzy inference rules for the left steering system
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90
Current Advances in Mechanical Design and Production, MDP-7 DST
:
(2)
Uo /2aDEC ,
where, u 0 , is the vehicle initial velocity at the time of brake application, and aDE C is the deceleration rate which can be obtained from: aOE C
= F B /mt
,
(3)
where F B is the total braking force and m t is the total vehicle mass. Assuming equal vertical loads on the four wheels and ignoring the dynamic load transfer due to braking, one obtains this simple equation aDE C
:
(4)
4FB(tire )~mr,
where FB(tire ) is the braking force per tire. Using the tire model and assuming 0.15 longitudinal slip the braking force per tire is about 80% of the vertical load applied i.e.: (5)
F B(tire)= O.8(mtg/4).
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Current Advances #t Mechanical Design and Production, MDP-7
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Table 2. Inference rules for the braking fuzzy system .
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4. SIMULATION, RESULTS AND DISCUSSIONS The developed driver model here is employed in this section throughout two simulated emergency maneuvers. The first driving situation simulates the case of an attempt to handle a sharp 90 degrees turn with relatively high speed at the turn entrance. Figure 6 shows the road turn, the starting point is 200 meters far from the turn start. At this point a speed of 45 m/s (about 160 km/hr) is assumed. The simulation includes the performance of a vehicle equipped with the fuzzy active-steering differential-braking control system developed in Zeyada et. al. [ 1]. Figure 7 shows the vehicle deceleration due to braking action. Figures 8 and 9 show the breaking and steering commands performed by the fuzzy driver model. Lastly, Fig 10 shows a 3-D plot for the braking command related to the vehicle position within the turn.
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Current Advances In Mechanical Design and Production, MDP-7
92
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The second simulation presented here is performed to simulate the situation where the driver wants to change Lane due to an accident occurrence in his lane ahead. A schematic diagram for this maneuver is shown in Fig. 11. To create a critical situation the initial velocity of the vehicle is assigned to 45 m/s (about 162 km/hr), and the initial distance between the vehicle and the accident blockage is assigned 50 meters. Figure 12 shows the vehicle path for different values of steering sensitivity gain (SSG). This is a premature to adjust the fuzzy driver model to suit various conditions encountered in different simulations. The steering and braking commands performed by the driver model are given in Fig. 13 and Fig. 14. Considering these plots, one can see that the fuzzy driver model managed to steer the vehicle throughout this critical situation with a performance close to what a human driver is expected to do.
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Current Advances in Mechanical Design and Production, MDP-7
Targeted lane
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However, a problem of this model is the erratic steering and braking behavior observed in the above characteristics. Despite the fact that this behavior could also be existed in a real critical driving situation due to the stressful nature of the maneuver, there are ways to train the fuzzy model to behave closely to an existing maneuvering performance observed by a human driver. Instead of manually tuning the fuzzy rules developed earlier in this chapter, a training procedure can be employed to tune those rules using input-output data pairs obtained from experimental observations. The resulting fuzzy system would give input-output characteristics closer to be experimental observations depends on the accuracy of the training procedure. For simplicity, a tuning process would be established for only the braking module of the fuzzy driver model. Artificial data pairs will be used here due to the lack of experimental measurements. A training scheme is now used to create a fuzzy braking system that performs according to the desired characteristics. The fuzzy model structure used for this purpose is the
.
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Current Advances in Mechanical Design and Production, MDP- 7 1.4
E0 U
D e s i r e d characteristics
F u z z y model output
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Sugeno [2] type where outputs of the fuzzy system are mainly represented by a linear combination from inputs. Using Matlab software, a hybrid training technique is used to train the system. Three training epochs where performed to lower least square error below 1E-5. The resulted characteristics of the tuned braking module are depicted in Fig. 15. 5. CONCLUSIONS On the basis of the above discussions, one can say that fuzzy inference systems can be very powerful tool when it comes to human reasoning. The introduced fuzzy driver model here is capable of guiding the vehicle in both lane following and collision avoidance situations. It provides the vehicle with appropriate steering and braking commands, and further training of this type of models might lead to performance features very close to desired ones. REFERENCES 1. Zeyada, Y., Kamopp, D., EIArabi, M. and EIBeheiry, E., "A Combined active Steering Differential-Braking Yaw Rate Control Strategy For Emergency Maneuvers", SAE Paper No. 980230, (1998). . Sugeno, M., "An introductory Survey of Fuzzy Control", Information Sciences, Vol. 36, pp 59-83, (1985). 3. Ghazi Zadeh, A., Fahim, A. and EIGindy, M., "Neural Networks and Fuzzy Logic Applications to Vehicle Systems", Int. J. of Veh. Des., Vol. 18, No. 2, pp 132-193, (1997) 4. Wuertenburger, M. and Isermann, R., "Model Based Supervision of Lateral Vehicle Dynamics", Proceedings of the American Control Conference, Vol. l, pp 270-280, (1994) 5. Nagai, M., Ueda, E. and Moran, A., " Nonlinear Design Approach for Four-WheelSteering System using Neural Networks" Veh. Sys. Dyn., Vol. 24, No. 4-5, pp 329-342, (1995). 6. Kageyama, I. and Pacejka, H. B., " On a New Driver Model with Fuzzy Control", supplement to Vehicle System Dynamics, Vol. 20, pp 314-324, (1991). 7. 7.Nuesser, S., et al., "Neuro-Control for Lateral Vehicle Guidance", IEEE Trans. On Microprocessors, Vol. 13, No. l, 1993, pp 57-66, (1993). 8. Xi, G. and Qun, Y.," Driver-Vehicle Environment Closed-Loop Simulation of Handling and Stability using Fuzzy Control Theory", Veh. Sys. Dyn., Vol. 23, pp 172-181, (1994). 9. Zeyada, Y., "Integrated Active Systems in Emergency Maneuvers for Automobiles", Ph. D. Thesis, Cairo University, Giza, Egypt, (1999)
Current Advances in Mechanical Design and Production Seventh Cairo UniversiO' hffernational MDP Conference Cairo, February 15-17, 2000
95
OPTIMAL ACTIVE SUSPENSION WITH PREVIEW FOR A QUARTER-CAR MODEL INCORPORATING INTEGRAL CONSTRAINT AND VIBRATION ABSORBER
Abduijabbar, Z.S. and EIMadany, M.M. Mechanical Engineering Department, King Saud University, Riyadh, Saudi Arabia E-mail:
[email protected] and
[email protected] ABSTRACT An optimal multivariable controller with preview has been designed for suspension control of vehicles. The controller takes the form of a linear quadratic regulator with supplementary states for added integral action. The effect of preview control on the performance of a quartercar model equipped with a passive vibration absorber is examined. The vibration absorber is used to reduce the axle vibration without sacrificing the ride comfort. It is assumed that the road irregularities ahead of the vehicle are measured, and this information is used to generate an enhanced control law in order to provide further improvements in the performance over that without preview. An optimal performance comparison of active systems with preview, optimally designed using full-state feedback with and without a passive vibration absorber is presented and discussed.
KEYWORDS Suspension Design, Vehicle Control, Preview Control. I. INTRODUCTION The design methodology of road vehicle passive suspensions has been well established, and an automotive engineer must make a trade-off between ride comfort and road holding. Active suspension technologies have been investigated and developed in order to improve the ride quality [ 1-3]. Unlike passive systems comprised of springs and dampers, active suspensions use force-generating elements, driven by an external power source. These elements can be made to respond to any set of measurements of the system parameters. Thus, the active suspensions can, in principle, continually supply and modulate the flow of energy, and can be adapted to the instantaneous operating conditions by changing their characteristics accordingly. However, the performance improvements of the active system over the passive ones are limited because of the lack of sufficient information about the incoming road input. This lack of information forces the designer to select the control laws that are acceptable to a large class of inputs, leading to systems that are optimally in an average sense. With look-ahead preview, road elevation information ahead of the vehicle is gathered and utilized in controlling
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Current Advances In Mechanical Design and Production, MDP-7
suspension actuators. Consequently, the required control force can be synchronized in an efficient way, leading to more relaxation of the trade-off between ride comfort and road holding [4-13]. It is well known that it is not the "optimal" nature of the design alone that makes the linear quadratic regulator (LQR) attractive but the fact that it provides the control engineer with a straightforward technique to obtain a multivariable control law. Although LQR is intended for application as a regulator in its classical form, manipulation of the performance index can extend its use to servo problem [14]. The addition of integral action on the controlled variable of the vehicle suspension deflection can increase the robustness of the suspension design through compensation of vehicle model mismatch and the elimination of steady-state error as a result of constant disturbing forces acting on the system. This paper deals with the synthesis of an optimal finite preview multivariable integral controller for active suspension system based on a three-degree-of-freedom vehicle model. The controller utilizes knowledge about approaching road disturbances to prepare the active suspension system for the incoming input. An integral constraint is included in the performance index to achieve better attitude characteristics. The suspension control problem is described and the optimal control law is presented. The results of the analysis is applied to a quarter car model equipped with a simple vibration absorber to suppress the unsprung mass vibration. 2. VEHICLE MODEL A three-degree-of-freedom quarter-car model is shown in Fig. 1. The sprung and unsprung masses are denoted by mm amd m2, respectively. The active control force is assumed to be in parallel with a passive suspension comprising of stiffness kl and damping cl. The tire stiffness is k2. A vibration absorber of mass m3 is attached to the unspring mass along with a passive suspension having a spring stiffness k3 and a damping c3.
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Current Advances in Mechanical Design and Production, MDP-7
The vehicle is assumed to travel at a constant speed over a random road surface, which is approximated by an integrated white noise. Hence, the vertical road velocity disturbance, vi, is modelled as a white-noise input and it is specified by E[vi (t)] = 0 and E[vi (tl) vi (t2)] = V8 (tl - t2), where E[.] denotes the expectation operator, 8(.) is the Dirac delta function, and V = 2nx 10 .4 m2/s 3. It is also assumed that the rate of change of road elevation is measured at the distance L in front of the vehicle, i.e., vi (x), xc [t, t + tp] is known where tp = L/v and v is the forward vehicle velocity. The equations of motion can be written in standard state variable form: =Ax+Bu+Ew
(1)
The state variables are the suspension deflection xl, the tire deflection x2, the sprung mass velocity x3, the unsprung mass velocity x4, the integration of the suspension deflection x5 - ~ x! dt, the absorber mass deflection and velocity x6 and x7, respectively. The matrices of the governing equation (1) are given by
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0
-2%10) 1
2%10) 1
0
0
0
_pl0)2
0)2
2~10)1Pl
1
0
0
0
0
0
0
0
0
0
1
0
0
-1
0
0
0
2~3o33
0
0)2
_2~30) 3
_2~10)lPl_2~3o)3P3
0-0)2p3-2~30)3p
3
t
B-J0 o 1 - o ,
0 0 0], t
E=[0
1 0
0
0
0
0] , a n d w = v i .
The symbols are defined as follows: u = - x reactive force ml 0) 3 = ~ k 3 / m3 = 23n
input; co I = ffk I / ml = 2n rad/s; co 2 = ffk2 / m2 = 20n rad/s;
rad/s; Pl = ml / m2 = 10; p3 = m3/m2 = 0.1; ~l = el / 2ml0)l = 0.3; ~3=
c3 / 2m30)3= 0.4. The controller design is based upon the minimization of a quadratic performance index including sprung mass acceleration, suspension working stroke, tire deflection, integral action, and control force,
98
Current Advances in Mechanical Design and Production, MDP- 7
Ltx Ix]]
=
EI~:] + qtx~ + q2x~ + q3x] + pu2 ]
=
E
TuT
(2)
R u
where o4 + q l
0
-2~!c013
2~1Ol3
0
0 0"
0
q2
0
0
0
-291o~ 3
0
(2%,co!)2-(2~,co,)20
0 0
o
o
Q~
N =[o 2
0 0 o
o
0
0
0
0
q3
0 0
0
0
0
0
0
0 0
0
0
0
0
0
0 0
0 -2~1o I 2~lc01 0
0
0] T
R=l+p ql, q2, q3 and p are weighting factors which reflect the designer's preference. In this study, the corresponding values are ql = 5x10 3, q2 = 7x10 7, q3 = 104, and p = 0. 3. O P T I M A L PREVIEW CONTROL LAW Consider the linear system described by Eq. (1), given that the matrix A is asymptotically stable. Assume that w(t) is an unknown a priori input with zero mean and w(x), x c [t, t + tp] is given deterministically. Consider also the performance criterion of Eq. (2) in which Q is a symmetric and nonnegative definite matrix, R is a positive definite scalar and such that Qn = Q - NR "l N T is a nonnegative definite. Then the problem of determining an input which minimizes the criterion of Eq. (2) is called an optimal preview control problem. It is to be noted that solutions to finite preview control problem for linear active systems have been obtained using spectral techniques and Wiener filter theory [4], dynamic programing [5], and calculus of variation [6]. The control law is given by u='Olx'G2 r where
G! =
(3)
R-l[ N T + BTp] and G2 = R "l B T
Here, P is the symmetric positive definite solution of the Riccati equation P An + ATp - PBR-IBTp + Qn = 0,
(4)
Current Advances in Mechanical Design and Production, MDP-7
with
99
A, = A - BR "l N T.
The vector r is given by
r(t) =
'i e
(5)
PDw(t + a)da
o
where Ac = A - BR "1 (N T + BTp) is the closed loop matrix which is asymptotically stable. The control law of Eq. (3) is composed of a feedback part-R "! (N T + BTp)x which is an often used optimal control strategy for active suspensions without preview, and a preview controller -R "1 B ' r which is a feedforward part, containing the information about the road input obtained from preview sensors. The feedforward part utilizes preview to smooth out the vehicle response. The closed loop system is described by = Acx - BR-tBTr + Ew
(6)
The power spectral density matrix for the system of Eq. (6) is given by S o (co) = VHDD* H * where
H=
(7)
[jcoI-A c ]-1, tp
D = - B R - I B T f e Aca PE eJC~
+ E,
o
and * denotes matrix conjugate transposition. 4. RESULTS AND DISCUSSIONS In this section, selected results of the simulation of the quarter-car model shown in Fig. 1 are presented and discussed. Optimal control theory has been employed to find the optimum control force that minimizes the performance index given by Eq. (2). By changing the weighting factors p, ql, q2, and q3, an infinity of optimal systems can be identified and consequently a wide range of system performance characteristics can be obtained. The wheelhop damping ratio is kept approximately at 0.3 for the different vehicle suspension systems considered. Table 1 shows the rms response of the quarter-car model to random road disturbances, with and without preview. The results are presented for a passive system, and a multivariable proportional plus integral (PI) controller which is based on state variable feedback and integral control of the suspension deflection. The PI controller is considered with and without passive vibration absorber.
1O0
Current Advances in Mechanical Design and Production, MDP- 7
Table 1. RMS Response of the Vehicle to Random Road Inputs
(a) Without Preview . Passive PI . 0.0800 0.0708 0.0096 0.0078 0.0033 0.0032
. ,
rms body acceieration, g rms suspension deflection, rn rms tire deflection, m
_
(b) With Preview PI + PI Absorber 0.0590 0.0560 0.0062 0.0062 0.0021 0.0020
.
.....tp- 0.2s
tp = O.Is ,
,
irms body acceieration, g . rms suspension deflection, m rms tire deflection, m
i
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.0507
.
9
.
..
0.0050 0.0019 ..
PI + Absorber 0.0570 0.0082 0.0031 .
.
.
.
Pi + Absorber 0.054 0.005 0.002
For the systems without preview, the PI controller shows an 18% reduction in the rms suspension deflection, while the rms values of the other variables are kept almost the same compared to passive suspension performance. However, the PI with vibration absorber provides a significant improvement in the fide comfort which is manifested in the reduction of the rms body acceleration by a 29% reduction in comparison to the passive system. The rms values of the tire deflection quantifies the loss of contact of tire from the road. It may be observed that the rms tire deflection and hence the road contact is the same for the three suspensions considered. Therefore, the PI with vibration absorber provides improvements in body acceleration and suspension deflection without diverse effect on tire deflection. For the active systems with preview, marked improvements in body acceleration suspension deflection and tire deflection compared to active systems without preview are observed. In Fig. 2, the power spectra of the body acceleration, suspension rattle space, tire deflection and control force are shown. The results are presented for systems without preview. In terms of body acceleration, substantial benefits of the addition of the vibration absorber can be observed particularly at frequencies close to the body and wheel-hop frequency. There is a point on the body acceleration characteristic corresponding to the wheel-hop frequency, which is known to be an invariant point, i.e., it cannot be changed regardless of the control law used as in passive and PI controller without absorber. However, the addition of the absorber eliminates such a constraint and an improvement in body isolation is obtained in the whole frequency range of interest. At the wheel resonance frequency, the presence of the absorber reduces tire deflection, but increases it in the low frequency range. The suspension deflections for the PI and PI + absorber are very close. With preview control, tp - 0.2 s, shown in Fig. 3, a substantial reduction in the required rattle space at the low frequency range is obtained, eliminating the drawback of the PI controller without preview. Marked improvements in the sprung mass acceleration are observed compared with passive and active systems without preview. Drastic reduction in the tire deflection is obtained in almost the whole frequency range. In the presence of preview control, the frequency domain performances of the PI and PI + absorber, in general, are very close.
Current Advances in Mechanical Design and Production, MDP- 7
I0 -i
10 4
,
.
,_
l 01
.
N E10 m
qn
k
~" I0.i C) (n o.
o rn
100 I0 i Frequency. Hz
10
10 i
i0 i
10 0 I0 i Frequency. Hz 10 "l N
N
N ~ I O "l
" ~ 10 -?
~ tn o.
'~\
~10 4
\ I0 4
~
t.
10
10o 10 ~ Frequency. Hz
.~
10 2
\
10 o 10 ~ Frequency, Hz
102
Fig. 2. Response power spectra without preview. (--- passive, active (PI), " active (PI + Absorber) 10 -~ N
~
,,,
10 ~
10 "4
I0
/
\
N
E 10 4
4
~
0in 10 4 Q.
(2 a.
Y
I0 "i
10 0 10 t Frequency, Hz
10e 10 t Frequency, Hi:
10=
10 "1 /%
,,,•E
lO-V
,-
,'.', /
10" J
.
10
r~ - I 0 -a n
10 ,,
10o 101 Frequency. Hz
10 2
.
,
10 o 10 ~ Frequency, Hz
Fig. 3. Response power spectra with preview tp - 0.2 s (--- passive, active (PI), active (PI+Absorber)).
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Current Advances In Mechanical Design and Production, MDP- 7
5. CONCLUSIONS Optimal multivariable controllers with and without preview, with and without passive vibration absorber have been designed for the ride control of road vehicles. The performance characteristics of such suspension systems are evaluated and compared with a passive suspension system. In case of no preview, the actively controlled suspension equipped with vibration absorber may be designed to provide excellent ride comfort without diverse effect on suspension travel or tire deflection. The presence of preview control allows significant improvements in the vehicle performance in terms of ride comfort, suspension travel, and road-holding ability. The active systems with and without vibration absorber give similar performance characteristics. ACKNOWLEDGEMENT The authors would like to thank the Research Center, King Saud University for supporting this research. REFERENCES 1. Hac, A., "Suspension Optimization of a 2-DOF Vehicle Model Using a Stochastic Optimal Control Technique", Journal of Sound and Vibration, Vol. 100, No. 3, pp. 343357, (1985). Chalasani, R.M., "Ride Performance Potential of Active Suspension Systems- Part I: Simplified Analysis Based on a Quarter-Car Model", ASME Monograph, AMD - Vol. 80, DSC-Vol.2, pp. 205-221, (1986). 3. EIMadany, M.M., and Abduljabbar, Z., "On the Ride and Attitude Control of Road Vehicles", Computers and Structures, pp. 245-253, (1992). 4. Bender, E.K., "Optimum Linear Preview Control with Application to Vehicle Suspension", Trans. ASME Journal of Basic Engineering, Ser. D., Vol. 90, No. 2, pp. 213221, (1968). 5. Tomizuka, M., "Optimal Linear Preview Control with Application to Vehicle Suspension Revisited", ASME Trans. Journal of Dynamic Systems, Measurement, and Control, Vol. 98, No. 3, pp. 309-315, (1975). 6. Hac, A., "Optimal Linear Preview Control of Active Vehicle Suspension", Vehicle System Dynamics, Vol. 21, pp. 167-195, (1992). 7. Huisman, R.G.M., Veldpaus, F.E., Voets, H.J.M., and Kok, J.J., "An Optimal Continuous Time Control Strategy for Active Suspensions with Preview", Vehicle System Dynamics, Vol. 22, pp. 43-55, (1993). 8. Pilbeam, C., and Sharp, R.S., "On the Preview Control of Limited Bandwidth Vehicle Suspensions", Proc. I. Mech. E., Vol. 207, pp. 185-193, (1993). 9. Abdel-Hady, M.B.A., "Active Suspension with Preview Control", Vehicle System Dynamics, Vol. 23, pp. 1-13, (1994). 10. EI-Demerdash, S.M., and Crolla, D.A., "Hydro-pneumatic Slow-active Suspension with Preview Control", Vehicle System Dynamaeis, Vol. 25, pp. 369-386, (1996). 11. Senthil, S., and Narayanan, S., "Optimal Preview Control of a Two-dof Vehicle Model Using Stochastic Optimal Control Theory", Vehicle System Dynamics, Vol. 25, pp. 413430, (1996). .
Current Advances in Mechanical Design and Production, MDP-7
103
12. Kok, J.J., van Heck, J.G.A.M., Huisman, R.G.M., Muijderman, J.H.E.A., Veldpaus, F.E., "Active and Semi-active Control of Suspension Systems for Commercial Vehicles, Based on Preview", Proceedings of the American Control Conference, Vol. 5, pp. 2992-2996, (1997). 13. Li, Qin, Yoshimura, T., and Hino, J., "Active Suspension with Preview of Large-Sized Buses Using Fuzzy Reasoning", Int. Journal of Vehicle Design, Vol. 19, No. 2, pp. 187198, (1998). 14. ElMadany, M.M., and Abduljabbar, Z.S., "Linear Quadratic Gaussian Control of a Quarter-Car Suspension", Accepted for Publication in Vehicle System Dynamics, (1999).
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Current Advances hz Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
105
DYNAMIC RIDE PROPERTIES OF A ROLL-CONNECTED VEHICLE SUSPENSION
Rakheja, S. ~ Ahmed, A.K.W. *, Liu, p.'t and Richard, M.J. t *CONCAVE Research Center, Concordia University, Montr6al, Qu6bec, Canada H3G 1M8 tPresently with Goodyear Tires, Akron, Ohio, U.S.A. tDepartment of Mechanical Engineering, Universit6 Laval, Qu6bec, Canada G 1K 7P4 ABSTRACT Static and dynamic properties of a vehicle suspension comprising hydraulic struts interconnected in the roll plane are investigated. The feedback effects due to fluid flow through the connecting pipes on the suspension stiffness and damping properties are derived and discussed. Fundamental properties of the interconnected suspension are compared with those of an unconnected suspension with and without an anti-roll bar, in terms of load-carrying capacity, suspension rate, roll stiffness and damping characteristics. The anti-roll performance of the interconnected suspension is analyzed for centrifugal acceleration excitations encountered during directional maneuvers. The ride quality performance is evaluated for excitations occurring at the tire-road interface. It is concluded that the interconnected suspension with inherent enhanced roll stiffness and damping characteristics can significantly improve the heave ride performance and limit the body roll motion. KEYWORDS Interconnected suspension, feedback damping, anti-roll suspension, vehicle ride, roll stability. 1. INTRODUCTION Vehicle suspension designers are challenged to achieve a compromise between the conflicting dynamic ride, handling and control performance requirements [1]. While enhancement of rollover threshold and directional control performance requires relatively stiff suspension, softer suspension is desirable to achieve good ride performance. Passive vehicle suspensions are thus invariably designed with soft springs coupled with auxiliary roll stiffness mechanisms to attain a compromise between ride, handling and control performance and rattle space requirements of the vehicle. Auxiliary roll stiffeners, achieved through antiroll devices, however may affect the vehicle ride in an adverse manner due to coupled vertical and roll dynamics of the vehicle. Inter-connected hydro-pneumatic suspensions with progressive rate stiffness characteristics and almost constant natural frequency offer considerable potential for ride amplitude control, self-leveling, anti-roll and anti-pitch control [1,2]. Such system may therefore, be tuned to offer an improved handling and control properties, while good ride quality of the suspension is retained. Although roll and ride properties of various passive, semi-active and active suspension systems have been extensively reported, such properties of interconnected suspension have been reported in only few published studies. Moulton and Best [3] proposed and analyzed a
106
Current Advances in Mechanical Design and Production, MDP-7
concept of an interlinked suspension system with rubber springs. Meller [4] proposed an interlinked hydro-pneumatic suspension comprising a self-energizing load-dependent intemal pump, energized by the relative movement between the unsprung and sprung masses, which facilitated the ride height control. Felez and Vera [5] investigated the ride and roll characteristics of passive and active interlinked hydro-pneumatic suspensions. The study demonstrated an improved anti-roll performance of the inter-connected suspension and instability problems associated with the active suspension. Tanahashi et al. [6] investigated an interlinked hydro-pneumatic suspension with three stage damping force modulation based on the vehicle response and road conditions. Rosam and Darling [7] presented the performance potentials of interconnected suspension coupled with an active roll control concept. In this paper, a nonlinear analytical model of a vehicle suspension, inter-connected in the roll plane, is analyzed to examine its properties in the vertical and roll modes. The interconnected suspension system is modeled incorporating the fluid compressibility, nonlinear damping characteristics, multiple chambers of pressurized fluid and gas, and the multidegrees-of-freedom (DOF) vehicle model. The feedback effects of inter-connecting fluid pipes on the vehicle roll stability and ride performance are established from the simulation results. The dynamic properties of the interlinked suspension with a beam axle are compared with those of an independent strut suspension to demonstrate its performance potentials. 2. ANALYTICAL MODEL OF AN INTERCONNECWED SUSPENSION The ride and roll performance of a vehicle suspension can be conveniently analyzed using the four-DOF roll plane model of the vehicle. Figs. 1(a) and 1(b) illustrate the roll plane models of a vehicle with interconnected and unconnected hydraulic suspension, respectively, in conjunction with a beam axle. The tires are modeled as a parallel combination of a linear spring and a viscous damper, assuming point contact in the roll plane, while the unconnected suspension employs an anti-roll bar. X~t(t ) and x~,(t) represent the excitations due to road roughness at the left-and fight-wheels, respectively. T,(t) is the roll moment excitation encountered by the sprung mass during a directional maneuver. The differential equations of motion of the four-DOF vehicle models can be expressed in the following matrix form:
§ tr( ,
try}
(1)
where [m] is the 4x4 mass matrix and {re } is the excitation force vector due to the tire-terrain interactions and roll moment. The force vector {f(z,~,t)} comprises forces due to suspension and tires. Vector {z}represents the coordinates for relative motions. 2.1. Ride and Roll Properties of Interconnected Suspension The interlinked suspension is realized by connecting the upper and lower chambers of the struts as shown in Fig. 1. The suspension is analytically modeled assuming turbulent fluid flows through the damping valves, laminar flows through the interconnecting pipes, polytropic process of confined gas, incompressible fluid, and negligible strut friction [9]. The fluid couplings between the two suspension struts yield a feedback effect that influences the static as well as dynamic characteristics of the suspension forces. The performance characteristics of vehicle suspension are strongly related to its static and dynamic properties, such as load-carrying capacity, bounce suspension rate, effective roll stiffness, and bounce and roll damping. The interconnected suspension model is thus analyzed to derive its static and dynamic properties [9]:
107
Current Advances in Mechanical Design and Production, MDP-7
L _ F -
L~
~s(t) _ -V
L,
L
j q
...........]l ~
,.j
............ ~1:2.
L, .....
,F .
X.j.(t)
Lr
.J
T
x'W"-
59:
Fr'
F,' ..... I Ktl
Ctl
0.(t)
Xi~(t) L
Ktr
.... ~_
Ktl
tr
Ctl
Xi,(t)
Xil(t)
J
0.(t)
Kt,
tX,,(t)
L ....
Lw, -I I- " Lwl Lwr -I (a) (b) Fig. I. Roll plane models of vehicle with (a) interconnected and (b) unconnected suspension. F
l-~.t
T
Load-Carrying Capacity: Assuming identical struts, the load-carrying capacity, defined as the load supported by the beam axle suspension at the design ride height, can be expressed as: W = 2N,(po -po)A
where N,
(2)
is the number of suspension struts used on each side, P0 is the nominal gas
pressure at the design ride height and po is the atmospheric pressure. Area A represents the piston rod area for an interconnected system, and the piston head area for an unconnected system. The load-carrying capacity of an interconnected suspension is thus considerably smaller than that of an unconnected suspension with identical struts and charge pressure. The nominal pressure or strut size of the interconnected suspension therefore must be appropritely selected to achieve load-carrying capacity identical to that of an unconnected suspension.
Suspension Rate: Assuming polytropic process of the confined gas, the spring rate of the axle suspension corresponding to the static ride height of the vehicle can be derived as [9]: K,,~ = 2N, n p~ A 2 (3) Vo where n is the polytropic exponent and v0 is the nominal gas volume of a single strut. Since A represents piston rod and head areas for interconnected and unconnected systems, respectively, the suspension rate of an interconnected suspension is considerably lower than that of an unconnected suspension.
Roll Stiffness: Static suspension roll stiffness can be derived from the restoring moment ( M ) developed by the suspension which is subjected to a static roll angle (~o). During roll motions, the left and right struts develop unequal forces leading to certain heave motion of the sprung mass ( x ). The roll stiffness can thus be expressed as: dM OM ,3M dx K~, . . . . + ~ ~ (4)
d~o
O~o
& rico
where dx/dtp can be computed from the corresponding static equilibrium equation. Solution of Equation (4) yields the roll stiffness corresponding to its static ride position: K~0 = K .0 e 2 (2a - l )2
(5)
108
Current Advances in Mechanical Design and Production, MDP-7
where g is half the suspension track, and a is piston head to rod area ratio, which is greater than 1. The roll stiffness of the unconnected suspension at design ride height is derived as: 0 2 K~0 = K=g , without anti-roll bar;
0
0
K@ = K ~ g
2
+ K s , with anti-roll bar
where K s is the auxiliary roll stiffness of the anti-roll bar. Equation (5) reveals that the static roll stiffness of the interconnected suspension can be effectively enhanced by selecting larger piston area ratios. Damping Characteristics: The hydro-pneumatic suspension interconnected in the roll plane yields additional damping due to fluid flow through the interconnecting pipes, and due to the coupling effects. The damping force developed in a strut is strongly influenced by the movement of the interconnected strut. The damping properties of the interconnected suspension can be explored for different vibration modes. In the bounce mode, the damping force can be expressed as:
f~, = , lel + (6) where c I and c 2 are damping parameters associated with turbulent orifice flows and laminar pipe flows given by cl = 0.5#43,/(Caa) 2 and c 2 = 128g/~ 2,/OrD4). A , , A, and a are crosssection areas of piston rod, interconnecting pipes and damping orifice, respectively, p and/l are fluid density and viscosity, and L is the pipe length. Ca is the discharge coefficient. In the roll mode, the damping force developed by a strut can be expressed as: f~, = c, (2a - 1)3 ilil + c2i
(7)
Equation (7) reveals that the damping force developed by a strut in the roll mode is always larger than that in the bounce mode due to the coupling effects of the left and fight struts. For an unconnected strut, the damping force is represented by the first term of Equation (6), where ct is the damping parameter associated with turbulent flows through the orifice. 3. COMPARISON OF SUSPENSION PROPERTIES
In view of ride and handling performance, the stiffness and damping properties of four different configurations of vehicle suspension are evaluated and compared to demonstrate the coupling effects of an interconnected suspension. The selected suspension configurations include: (i) conventional unconnected suspension (UC); (ii) unconnected suspension with anti-roll bar (UCR); (iii) interconnected suspension I (IC-1); and (iv) interconnected suspension H (IC-2). The parameters of different suspensions are selected to achieve identical load-carrying capacity of 14110 kg at design ride height. The suspension rate and roll stiffness of IC-I configuration are selected as 960 kN/m and 470.4 kNm/rad (ot=1.125). The IC-2 suspension, however, is configured to yield low suspension rate (610 kN/m) and high roll stiffness (735 kNm/rad) by selecting a=1.284. The static properties of the four suspension systems are evaluated and compared as functions of the relative vertical and roll displacements. Fig. 2 illustrates the vertical and roll stiffness properties of the four suspension configurations as a function of the relative vertical and roll displacement across the struts, respectively. All the suspensions exhibit progressively hardening vertical rate due to the force-deflection characteristic of gas springs. Unconnected suspension with and without anti-roll bar, and IC-I suspension yield identical vertical rates over the defection range of _+0.03 m, whereas the IC-2 suspension yields lower vertical rate.
109
Current Advances in Mechanical Design and Production, MDP-7
The roll stiffness rates of the interconnected suspensions are considerably larger than those of the unconnected suspension, due to the coupling effects of the interconnected struts. The roll stiffness of unconnected suspension is considerably enhanced by the anti-roll bar. The roll stiffness of all the suspension configurations exhibits progressively softening property. 1.,1~0 t .2QO ~
! ..
1.000
..........'--::- :;:i:i-:iii.::i.-:?i;iii
i ~
!~ l
l
0
,~,
,l
.
I
JL
.__
0
...... '
DEFLECTION (fit
.....
I
8PRUNG MASS ~
.....
i
I, , 0.~
!
"
).08
ANGLE (rid)
(a) Vertical suspension rate (b) Roll stiffness Fig.2.Vertical and roll stiffness properties of different suspension configurations. Beside stiffness, the suspension damping plays an important role in the ride and handling characteristics of a vehicle. The relative vertical and roll damping properties of the suspension configurations are thus evaluated assuming constant area damping orifices. Figs. 3 illustrates the damping forces of a single strut in bounce and roll modes. Although the suspension systems are configured with identical orifice areas, the resulting damping properties of various suspension systems are considerably different as illustrated in the Figures. Since anti-roll bars do not contribute to damping properties of a suspension, the damping properties of the two unconnected suspensions are identical. Fig. 3 demonstrates that the bounce mode damping force of IC-1 suspension strut is similar to that of the unconnected suspension, while the damping force of lC-2 strut is considerably lower. The damping force of an unconnected suspension strut in the roll mode is identical to that in the bounce mode. The roll mode damping forces due to the interconnected struts in all the configurations, however, increase considerably due to feedback effect. The interconnected configurations thus offer significantly larger damping force in the roll mode. 60
,|
l
I
......
80
..."Y"
~
,,..Y ...... ...""""
,J ~.,..o
, , ~ . I "r
10 O0
0,0$
~1
0,1S
I l m t l r VEI.~:~I"r (mtl)
0.|
r
0.(:6
O.T 0..16 8 r o U T Via.OOZe Ovr
0.1
O.S
(b) Roll mode (a) Bounce mode Fig.3. Damping force developed by a single strut in bounce and roll modes. 4. PERFORMANCE OF THE INTERCONNECTED SUSPENSION The equations of motion of the four-DOF vehicle models are solved to determine the performance characteristics of the four different suspension configurations. The simulation parameters used in the study correspond to a modern highway bus, and are given by:
110
Current Advances in Mechanical Design and Production, MDP-7
ms=1411Okg,
ls=19300kgn~, I,,=381~_:_~ kgrn2, ee=e,.=O.7m
m..=3616kg,
k,t = k,r =37548 kN/m, C,e =c,, =9500 Ns/m, h.z =0.62 m, e,, =ew,. =l.03m
The roll and ride properties of the interconnected suspension system are evaluated and compared with those of the vehicle models employing unconnected suspensions with and without anti-roll bar. The roll performance is analyzed in terms of the sprung mass roll angle response of the vehicle subject to a lateral acceleration excitation encountered during a directional maneuver. The dynamic ride performance is established in terms of both heave and roll shock isolation characteristics of the suspensions subject to road excitations. 4.1. Roll Response Performance The roll response of the vehicle employing different suspension systems is investigated under two different lateral acceleration excitations encountered by the sprung mass: (0 rounded step acceleration encountered during a steady turn; and (ii) transient lateral acceleration encountered during a typical lane-change maneuver [9]. The roll angle response of the sprung mass subjected to rounded step and transient lateral acceleration excitations are shown in Fig. 4. The results under steady turning acceleration excitation show that the steady state roll angle response of the sprung mass with unconnected suspension (0.065 radians) is quite large due to its low roll stiffness. The roll angle response approaches a steady value of 0.038 radians, when interconnected as well as anti-roll bar suspensions are employed. The peak roll angle response of the vehicle employing interconnected suspensions, however, are lowest due to their high roll mode damping. The results further reveal that the roll oscillation frequencies are also different reflecting the difference in roll stiffness as presented in Fig. 2. For the interconnected suspension systems, faster decay rates of the sprung mass roll response is also observed due to larger damping rates in the roll mode.
I-----
I oc u,,c.,,c_ I
o.1
l~
I~
O.O4
o
1
|
s 111,E(a)
4
8
e
a. Rounded step lateral acceleration.
gt)
0
!
l
II
4
$
t
?
9~ o0 b. Transient lateral acceleration.
Fig.4. Roll angle response of sprung mass under lateral acceleration excitations. The roll response characteristics of the sprung mass under transient lateral acceleration excitation corresponding to a typical lane change maneuver reveal that the suspension system interconnected in the roll plane can effectively limit the body roll motion. The anti-roll bar suspension also demonstrates its superior anti-roll capability. A comparison of the peak response amplitudes of interconnected suspensions further reveals the effect of roll mode damping in reducing the body roll. The effect, however, is not as significant as that observed for step response.
111
Current Advances in Mechanical Design and Production, MDP-7
4.2. Ride Response Performance Dynamic ride characteristics of the vehicle employing unconnected, unconnected with antiroll bar and interconnected suspensions are analyzed in terms of transient vertical and roll response of the sprung mass, when subjected to half-sine displacement excitations at the tireterrain interface. The vertical shock attenuation characteristics are evaluated for an in-phase excitation, while the roll response is evaluated for an out-of-phase excitation. Fig. 5 presents the vertical displacement and acceleration response of the sprung mass employing different suspensions, subjected to an in-phase half-sine bump input. As the results show, the peak displacement response of the IC- 1 suspension is similar to that of the unconnected suspension and considerably larger than that of the IC-2 suspension. The large peak response of unconnected and IC-I suspension is attributed to the high bounce mode damping of these suspensions. All the suspension configurations, except the IC-2, exhibit sprung mass bounce resonance near 1.2 Hz, while the corresponding frequency of the IC-2 suspension is near 1 Hz. These results further show that the IC-2 suspension produces nearly 50% of the acceleration response of the unconnected and IC-1 suspensions, under half-sine bump input.
o.oo4
| IC/I IC[~..IC- L . . _
O.OOG I 0.00~ o.00t 0
1
la
--
Io i
/~
I
1 ~
w ~'~o
~
~
~,aE(,1 s
....
o
.......
............. ". . . . . ~ -
(~ 4
8
e
0.~
o.~
o.~
'n~uE1,) o.~
0,~
o.~2
o.~4
a. Displacement b. Acceleration Fig.5. Vertical displacement and acceleration response of the sprung mass of the vehicle under in-phase bump excitation. The roll angle and roll acceleration response characteristics of the sprung mass of the vehicle employing different suspensions, subjected to an out-of-phase bump, are compared in Fig. 6. The results clearly reveal that the peak roll displacement and acceleration response of the vehicle with IC-l suspension is considerably large due to its high suspension damping in the roll mode. The roll displacement response of the vehicle, however, decays relatively rapidly. The results further demonstrate that the anti-roll bar has no significant influence on the roll ride response since the body roll angle is considerably small under such an excitation. Although the peak roll response of the IC-2 suspension is slightly larger than that of the antiroll bar suspension, its roll response decays at faster rate. It should be noticed that the IC-I suspension, which yields superior anti-roll performance, results in excessive roll acceleration of the sprung mass. It is thus concluded that similar to suspension stiffness, high suspension damping is desirable for good vehicle handling, while lower damping is preferred for enhancement of vehicle shock attenuation performance. 5. CONCLUSIONS A vehicle system roll plane model is developed and investigated to examine the fundamental properties of a roll connected hydro-pneumatic suspension. The properties are evaluated in terms of load-carrying capacity, suspension rate, roll stiffness and damping characteristics.
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Current Advances in Mechanical Design and Production, MDP-7
The roll and ride response characteristics of a highway bus employing the interconnected suspension is evaluated for lateral acceleration excitations encountered during directional maneuvers and for transient excitations occurring at the tire-road interface. It is shown that the interconnected suspension has inherent enhanced roll stiffness, which can be effectively controlled by the roll stiffness amplification factor determined by the strut configuration. The suspension rate of the interconnected suspension exhibits a progressively hardening property, while the roll stiffness exhibits a progressively softening characteristic. The interconnected suspension also yields higher damping rate in the roll mode than in the bounce mode. The simulation results reveal that the interconnected suspension with enhanced roll stiffness and damping can significantly restrict the body roll motion for improved vehicle handling performance. The larger roll stiffness and damping, however, deteriorate the roll ride performance of the vehicle. The property of relatively large roll damping of the interconnected suspension is a positive factor for restricting body roll, but a negative factor for good vehicle ride quality. The interconnected IC-1 suspension with enhanced roll stiffness and reduced suspension rate can provide a better compromise between ride quality and handling performance of a vehicle. o.o~
I
I
o.ool
"~
:-,
I
o
~ fo.oo~)
[0: 0
1
2
$ TME (a)
4
S
11
0
J
0.C~ 0.G4 0,08 0.~ 'flUE Iq
0.1
Gll
0.1'4
b. Acceleration a. Displacement Fig.6. Roll displacement and acceleration response of the sprung mass of the vehicle under out-of-phase bump excitation. REFERENCES 1. Bank, T. A., "Some ABC's of Air Spring Suspensions for Commercial Road Vehicles," SAE paper No.800482.,(1980). 2. Chu, Y. and Li, Z., "The Dynamic Response of Vehicles with Hydro-gas Suspension to Roadway Undulation," ACTA Armamentari, No.2, pp. 30-42, (1984). 3. Moulton, A.E. and Best, A., "Rubber Springs and Inter-connected Suspension Systems," Engineering Design Show Conference, Paper No. 15a, (1970). 4. Meller, T., "Self-Energising, Hydropneumatie Levelling Systems," SAE No. 780052. 5. Felez, J. and Vera, C., 1987, "Bond Graph Assisted Models for Hydro-Pneumatic Suspensions In Crane Vehicles," Veh. Sys. Dyn., Vol.16, pp. 313-332, (1978). 6. Tanahashi, H., Shindo, K., Nogami, T. and Oonuma, T., "Toyota Electronic Modulated Air Suspension for the 1986 SOARER," SAE Paper No. 870541, (1987). 7. Rosam, N., and Darling, J., "Development and Simulation of a Novel Roll Control System for Interconnected Hydragas Suspension", Veh. Sys. Dyn.,V.27, pp. 1-18,(1997). 8. Mayne, R. W., "The Effects of Fluid and Mechanical Compliance on the Performance of Hydraulic Shock Absorbers," ASME J. of Eng. for Industry, pp. 101-106, (1974). 9. Liu, P.J., An Analytical Study of Ride and Handling performance of an Interconnected Vehicle Suspension", M.A.Sc. Thesis, Concordia University, Canada, (1994).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
113
BILINEAR CONTROL THEORY OF SMART DAMPING SYSTEMS
EI-Beheiry, E.M. Lecturer, Department of Production Engineering and Mechanical Design, Faculty of Engineering, Menoufia University, Shebin EIKom, Egypt
ABSTRACT Methods for optimal and sub-optimal bilinear control of smart dampers are derived in this article. The smart dampers are considered in the form of (CVD) continuously variable semiactive dampers which are capable of adapting their damping forces to reasonably match fullor limited-state control (inputs) forces generated by broad-band actuators. During the operation of the smart dampers, it is assumed that some other locations in the vibratory system are controlled by simultaneous broad-band force generators. The System is then a piecewise (bilinear) vibrator in which the control inputs appear additively and multiplicatively. The chosen performance index is constrained by (i) the inability to measure all the system state variables that are necessary for the operation of the broadband control input and (ii) the necessity to consider the switching states of the smart damping elements. A bilinear control theory is first developed and then applied to an automobile model of 4-DOF having front suspension of smart damper and rear suspension of broadband ideal actuator. Performance comparisons with (FLSC) full-state control and (LMSC)limited-state control designs show the effectiveness of the (FLBC) full-state bilinear control and the (LSBC) limited-state bilinear control designs derived in this work. KEYWORDS Bilinear Control, Output Feedback, Vibration Control, Vehicle Suspensions. 1. INTRODUCTION Active suspensions are close to be in mass production in the coming few years, s e e Hac et al. [3,4]. Several studies like those made by Kamopp and Margolis [7], Kimbrough [8] and Youn and Hac (9) have shown that adaptable suspensions of variable structure in the sense of damping and stiffness elements might lead to a remarkable vibration control of vehicles on roads of varying surface conditions. These systems are the ones we call "smart" in this work because they are always capable of adapting themselves to the designer's desires of performance. However, when we come to the optimization of adaptable semi-active dampers the task is not as easy as the calculation of optimal control laws of active suspensions. Semiactive dampers are rapidly switched in nature and therefore they have strongly nonlinear dynamic behavior. Many attempts have been made for the optimization of semi-active dampers, but the contribution of researchers in this area is not as much as their contribution in the optimization of active suspensions. EIBeheiry [1], Hac and Youn [2], Hedrick et al. [5] and Hrovat, et al.[6] have all considered the optimization of semi-active dampers with/without preview control algorithms.
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This paper addresses the problem of developing optimal and sub-optimal bilinear control methods for the design of adaptable semi-active dampers, which simultaneously operates along with ideal broadband actuators mechanized at different locations in one system. The problem is complicated by the inability to measure all the system state variables that are necessary for optimizing the force generated by the actuator. This situation clearly appears in vehicles where tire deflections represent the most difficult parameters to measure or to estimate. 2. A P P L I C A T I O N M O D E L AND F O R M U L A T I O N The model of an in-plane 4-DOF half-car model is schematically shown in Fig. 1. mb is the sprung mass and db is the sprung mass pitch moment of inertia. If Yb and Ob represent the bounce and pitch displacements of the sprung mass, respectively, then Yl = Yb + llOb and Y2 = Yb-120b will be the sprung mass displacements at the front and rear suspension connections, respectively. Masses m I and m 2 represent front and rear axles; their absolute displacements are Y3 and Y4, respectively, k 1, k 2 and Cl, c2 denote stiffness and damping factors of passive front and rear suspension, respectively. Linear springs k 3 and k 4 represent front and rear tire stiffness. Active control forces at the front and rear suspensions are denoted by u I and u 2 , respectively, l I is the distance from the vehicle C.G. to the front suspension and 12 is the distance from the vehicle C.G. to the rear suspension. Yol and Yo2 are the road elevations imparted at the front and rear tire. Values of both basic vehicle and optimized passive suspension parameters are shown in Table 1. The model equations of motion are:
mbf~b + kl (Yl - Y3 ) + Cl (J'i - Y3 ) + k2 (Y2 - Y4 ) + c2 (J'2 - Y4 ) = Ul + u2 JbOb + klll (Yl - Y3 ) + ClII ()I - . P 3 ) - k212 (Y2 - Y 4 ) - c 2 1 2 ( ) 2 - J'4 ) = llUl -12u2 (I) ml.P3 + k3 (Y3 - Yol ) + kl (Y3 - Yl ) + Cl ()'3 - .Vl ) = -Ul m2.P4 + k4 (Y4 - Yo2 ) + k2 (Y4 - Y2 ) + c2 (3;'4 - JP2 ) = -u2
Table 1. Basic and optimized vehicl e parameters Basic vehicleparmmters
Yv
Y2
1 2 ~ l
mb = 5747 kg
!
db = 7689 kg.m 2 n~ =m2 =59.5 kg k3 =
~
=
190000N/m
I I = !.38 m
~ ~U Au2omor
Actuator
kl, Cl
Ul
,.
12 = 1.36 m
0pt~
paraS, ers
r kI = 10202N/m k2 = 10114N/m
r4 ~ ~ ]
Yo2
K~ ~ .
T Yol
cn = 1 5 0 0 N M m
c2 = 1500N.s/m
Fig. 1. A 4-DOF in-plane half car model Introducing the following state variable, control input and disturbance input vectors
Current Advances in Mechanical Design and Production, MDP-7 t
x(t)=[x,,
x2.........
t
x81,
u(t)=[u,,
115 t
u 2] , w ( t ) = [ w , ,
(2,3a,b)
w 21
x! =(Yl -Y3),
x2 =(Y2 - Y 4 ) ,
x3 =(Y3 - Y o l ) ,
x4 =(Y4 - Y o 2 ) ,
X5 = Yl,
X6 = P2,
X7 = ))3,
X8 = ))4,
(4)
equations (1), (2) and (3) can be reformed in a linear state space form as follows
(5)
x(t) = Ax(t) + BlUl (t) + B2u2(t)+ Dw(t), where A,
B 1 , B 2 and D are the constant state space matrices, w(t) is a vector of road white
noise inputs where w ! = Yol and w 2 = Yo2 .In this study, the limited-state control forces at the front or the rear suspensions are assumed to be of local-state measurements, i.e., the feedback signals are confined to only local signals at each actuator's location. One can say that the limited-state control forces are subjected to control structure constraints such that : u!=KlMIX,
(6,7)
u 2 =K2M2x
where K 1 and K 2 are constant feedback gain matrices, M 1 and M 2 are local measurement matrices of dimensions 3 • 8. In the case when a (FCVD) front continuously variable semiactive damper is used at the front suspension, the control force u I (t) is given by
(8)
U l(t ) = v I (t)(x5 - x 7 ),
where v 1(t) is a variable damping coefficient that can be varied within a given range of values (9)
v imin < v I (t) _ dm~. r dm~"- d _ 0 then r = ~.,., if 2~, < 0 then r = ~.... The case where 2~, = 0 is more complex to analyse, however this singular situation can be clarified through the manipulation of the optimality conditions [13], leading to the conclusion that when 2~, = 0 then ~bp = 0 and Vp remains constant. Hence, a time-optimal planar trajectory consists of a sequence of maximum bank angle turns and straight line segments.
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Current Advances in Mechanical Design and Production, MDP- 7
3. OPTIMAL T R A J E C T O R Y GENERATION In this study, a simple case with two aircraft is considered for trajectory generation, as shown in Fig. 2, with the following considerations 9 9 The cartesian co-ordinate system is oriented with the Leader at origin, and flying along positive x-axis direction (heading angle of 900). 9 The Pursuer aircraft may have any position and heading angle in the horizontal (x, y) plan. 9 The aircraft are flying at the same constant speed (V= VL=Vp).
-
!
YP
Ve
! !
xe
L~X p,-
VL
Fig. 2. Relative horizontal position of two aircrafts 3.1. Trajectory Elements It has been shown in the previous section that all optimal trajectories of the Pursucr consist of sequences of arcs and straight line segments. The set of optimal maneuvers can be divided into subsets depending upon the initial relative positions and heading angles of the two aircraft" 9 The Pursuer is within the protected zone around the Leader and its first providence is to get clear from the Leader. 9 A direct maneuver is feasible : the Pursuer can arrive behind the Leader at a distance greater than or equal to minimum separation distance in minimum time. 9 No direct maneuver is feasible : it covers situations where the Pursuer must delay its trajectory so that minimum separation is effective. So, the optimal trajectory is composed of different elements organised in sequence. Dynamic Programming can be used here to globally optimise the parameters of each element. The algorithm for an optimal trajectory generation is given as" Step 1. Compute the relative distance 'd' between the two aircraft. Step 2. Determine if d < din,,. If no, go to step 3. Otherwise, determine the aircraft trajectory with left as well as right turn. The selected turn direction is one which results in larger value of the minimum in between distance during maneuver. Step 3. Determine if a direct maneuver is feasible. If yes, go to step 4. Otherwise, determine the aircraft trajectory with left as well as fight turn till the miss-distance becomes equal to the minimum separation distance behind the Leader. Afterwards, the aircraft flies straight ahead with the present heading till the point a direct maneuver becomes feasible. Step 4. Determine the aircraft trajectory with left as well as fight turn till the point from where convergence to the Leader's route may be possible. Step 5. Determine the straight segment of the aircraft trajectory for convergence towards the Leader's route. Step 6. Determine the aircraft trajectory for joining the Leader's route. Step 7. Out of generated trajectories (two solutions are possible for step 2 and step 3), that one is selected with which the Pursuer joins the Leader's route in minimum time.
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Current Advances in Mechanical Design and Production, MDP-7
Hence, following the above algorithm, the Pursuer trajectory can be built up from a set of basic parameterized maneuvers : 9 Avoidance Turn : tA, time at the end of Avoidance Turn; grA, final value of heading angle attained at the end of Avoidance Turn. 9 Distancing : tD, time for Distancing. 9 Redirection : tn, time for Redirection; VR, final value of heading angle attained at the end of Redirection. 9 Convergence :tc, time for Convergence. 9 Joining" tj, time for Joining. Some examples of generated optimal trajectory parameters are presented in Table 1 and the corresponding trajectories are shown in Fig. 3.
Table 1. Some examples of the optimal trajectory parameters
1 2 3 4 5
20.0 '20.0 30.0 -7.0 7.0
20.0 20.0 30.0 7.0 7.0
225" 135 90 135 225
2 0 !10 22.4 36.5
222 -245 166.6 276.5
52 0 21 0 0
125 31.9 148 83.5 186.34
46.4 180 36.1 48.8 13.6
0 17.6 0 0 0
Fig. 3. Examples of optimal rejoinder trajectories
31 63.8 38.1 29.1 54.2
211.5 133.3 317.3 135 277
17.493 33.987 24.545 14.197 47.274
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Current Advances in Mechanical Design and Production, MDP- 7
4. ON LINE NEURAL TRAJECTORY GENERATION The Leader may modify at any time its guidance parameters (speed, heading and flight level) in accordance with new atmospheric conditions (wind and temperature) or following instructions issued by the traffic control service. Thus on line trajectory generation is a necessity here. During this on line process it becomes impractical to calculate the optimal trajectory at each instant of time, in fact the need is to have some solution which could be immediately displayed to the pilot or taken into account by the autopilot. A neural network fulfils this requirement as it can be trained to memorize input-output relationships for a system and afterwards can be used as an interpolator to generate outputs corresponding to current inputs. Since the early 1990s, neural networks have been used in a variety of applications and there has been a growing interest in using neural networks to solve optimal guidance problems [ 14]. 4.1. Training Data The optimal Pursuer trajectories are generated for many different initial conditions. The overall input/output structure of the program used for this purpose is shown in Fig. 4. Inputs:
9 Initial position and heading angle of the Pursuer (xp, yp, r Outputs: 9 Avoidance tum: tA, ~a, da (da=0 for left turn and da =1 for right turn) 9 Distancing'to 9 Redirection:ts, ~s, dR 9 Approach : tc 9 Joining 9tj Input situations considered for training are xp(km) : - 100, -50, -30, - 10, -4, -2, 0, 2, 4, 10, 30, 50, 100 ye(km) 1O 0 , 5 0 , 3 0 , 1 0 , 4 , 2 , 0 : 9(o)0 o , 2o, 4 o, 6 ~......... 360 o. Since the infinite set of optimal trajectories is symmetric about the x-axis, the training has been performed only for positive y co-ordinate values of the initial position of the Pursuer. Only time required for each phase of maneuver and turn directions have been used for training as corresponding heading angles can be calculated analytically. The general structure of the neural networks used in this application is classical [15] and is composed of three layers. The transfer functions associated with the neurons are selected as hyperbolic tangent functions. The Levenberg-Marquardt's training method has been used here, it combines elements of steepest descent and Gauss-Newton methods. The general structure of the designed neural networks is given in Fig. 6. Each trajectory parameter is computed by a separate neural network to get a good accuracy in terms of the sum-squared error between the network outputs and the desired outputs. Table 2 summarizes the hidden layer composition of each neural
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Current Advances in Mechanical Design and Production, MDP-7
network designed for trajectory generation. Training is made efficient by scaling the training data, so that it falls in the range [- 1 1]. The input-output data pairs have been used in a random manner in order to improve efficiency and speed of the training process [ 15]. Table 2. The construction of hidden layers NeuralNetworks
~qumberof neuronsin ....... the hidden!a~er Time for AvoidanceTurntA(sec) 40 Directionof AvoidanceTurnd~ 40 Time for Distancin[tD(sec) 35 Time for Redirectionta (sec) ........ 40 Directionof Redirection.dR 30 -- " Time for Convergencetr (sec) 20 Timefor.Joinin~tl(sec) 5
input
. Hidden
Laver
Output
L _ . J ~
Layer
_.a~
..a'
Fig.6. The structure of neural networks
5. SIMULATION RESULTS AND CONCLUSIONS A simulation study has been performed considering two Airbus A300 aircraft. It has been supposed that the Leader aircraft makes a right tum to take a new constant heading. The guidance system of the Pursuer makes use of a neural trajectory generator to define, every second, new references (either a turn rate or a constant heading) for the autopilot. The guidance laws implemented in the autopilot are classical superposed PID loops (a fast piloting loop and a slower guidance loop) similar to those encountered in aircraft of this class [ 16]. 8o
Y (kw)
gllll~
dlt~lkts)= .avsonl
~V (km)
1~1.4t.e?4 (.)s
o.]Js,
t]~aa.I
~111~ (~,): d| tlYmlk till
Nwtq~at
r
r i iill
11,143
O.Ot| tls aV. r
SS.*
6
- ~JoN i
\ x (,.) -eo
-40
o
9
x (k.) .
4O
( -eo
eO
.
-44)
:3S.O0 I~-
Crt):
solge
ii
o
c.=:
dD*=,~,):
I~t_e
~1 r vw,Q z~mr
(.)~
C*ts),
(ft)=
9,stg
-?.re4 i14.19
1~6.g0 4*e.~O )Ogga
Fig. 7a. Simulation without Leader's intent information
os* r ~,,,~o zMr
(*ts)~
(ft)=
13},gO t~4.tt
46e.91 10998
o (.): dDt(,v'.): dlse (n):
~l_a (-)-r (-): vlero s~r
(kill-*
(ft)=
9
O.OOe lSeY.: s:S.O0 X:4,99 4141. ti
soq)9~l
Fig. 7b. Simulation with Leader's intent information
Simulation results are shown in Fig. 7a and 7b. Initial position of the Leader is taken as coordinate (0, 0) with a heading of 90 ~ The Pursuer is initially at 40 km behind and 40 km North of the Leader, with an absolute heading of 90 ~ In the case of Fig. 7a, it is assumed that the Pursuer has no knowledge of the Leader's flight plan. With the activation of the relative guidance mode, the Pursuer makes an initial right turn (Redirection) and starts Convergence. When the Leader modifies its heading to 135 ~, the Pursuer changes progressively its own heading (Redirection) and then after convergence turns right (Joining) to follow the Leader. This results in a very swaying trajectory, which is much undesirable in ATC standards. While
iii~il ~II~I~I~I~ i~iii
~i ~
~i
i i i i i i ii~i ~ ii//ii~ i~i i~ ~i~ ~:
i~!ii~ ~i ~!~
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Current Advances in Mechanical Design and Production, MDP- 7
in the case of Fig. 7b, the Pursuer knows exactly the intent of the Leader so, the pursuit trajectory can be generated taking into account the final route of the Leader. In the second case, the Pursuer avoids excessive maneuvering while the pursuit time is smaller than the previous case. Extensive simulation experiments show that the proposed approach provides an efficient device to cope with relative guidance of civil aircraft. REFERENCES 1. RTCA, Inc., Report of the RTCA Board of Director's Select Committee on Free Flight, Washington, (1994). 2. Krozel, J., and Peters, M., "Conflict Detection and Resolution for Free Flight", Air Traffic Control Quarterly Journal, Special Issue on Free Flight, 1997. 3. Erzberger, H., and Lee, H.Q., "Optimum Horizontal Guidance Techniques for Aircraft", Journal of Aircraft, Vol. 8(2), pp 95-101, (1971). 4. Slattery, R. and Zhao, Y., "Trajectory Synthesis for Air Traffic Automation", Journal of Guidance, Control, and Dynamics, Vol. 20(2), pp 232-238, (1997). 5. Shapiro, I., and Ben-Asher, J.Z., "Near-Optimal Horizontal Trajectories for Autonomous Air Vehicles", Journal of Guidance, Control, and Dynamics, Vol. 20(4), pp 735-741, (1997). 6. Grimm, W., and Hans, M., "Time-Optimal Turn to a Heading: An Improved Analytical Solution", Journal of Guidance, Control, and Dynamics, Vol. 21 (6), pp 940-947, (1998). 7. Clements, J.C., "Minimum-Time Turn Trajectories to Fly-to-Points", Optimal Control Applications and Methods, Vol. 11, pp 39-50, (1990). 8. Betts, J.T., "Survey of Numerical Methods for Trajectory Optimization", Journal of Guidance, Control, and Dynamics, Vol. 21 (2), pp 193-207, (1998). 9. Guelman, M. and Shinar, J., "Optimal Guidance Law in the Plane", Journal of Guidance, Control, and Dynamics, Vol. 7, No. 4, pp 471-476, (1984). 10. Yuan, Pin-Jar., "Optimal Guidance of Proportional Navigation", IEEE Transactions on Aerospace and Electronic Systems, AES-33, Vol. 33(3), pp 1007-1011, (1997). 11. Hale, F.J., 'Introduction to Aircraft Performance, Selection and Design", Wiley, New York, 1984. 12. Shahzad, M., Slama, J.G., and Mora-Camino, F., "A New Approach for the Automation of Relative Guidance of Aircraft", 13th International Conference on Systems Engineering, Las Vegas, USA, Aug. 1999. 13. Bryson, A.E., and Ho, Y.C., "Applied Optimal Control", Hemisphere Publishing Corporation, Washington, D.C., (1969). 14. Youmans, E.A., and Lutze, F.H., "Neural Network Control of Space Vehicle Intercept and Rendezvous Maneuvers", Journal of Guidance, Control, and Dynamics, Vol. 21, No. 1, pp 116-121, (1998). 15. Demuth, H., and Beale, M., "MATLAB Neural Network Toolbox", Math-Works Inc., Natick. MA, (1994). 16. Mora-Camino, F., "Syst6mes de Conduite Automatique et de Gestion du Vol", Ecole Nationale d'Aviation Civile, (1995).
Current Advances in Mechanical Design and Production Seventh Cairo Universi O' International MDP Conference Cairo, February 15-17, 2000
131
SIMULATION OF TURBULENCE-INDUCED VIBRATION OF LOOSELY SUPPORTED HEAT EXCHANGER TUBES
Hassan, M.*, Dokainish, M.** and Weaver, D.** *PhD Candidate, ** Professor, Department of Mechanical Engineering McMaster University, Hamilton, Ontario, Canada, L8S-4L7 ABSTRACT Flow-induced vibrations of exchanger tubes result in tubes impacting and rubbing against their supports which leads to tube failure due to fretting wear damage. Fretting wear is correlated to the tube response obtained by analytical techniques. In this paper, a finite element model is presented and utilized to analyse the nonlinear dynamic behaviour of heat exchanger tubes subjected to crossflow turbulence. The tube/support interaction model that accounts for the effect of various tube parameters, such as clearances, friction at the supports, support stiffness, and damping has been incorporated in INDAP (Incremental Nonlinear Dynamic Analysis Program). An extensive parametric study was conducted. Simulations of the tubes with lattice-bar supports were carried out. The effect of the permanent tube-support preload arising from the crossflow drag and the tube-to-support clearances on the tube response is investigated. The results obtained show that the tube exhibits two different responses in the drag and lift directions. Tube response, impact force, and contact ratio are effectively presented in a dimensionless form. KEYWORDS Nonlinear Dynamics, Flow-induced Vibrations, Finite Element Analysis, Impact. INTRODUCTION Tube and shell heat exchangers used in nuclear and chemical power plants contain bundles of tubes exposed to parallel and/or cross-flow. Each of the tubes is supported at a number of locations to stiffen the structure and hence reduce the vibration amplitude due to fluid excitation. Supports are typically assumed to provide perfect pinned boundary conditions forcing vibration nodes at the support locations. This type of support is called support-active (SA). With this assumption, simple beam analysis can be used to predict the tube natural frequencies as well as tube responses to different excitations. However, in practice, clearances have to be allowed at the support locations to provide the manufacturing tolerances required for tube/support assembly, to allow for thermal expansion, and to facilitate construction. These clearances are a source of nonlinearity in the tube's boundary conditions. Clearances can allow tubes to vibrate freely in the support space without contacting the support. This type of support is called support-inactive (SI). Tubes with support-inactive can be susceptible to fluidelastic instability forces due to their low natural frequencies. Moreover, clearance allows impact and rubbing between tubes and their supports to take place at the support locations which can lead to tube fretting wear. In extreme cases, tube failures occur due to fatigue or thinning and splitting at mid-span as a result of tube-to-tube clashing. However, most plant-shutdowns are attributed to failures due to fretting wear at the tube supports. Problems typically arise in U-bend regions where the tube's natural frequencies tend to be low due to the ineffectiveness of supports, or in areas producing localized high velocities such
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Current Advances in Mechanical Design and Production, MDP- 7
as entrance and exit nozzles. U-tube supports need special design consideration due to the tubes' curvature. Various types of supports are being used in the U-bend region. Weaver and Schneider [ l ] reported a variety of support geometries suitable for the U-tube supports. Anti-vibration bars (AVBs) have shown effectiveness in supporting the tubes in the U-bend region by minimizing the risk of failure due to fluidelastic instability. There is virtually no reference in the literature to the dynamics of tubes with lattice-bar support configuration (triangular array). This type of tube-support arrangement is considered highly nonlinear and its dynamics are not well understood. Excitation due to fluid flow can be categorized as (a) turbulent buffeting, (b) Strouhal periodicity, (c) fluidelastic instability, and (d) acoustic resonance. The fluidelastic instability mechanism often leads to failures on a short term basis. However, turbulence buffeting can cause low amplitude tube motion resulting in a long-term fretting wear or fatigue. Taylor et al. [2] attributed the tube response found in the U-bend region of operating steam generators to the turbulence-induced forces and not to the fluidelastic instability forces. In addition, experimental investigations carded out by Iwase et al. [3] showed that when the tubes are supported according to the proper design specifications, they are only subjected to the random excitation mechanism. In addition, progressive long term wear and chemical cleaning processes cause enlargement of the tube-to-support clearances which in turn alters the support effectiveness. To increase the design reliability and to assess the service life of heat exchanger components, a study of the tube support-interaction behaviour and associated tube dynamics has to be established. Analytical tools are required to calculate the tube response and tube support interaction of actual geometries. To date, the determination of tube response and impact forces has usually been restricted to specific flow and tube conditions that resemble some practical cases. There is a need for a generalized and a systematic analysis which gives a better understanding of the nature of nonlinear tube dynamics. TUBES/SUPPORT INTERACTION Analysis of vibrations of tubes with loose supports is a nonlinear boundary condition problem. This nonlinearity in the boundary conditions results in a system with an unknown set of natural frequencies which is the main parameter used in assessing the design adequacy. The impact-sliding behaviour of heat exchanger tubes at their supports is very complex due to its dependence on the tube/support geometry as well as the fluid excitation modelling. In this paper, INDAP was used to simulate the nonlinear dynamics of tubes in crossflow. INDAP is an in-house general purpose finite element program capable of solving a large variety of nonlinear dynamics problems. The code structure and organization are described in [4]. The tube/support interaction model was implemented in INDAP to simulate the nonlinear dynamics of multi-span heat exchanger tubes. INDAP simulates tube response to external multi-sinusoidal or random excitations. The beam equation of motion is discretized in space into finite elements and in time by Newmark's method. The resulting set of equations of motion can be written as:
Equation 1 represents equations of the unknown vectors {//}, { ~ }, and { u } representing all possible displacements and their temporal derivatives. [M], [C], and [K] are the global mass, damping, and stiffness matrices respectively and { Fe(t) } is the external fluid excitation due to crossflow. The system matrices are, in general, nonlinear due to contact. The nonlinear contributions due to the contact are added to the system equations through a displacement and velocity dependent external force (Pseudo-force). The contact forces are then calculated based
Otrrent Advances in Mechanical Design and Production, MDP-7
133
on the support geometry and tube response. The Pseudo-force vector for the dynamic equation is given by" {F.vt}
(2)
= - [ C x t ] {ti} - [ K NL ] {U}
Hence the equation of motion becomes: (3)
[ M ] {//} + [ C t ] {ti} + [ K t ] {u} = {F(t)} + {FNL(t)}
where subscript L refers to linear parameters and subscript NL refers to nonlinear parameters. Tube/support impact is modelled by introducing a spring and a damper to the system. When contact takes place, the tube wall and the support both deform and the impact force (F~) results. This impact force is given by the product of the equivalent tube/support stiffness and the normal tube/support overlap. Impact forces are not known since they are dependant on the displacement and the velocity of the tube. Therefore, an iterative procedure is used to calculate the time history of displacement and impact forces. Modelling of impact forces is very dependent on the contact configuration dictated by the support geometry. The mathematical modelling of the fiat-bar support used in this analysis was described in detail by Yetisir and Weaver [5]. SYSTEM DESCRIPTION Figure I shows a finite element model that consists of a straight tube fixed at one end and loosely-supported at the other end. Three tube configurations were used in this analysis. In the first configuration, the overall tube length is 2 m with an outside diameter of 0.015 m and a wall thickness of 0.0015 m. The tube mass per unit length and Young's modulus are 0.7397 kg/m and 230 GPa respectively. . . . . - -
x/L = 1.0
Fig. 1. Finite element model of the cantilever tube The tube is loosely supported by a lattice-bar support which consists of a group of bars arranged to form a diamond shaped support clearance. Lattice-bar supports can be categorized according to support offset as shown in Fig 2 where d is the tube diameter, Cr is the radial clearance of a centred tube, and e is the offset of the supports. Tubes are discretized into twenty elements. Each element is a twelve DOF beam element. d
a) No offset
b) With offset
Fig. 2 Lattice bar supports
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Current Advances In Mechanical Design and Production, MDP- 7
TURBULENT EXCITATION FORCING FUNCTION Tubes were subjected to random excitation forces in two orthogonal directions arising from turbulence in the flow field. Turbulence forces are usually characterized by their power spectral density (PSD). The bounding spectra given by OengOren and Ziada [6] gives the PSD of the fluctuating forces S r ~ ) . The PSD presenting the turbulence must be translated into a force versus time record which has the same statistical properties of the actual forcing function [7]. This is accomplished by presenting the fluctuating forces per unit length by Fourier series. The power spectral density of the fluctuating forces in the drag and lift directions are assumed to be the same. Fluctuating forces in the drag direction are composed of a steady state component representing the steady drag and a random component representing the fluctuation due to turbulence. The fluctuating forces were imposed as fully correlated in space. However, excitation forces for the two orthogonal directions (lift and drag) were fully uncorrelated. RESULTS Twelve radial clearance values ranging from 0.001 mm to 1 mm were examined. Figure 3-a shows the contact ratio (Ct) of the tube with lattice-bar support for various flow velocities. The contact ratio is defined as the ratio of the contact duration to the total time. For all flow velocities, increasing the clearance (Cr) decreases the contact ratio. The steady drag component keeps the tube in continuous contact with the support for small clearances. Further increase in the clearance results in a sharp decrease in the contact ratio then it levels off asymptotic to zero. For any clearance size, increasing the flow velocity increases the contact ratio. Data can be effectively displayed in dimensionless form, as shown in Fig. 3-b. This is accomplished by normalizing the clearance by the RMS support inactive resultant response (drs~). RMS supportinactive response is the linear response of the tube without support at the end. RMS impact force (F~mp)depends on both the level of the impact peaks and the contact duration. Figure 4-a depicts the RMS impact force values versus clearance for various flow velocities. RMS impact forces decrease linearly as support clearance increases, then levels off asymptotic to zero. For any clearance size, increasing the flow velocity increases the RMS impact force. The dimensionless RMS impact force is presented as F=FimJ(Ftur L). F, Fturand L are the dimensionless RMS impact force, RMS distributed turbulence force per unit length, and the tube length respectively. Using these dimensionless parameters, the RMS impact forces collapse on one single curve for the same support configuration, as shown in Fig. 4-b. I
Ilell Ilell
I
Ib ell ~
II
9 Confi 8. I - 4 m/s 9 Config I - 6 m/s Config l - $ m/s
...06
04
0.~ 2
~
Confilt Coati| --.i-- Coatis
00
.....
0
OI
0.2
03
04
05
06
07
011
Cr [mm]
a) Contact ratio versus clearance
09
0
0 I
02
03
0.4
05
06
0.7
0.8
0.9
Cr/drsi
b) Contact ratio versus the dimensionless clearance
Fig. 3. Effect of clearance on the contact ratio
Current Advances hr Mechanical Design and Production, MDP-7
135
O4 035 03 025
~t
l
d'mO
. co,,f,s I.~m/s 9 Config 9 Config
I - 6 m/s I - 8 m/s
At e
f
J015 01 005 0
o 0
01
02
03
04
05
06
07
Og
07
0q
Cr [rnrnl
a) R M S Impact forces versus clearance
og
09
Cr/drst
(b) Dimensionless Impact forces versus the dimensionless clearance
Fig. 4. Effect of the clearance on the RMS impact forces Figure 5-a shows the RMS tube response (dy)in the lift direction as a function of the support clearance for various flow velocities. Initially, increasing the support clearance causes the RMS tube response to decrease up to a certain support clearance value. Then the RMS tube response increases gradually. At large support clearances, the RMS response is expected to approach the RMS response of the tube with support-inactive. The point of minimum RMS response, however, shifts to a larger clearance as the flow velocity increases. For any clearance, increasing the flow velocity increases the response level. The RMS lift response and support clearance are normalized by the RMS lift and the resultant support-inactive response respectively. Data for various flow velocities are represented by a single curve using these dimensionless parameters (Fig. 5-b). Figure 6-a shows the RMS mid-span tube response (dz) in the drag direction as a function of the support clearance for various flow velocities. The RMS tube response increases linearly as the clearance increases. This behaviour is maintained up to a certain clearance after which the response approaches the RMS tube response with support-inactive. As both axes are replaced by dimensionless parameters, a single curve can be obtained, as shown in Fig. 6-b. Figure 7 shows a sample of the PSD of the tube response with zero-offset lattice supports. The spectra presented are for different clearances and all with the same flow velocity (4 m/s). For a small clearance (Fig. 7-a) the PSD of the tube response contains peaks at frequencies which correspond to the natural frequencies of the fixed-hinged configuration. For larger clearances (Fig. 7-c). the PSD of the tube response contains peaks at frequencies which correspond to the natural frequencies of the fixed-free configuration. Figure 7-b depicts the PSD of the tube response for an intermediate clearance where the mode switch takes place. Moreover, the point at which dimensionless lift and drag response change their trend corresponds to the dimensionless clearance at which mode switch takes place. The tube response spectra for the case of the support with a 5% offset of the tube length (e/L=0.05) are shown in Fig. 8. For small clearances (Fig. 8-a), the PSD contains peaks at frequencies corresponding to a combination of two linear modes. These modes correspond to the pinned configuration at the support locations. For larger clearances (Fig. 8-c), the PSD of the tube response contains peaks at frequencies corresponding to the natural frequencies of the fixed-free configuration. Figure 8-b depicts the PSD of the tube response for an intermediate clearance where the mode switch takes place.
Current Advances in Mechanical Design and Production, MDP-7
136
4O
~
C o n f i S . I . 4 m/s !
35 ~ ' 4 - 30
I
Co,s
- 6 m/s[
* 9
o8
Confi$. I - 4 mIs I Confij I - 6 m/si Config. I - 8 m / s I
(I 7
--
.I . . . . . . . .
.
.
.
.
/
..
/
$
o 6
04 (! 3 (I 2
J m o ~m6 om~. 9
m m
O! . . . . . . . . . . . . . . . .
0
OI
0.2
03
04
,
........
05
06
,,,, 07
,
l!
~,-.,
08
09
0 I
1)
0 2
0.]
0 $
0.4
06
0 7
08
b) Dimensionless RMS tube response versus the dimensionless clearance
a) RMS Tube response versus clearance
Fig. 5. Effect of the clearance on the RMS tube response in the lift direction ,oo - ' - - c - , , 700
+
t-..~.i
C o o 5 s I - 6 rids i
.
I
1"'"i
c,,.r. - 6 mill ComGI. Cemfi| I - II m/s
09
6O0 ..
0 II
500;
~
E-
l) 7 1|6
30O
mS
04
20O
03 100 ~
02 Ol
02
03
04
05 0
06
07
08
#
p.
fll
0
0
09
Ol
. . . . . . ,'. 112 I)3
....... : ..... 04 0.~ 0.6
tram]
: . . . . 0.7 0.11
Cr/drsi
b) Dimensionless tube response versus dimensionless clearance
b) RMS tube response versus the clearance
Fig. 6. Effect of the clearance on the RMS tube response in the drag direction
o
enxluen~
1o
a) Cr = 0.001 mm
do
g
to me oo e~iuency
qao m
~m
/m
40
Io
to
",oo
~10
1~0
1to
~OO
frequency
b) Cr = 0.7 mm
a) Cr = 1.0 mm
Fig. 7. Response spectra of the tube with lattice-bar support (0% offset) tO*
o
~,
,0
m
m
.o
u.
,.o
0
Cr/~i
Cr ( m m l
~
a) Cr = 0.001 mm
,m
~,
-;
~~,
&
/,
m,
d.
, ; , " ,i,
b) Cr = 0.7 mm
,io
~0
. r
o
~0
.
,,
.
.
m
.
.
m
.
~,
.
,10.,
c) Cr = 1.0 mm
Fig. 8. Response spectra of the tube with lattice-bar support (5% offset)
-
.
9
Current Advances in Mechanical Design and Production, MDP-7
137
To investigate the applicability of scaling the dimensionless results reported herein, additional configurations with different tube lengths and diameters were analysed. These configurations are designated as configuration 2 and 3 respectively. The geometric properties are listed in Table I. Configuration
Tube Length mm
Outer diameter mm
308
6.35 15.88
Lineal density kg/m
Modulus of elasticity GPa
3.86
0.16912
105
14.25
0.32
106
Inner
diameter mm ......
2
..........
........
617
..
Table. l Geometric and material properties of test cases Figure 9-a-d show contact ratio, dimensionless RMS impact forces, lift response, and drag response for all configurations. The general behaviour of the response is unchanged and an excellent agreement with the first configuration is obtained. 04
Config Config Config
~~i~o - _
9 Config *
I I I 2
-
4 6 I; 6
m/s m/s m/s m/s
035 o
03
Config 3 - 6 m / s
i. O6 %. .E
04
025 02
015 Ol .,o~
9
. . . . . . . . 04
06
04
I
Og
06
Cr.drst
9
08
9
_,
.....
I
-.~A, 12
,
14
Cr/drsi
a) Contact ratio
b) Impact Forces
09
08 07
9 *
O6
~,
Config. Config. Config. Config. Config.
I I I 2 3
6 $ 6 6
I - 4 m/s Config I - 6 m/s C o n f i s ! - I! m/s
Config
- 4 m/s]
-
v~'L=O 5
m/s| m/s[ m/s[ m/s]
= o
Config :) - 6 m/s Config 3 - 6 m / s 0
U
05 04
O3
,o
02
~-~-
01
=,-
.
eo
9
r
0
0
0.1
0.2
0.3
0.4
O.S
Cr,drsi
C) Lift response
0.6
03
0.8
0.9
I) 3
f) 4
05
06
Cr/drsi
d) Drag response
Fig. 9. Dimensionless tube response
0 7
0.8
l) 9
138
Current Advances In Mechanical Design and Production, MDP- 7
CONCLUSION Simulations for a loosely supported heat exchanger tube subjected to crossflow turbulence excitation were presented. The effect of the clearance which exists between the tube and the lattice-bar supports was investigated. The study indicates that increasing the support clearance results in decreasing the impact forces and the contact ratio. On the other hand, lift response of the tube slightly decreases for a range of small clearances then increases producing a point of minimum response at a certain clearance. Drag response increases linearly as clearance increases. The tube response, impact force, and contact ratio were effectively represented in a dimensionless form. It was demonstrated that the results can be scaled to predict the nonlinear response of tubes with different geometrical and material properties and subjected to different flow conditions. The proposed dimensionless parameters are of practical interest since the tube nonlinear-response can be correlated to random excitation due to turbulence. The proposed dimensionless parameters can be used as a guideline for design purposes and for assessing the in-service heat exchangers. Ultimately the method would be extended to account for both turbulence and fluidelastic excitations. REFERENCES 1- Weaver, D. S., and Schneider, W.,"The Effect of Flat Bar Supports on the Crossflow Induced Response of Heat Exchanger U-Tubes," ASME Journal of Pressure Vessel Technology, Vol. 105, pp 775-78 l, (1983). 2- Taylor, C., Boucher, K., and Yetisir, M.,"Vibration and Impact Forces Due to Two-Phase Cross-Flow in U-bend Region of Nuclear Steam Generators," Flow Induced Vibrations, Bearman, P. (ed), Balkema, A., Rotterdam, pp. 401-412, (1995). 3- Iwase, T., Sunami, T., Matsutani, K., Nakamura, T., Mureithi, N., Tsuge, A., Watanabe, Y., Tomomatsu, K., and Takaba, O.,"Flow-Induced Vibration of a Tube Array in the Inlet Region of A High Performance Steam Generator (Part l : Turbulence Induced Vibration)", ASME Winter Annual Meeting, Dallas, Texas, Vol. l, pp 265-272, (1997). 4- Dokainish, M. A.,"Incremental Nonlinear Dynamic Analysis Program: Verification Manual," INDAP Manual, McMaster University, Hamilton, Ontario, Canada, (1987). 5- Yetisir, M., and Weaver, D. S.,"The Dynamics of Heat Exchanger U-Bend Tubes With FlatBar Supports," ASME Journal of Pressure Vessel Technology, Vol. 108, pp 406-412, (1986). 6- Oengoeren, A., and Ziada, S.,"Unsteady Fluid Forces Acting on a Square Tube Bundle in Air Cross-Flow", Symposium on Flow-Induced Vibration and Noise, Vol. l, pp. 55-74, (1992). 7- Shinozuka, M.,"Digital Simulation of Random Process in Engineering Mechanics with the Aid of FTT Technique," Stochastic Problems in Mechanics, University of Waterloo Press, pp. 227-286, (1974).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
139
EFFECT OF PORT PLATE SILENCING GROOVES ON PERFORMANCE OF SWASH PLATE AXIAL PISTON PUMPS Kassem, S.A.* and Bahr, M.K.** *Professor, **Graduate student, Mechanical Design and Production Department Faculty of Engineering, Cairo University, Giza 12316- Egypt.
ABSTRACT In this paper a mathematical model is developed for the kinematics, piston chamber pressure and output flow rate of swash plate axial piston pumps with conical cylinder blocks. Software is developed to study the effect of the port silencing grooves on these variables. The study led to a proposed geometry of the suction and delivery silencing grooves that causes gradual rise and drop of the piston chamber pressure, with cavitation avoided, for a pump with certain dimensions. The recommended groove dimensions render less fluctuation in the output flow rate for a nine-piston pump. The developed model can be used to conduct a similar study for any swash plate axial piston pump. KEYWORDS Swash Plate, Axial Piston Pump, Chamber Pressure, Silencing Grooves, Instantaneous Flow Rate. 1. INTRODUCTION Various aspects of the static and dynamic characteristics of constant and variable geometric volume swash plate axial piston pumps were investigated during the last three decades. The variation of cylinder pressure during one complete revolution of the cylinder block was studied and shown to be much influenced by the geometry of the silencing grooves existing at the ends of the suction and delivery ports [ 1,2,5,6]. Effects of sloping and non-sloping, square or semi-circular silencing grooves on the variation of cylinder pressure were investigated theoretically [5] and shown to yield poor characteristics, especially when the initial depth is large or the slope is small. Triangular cross-section grooves were found to create low reverse flow and less overshoots and undershoots of cylinder pressure [ 1-6]. When the pump suction pressure was atmospheric, cavitation in the chamber was recorded, in some cases for very for small intervals. The investigations dealt mainly with pumps with cylindrical cylinder blocks, where the piston line of stroke is parallel to the pump axis of rotation [1-6]. Pumps with conical cylinder blocks are now widely used in both industrial and mobile applications. In these pumps the piston line of stroke is inclined to the pump axis of rotation in order to reduce the piston inertia force that tends to detach the piston from the slipper pads or the swash plate. Such a technique allows driving the pumps at higher speeds, which increases the pump specific power.
140
Current Advances in Mechanical Design and Production, MDP- 7
Edge and Darling [ 1] investigated the cylinder pressure transients in pumps with sliding valve plate using a developed model for axial piston pumps of cylindrical cylinder blocks which incorporates bi-directional port plates with sloping semi circular cross section silencing grooves. The pump suction pressure during their analysis was boosted while the delivery pressure was constant. The evaluated cylinder pressure overshoot and undershoot was considerable. The drop of cylinder pressure during the start of suction stroke is evidently not suitable for pump operation, if the pump suction pressure is nearly atmospheric; the case widely met in industrial applications. Modification of silencing grooves to suit this case was not studied. Marring [2] investigated the variation of the pump driving torque assuming a shape for the silencing grooves that causes linear variation of the cylinder pressure between the valve plate delivery and suction ports at a certain value of the delivery pressure. At the other values of the delivery pressure, the assumed shape of the grooves was shown to yield noticeable overshoots and undershoots in the cylinder pressure. Kaliafetis and Costopoulos [3] studied both theoretically and experimentally the static and dynamic characteristics of a standard variable geometric volume swash plate pump with constant pressure regulator. Modeling and designing a variable geometric volume axial piston pump was carried out in [4], and the effect of some design parameters on the pump performance was studied theoretically. Effect of the suction pressure, rotational speed, shape and sealing condition of the valve plate on cavitation in an axial piston pump was studied experimentally by Atsushi [5]. Calculation of the moment acting on the swash plate is presented in [6]. In the present work, a comprehensive theoretical study is carried out to investigate the effect of the triangular silencing grooves dimensions on the piston chamber pressure and pump flow rate fluctuation for a nine-piston swash plate pump with conical cylinder block, in order to reach a recommended port plate configuration. 2. PUMP DESCRIPTION Figure 1 shows schematically the pumping mechanism of the investigated swash plate axial piston pump, which has an odd number of pistons nested in a circular array within the block at equal intervals. The cylinder block is rotated by means of the pump driving shaft, and is held tightly against the valve plate under the effect of the supply pressure and the force of the cylinder block compression spring. A ball-and-socket joint connects the end of each piston to a slipper pad, which is kept always in contact with the swash plate. During cylinder block rotation, each piston periodically passes over the discharge and intake ports on the valve plate. Since the slipper pads are held against the inclined plane of the swash plate, each piston undergoes an oscillatory motion in and out of the cylinder block. As any piston passes over the intake port, it withdraws from the cylinder block and fluid is drawn into the piston chamber. As the piston passes over the discharge port, it advances into the cylinder block and pushes the fluid out of the piston chamber. The pivoting point, around which the swash plate swings, is the center of the pitch circle of the pistons spherical heads when the swash plate angle of inclination is zero. 3. PUMP MODELING 3.1 Pump Kinematics The displacement Sk of the kt~ piston as a function of its angle of rotation Ok, measured from the top dead center, can be deduced as follows. Take the swash plate pivoting point as the origin of the initial frame of reference of the pumping mechanism, with Z0 coinciding with the axis of the driving shaft and Y0 the axis around which the swash plate is swinging, as
Current Advances in Mechanical Design and Production, MDP- 7
141
shown in Fig. 2. The figure shows also the other needed five frames of reference XIYIZ~ to XsYsZs. Five steps of rotations and translations are then carried out to get the coordinates
142
Current Advances In Mechanical Design and Production, MDP-7
of the k th piston spherical head center at the angular position Ok relative to the initial frame of reference. These coordinates must satisfy the equation of the swinging plane, which is inclined by an arbitrary angle ot to the vertical plane. The first transformation matrix Tol represents the transformation from the initial frame of reference to the first one, which governs the translation along the Z0 axis for a distance Lt. The matrix Tt2 represents the rotation around Zt by an angle Ok. The matrices T23, T34and T45 represent the translation along the X2 axis a distance R2 ( R2 = 0.5 D2 ), the rotation around Y3 through an angle [3, and the translation along Z4 for a distance-L3, respectively, the values of the transformation matrices are given in Appendix 1. Now,
[cos0~ co~
Tos=To,xTI2xT2,xT34xT4,=/si~cosl3
-si~
cosO
IS,ore
0
0
-cos0~ sin{3 -sin0k si~ co~ 0
cos0 sinl3+ cos0k'] L3ksin0kco + sin0k|
and relativeto the initial frameof reference: Xsk = L3k COS0k sinl~ + R2 cos0~
(l)
Zsk =-- L3k cosl3 + L,
(2)
These coordinates must satisfy the equation of the swash plate inclined surface (3)
x 0 tan a + z o = 0
Substituting the values of XSkand Zsk from equations 1 and 2 into equation 3 results in, L3k = ( R z cos Oktan ot + L, )/(cos [3- cos Ok sin [3tan ct)
(4)
The displacement sk of the kth piston given by
(5)
s k = L3k - L I / cos [3 where : Ok = tot + 2n (k-l)/z
and
[3 = tan "~0.5(D,- D2)/L ~
A program was developed to calculate the displacement sk of any piston as function of Ok using the aforementioned relations. The program shows simultaneously, in an animated way, the swash plate and the piston movements during one revolution of the drive shaft, and plots the piston displacement, velocity and acceleration against 0 k for any constructional parameters of the pump. 3.2 Piston Chamber
Pressure
The variation of each piston chamber pressure during one complete revolution of the pump shaft is derived assuming that the pump rotational speed is constant, the pump suction and delivery pressures are constants, the inertia effect of the oil column inside the piston chamber is negligible, and that the total instantaneous leakage flow rate out of the piston chamber QLk is proportional to the piston chamber instantaneous pressure Pk. Applying the continuity equation to the control volume of the piston chamber shown in Fig. 3, we get:
Current Advances in Mechanical Design and Production, MDP-7
where
and
143
dsk Pk Vck dpk Osk + Ap "-~'- = Odk + ~ + ~ " R L B dt
(6)
Qdk =Cd" Adk~][2[(Pk-Pd)]/P]" sign( Pk "Pd )
(7)
Qsk = Cd" Ask~/[2i(Ps Pk)t/ P]" sign (Ps "Pk )
(8) (9)
Vck = Ap (0.5 k - s k) + Vo
To evaluate the values of Adk and Ask at any Ok, the dimensions of the port plate suction and delivery ports should be known. Knowing the variation of Adk and Ask with Ok, and for given constructional and operational parameters, the value of each piston chamber pressure pk and delivery flow rate Qdk at each Ok can be calculated using equations 6 to 9. Now, the total pump delivery flow rate Q at any Ok is given by z
Q = ~-'~Qdk
(lO)
k---!
4. PUMP SIMULATION A software package based on Matlab was developed to calculate first the displacement, velocity, and acceleration of the piston at any angular location Ok using equations 1 to 5. The program plots the piston displacement, velocity, and acceleration versus Ok. This was carried out for a 9-piston pump with the following dimensions in mm; D!=71.75, D2=60.20, D3=54.70, dp=17.00, L1=76.60, L2=66.10, Lc=57.30, Lp=59.10 and 0p=10 ~ (semiangle subtended by the cylinder port ) when running at 1450 rpm. Simulation results are shown in Fig. 4, which shows that the motion is nearly harmonic. The piston motion is simple harmonic only when 13= 0. The developed software also calculates the piston chamber pressure and the pump delivery flow rate. For this purpose, the shape of the suctionand delivery ports should be known. Figure 5 shows the shape of a port plate, which is of practical importance. The two silencing grooves at the ends of each port are triangular, with two equal sides that meet at right angles. The angles at which the silencing grooves start and end, as well as the other angles of interest on the port plate, are also shown in this figure. A separate program first calculates and plots the values of the ports areas Adk and Ask for one complete revolution of the cylinder block. The calculated areas are saved in data files to be read when needed, to reduce the required memory size and computational time when pump simulation is in run. The program allows changing any angle on the port plate using a user interface. Figure 6 shows this interface and the variation of the porting areas for the shown values of the angles. With the piston motion determined, and Adk and Askcalculated, the pump variables concerned with can be calculated by solving equations 6 through 10 numerically. This has been carried out for the pump with the aforementioned parameters. The obtained results are presented in Figs. 7 to 11. It is to be noted that the simulation process is terminated when the pressure Pk is of any value less than zero, since cavitation would then develop within the piston chamber, and the mathematical model would be invalid.
144
Current Advances in Mechanical Design and Production, MDP- 7
Fig. 6. User interface for calculating port areas. Figure 7 shows the variation of chamber pressure with Ok for three values of the groove angle to. It shows that the chamber pressure increases with pump rotation and overshoots the delivery pressure, then it decreases to the value of the delivery pressure. This can be explained as follows. At the early stages of the delivery stroke the piston chamber is isolated from both the suction and delivery ports. During this stage, the piston advances in its chamber and compresses the fluid causing its pressure to rise at a rate depending upon the piston velocity, piston area, chamber volume, oil bulk modulus and leakage flow rate. After a certain angle of rotation the piston chamber communicates with the delivery port through the silencing groove. Fluid flows from the delivery port into the piston chamber if the delivery pressure is higher than the chamber pressure, and in the opposite direction if the delivery pressure is less than the chamber pressure. In either case the flow rate depends upon the groove dimensions and the differential pressure. The chamber pressure during this stage can be controlled by the proper choice of the dimensions of the grooves and the angle through which this precompression period; namely 0 = 01d - 0p, occurs. When the chamber is fully connected to the delivery port, the chamber and delivery pressures are equal. Figure 7 shows that with narrow and shallow grooves, i.e when to is small, the overshoot in the chamber pressure is high. When the piston approaches the end of the delivery stroke, the chamber pressure is seen to rise again above the delivery pressure, since the silencing groove area is now decreasing with cylinder block rotation till it attains zero, while the piston is still forwarding in the chamber compressing the oil inside it. Maximum chamber pressure occurs when 0 = 180~ During the initial stages of the suction stroke, the chamber pressure is seen to decrease to values even less than zero, indicating the occurrence of cavitation. Cavitation conditions are worse when to is small, since the notch effective area through which suction occurs increases in this case at a low rate. This would not allow filling the chamber with fluid at the appropriate rate which copes with the piston velocity. Cavitation occurrence clearly depends on the silencing groove angle to and the angles at which the silencing grooves start and terminate. A wide and deepo notch is seen to be better regarding the variation in the chamber pressure. Increasing to from 5 to 15~ is seen to improve the variation in the cylinder pressure, which would be reflected positively on pump performance and noise. It also improves the fluctuations in the output flow rate. For the values of to higher than 15" ( results are not presented here ) no further improvement was recorded. On the other hand, even for to - 15" cavitation is seen to occur at the start of the suction stroke. To eliminate this, the effect of the other constructional parameters on the chamber pressure are investigated. Figure 8 depicts the effect of the angles 0~d and 0~s on the piston chamber pressure, when the notch length is constant and to = 15". Increasing 010 and 0ms while keeping the valve plate symmetrical is seen to be of detrimental effect on the chamber pressure and the output flow rate. Several computational runs were
Current Advances in Mechanical Design and Production, MDP-7
145
carried out to find the valve plate geometrical parameters which yield performance. Some of these results are presented in Figs. 9 to 11. Figure 9 cavitation can be avoided when 0nd = 0is = 10 ~and 02d = 0:s = 20 ~ , for a delivery 10 MPa. Keeping 01d = 0is = 10 ~ and increasing the notch length; namely 02d and to cause cavitation at the beginning of the suction stroke as seen in cases 8 and 9.
acceptable shows that pressure of 02s, is seen
When the pump supply pressure was increased to 20 MPa and 30 MPa, the assumed geometry of the grooves in case 7 yielded poor performance. Better perform.ance at the various delivery pressures was recorded for the following angles: q~ = 15* 0nd = 15,02d = 25", Ois = 10* and 02s = 19", and the silencing grooves at the ends of the ports are eliminated, with Oad = O 4 d - 165* and 03s = 04s = 169 , as shown in Fig. 10. In this case, however, the fluctuation in the output flow rate is seen to be high when the delivery pressure is 30 MPa. If this is not acceptable, the dimensions given in Fig. I 1 can be adopted since the resulting flow rate fluctuations are less. Case I tp:5 ~
Case 1
= 9,,o
I
Case 2 q ~ : 100
,, 9
,
'
Case 3 q ~ : 15 ~
01d : eli
=
15
Cne 3
ii-,
',
Max Overshoot
:-"
~
.,:;,
~so
j,. ,,-
9171%
i
'_f(v - u) with,
and
VvaH,
(2)
a(u, v) = ~oij(u)eij(v)dxdydz,
(3)
J(u, v) = ~c - ~toN(u)~vrlds ,
(4)
f(v) = ~ ~ vidxdydz + -,[ ti v ~ds
(5)
Adopting the VI approach, the mathematical formulations describing frictional contact in elastic and elasto-plastic solids have been developed and the necessary solution techniques have been implemented. In addition, the new approach has been applied successfully to a variety of engineering problems involving contact in gears, aeroengine disc assemblies, robotic dextrous grippers and simulation of metal forming processes. In the example that follows, we demonstrate the advantages offered by the VI approach over that of traditional contact elements. 4.2 Verification Example Let us consider the problem of frictionless contact between an elastic cylinder and a rigid foundation, as depicted in Fig. 12(a). The problem is solved using the newly developed VI approach as well as using a commercial finite element package employing a traditional contact element. This element employs a penalty method with implicit contact constraints iterations and generally requires the user to supply two penalty parameters representing the normal and tangential contact stiffnesses, K. and K,, respectively. In this case, however, the problem was solved using different values of K. only. The results, which are summarised in Fig. 12(b), show the normal contact stress distribution in the contact zone obtained by VI in comparison with the theoretical Hertz solution and contact element predictions. The figure shows that the accuracy of results obtained using contact elements are governed by the choice of the penalty
Current Advances in Mechanical Design and Production, MDP- 7
171
parameter Kn. The figure also reveals that only when Kn is of the order of l 0 s N/m or higher does the results of contact element approach those obtained by VI.
Fig. 12. Contact between an elastic cylinder and a rigid foundation: (a) discretized geometry of cylinder, and (b) variation of normalised contact stress versus contact displacement.
5. CONCLUSIONS This study reveals the following: 1. the maximum stress occurs at and just below the lower contact point along the bottom tooth of a compressor disc, 2. the stresses vary through the disc thickness and cannot be predicted by a two dimensional analysis, the straight slot geometry experiences maximum stresses at the outer surfaces, 4. the coefficient of friction influences the peak stress value, 5. in the skew model, relatively large stress variations exist at the contact region as well as through the thickness, 6. the maximum-minimum principal stress crack tracking criterion can reliably predict the general direction of the propagating cracks into the dovetail region, 7. for the dovetail geometry considered, crack propagation will take place across the finger region and not through the disc, thus containing possible fretting damage, and 8. the newly developed VI approach is highly effective in modelling contact. ,
ACKNOWLEDGEMENT The author wishes to thank Dr. N. EI-Abbasi for his assistance with the preparation of this article.
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REFERENCES
1. Larsen, J.M. and Nicholas, T., "Cumulative-Damage Modeling of Fatigue Crack Growth in Turbine Engine Materials", Engineering Fracture Mechanics, 22, pp. 713-730, (1985). 2. Singh, G.D. and Rawtani, S., "Fir-tree Fastening of Turbomachinery Blades", Int. J. Mech. Sci., 24, pp. 377-384, (1982). 3. Liu, A. F. and Ekvall, J.C. "Materials Toughness and Residual Strength of Damage Tolerant AircraR Structures", ASTM, STP 486, pp. 98-121, (1970). 4. Meguid, S.A., "Engineering Fracture Mechanics", Elsevier Applied Science, London, (1988). 5. Kenny, B., Patterson, E., Said, M., and Aradhya, K., "Contact Stress Distributions in a Turbine Disc Dovetail Type Joint- A Comparison of Photoelastic and Finite Element Results", Strain, 27, pp. 21-24, (1991 ). 6. Nurse, A.D., and Patterson, E., "Experimental Determination of Stress Intensity Factors for Cracks in Turbine Discs", Fatigue Fract. Engng Mater. and Struct., 16, pp. 315-325, (1993). 7. Parks, V.J., and Sanford, R.J., "Experimental Stress Analysis of the TF-30 Turbine Engine Third-stage Fan-blade/Disc Dovetail Region", NRL Report, NRL 8149, (1977). 8. Parks, V.J., and Sanford, R.J., "Three-dimensional Photoelastic Stress Analysis of the Dovetail Region of the TF-30 Turbine Engine Third-stage Fan, NRL Report, NRL 8276, (1978). 9. Ruiz, C., Post, D. and Czamek, R. "Moir6 Intefferometric Study of Dovetail Joints", ASME J. Appl. Mech., 52, pp. 109-114, (1985). 10. Papanikos, P., and Meguid, S.A., "Theoretical and Experimental Studies of FrettingInitiated Fatigue Failure of Aeroengine Compressor Discs", Fatigue Fract Eng. Mat. Struct 17, pp. 539-550, (1994). 11. Papanikos, P., "On the Structural Integrity of Dovetail Joints in Aeroengine Discs", M.A.Sc. Thesis, University of Toronto, (1992). 12. Papanikos, P., Meguid, S.A., and Stjepanovic, S., "Three Dimensional Nonlinear Finite Element Analysis of Dovetail Joints in Aeroengine Discs", Finite Elem. Anal. Des., 29, pp. 173-186, (1998). 13. Hamdy, M.M., and Waterhouse, R.B., "The Fretting Wear of Ti-6AI-4V and Aged Inconel 718 at Elevated Temperatures". Wear, 71, pp. 237-248, (1981). 14. Erdogan, F., and Sih, G.C., "On the Crack Extension in Plates under Plane Loading and Transverse Shear", Journal of Basic Engineering, 85, pp. 519-527, (1963). 15. Refaat, M.H., and Meguid, S.A., "On the Elastic Solution of Frictional Contact Problems using Variational Inequalities", Int. J. Mech. Sci., 36, pp. 329-342, (1994). 16. Refaat, M.H., and Meguid, S.A., "A Novel Finite Element Approach to Frictional Contact Problems", Int. J. of Num. Meth. in Eng., 39, pp. 3889-3902, (1996). 17. EI-Abbasi, N., and Meguid, S.A., "Large Deformation Analysis of Contact in Degenerate Shell Elements", Int. J. Numer. Meth. Engrg., 43, pp. 1127-1141, (1998).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-I 7, 2000
173
EFFECT OF VOID GROWTH ON THE PLASTIC INSTABILITY OF UNIAXIALLY LOADED SHEETS Saleh, Ch.A.R. * and Ragab, A.R.** Assistant Professor, ** Professor, Department of Mechanical Design and Production, Faculty of Engineering, Cairo University, Giza, 12316-Egypt
ABSTRACT Plastic instability for uniaxially loaded bars for materials undergoing void growth is investigated. These voids may already exist initially in the material or initiate earlier with deformation at sites of second-phase hard particles. A simplified version of GursonTvergaard's yield criterion, amenable to analytical derivation is used in the analyses. The predicted limit strains compared with that obtained experimentally show the same trend. Limit strains at localized necking is also determined for sheet with a defect in the form of a voided zone inclined to the loading axis. It is found that necking will not develop in the defected zone unless the initial void volume fraction combined with the appropriate angle of inclination produces limit strain less than that predicted by Hill. Other wise necking will occur according to Hill's condition. KEYWORDS Limit Strains, Necking, Plastic Instability, Void Growth. I. INTRODUCTION Limit strains are often determined analytically by considering plastic instability where uniform plastic deformation gives way to one in which deformation becomes localized. Necking in a tensile bar is the most familiar example, where the analysis results in Consid6re's condition of uniform tensile strain c u equal to the strain-hardening exponent in the law
o=Kc"
(I)
i.e.c u = n . This limit strain defining the onset of diffuse necking is identified by the maximum tensile load condition which has been applied by Swirl [ 1] to include other biaxial straining conditions. A further development by Hill [2] showed that localized necking in a sheet subjected to uniaxial tension is identified by assuming maximum load condition together with plane strain condition at a specific plane. Hill's analysis defines a limit strain ors a = 2n at a plane inclined 54 ~ 44' to the tensile axis using the simple power law; eq. (1). In this work the effect of voids and void growth on the plastic instability of voided metals are displayed using a yield criterion for voided material. This surpasses the shortcomings that conventional plasticity theories and their associated flow rules do not rigorously fit the
174
Current Advances in Mechanical Design and Production, MDP-7
analysis of plastic deformation of voided (porous) solids, for which the condition of volume constancy as well as the independence of yielding on the hydrostatic stress component are not valid. 2. CONSTITUTIVE MODEL FOR VOIDED SOLIDS Voids may be found in metals as in sintered powder compacts or exist at grain boundaries as a material defect. They also may be nucleated around inclusions or at second hard phase particles. With excess plastic deformation, voids link-up to form macroscopic defects. Gurson [3] in 1977 suggested a yield condition for voided solids, expressed as
]j
(2)
where UM is the effective stress of the matrix material, ~ and Om are the apparent effective and mean stresses respectively. The void volume fraction Cv defined as the ratio of the volume of voids to the total volume of porous solids, could be determined by microscopic examination for specimens cut from the deformed sheet metal or from measurements of the relative density change with strain. The above mentioned methods used to evaluate the initial void volume fraction, showed that the void volume fraction ranges from 1x 10"4to 5x 10-3 for conventional metallic alloys [ 4 and 5]. However it could reach 0.2 for sintered powder compacts [6 ]. For superplastie alloys the current volume fraction of a value 0.3 is not uncommon at tensile elongation in excess of 500% [7]. Gurson's model is modified by Tvergaard [8] to the following form
(3) The parameters q l and q2 were introduced to bring predictions of the model into closer agreement with full numerical analysis for materials with periodically distributed voids. It is found that assigning a value in the range of 1.5 to 2.5 to ql and taking q2 equal to unity improves agreement between experimental results for several different powder materials with numerical simulation of plastic flow of porous ductile solids [9]. A simplified version of Gurson-Tvergaard's amenable to analytical derivations has been suggested [10] by expanding the hyperbolic term and neglecting the terms of (3q2Om/2~M) of order higher than 2. Since the loading is uniaxial or biaxial, it is expected to obtain results identical to that when using the original modified Gurson's model [11] i.e. eq. (3).The effective matrix stress is thus related to apparent stress according to the simplified GursonTvergaard's model by =
+9qlq t 0'
1 O-qlCv)
2Cv(l' m
(4)
For uniaxial loading this relation becomes
/ 1+ q~q~ev 4 cM-
(l-qlCv)
~l
(5)
Current Advances in Mechanical Design and Production, MDP-
7
175
The matrix strain can also be related to the apparent strain ct by substituting equation (A-9) into eq.(A-3) (see Appendix A) for Xxy= 0 to give that
(
(1-q,Cv) 1+ q~q~Cv E:M= ( l - C v )
c,
(6)
4
The current void volume fraction corresponding to a certain level of cl is obtained by integration of eq.(A-3) ( for uniaxial loading and after substitution of dL by dCv from eq. (A-
8)) as e,
lln
fray/' [ ,q (C'v-i i ) ~Cv. -11
=3 [[,C )
qlq2
(7)
L
In this work the matrix material is assumed to obey the power law --
nM
6 M =KM(E)M
--
YM
(e)M
(8)
where the subscript M refers to the matrix material, i.e. KM,n M and 7M are the strength coefficient, strain-hardening exponent and the strain-rate sensitivity of the matrix material respectively. These parameters are assumed to be initially known. 3. TENSILE L I M I T STRAIN OF UNIAXIALLY LOADED BARS
In the following the plastic instability of an axially-loaded bar having a heterogeneous distribution of voids shown in Fig. 1 is analyzed. The initial void fraction of voids for zones (i) and (u) are C vo' and C vo,. respectively where zone (i) has a higher concentration of voids than the average, zone (u) i.e. (Cvo, > Cvo, ). The zone of
.Zone (i), Cvoi bo O o
P ,qt--
O o o
,--)~ p
.o. ~ . . ~ne (u); Cvou=0
higher void concentration is assumed to be extended in a plane normal to the applied Fig. 1 - Schematicdrawing for a specimen with an initial load. As the matrix material supports the void inhomogeneityunder uniaxial loading. load P, the equilibrium requires that P = OM A M = OM AM, (9) where AM is the cross-sectional area of the matrix material. If the original apparent crosssectional area is Ao, and AoM is the original cross-sectional area of the matrix material, then eq. (9) may be written as OM=e-CU=AoM. = O'M,e -eMiAoM' Since the following relation holds approximately between areas AoM ~ A o ( 1 - C v o ) hence,
(10)
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176
If the matrix material - deforming at constant volume - behaves according to eq. (8), then substitution of eq. (8) into eq. (11) and the integration give ~....//~ - ~ . / / ~ dc M, r E ; M, e
=
F ( l - C v o , ) ] ~'*~' .~/ ,.//, [ j e~'ed~ M l
(12)
The calculation proceeds by determining the current void volume fraction C, corresponding to an apparent longitudinal strain c i from equation (7). Then the matrix strain CMbeing expressed by equation (6) allows the determination of the value of CM. and hence c u by integration from equation (12). The limit strain c= is defined at the condition ds=/dc i --->0 i.e. when deformation becomes localized in the heavily voided zone (i) Fig. 2 shows limit tensile strains in uniaxially-loaded bars with different values of Cvo where zone (u)is considered to be free of voids. 3.1 Influence of Strain-Hardening Exponent The effect of both the initial void volume fraction and the strain-hardening exponent on tensile limit strain are shown in Fig. 3, where zone (u) is taken to be free of voids. The experimental points -describing the strain at fracture- are shown in the figure are for 1% Cr, 0.25% Mo cast steel [12]. It can be shown that the limit strain decreases with increasing the initial void volume fraction. Also -as expected- the limit strain increases with increasing the strain-hardening exponent (riM). 0.5 . 0.3 -
n = 0.3,]k= 0.01
Cv~
1
9Experimenlal;Caststeel [12 ]
~
0.4
q,-'q; 1
y=O.Ol,q,= q= I
=i
'
,~ .=_ I
0.3
0.2
e~ t.., ,,..,
.,., I
o.,[
0.1
0.0 0.0
,,
0.1
0.2
0.3
Strain "r Fig2- Limit strains in an tmiaxialy loaded rod with initial void inhomogeneity as predicted by Gurson Tvergaard's model.
0.0 0.00
I
I 0.02
,
I 0.04
0.06
Initial VoidVolumeFraction- C
0.08
0.10
VO
Fig. 3-Comparison between experimental fmchu~ strain and the calculatedlimit strain at necking for different values of initial void volume fraction.
3.2 Influence of Strain Rate-Sensitivity In order to examine the strain rate-sensitivity, the material is considered to be perfectly plastic i.e. nM = 0. In this ease the integration of eq. (12) gives
Current Advances in Mechanical Design and Production, MDP- 7
eu=-'/ln(l-Cvo
'
-
177
(13)
+
Again for each value of •i ( the apparent strain in zone ( i ) ) , the current void volume fraction Cv and hence the corresponding matrix strain is defined using eqs. (7)and (6) respectively. Thene u is
35 . . . . . 9 Zircaloy-4
--
C,
//
* T P bi --6S nA IE- u4 lVe c l i r
n:0
2s/
9
/ ~/_ ~-i ~ P2
Load vertex l
2
4 3 Corrected Original Corrected -P2 0 3PI -P2
O' 1
Original 0
Corrected 0
Original 3P!
Corrected 3P 91
Original 3PI -P2
G2
0
0
0
0
0
0
0
0
t~ 3
0
0
0
0
0
0
0
"P2
_
,
0.1,0.2,0"3 are the principal stresses
6. DISCUSSION The ECM results agree with both EP and corrected Stein solutions particularly for higher ratios of P2/PI. Compared with the full EP solution, the ECM technique has the advantages of being simple, modest in its CPU and computer storage requirement. As can be seen from Fig. 8, remarkable differences are noted between the above family of solutions and both the original Stein solution and Belytschko solution. It was noticed that original Stein solution is correct only for the case of uniaxial traction. In Belytschko solution, a mesh containing only 26 elements with only 4 elements at the hole were used. Clearly, this is a very rough mesh in comparison to the meshes adopted here. Figure 9 illustrates the shakedown domain as obtained by all previously mentioned solution techniques. As seen from the figure, there is very good agreement between the elasto-plastic
190
Current Advances in Mechanical Design and Production, MDP-7
and the corrected Stein solutions. The strong agreement between these two solutions indicates that the correction made to the original Stein solution is reasonable. To verify the validity of the obtained solutions, two loading points were selected. The first point lies at Pn=P2=0.75Y, i.e. outside the domain obtained by Stein and the second point lies at P n=P2=l.05Y, i.e. outside the domain obtained by both the EP and ECM solutions; see Fig. (8). The problem was solved at these points for 10 consecutive cycles using EP method and the resulting stress strain curves are shown in Figs. 10(a), 10(b) respectively. In Fig. 10 (a), the structure will move along the path 2-3 after the first cycle, which means that the behavior is completely elastic. This contradicts with the original Stein's solution [5] and agrees with the EP, ECM, and corrected Stein solutions developed in this paper. Figure 10(b) shows that the structure will exhibit cyclic plasticity once the loading point exceeds the shakedown domains obtained by EP, ECM and corrected Stein solutions obtained here. 7. CONCLUSIONS Analyses of the shakedown domain of an infinite plate with a central hole by a number of numerical and analytical techniques have shown the following: 9 Care has to be taken when constructing the principal stress vector in the analytical method proposed by Stein (1994) to avoid misleading results. 9 Both the elasto-plastic and corrected Stein solution yielded almost identical results 9 There is a fair agreement between shakedown domains obtained by the elasto-plastic and elastic compensation method. 9 The elastic compensation method has shown to be simpler due to its modest computing requirements. Artificial means for modifying Young's modulus should, however, be sought and incorporated into the ECM. 8. REFERENCES 1. Megahed, M. M., "Cyclic Plastic Behavior of Structural Components" International Journal of Mechanical Engineering Education, Vol. 10, No. 4, pp.235-257, (1982). 2. Timoshenko, S. P. and Goodier, J., "Theory of Elasticity", McGraw-Hill, New York, (1951). 3. Chakrabarty, J., "Theory of Plasticity", McGraw-Hill, (1987). 4. Belytschko, T., " Plane Stress Shakedown by Finite Elements", International Journal of. Mechanical Sciences, Vol. 14, pp. 619-625, (1972). 5. Stein, E. and Huang, Y., "An Analytical Method for Shakedown Problems with Linear Kinematic Materials", International Journal of Solids Structures, Vol. 31, No. 18, pp. 2433-2444, (1994). 6. Moustafa, A. A., "Mechanical System Design", Cairo University Press, (1996). 7. Mackenzie, D. and Boyle, J.T., "A Simple Method for Estimating Shakedown Load for Complex Structures ", Trans. ASME, Journal of Pressure Vessels Technology, Vol. 265, pp. 89-94, (1993). 8. Mackenzie, D. and Boyle, J. T., "An Iterative Elastic Procedure for Estimating Lower Bound Limit Loads" ASME, Journal of Pressure Vessels Technology, Vol. 230, pp. 129134, (1992). 9. Mackenzie, D. and Boyle, J. T., "A Method of Estimating Limit Loads by Iterative Elastic Analysis", International Journal of Pressure Vessels Technology, Vol. 53, No. 1, pp. 7795, part I, pp.77-95, part II, pp.97-119, part III, pp.121-142, (1993).
Current Advances in Mechanical Design and Production, MDP- 7
191
10. Marriott, D. L., " Evaluation of Deformation or Load Control of Stress Under Inelastic Conditions Using Finite Element Stress Analysis", International Conference of Pressure Vessels and Piping, ASME, Pittsburgh (1988). 11. COSMOS/M V.I.75A manuals, Structural Research and Analysis Corp., SRAC, (1996). 12. Mohamed, A. I., Megahed, M.M., Bayoumi, L. S., Younan, M. Y. A., "Applications of Iterative Elastic Techniques for Elastic-Plastic Analysis of Pressure Vessels", ASME, Journal of Pressure Vessel Technology, Vol. 121, No. 1, pp. 1-6, (1999). 13. Zarka, J. et al., " A Practical Method to Determine Elastic or Plastic Shakedown of Structures"- Simplified methods in pressure vessel analysis, ASME/CSME Pressure Vessels and Piping Conference, Montreal, pp. 47-60 (1978).
Current Advances in Mechanical Design and Production, MDP-7
192
L
y
X
~
+++,rI+;++++
x
1% Fig. l(b). Elastic stress distribution around the hole under uniaxial horizontal traction
Fig. l(a). Schematic representation of the plate problem -
~.
1.0
"~
o.8
9
.......
--~ -' I l.~mit Load (Bclyl.~hko.
i--
i
-
1972 !
Shakedo~ d(wnam ( Stem. I ~)4 ! Shakedo~vl loads ( Belylschko. I r
9
)
I
I,,, ,w,,
"'~
0+5
~
~ o.... 9
i
o "~
9 ;,
I
: ~
I /
i
0
1
2
0.0 0.3 0.5 0.8 1.0 N o r m a l i z e d h o r i z o n t a l traction, PI IY
3 4 Time (t)
5
6
Fig. l(d). Cyclic loading on the plate
Fig. l(c). Current solutions for limit and shakedown domains
I
"
f
t
'
.
i
Y iiiiii
Fig. 2. FE mesh for the elasto-plastic solution.
........
Z -=-+L-~
Current Advances in Mechanical Design and Production, MDP- 7
~
,'
i
---,-7-
193
I~iJ7/YYV7 ///>'I I_15 I//.J ///./b4 I I I I / / /////LA/] I I I I/,//.///././//
~, 1.0
._r 0.8 .i
l I / ! //J////V i// 17 I I ////,I
.,., 0.6 t_
;" 0.4
[I#
/ tllil/
,,,,
J, ,V/.
t l l-I
-i1/i//
-
/ /
/
t
/ .t
./
-
0.2 0.0
-
().0
i
l
,
i
i
().2 I).4 0.6 0.8 1.0 N o r m a l i z e d h o r i z o n t a l traction, Pt/Y
Fig. 5. FE mesh for the E C M solution
Fig. 3. S h a k e d o w n domain according to the elasto-plastic solution
Elaslo-plaslic solution
~. i
r I I
+----i I
Input I Model Geometry & IBoundaryCon ditions !..... t~
II
s ._:
!
~=:
.'
.ll -'li"
0.6
. .
ii.m dm
@.
il ~
II
,
Write Output
IEffective Filel I.... m Stress ____I
e
02 t
- -
E
I
Modit~' "E"
accordini;to ECM
on
c 0
.
.e 0.8
I~
e
6_0 B
Iterative elastic solution
l
START
I IL~
1.0 ~ . . . . . . . .
=50%, o0R /Y = -0.46 for m- 0.5. For m=0.45, o0R /Y is-0.52, which is 13% greater, whereas for m=0.55, o0R/Y is -0.42, which is 9~ smaller. 4.4. Comparison between Experimental and Predicted Results
Results using the original and modified models are compared with the experimental residual stress distributions obtained using the electro-chemical boring technique, as shown in Fig. 9. A fairly good accuracy is generally observed between the experimental results and the modified model, which indicates the validity and accuracy of the modified model. 5. CONCLUSIONS Based on the above results and discussions, the following conclusions can be drawn: a) There is a strong dependence of A~0P on m; use of exact value of m would contribute significantly to the accuracy of computation. b) The change of Y*/Y with plastic strain has a significant effect on the predicted residual hoop stress, especially for low overstrain values. This effect was found less pronounced for greater radius ratios. c) A fairly good agreement is obtained when comparing the results of the modified model with the experimental results obtained using the ECB technique. REFERENCES 1. Hill, R., "The Mathematical Theory of Plasticity", Clarendon Press, Oxford, (1950). 2. Parker, A., Underwood, J., Throop, J., and Andrasic, C., "Stress Intensity and Fatigue Crack Growth in a Pressurized, Autofrettaged Thick Cylinder", ASTM STP 791, American Society for Testing and Materials, Philadelphia, pp. I-216 - 1-237, (1983). 3. Chen, P., "The Bauschinger and Hardening Effect on Residual Stresses in an Autofrettaged Thick-Walled Cylinder", Journal of Pressure Vessel Technology, Vol. 108, pp. 108-112, (1986). 4. Gamer, U., "The Expansion of Elastic-plastic Spherical Shell with Non-linear Hardening", Int. J. Mech. Sc., Vol. 30, pp. 415-426, (1988). 5. Megahed, M. and Abbas, A., "Influence of Reverse Yielding on Residual Stresses Induced by Autofrettage", Int. J. Mech. Science, Vol. 33, pp. 139-150, (1991). 6. Perl, M. and Arone, R., "An Axisynunetric Stress Release Method for Measuring the Autofrettage Level in Thick-Walled Cylinders-Part I: Basic Concept and Numerical Simulation", Journal of Pressure Vessel Technology, Vol. 116, pp. 384-388, (1994). 7. Cheng, W., and Finnie, L., "Measurement of Residual Hoop Stress in Cylinders Using the Compliance Method", Journal of Engineering Materials and Technology, Vol. 108, pp. 87-92, (1986). 8. Aref, N., Senbel, H., Abdel-Kader, M., EI-Maddah, M., and Megahed, M., "Measurement of Residual Stresses in Autofrettaged Thick-Walled Tubes Using Electrochemical Boring Technique", 70, AMME Conference, Military Technical College, Cairo-Egypt, pp.145155, (1996). 9. EI-Shaer, Y. I., Mostafa, M. M., and Abdel-Kader, M. S., "On Modeling Autofrettage Residual Stresses", 8th Int. ASAT Conf., Military Technical College, Cairo, Egypt, (1999).
Current Advances in Mechanical Design and Production, MDP-?
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10. EI-Shaer, Y. I., "Effect of Residual Stresses on Fracture Toughness and Fatigue Crack Growth in Thick-Walled Tubes", M. Sc. Thesis, Military Technical College, Cairo, Egypt, (1998). 11. Aref, N., " Residual Stress Assessment of Elastic-Plastic Thick-Walled Tubes Using Electro-Chemical Machining", Ph. D. Thesis, Military Technical College, Cairo, Egypt, (1998). 12. Kovac, M., Miyano, Y. and Woo, T. C., "Residual-Stress Measurement in SS 304 Seamless Tube", Experimental Mechanics, pp. 209-213, (1989). 0.3 0.2 ~
/
0.1 o
a. -0.1 O O J= -0.2 m
//
m
-o.3
/
/
/
t
i
A
Experiment Megahed and Abbas
-----
Chen Hill
"O -O.4 o N
m -0.5 0
z
!
-o.6
-0.7
.0
1.2
1.4
1.6
1.8
2.0
R a d i u s ratio, r/a
Fig. 3. Experimental and predicted residual hoop stress, b/a=2, 0--49%
o
Experiment Original model Modified model
0.8 *>" 0.6 -
=
0.4
G
I
o
i
0
)
~
(%)
2
P Fig. 4. Variation of Y*/Y ratio with monotonic plastic strain
Current Advances in Mechanical DesiEn and Production, MDP-7
202
0.6
r r
n, ~0.4 t~
r
.=
~9 0.2
Q. O O J: --
0 1.1
Nm "O m
~
1.2
1.:
1.5
1.6
1.7
1.8
1.9
[ bla=2 ]
-0.2
Q
.N ,,m, tl
~-0.4 O Z
...... Original Megahed-Abbas Procedure ,,.
-0.6
Modified Megahed-Abbas Procedure Radius ratio, rla
Fig. 5. Effect of Y*/Y representation on the normalized residual hoop stress
(I-=0.5, b/a=2) 0.25
~
o.2
[
e m
bla=2, r - B - m=.45
m
D. O O .r
m=.50
0.1s
--e- m=.55
imUw (/) t~ Q.
I
0.1
"O
~1 o.os 0 e~
. . . . 1
1.02
1.04
~~l~__ 1.06
1.08
1.1
1.12
m
.m.
m
-
1.14
1.16
1.18
1.2
Radius ratio, rla Fig. 6. Effect of m on ~ur r (Y*/Y --0.61, b/a=2, ~------~0%)
Current Advances in Mechanical Design and Production, MDP-7
203
0.3 0.2
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204
Current Advances in Mechanical Design and Production, MDP-7
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
205
A MATHEMATICAL MODEL FOR THE STUDY OF THE
DYNAMIC-VISCO-ELASTIC CONTACT PROBLEMS Ali-Eldin, S.S.*, Aly, M.F. ** and Mahmoud, F.F. *
*Department of Mechanical Design and Production Zagazig University, Zagazig ,Egypt **National Center for Nuclear Safley and Radiological Control Atomic Energy Authority, Egypt ABSTRACT The main objective of the present research work is to develop a computational model to simulate the visco-elastic dynamic contact problems. The proposed mathematical model exploits the incremental convex programming in the framework of a finite element model. The main concern of the model is to study basically the effect of the internal material viscosity on the dynamic characteristics of the system such as rate of decaying and response amplitude. Rayleigh damping model is adopted for simulating the internal material viscosity. For model verification and testing two examples of different natures are solved. KEYORDS Dynamic Contact, Integration
Viscoelasticity, Convex Programming, Active Constraints, Direct
I. NTRODUCTIOIN Mechanical contact is of particular interest in numerous engineering applications ranging from metal forming, nuclear reactors, machine design to soil-structure and structure-structure interaction under dynamic excitation. The complexity of a contact problem is in three aspects. The first one is the nonlinear boundary condition of the contact region. The second aspect is how to describe truly the friction phenomenon on the contact surface. The last aspect is the non-linearities of material behaviors and geometric deformation. The importance of the contact phenomena has brought about extensive research work over the years. At present, there are three categories of study methods: analytical, iterative, and mathematical programming method. The analytical method can give a closed form solution and is convenient in practice. Nevertheless, the fitting range for the analytical method is much smaller and only a few problems with regular configurations and special loads can be solved. In the iterative method, the fundamental principle is to take the contact boundary conditions into the finite element equations and then adopt a trial and error method to solve the problem. The third is the mathematical programming method, whose fundamental principle is to turn the contact problems into a special example of mathematical programming. In general, the contact problem can be treated as the minimization of the systematic potential energy under some contact constraints.
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Current Advances in Mechanical Design and Production, MDP-7
In the aspect of dealing with contact constraints, the existing methods can be mainly classified into two major branches. One is the Lagrange multiplier method, suggested by Carpenter, et al [1] and Taylor, et al. [10], which can be explained as a hybrid variational principles. Another is the penalty function method, which can be thought of as a kind of artificial element, Hunek, [3]. Mahmoud et al. [6] have developed an incremental convex programming model for solving mathematical programming models consisting of a quadratic objective function and a set of linear inequalities, with non zero free coefficients and the design variables could have positive or negative signs. This developed model has been applied on non-conformal contact problems, Mahmoud et al. [5]. Recently, Mahmoud et al. [7], extended the incremental convex programming model to deal with the inequalities of zero free coefficients, which corresponds to the conformal contact , and applied on multiphase frictional contact problems. The present work extends the model to deal with elasto-dynamic contact problem where both inertia and material damping are taken into account. 2. DAMPING MODEL Damping exists in most mechanical systems or structures. Damping predominantly controls the dynamic response and attenuation of vibration. Therefore, using an assumption about damping in the analysis is a justified compromise. Even for experimental purposes, the assumption of damping has to be incorporated so that the obtained data can be processed. There are several damping assumptions available, which are based on single degree of freedom system and extended to multi degree of freedom systems. For single degree of freedom system, damping theories are established to include; for example, viscous damping, coulomb damping and structural damping. A dashpot is frequently idealized as a viscous damping element, in which the damping force is assumed to be proportional to the relative velocity between the two ends of it. The relative motion of surfaces sliding against each other causes coulomb damping. Structural damping caused by the relative motion of surfaces sliding against each other's, it is introduced to model energy dissipation of continuous systems. For a continuous system, energy dissipation within materials is supposed to be caused by thermoelasticity and grain boundary. In dealing with a multi degree of freedom system, all the above damping assumptions are applicable. Amongst these, viscous damping and structural (hysteretic) damping models are most commonly used. Due to the problems of obtaining a system's damping information, there is no practical way of forming the physical damping matrix by using the finite element method The damping properties of a system can vary according to the structure design or the material used. In the analysis of the dynamic response of any system the damping information is essential for satisfactory results. The common Rayleigh damping [8] given by
* , k=0
is a reasonable approximation for small levels of damping, where p is an integer. Using the Rayleigh will consequently result in the existence of real modes that can decouples the system's equations of motion. The simplest case is that consisting of only two terms such as:
[cl-.o [zl +..[-1. where,
ao and a l are
arbitrary constants.
(2)
Current Advances in Mechanical Design and Production, MDP-7
207
The common Rayleigh damping described by Eqn.(1) is proposed by Caughey [8], for which the two-term model (2) is only a special case when p=2, where aoand at are the material and structural damping coefficients, respectively [2]. They are typically determined from two given damping ratios corresponding to two unequal natural frequencies of vibration. If natural frequency, oi and a~ corresponding modal damping ratio, ~i are selected, a0 and at should satisfy the following relation :ao
at
~'s = 2oh + m2 a~, , where i -- 1,2
(3)
3. STATEMENT OF THE DYNAMIC CONTACT PROBLEM Consider two elastic bodies, as shown in Fig. 1,being subjected to dynamic load. Each of the two bodies occupies a bounded domain f2 in RN,N = 2,3 with open interior if2. Assuming that the material of the two bodies is isotropic and homogeneous and the contact between adjacent surfaces is frictionless. Figurel depicts the contact system and illustrates the different types of boundaries.
r~o _ , t ~ , l ,
FD Fig.l. Contact of two bodies
3.1. Strong Form of The Initial- Boundary Value Problem The problem is considered as an initial boundary value problem The deformation is governed by the following equation of motion of the system, boundary conditions, and initial conditions, which are known as the strong form of the problem (i) Equilibrium equations:.. Gij,j + fbi = P u i + ~ u i where, the constitutive equation is given by Gij = Eijkl.Uk,I ii) Initial and Boundary conditions:a) Initial conditions
(5)
b) Displacement boundary conditions
u(x,O) = 0 9 J u(x,O) = 0
(4)
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c) Loading boundary conditions:crijnj= qi 'V' xE FF
d) Contact boundary conditions :un-g. 30
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1 00
10 00
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Current Advances in Mechanical Design and Production, MDP-7
200
::L
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1.0
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
253
ACOUSTIC EMISSION DETECTION OF MICRO-CRACKS INITIATION AND GROWTH IN POLYMERIC MATERIALS Abo-EI-Ezz, A.E. Associate Professor, Force and Material Metrology Department National Institute for Standards, P.O.B. 136 Giza, Egypt.
ABSTRACT In this study, an Acoustic Emission (AE) monitoring system was developed and employed to detect the AE signals originating at the near vicinity of a single edge crack in a PMMA sheet. Different loading modes were applied in one loading spectrum on each test specimen. Throughout the entire loading spectrum the AE signals were recorded simultaneously with the load. The relationship between the crack advancement and AE signals was discussed. Examination of the fracture surface of test specimens revealed surface marks due to microcrack advancement. The number and the size of marks were matched with that of AE signals, which support the validity of AE technique. KEYWORDS Crack Detection ,Cracked Polymer, Crack Propagation, Crack Initiation, Cyclic Loading, Acoustic Emission, Stress Intensity Factor. I. INTRODUCTION The Acoustic Emission (AE) signals are stress waves. They are generated at critical sections of the stressed materials as a result of micro-cracks advancement or deformation process. The theme is very important as a non-destructive testing technique for observing critical parts in the components of structures or machinery. Utilizing the AE signals produced by the stressed components during normal working enables us to monitor the condition of the important element continuously. When the material is loaded or strained, it may produce a certain acoustic characteristic pattern. If the component begins to fail, its acoustic pattern will change [1]. The detection of such change may serve as a warning for the replacement of the component before it causes sudden failure of the entire structure. The AE technique has been widely applied to hard materials such as metallic alloys [1], concrete and ceramics [2, 3]. However the applications of the technique to polymeric materials or polymer composite materials are very few [4]. In polymers the amount of released energy due to the advancement of micro-cracks is relatively small and hence the AE signals are weak. It should be noted that the amount of released energy depends on the elastic modulus of the material, which is very low in polymers as compared with those of metals (several ten times smaller than those of metals). The differences in the amount of the released energy are thought to be in the same order of the difference in Young's modulus [5]. In
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Current Advances in Mechanical Design and Production, MDP-7
addition to that, the attenuation of these waves in polymers in the frequency range of AE signals, from 10 kHz to 500 kHz, is very large compared to that in metallic materials. These two factors cause some difficulties for detecting AE signals in polymeric materials. There are several non-destructive testing techniques in use to detect material deformations. The strain gauge technique is the most common one, since it could be used with different types of engineering materials. However, the strain gauge should be placed very close to suspected defects, which are unknown in many cases. The small angle X-ray scattering method is another detecting technique that could be used with a thin film placed into vacuum. [6]. While the scattering of visible light (laser beam)could be used only with transparent materials [6]. There are two other new techniques, magnetic and electric field techniques, which are in use for metallic materials [7]. This work has been carried out aiming at implementing the AE technique to detect the initiation and growth of cracks in a polymeric material using the AE signals originating at the near vicinity of a single edge crack. 2. EXPERIMENTAL Single edge notched (SEN) specimens were prepared from cast sheet of Poly-methylmethacrylate (PMMA). A sharp edge notch was generated on one edge of the specimen by chisel impact followed by fatigue pre-cracking (vibration crack) [8]. This procedure resulted in well-finished single crack with the desired length. The specimen geometry was 200 mm in length, 40 mm in width and 4.8 mm in thickness. An Instron-type universal-testing machine was used for specimen loading with a constant cross head speed of 0.02 mm/s. This very slow speed was selected to minimize the mechanical noise from the machine. To investigate the condition at which the AE signals may be studied in stressed PMMA, each specimen was loaded under three different loading modes in one loading spectrum. The modes are monotonic tensile loading, cyclic loading with continuous increase of mean load and finally cyclic tensile loading with a constant mean load. The tensile loading (initial setting load mode) was first applied to a certain stress intensity factor [9] below Kii (Kli is the stress intensity factor for slow crack initiation) to avoid crack propagation. The second, cyclic triangle waveform with continuos increase of mean load was applied (by increasing upper and lower loading stresses every cycle) until the value of Kli is reached. In the third loading mode, the cyclic loading with the constant mean load was continued until final fracture occurred. The AE detection was made via three identical AE sensors of resonance type (190 kHz resonance frequency for each). The sensors were fixed, on the surface of the specimen, at three different positions from the crack tip as shown in Fig. 1. This arrangement was developed as an attempt to distinguish AE signals (true signals) from untrue signals (false) which may be produced from various types of noises, mechanical or electrical noise. Each signal detected by the three AE sensors 1, 2 and 3 were amplified (100 dB in each channel) in three sets of amplifiers; pre and main amplifiers. The output of the AE system was stored in two digital memories A and B, of two channels each, and then displayed on two oscilloscopes A and B, of two channels each. The output of the two sensors I and 2 were stored in the digital memory A and then displayed in oscilloscope A. While the output of the two sensors 1 and 3 were stored in digital memory B and then displayed in oscilloscope B. A schematic diagram for the AE measuring system is shown in Fig. 1. Using such a system would help taking the right decision about the true and false signals.
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Current Advances in Mechanical Design and Production, MDP-7
Machine Fixed Head --~
~
Force Transducer
+
Test Specimen
,+
L~___ ~re-am'l
2
TV Camera rec,
C ' T'ol 1
Light ]
Load-Time Load-Signal Recorder
~
,,m
- [M"i~n'am,,P',~ -
!Pre"ampI
fiTwo
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]
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Output of sensors 1,2 from oscilloscope A
Output of sensors 1,3 from oscilloscope B
...,.
l 0 ~tm) the crack proceeds through the crazed region producing the heavy marked striations. At this level of loading the AE signals are not always detectable
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Current Advances in Mechanical Design and Production, MDP-7
because the tips of craze and crack advance rather slowly in a plasticized material without a sudden release of stored energy. On the other hand, at high stress levels (near K1i)the advancement of craze and crack takes place almost at same time releasing high energy accompanied by strong AE waves. The AE signals are always detectable at this condition. It may be concluded that the AE method is applicable for the detection of an advancing of micro cracks of an order of a few tens of microns in brittle polymers. ACKNOWLEDGMENTS: The author is grateful to Prof. K. Takahashi, Director of Research Institute For Applied Mechanics, Kyushu University, Japan and to Japanese government for giving him the opportunity to do part of the experimental study at Kyushu University. REFERENCES: 1. Williams, R.V., "Acoustic Emission", Adam Hilger LTD, Bristol U. K. (1980). 2. Ohtsu, M., Shigeishi, M. and lwase, H, "AE Observation in the PuU-out Process of Shallow Hook Anchors", Proc. of JSCE, No 408/V-11 (Concrete Eng. and Pavements) Aug., pp. 177- 186, (1989). 3. Shigeishi, M. and Ohtsu, M. "Observation of Mixed-mode Fracture Mechanism by SIGMA-2 D", J. of Acoustic Emission, Vol. 11, No. 4, pp. S 57- S 63, (1993). 4. Nagoh, S., Choi, N. and Takahashi, K., "Acoustic Emission in SMC Specimens with Various Notch Trip Radii", RPPJ, Vol. 32, pp. 311- 314, (1989). 5. Shirouzu, S., "Studies on Acoustic Emission in Polymers", Master Thesis, Kyushu Uni. Japan, March (1984). 6. Zhurkov, S.N. and Kuksenko, V.S., " The Micro-mechanics of Polymer Fracture", Inter. J. ofFrac. Vol. 11, 4, pp. 629- 939, Aug. (1975). 7. Winkler, S. and Winkler, W.D., "Field Emission in Charpy Tests", Roell-Amsler Symposium, May (1998). 8. Abo-EI-Ezz, A.E., Abd-el-hakeem, H.M. and Takahashi, K., "A Study of Fatigue crack Propagation in Glassy Polymer by the Optical Method of Caustics", JSME Int. J., Series A, Vol. 37, No. 4, pp. 466-471, (1994). 9. Tada, H., Paris, P.S., and Irwin, G.R. "The Stress Analysis of Cracks", Handbook, Del Res. Co., Hellertown, Pennsylvania, p. 2.10, (1973). 10. Begueiin, P, "Experimental Approach of Mechanical Behavior of Polymers Under High Rate of Loading", Report No. 1572, Ecole Polytechnique Federale, Lausanne, pp. 1-26, (1996).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
261
EVALUATING THE STRESS CONCENTRATION DUE TO ELONGATED DEFECTS IN WELDED AREA
Abd EI-Ghany, K.M.', EI-Mahallawi, I.'" and EI-Koussy, M.R. "** Cairo University Center for Advanced Software Development Associate Professor, Department of Mining, Petroleum and Metallurgy Professor, Department of Mining, Petroleum and Metallurgy Faculty of Engineering, Cairo University, Giza, 12316 - Egypt
ABSTRACT In welded pipes, a welding defect can cause a disaster. Among the commonly known welding defects, incomplete penetration and fusion, porosity and slag inclusions are the most common to affect welding strength. The acceptance and rejection criteria of discontinuities are the roles of the standard codes such as API, ASME, ASTM, AWS, etc. The standard specifications are restricted and force the inspection engineers to reject or re-weld pipes that may be "Fit for their Purpose". In many cases, these pipes may contain welding defects and discontinuities that produce stress concentration values that may not affect the service life of the material. This paper describes the employment of three-dimensional linear static finite element analysis to find out the values of stress concentration that develop due to the presence of elongated volumetric discontinuities in the welded region. Also, it seeks for the relations among the length and the diameter of the discontinuity, the effective weld and wall sizes and the triaxial stress region induced around each discontinuity. The finite element software package I-DEAS [ 1] was used for this work. About 200 simulation cases were modeled and solved by the finite element software. The modeled cases were identical in the geometry, applied stresses, boundary conditions and meshing size and type, but they varied in weld thickness and discontinuity length and diameter. The applied stresses were biaxial to simulate the welded piping applications and uniaxial to simulate the general purpose applications. The results indicate that, I) The average stress concentration factor when the discontinuity length is parallel to the maximum applied stress is about two, and about three in the normal direction, (yon Mises stress), 2) The acceptance and rejection criteria of the codes AWS D 1.1 and ASME section V (for boilers and pressure vessels) are conservative to some extent, 3) The calculated values were found closer to the criteria o f A P I - 1 1 0 4 and 4). The effect of the triaxial stresses induced around the curved surfaces of the discontinuities is remarkable and should be considered while judging the discontinuities effect. KEYWORDS Welding, Discontinuities, Finite Element Analysis, Concentration, Acceptance and Rejection Codes
Standard Specifications, Stress
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Current Advances in Mechanical Design and Production, MDP-7
1. WELDING DISCONTINUITIES The steel pipes are likely to suffer from welding imperfections and discontinuities that affect the pipes and the whole structure strength and resistance to the applied stresses and environmental conditions. For example, a welding discontinuity only a few centimeters long can reduce the strength of the pipe by a factor of 5 and lead to disastrous failure [2]. The welded pipe inherits its durability and long life from the following factors: 1. The material properties of the base metal, weld metal and the heat affected zone. 2. The pipe design, dimensions and wall thickness, joint design and weld throat size. 3. The proper selection of welding procedure specification (WPS), Procedure Qualification Record (PQR) and qualified welders. 4. The type, size, and position of welding imperfections and discontinuities in the welded region and in the base metal. Welding discontinuities are classified in the documentation of ISO 6520 and the parameters characterizing flaws are defined in the documentation of IIW/IIS-340-69 [3] into three general types: - Linear (one dimensional) discontinuities, such as slag lines. - Planer (two dimensional) discontinuities, such as cracks, lack of fusion, lack of penetration and other like flaws. - Volumetric (three-dimensional) discontinuities, such as porosity, inclusions and similar flaws. - Imperfect shape, such as reinforcement, angular distortions, misalignment and similar flaws and geometric discontinuities. The determination of the presence, dimensions and position of each type of these discontinuities is the matter of using non-destructive examination techniques such as radiographic, ultrasonic, eddy current, magnetic particle testing, etc. While the cumulative damage due to their presence in the welded region should be carefully evaluated, referring to the published analytical and experimental studies for fracture mechanics [4,5]. On the other hand, the presence of volumetric discontinuities such as porosity and inclusions produces stress concentration around the discontinuities. The apparent material damage occurs when the elevated stress level due to stress concentration exceeds the yield strength of the material. In the ideal cases and under only static monotonic stress, a plastically deformed layer of material covers the discontinuity with no further damage occurring. However, this plastically deformed layer is not desirable under cyclic stress because it can host finite cavities that grow and form dangerous planer defects and cracks (Fig 1.) 0"00"
~
(a) Cyclic Load
0. 13" 0. Plastic deformation region. 0
(b) Static loading
Fig. 1. The behavior of the discontinuity under cyclic and static stresses.
Current Advances in Mechanical Design and Production, MDP-7
263
2. FITNESS FOR PURPOSE CRITERIA The techniques of non-destructive examination provide the engineer with the necessary tools to judge if the welding region contains discontinuities or not. The acceptance and rejection criteria for a discontinuity are the role of the standard codes and specifications such as API, ASME, ASTM, AWS, etc. The implemented acceptance criteria are based on empirical data and experimental case studies, as a result, they may be conservative in some applications in order to gain global acceptance and reliability for their proposed scope of engineering applications. The criteria of Fitness for Purpose of a product means that the product containing discontinuities has sufficient strength and that it does not fail prematurely during service. It is assumed, of course, that the product in not being misused and that it is eventually maintained or renovated when the stipulated lifetime has passed. Any pipe should be designed, manufactured, inspected, and evaluated in such away that it is fit for its intended purpose. The introduction of the Fitness for Purpose criteria is linked to the progress in material science, welding processes and non-destructive examination techniques, in addition to the advanced computer and stress analysis systems. This will allow for the acceptance of larger discontinuities and save the money and time consumed for repairing or rejecting the weldments. 3. WELDING RELATED STANDARD SPECIFICATIONS The standard codes define the quality assurance and quality control of any manufactured product. For example, the standard acceptance codes relevant to welding process define the whole process from selecting the welding consumables (e.g. BS 639, DIN 1913, ANSI/AWS A5.1-81), welding procedure approval (e.g. BS 4870, ASME Section IX), welder approval (e.g. BS 4871, BS 4872, ASME Section IX, DIN 8560) and finally the quality of a completed fabrication (e.g. BS 5500, ASME Section VIII, AWS Structural Welding Code) [6]. Table 1, summarizes the specified acceptance criteria for the elongated and rounded discontinuities (elongated discontinuity is one in which its length exceeds three times its diameter), where t = wall thickness, O.D = pipe outer diameter, L = Discontinuity length, W = Discontinuity width and D = discontinuity diameter.
Table 1. The maximum allowable discontinuities dimensions
Standard Code
API 1104
ASME
AWS
Allowable elongated dimension For O.D > 60.3 mm L
2.70 2.8S
50
-
I,
4
I
12
Discontinuity
30 Length
Fig. 4. Stress concentration factors vs. the Fig. 5. Stress concentration factors vs. the discontinuity length for different wall thickness, discontinuity length for different wall thickness. Results from the first group of runs. Results from the second group of runs.
3.t
i
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8
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I 0.8
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0.4
1.|
0.8
Discontinuity Diameter
Fig. 6. Stress concentration factors vs. the Fig. 7. Stress concentration factors vs. the discontinuity diameter for different wall discontinuity diameters for different wall thickness. Results from the first group of runs. thickness. Results from the second group of runs.
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Fig. 8. Stress concentration factor when the discontinuity is deep and on surface (t-12.5 mm and D--0.8 ram.) under the two stress cases.
2
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270
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Current Advances in Mechanical Design and Production, MDP-7
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Table. 2. The values of stress components resulted from the first group of runs where the discontinuities were located parallel to the maximum applied stress. The shown results are only for wall thickness = 6.4 mm. 'Defect Diam. 0.4
Defect . . . . .
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
271
MECHANICS OF FRACTURE IN FIBROUS METAL MATRIX COMPOSITES
Bahei-EI-Din, Y.A. and Elrafei, A.M. Structural Engineering Department Cairo University, Giza, Egypt
ABSTRACT An experimental and analytical study of crack-tip plastic deformation fields and fracture of three unidirectionally reinforced metal-matrix composites, B/As FP-Ae203/As and SiC/Ti is described. Behavior of the first two materials with a 6061 aluminum matrix was evaluated at room temperature, while that of the titanium matrix composite was evaluated at 650 ~ The experiments revealed the existence of discrete plastic zones at the notch tip, which cause significant stress redistribution in the notched part before failure. Computed with a modified finite element scheme which incorporates the experimental observations, the stress field suggests that failure of the fracture specimens is controlled by a stress ratio between the maximum axial normal stress in the unnotched ligament and the ultimate strength of the composite. KEYWORDS Fibrous Composites, Fracture, High Temperature, Experiments, Finite Elements. I. INTRODUCTION It is well established that standard or modified fracture mechanics approaches could not in general be used to predict the fracture strength of notched metal-matrix fibrous composite materials [1]. The difficulty lies in the development of inelastic strains which cause substantial stress redistribution and alter the magnitude of the stress concentration factor. The effect of the inelastic response on the fracture behavior of a notched part depends on the yield strength and the strain-hardening rate of the matrix. For example, long plastic zones are found in notched boron-aluminum specimens in the as-fabricated condition [2,3], but not in the overaged T6 condition [4]. While the fracture behavior of this composite system in the latter condition can be estimated from the elastic field, it must be founded on a continuum plasticity theory of the composite material for the former condition. Moreover, the effect of loading rate is significant at high temperature and must be considered in realistic estimates of the fracture load. Attempts to predict the fracture strength of notched polymer and metal matrix composites have centered mainly on the shear lag model with various modifications [5-7]. The present work utilizes the finite element method with feed back from experiments to study the fracture behavior of metal matrix composites. Three unidirectionally reinforced composites are considered, B/Ag, FP-Ag203/Ag and SiC/Ti. Behavior of the first two materials with a 6061 aluminum-matrix was evaluated at room temperature, while that of the
272
Current Advances In Mechanical Design and Production, MDP-7
titanium-matrix composite was evaluated at 650 ~ The experiments on as-fabricated and annealed B/At specimens were conducted by Babel-El-Din et al. [3,4] and revealed the existence of discrete plastic zones at the notch tip, which blunt the tip and cause significant stress redistribution in the notched part before failure. This led to a fracture criterion [8] which is examined here for the FP/At and SiC/Ti specimens. The stress and strain fields in the tested specimens were computed with a modified finite element scheme which incorporates the experimental observations, and the elastic-plastic and/or rate-dependent behavior of the matrix. The analysis predicted long plastic zones of length similar to that observed experimentally in the as-fabricated and annealed B/At composite specimens. The plastic zones computed in the FP/At composite were much shorter, but sufficiently long to blunt the notch tip and modify the local stresses. Slow loading and the low matrix yield strength found at high temperature caused substantial inelastic strains at the notch tip in the SiC/Ti composite as well. In all three composite systems, the computed stress field suggests that failure of the fracture specimens is controlled by a stress ratio between the maximum axial normal stress in the unnotched ligament and the ultimate strength of the composite. The experimental program is described in Section 2. Analytical evaluation of the local stresses in the notched specimens using the finite element method is given in Section 3 together with results for the J-integral and G~ energy release rates and validation of the fracture criterion proposed in [8]. 2. EXPERIMENTAL PROGRAM 2.1 Materials and Testing Procedure The experimental results presented here include three different materials, one is a B/At composite which has been tested earlier by the first author and his coworkers [3,4] and is used here for comparison, and the other two are FP/At and SiC/Ti. All materials were fabricated by diffusion bonding of unidrectionally reinforced plies using hot isostatic pressing at a certain temperature and pressure. The matrix material for both the B/At and FP/At composites is 6061 aluminum, and for the SiC/Ti composite is a Timeta121S titanium. The fiber diameter for the boron and the Sigma SiC filaments is 142 and 100 tim, respectively. The number of plies is six for the B/At composite and four for the SiC/Ti composite, which amounts to a laminate thickness of 1.067 mm and 0.635 mm, respectively. On the other hand, the alumina (FP) fiber has a diameter of only 20 tim. The specimen thickness is 2.5 ram. The fiber volume fraction cf estimated by the manufactures of these materials and verified from micrographs is 0.5, 0.35, and 0.3, for B/At, FP/At and SiC/Ti, respectively. All composite panels were obtained in the as-fabricated condition. Specimens were cut from the composite panels with a diamond saw. The specimen length was the same as the side length of the composite panel; no saw cuts were made perpendicular to the fiber. A center notch was made in each specimen, in the direction perpendicular to the fibers, using the electrostatic discharge machining technique. These notches were typically 0.2-0.4 mm wide. Tables 1-3 show specimen and notch dimensions, where W is the specimen width and 2c is the notch length. Several unnotched specimens were also used for strength measurements. The ends of all specimens were bonded to thick annealed aluminum tabs. The free length between the grips was 203 mm for the B/At, 87 mm for the FP/Al and 127 mm for the SiC/Ti.
Current Advances in Mechanical Design and Production, MDP-7
273
T a b l e 1. C e n t e r - n o t c h e d , B/A~ c o m p o s i t e s p e c i m e n s tested at r o o m t e m p e r a t u r e [3,4]. i=
i
Specimen
W (mm)
AI
25.3
A2 A3 A4 A5 A6 A7
25.3 25.5 25.7 24.4 25.2 25.6
A8 A9 AI0
BI B2 B3 B4
B5 B6 B7 O1 02
'
,
,,
,
t~ (MPa/s)
0.204 I'
37.0
25.3 25.5 25.5
0.708 0.703 0.706
37.0 37.0 37.0
322 329 295
50.8' 50.9 50.6 50.5
0.302 0.301 0.503 0.504
18.5 18.5 18.5 18.5
586 618 455 456
II
18.5 18.5 18.5 37.0 37.0
0.197 0.197 0.194 0.401 0.401 0.402 0.601 0.602 0.602
,
Glig(MPa)
883 . . . . . . . . . .
863 831 642 630 396 454
0.702 0.700 0.701 0.112 0.I 15
,,
Oult(MPa)
(Yult(MPa)
37.0 37.0 37.0 37.0 37.0 37.0
25.4 25.3 25.4 25.3 25.3 25.2 25.4 25.4 25.3
L
i
0.200 0.199 0.395 0.416 0.606 0.592
50.9 50.8 50.9 25.3 25.4
03 04 05 06 07 08 09 OI0 Ol I
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2c/W
859
425
602
37.0 37.0 37.0 37.0 37.0 37.0 37.0 37.0 37.0 I
I
1071
862 917
987 1007 963 1041 1033
295 919
769 753 788 568 557 555 368 378 364
1059
840 884 915 919
456
560
370 I
986 i
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ill ii
1037
958 938 978 948 930 928 922 950 915
770
1075 1070
1103 1108 1003
315
294 302 288 924 914
'
1079 1037 1061 1079 1005 1113
636
'"
~lig (MPa)
1109
958
935
929
T a b l e 2. C e n t e r - n o t c h e d , F P / A / c o m p o s i t e s p e c i m e n s tested at r o o m t e m p e r a t u r e . ii
i
iil
_
Specimen
W (ram)
2c/W
Pl P2 P3 P4 P5 P6 P7 P8
22.') 22.7 22.7 22.7 22.7 22.7 22.7 22.7
0.1 0.1 0.3 0.3 0.5 0.5 0.7 0.7
t~ (MPa/s)
i
i
i
i
i
i
i
i
i
i
i
ii
i
j
i
i
O'uit (MPa)
~ult (MPa)
O'lig(MPa)
~lig (MPa)
316 237 167 211 139 140 88 78
277
351 263 239 301 278 280 293 260
307
.... 33.0 33.0 33.0 33.0 33.0 33.0 33.0 33.0
189 140 83
270 279 277
T a b l e 3. C e n t e r - n o t c h e d , S i C / T i c o m p o s i t e s p e c i m e n s t e s t e d at 650 ~ 9
Specimen ........S i
2c/W
t~ (MPa/s)
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0.096
0.50
560 . . . . . . . . . .
6i9
0.50 0.50 0.50 0.50 0.50 0.50 50.0
366 373 373 260 252 275 402
518 528 528 527 511 556 573
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0.114 0.102
$4 $5 $6 $7 $8 $9
19. I 19.1 19.1 18.4 18.5 18.8
0.293 0.293 0.294 0.507 0.507 0.505 0.298
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i,iii
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537 523
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(Ylig(MPa)
W (ram)
540
606 582
371 262 ,,,,
402 .... ,,
,
~lig (MPa) 602 525 531 ,,j
573
274
Current Advances in Mechanical Design and Production, MDP- 7
Before testing, each specimen was installed, and symmetrically aligned with flat grips which were then attached to the testing machine. Loading was applied by a 50 kN MTS testing machine, in simple tension, at the rate of about I kN/min for the B/At and FP/At specimens, and 36 kN/min and 0.36 kN/min for the SiC/Ti specimens. All specimens were loaded up to failure, and the ultimate load was recorded. The B/Ag and FP/At specimens were tested at room temperature, while the SiC/Ti specimens were tested at 650 ~ Heating of the SiC/Ti specimens was achieved by a steel susceptor tube that enclosed the specimen and was heated by direct induction using a 5-kW solid-state heating unit. The susceptor tube was 25.4 mm in diameter and 100 mm long. This allowed cooling-water pipes to be installed adjacent to the grips. Temperature measurements and control were performed with thermocouples. A uniform temperature was maintained in the 50.8 mm center zone, while the grips were kept at room temperature.
2.2 Experimental Results Notched specimens of all three materials failed by non self-similar fracture of the unnotched ligaments. Fracture of the B/At and FP/At was accompanied by long splits at the notch tips in the fiber direction. This was preceded by formation of discrete, plastic shear zones at the notch tips which were detected for the B/At specimens by Bahei-EI-Din et al. [3,4] using a bar code technique. Splitting in the fiber direction was not observed in the notched SiC/Ti specimens, but formation of plastic zones at the notch tips is expected to take place prior to fracture as predicted analytically in Section 3. Figure 1 shows a photograph of specimen $7, Table 3, a~er failure. Tables 1-3 list the strength magnitudes found for the fracture specimens. The overall stress at failure ault and the net ligament stress at failure at~g are given for each specimen. Also given in Tables 1-3 is the average ultimate stress Uu~t and average ligament stress UJis at failure for specimens with a constant nominal 2c/W. The results in Table 1 indicate that, in principle, the measured fracture strength is unaffected by the specimen width or by annealing. The loading rate appears to have a small effect on the fracture strength even for the SiC/Ti specimens tested at 650 ~ For a given notch length to specimen width ratio, the measured fracture strength is substantially different for the three tested materials since failure is eventually controlled by the fiber strength. Since deformation of the tested specimens from all three materials had similar features, their fracture strength is best compared on a relative strength scale in which the unnotched strength is the normalizing factor. The measured ultimate strength for the three composite materials listed in Tables 1-3 are normalized by their respective unnotched tensile strength and plotted versus the 2c/W ratio in Fig. 2 for the B/Ag and FP/Ag specimens, and in Fig. 3 for the SiC/Ti specimens. At room temperature, the unnotched tensile strength was measured experimentally at 1659 MPa for the B/Ag composite and 451 MPa for the FP/Ag composite. The ultimate tensile strength for the SiC/Ti composite material tested at 650 ~ is 804 MPa. Apart from experimental scatter,
Current Advances in Mechanical Design and Production, MDP-7 1.0: O.9
~
0.8
'-' ~I
o.7
IZ c
s~
0.5
c ~,~
0.4
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A
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275
B/AI-O, Cf = 0.5, W = 25.4 mm
9
FPIAI, c4 = 0.35, W = 23.0 mm
~
- - - - - r = 11 - 2c I W) ~ ~4P~.._ ------- r = 0.57 - 0.56 (2cNV) + 0.43 e ('le (2cNV))
~,~,~,
(correlation coefficient = 0.982)
0.0
0.1
0.2
0.3
Notch
0.4
0.5
0.6
Length~SpecimenWktth,
0.7
0.8
2c/W
Fig. 2. Measured relative strength reduction caused by a center notch in B/Ae and FP/A/specimens. 1.0;
.r
"
o~ C
0.9
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0.8
~
~
o.6 o.5
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or
=~ ~
.~
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o.1
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- - - - - r=(1 - 2 c l W ) --..--- r = 0.65 - 0.65 (2c/W) + 0.34 e('14 (2c~) (correlation coefficient = 0.996) ,
I
0.1
,
h
0.2
,,
L
0.3
,
~
0.4
,
I
0.5
,
I
0.6
,
I
0.7
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,
,
0.8
Notch Length/Specimen Width, 2c/W
Fig. 3. Measured relative strength reduction caused by a center notch in SiC/Ti specimens.
which is largest for FP/At', the data points in each of Figs. 2 and 3 fall into a narrow range. Moreover, the fracture strength measured for specimen Q1, Table 3, at a high loading rate also falls in the range of the S-specimens tested at a slow loading rate. This suggests that predictions of the relative strength can be derived from a single model of the fracture process, which is a function only of the 2c/W ratio and the material properties. As suggested by Fig. 2, and on a relative strength scale, the fiber material does not play a role in the fracture criterion other than restricting plastic shear deformations at the notch tip to a narrow zone that is parallel to the fiber direction. The dashed line shown in Figs. 2 and 3 indicate the variation of the net ligament strength ratio with 2c/W. The reduction from the net ligament strength is due to stress concentrations caused by the notches. Specifically, plastic zones emanating from the notch tips in the fiber direction induce additional local stresses and cause the reductions in overall strength of the notched specimens from the net ligament strength values [4,8].
276
Current Advances in Mechanical Design and Production, MDP-7
3. ANALYSIS
3.1 Finite Element Model The local stresses in the tension specimens containing a center notch were evaluated with a modified finite element analysis which recognizes the effect of the discrete plastic zones present at the notch tips on the local stresses [4,8]. The finite element domain is shown in Fig. 4. It represents one quadrant of the notched specimen. Displacement boundary conditions derived from symmetry of the specimen geometry and the applied load with respect to the planes x l = 0 and x2 = 0 were applied together with a uniform displacement condition at the boundary x2 = L/2, where L is the specimen length.
l
1~
uniform
displacement
"'
,)
F,
L/2
g ..J a,
t~,:
-
.~ .f
Xl
The finite element solution was simplified by Fig. 4. Finite element domain and limiting plastic or viscoplastic deformation in boundary conditions of centerthe domain to one row of elements which were notched specimens. designated to represent the plastic shear zones. All other elements were assumed to remain elastic. This was suggested by the existence of two plastic deformation modes in metal matrix composites, one is matrix-dominated and the other is fiber-dominated [9]. Stresses outside the plastic zone promote the fiber-dominated mode in which the inelastic strains are negligible compared to the elastic strains. This was confirmed by a nonlinear finite element solution of the whole domain for one B/Ag specimen and one SiC/Ti specimen, which provided local stresses that are identical with the stresses computed for the modified domain of Fig. 4. Finite element solution of the B / A t and FP/At specimens was obtained by the ABAQUS program [10]. In the elastic range, the composite was regarded as a homogeneous, transversely isotropic medium. Elastic-plastic response of the plastic zone elements was specified with the help of Hill's anisotropic yield criterion. A rail shear test provided the stress-strain response of the composite materials. The initial yield stress in shear was measured at 40 MPa and 34 MPa for the B/At composite in the F and O condition, respectively, and 49 MPa for the FP/At composite. Finite element analysis of the SiC/Ti specimens on the other hand was performed with the VISCOPAC program [ 11 ] which derives viscoplastic response of the plastic zone elements from a rate-dependent, micromechanical model of fibrous composites. Properties of the Timetal 21S matrix measured by Dvorak et al. [12] at 650 ~ provide initial yield stress in shear of 26 MPa. Refinement of the stress field computed at the notch tip followed two slightly different procedures. In the SiC/Ti specimens, a refined mesh was selected in the vicinity of the notch tip such that the element dimension in direction of the xl-axis is in the order of the width Wry of a representative volume element (RVE) of the composite. The latter was evaluated from a hexagonal composite cylinder containing a fiber of diameter df as Wry = df ~/(2~t/3~/(3)cf). For df = 0.1 mm and cf = 0.3, we find Wr~ = 0.2 ram. Actual meshing of the domain in Fig. 4 provided elements of width 0.25 mm at the notch tip. The finite element mesh was also refined in the fiber axial direction particularly in the vicinity of the notch tip.
Current Advances in Mechanical Design and Production, MDP-7
277
Calculation of the local stresses in the B/Ae and the FP/Ag specimens on the other hand followed the procedure described by Bahei-EI-Din et al. [4,8] which combines the numerical solution provided by the finite element method, and an analytical solution of the notch tip stress field. The latter is obtained from solution of an elastic half-space under boundary shear tractions equal to the shear flow stress generated in the plastic zones. The half space represents the unnotched ligament ahead of the notch tip, and the shear tractions are applied at the boundary xl = c, starting from the notch tip and extend in the x2-direction for a predetermined distance. Details of this method are provided in references [4,8]. In this case, however, the finite element domain shown in Fig. 4 is modified by evacuating the plastic zone elements within the predetermined distance at which the shear tractions are applied in the analytical solution. The local stresses are then determined by superposition of the numerical and analytical solutions. 3.2 Finite Element Results The length of the plastic zones R was determined by identifying the first element in the plastic zone region, Fig. 4, for which the shear strain is smaller than the elastic limit. The latter was evaluated from the shear rail test for B/Ag and FP/A~, and estimated from the Mori-Tanaka [13] micromechanical model for SiC/Ti. Bahei-EI-Din et al. [3,8] detected long plastic zones in the B/Ae specimens using the bar code technique, and computed the plastic zone length R in the order of 3-7 times half the notch length c. Similar plastic zones were computed for the SiC/Ti specimens with the R/c ratio of 3-5. The plastic zones computed for the FP/Ae specimens however were much smaller with R/c in the order of 0.4 to 2. In any case, the computed energy release rate for the tested specimens indicates that the plastic zones were sufficiently long to blunt the notch and modify the elastic stress field.
It was shown by Bahei-EI-Din et al. [4,8] that the elastic energy release rate GI computed for the B/Ae fracture specimens at the measured ultimate load is different from the J-integral, and neither quantity can be considered a material property since it varies with the 2c/W ratio. The difference between G! and the J-integral is attributed to the plastic zones which modify the elastic field in the notched specimens. Our calculations for Gi and the J-integral for the SiC/Ti specimens are shown in Fig. 5. Here too, the plastic zones result in a J-integral that is quite
20 A E ~ Z v
~ ID
~ llJ
-~
18
SiC/Ti, cf = 0.3
650~
0
J-integral ,
alh
16 14 12 m
10
a""
8
B
4 uJ
9
r'l 0
C
o.s.P~/. 50 UPa/s
2 0
,
0.0
|
0.1
J
I
0.2
,
I
I
0.3
0.4
,
|
O.S
9
9
0.6
2c/W
Fig. 5. J-integral and elastic energy release rate G! computed at fracture stress for SiC/Ti specimens.
278
Current Advances in Mechanical Design and Production, MDP-7
different from the elastic energy release rate. The J-integral was not computed for the FP/Ar specimens, but the Gt values, in the order of 1.4 to 3.7 N/mm, were dependent on the 2c/W ratio. Pending verification with more experiments, it can be concluded that the critical energy release rate of the notched unidirectional metal matrix composites tested here is geometry dependent, and is substantially different from the elastic energy release rate. Bahei-EI-Din et al. [8] examined the local stresses computed at the fracture load for the center-notched B/At' specimens listed in Table 1 and suggested that the onset of fracture is controlled by a critical stress ratio of the normal stress in the unnotched ligament at the notch tip and the ultimate strength of the composite material. When the local normal stress was computed at the notch tip as an average over the width Wryof a representative volume of the composite material which contains a single fiber, the critical stress ratio was found to be 1.0, and independent of the speeimen's width. Moreover, the same critical stress ratio was computed for both the as-fabricated and annealed specimens, and for B/At' specimens with a center hole and edge notches [4]. This fracture criterion is expected to be valid only for unidirectional composites with a low yield stress in shear such that discrete plastic shear zones can be formed and blunt the notch tip. For example, Bahei-EI-Din et al. [4] found that the proposed fracture criterion was not applicable for 6061-T6 At'/B specimens with shear yield stress of 74 MPa and flow stress of 188 MPa. The fracture criterion was examined for the FP/At' and SiC/Ti specimens of Tables 2 and 3. Figure 6 shows the local normal stress computed in the SiC/Ti specimens at the fracture loads and normalized by the unnotehed strength of 804 MPa. Two groups of points are shown for these specimens, one computed with the local stresses averaged over a representative volume of width Wry = 2.5 dr, and the other for Wry= 8.0 dr, where dr = 100 ~tm. This result suggests that although the magnitude of the critical stress ratio varies with the selected RVE width, it is a constant, within an experimental scatter, for 2e/W > 0.3. The lower values found at 2c/W = 0.1 suggest that the elastic field is dominant as also indicated by the close values of the J-integral and GI, Fig. 5, for this crack length to specimen width ratio. Also shown in Fig. 6 is the ligament stress, normalized by the unnotched ultimate strength. The difference between the total local stress and the ligament stress is the stress caused by the plastic zone shear stress [4,8]. Figure 7 shows the local normal stress computed at the fracture load for the FP/At' specimens normalized by the unnotched strength of 451 MPa. Here, the stress ratio is plotted against the width of the representative volume over which the local stress is averaged, normalized by the fiber diameter. For a selected Wr, value, the stress ratios corresponding to the fracture load fall within a narrow band that is consistent with the experimental scatter, and suggest that fracture in the FP/At' specimens is also controlled by a critical stress ratio. The implication is that the fracture strength of notched, unidirectional composites which develop significant plastic deformations at the notch tip can be predicted analytically by comparing the local stresses averaged over a selected representative volume to a critical value that is considered a material property. The latter is evaluated analytically for a notched specimen of the material at the overall fracture stress determined experimentally.
Current Advances in Mechanical Design and Production, MDP-7
SiC/l'i, cf = 0.3
2,50
~
total stress, Wn/df = 2.5
650Oc
2.25
0
279
.00
~
1.75
o9 -~
1.25
total stress, Wrvldf = 8.0
1.50
~
-
-
,
1.00
0
z --
0.75
o .J
0.50
~'-"-'-'~-0
-
ligament stress
[ m [ 2 O v ' ' O s M O I ~5/0$M ' ] O "~ J
0.25 0.00 0.0
1.,i
,
l
0.1
0.2
.
|
,
,,
0.3
I
i
I
0.4
,
,
* 0.6
0.5
2c I W
Fig. 6. Local normal stress ratio computed at fracture stress for SiC/Ti specimens. 2.2
FP/AI, c,f = 0.35
2.0 0
m n,
1.8 1.6
u) m
1.2
m
1.0
o
0.8
1.4
Z "~ o _j
0.6 0.4 0.2 0.0
r
oA~
0
J
|
I0
20
.
i
_
30
,
40
,
9
~n
o~
i
50
,
,
60
,
r
,
70
,
|
80
,
I
90
,
*
100
,
|
110
RVE Width / Fiber Diameter
Fig. 7. Local normal stress ratio computed at fracture stress for FP/AI specimens.
4. C O N C L U S I O N S
The results confirm that the onset of failure in center-notched, unidirectionally reinforced B/Ag and FP/Ae tested at room temperature, and SiC/Ti tested at a high temperature is controlled by a critical stress ratio of the normal stress in the urmotched ligament and the ultimate strength of the composite material. They also confirm that the fracture process is not self-similar; long plastic shear zones form at the notch tip before fracture but no such zones appear along the fracture surface. Formation of the plastic zones blunts the notch tip and modifies the local stresses which can be computed analytically by a modified finite element scheme. This is likely to take place in fibrous composites with a low yield strength, which favor matrix-dominated plasticity such as those reinforced by fibers of large longitudinal shear stiffness such as the boron, silicon-carbide and alumina (FP) fibers.
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ACKNOWLEDGEMENTS The experiments and ABAQUS computations were conducted by YAB while visiting at Rensselaer Polytechnic Institute. Financial support from the Office of Naval Research and the Air Force Office of Scientific Research is acknowledged. REFERENCES 1. Reedy, E.D., "On the Specimen Dependence of Unidirectional Boron/Aluminum Fracture Toughness", Journal of Composite Materials Supplement, Vol. 14, pp. 118-131 (1980). 2. Post D., Czamek R., Joh, D. and Guo, Y., "Deformation of a Metal Matrix Tensile Coupon with a Central Slot: An Experimental Study", Journal of Composites Technology and Research, Vol. 9, pp. 3-9 (1977). 3. Dvorak, G.J., Bahei-EI-Din, Y.A. and Bank, L.C., "Fracture of Fibrous Metal Matrix Composites - I. Experimental Results", Engineering Fracture Mechanics, Vol. 34, pp. 87-104 (1989). 4. Bahei-EI-Din, Y.A. and Nigam, H., "Fracture of Fibrous Metal Matrix CompositesIII. Effect of Imperfection Geometry and Heat Treatment", Engineering Fracture Mechanics, Vol. 37, pp. 1207-1232 (1990). 5. Naim, J.A., "Fracture Mechanics of Unidirectional Composites", Journal of Reinforced Plastics and Composites, Vol. 9, pp. 91-101 (1990). 6. Dharani, L.R., Venkatakrishnaiah, S. and Dupuy, R.A., "Effect of Specimen Width on the Fracture of Unidirectional Metal Matrix Composites", Composites Science and Technology, Vol. 45, pp. 117-123 (1992). 7. Tsai, W.T. and Dharani, L.R., "Non-Self-Similar Fiber Fracture in Unidirectional Composites", Engineering Fracture Mechanics, Vol. 44, pp. 43-49 (1993). 8. Bahei-EI-Din, Y.A., Dvorak, G.J. and Wu, J.F., "Fracture of Fibrous Metal Matrix Composites - II. Modeling and Numerical Analysis", Engineering Fracture Mechanics, Vol. 34, pp. 105-123 (1989). 9. Dvorak, G.J. and Bahei-EI-Din, Y.A., "A Bimodal Plasticity Theory of Fibrous Composite Materials", Acta Mechanica, Vol. 69, pp. 219-241 (1987). 10. ABAQUS Finite Element Program, HKS, Inc., Rhode Island. 11. Bahei-EI-Din, Y.A., "Finite Element Analysis of Viscoplastic Composite Materials and Structures", Mechanics of Composite Materials and Structures, Vol. 3, pp. 1-28 (1996). 12. Dvorak, G.J., Nigam, H. and Bahei-EI-Din, Y.A., "Isothermal Fatigue of Sigma/Timetal 21S Laminates: I. Experimental Results", Mechanics of Composite Materials and Structures, Vol. 4, pp. 113-130 (1997). 13. Moil, T. and Tanaka, K., "Average Stress in Matrix and Average Elastic Energy of Materials with MisfiRing Inclusions", Acta. Metall., Vol. 21, pp. 571-574 (1973).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
281
ON SIMULTANEOUS FAILURE OF CROSS-PLY AND ANGLE-PLY COMPOSITE LAMINATES Khalil, M. *, Bakhiet, E. * and EI-Zoghby, A. **
* Assistant Professor, Aircraft Engineering Department Institute of Aviation Engineering and Technology, Embaba Aerodrome, Giza, Egypt. ** Associate Professor, Mechanical Design and Production Department Faculty of Engineering, Cairo University, Giza 12316-Egypt.
ABSTRACT This paper presents a particular class of bidirectional laminated composites where all layers fail simultaneously. The loading conditions, laminate configurations and the corresponding failure envelopes for simultaneous failure are obtained for cross-ply and angle-ply laminates under in-plane loads. The stiffness and strength characteristics of the laminates are discussed based upon the concept of lamination parameters. The optimal laminate configurations to maximize the in-plane strength are also obtained using the Tsai-Wu criterion as a first ply failure strength criterion. The failure surfaces of the studied laminates in stress space and in strain space are drawn and the optimal configurations are then determined. It is shown that the existance of laminate configurations for simultaneous failure depends mainly on the loading conditions. KEYWORDS Cross-ply Laminates, Angle-ply Laminates, Simultaneous Failure, In-plane loads, Lamination Parameters, Failure Strength. 1. INTRODUCTION Composite materials have become an increasingly attractive alternative to metals for structural applications where high strength-to-weight and stiffness-to-weight ratios are required. These materials are strong, durable and damage tolerant. They meet design and certification requirements and offer significant weight advantages. This work mainly concentrates on the strength and stiffness of symmetric bidirectional composites, namely cross-ply and angle-ply laminates, which are orthotropic and hence easy to handle. When used in the fight way, they may offer weight advantages over quasi-isotropic lay-up [ 1]. The failure strength characteristics of these laminates depend highly on the fiber direction of the individual plies. Thus, it is important to tailor laminate configurations for enhancing the strength of composite structures. For the analysis of the required stiffness and strength of laminates, the use of the lamination parameters is convenient. Some work on the optimum design of laminates using lamination
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parameters have been made, such as the evaluation method of the in-plane stiffness [2], the stability optimization of laminated composite plates [3], the strength optimization of laminates [4] and the optimum design of laminated cylindrical pressure vessels under both stiffness and strength constraints [5]. The present paper first shows the failure surfaces of cross-ply and angle-ply laminates using the Tsai-Wu failure criterion. The optimal laminate configurations which give the highest failure strength of the studied laminates are determined. Based on the lamination parameters [4-6], the loading conditions and the laminate configurations for simultaneous failure are then derived under in-plane loads. The corresponding simultaneous failure envelopes for Carbon / Epoxy composite laminates are drawn and the results are finally analyzed. 2. STRENGTH OF BIDIRECTIONAL LAMINATES 2.1 Failure Criteria The quadratic failure criterion of Tsai-Wu is based on firing an ellipse to the experimental failure strengths of a unidirectional lamina. The form of the equation accounts for interaction between the stresses causing failure. As in most laminate failure criteria, each ply in the laminate must be interrogated separately in order to determine if failure has occured. In this paper, first-ply failure is adopted in contrast to a progrssive failure model [7]. The axis system convention adopted here is shown in Fig. 1. The x, y and s subscripts denote properties in the ply axis system, and 1, 2 and 6 denote properties in the laminate axis system.
Fig. 1. Laminate and ply axis systems The quadratic failure criterion in stress-space takes the form: F~.cri + F#oio j
=
1
( i , j = 1, 2, 6)
(1)
where the strength coefficients F# and F i are given in Ref. [8]. On the other hand, stating the failure criteria in terms of strain is convenient. In strain space the failure envelopes stays fixed even if the ply ratios of the laminate are changed. The strain limits of a ply are independent of the laminate stiffness [7]. The failure criterion in strain-space can be rewritten as:
Current Advances in Mechanical Design and Production, M D P - 7
Gis i § Gijsis j - 1
( i,j = 1, 2, 6 )
,
283
(2)
where the G's are material constants determined by the reduced stiffness Qij [6]. Introducing the strength ratio, R, for a unidirectional composite: R =
cr allowable = 8 allowable
applied
(3)
F,applied
The failure criterion can be solved for R
[co , j]R
(4)
Because of the quadratic form, there will be two solutions R + and R- corresponding to strain vectors piercing the failure envelope in opposite directions. The in-plane strength of a unidirectional laminate will have multiple strength ratios; one set (R + and R- ) for each ply orientation. The ply with the lowest strength ratio fails first. Two factors control the ply failures in a laminate: the in-plane compliance and the specific ply orientation. There is an envelope for each ply orientation; the first ply failure (FPF) envelope is the innermost boundary of the superposed ply failure surfaces. The plies with higher stress ratios will fail later, when the externally applied stress is increased. This successive ply failure progresses until the last ply failure (LPF) occurs [6]. Figure 2 shows the failure surfaces in strain space for T300/5208 graphite/epoxy unidirectional composites [4]. (2' 10- 3 10
-30
10-3
o
Fig. 2. Failure envelopes in the strain space for T300/5208 composites 2.2 Failure Surfaces of Cross-Ply and Angle-Ply Laminates The failure envelopes in normal stress resultant space - or the principal stress plane- for various cross-ply configurations are shown in Fig. 3. Basically quadrants I (tension-tension) and III (compression-compression) are similar, and there is similarity between II (compression-tension) and IV (tension-compression).
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For clarity purpose, the [03 / 902 ] laminate is taken as an example, its envelopes are taken in bold lines, the upper ellipse presenting the 90 ~ plies, the lower one the 0 ~ plies. The shaded area is the FPF envelope [1].
NU/h 3OOO0]
[NLARG(MINTOF [03/902]ENVF..LOP($.OUAORANT I
r Nw/h
1 O0 0
Fig. 3. [0 m /90 n ] Cross-ply failure envelopes in stress space, m/n ratio in 20%-steps
If simultaneous failure of all plies is an optimum condition for a laminate design, it is possible to achieve it only in the quadrants I and Ill where the two failure envelopes intersect. Of all possible stress ratio vectors in both quadrants, those to the intersecting points are the longest, which means an optimum condition for laminate design. There is no coincidence of failure envelopes in quadrants II and IV and consequently no simultaneous failure. But as before the optimum design is characterized by the longest stress ratio vectors. For a given stress ratio, the cross-ply arrangement whose inner failure envelope is the most distant from the origin of the coordinates is the optimum. There is no FPF for angle-ply laminates as all plies must fail simultaneously with the absence of shear. The failure envelopes are narrow, making angle-ply configurations rather susceptible to failure for small changes in stress ratio or ply angle. An optimum ply angle is found in quadrants I and III. However, angle-ply laminates behave poorly in quadrants II and IV as compared to cross-ply laminates [1]. 3. SIMULTANEOUS FAILURE LAMINATES 3.1 Strain Condition for Simultaneous Failure
We consider the laminated composites under uniform in-plane strains (r axis system convention shown in Fig. 1, the strain transformation in double angle function for the layer with the fiber direction 0 can be expressed as [6]:
Current Advances in Mechanical Design and Production, MDP-7
t 6 x t II cos20 % = 1 -cos20 cs 0 -2sin20
2sin20 l i p ] -2sin20/J q 2cos20.][rJ
285
(5)
where p=(61 + e 2 ) / 2 (6)
q = ( 6 ! - ~r2)/2 r = 6 6/2
It is shown that the strain components are independent of the fiber direction 0 when q = 0 (61 = 62 ) and r = 0 (e 6 = 0). Then, the strain and stress components become: ~'! =~162=6"x =~
,
O'x =(Qxx +Qyy)o~x
,
66 = 0
O"r =(Qxy +Q.yy)oex
(7)
o"s = O0 where Qij is the reduced stiffness [6]. It can be seen from Eq. (7) that all layers fail simultaneously for the laminate satisfying the relations of q = r = 0. Such laminate can be considered as possessing the configuration for simultaneous failure. 3.2 Condition of Lamination Parameters for Simultaneous Failure
We consider the laminated composites under in-plane loads (N t ,N2,N6) as shown in Fig. 1. Transforming the original coordinates (1-2) into the principal coordinates (x-y)where the shear component N 6 vanishes, the principal loads become [4]:
NI =(NI +N2)/2 + [(N,-N2) 2/4+N~I '2 Nil =(N, +N2)/2 - [(N,-N2) 2/4+N261 '2
(8)
where the angle, tx, between the principal and the original coordinates is defined by: tan2a = 2N6 / ( N I - N 2)
(9)
Assuming the laminate to be symmetric with respect to the mid-plane, the bending-extension coupling vanishes, and the in-plane stress-strain relations of the laminate in the principal coordinates will be given by:
t = At2 A22 A26 c2
AI6 A26 A66 ~'6
(1o)
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The in-plane stiffness in Eq. (10) can be rewritten as follows:
All AI2 ,'122 ,=h d6~
"1 o
~:1 0
1
-
0
,A2~
~1 0
~X2 -~2 ~:2
-~2
0 O" Ui ! 0 U2 0
0
0
1
(11)
U4
0 ~212
~:4
0 0
0 ~3/2
-~4
0 O
U5
where h is the thickness of the laminate and U i are the stiffness invariants [4, 5]. For the symmetric laminate, the lamination parameters ~:i are given by: 1 h/2
n
j" = -h -hi2c~176
1 h/2
n
: j'100 ~tm, while a minimum of 1.2 ~tm was revealed for the rolled alloy. A significant improvement in the mechanical properties at room temperatures were accomplished by ECAE processing in comparison with conventional processing. KEYWORDS Weldalite Al-alloys, ECAE Processing, Conventional Processing T4, T6 Tempering, Room Temperature Mechanical Properties, Microstructure Evolution I. INTRODUCTION Aluminum alloys went through series of developments for aircraft and aerospace applications since 1970s. The demand for ultrahigh strength without prior cold working (stretching), very strong and rapid natural aging capability, good fracture toughness at room and cryogenic temperatures, good weldability, and stress corrosion cracking resistance led to the development of AI-Cu-Li base alloys by late 1980s. These alloys are called Weldalite alloys which were first designed by Martin Meriata, and Reynolds Metal Company (RMC). Weldalite alloys were designed for applications such as cryogenic tankage on space launch systems where fuels are contained at -196~ (77K). Weldalite alloys are characterized by their high Cu:Li ratio with Cu content > 2.5 wt%, and Li-content < 1.6 wt%, in addition to other alloying elements such as magnesium, silver, and zirconium [1-3]. Weldalite alloys contain controlled amounts of different alloying elements in order to achieve the required ultrahigh strength and good fracture toughness for cryogenic aerospace
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applications through the precipitation of different second phase particles. Examples of which are ~5/(AI3Li), [3] 0/(AI2Cu), TI(AI2CuLi), T2(AI6CuLi), and TB(AIzsCusLi2) phases, S/(AI2CuMg), and [3/(Al3Zr) which all precipitate at different morphologies and along different planes and directions [3-5]. Commercially extruded plates and rolled sheets of AlCu-Li base alloys are typically subjected to a cold stretch operation prior to natural (T3 temper) and artificial (T8 temper) aging to introduce dislocations that act as preferable sites for the nucleation of the strengthening 0/, TI, and SI phases. Since ingot Al-base alloys are mostly formed superplastically at relatively high forming temperatures (~0.8Tm) and relatively low forming strain rates (10"3 to 10.4 s"~) lots of efforts were done aiming towards the increase and/or decrease of the SPF strain rates and temperature, respectively and hence achieving cost effectiveness. According to Langdon et al., 16] grain refinement to the sub-micron level offers potential advantages in increasing the overall ductility while increasing strain rate and/or decreasing the SPF temperature. Subjecting cast metals to severe plastic deformation through homogeneous simple shear can produce SMG structure. Equal channel angular extrusion (ECAE) was found to be effective in producing SMG of 0.2-0.3 ttm [71. Accordingly, characterization of the room temperature tensile properties of the ECAE processed Weldalite alloys versus sheets processed via rolling conducted in comparison with previously reported room temperature properties of similar alloys in the T4 and T6 temper conditions investigated by the different researchers. 2. MATERIALS AND EXPERIMENTAL PROCEDURE Weldalite 2095 and 2195 alloy ingots were direct chill (DC) cast at RMC, Richmond VA. The 2095 and 2195 alloys were received in the as-cast condition in the form of plates, and a third alloy was also investigated which was 2095 Weldalite received in the form of superplastic dynamically recrystallized sheets. These sheets were conventionally processed by rolling by RMC (thermomechanical treatment details are classified), and was selected to compare ECAE with conventional processing techniques. Measured compositions o f th e investigated alloys were determined using wet chemistry analysis as given in Table 1. The as-cast alloys were homogenized, solution heat treated (SHT) at 510~ (783K) for 70 min followed by water quenching (WQ) then peak aged at 160~ (433K) for 20 h. Billets with initial dimensions of 2.5x2.5x12.7 cm 2 from the peak aged plates were plastically deformed using ECAE. In ECAE processing, uniform deformation is imposed throughout massive billets without a change in the cross-sectional area. Billets are subjected to shear deformation by extruding them through 2-channels of equal cross-section area intersecting at an angle of 7t/2, as shown in Fig. 1 [7,81. Table 1. Chemical Composition of As-Cast Weldalite Plates and SP Rolled sheets ALLOY
Thickness (in)
209"5as-east plate
0.89
LI 0.92
ca
Mg
Zr
4.18
0.39
0113
0.32
0.02
0.07
"0.44
0.026
01079
219"5as-cast plate
0.77
0.89
3.89
0.28
0.11
2095 SP sheet
0.063
1.08
4.52
0.31
0.15
Ag
0.34 .
.
.
.
Ti
Zn
0 . 0 2 8 0.065
.
Some minor additions: Mn, Fe, and Si less than 0.06 wt% each
The extrusion operation can be repeated several times, since there is no dimensional changes encountered by the material after each pass. Moreover, different SMG structures and textures can be created by changing the plane and direction of shear at each pass. Routes A is when the
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359
orientation of the billet is unchanged at each pass, and route C is when the billet is rotated 180~ about its extrusion axis after each pass. The ECAE processed billets via routes A and C were extruded 4 passes at 350~ (623K) followed by 4 passes at 200~ (473K) with a total effective strain of 9.36 at 0.5 crn/s cross head speed. Post forming heat treatments were performed to investigate the room temperature mechanical properties of the ECAE versus conventional processing of the Weldalite alloys in the T4 and T6 temper conditions. Post forming heat treatment was conducted by SHT of the as-extruded and as-rolled alloys at 510~ (783K) for lh followed by WQ then natural aging at room temperature for >1000 h, (T4 temper). Peak strengthening (T6 temper) was achieved by monitoring the artificial aging response of the solutionized billets and sheets at 180~ (453K) for different time intervals up to 30 h. For the characterization of ECAE processing for grain refinement in comparison with conventional methods, different analysis such as hardness measurements, room temperature tensile properties in the T4 and T6 temper conditions and microstructural evolution were conducted. Rockwell Hardness (RB) was performed under loads of 100 Kg during 15 sec impressions with an indenting steel ball with 1/16 in diameter. For the extruded samples, hardness measurements were done on a plane sectioned parallel to the flow plane in the middle of the billet. Yield strength (or), ultimate tensile strength (OUTS),and % elongationsto-fracture (%EF) were determined for the ECAE and SP sheets respectively, in the T4 and T6 temper conditions. An MTS tensile testing machine with 20 kips loading capacity was employed. Specimens sliced from the deformed billets parallel to the extrusion flow plane, and from the rolled SP sheets parallel to the rolling direction were machined for the tensile testing. The tensile properties of each condition were based on average results obtained from a minimum of 2-specimens. Microstructural analysis was conducted through the employment of OM and TEM to observe the effect of ECAE processing on the size shape of the produced microstructure under bright field. In addition, particle morphology within the grains and at grain boundaries was also analyzed. TEM analysis was conducted using a JEOI JEM-2010 TEM at high voltages of 200 kv. In order to determine the sub-cells, subgrain, and grain sizes for the as-cast and processed alloys linear intercept method was employed using OM and TEM images, respectively. A double-jet electropolishing technique was employed for the final thinning of the disc. The electrolyte used for the final thinning of the samples was a 25% nitric acid and 75% methanol mixture. Final thinning was performed at an operating voltage of 12 volts, which corresponds to a value of current between 0.275 and 0.3 A and at an operating temperature of-15~ (158K)[9]. An Energy Dispersive Spectroscopy (EDS) attached to the JEOl JEM-2010 TEM was utilized for the composition analysis of the different second phase particles for the ascast, as-extruded, and as-rolled structures. In addition, diffraction patterns were obtained for the different samples to derive useful information about the different second phase particles precipitating within the structure in the SHT, T4, and T6 temper conditions, due to the fineness of the phases. For the identification of the different second phase particles the EDSIM program was utilized through matching the spot patterns generated for the different particles by the TEM with those simulated by the program. 3. EXPERIMENTAL RESULTS 3.1 Hardness Measurements The ECAE processed alloys exhibited an extremely rapid natural aging response when aged at room temperature over time intervals up to 1000 h. Figure 2, shows RB-variation with aging time at room temperature for the ECAE processed billets processed via 4C, 350~ (623K) + 4C, 200~ (473K) versus the rolled alloy. Most of the hardness increase for the ECAE
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processed alloys occurred over the first 24 h of aging time. On the other hand, rolled sheets displayed more gradual increase in hardness-values up to > 1000 h of aging time. Figure 3, shows the variation of RB-values with time for 2195 alloy ECAE processed via routes A and C where route C displayed higher hardness values compared to route A at all aging intervals. Similar behavior was exhibited during artificial aging (T6 temper) at 180~ over time intervals from 25 min up to 30 h. RB variations with aging time at 180~ for the ECAE processed alloys versus the rolled one are shown in Fig. 4. It was revealed that after 15 to 30 min of artificial aging, a slight decrease in hardness occurred. A rapid increase in hardness of the ECAE processed alloys took place up to 4 h of artificial aging followed by gradual and slight increase up to 1Oh of aging time. Artificial aging produced maximum hardness-values at the top of the RB-scale of 95.6 and 92.5 for 2095 and 2195, respectively which occurred at technological practical times of 12 h. Conversely, the SP rolled 2095 sheets exhibited gradual increase in RB-values up to 24 h with a highest displayed value of 93.8 occurring at 28 h. Similar to natural aging behavior, route C exhibited higher artificial aging response than route A.
3.2 Room Temperature Tensile Properties Table 2 lists the ours, and %EF experimental values for 2095 and 2195 ECAE processed alloys via routes A and C versus the 2095 rolled sheets in the T4 and T6 temper conditions. In addition, other results of previously published data by Pickens et a/.,[31 on AI-(4.5-6.3)Cu1.3Li-0.4Ag-0.4Mg-0.14Zr. Weldalite alloys conventionally processed via rolling or extrusion are listed in the table for the sake of comparison [3]. Note that, the Weldalite alloys were naturally aged (T4) for 1000 to 1200 h at room temperature, and artificially aged (T6 & T8 with prior stretching) at 180~ for 12 h. Superior tensile properties were displayed by the ECAE processed alloys regardless of Cu and Li-contents. Moreover, route A processing exhibited a noticeably higher %EF than route C with a slight decrease in UTS. Table 2. Room Temperature Tensile Properties of Weldalite alloys Alloy composition& processing Cony.i~xtrusion 049(6..3 Cu-.2Li)'" Cony. Extrusion2094 (4.6 Cu-l.2Li)'" Rolled2094 Shee.ts (4.89 Cu-1.3Li)'" Rolled SP sh_eets2095 (4.3Cu-l.lLi)" ECAE.2195(3.89 Cu-0.89L!),RouteA" ECAE 2195 (3.89 Cufl.89L!), RouteC" ECAE 2095(4.18 Cu-l.0Li), Route A~' ECAE 2095 (4.18 Cu-l.0Li), RouteC"
"Investigated alloys,
T6 UTS(MPa) %EF 690 4.3 614 10.5 665 5.2 18 715 6.9 15.6 707 12.2 25.3 715 !1 24.0 741 10.5 764 9.1 1 2'1.2
T4 UTs(MPa) %EF 593 16 ,
544 532 531 543
. 591
T8 UTS(Mpa) %EF
,,
712 ,
712
5.1 ,,
6-9
,
""Previously published [3]
,.
UTS units in MPa
3.3 Microstruetural Analysis In order to derive valuable explanation for the superior mechanical properties exhibited by the ECAE processed alloys in comparison with those conventionally processed, it was necessary to study the microstructural evolution and particle morphology of the deformed and post heat treated alloys. This section will be included in the discussion
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4. DISCUSSION 4.1 As-Extruded and As-Rolled Mierostructures
Optical microscopy investigation revealed that the as-east average grain sizes of 2095 and 2195 alloys were 123 and 78 ~tm,respectively. TEM investigation of the ECAE processed Billets deformed at an effective total strain of 9.23 (4 passes, 350~ (623K)+ 4C, 200~ (473K)) revealed the formation of SMG structure. Figure. 5 shows TEM photomicrographs of the as-extruded 2095 billets ECAE processed via routes A and C, and the as-rolled SP sheets. Examination of the ECAE processed alloys via route A revealed the formation of fragmented structure made out of elongated sub-cells about l x0.3 ~tm2 (marked A&B), and equiaxed subgrain structure about 0.25 ~tm (marked C) as shown in Fig. 5a & b. Formation of well defined subgrains and high dislocation activity within others, is an indication of energetic structure (marked B, Fig. 5b) was evident. Examination of the billets ECAE processed via route C revealed the formation of similar structure to route A except for the disappearance of the elongated sub-cells which were replaced by a fine equiaxed subgrains and dislocation cells about 0.18 ttm in average size, (Fig. 5c). A very limited fraction of a dynamically recrystallized subgrain structure about 0.2 ~tm in average size with extinction contours at their boundaries was also observed (marked by arrows, Fig. 5d). The 2095 as-rolled SP sheets examination revealed the presence of equiaxed grain structure with well defined, sharp, and relaxed boundaries about 1.5-2 ~tm in average size and with little dislocation activity within the grains (Fig. 5e). Coarse particles with different compositions were identified in the ECAE and rolled alloys using EDS, such as 8(AILi), 0/(AI2Cu), TI(AI2CuLi), Tz(AI6CuLi), Ta(AllsCusLi2), S/(AI2CuMg), and 13/(AI3Zr)phases. These phases were present within grains and mostly at grain boundaries and triple points with different sizes, volume fraction and distribution. 4.2 Solution Heat Treated Microstructures
TEM analysis of 2095 and 2195 billets processed via ECAE similar condition revealed the formation of a roughly equiaxed-SMG structure with an average size of 0.98 and 1.1 lain respectively. Figure 6 shows a bright field TEM photomicrograph of 2095 billets processed via route C + SHT + WQ. Some of the grains revealed relaxed well defined boundaries, while others showed extinction contours at some of their boundaries which was indicative of an energetic structure. In addition, the TEM photomicrographs revealed the presence of the 13/(AI3Zr) dark spherical precipitate within the grains and at grain boundaries with an evidence of grain boundary pinning (marked by small arrows, Fig. 6a) [10, 11]. Higher Zr-content in the 2095 alloy compared with that of 2195 ECAE processed alloys (Table 1) resulted in the precipitation of higher volume fraction of 13/-phase and hence finer grain structure. Examination of a SAD pattern corresponding to the [112]matrixzone axis, revealed no evidence of the presence of any phases in the matrix with an exception of 8/-phase superlattice reflections and that of 13/-phase, marked by arrows in Fig. 6b. Both phases have LI2 structure with cube-on-cube orientation relationship with the matrix with (111)S/orl3///(111)A~ [3, 12]. The absence of the coarse second phase particles formed in the structure during the extrusion process, indicates their dissolution during the SHT process which was intended. Examination of 2095 SP sheets processed via rolling in the SHT condition, revealed the development of a well defined and relaxed grain structure with an average size of 4 ~tm. Fig. 7a, shows a bright field TEM image of 2095 SP rolled sheets SHT. Similar precipitations to that of the ECAE processed alloys in the SAD pattern corresponding to the [110]m~ix zone axis was evident, (Fig. 7b).
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4.3 Natural Aging The rapid natural aging response of ECAE processed alloys in the T4 temper condition (Fig. 2) indicates that most of the second phase particles which are responsible for strengthening has precipitated within the first few hours of aging at room temperature which agrees with the observations made by Pickens et al.,[101. It is suggested that, the equiaxed grain structure developed by route C processing could be providing the structure with higher resistance to deformation than route A which develops elongated grain structure in the direction of flow (parallel to the to the tensile axis of deformation) as shown in Fig 5a. This explains the lower natural aging hardness and strength values displayed by route A versus C, and the significant increase in ductility (Table 2). Therefore, route A processing provides the material with excellent combination of strength and fracture toughness which are the major mechanical properties required for cryogenic applications. Work done by Pickens et al.,[3] on Weldalite 049 (6.3%Cu) extruded plates displayed almost equivalent strength in the T4 temper and significantly lower ductility than that obtained for 2095 (4.18% Cu) billets ECAE processed via route C (Table 2). Considering the difference in Cu-content between the conventionally extruded Weldalite 049 and the ECAE processed alloys, a significant improvement in the room temperature tensile properties was accomplished by ECAE processing. The observed faster natural aging response of the 2095 ECAE processed billets (Fig. 2). could be explained by the higher Li, and Cu-contents in the 2095 compared to 2195 as-cast plates (Table 1). Figure 8 shows low and high magnification TEM photomicrographs of 2195 ECAE processed billets in the T4 temper condition. 8/ and 13/ precipitates were evident by the presence of the superlattice reflection spots in the SAD pattern corresponding to [ 100]m~x zone axis (marked by large arrows, Fig. 8b). In addition, 0/ (AI2Cu) phase, which presence was evident in the structure of the ECAE and rolled alloys in the T4 temper condition, was manifested by the vertical and horizontal streaks for the two variants lying parallel to the beam direction (marked by small arrows, Fig. 8b). Similar observations were made by Kumar et al.,[5] who reported the precipitation of 8/ (AI3Li), and 0/ (AI2Cu) phases in the T3 and T4 temper AI-Cu-Li-Ag-Mg-Zr alloy. Although, the 2095 alloy processed via ECAE had lower Li and Cu-contents (0.92Li-4.18Cu) than that of 2095 SP sheets processed via rolling (l.08Li-4.52Cu), the ECAE processed billets exhibited higher and faster natural aging response than the rolled alloy (Fig. 2). This could only be related to the microstructure developed in each processing condition. Since the 8/(AI3Li) precipitation is oRen associated with dislocations,J131 it could be assumed that higher dislocation density and activity retained in the structure of the ECAE processed billets even after SHT, compared to that of the rolled sheets resulted in more rapid aging response and hence higher exhibited mechanical properties. This was evident by comparing the SHT structures of the rolled 2095 alloy (Fig. 7) with that of 2195 ECAE processed one (Fig. 8a) in the T4 temper condition. 4.4 Artificial Aging Artificially aged samples processed via ECAE displayed hardness values at the top of the RBscale, (Fig. 4). The observed drop in hardness within the first 30 rain is attributed to the reversion taking place by the dissolution of the 8/ (AI3Li) phase which precipitated during the quenching process and during sample preparation. Similar observation was reported by Kumar et al.,[5] and Gayle et al., [14]. The T6 temper billets and sheets were characterized by a higher YS, UTS, and lower ductility than the T4 terrier condition (Table 2), which was consistent with the investigations made by Kumar et al. This is due to the different second phase particles that precipitate in the structure of the T6 and which contribute into the high
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strengthening effect [3, 141. Work done by Pickens, et al.,131 on Weldalite 049 2094 (4.65%Cu) and 049 (6.3%Cu) conventionally extruded plates revealed lower strength/ductility combination in the T6 and T8 temper than that obtained for 2095 (4.18% Cu) billets processed via route C in the T6 temper condition (Table 2). This is indicative of the outstanding room temperature mechanical properties obtained by ECAE processing compared to that of the conventional methods (rolling and extrusion), regardless of the difference in Cu and Li-contents. Billets processed via routes A and C revealed extreme grain size stability against grain growth after artificial aging. A maximum average grain size of 1.2 and 1.07 ~tm were obtained for 2195 and 2095 ECAE processed billets via route C in the T6 temper conditions, respectively. Figure 9 shows TEM photomicrographs of 2195 billets ECAE processed via route C in the T6 temper condition. On the other hand, an average grain size of 4.2 ~tm was obtained for the rolled 2095 SP sheets (Fig. 10). Examination of the TEM photomicrographs for the ECAE processed billets in the T6 temper condition revealed the development of a relaxed grain structure about 1 ~tm in average size (Fig 9a). A well defined wide boundaries with preferred precipitation of Ti-phase (marked by arrows), while other sharp boundaries were precipitation free were observed (Fig. 9b). Note that, no PFZs were observed close to grain boundaries. Investigation of the second phase particles responsible for strengthening of the T6 temper alloys revealed the presence of similar phases to those reported by Pickens et al.,[3, 141, Huang et al.,[15], and Langdon et al.,[16l. Figure 9b shows a SAD pattern corresponding to the [112]matrixzone axis which revealed the following (1) Presence of one variant of Tl-phase normal to zone axis and a second 19.5~ from the zone axis which were represented by the streaks in between the spots (marked by arrows) and elongated spots (marked a), respectively, and the other 2-variants of Tl-phase make 61.9 ~ with the zone axis which made them steeply inclined and which were also represented by the elongated spots, (2) Presence of faint spots from the superlattice reflections of 5/-phase (marked b), (3) Presence of faint streaks on the t420] mauixdirections for [112]matrixzone axis (marked c), was an evidence of the presence of S-phase. Examination of other SAD patterns also revealed the presence of Ti, 8/, I3/, S/ phases, and the absence 0/phase, which dissolution in favor of Ti-phase is suggested 13-51. Similar second phase particle morphology was observed for 2095 rolled alloy except for the absence of evidence for S/phase, which in turn resulted in the formation of PFZs as marked by the arrows in Fig. 10a and b. Previous studies revealed that an optimum mechanical properties can be achieved only when the material is stretched prior to aging. According to Sanders, S/ [3] and TI-phases precipitate both at high and low angle grain boundaries while being the dominant phase. Consequently, the absence of S/ in the structure of the T6 temper billets or sheets whether processed via ECAE or rolling should be expected since no stretching was involved prior to aging. Evidence for the precipitation of S/ in the structure of the T6 billets processed via ECAE, is consistent with the behavior of a T8 temper condition; although stretching was not involved. This can only be explained by the high dislocation activity retained in the structure even after one hour SHT at 510~ and 12 h exposure to 180~ during artificial aging, knowing that S/ precipitation is usually associated with dislocations 13, 14]. Accordingly, the combined precipitation of the S/ and the Ti-phases in addition to the fine grain structure resulted in an ultra-high strength and ductility of the processed alloys at room temperature.
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Current Advances in Mechanical Design and Production, MDP-7
CONCLUSIONS From the current research analysis, ECAE processing has proven its capability of developing a SMG structure 100 lam. Route C processing produces equiaxed SMG structure slightly finer than that produced by route A. ECAE also Proved its capability of producing extremely energetic structure after post forming HT which resulted in a rapid aging response through the precipitation of 8/, 131,and 01 phases in the T4 temper condition and 8I, 131, and Tin,and S/ phases in the T6 temper condition. In addition, the elimination of PFZs near grain boundaries in the T6 temper condition by S/phase precipitation without prior stretching was evident. Finally, ECAE processing has significantly enhanced the ambient temperature mechanical properties of Weldalite alloys in comparison with conventional processing techniques and hence is most suitable for cryogenic structural parts in aerospace, where ultra-high strength and fracture toughness are demanded. REFERENCES 1. Moore, W. M., Zelin and Chaudhury K., "Superplasticity and Superplastic Forming", eds., Ghosh, A. and Bieler, R., The Minerals, Metal and Materials Society, Denmark, pp. 267275, (1995). 2. Cho, A., Greene, M., Fielding, P. and Skillingberg, H., "Report Reynolds Metal Co.", Richmond, VA, pp. 3-8, (1995). 3. Sanders, T. H., "In Aluminum-Lithium Alloys II", eds., Sanders, T. and Starke, E. A., The Metallurgical Society of AIME, Monterey, CA, pp. 1- 15, (1984). 4. Lavernia, E., Srivalsav, T., and Mohammed, F., J. of Mat. Sci., Vol. 25, pp. 1137-1158, (1990). 5. Kumar, K. S., Brown, S. A., and Pickens. J. R., Acta Metar., Vol. 44, No. 5, pp. 1899, (1996). 6. Langdon, T. J., and Pickens, J. R., In Aluminum-Lithium Alloys V, VA, March 27-31, pp.691-700., (1989). 7. Segal, V., "In 1st International Conference on-Proc. Mat. For Prop." eds., Henein, H. and Oki, T., TMS, Honolulu, pp. 611-613, (1993). 8. Ferrasse, S., Segal, V., Goforth, R., and Hartwig, T., "Development of SubmicrometerGraind (SMG) Microstructure in Aluminum 6061 Using Equal Channel Angular Extrusion", (in press). 9. Goodhew, P. J., Mater. Sci., pp. 153, (1973). 10. Zaidi, M., and Wert, J., "Treatise on Materials Science and Technology", eds., G. Mahon and R. Ricks, PA, pp. 139 (Personal collection, Salem, H.)., (1989) 11. Humphreys, F., and Hatherly, M., "Recrystallization and Related Annealing Phenomena", British Library Cataloguing Publication Data, Pergamon, pp. 167, (1995). 12. Skorlzki, B., Shiflet, G., and Starke, E. A., J. Metallurgical and Materials Transactions A, Vol. 27A, pp. 3431-3444, (1996). 13. Ahmed, M., and Ericsson, T., Scripta Metall., Vol. 19, pp. 457264, (1985). 14. Gayle, F., Heubaum, F., and Pickens, J., In Aluminum-Lithium V, eds., T. H. Sanders and E. A. Starke. Jr., Materials and Component Engineering Publication Ltd., Birmingham UK, pp. 701-711, (1989). 15. Ashton, R. F., Thompson, D. S., Starke, E. A., Jr., and Lin, F. S., Aluminum-Lithium III, eds., C. Baker, P.J. Gregson, S. J. Harris, and C. J. Peel, Institute of Metals, London, pp. 57-65, (1986). 16. Kumar, S., Mcshane, H. B., and Shepherds, T., Materials Science and Technology, vol. 10, pp. 162-170, (1994).
Current Advances in Mechanical Design and Production, MDP. 7
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366
Current Advances in Mechanical Design and Production, MDP-7
Fl&
5 TEM Photomicmvphs for ECAE and Rolled alloys, (a) and @) Low and high magnifications for 2095 ECAE pmctsSett alloys via Mute A with elongatcd (marked A a B) and cquiaved (marked C) structural regions, s h m n g
dislocation aclivlty (marked B). And a SAD pattern carresponding to zone aurs. (C) L (d) Via mute C, wth extinction contours along gram boundaries And a SAD pattern corresponding to [ 112]~lzone axis. and (e) 2095 shccts pmcssed via rolling where some distmtion actiww show around p’ dark spherical precipitates. h d a S A D paitern aonespondinR to [ 1 ionc axis.
Current Advances in Mechanical Design and Production, MDP-7
Flpl. 6 TEM Photomicrographs for 2095 ECAE billets processed via m i t e C M e r SHT at 5 10°C for l h followed hy WQ. (a) SMF situcturc < t pm. Large B ~ O I Y Sshow cxtinaion contours along some o l the grain boundanes mdicahve of energetjc structure. and small amrvs show the dark p-phase pfnning grain horindanes. &I) shows a SAD pattern corrcsponding to II12]hl 7onc axis with riipcrlnnicc rcflections for fil andlor P’-phnse
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
381
CONVENTIONAL VERSUS THIN SLAB CASTING: A NUMERICAL SIMULATION APPROACH FOR THE COMPARISON OF MICROSTRUCTURAL PROPERTIES
Youssef, Y.M.', Megahed, G.M.'" and Lee, P.D."" Alexandria National Iron and Steel Co. (ANSDK), Alexandria Tabbin Institute for Metallurgical Studies, Cairo. Dept. of Materials, Imperial College of Science, Technology and Medicine, UK **
ABSTRACT Due to improved efficiency and reduced environmental impact, steel producers are exploring the feasibility of thin slab casting to replace conventional continuous casting. A numerical simulation of the solidification process, including thermal and microstruetural predictions, was developed and applied to compare the quality of the microstructures in these competitive processes. Through appropriate assumptions, the model was optimised to run quickly on a PC, allowing fast and inexpensive comparisons to be made. The predicted temperature time history was used to calculate the microstructural features of both secondary Dendrite Arm Spacing and carbon microsegregation across dendrite arms. The model was validated by comparing the model results with published thermal data for conventional slab casting, together with calculated and measured data of secondary Dendrite Arm Spacing for conventional and thin slabs. Experiments were also performed to physically simulate the thin slab casting process, providing data for the heat transfer coefficients and a means for further validation of both the thermal and microstructural predictions. The validated model was used to compare the conventional and thin slab casting processes for the production of two steel grades, AISI 1008 low carbon and AISI 1026 medium carbon steels. It was found that a more refined structure was obtained rising the thin slab process, leading to improved quality in the final product. KEYWORDS Continuous Casting, Thin Slab Casting, Modelling, Microstructural Simulation, Steel. I. INTRODUCTION Steel companies are under continuous pressure to reduce operating costs whilst improving product quality. To achieve this goal a number of new manufacturing methods have been developed that provide an excellent product whilst benefiting from the economic savings of a reduced number of production steps compared to conventional processes. Thin slab and strip casting are two examples of technologies that both reduce the number of processing steps whilst combining others into a continuous line operation. Therefore, both methods produce cast material nearer to the dimensions of the final product with reduced material loss, energy consumption and manpower. Such technologies have become known as near net shape
382 casting
Current Advances in Mechanical Design and Production, MDP-7
[1,2].
Near net shape casting of steel is currently a subject of world-wide activity
because of its expected advantages concerning investment costs, energy consumption and saving of numerous costly process steps in the production of flat products [3]. From the viewpoint of material properties, the refinement of the microstructure in the as-cast state, namely: secondary Dendrite Arm Spacing (DAS) has one of the most beneficial effects [4]. This effect cannot be ascribed to the reduction in casting thickness only, which decreases in the sequence thin slab casting, strip casting and thin strip casting, but also depends on other process parameters such as metal-mould heat transfer and casting temperature. To determine the significance of these parameters upon the as cast structure, simulations of heat transfer and solidification structure can be applied. The mathematical simulation of conventional continuous casting is now well established using either finite difference/volume [5-7] or finite element [8] methods. Recently, several authors have also simulated near net shape processes [9-11], however, one of the most significant parameters, the heat transfer coefficient at the mould-metal interface, has not been as well characterised. Furthermore, the coupling to the prediction of microstructural properties has not been widely applied. The present work is aimed at producing a combined model that can provide a simulation tool that can predict the microstructural features for a range of continuous casting processes for quick comparison, even on a personal computer. The assumptions required to produce an effective fast tool for process comparison together with the model implementation are presented first. An instrumented mould developed to both determine the boundary conditions and validate the model is then outlined. Finally, the application of the programme for the comparison of thin slab processes (cast thickness ranging from 20 to 80 mm) to conventional slabs (normally 200 to 270 mm thickness) is described. Although two microstructural features were compared, secondary Dendrite Arm Spacing and microsegregation, only the DAS values will be discussed in this paper. 2. EXPERIMENTAL METHODS A static mould with an adjustable thickness and the ability to apply varying pressures was designed and constructed in order to measure the heat transfer coefficients over a range of conditions and to produce samples for model validation. The experimental apparatus consisted of a static water cooled copper mould that simulates the mould cooling zone for different thin slab thicknesses by varying the distance between the two mould sides. The slab to mould heat transfer coefficient was determined using thermocouples embedded in the copper mould. Castings of different steel grades were made, and optical emission spectrometry was used to verify the composition whilst the microstructure was examined using optical microscopy. Further details of the experimental apparatus and methods are given by Youssef [12]. 2. MATHEMATICAL MODELLING 3.1 Solidification Model Theory The high heat capacity and low thermal conductivity of steel combined with the high casting speeds in thin slab casting (5 to 6 m/min), make it a reasonable assumption that the heat transferred by conduction in the axial (z) direction is negligible compared to the heat
Current Advances tn Mechanical Design and Production, MDP-7
383
transferred by the bulk motion of the strand. Furthermore, in thin slab casting the thickness of the slab is very small compared to the width. Therefore, by using a moving frame of reference where a slice is followed along the caster, only the heat transfer through the thickness need be simulated. Using these assumptions, the transient heat transfer and solidification in continuous casting of steel slab can be simulated by the Fourier equation in one dimension: 8 k(T) 8T
~x
OH
-~-- = p ( r ) - ~ -
(I)
where T is the temperature, k is the thermal conductivity, is the density, t is the time, x is the through thickness dimension and H is the enthalpy. In order to include the latent heat of solidification, the enthalpy-temperature relationship takes the following form: H = Cpr + L , r >_ 7'1
(2)
H = CpT + L ( I - f~), T~ < T < T~
(3)
H = C p r , r t~
Fig. 1. Micrographs of explosively compacted coarse copper powder at 5.6 GPa for three different positions
418
Current Advances in Mechanical Design and Production, MDP-7
Figure 1 shows three micrographs of an explosively compacted copper powder at 5.6 GPa for three positions along the compact height. These microstructures reveal the differences in their morphology along the axis of the specimen. Fig. I a shows rearrangement of the particles with little plastic deformation. Heat generated in the particle surfaces and the compressive effect of the shock wave result in neck formation and interparticle bonding. The neck formation is very clear in Fig. lb. Fig. l c clarifies the melting zones around the particle surfaces lead to welding. The melting mechanism could be explained as follows. Small areas of contact between particles causes thermal build-up, thus raising the temperature until local melting begins in region of intimate contact. The liquid metals expand outward and fill spaces between the solid particles. A necked region of molten material form with a concave surface due to the surface tension between the solid particles. This is clearly seen in Fig. 1 b and c. These results are in consistent with one of the mechanisms proposed by Morris [3]. Figure 2 supports this phenomenon by using polypropylene granules. It can be seen from the figure that jetting between two granules has been achieved. This jetting means that, melting at the granules surfaces and inter-granular bonding takes place.
Fig.2. Optical micrograph for polypropylene compact Figures 3&4 show the variations of density and micro- Vicker's hardness with the compact length at shock wave pressures of 5.6 GPa and 3.2GPa for coarse and fine copper powders respectively. These properties have been measured for sound compacts on their outer surfaces. The expected increase in the compact hardness and density with increasing shock wave pressure was found. It is clear from figures 3a & 4a that the as-consolidated density ,green density, is affected by the height of the compact. It can be seen that the relative density, i.e. the ratio of the green density of the compact to the density of solid material, increases from top to bottom along the compact axis. These results may be due to the build-up of the shock wave pressure during the wave propagation after detonation [2,4]. It can be noticed also that, at lower pressure, 3.2 GPa, the maximum relative density achieved equal to 94%, while
Current Advances in Mechanical Design and Production, MDP- 7
419
for higher pressure, 5.6 GPa, has been 98.7%. Under the higher initial shock wave pressure, 5.6 GPa, a decrease of the relative density has been observed at the lower end portion, (see Fig. 1). This may be due to the debonding effect of the strong reflected tensile shock wave at high pressures. 2 2 0 ~,
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Fig. 4. Variations of density and hardness with the compact positions for fine copper powder The micro-Vicker's hardness measurements shown in the figs. 3b& 4b show that the hardness of the compacts increases with compaction pressure and proceeding wave distance (through positions 1-5) from an initial value of about 170 VHN to about 210 VHN. This set of data, however, shows leveling off at lower shock wave pressure ( 3.2 GPa) but under higher pressure (5.6 GPa) the hardness values tend to decrease as the compact approaches its limiting value of density (which is less than the theoretical density). This can be correlated with the pressure build-up and increasing the particles interior temperature. Since the particle surface temperature and consequently the mean temperature of the compact depend upon the induced pressure and particle deformation. Mammals et al [9] reported that, a considerable softening of material takes place under shock compaction at very high pressure used, these results are in good agreement with those which have been reported under different shock wave pressures. Therefore, to get nearly constant density and hardness along the whole compact length it
420
Current Advances in Mechanical Design and Production, MDP- 7
should take into account the effect of pressure build-up during the wave propagation. For this, further investigations should be carried out to overcome this problem. Figure 5 shows an optical micrograph and hardness variations for the portion no. 4 (see Fig. 1) which has higher mean values of density and micro-hardness (see Figs. 3 and 4). Microhardness measurements performed on the central region of the particles, revealed a microhardness of 208 + 5 HV, where that performed on the interpanicle boundaries revealed a microhardness of 223+ 3 HV. This means that the microhardness of the interparticle boundaries is always higher than that of the particles surfaces. The increase of the microhardness at the interparticle boundaries is probably due to the melting of these boundaries which is rapidly solidified. This may be another fact supporting the occurrence of interparticle bonding.
Fig. 5. Optical micrograph showing the microstructure of compact in portion no. 4 of the copper powder compact and microhardness on the particle surfaces (Ps) and on interparticle boundaries(l b) On the other hand, it can be noticed from the figures that the compacts obtained from coarse copper show higher values of relative density (98.7 %) than that achieved for fine powder (96%) under same compaction pressure. This may be due to the ability of coarse powder to deform plastically and good bonding between particles obtained consequently. CONCLUSIONS The purpose of the present work was to detect the effect of shock wave proceeding distance on the properties of the obtained compact. Strain localization at the particle surfaces leads to adiabatic heating and melting zones, jetting and interparticle bonding have been identified. Micro-Vickers hardness, density and morphology of the compacts produced indicate that the great effect of the proceeding wave distance. This correlated to the pressure build-up through the propagating shock wave distance. The results also show that the particle size has significant influence in the compact density. The product homogeneity along its length is needed to further investigation to overcome the pressure build up effect.
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REFERENCES 1. Torsten, R., Prummer, R., and Roll, W. "Explosive Compaction of Nanosized TiNPowders", Material Sci. Forum Vols. 235-238 pp. 285-290 (1997). 2. Hegazy, A. Abousree, "Some Aspects Of Implosive Powder Consolidation", 5th Int. Conf. On Prod. Eng. and Design For Development FEDD5, April 28-30, pp. 409-417(1998). 3. .Morris, D.G., "Bonding Process During Dynamic Compaction of Metallic Powders", Materials Sci. and Engineering, 57 pp. 187-195 (1983). 4. Staudhamer, K.P. and Jhonson, K.A., "Controlled Powder Morphology Experiments in Megabar 304 L Stainless steel Compaction", In Metallurgical Applications of Shock Wave and High Strain Rate Phenomena, ed. By Murr, L.E. et al., Marcel Dekker, INC, New York and Basel, pp. 149- 166 (1986). 5. Dogan, C.P., Rawers, J.C., Gorier, R.D, and Korth, G., "Mechanical Processing, Compaction, and Thermal Processing of Alpha- Fe Powder", Nanostructural Materials V4 n 6 pp. 631-644 (1994). 6. Kochsiek, D., Prummer, R., and Bnmold, A.,"Synthesis of Intermetallic Aluminides by Explosive Reaction Pressing", Metall Vol.49, Jahrgang, No. 3 pp. 168-172 (1995). 7. Sivakumar, K., Prasad, K.S., Bhat, T., Balakrishana, and Ramakrislman, P., "Microstructural Characteristics Of Shock Consolidated 2124 AI Alloy Compacts", J. of Mat. Sci. V32 n 19 pp. 5271-5278 (1997). 8. Rawers, J., Gorier, D., and Korth, G., "Consolidation, Mechanical Properties, and Phase Stability Of Mechanically Alloyed Fe-N Powder Compositions", ISIJ International VOL.36 n 7 pp. 947-950 (1996). 9. Mamalis, A.G., Gioftsidis, G.N., and Prohaszka, J., "Manufacturing of Thin Rectangular Plates By Explosive Compaction Of Copper Powder", J. of Eng. Manufacture, 204 pp. 237-247 (1990). 10. Thomas, T., Bensussan, P., Chartagnac, P., and Bienvenu, Y.,"Dynamic Compaction of Copper Powder: Experimental Results and 2D Numerical Simulation" In "Shock Wave and High Strain Rate Phenomena in Materials" ed. Meyers, M.A., Murr, L.E., and Staudhammer, K.P., Marcel Dekker, Inc., New York, pp. 433-441 (1992). 11. Schwar'z, R.B., Kasiraj, P., Vreeland, T.V. and Ahrens, T.J., "The Theory for the Shock Wave Consolidation of Powders", Acta Metall. Vol. 32 n 8 pp. 1243-1252 (1984). 12. Jaramillo, D., Hinojosa, G., Hallen, J.M., Balmor, N., Inal, O.T., "Mechanical and Microstructural Characterisation Of AI-SiC Composite Material Obtained By Dynamic Compaction", In 1st Int. Conf. On Ceramic and Metal Matrix Composites, San Sebastian, Spain (1996). 13. Hegazy, A. Abousree, and Hegazy, A.A., "Explosive Compaction of Metallic and Polymeric Powder Mixture", In Proc. Of 4th Cairo University, MDP Conf., Cairo, pp. 7986 (1988). 14. Bystrzycki, J., Paszula, J., Trebinsk, R., and Varin, R.A., "Microstructure and Interface Behaviour ofNi/Ni AI Composites Produced By The Explosive Compaction of Powders", J. Materials Sci. Vol. 29 n 23 pp. 6221-6226 (1994).
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Section IV
MANUFACTURING PROCESSES
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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NEW TRENDS IN CIM: VIRTUAL MANUFACTURING SYSTEMS FOR NEXT GENERATION MANUFACTURING
Choi, B.K." and Kim, B. H. * Director of the VMS Lab Department of Industrial Engineering, KAIST, Taejon, South Korea e-mail:
[email protected] ABSTRACT The paper provides a structured review on recent developments and new trends in CIMrelated technologies and business strategies, and presents a new human-centered CIM framework called Human-Centered Virtual Manufacturing Systems which is suitable for meeting the requirements of Next Generation Manufacturing. KEYWORDS CIM, Computer and Human Integrated Manufacturing (CHIM), Intelligent Manufacturing Systems (IMS), Holonic Manufacturing Systems (HMS), Virtual Manufacturing System (VMS), Agile Manufacturing (AM), Next Generation Manufacturing (NGM), HumanCentered Systems (HCS). I. INTRODUCTION The acronym CIM is one of the most pervasive, but somewhat illusive, concepts in modem manufacturing. Since early seventies, a large variety of CIM architectures (or frameworks) have been proposed, and several academic journals beating the term "Computer Integrated Manufacturing" have been published including the Int. Jr. of CIM (IJCIM). A typical CIM architecture [1] is shown in Figure 1. In order to explore new trends in CIM, it is first necessary to establish a common view of CIM. In Figure 1, a CIM system is viewed as a structure consisting of three areas - Product and Process Design (PPD), Manufacturing Planning and Control (MP&C), and Production Process - that are linked together via an Information Technology (IT) infrastructure. CAD/CAM, MRP (Manufacturing Resource Planning), and FMS (Flexible Manufacturing System), respectively, were regarded as the representative technology in each of the three areas. Thus, CIM is not regarded as a "tangible" technology by itself, but is viewed as a competitive strategy or as a manufacturing paradigm. The two views of CIM are complementary, however, since the former defines the goals of CIM while the latter prescribes means to achieve them. In the paradigm view, the main theme is integration (and computerization) of its subsystems [2]. An implication of the paradigm-view of CIM is that the whole manufacturing system, from order picking to selling, is regarded as a complex machine requiting little human interference
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[3]. Along this direction of the CIM paradigm, the concepts of Intelligent Manufacturing System (IMS) were evolved. Especially, the Japanese initiative on the IMS Program [4] had fueled worldwide interests in IMS. By pursuing this direction to its extreme, there have been proposed various "new manufacturing paradigms" such as Bionic and Genetic Manufacturing, Fractal Manufacturing System, and Holonic Manufacturing System (HMS) [5]. A common theme of these paradigms is manufacturing with minimal human intervention, which may be characterized as Computer-Centered CIM.
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Fig. 1. A Typical CIM Architecture [1] In the mean time, failure with CIM and advanced manufacturing technology (AMT)had widely been reported, and it was recognized that the failure was mainly due to the neglect of human and organizational factors. As a result, different kinds of new manufacturing paradigms, such as Human Integrated Manufacturing and Computer and Human Integrated Manufacturing (CHIM), were emerged [3]. A common theme here is "manufacturing with human involvement", which is often called Human-Centered CIM. More recently, general concepts of Human-Centered Systems had been elaborated [6]. Another line of development related to the technological aspect of CIM is the concepts of Virtual Manufacturing (VM). The idea of a VM-system is proposed in order to integrate various kinds of virtual manufacturing [7], and the VM-concept is employed to achieve a total manufacturing integration [8]. The VM-concept is a "neutral" CIM-concept because it may be used as either a skill supporting or skill substituting means (or both). A CIM-concept is called "human-centered" if it's purpose is to support human skills, and is called "computer-centered" if it aims to substitute human skills. A line of development related to the strategy-view of CIM is the concepts of Agile Manufacturing (AM) and Next Generation Manufacturing (NGM). The concept of AM is built around the synthesis of a number of enterprises that each has some core skills which they bring to a joint venturing operation [9]. The term "a number of enterprises" in the above sentence is often called a Virtual Enterprise [ 10] or Extended Enterprise [ 11]. The concept of NGM is similar to that of AM, but it provides a comprehensive strategic framework as well as a set of requirements or goals (that has to be satisfied by a CIM system).
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The purpose of this paper is to (1) provide a brief but structured review of the recent trends in CIM-related concepts and technologies and (2) propose a new CIM framework, which we call a "Human-Centered VMS" framework, to meet the manufacturing requirements in the 21st Century. 2. OVERALL TRENDS IN THE PARADIGM-VIEW OF CIM 2.1. Trends in Computer-Centered CIM The paradigm-view of CIM, in which the whole manufacturing system is regarded as a complex machine requiring little human interference, had led to the concepts of IMS. Even though the original vision of IMS was to better harmonize human beings and intelligent machines, most of the IMS research activities have been focused on computer-centered systems [4, 12].
As a result, some researchers regard Holonic Manufacturing System (HMS) as the enabling technology for implementing IMS. For example, [ 13] states that "IMS in general and HMS in particular are new concepts in the Manufacturing arena. They differ from conventional Manufacturing Systems - even advanced ones - in their inherent capability to adapt to changes without external intervention". HMS aims to develop a holonic organization as a means to realize the IMS vision. The research in HMS has resulted in some prototype systems in the areas of process sequencing and scheduling and autonomous co-operative control [5, 14]. 2.2. Trends in Human-Centered CIM As discussed earlier, failure with CIM and advanced manufacturing technology (AMT) had widely been reported, and it was recognized that the failure was mainly due to the neglect of human and organizational factors. Furthermore, in the case of Flexible Manufacturing System (FMS), it was realized that running the system required close attention from qualified people. And much of the responsibility for the efficient daily operation of the FMS was shown to depend less on the technology and more on the persons attending it [15]. It was further recognized that human skill and its application be optimized in harmony with AMT [16] and that system productivity depended directly on human skill and the human element in AMT was responsible for vital flexibility [17]. These observations, together with social science influences, had led to the concepts of Human-Centered Design (of CIM components)and Computer and Human Integrated Manufacturing (CHIM) [3].
Human-Centered CIM-components should 1) provide the user with the opportunity to learn and develop skills and knowledge, 2) facilitate the maximization of operator choice and control, 3) integrate the planning, execution and monitoring components of the work, 4) encourage social communication and interaction between its users. And empirical evidence shows that companies tend to move toward the CIM paradigm first and experience low performance, and then shift from CIM to the CHIM paradigm [3]. 2.3. Trends in Neutral CIM The concept of Virtual Manufacturing (VM) represents a technological aspect of CIM, and is defined as the concept of "executing manufacturing processes in computers as well as in the real world" [l 8]. The purpose of VM is to achieve a total manufacturing integration [8] by integrating various kinds of manufacturing in virtual as well as real domain.
In this ' view [ 18], a manufacturing system is decomposed into a real physical system (RPS)
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and a real information system (RIS). An RPS is composed of substantial entities that exist in the real world. An RIS involves activities of information processing and decision making in manufacturing. Then, it is defined that: 1) Virtual Physical System (VPS): A computer system that simulates the responses of the RPS, 2) Virtual Information System (VIS): A computer system that simulates the RIS and generates control commands for the RPS, 3) Virtual Manufacturing System (VMS) consists of a VIS and a VPS. The concept of VMS was proposed to achieve a total manufacturing integration by integrating various kinds of manufacturing in virtual as well as real domain [ 19]. 3. OVERALL TRENDS IN THE STRATEGY-VIEW OF CIM With ever-increasing global competition, sustained technological gains, and the need for rapid response to customer demands, manufacturing industries are constantly seeking for new business strategies as a way of survival. As a result, we have witnessed such buzzwords as Just-in-time (JIT), Flexible Manufacturing, Lean Production, Agile Manufacturing (AM), and Next Generation Manufacturing (NGM). In order to define strategic requirements of future CIM, it is necessary to understand these "paradigm shifts in manufacturing".
3.1. Agile Manufacturing The concept of Agile Manufacturing (AM) is built around the synthesis of a number of companies, each of which has some core skills or competence brought to a joint venturing operation, based on the each partner's facilities and resources [9]. The "synthesis of a number of enterprises" in the above sentence is often called a Virtual Enterprise [10] or Extended Enterprise [ 11 ]. Using the virtual enterprise concept, agility is about rapidly reconfiguring the whole supply chain, if necessary, to meet the current and emerging market needs. The concept of AM is more characteristic of project shops and job shops (where high variety, high uncertainty of "what the next specification will be" are the operations orders of the day [20]) and is about rapidly reconfiguring the whole supply chain to meet the market needs. But it places a particular emphasis on the participation of people [21 ].
3.2. Next Generation Manufacturing The concept of Next Generation Manufacturing (NGM) is similar to that of Agile Manufacturing, but it provides a more comprehensive strategic framework. Key characteristics of NGMS (or NGM System) were first developed at an NGMS Program Definition Workshop conducted by CAM-I t in February 1994 for twenty large manufacturing companies from Europe, Japan, and the United States [10]. The key characteristics of NGMS identified during the workshop are 1) support for virtual enterprise, 2) customer focus and business centered, 3) Re-configurable, adaptable, and flexible in response to customers, 4) Global support for design and production, 5) information and knowledge based, human intelligence oriented, 6) modular to support distribution/autonomy, but cooperative to achieve enterprise goals, 7) environmentally aware. To meet the above requirements, an NGMSarchitecture called "The NGMS Conceptual Pyramid" is proposed However, a more comprehensive effort was launched in 1995 involving a team of nearly 500 experts to sculpt a vision of future manufacturing in the United States, and an NGM Report of over 600 pages was prepared in early 1997 [ l l ]. The purpose of the project is to develop a t Consortiumfor AdvancedMfg. Integration(CAM-I): 1250E CopelandRoad, Arlington,TX 76011,U.S.A.
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broadly accepted model of future manufacturing enterprises, and recommend actions that manufacturers can use to attain world-class status. The concept of NGM is based on the concept of Extended Enterprise. It is a group of companies that collaborate to create and support timely and cost-effective services and/or products, which requires (1) sharing of knowledge and resources, (2) collaboration to create products and services, and (3) seamless integration. The two key words of the NGM attributes are responsiveness and teaming.
3.3. Human-Centered Systems Traditionally, the theme of technology development has been "Science Finds, Industry Applies, Man Conforms". But, in February 1997, participants at an NSF Workshop discussed a very different theme: "Humans Live, Science Finds Out How, Technology Conforms" [22]. The topic of the workshop was "Human-Centered Systems: Information, Interactivity, and Intelligence", and the goal was to define this emerging interdisciplinary field and articulate research, education, and infrastructure needs to support work in this area [ 11 ]2. The paradigm change, from the technology-driven approach to the human-centered approach, is depicted in Figure 2
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Fig. 2. The Paradigm Change in HCS [1 l l In the HCS concept, the human's creativity can not be developed and utilized unless he 1) feels of "mastery" over the problem while using the tool, 2) is able to "achieve" a result that satisfies him, and 3) feels that "he" has solved the problem. In addition, there is a strong motivation for collaboration because of the augmentative, integrative, and debative nature of the work. And there are strong social incentives that knowing that others depend on you may motivate you to do your part and that working in a team is more fun and offers possibilities for friendship. Thus, a HCS should be designed to support human creativity and collaboration. 4. HUMAN-CENTERED VMS FOR NGM In this section, we will propose a new Virtual Manufacturing Systems Framework to meet the requirements of NGM in a human-centered way. The new framework is termed a HumanCentered VMS.
4.1. Scope of the Human-Centered VMS Since the term "manufacturing" has different meanings depending on the context, so does the VMS. Thus, it is necessary first to define the scope (or domain) of the VMS to be proposed in this paper. As depicted in Figure 3, the top-level model of the Virtual Enterprise (or Extended Enterprise) is called a NGME (Next-Generation Manufacturing Enterprise). The "Product Related Processes" portion of the NGME is called a NGMS, which consists of 1) Product & Process Development, 2) Production, 3) Logistics, and 4) Post-Sales Services. -"The document is available at http://www.ifp.uiuc.edu/nsfhcs/
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We call the NGMS in Figure 3 a "wide" manufacturing system. The four sub-functions in the NGMS are a generic grouping. However, we have developed a more structured model of the same manufacturing system, as depicted in Figure 4. In our model, there are five subfunctions: Order Control System (OCS), Manufacturing Execution System (MES), Product Design System (PDS), Manufacturing Preparation System (MPS), and Factory. In this model, the Logistics function and the Process Development function in the NGMS model may be handled by the MES, and the Post-Sales Service function can be handled by the Order Control System.
Fig. 4. A Structured Model of Wide Manufacturing System The entire "wide-manufacturing" function of Figure 4 may be handled either by one company or by a "group of company" forming an extended enterprise. In the latter case, which is becoming more common these days, the Product Design function may be handled by more than one Design Company and the rest of the functions by another group of companies, as shown in Figure 5.
Fig. 5. An Extended Enterprise Model for the Wide Manufacturing System
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In this "extended enterprise" model, a Design Company consists of three sub-systems: Order Control System, Design Execution System (DES), and Product Design System (PDS). And, a Manufacturing Company has four sub-systems - Order Control System, Manufacturing Execution System (MES), Manufacturing Preparation System (MPS), and Factory-, which constitute a "narrow manufacturing system". From now on, we will focus on the narrow manufacturing system, the Manufacturing Company in the extended enterprise model of Figure 5, which becomes the domain of the Human-Centered VMS to be described shortly. 4.2. Architecture of Human-Centered VMS
Our goal is to design and operate a manufacturing company so that it can serve both external customers (or the society) and internal customers (i.e., its employees). To meet this goal, we propose a Human-Centered Virtual Manufacturing System that will (1) expand human capabilities, (2) support human creativity, and (3) support collaboration among the people so that the responsiveness and teaming of the NGM Company are maximized. We define a VMS as a "cyberspace structure of interface, planning, and control mechanisms to support human decision-making via monitoring & simulation of actual manufacturing situations through modeling of all activities and resources in a physical manufacturing system". Furthermore, we regard a (narrow) manufacturing system to be composed of 1) a hardware system including tools, machines, facility, materials, parts, and products, 2) people including organizational structures, 3) a VMS, and 4) others including paper documents, company policies, etc. In this sense, the VMS is also a"real" part (as a decision support system) of the manufacturing system. ....
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Shown in Figure 6 is a reference model of the Manufacturing Company in Figure 5, which we called a narrow manufacturing system. In this model,
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OCS (Order Control System) consists of Marketing, Bidding & Negotiation, Delivery Control, and Post-Sales Services functions. MES (Manufacturing Execution System) consists of Shop-level Process Planning, Loading Scheduling, Progress Control, and Logistics functions. MPS (Manufacturing Preparation System) consists of Process Planning, Tooling, P/G, and Exception Handling functions. The Exception Handling function is responsible for facility maintenance, ECO (engineering change orders), trouble shooting, etc. The Factory has machining, inspection and assembly stations as well as transportation and storage facilities. The proposed VMS framework supports two types of VMS: Prototypic and Operational VMSs. As depicted in Figure 7, the Prototypic VMS is used as a virtual prototyping tool for designing a new manufacturing system. This concept is used widely in designing anew product, which is often called Digital Manufacturing. An Operational VMS may be obtained from the Prototypic VMS (for a new manufacturing system) or constructed from the scratch for an existing manufacturing system.
The Human-Centered VMS concept is geared to the Operational VMS. The Architecture of our Human-Centered VMS is presented in Figure 8. The Humans in the Physical Manufacturing Domain are hooked up to the VMS, and there are "high- speed" Data Transfer Lines connecting the VMS to the physical Factory and to the outside world (via Internet). The VMS shown at the bottom of Figure 8 has four "virtual" sub-systems - Virtual MES, Virtual MPS, Virtual Factory, and Common DB - as well as User Interface Control and Data Handler. The overall structure of the VMS looks similar to the CIM Architecture of Figure 1, but its concept is quite different. In the case of CIM architecture, integration is achieved in a physical domain (with or without human involvement). In the VMS concept, integration is achieved in a virtual domain to serve the humans in the physical domain where teaming and human creativity are more important. That is, the role of VMS is to expand the capabilities of humans and support their creativity and collaboration efforts. 5. DISCUSSIONS AND CONCLUSIONS Proposed in the paper is a Human-Centered Virtual Manufacturing System that is compatible with current trends in CIM and is suitable for meeting the requirements of the NGM (Next Generation Manufacturing) and HCS (Human-Centered Systems) concepts. Even though there remain some technical and theoretic issues in modeling and simulation (of products, machines, manufacturing processes, planning and control activities, etc.), there are available various kinds of commercial VM technologies.
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However, in order to realize the proposed VMS concept, a number of technical and social issues have to be resolved. One major issue in designing a Human-Centered VMS is how to allocate functions among humans and computer systems. Another key issue is how to design a human-oriented software system to capitalize on and accommodate human skills in perception, attention, and cognition, while minimizing the opportunities for and effects of human error [23]. Also the Clean Interface Concept [24] and the Machinery Station Concept [21] have to be developed further, and the general perception that "human operators are sources of error and unpredictability" is likely to be a huddle. The proposed Human-Centered VMS concept is a new one and has to be validated and seasoned under real manufacturing environments. The VMS Lab3 at KAIST (Korea Advanced Institute of Science and Technology) has been focusing on developing "component technologies" and sub-systems of VMS as well as integration methods, and is fully devoted to continuous research efforts along these directions. It is one of the National Research Laboratories endowed by Korean Government.
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In the Virtual MPS area (of Figure 8), a virtual machining software system has been developed based on the unified CAM system architecture proposed by the first author of this paper [25]. In the Virtual MES, a Gantt Chart based MES system has been developed [26]. These two software systems, named Z-Master| and mold-MES | respectively, have been commercialized. We have also been developing a Virtual Factory software, named VFS| based on the modeling scheme proposed by the first author and his former students [27, 28]. Shown in Figures 9, 10, and 11 are screen images of these software systems.
ACKNOWLEDGEMENT The research was supported by The Ministry of Science and Technology of Korea through its National Research Lab Program. REFERENCES 1. Gunn, T.G., Mfg. Competitive Advantage - Becoming a World Class Manufacturer, Ballinger Publish (1987). 2. Choi, B.K., "CIM in the Four Dragons", IJCIM, Vol. 5(4), pp. 199-200, (1992).
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3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
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Sun, H. and Frick, J., "A Shift from CIM to CHIM", IJCIM, Vol. 12(5), pp. 461-469, (1999). Hayashi, H., "Intelligent Manufacturing", IEEE Spectrum, Vol. 30(9), pp. 82-84, (1993) Brussel, H.V., et al, "Ref. Architecture for HMS", Computers in Industry, Vol. 37(3), pp. 255-274, (1998). NSF, "Reports from Break-out Groups (BOGs)", NSF Workshop Final Report, pp. 21-91, (1997). Iwata, K., et al., "A Modeling and Simulation Architecture for Virtual Manufacturing Systems", Annals of CIRP, Vol. 44, pp. 399-402, (1995). Kimura, F., "Product and Process Modeling as a Kernel for Virtual Manufacturing Environment", Annals of the CIRP, Vol. 42, pp. 147-150, (1993). Kidd, P.T., "Agile mfg.: a Strategy for the 21st Century", lEE Colloquium on Agile mfg., pp. 1/1-1/6, (1995). Jordan, J.A. and Bunce, P.M., "Requirements for Next Generation Manufacturing Systems", IEEE/CPMT Int"l Electronics Manufacturing Technology Symposium, pp. 247252, (1994). NGM Project Report, Agility Forum, (1997). Valckenaers, P., "Intelligent Mfg. Systems", Computers In Industry, Vol. 37(3), pp. 169170, (1998). Sousa, P., and Ramos, C., "A Distributed Architecture and Negotiation Protocol for Scheduling in Manufacturing Systems", Computers in Industry, Vol. 38(2), pp. 103-113, (1999). Leeuwen, E.H. and Norrie, D., "Holons and Holarchies", Mfg. Eng., Vol. 76(2), pp. 8688, (1997). Martensson, N., Martensson, L. and Stahre, J., "Human-Centered Flexible Manufacturing System in Machining and Assembly", Human-Intelligent Based manufacturing, SpringerVerlag, pp. 67-82, (1993). Murphy, S., "Human-Centered CIM Systems" ESPRIT 88, Putting the Technology to use, Proceedings of the 5th Annual ESPRIT Conference, Brussels(BE), pp. 1615-1629, (1998). Slatter, R.R., et al, "Human-Centered Approach to the Design of Advanced Manufacturing Systems", Annals of the CIRP, Vol. 38, pp. 461-464, (1989). Onosato, M., et al., "Development of a Virtual Manufacturing System by Integrating Product Models and Factory Models", Annals of the CIRP, Vol. 42, pp. 475-478, (1993). Iwata, K. et al., "Virtual Manufacturing Systems as Advanced Information Infrastructure for Integrating Manufacturing Resources and Activities", Annals of CIRP, Vol. 46, pp. 335-338, (1997). Harrison, A., "From Lean to Agile Manufacturing", lEE Colloquium on Agile manufacturing, pp. 1-7, (1997). Tu, Y., "Production Planning and Control in a Virtual One-of-a-Kind Production Company", Computers in Industry, Vol. 34, pp. 271-283, (1997). Talbert, N., "Toward Human-Centered Systems", IEEE Computer Graphics and Applications, Vol. 17(4), pp. 21-28, (1997). Chappell, S.L., "Certification of Usability: A Process for Creating a Human-centered System", Proc. of the digital avionics systems conference - 17th DASC, pp. E1 l/l-El 1/8, (1998). Jerard, R.B., "FACILE: A Clean Interface for Design and Fabrication Intemet-Aided Design, Manufacturing and Commerce", 18th Computers in Engineering Conf., Atlanta, Sep., (1998). Choi, B.K., et al, "Unified CAM System Architecture for Die and Mold Manufacturing", CAD, Vol. 26(6), pp. 235-243, (1994).
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26. Choi, B.K., Kim, D.H. and Hwang, H., "Gantt Chart Based MES for Die and Mold Manufacturing", Proc. IFIP WG 5.7 Conf. On Managing Concurrent Manufacturing, Seattle, USA, pp. 105-114, (1995). 27. Choi, B.K., Han, K.H. and Park, T.Y., "Object-oriented Graphical Modeling of FMS", Int. Jr. ofFMS, Vol. 8(2), pp..159-182, (1996). 28. Park, T.Y, et al, "An OOM framework for AMS", IJCIM, Vol. 10(5), pp. 324-334, (1997).
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
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INTELLIGENT MACHINING SYSTEMS - CHALLENGES AND OPPORTUTINTIES
Teltz, R. ~ and Elbestawi, M.A.** ** Professor and Chair, Department of Mechanical Engineering Research Engineer, Intelligent Machines and Manufacturing Research Center (IMMRC) Faculty of Engineering, McMaster University, Hamilton, Ontario, Canada Email: elbestaw~mcmaster.ca and n_'
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ABSTRACT Machine tool control systems typically involve servo motion and discrete input/output control. In this manner they only indirectly control the metal cutting process and consequently the economic and efficient production of discrete parts. Intelligent Machining Systems attempt to more directly control machining processes by incorporating models of the cutting process and real time sensor data. Operational parameters such as cutting speeds and feeds are formulated to directly address goals of economy and product quality. Intelligent Machining Systems integrate technologies related to sensing, modeling, control and monitoring. The process variables involved include tool condition, process cost, rate of production, and part quality. More recently, the use of Open Architecture concepts in CNC machine controllers has allowed a greater realization of complex Intelligent Machining Systems; particularly with regards to systems which involve the control or monitoring of multiple process conditions. This paper describes research performed in the IMMRC labs relating to modeling, monitoring and control for Intelligent Machining Systems (IMS). Applications in milling and turning are described. Key points regarding the challenges and opportunities for future research are discussed. KEYWORDS Intelligent machining systems, machine tools, metal cutting, sensing, modeling, controls 1. MODELING FOR THREE AXES MILLING OF DIES AND MOLDS
The Role of Process Modeling in Intelligent Machining Systems Accurate models and simulation systems of metal cutting processes are used to evaluate operational parameters (ie: feeds, speeds, depth of cut) with respect to critical process constraints such as cutting loads, cutting stability, produced part features, etc. In this regard, process modeling is a major component of Intelligent Machining Systems for optimization algorithms and for the generation of base line data for actual process evaluation. There are several aspects of metal cutting that make process simulation complex. The first involves the accurate calculation of the geometry of interaction between the tool and workpiece - an issue compounded by the use of tooling and work pieces with complicated profiles. Moreover, for milling of dies and molds in particular, tool paths can involve continuously varying immersion geometries, with rough, semi finish and finish passes. This introduces the need to manage very large amounts of data. Added to all of this is the requirement for inclusion of the effects of machine tool dynamics due to their critical effect on chip geometry. Other issues include model calibration where for example, variation in cutting velocities along the tool edge profile require a nonlinear calibration.
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Research at the IMMRC has addressed several aspects of these problems, with the particular goal of making simulation systems more accurate in their predictions and more applicable to the needs of Intelligent Machining Systems. The follow sub sections discusses some of this research.
Cutting Edge Representation The primary type of tooling used today to manufacture dies and molds includes ball and flat end mills. However, the specific cutting edge geometry varies significantly within this class of tooling. While the majority of published research has dealt with simple cutting edge designs (ie. fixed diameter, zero helix, constant helix, constant pitch), the problem remains in having a generalized modeling procedure to represent a varied cross-section of tooling, including form tools. Recent work at the IMMRC has attempted to generalize the representation of the cutting edge. Imani [ 1] and Abrari [2] use a cubic piecewise NURBS curve [3] to represent the tool cutting edge profile for a constant lead/pitch ball end mill. A key advantage this representation is that there are standard techniques available for manipulating the location and orientation of a NURBS curve. Thus, virtually any cutting edge profile can be used by the simulation allowing for example, the tool profile itself to be a manipulated variable in an optimization strategy of the Intelligent Machining System. A p'" degree NURBS curve is defined by: I!
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~ i _ o Ni.v ( u )o)i where the Pi are the control points, the t.o, are the weights and the Ni.p(U) are the pth B-spline basis functions. The set of three dimensional control points P~ along the cutting edge profile are typically determined from either of 1) an analytical expression for the edge profile, 2) a tooling drawing, or 3) discrete three dimensional points measured by a coordinate measurement machine.
Part Local Surface Topology Representation The surface topology of a die or mold is typically composed of many complex surfaces. This complexity is a result of both the desired part form in addition to any "marks" left on the surface from previous machining operations (eg: roughing, semi finishing). To simulate the machining of even moderately sized parts, the required storage of surface data can quickly become unwieldy. To overcome this problem, the developed simulation systems uses only the local topology of the part in the vicinity of the tool. Figure I depicts this procedure. The "Control Space" shown in the figure is chosen to consider all surfaces within possible contact of the tool. A simple bounding box intersection algorithm is used to determine which tool path segments contribute to the local surface topology of the part in the vicinity of the current tool position. In the case of secondary operations (ie: semi finishing or finishing) all previous operations are reconstructed to provide the surface data seen by the current pass. In this manner the need to store vast amounts of surface data is avoided.
Mechanistic Force Model The mechanistic force model has been formulated following those models presented by Imani [ l ] and Yucesan [4]. The modeling approach taken was to first develop the cutting edge geometry, then to develop the cutting force expressions. The cutting edge geometry for a ball end mill is shown in figure 2. The coordinate values of any point P~on the cutting edge can be determined from the NURBS curve representation, if given the corresponding parameter u,. The parameter u~, normalizes points at equal spacing along the rotational axis (z axis) of the tool. Given features of the geometry in the locale of point Pi, the instantaneous
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(2)
Over a discrete, infinitesimal length of the cutting edge segment, dS, the instantaneous chip thickness, tc, at any angular position, 0,, and axial depth, z, is given by
t,(O,z)= R2(z)+ f,,sinO-[R,2(z)-
f,2cos20]'/"
(3)
where fn is the feed per tooth, and R~, R_~are the tool radii at the beginning and end of dS respectively. The differential contact area dAr is given by, dz
dm r = 2t ( O , z ) ~ COS j~:
(4)
where K,, and Kt, are the pressure and friction cutting coefficients on the rake face. The differential normal and frictional edge forces, dE,,,, and are calculated as"
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Id .rl= Knr " d A r
IdPfrl: Kfr " d A r
(5)
(7)
where K= and Kfe are the pressure and friction cutting coefficients acting on the cutting edge. The instantaneous force acting over the length of the active cutting edge is given as the summation of the differential normal and friction forces acting on the rake face and cutting edge.
F,( O) = Fnr( O) + Fir(O)+ Fne( O) + Ff,( O) --
(9)
{Knr'[--~r(O,z)~" K[r'nfr(~r
+ K,, . [ - ~ , ( O. z ) + Kre . ~r,( O. z ) ] . dS } Calibration The coefficients K=, Kfr and K=, Kf, are functions of cutting edge geometry, cutting speed, and feed per tooth. Various researchers have addressed the problem of calibrating these coefficients to experimental data; some use averaged values over a range of cutting conditions [4], while others have considered only the variation with cutting speed for example [ 1]. It should be noted that the subject of calibration is an ongoing research issue for metal cutting process modeling. The method proposed by the work presented here expresses the variation of pressure and friction coefficients as an exponential function of the cutting speed, the helix angle and the feed: K~ = e "'§ (10) where ~ is the instantaneous cutting coefficient, and V~, 13~are the instantaneous cutting speed and helix angle respectively. The coefficients a l, a2, a3 are functions of the instantaneous feed per tooth. Figure 3 shows the comparison of some experimental results with the calibrated force model. The calibration has been shown to provide good correlation with experimental data over a wide range of cutting conditions. Force X fl~:
o
' Force' Y fpt:
0.0045"
.
.
.
.
.
0.0045"
.~oo .3oo a.
.400 -500 120
-
140
Exp
160
180
~RefCo~
200
-
220
240
120
- Exp
A~Coeff
Force Z fpt:
120
- - Exp
140
~
180
200
(Oegrees)
Ref Coef
~
fpt:
180 200 (I)egrNs)
Ref Cool
160
220
240
~ Avg Coeff
Resultant Force
0.0045"
160 ~e
140
-
220
Avg Coefl
240
0.0045"
100 0 120
- E xp
140
160 1tl0 200 Angle (Degrees) ~
Ref Cool
220
240
.... Avg Coeff
Figure 3. Calibration results for the three axes ball end milling model (material: Titanium, tool 3/4" solid carbide ball end mill).
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2. PROCESS M O N I T O R I N G USING FUZZY NEURAL N E T W O R K S
Machining Process Monitoring Modem techniques for automated machining process monitoring include expert systems, neural networks, and fuzzy pattern classification. Indirect measurements such as cutting force and vibration signals are typically used however, the measured alone signals do not often give sufficient information to discriminate between the different tool conditions. Accordingly, further processing stages is required to yield useful features which are more sensitive to tool conditions, but insensitive to noise. The problem of decision making relies on criteria relating the variables in the feature space to the classification space. Typically. parameters (weights) in a matrix form are used for this mapping, such as the weights in a linear classifier or in a back-Propagation neural network. In more sophisticated implementations, multi-layered neural networks are employed, which consists of nonlinear connections between the inputs and the outputs.
Implementation of the Fuzzy Neural Network As an alternative to typical neural networks, the work discussed here [5] proposes a fuzzy neural network approach for tool condition monitoring. Using this combination of technologies, it is possible to exploit the parallel nature of the classification and to provide a fuzzy reasoning approach. Different from matrix-type decision making, a tree structure is used for reducing unnecessary interconnections within the networks. This approach results in simpler and faster training and classification. An example neural network that utilizes fuzzy logic is shown in Figure 4. The input layer, FA, has processing elements, for each of the n dimensions of the input pattern Xk(feature vector). The parameters of the probability distributions represent the connections to the hidden layer. The hidden layer FB, consists of neurons that use fuzzy classification to address subsets of the original data set and any necessary information from other neurons. A membership function is used for the classifications at neurons, the interconnections within the hidden layer and the connections to the output layer. The neurons of the output layer, Fc, represent the degrees to which the input pattern Xk, fits within the each class (tool conditions). The output decision may be either "soft" or "hard" depending on the requirements.
Figure 4. Structure of the Fuzzy Neural Network.
Figure 5. Example of Maximum Partition.
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Current Advances In Mechanical Design and Production, MDP- 7
Learning and Classification
In the learning phase, the hidden layer is constructed based on available learning samples. A key operation here is determining the "the Maximum Partition", Xp = Ap + Bp, where Ap is the set which contains the samples that belong to certain tool conditions and Bp is the set of all the remaining samples. The partition is determined from probability distributions of the data, using a technique referred to "Multiple Principal Components"[5]. The monitoring index associated with a resulting partition is defined as "the pivot index" and is applied as a major input of a neuron in the hidden layer. Figure 5 gives an example of the Maximum Partition of a 2-index, 3-class problem. By studying the distribution of the samples from classes A, B, and C, we notice that 12gives the maximum partition of the learning set for class C with respect to the other classes. Here, 12 is chosen as "the pivot index" at this neuron for the maximum partition. The maximum partition in multiple directions at neuron p generates two new neurons: one with the learning set Xp and the other with the learning set Bp = X~ - Ap. In this manner the distribution of learning samples is used to measure the supporting strength of the partition. Implementation of this procedure in algorithmic form results in an automated learning procedure - also referred to as unsupervised learning capability. Classification is performed by searching a path of neurons through the network, leading to an output decision. At each neuron in the hidden layer, the search is directed by the fuzzy membership grade at that neuron, using the membership function. This information propagates through the hidden and/or the input layer to the output layer where the final decision is given. Experimental Verification
Cutting tests were performed using 10 Hp NC lathe, instrumented with a cutting force dynamometer mounted under the tool holder (Fx, Fy, Fz), two accelerometers located on the tool holder (Ax, Ay), and a spindle current sensor for cutting power, Pw. All signals were low pass filtered at 1 kHz, and then were sampled at 2 kHz. The work pieces were AISI 1014 steel bars and the cutting tools were ISO K21 and K6 grade carbide inserts. Fifty two cutting conditions were tested as folows: cutting speeds 96 to 322 m/min, feeds 0.024 to 0.246 mm/rev., and depths of cut 1.2 to 3.5 mm. Five different tool conditions were considered:, sharp tool (no chipping, flank and crater wear 0.04mm2). Table I summarizes the results obtained from those tests. For comparison, the results obtained using a well-known feed-forward neural network trained by back-propagation (BPNN) [6]. For the various parameters of the BPNN algorithm, those chosen for comparison represent the best results obtained. The Fuzzy Neural Network was found to outperform the BPNN approach. Test
Classification Case
Success Rate FNN
Success Rate BPNN
Self Classification
94.7 %
89.3 %
Different Record, Same Cutting Conditions
89.3 %
80.0 %
94.7 %
84.0 %
, ,
.
.
.
.
.
.
.
,
Self Classification .
.
.
Different Record, Same Cutting Conditions
.
.
.
84.0 %
.
70.7 % .
Self Classification
96.0 %
Different Cutting Condition 80.0 % Table 1. Results of Fuzzy Neural Network
.
.
86.7 % 69.3 %
.
,
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3. INTELLIGENT, OPEN A R C H I T E C T U R E C O N T R O L IN TURNING Open architecture machine tool control is recognised as an enabling technology for advanced machining systems. The full realization of process control in machining depends on flexible and open interfaces to critical machining functions, In this manner, Open Architecture Control (OAC) facilitates the implementation of complex Intelligent Machining Systems. The IMMRC OAC [7]is based on a distributed, peer to peer functional model of the entire machine control system. Industry standards are used for hardware, software, and communication media. Machine motion and I/O functionality is provided by a module referred to as the 'Machine Server'. Generalized services support command, data and timing mechanisms between the Machine Server and 'client' applications (ie: IMS modules). Figure 6 depicts the relationship between IMS modules and the OAC Machine Server. FEED, SPEED, CLIENT APPLICATION
,'......
..~'""-.
LAYER
/ I -F~
MACHINE SERVER
7 -Chatter Monitoring& /
_
Figure 6. The OAC Machine Server and its Relationship to Other System Modules.
-
.FORCE,
t,/IBRA~O~
k.J -Tool Condition Monitodngl_ MEAS. / Control
IN PASS CONTROL ............
=~
~chine Server Module
Regulation
/ I TOOL WEAl
-Vision Wear ieasuremen~ ~ MEAS. -Process Simulation& ~ -Planning ,|
i
PER PASS CONTROL
-Trend model coefficients i MONITORIN( I-Recalibrate models !- RESULTS
l
-Initiatemonitoring
-Trend performance MULTIPASSCONTROL
F
Figure 7. Intelligent Machining System for Multipass Turning
A comprehensive Intelligent Machining System has been developed and demonstrated on the OAC. Figure 7 depicts the system for optimization and control of multi pass turning. The temporal aspects of the multi pass problem are addressed by the three feedback loops depicted. In pass control addresses time critical events related to the safety and integrity of the current pass. Tool breakage, chatter, and force regulation are addressed. Per pass control provides a plan of tool paths and feeds to provide constant cutting forces for the next pass. The model used for planning and prediction of cutting forces incorporates tool wear measurements acquired between passes by an on machine vision system. Multi pass control records and models longer term process behaviors. Lower levels of control are compensated; models are recalibrated and parameters are adjusted. In pass control tasks are performed by C language routines implemented at the real time level of the OAC structure. The time critical nature of these activities is supported here by a real time operating system and low latency access to sensor data. The per pass control tasks include tool wear measurement and image data processing cycles, and feed schedule calculations. These algorithms execute at the non real time level of the OAC and, in the case of the feed scheduling tasks, use routines
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from a commercial solid modeling package (ACIS, by Spatial Tech. Inc.). Multipass control is realized using CLIPS (C Language Integrated Production System), an open source expert system. The system described here is detailed in the references given. The discussion here is provided to underscore the advanced and seemingly limitless possibilities for optimization of machining processes offered by the open architecture control platform. The concept of Intelligent Machining systems, and moreover the adoption of such technologies in industry clearly stands to benefit from further development and proliferation of the OAC systems. 4. DISCUSSION AND CONCLUSIONS This paper has attempted to give an overview of research related to the development of Intelligent Machining Systems at the IMMRC. Some advancements in process modeling was described for die and mold milling applications, tool condition monitoring using hybrid fuzzy logic and neural network technologies, and open architecture machine tool controls. These three areas emphasize the broad scope of focused research required to further the larger realization of Intelligent Machining Systems. Many challenges remain before such systems become commonplace in industry. Slow, but deliberate movements within the machine controls sector is advancing the proliferation of Open Architecture machine control. These platforms are the practical basis of Intelligent Machining Systems in that they allow the other technologies - modeling, monitoring, and controls to be integrated into the production machining environment. In this regard, there can be expected an increasing number of opportunities for these Intelligent Machining technologies to manifest tangible benefits for manufacturers in the future. Several challenges remain however, before such benefits are truly seen by practicing manufacturers. Process modeling methods must become more accepted as part of commercial CAM installations. Future engineers and manufacturers need to appreciate the potential of such tools, and be ready to rely on their predictions for process development. This underscores the need for further research towards more accurate modeling strategies, more comprehensive models (eg: the inclusion of fixture and part dynamics), and the development of simpler and better calibration schemes. Monitoring systems must also progress forward to provide higher reliability and sensitivity to process conditions. False alarms cannot be tolerated in high volume machining systems. Perhaps a greater understanding of their limitations by manufacturers would promote their use today in proper applications. In this regard, the mind set of end users of Intelligent Machining Technology remains perhaps the greatest challenge to overcome. -
REFERENCES 1 - lmani, B., "Model Based Die Cavity Machining Simulation Methodology", Phd Thesis, McMaster University, (1998). 2 - Abrari, F., "MultiAxis Milling of Flexible Parts", Phd Thesis, McMaster University, (1998). 3 - Piegl, L., Tiller, W., "The NURBS Book", Springer, 2"a Edition, (1997). 4 - Yucesan, G., Altintas, Y., "Mechanics of Bali End Milling Process", ASME Journal of Manufacturing Science and Engineering", Voi 64, pp. 543-55 I, (1993). 5 - Li, S., Elbestawi, M.A., "Tool Condition Monitoring in Machining by Fuzzy Neural Networks", ASME Journal of Dynamic Systems, Measurement and Control, Dec. 1996, Vol.55, pp 1019-1034, (1994).
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6 - Lippmann, R.P., "Pattern Classification using Neural Networks", IEEE Communication Magazine, pp 47-64, Nov., (1989). 7 - Teltz, R., Elbestawi, M.A., "Design Basis and Implementation of an Open Architecture Machine Tool Controller", Proc. of the 25th NAMRC, Lincoln Nebraska, (I 997).
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Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
447
CAM SYSTEM FOR EFFICIENT GENERATION OF PART-PROGRAMS FOR WIRE-EDM EI-Midany, T.T. ~ EI-Keran, A.A. ~ and Radwan, H.T. ~176
'Professor, *'Assist. Prof., Prod. Eng. and Mech. Design Dept. '"Production Engineer. Faculty of Eng. P.O.B. 2 Mansoura University 35516, Egypt
ABSTRACT In the recent years wire-EDM became a necessity for many engineering applications, particularly in the dies making. Design of a good CAM system as an aid for making dies or other wire-EDM applications, became a challenging job with increased use of complex part geometry. In this paper a proposed CAM system is designed to overcome the limitations of the existing CAM systems. The proposed CAM provides automatic correction for maximum taper angle, and automatic creation of the programmed fixation stop. Programming time has been reduced using the automatic tool path creation, postprocessing, trace checking feature, and simulation check. Enhanced two-axis programming capabilities include, multiple operation automatic lead-in and lead-out generation, automatic comer filleting for internal, external, or all sharp intersections. One directional trim cut can be written as a main program which calls a separate sub-program NC file for each trim cut to reduce the NC program size. No-Core cut capabilities are also added to avoid the machine damage. Great saving in programming time, user effort, and the precision for all steps have been achieved. KEYWORDS Wire EDM Programming, CAM System, No-Core Cut, Taper Cutting, Trim Cut, NC. 1. INTRODUCTION Wire cutting process is widely used for making stamping dies, turning form tools, templates, extrusion dies, progressive dies, and prototype production. The wire-EDM programming start with a manual method [ 1]. With the increased use of NC systems and growth in complexity of production the manual programming became a tedious work and time consuming. The manual method depends to a great extent on the experience of the operator, which leads to many human errors. So, the part programmer was no longer able to calculate efficiently the required tool path, and the use of CAM systems as an aid to part programming became a necessity [2,3]. Automatic generation of standard radius, sequence of wire path, fixation stability, and No-Core cut are examples for developing wire-EDM programming. When developing CAM software, one must consider how the interaction between the user and the program will take place. The portion of the software that provides this link is referred to as the user interface. The interface should provide a convenient means of moving through the various options of the program. In addition to the selection of program options, the interface must provide error
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Current Advances In Mechanical Design and Production, MDP-7
detection and recovery as well as the ability to cancel any operation midway through the process. Other desirable attributes of the user interface include making it easy to enter the information, providing graphical feedback to user inputs (such as blinking of selected entities), and preventing unpleasant surprises such as program crashes. Certainly, the items listed above make the construction of good user interfaces a challenging job. Thus, the objective of this work is to design a CAM system to sequence the wire path in a way to minimize the machining time, and generating the standard radius, maintaining fixation stability and optimizing the wire curing process. Some of the wire-EDM techniques such as, making the conical curing, trim cuts, and calculating the optimum location for the threading/breaking point of the wire, are also achieved through the proposed system. Through different applications of the proposed system, the results show a great saving in programming time, user effort, high precision for all steps of the curing process, particularly in the compound dies, prototype, and form tools. 2. THE INFORMATION FLOW THROUGH THE PROPOSED SYSTEM The information flow through the proposed system is illustrated in Fig. 1.
3. THE PROPOSED SYSTEM FUNCTIONS The proposed system was designed to satisfy the following factors:
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a. Sequencing the wire cutting motion in a way to minimize the machining cutting time. b. Considering the fixation stability to avoid the drop of the work piece. c. Analyzing the part-geometry and adding special features to the contour to improve the quality of the cutting process. d. Optimizing the wire cutting parameters for individual part-elements according to the part-features and technology data. e. Graphically simulating the cutting process of the spark erosion to verify the cutting sequence. f. Generating NC part-programs directly from CAD geometry database. 4. THE PROPOSED CAM SYSTEM The proposed system contains two processors for CAD/CAM calculations. Geometry processor for CAD database calculations and Technology processor for CAM and technology aspects calculations 4.1 Geometry Processor
The geometry processor analyses the extracted part-geometrical elements, and decides an optimized cutting sequence. The criteria for an optimized sequence are the fixation stability for the remaining sheet and the cutting time. During this step, part geometrical elements are processed according to the following steps: 1 Filter part-geometrical elements by removing ~emoveoverlapped ~ g ~" = E "the overlapped elements and joining together any )o]nCl--ose'Elements ) ~ ~ ) close vertices if the distance between them is less ......... than kerf width, as shown in Fig. 2. Fig. 2.Geometry Filter 2. Convert the individual geometrical elements to polygons by sorting and joining together any two identical vertices. Then identifying all generated polygons as closed or open polygons. 3. Generate a vertex searching space 0 sTART and END vertices of all open ~ I olygons. list that contains start and end vertex All vertices of all closed polygons. coordinates of all open polygons as well as all vertex coordinates of all closed .... Fig: 3"Sear.rh Lis t Generation . ~ , ~ J polygons, Fig. 3. 4. Sequencing the cutting motion by selecting from search space list the polygon that has the closest vertex to the previous tool (wire) position, taking into consideration that if the selected polygon includes other polygons, then skip this selection. If the selected polygon includes other polygons, then all inner polygons must be machined before the outer polygon. Remove the selected polygon from search space list and repeat the step no. 4 until the end of all nesting parts. Insert standard radius for the sharp comers, as shown in Fig. 4. 5. Read the standard radius value from cutting libraries; the option default value equal to 2 mm. Searching for sharp comers, and making grips around them, and then filtering. 4.2 Technology Processor
The technology processor analyses the data stored in the wire cutting libraries, which is available for modifying from the "Wire Option" dialogue box, and decides the optimum cutting conditions. During this step, cutting conditions are processed according to the following steps:
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Current Advances in Mechanical Design and Production, MDP. 7
_~
C]
f:
Select from search list the polygon, which
L
[
has the closest vertex to origin point, and make this
vertex, the START vertex. ,,
polygon was closed
'~
START
Set Origin Point = END Vertex
Sequencing _ ~ Yes Remove this polygon temporarily fiom search list
if this polygon includes another
N~ I Set origin point = START vertex
l
No
iq-oo,o,nd 1
Fig. 4.Sequencing Calculate for the polygon area, if the polygon area < 3 mm, the technology processor automatically adds the codes for No-Core cut in the NC file (*.cnc). 1. Add trim cut, correction, and the taper angle in the technology and NC files [9,10,11]. 2. Make the multiple cuts Fig. 5 and employing the 9 P o l a r arra~" ,'+ dimensions (mm/inch) in the job file. 666 3. Determine the best location of the 9 RectanrularArrav: ~ ~ threading/breaking point of the wire. Store all cutting files, NC file, position, technology file, and Fi~. 5.Makinl[ Multiple Cuts job file, as shown in Fig. 6. 4 44
4 4~
5. THE PROPOSED SYSTEM PART-PROGRAMS
5.1 Position File Position file contains the command for positioning the machine head for x-y and z-axis without wire and cutting. 5.2 Numerical Control File This file contains the elements of the drawing in the ISO code format. The basic structure of this file contains lines ~ G01, curves C C W ~ G03, curves in C W ~ G02, stop ~ MOO, [12] etc. 5.3 Technology Data File The cutting libraries use this file to define the technology functions.
Current Advances in Mechanical Design and Production, MDP- 7
............ ~ Determine ThreadingPoints ~ ~
r
1
451
~ "Point
-'-~nd Point
Punch
i,l Select the Outer Point
Select the Inner [ Point
Fig. 6.Determination Threading/Breaking point 5.4 Job File In order to execute the erosion of a workpiece the machine needs the job file. This file is used to call the three files, position, NC and technology file.
6. HIGH-PRECISION TAPERING In many cutting dies, the taper is only included to allow guidance and ejection of the cut part (slug) away from the machine. A medium level of taper accuracy may be often enough to ensure this function. However, a number of applications require precise tapers. These include, dies for powder materials, dies having small taper angle, fine-blanking dies, resharpening of chipped tools and milling cutters, and electrode geometries for die-sinking EDM of die and mold cavities. For thin sheet material, higher taper accuracy is needed to ensure that the cut parts are not bent during ejection. To obtain high accuracy on the cut parts, it is necessary for the wire to have the best possible mechanical properties when it enters the machining area [ 13]. The wire must be selected and treated in such a way that it does not plastically deform during its passage through the wire drive system and through the guides [ 14,15]. 7. SIMULATION CHECK Simulation aims to make the best use of resources by helping the user to decide what is needed, and where is the best position to locate it. The proposed system uses the simulation check to save hours of machining through few minutes of verifying check, and to save the machine time for the actual cutting. 8. PROPOSED SYSTEM APPLICATION 8.1 manufacturing of Compound Dies The proposed system is able to machine compound dies in one path, instead of two programs (one for blanking punch and the other for piercing die), as shown in Fig.7. (Refer to appendix
A for the list ofoutputfiles)
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Current Advances in Mechanical Design and Production, MDP-7
Fig. 7.Manufacturing of Compound Dies 8.2 Prototype Production Prototype production is one of the main applications of the wire-EDM. The required number of the prototypes in this example is five. These are fired by rivets and machined by stack machining, Fig. 8. (Refer to appendix B for the list of output files) 8.3 Manufacturing of Form Tools Manufacturing of turning form tools is very important, particularly for automatic and semiautomatic lathes in mass production, using a suitable fixture equipped to the machine to produce clearance, Fig 9.
9. CONCLUSION The application affirms that the time saved through the proposed CAM system starting from the initial calculation to the generation of the NC file is about 50 % or more compared with EasyCut CAM software. By using the proposed CAM system, sequencing the wire cutting motion in a way to minimize the machining time, can be accurately and simply performed. Suitable wire cutting techniques are selected based on the geometry specifications. The type of polygon (punch/die) is selected by an automatic method based on the fixation stability and the geometry specification. The location for the threading and the breaking point of the wire is selected by an automatic method to attain the optimum loeation._Adding taper angle, trim cut, and wire-offset are achieved by an automatic method. Programming time has been reduced using postprocessor. Making NC programs from minimal information. The proposed CAM system has been used to produce several practical workpieees at private sector in the 10th of Ramadan City / Egypt. REFERENCES Herbert, Y.W., "Manufacturing Process", pp. 303-323 Prentice-Hall, Inc. Englewood Cliffs, New Jersey, USA, (1997).
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2. Jain, K.S., "Manufacturing Engineering Technology", 2nd Ed., Addison Wesley Publishing Co. Inc., (1991). 3. Fuller, J.E., "Electrical Discharge Machining", Vol. 16, Metal Handbook, 9th Ed., (1988). 4. User Manual, "AutoLISP", for AutoCAD R. 10-14, USA, (1996). 5. User Manual, "A-CAD R. 13 for Microsoft Win 4.0 (x86)", Autodesk, Inc., USA, (1996). 6. Beltrami, I. and et al, "Simplified Post Processor for Wire EDM", Vol. 58 No. 4, pp. 385389, Journal of Materials Processing Technology, AGIE Ltd., Losone, Switzerland, April 15th (1996). 7. AGIECUT Sprint Operating Manual; Published by AGIE Industrial Electronics Ltd., Losone, Locamo, Switzerland, April (1995). 8. "EasyCut CAD/CAM System" Software Package, AGIE Industrial Electronics Ltd., Losone, Locamo, Switzerland, Feb. (1991). 9. Noaker, P., "EDM Gets with the Program", Manufacturing Engineering Vol. 110, No. 6, pp. 55-57, June (1993). 10. Kozak, J., and etal, "Material Removal In WEDM of PCD Blanks" Journal of Engineering For Industry ASME Vol. 116, No. 3, See. B, pp. 363-369, August (1994). 11. Luo, Y.F., and Chen, C.G., "Effect of a Pulsed Electromagnetic Field on the Surface Roughness in Super Finishing EDM", Precision Engineering Vol. 12, No. 2, pp. 97-100, April (1990). 12. EI-Midany T.T., Elkeran A.; and Radwan H.T., "CNC Automatic Programming System for Wire-EDM"; The Twelfth International Conference on Industrial and Engineering Applications of Artificial Intelligence and Expert System (lEA / AIE-99); Cairo; Egypt; May 31 st-June 3 rd (1999). 13. David, T., "EDM Software", Manufac. Engineering Vol. 112, No. 6, pp. 57, June (1995). 14. Bally, F. and Buemi, C., "High Precision Tapering Made Easy", American Machinist Vol. 140, pp. 33-35, June (1996). 15. Michael, T., "Leasing Lets EDM Machines Go To College", American Machinist Vol. 138, July (1994). APPENDIX A. List of proposed system output files for compound dies A.I Sample List of NC file (examplel.cnc) $Proposed system Ver 1.01 cod(iso) ldr(6) lit(%) N001 G01 X+056678 Y+ 104772 M22 :00002 M63 . o ,
N043 M02 A.2 Sample List of Technology data file (example l.ted) % I 13 12 11 00 00 00 14 (AEGDIR_3) TOO 15 05 19 002 02 02 02 002 002 02 02 . . .
M02 A.3 Sample List of position file (example l.pof) cod( POSITION File )
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Current Advances in Mechanical Design and Production, MDP-7
lit(!) Pos O1 PAX+056678 PaY+I05348 !
$$ Position Without Wire
= 168.0207
A.4 Sample List of job file (example l.job) Dimensions COMPLETE MM POF CAProgram Files~Vlicrosoft Visual Basic\pof~xample(1).pof TEC C:LProgram Files~4icrosofl Visual Basic\ted~Example(l).ted CNC sav RETRY FROM : 9 TIMES WITH TOO TECH SKIP STP IF NOT THREAD STOP CNC CAProgram FilesLMicrosoR Visual Basic\CNC\Example(l).cnc EXEC STP EXEC WIE EXEC TO1 EXEC SO1 FROM 001 TO 999 FIN
APPENDIX B. List of proposed system output files for prototype production B.I Sample List of NC file (example2.cnc) $Proposed system Ver 1.01 cod(iso) ldr(6) lit(%) N001 GO1 X+028000 Y-015074 M22 N043 M02 B.2 Sample List of Technology data file (example2.ted) % I 13 12 11 00 00 01 14 (ASGDIR_7) D03-178 D04-178 M02 B.3 Sample List of position file (example2.pof) cod( POSITION File ) lit(!) Pos 02 PAX-001050 !
$$ Position Without Wire
-- 127.6244
B.4 Sample List of job file (example2.job) Dimensions COMPLETE MM POF CAProgram Files~d~1icrosoR Visual Basic\pof~Example(2).pof Fin
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, Februat3, 15-17, 2000
455
ON THE PREDICTION OF SURFACE ROUGHNESS IN TURNING USING ARTIFICIAL NEURAL NETWORKS
El-Sonbaty, I." and Megahed, A.A.*" * Assoc. Prof. and ** Graduate Student Mechanical Design and production Engineering Department, Faculty of Engineering, Zagazig University, Zagazig, Egypt.
ABSTRACT The aim of the present work is to use the technique of artificial neural networks, with backpropagation routine, for the prediction of surface roughness of workpieces produced by turning process. To provide the data required to train the networks, specimens were turned at different cutting conditions using tools having different pre-machining tool-flank wear. The vibration level during machining, and the after-machining tool-flank wear were measured and the surface roughness parameters of turned specimens were assessed. Several neural networks were trained with changing both of the network structure and the number of training samples. The best of these networks were selected to be used for the prediction of surface roughness parameters, vibration level and tool wear. From the results obtained it may be concluded that the developed neural networks permit acceptable outputs for the prediction of the investigated surface roughness parameters, vibration level and tool-flank wear. Also, based on these networks, an integrated system for continuous prediction of surface roughness was developed, and a computer program was designed for selecting the appropriate machining conditions required to achieve desired values of surface roughness parameters. KEYWORDS Surface Roughness, Vibration Level, Tool-flank Wear, Turning, and Neural Network. I. INTRODUCTION Surface roughness is a factor of great importance in the evaluation of workshop production and considerable attention is now being focused on its measurement as a means of quality control [l]. Several attempts were performed to predict the surface roughness in turning operation by using either the mathematical models or the artificial neural network technique. As an example of the mathematical models, Mital and Mehta [2] generated surface finish prediction models. All proposed prediction models take into account cutting speed, feed, cutting tool nose radius, and their interactions. Fei and Jawahir [3] presented a methodology for predicting surface roughness in turning through a fuzzy knowledge-based system for a given set of cutting conditions, work material, tool insert type and tool geometries (including chip groove geometries). EI-Baradie [4] developed a mathematical model which uses cutting speed, feed rate and nose radius as inputs for predicting surface roughness of machined components in turning gray cast iron. Arsecularatne, Fowle and Mathew [5] described an
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Current Advances In Mechanical Design and Production, MDP- 7
analytical approach to predict the chip flow direction, cutting forces and surface roughness under finishing conditions. The major weakness of the mathematical model systems lies in the random errors contained in the machining response data. It has usually been found that metal cutting data are susceptible to an appreciable amount of scatter. This is due to the non-homogeneity of the material being cut and its effect on the measurement of the machining response data, as well as due to the errors in the measuring technique itself [4]. So, another method was used based on using the artificial neural network technique. The primary advantage of the artificial neural networks approach over the mathematical models is that the neural networks can model very complex relationships and yet are relatively insensitive to noise included in the machining data [6]. Chryssolouris and Guillot [6] developed neural networks to predict surface roughness (Re parameter), spindle power, noise level, and chip merit mark in orthogonal turning operation using process parameters (feed rate, cutting speed, rake angle, width of cut, and flank wear) as inputs. Matsumura, et al [7] proposed an autonomous turning operation planning, to minimize machining cost by predicting tool-flank wear analytically and surface roughness with a neural network. Darenfed, Miao and Boudreau [8] developed a multi-layered learning network for predicting surface roughness using cutting parameters (cutting speed, feed, depth of cut, and nose radius) as inputs and one surface roughness parameter as an output, as a performance criterion of learning networks. Aboul-Nour, et al [9] used the artificial neural networks to predict the surface roughness (1~, Rt and Rz parameters) of products as a representator for the machining performance in milling operation. Fang and Yao [10] developed neural networks for on-line prediction of machining performance (characterized by surface roughness, cutting forces, chip breakability, and dimensional deviation due to tool wear) during tool wear progression using a single sensor. The tool wear influences the surface roughness of the workpiece and the value of surface roughness is one of the main parameters used to establish the moment to change the tool in finish turning [11]. Tool-flank wear was predicted either analytically [12-14], or with the neural networks [9,15]. The present work is an approach to predict the surface roughness, in turning operation, using the feed-forward artificial neural networks (ANN) technique trained with the backpropagation routine. The inputs to the neural network of surface roughness prediction are cutting speed, feed, depth of cut, nose radius, pre-tool-flank wear, and vibration level. The tool-flank wear and the vibration level are predicted using other neural networks. An integrated system, comprising the three developed neural nets, is aimed to be made for continuous prediction of surface roughness. Also, a computer program is designed for the selection of the most suitable machining conditions, which are required to achieve a desired value of surface roughness parameters. 2. EXPERIMENTAL WORK In order to train the neural network, input-output pairs must be collected. In the present work input-output pairs required to train the developed neural networks are collected through the experimental work. The results of this experimental work have been used to train, evaluate, and test the developed neural networks. Sixty workpieces were prepared, each with 80 ram. diameter and 260 mm. length, from steel bars having the composition; 0.39% C, 1.5% Mn, 0.12% S, 0.025% P, 0.21% Si and 0.07% Cu, with a hardness value of 170 HB. The used length of the workpiece in machining and
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measurements (effective length) was 200 ram. and the remainder 60 ram. was used for clamping. In the machining process, each effective length was divided into five equal regions (40 mm. each) which represent five different samples. The experimental results were gained by dry finish turning of the sixty workpieces (i.e. 300 samples) using coated cemented carbide throw-away square tips (grade TNI50) with three different nose radii (r = 0.4, 0.8 and 1.2 ram.). The machining was implemented on a Colchester 7.5 HP lathe. Four different cutting speeds (V = 130, 180, 250 and 350 m/min.), four feeds (f = 0.06, 0.08, 0.10 and 0.12 mm/rev.) and two depths of cut (d = 0.5 and 0.8 ram.) were selected as machining conditions. Each workpiece was machined with fresh (new) tip. For each sample the maximum values of tool-flank wear, before machining (Vbt) and after machining (Vb2), were determined using an optical microscope (type 2158) with vernier resolution of 0.01 mm. During machining of each sample the vibration level of the cutting tool (V.L.) was measured using an integrated vibration meter (type 2513). After machining each workpiece, the surface roughness of each sample was assessed by using "Taylor Hobson Surtronic 3+" surface roughness measuring instrument with its software (version 1992) taking a cut-off value of 0.8 mm. and a traversing length of 4 mm. Thus, after the finishing of the experimental work, the gained results were; 300 values for vibration level, 300 values for tool-flank wear (for each of Vbl and Vb2), and 300 values for each of seven surface roughness parameters (six vertical parameters; Ra, Rq, Rt, R3z, Rp, Rv and one horizontal parameter S). For the purpose of testing the developed neural networks, other four samples were machined with machining conditions which are different from that on which the neural networks were trained. Beside these four samples, six samples were selected from the above 300 samples to permit test sets used to check the ability of the developed neural networks to be considered as a generalized one. 3. RESULTS AND DISCUSSION 3.1. The Developed Neural Networks In order to develop the required neural networks, from all of the samples that were machined in the experimental work, samples were selected randomly as; 240 samples for training patterns, 54 samples for evaluation patterns (which were used as a training stoppage criterion [16]), and 10 samples were used as test patterns. Table 1 shows a sample of the results of the experimental work that were used to train, evaluate and test the developed neural networks. Actually, each row in the Table presents the results of one sample of a machined workpiece. Table 1. Sample of the results of the experimental work i
Machining conditions No. l 2 3 4 5 6 7 8 9 10
V --
f
d
r
m/min,
mm~-
mm
350 250 130 180 130 250 180 350 350 130
():'06 0.08 0.10 0.12 0.10 0.06 0.08 0.12 0.06 0.06
0.~ 0,4 0.5 0.4 0.5 0.4 0.5 0.8 0.5 0.8 0.5 1.2 0.5 1.2 0,5 1.2 0.8 0.4 0.8 0.8
mm
q
Surface roughness parameters 0tm.)
Vbl
V.L.
Vb2
ram.
dB
ram.
Ra
0.09 0.13 0.12 0.14 0.09 0.11 0.09 0.11 0.13 0.20
0.9'7 1.40 !.79 0.77 1.69 0.73 0.77 0.7! 0.99 0.99
Q.O0 0.11 0.11 0.14 0.00 0.11 0.09 0.10 0.11 0.18
109,0 ll0.0 108.5 109.0 109.0 109.0 108.0 109.5 !11.0 111.5
Rq
Rt
1.21 .6,70 1.71 10.1 2.15 10.4 1.03 5.50 2.10 12.9 0.95 7.60 1.00 7.20 0.90 5.10 1.23 6.20 1.30 9..00
Raz
P',r
4.50 3.90 5.40 5.50 7.60 ~6.20 4.10 ~3.30 6.80 6.40 3.20 4.80 3.70 3.20 3.10 .3"10 4.90 3.60 4.30 3.50
Rv
S
2.80 27.58 4.60 20.64 4120 22.09 2.20 29.22 6.60 26.35 2.80 28.18 4.00 21.10 2.00 26.65 2.60 20.67 5.50 27.24
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Fig. 1 shows the schematic diagram of the neural net for surface roughness prediction. The output is the surface roughness which is assessed by seven different parameters; Ra, Rq, Rt, R3z, Rp, Rv and S. For each parameter, ten neural networks were developed using two network structures (5 and 10 hidden units) with five different numbers of training samples (70, 100, 150, 200 and 240). Fig. 2 illustrates the neural net for vibration level prediction. Fifteen neural networks were developed using three network structures (5, 10 and 15 hidden units) with five different numbers of training samples (70, 100, 150, 200 and 240). Fig. 3 shows the neural net for tool-flank wear prediction. Fifteen neural networks were developed using three network structures (5, 10 and 15 hidden units) with five different numbers of training samples (70, 100, 150, 200 and 240). From the all developed neural networks, nine networks were selected as the best nets developed for prediction of 1~, Rq, Rt, R3z, Rp, Rv, S, V.L. and Vb2. The criterion of selecting the best net is the minimum root mean square of the differences between the measured and the predicted values obtained from the net (Es), applied for the test data sets. Table 2 shows the best neural network of every predicted output, each associated with the number of training samples, the number of evaluation sets, the number of training iterations and the test sets error (Es). From this Table, with respect to surface roughness parameters the prediction of which is the main goal of this work, it is found that the prediction of R, gives the smallest test sets error (0.023 ttm.). So, the P~ parameter can be considered as the best parameter for surface roughness prediction in the present work. Figs. 4 to 12 present the measured and predicted values of the nine best developed neural networks. From these figures it can be seen that, for the most of the developed nets, there is a small error between the measured and predicted values for both training and evaluation sets, while for the test sets there is a relatively larger error (from-54.59% to 57.52%). For the surface roughness patterns, these errors may be attributed to the randomness in the obtained results which usually comprise the random portion of the surface roughness assessment. However, it can be said that the best developed neural networks permit acceptable results for the prediction of the studied outputs irrespective of the existence of some errors which can be minimized by increasing the number of inputs to the networks.
Input Information
Input Layer Units
Hidden Layer Units I
Cutting speed Feed of cut Nose radius
Output Layer Unit
Output Information Workpiece surface roughness
Pre-tool-flank
Fig.l. Schematic diagram of the neural network for surface roughness prediction.
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Input Information
Input Layer Units
Hidden Layer Units
Cutting speed
Output Layer Unit
Feed
Output Information
Depth of cut [
Nose radius Pre-tool-flank wear
Fig.2. Schematic diagram of the neural network for vibration level prediction.
Input Information
Input Layer Units
Hidden Layer Units
I
Cutting speed
Feed
O,
._...._.__._.__..]_~~ Depth of cut [
Layer Unit
Output Information After-machining tool-flank wear
Nose radius
[ Machining time
[ Pre-tool-flank wear Vibration level
Fig.3. Schematic diagram of the neural network for tool-flank wear prediction. Table 2. The best developed neural networks
Output
Net structure
R. R~ R3z ..
R~ S V.L. Vb2
6-5-i 6-5-1 6-10-1 6-5'1 6-5-1 6-5-1 6-5-1 5-15-1 7-5-1 .,
,
. .
Number of training samples 1o0
Number of evaluation sets 25
~oo
25
150 150 100 100 70 240 70
35 35 25 25 15 54 15 ,..
Number of training iterations 8070 8620 1759 87790 133159 ..... 72119 1970 11538 98895 i
Test sets error (E,)
ii
. ,
o o23 0.04.,6pm. 0.,141 pm. 0.065 pm. 0.119 pm. 0.090 pm. 0.053 pm. 0.024 dB 0.104 mm. i
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Current Advances in Mechanical Design and Production, MDP-7
E :~3
[] + 9 --
E 3
Training data set Evaluation data Test data set O~timum relation
+D
43
~2
~2
~
t~
0
1 2 3 Measured values (~ m)
9
+
4
0
Fig. 4. M e a s u r e d vs P r e d i c t e d values of 1~.
1 2 3 Measured values (ttm.)
4
Fig. 5. M e a s u r e d vs P r e d i c t e d values of Rq.
O Training data set + Evaluation data :
~, 20
-7,.I E
9 Test data set - - Optimum rdatiou
~
12
.~
8
~
4
m
r~ o
.o
0
~ 0
4 8 12 16 20 Measured values (pm)
0
24
Fig. 6. M e a s u r e d vs P r e d i c t e d values of Rt.
2 4 6 8 10 Measured values (ttm.)
Fig. 7. M e a s u r e d vs P r e d i c t e d values of R3z.
12
12
~'. 10 E ::l.
~.10 E ::L
+ Evaluation data se 9 Test data set ~-- O~timum relation
8 ~
6
~
4
>
6
~
4
o,~
2
12
w o
~% FO
2
0 0
2 4 6 8 10 Measured values (tim.)
12
Fig. 8. M e a s u r e d vs P r e d i c t e d values of Rp.
0
2 4 6 8 10 Measured values (~tm.)
Fig. 9. M e a s u r e d vs P r e d i c t e d values of Rv.
12
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Current Advances in Mechanical Design and Production, MDP-7
70 ~. 60 E "-"50
116
[] Training data set + Evaluation data 9 Test data set - - O m i m u m relation
-uI
112
~ 40 ID
,
~ 30
+
~9 108
o-,
~., 20 10 10
20 30 40 50 60 Measured values (lam.)
70
Fig. 10. Measured vs Predicted values of S.
0.3 E E o'J
q~
104 104
108 112 Measured values (dB)
116
Fig. 11. Measured vs Predicted values of V.L.
El Training data set 9 + Evaluation data set 9 Test data set - - O o t i m u m relation 9
0.2
>.
.u
0.1
0.0 0.0
0.1 0.2 Measured values (ram.)
0.3
Fig. 12. Measured vs Predicted values of Vbz. 3.2. The Continuous Prediction of Surface Roughness With the aid of the best obtained developed neural networks the surface roughness, the toolflank wear, and the vibration level, resulting from machining of a workpiece at certain machining conditions, can be predicted. Each one of the outputs of these networks may be obtained separately by preparing the inputs for the networks and getting the predicted results (direct prediction). One of the inputs of both the surface roughness and tool-flank wear networks is the vibration level. Therefore, before using both of these networks, the vibration level should be predicted. If the same cutting tool will be used to perform another operation, after the operation under study, the after-machining tool-flank wear must be predicted to complete the inputs for the prediction of the surface roughness of that operation. Therefore, the integration between the networks of the prediction of vibration level, surface roughness and tool-flank wear is deemed to be urgent in order to establish a system for continuous prediction of surface roughness. Fig. 13 shows the flowchart of such integrated system which is capable to predict the surface roughness continuously without the need to measure the vibration level or the tool-flank wear.
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Start ] 1~...
Yes
.
.
.
Pre-tool-flank wear=0
Enter the pre- / tool-flank wear "-T
.
h.~.
~..
[ Choose the machining conditions for finish turning ] Get the vibration level during the operation' (using the neural net of Fig. ,2.) Get the surface roughness (using the neural net of Fig. 1.) Get the alter-machining tool-flank wear (u (using the neural net of Fig. 3.) .
/
.
.
.
Print the surface / roughness
Yes
No
Yes
[ Set the pre-tool-flank wear--Vb2 I ,,
]
Fig. 13. Flowchart of integrated system for continuous prediction of surface roughness. A comparison between the direct and continuous systems for predicting the surface roughness parameters was made using the results of 30 samples selected randomly from the samples of the experimental work. For each case the error, which is the difference between predicted (obtained) value and the measured (nominal) values, was determined. Then, the mean value and the standard deviation (S.D.) for the samples were estimated and presented in Table 3. From this Table, it can be noticed that most of the mean values of errors for continuous prediction system are smaller than that of direct prediction system, and most of the values of standard deviation of errors are higher for continuous prediction. This may be attributed to the accumulation of the dispersed errors in the pre-predicted values of vibration level and toolflank wear. It is clear that, the continuous prediction, however, can offer a reasonable mean for prediction of surface roughness of any number of continuous machining operations without the need to measure the vibration level or the tool-flank wear.
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Table 3. Comparison between the errors resulting from direct and continuous predictions of surface roughness parameters
Roughness Parameter
!
R~
........
P~ R3z
P~ Rv
Direct prediction Error =predicted-measured Mean (~tm.) s. D. (jam.) 0.109 0.284 0.038 0.289 -I.105 2.716 0.126 1.046 -0.390 1.609 -0.207 1.004 0.841 4.22 i ,
. . . .
i
Continuous prediction' Error--predicted-measured Mean (~m.) ,S.D. (gm.) 0.105 0.297 0.030 0.293 -I.033 2.746 0.169 1.150 -0.316 !.654 -0.203 1.042 1.254 4.211 ....
i
3.3. Selection of the Appropriate Machining Conditions After the neural networks of surface roughness prediction have been constructed, it is desirable to perform the reverse process, i.e. selecting the appropriate machining conditions (cutting speed, feed, depth of cut and tool-nose radius) that lead to a desired value of workpiece surface roughness. Based on the best developed neural networks of the different outputs, the steps of the selection process are as follows (Fig. 14): - Enter the workpiece length, workpiece diameter, pre-tool-flank wear and the desired surface roughness parameter with its value (R). - Choose the machining conditions that are suitable for finish turning process on the used machine tool (from machining literature) to get the input patterns (N). - Present these chosen machining conditions and the pr-tool flank wear as inputs to the vibration level neural network, and get the vibration level. - Take the value of the predicted vibration level and feed it as the sixth input to the neural network for surface roughness prediction, and get the workpiece surface roughness (S.R.). - Determine the machining time and feed it to the neural network of tool-flank wear prediction, and get the after-machining tool-flank wear. - Choose the patterns that fulfill the condition; S.R. _ 0.15 was commonly used in finite element calculations for sheet metal forming [9,10,15]. It should be emphasized that the coefficient of friction obtained by the above mentioned inverse approach is an average generalized coefficient of friction, that describes the overall process. In fact, there are experimental evidences I16-17], that the value of ~t changes from one deformation zone to the other and could be affected by the strain, strain rate and contact pressure which change during the process. careful analysis revealed that there is no significant differences between the predicted drawing forces when using one layer or two layers. Hence, one layer of elements is suitable to model the blank properly. No much difference in the predicted punch force when either 40 element or 80 elements are used. However, the 40 element model shows some oscillations towards the end of stroke. Nevertheless, the trend of this model seems to be correct, as seen from Fig. 5. Such oscillations are usually acceptable in finite element analysis [15]. Thus to keep the CPU time and hard disk storage within practical limits, it was decided to utilize the one layer, 40 elements CAX8 element model throughout the present study.
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472
Type
of
[
etements
~
I
[
1 ...... OD(4I 1
CAX4
C,4X4FI
FrlcUon coefficient:
Num~oer o~: e t e m e n t s
Numioe, o , c l a y e r s
|
1
o,1
]
[
1
'"~ l
~
i
~ "
1
r---
1
STY.' St37 ....
C.AX8
;
4e el t , m ~ t
l
I_ F.~ct~e c D ~ t
~
Lt lU~ ~
Figure 3 THE PLAN OF THE MODEL VARIABLES ADJUSTING RUNS.
45
40
~
3o ~
"~ 25 ~7,
15
XSCrNii89,BHF = 90.8 kN
~.~: 0~~;--,~ ~ ~ , , ,
10
Recommended value 0
~,
-5 -10 O.(X)
i
i
i
i
i
0.05
i
0.10
i
i
i
i
I
0.15
i
I
0.20
Coefficient of friction, g Figure 4 ADJUSTING THE COEFFICIENT OF FRICTION.
i
0.25
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473
4. VALIDATION OF THE MODEL To validate the finite element model and the simplified inverse technique developed in the previous sections complete deep drawing processes for hemispherical cups made of steel St37 and steel X5CrNi 189 are analyzed. Figure 5 indicates that the predicted punch force agrees very well with the experimental values [12] throughout the whole drawing process, and not just at maximum punch force (which is the criterion used for the model adjustment). Note that in the example of steel XSCrNi189 a blankholder force of 90.8 kN is used, which is different from the one used in the previous section. This suggests that ~t = 0.25 is also valid under this blankholder force. Figure 6 shows the effect of changing the BHF on the maximum drawing force. The agreement between the finite element prediction and experimental results is obvious. As shown in Fig. 7, more thinning is obtained at higher BHF. However for St37 steel the maximum thinning is predicted relatively nearer to the drawbead, whereas for X5CrNil89 steel the maximum thinning is observed near to the punch zone, Figs. 8 and 9. For the values of the BHF considered the thinning of XSCrNi189 steel seems to be less affected by the change in BHF in comparison with St37 steel, because of the higher n-value of X5CrNil89 steel. Good agreement between experimental and numerical predictions can be seen in Figs. 8 and 9. In fact, the finite element results, with the adjusted values of the variables, could represent the complexity of the strain distributions at different zones of the cups, which proves the validity of the simplified inverse technique developed in the study. 5. CONCLUSIONS A simplified inverse technique has been developed to adjust the finite element model parameters, namely number of elements, number of layers, element type and generalized coefficient of friction. In this technique, the minimized difference between the numerical and experimental maximum drawing force was obtained through an experimental/computer-aided optimization by simulation approach. The optimized model was validated by comparing the predicted drawing force at different punch stroke, and the strain distributions with those obtained experimentally under different blank holder forces. The simplified inverse technique has shown that the adjusted model with one layer of 40 elements of 8 noded axisymmetric solid element (CAX8) was good enough to simulate the blank during the cup forming process. The global generalized coefficient of friction has been found to be 0.25 and 0.22 for steel X5CrNi 189 and St37 respectively in case of conventionally lubricated blank and tool. It should be emphasized that the optimal values of the model parameters could be affected by the finite element formulation and the numerical techniques used to handle contact boundary conditions. However, the presented approach is general and the obtained results could be used as guidance.
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474
250
I
oa~
t~
i Mat. X5CrNi 189 2110 84 - B H F = 90.8 K N Coefficient of Friction 0.25
I
I00
i
I........... ,
!....
i "~,
L"-
I
__:
'~ . f l ~ ~ i
Trend,
i~
I
i
I
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i0
20
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I
9
30
,
--
Numerical
[
[
=,
~
~
~
E xpe,rimental [121 ~
9o
(%) ~
Relative Stroke, (YIY,,~)* 100 Figure 5 V A R I A T I O N O F T H E P U N C H F O R C E W I T H S T R O K E .
~Z
U, o
_
250
2o0
i
i
..
......
i
'
.,~
-_..~J~:-
"-'--
-
.... !
Mat. XSCrNilg9
"~.
150
100
Mat. S137
9
-
-
1~
-
"
.
r
0
II
-
6O
imental[ 121 Numr r i c a l I ' Expe,
. . . . . . . .
'
'
14
80 100 120 Blankholder Force, B H F , k N
F i g u r e 6 E F F E C T O F B H F O N TIIE M A X I M U M
DRAWING FORCE.
-0.15
|
,...1,_
0
E~
".
Mat. X5CrNi ! 89 - - - -
-0.20
-0.25 Mat. St37
'N)
-0.30 ~>
I
I
-0.35 60
@0
I 100
-
!
120
14
Blankholder Force, BHF, kN
Figure 7 EFFECT OF BHF ON THE MAXIMUM PREDICTED LOGARITHMIC T H I C K N E S S S T R A I N IN T H E D O M E A R E A .
Current Advances in Mechanical Design and Production, MDP-7
-
-
ll~i~l~illl
-
lUlilUllllII......IllUilU/ill / Blankholder force -- 69.3 kN [
~
I
/
1 ] ~oo,~ TI and D(t) > O. For more details of Equation (2) the reader is referred to Enright and Frangopal (1998). The time-variant reinforced concrete strength, Mp(t), can now be evaluated using the conventional design equations in [3].
and
a=(nAs fr)/~.85 fc b )
(4)
Note that As= xD(t)2/4. The reinforcing steel and the concrete strengths are fy and fc, respectively. The number of reinforcing bars is n. The effective depth and the width of the beam are d and b, respectively. For the current example, the values of different parameters were chosen as fy = 40 ksi, fc =3 ksi, d = 27 in. and b = 16. Using Equations 2 through 4 the random time-variant strength, Mp(t), can be estimated. Using the previously mentioned statistical values of rco,~ and 7'1 and a Monte Carlo simulation technique, different strength values for different reinforced concrete beams can be simulated. Thus, a discrete time-variant reinforced concrete strength, x# can be evaluated from the Monte Carlo simulation of the continuous strength Mp(t). Figure 1 shows the time-variant strength, xo -- M p(t) for different reinforced concrete units. It is assumed that the strength/degradation data of 20 beams are monitored for 40 years. Note that these data are assumed to be independent as discussed in Ettouney and Elsayed [ 1999]. The Maximum Likelihood method was utilized to estimate the parameters of the model as given by Eq(1 ) wI
L(y,a,b,t)=I" m
[ ( Y )., ,-i b exp( - at ~)
n~
1-I I-I
,=~ j=~
u
r-I exp(
xo.
x~ b e x ,,( .. -at5 )
(5)
-
where m is the number of years, n, is the total number of degradation data in a year i and x# is the strength of unitj in year i. Taking the logarithm of Eq.(5) we obtain: m
•
m
i=l
t=1
i=!
lnL=~n, lny-~'n, lnb+~'n,at,+ (6) (y' - 1)In x o i=1
j=l
"=
x~ j-t b e x p ( - a t , )
Equating the partial derivatives of Eq.(6) with respect to 7', a and b to zeros and solving the resulting equations using a modified Powell hybrid algorithm and a finite difference approximation to the Jacobian yields: a=0.12, b-11346x 107 and y=1.49.
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559
Substituting the values of q(t) and g(t) into Eq.(l), the reliability function can then be determined as _x r
Rx(t) = P ( X > x; t ) = e x P [ b e x p ( _ a t ) | _
or
R~(t) = exp[
X 1-49
11346598exp(-0.12t) ]
(7)
The reliability for different threshold values of the strength is shown in Fig. 2. The time to failure for threshold values of 4800, 4000, 3500, 3000, and 2500 are 25.04, 27.25, 28.88, 30.76, and 33.0 years The mean residual life of the beams can be obtained as follows: The mean residual life (MRL) function e(t) of a non-negative random variable T with reliability function Rx(t) is
e(t) = E(T - t / T > t) = R~ (t)-' ~R x (u)du t
The mean residual lives for different degradation threshold values are shown in Fig.s 3. REFERENCES: 1. Blake, I.F., and Lindsey W.C., "Level Crossing Problems for Random Processes," IEEE Transactions on Information Technology, Vol. IT-19, No.3, pp. 295-314, (1973). 2. Carey, M.B. and Koenig R.H., "Reliability Assessment Based on Accelerated Degradation: A Case Study," IEEE Transactions on reliability, Vol. 40, No.5, pp. 499506, (1991). 3. Chick, S. E. and Mendel, M.B., "An Engineering Basis for Statistical Lifetime Models with an Application to Tribology," IEEE Transactions on Reliability, Vol. 45, No. 2, pp. 208-214, (1996). 4. Chuang, S. L., Ishibashi, A., Kijima, S., Nakayama, N., Ukita M., and Taniguchi S., "Kinetic Model for Degradation of Light Emitting Diodes," IEEE Journal of Quantum Electronics, Vol. 33, No. 6, pp. 970-979, (1997). 5. Ditlevsen, O. and Madsen, H.O., Structural Reliability Methods, John Wiley, (1996). 6. Domine M., "Moments of the first passage time of a Wiener process with drift between two elastic barriers," J. Appl. Prob., Vol. 32, pp. 1007-l 013, (1995). 7. Domine M., "First passage time distribution of a Wiener process with drift concerning two elastic barriers," J. Appl. Prob., Vol. 33, pp. 164-175, (1996). 8. Doksum, K. A. and Hoyland, A., "Models for Variable-Stress Accelerated Life Testing Experiments Based on Wiener Processes and the Inverse Gaussian Distribution," Technometrics, Vol. 34, No. l, pp. 74-82, (1992). 9. Eghbali, G., "Reliability Estimate Using Accelerated Degradation Data," Ph.D. Dissertation, Department of Industrial Engineering, Rutgers University, Piscataway, NJ, (1999). 10. Elsayed, E. A., Reliability Engineering, Addison-Wesley, (1996). 1I. Enright, M. P. and Frangopol, D. M., "Probabilistic Analysis of Resistance Degradation of Reinforced Concrete Bridge Beams Under Corrosion," Engineering Structures, Vol. 20 No. I l, pp. 960-971, (1998).
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12. Ettouney, M., Elsayed, E. A., "Reliability Estimation of Degraded Structural Components Subject to Corrosion," 5~ ISSAT International Conference on Reliability and Quality in Design, August 11-13, Las Vegas, Nevada, pp. 291-295, (1999). 13. Feinberg, A.A. and Widom, A., "Connecting Parametric Aging to Catastrophic Failure Through Thermodynamics," IEEE Transactions on Reliability, Vol. 45, No. 1, pp. 28-33, (1996). 14. Ioannides, E., Beghini E., Bergling G., Goodall J. and Jacobson B., "Cleanliness and its importance for bearing performance," Ball Bearing Jouma1242, pp. 8-15, (1993). 15. Lu, J. C. and Meeker W. Q., "Using Degradation Measures to Estimate a Time-to-Failure Distribution," Technometrics, Vol. 35, No.2, pp. 161 - 173, (1993). 16. Lu, J., Park J. and Yang Q., "Statistical Inference of a Time-to-Failure Distribution Derived From Linear Degradation Data," Technometrics, Vol. 39, No.4, pp. 391-400, (1997). 17. Meeker, W. Q. and Hamada, M., "Statistical Tools for the Rapid Development & Evaluation of High-Reliability Products," IEEE Transactions on reliability, Vol. 44, No.2, pp. 187-198, (1995). 18. Meeker, W.Q. and Escobar L. A. and Lu C.J., "Accelerated Degradation Tests: Modeling and Analysis," Technometrics, Vol. 40, No. 2, pp. 89-99, (1998). 19. Melchers, R. E., Structural Reliability, Ellis Horwood Limited, (1987). 20. Nelson, W., "Analysis of performance-degradation Data from Accelerated Tests," IEEE Transactions on reliability, Vol. R-30, No.2, pp. 149-155, (1981). 21. Pieper, V., Domine, M., Kurth, P., "Level Crossing Problem and Drift Reliability," Mathematical Methods of Operation Research, Vol. 45, 355-375, (I 997). 22. Qureshi, F. S. and Sheikh A. K., "A Probabilistic Characterization of Adhesive wear in Metals," IEEE Transactions on Reliability, Vol. 46, No. 1, pp. 38-43, (1997). 23. Stock, D., Vesely W. E. and Amanta P.K., "Development and Application of Degradation Modeling to Define Maintenance Practices," NUREG/CR-5967, (1994). 24. Tydeman, M. S. and Kirkwood T. B. L., "Design and analysis of accelerated degradation tests for the stability of biological standards I. Properties of maximum likelihood estimators," Journal of Biological Standardization No. 12, pp. 195-206, (1984). 25. Tseng, S. T., Hamada M. and Chiao C. H., "Using Degradation Data to Improve Fluorescent Lamp Reliability," Journal of Quality Technology, Vol. 27, No. 4, pp. 363369, (1995). 26. Yang, K., "Robust Design and Reliability Engineering, An Integrated Approach," Proceedings of Ford 2000 Conference on Integration of Quality Methods, Reliability, and Robust Design, Nov. 17-18, 1994, Fairlane Club, Dearborn, Michigan, pp. 141-185, (1994).
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s=2500
._
0,8
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-
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.
-, ,, , \
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/
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/
s=3500 s=4000 s=4800
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lib
0 0
10
20
30
40
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Fig 2. Reliability vs. time for different strengths
50
60
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Mean Residual Life
I ~=4~~
40 35
9
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40
50
Current Advances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
563
FEATURE RECOGNITION ALGORITHM FOR PROCESS SELECTION McCormack, A.D. and Ibrahim, R.N.
Dept of Mechanical Engineering, Monash University GPO Box 197 Caulfield East, 3149 Vie, Australia
ABSTRACT The function modeller presented in this paper was designed to extract product features for automatic process selection in an independent application. This modeller works on principles of Boundary Representation (B-Rep), specifically Euler formula and Attributed Adjacency Graphs and Constructive/De-Constructive Solid Geometry (CSG) modelling. These models are reviewed and presented as introductory material to this paper. The product CAD data that is used within these principles is obtained from a neutral STEP file format, ensuring its compatibility amongst existing CAD programs. A recursive checking method, utilising volume decomposition of a CSG model, is employed to ensure that a valid product model is developed. KEYWORDS Feature Recognition; Process Planning; Attributed Adjacency; Feature Extraction 1. INTRODUCTION Recent progresses in CADCAM research, particularly in three dimensional product representation, have created new opportunities for the previously difficult task of automatic generation of manufacturing process plans. The generative approach to process planning is an example of these relatively new manufacturing philosophies. This approach allows for new process plans to be generated for each component using decision logic and precoded algorithms. To enable automatic selection of process plans for an individual product, the Computer Aided Process Planning (CAPP) system must be able to extract detailed information from the product description. These components include geometry, dimensions, tolerances and surface conditions. Included in CAPP systems is the solid modeller, which consists of a interface for the extraction of shape information, data structures to represent the CAD file, and an algorithm to generate the shape information for the specified application. The data representation scheme is an integral part of the solid modeller and there are generally two recognised models [l]; boundary representation (B-REP) and constructive solid geometry (CSG). The boundary representation scheme is based on the concept that a solid object can be considered bounded by a series of faces. These faces are then made up of a series of edges, which consists of a series of vertices or points. This representation model basically breaks down the solid model in a series of cartesian points and their connectivity with each other.
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The relationships between the individual faces of the product are stored and make up an integral part of the boundary representation model. The topological relationships between the various features of a product can often directly determine the feasible process sequences [12]. There are typically four types of feature relations defined: intersections between protrusion, adjacency between protrusion or normal adjacency, adjacency otherwise, and intersection between protrusion or depression. Representing a product under the CSG method results in a collection of basic three dimensional solids, such as cubes or prisms. Their orientation inside a 3-D space is utilised to combine or subtract the primitive solids from each other to form a completed solid model. CSG works with a forward planning approach [6], but the steps can be worked in a reverse order. This requires the use of simulated stock work piece where basic three dimensional solids are removed.This process can be linked to a feature-based design technique known as volume-decomposition, where the solids removed correspond to a machine process. There are four major phases to a volume decomposition process [5]" 1. Determine the expected volume that will be removed. This should be seen as the raw stock minus the finished product. 2. Partitioning the removed volume into individual and basic three dimensional solids. 3. Combine adjacent cells for typical machining features. 4. Match machining features to typical operations. The algorithm being presented in this paper follows these steps when creating a secondary product model. The extraction of, the previously mentioned, machining features that are characteristic of a certain application is the key to an efficient process planning task. This can dictate the actual process selection and consequently the process sequencing, which is essentially the body of a process planning task. 2. REVIEW OF RELEVANT WORK There have been many documented cases of automated process planning systems in the past. The GT and CAPP system [2], utilises a group technology code to identify the features. Upon identification of the features, the processes are then selected and sequenced according to a typical algorithm of decisions, or rules. The disadvantages of a system based on a GT code are that the coding system must go into great detail, and involve a very lengthy searching process, to ensure the process plan will be valid for all products. Another notable example is the EXCAP system [7], which was designed to generate process plans for rotational components using both group-technology codes and specific generative language approaches. This was expanded further into the CADEXCAP system [8,9], which integrated EXCAP with CAD systems using a neutral file exchange system. Neutral file exchanges are extremely important in CAPP systems as they ensure compatibility amongst various CAD systems. Solid modeling is a concept that is not oRen considered along side rotational product modeling as this is often a two dimensional field but the CADEXCAP system utilises the product profile to extract features. It is now in three dimensional areas, which are not often required in rotational products, that the new solid modeling techniques benefit the process planner the most. The process selection algorithm being presented has been developed from a generative point of view, meaning that it creates a new process plan for each product. As was previously
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discussed, there are generally two recognised forms of product representation, B-Rep and CSG, and this algorithm seeks to create a solid model based on both techniques to utilise the advantages of both, while simultaneously compensating for their failings. It is important to note that the algorithm does not create two product models. The secondary CSG solid model is merely a way of verifying the proposed process plan. The main disadvantage of solid model constructed under CSG is there is no explicit information about faces, edges or vertices of the object, which makes it less suitable for application such as assembly. It is for this reason that the product model used in the algorithm being presented here is developed as a B-Rep model. A solid model also created under a boundary representation model can be easily recognised as unique if its geometric data and topological relationships are identical, therefore it is an excellent way of comparing models. 3. SELECTION ALGORITHM In this algorithm, data is obtained from a CAD file in STEP format via an interface. A B-Rep product solid model is developed to extract features using topological relationships and geometric data. Machining processes are linked to the features, via a machine dependent database and a CSG model, created using volume decomposition methods, tests the features as they are removed from a raw-stock model. The details of this algorithm are discussed below. This initial model that the algorithm creates is in the B-Rep form. It needs to read the topological and geometric data of the product from the CAD file and create a simple array for storing the data. As it is not possible for two different shaped objects to have the same topological relationships and geometry data, this is a very simple way of interpreting product data. When examining a product representation in B-Rep form, there will be various data arrays existing for each individual feature. For example, each face of the product will join another face at an edge on some point of the model. This relationship can be recorded in the form of a topological graph [1,3,10]. However, a more convenient fashion is a simple matrix with each face listing and showing the faces to which it is connected. If a search is done through the CAD file, we can compare the various topological matrices and geometric data to a database containing features typical to the application and machining environment. One important aspect of the geometrical data that will have a large influence on the recognition of features is the angles at which joining faces make with each other, also known as convexity or concavity. 4. MODIFIED ATTRIBUTED ADJACENCY Under a graph representation, or matrix structure, the face adjacency and relationships are stored, but the information regarding the type of relationship is also of great importance when feature recognition is required. The Attributed Adjacency Graph [13,14], which is also representable under matrix notation, is a way of storing the type of relationship. Often the relationship is given an attribute of 0, 1 or 9, which can represent data such as convex or concave relationship, parallel relationship or perpendicular relationships. The algorithm being presented in this paper utilises a modification on the Attributed Adjacency Graph/Matrix by further classifying the attribute of the relationship. For example, in one half of the matrix a convex relationship between two faces may be stored, but in its reflected element the actual angle is stored. Other data regarding the relationship may include noting that the interaction is between a cylindrical opening and a planar face. The data proves to be extremely useful when
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identifying features. An example would be the ability to immediately distinguish between a square slot, a dovetail slot, and a V slot. All of these slots would have otherwise been recognised as a sequence of three planar faces with concave relationships. Joshi and Chang [15] developed a heuristic for extracting sub-graphs/matrices from the Attributed Adjacency Matrix that is based on the statement: a face that is adjacent to all its neighbouring faces with a convex angle does not form part of a feature. This allows the features specific to the application to be extracted so that each individual machining operation can be identified. To separate an individual sub-graph/matrix from the modified attributed adjacency matrix we classify the end of a depression feature by the presence of a convex edge. The end of a protrusion feature is thus found via the presence of a concave edge. This extraction process is simplified by using the Euler equation [4] to determine whether the current feature is a passage or non-passage feature. The Euler formula validates all vertices, edges and faces against various criteria dependent on the class of the object. For example, in its simplest form the Euler equation determines if an object is valid, in that all faces are simply connected, and that is has no holes through it. This decision is shown in Figure 1, which depicts the decision flow of the algorithm.
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A passage feature is then given a genus rating [3] to determining the level of feature complexity. Genus, or genera, are other criteria with which the Euler equation must be compared to when dealing with non-simple objects, such as spheres or cylindrical through holes. Passage features are often in the form of holes, blind or open, so the process selection phase is quite simple. This genera data originates from the geometric data, which can indicate a relationship between a cylindrical opening and a planar face in a very simple manner. When non-passage features are found, they must typically be examined based on their
convexity or concavity, which can be determined from their individual topological relationships with other features. This is basic classification for non-hybrid features in that they are seen as either protrusions or depression. Simple features, such as protrusions or depressions, are stored in a features database, which the algorithm accesses in part with selecting the required machining process. Hybrid features of the non-passage type, which are not automatically recognised and attributed to a common machining process, must be broken down into individual features and then re-compiled to develop a small process sequence. An individual feature may be in the form of single face that requires only one machine pass. If the process sequence is deemed successful then the hybrid feature needs to be stored, in the form of a topological relationship and geometric data, to ensure future recognition. Once the individual features sub-graphs have all been identified, they must be indexed based on complexity. An obvious example would be that a counter-sunk thread could not be tapped until an initial basic drilling operation was performed. This sequencing is based on data stored in the features database and forms the foundations for the sequencing of the individual processes. The precedence of a feature by another indicates that it should be produced first in order to satisfy the algorithm. The selection of each machining process for each individual feature is made from data available in a database that is machine specific. This process entails patternmatching techniques between the individual feature sub-graphs and the process/feature database. The volume removed by each machining process that exists in the case-specific machining environment is extremely important to the validation of the product model and is contained in this machine tool database but is not used here. Once all features have been indexed, the dimensions, symmetry and orientation of each individual feature are examined simultaneously and compared to the current tool data to ensure machining compatibility. Symmetry of the features is examined only to simplify the process path selection. However, non-symmetrical feature can still be used. The accessibility should be viewed as the starting point for the individual feature's process path and will update the sequencing of each process. The determination of the approach direction was investigated by Aldak,hilallah and Ramesh [ l l ]. They proposed that a system should develop a set of all the directions from which a tool can approach a face. For example {+x,-x,+y,-y,+z,-z} would be a set possible for most three dimensional products. From this set, the system needs to interpret a set of feasible directions from which a face can be machined. A further refining process involves examining the following and previous processes approach path so that the loading and unloading of the workpiece is minimised.
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5. PROCESS SELECTION The machining data is now obtained and a product model is developed based on raw stock data and material removal of the current tool. An extension on the constructive solid geometry technique is utilised to form a trial of the product. This process is known as volume decomposition and is an excellent way of creating a secondary solid model as it refers to the volume removed by each process as an individual 3D solid. Each specific volume removed is constructed based on geometric constraints from the B-Rep model and from data that is unique to the required machine and tool. This produces the physical characteristics of each primitive and the orientation of the primitive has already been determined by the calculation of the tool approach direction. Since the original boundary representation model contains explicit geometric location data, a reference point for the three-dimensional primitive is then copied directly from the B-Rep model to the new CSG model. This reference point then positions the primitive solid according to original B-Rep model and the orientation is determined from the tool approach direction. Each solid is then subtracted from a user defined stock piece. Interactions between solids are irrelevant as all solids are being subtracted from the basic stock solid. This new CSG model has now essentially been created by an internal "design by features" function. Upon removal of each three-dimensional solid, a check is performed on the current feature and it is converted into a boundary representation model. This model is updated with each new process and is the basis for the recursive checking system. The conversion to B-Rep depends greatly on the geometric data of the machining processes stored in the feature/process database. What this means is that the algorithm can compare the topology matrix and geometry data of the newly removed feature and compare it to the corresponding feature in the original solid model. If the feature that is being proposed does not match the required original feature, the algorithm loops back to the initial feature indexing and evaluates the feature according a variant on the Euler formula. The Euler formula can be presented in various forms and these are stored to ensure that the feature checking process is as accurate as possible. If a problem arises in feature identification, this secondary CSG solid model can be quite useful for user prompting as this is an area which B-Rep often lacks [13]. Now that all processes have been synthesised on this trial, the geometric and topological data of the completed model are compared to the actual product model and the machining database to give an approximation of the final tolerance on the model. Appendix A depicts a simple example of the algorithm at work on a simple slotted workpiece. It shows the logical steps that the algorithm presented in Figure l would follow to produce a complete process plan. 6. CONCLUSION The presented algorithm developed using B-Rep and CSG techniques was used to determine the features for different products. In the algorithm, these techniques were employed to utilise the advantages of both in that the complexity of the product being modeled can be increased and the time taken to recognise the feature can be reduced. This ability is derived from the modified Attributed Adjacency Graph/Matrix. With a simple method for storing data regarding product features and their related machining processes, the process sequencing can be performed at a quicker speed also.
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REFERENCES 1. Bronsvoort, W.F. and Jansen, F.W. "Feature Modelling and Conversion- Key Concepts to Concurrent Engineering", Computers in Industry, 21, pp. 61-86, Elsevier, (1993). 2. Joshi, S.B. et al, "Design, Development and Implementation of an Integrated Group Technology and Computer Aided Process Planning System" Volume 26, Number 4, IIE Transactions, July, (1994). 3. Rembold, U., Nnaji, B.O. and Storr, A. "Computer Integrated Manufacturing and Engineering", Addison-Wesly, (1994) 4. Mortenson, M.E., "Geometric Modelling", John Wiley & Sons, (1996) 5. Shirur, A., Shah, J.J., and Hirode, K., "Machining Algebra for Mapping Volumes to Machining Operations for Developing Extensible Generative CAPP", Journal of Manufacturing Systems, Vol. 17/No. 3, (1998). 6. Chang, T.C., Wysk, R.A. and Wang, H.P., "Computer-Aided Manufacturing 2nd Ed.", Prentice Hall, (1998). 7. Kalta, M., Davies, B.J., "Product Representation for an Expert Process Planning System for Rotational Components'; International Journal of Advanced Manufacturing Technology, Vol. 9, pp. 180-187, (1994). 8. Kalta, M., Davies, B.J., "CADEXCAP: Integration of 2D CAD Models of Turned Components with CAPP" International Journal of Advanced Manufacturing Technology, Vol. 8, pp. 145-159, (1993). 9. Kalta, M., Davies, B.J., "Guidelines for building 2D CAD Models of Turned Components in CAD-CAPP Integration", International Journal of Advanced Manufacturing Technology, Vol. 8, pp. 285-296, (1993). 10. Shah, J.J., Mantyla, M. and Nau, D.S., "Advances in Feature Based Manufacturing", Elsevier, (1994). 11. Aldakhilallah, K.A. and Ramesh, R., "An integrated framework for automated process planning: design and analysis", International Journal of Production Research, Vol. 36, No. 4, pp. 939-956, (1998). 12. McMahon, C.A. et al, "Representation and reasoning in computer aided process planning ", Proceedings from the Institute of Mechanical Engineers, Vol. 211, Part B 13. Singh, N., "Systems Approach to Computer-Integrated Design and Manufacturing", John Wiley & Sons, Inc., (1996). 14. Zhang, C., Chan, K.W. and Chen, Y.H., "A Method for Recognising Feature Interactions and Components within the Interactions" International Journal of Advanced Manufacturing Technology, Vol. 13, pp 713-722, (1997). 15. Joshi, S. and Chang, T.C., "Graph-based Heuristics for Recognition of Machined Features from a 3D Solid Model", Computer-Aided Design, Vol. 20, Number 2, pp. 58-66, March, (1988).
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APPENDIX A
#1
#2
9 9 9
9
The above part geometry is employed as an example of the algorithm presented. CAD file interfaced Geometric and topological data extracted Euler verification: V-E+F=2*(S-G)+R Therefore 16-24+ I 1=2"(14))+ I and thus the object is valid under boundary representation The Attributed Adjacency Graph/Matrix stores the type of relationship between faces.
Feature Identification A Convex relationship between three faces of equal length indicates a slot. Slot immediately recognised as a V-slot. The modified AAG and AAM for this slot is shown below. In matrix form, the 1350 between two faces would be stored in the symmetrical unit of the matrix.
.
r, Ft 9
.
9 Connectivity between faces examined using AAG. 9 Based on attribute, further data regarding the relationship is stored, creating the modified Attributed Adjacency Matrix. 9 An edge created by two planar faces requires the angle between the two faces to be stored. A Blind hole requires reference face to the hole opening (ring) relationship to be stored.
0
9 The Blind hole has a reference face that is recognised as face 02 of the V-slot. 9 V-slot feature is indexed first. Therefore machining of this feature will occur before machining of blind hole. 9 Blind hole is indexed 9 Geometric data compared with process/feature relationship database to determine accessibility.
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.
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Verification Tool path is essentially generated from slot length, opening, and blind hole centre point. Volume Decomposition: solids subtracted from cubic primitive.
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Current Advances in Mechanical Design and Production
Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
571
DEVELOPMENT OF A GENETIC ALGORITHM BASED ON FUZZY LOGIC SETS FOR SOLVING FACILITY LAYOUT PROBLEMS
Ramadan, M.Z.* and Abou EI-Ez, S.R.S.** *Associate Professor, **M. Sc. Candidate Prod. Eng. Dept., Faculty of Engineering, Helwan University, Helwan, Cairo, Egypt. Email:
[email protected] ABSTRACT Facilities layout is concerned with how to arrange block areas within the facility. Traditionally, this problem has been approached through the use of analytical tools ranging from scaled templates to computer-aided layout algorithms. The latter algorithms are characterized by using the relationship charts or preference charts. The analyst is typically uncertain about what these inputs should be. In fact, one of the real difficulties in developing such models for layout design is the natural vagueness associated with the model inputs. The use of fuzzy logic relationships is one approach for handling such kind of inexact vague data. Therefore, this paper presents a genetic algorithm based on fuzzy logic sets for solving facility layout problems. Better solutions that satisfy multiple objectives are produced by implementing genetic operators to the proposed gene structures. Significant better quality layouts that are obtained by employing this approach are compared to test problems available in the literature. A numerical example is given to illustrate the proposed approach and to show its capability to deal with multi-criteria problem effectively. KEYWORDS Facility Planning/ Layout; Multiple Criteria Programming, Fuzzy Rule-based System; Genetic Algorithms; Search Strategies. 1. INTRODUCTION Facility layout deals with the selection of the most efficient layout of physical departments based on their inter-relationships in production and service organizations in order to operate costs effectively. The design of the facility layout of such organizations is of tremendous importance for its effective utilization. This fact was emphasized in Tompkins and White [1], with the authors pointing out that 20-50% of the total operating expenses in manufacturing are attributed to material handling and layout related costs. The use of effective methods for facilities planning can reduce these costs by at least 30%. They listed several layout design objectives such as: 1) improve materials handling; 2) utilize people, equipment, space, and energy; 3) minimize capital investment; 4) provide employee safety and job satisfaction; 5) be flexible and promote ease of maintenance.
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Traditionally, this problem has been dealt with the use of analytical tools ranging in sophistication from scaled templates to computer-aided layout algorithms. According to Islier [2], facility layout algorithms can be classified into two main categories: optimal (like total enumeration, branch and bound, or curing plane)and suboptimal methods. Optimal methods are not computationally feasible for even moderate size problems. This fact has led researchers to focus on developing heuristic algorithms. These algorithms are classified as: 1) construction methods, 2) improvement methods, 3) hybrid methods, and 4) graph theoretic methods. These layout approaches start with a collection of data that are either quantitative such as from-to-charts, or qualitative as relationship-charts. Regardless of this type of data, there is an element of vagueness in it [3,4]. The traditional layout techniques treat these inputs as exact. The analyst is typically uncertain about what these inputs should be. The use of Fuzzy methodology is one approach for handling inexact vague data and yet to work in a mathematically strict and rigorous way [5]. On the other hand, Elwany et al. [6] pointed out that the facility layout problem was treated traditionally as a single objective, single criterion problem. It could be solved by applying any simple management science technique. In the real life, problems contain a considerable number of variables, and a number of conflicting constraints that make the optimization of the problem a difficult task. John Holland [7] introduced genetic algorithms (GAs) that have received a great deal of attention in the recent literature, because of its highest capability to deal with multi-criteria combinatorial optimization problem and that they do not rely on the analytical properties of the function to be optimized. Moghaddam & Shayan [8] used GAs to solve Quadratic Assignment Problem formulation of equal and unequal sized facility layout problems. Islier [3] presented a GAs based model for dealing withmulti-criterea facility layout problem. Rajasekharan [9] proposed a GAs based strategy for solving facility layout problem in flexible manufacturing system. The previous literature indicated that better layout solutions can be obtained using GAs for solving the facility layout problem. No one tried to have both techniques in one model. Therefore, in this work, GA is used to optimize facility layout problems based on fuzzy rule-based outputs. 2. PROPOSED APPROACH: 2.1. The fuzzy sets representation: The fuzzy logic, or fuzzy set theory, aims to represent fuzzy concepts in an understandable form. It links natural language with reasoning computing system (e.g., soft computing system) through the use of linguistic variables and quantifiers. Linguistic variables can represent words such as "age", "tall", "hard", "street length", or "beauty". In addition linguistic quantifiers such as "young", "some", "many", "less than", or "average" are quantifiers as fuzzy subsets of the real line which correspond to imprecise values of an amount. The variables and quantifiers are mapped to fuzzy membership functions. Suppose a grade of membership measure fA(u) -- 0.6 means that u is an element of set A to a degree of 0.6 on a scale where zero is no membership at all, and one is a complete membership.
The main components of the fuzzy decision-making model are shown in Fig. 1. Numerical (crisp) values or fuzzy expressions (results of subjective evaluation or estimation) are allowed to be the process input. The main part of the fuzzy model is a knowledge base with a set of linguistic rules and definitions of fuzzy sets describing the system to be modeled. A fuzzy output value is computed on the basis of these rules using a certain inference method.
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Then this fuzzy output value can be transformed into a numerical value (so-called defuzzification) or approximated to one of the fuzzy, or linguistic values that have been defined for the output variable. This approximation (defuzzification process) can be accomplished by means of calculating the center of area between fuzzy sets. The following steps are description of how to obtain a fuzzy model, as described above, in a systematic way: Step 0. Select a set of input linguistic variables that are available. Step 1. Select a set of output linguistic variables whose crisp values are needed. Step2. Determine the membership functions for all linguistic labels of the linguistic variables. Step 3. Develop a knowledge base of fuzzy rules. Step 4. Select fuzzification technique in which crisp inputs are converted into fuzzy representations (i.e., suitable linguistic of fuzzy sets). Step 5. Develop a knowledge base of fuzzy rules. Step 6. Develop a fuzzy inference strategy applying compositional rules of inference (CRI) using IF-THEN rules. Step 7. Select defuzzification technique in which the propagated fuzzy representation is converted to a set of crisp output values. i
Fuzzy Database !
Crisp i ~ . ~
Rules
.~ Fuzmficat~on Interface
Fuzzy inputs
~ Fuzzy Inference] Inference ,~ 9
,,,[
Crisp output
l i
Fuzzy output l llll ii
Engine
f
Fig. 1. Architecture of typical fuzzy system. 2.2. The genetic algorithm representation Genetic algorithms are basically search techniques. They imitate the natural process of evolution by processing towards the optimum. In a genetic algorithm, an individual chromosome (any feasible solution to the problem) is the basic element of the population. These chromosomes are combinations of facilities numbers, known as genes. The best chromosomes are selected by the roulette wheel principle to be parents. Three genetic operators known as crossover, mutation, and inversion are applied to those parents to generate new offsprings. In this model, population size is five times the number of the facilities (length of the chromosome) in the layout. The crossover, mutation, and inversion probabilities are: 0.9, 0.01, and 0.05 respectively. The maximum number of generated population is 120. The proposed genetic algorithm is established as follows"
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Step 0. Initialize randomly a population size of chromosomes equal to five times the number of the facilities (n). In addition, each chromosome (as a potential solution) is represented by a decimal string of length equal to the number of the facilities. Step 1. Calculate the fitness value for each chromosome fv i (i-l,..., population size). Step 2. Find the total fitness of the population using the sum of the distances times the pop-size
relationships for each pair of facilities, F - ~ fv i ill
Step 3. Calculate the selection probability for each chromosome, Pi = fvi / F. i
Step 4. Calculate the cumulative probability for each chromosome, q = Z pj i
Jffil
Step 5. Develop the new population by spinning the roulette wheel population-size times for selecting the next chromosomes in the following way: a) Generate a random number r from the range { 0, ..., 1}. b) If r < q l then the first chromosome is selected; otherwise the ith chromosome is selected such that qi-I < r _0, a reduction in intercell moves of t-I
value G can be made by interchanging the K sequence of machine pairs from the "interchange list". After this is done, the procedure is repeated with another pair of cells till all the cell pairs are considered. The coverage of all the possible cell pairs is called a "pass". The passes are repeated until the pass gain G = zero. The original KL heuristic partitions a set of even number of nodes into two equal partitions minimizing the cost of arcs cut. The KL heuristic has also indicated an approach for the case where the number of partitions is not fixed but is to be equal to or greater than a specified value. In the case of cell formation, there is no constraint on the number of partitions (cells) and the inequality of cell sizes is permitted. The KL heuristic is based on swapping equal sets of nodes. In the present work, the KL heuristic has been modified as follows: 1. The number of partitions is not fixed and can be interactively controlled by the user. 2. The partition sizes have been manipulated to be unequal, by introducing dummy machines. 3. The modified KL heuristic is not restricted to even number of nodes.
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4.
Unequal sets of nodes can be swapped by including dummy machines in the swapped sets. 5. Further improvement is introduced to the KL final cellular configuration as follows: 9 An attempt is made to reduce exceptional part-operations by reassigning them to the primary cell to which the associated part is assigned. This attempt is limited by the individual machine utilization or usage constraint; and the availability of a machine that can perform the exceptional part-operation to be reassigned to the primary cell. 9 An attempt is made to reallocate the machines that process part-operations on parts for which the primary cell is not the one in which the machine is currently allocated. The pseudo code of the modified KL heuristic is shown below. Modified KL Pseudo Code:
Begin Until Pass Gain=O Do (* each outer loop iteration is called a "Pass" *) For c l = l to C-1 Do (* c I & c2 are cell indexes *) For c2=cl + 1 to C Do Compute g.lg~ - ~ ~2 for each machine pair. Select nl e cl and n2 E c2 such that g.l..2 _c~.c2 is maximum. Append nl & n2 to the exchange list. Update the gain values g
Choose K to MAX G =
~ g, ~=l,g,e( Interchange List )
Implement the sequence o f K exchanges. Update gains of all neighbors of switched machines End Until Implement the part-operations reallocations. Implement the bottleneck machine reallocations. End
The CFGP method can be used as a generator of alternative cellular designs. Several initial partitions using other heuristics could be used. The user-controlled parameters such as , V, and C can be used to generate alternative cellular configurations. This can assist the decision maker in selecting the suitable alternative. 3. ILLUSTRATION A hypothetical example is used to illustrate the CFGP heuristic. There are 9 machine types consist of 16 individual machines. The individual machine utilization is calculated based on their processing capabilities and the current operation assignments. Table 1 shows the individual machine utilizations and the machine types. Table 2 shows the desirability measure values for each machine pair (the results of stage 1). The entries of this matrix represent the arc weights of the mc-mc graph. The machine utilization or usage is used as a node size in the local refinement KL heuristic. Stage 2 results are given in Table 3. The final partitions after applying stage 3are given in Table 4.
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Table 1. Individual Machines Data Individual MC i
2 3 4 5 6 7 8 9 10 II 12 13 14 15 16
MC Type
Utilization
!
o.588
! 2 2 3 3 4 5. . . . . 5 6 6 7 7 8 9 9
0.214 0.712 0.654 0.200 0.591 0.226 0.317 0.590 0.225 0.217 0.273 0.180 0.378 0.521 0.097 .....
Table 2. Desirability Measures of Individual Machines 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.0() 0.00 0.46 0.00 0.78 1.00 0.65 0.00 0.00 0.00 0.00 0.28 1.00 0.33 0.00
2 3 0.00 0.00 0.00 0.00 0.00 0.00 0.73 0.00 0.00 0.00 0.27 0.00 0.38 0.00 0.36 0.45 0.00 1.00 0.00 0.26 0.73 0.47 0.27 0.00 0.38 0.00 0.27 0.00 1.00 0.00
4 5 0.46 0.00 ~0.00 0.73 0.00 0.00 0.00 0.00 ' 0.74 0.00 1.00 0.00 0.30 0.35 0.00 0.70 0.00 0.00 0.00 0.00 0.00 1.00 0.86 0.00 1.00 0.30 0.80 0.00 0.00 1.00
6 0.78 0.00 0.00 0.74 0:00 0.73 0.65 0.00 0.0() 0.74 0.00 0.57 0.73 0.53 0.00
7 1.00 0.27 0.00 1.00 0.00 0.73 0.65 0.00 0.00 0.00 0.00 1.00 0.92 1.00 0.00
Table 3. Initial Cellular Configuration Partition / Cell I . 2
Individual Machines 1'3'4'5'7'10'14,!.6 t__ ..2,6,8,9, I l, ! 2,.13,15
. 8 0.65 0.38 0.0(i 0.30 0.35 0.65 0.65 0.00 0.00 0.00 0.35 0.00 1.00 0.00 0.00
9 0.00 0.36 0.45 0.00 0.70 0.00 0.00 0.00 0.10 0.00 0.80 0.00 0.00 0.00 1.00
10 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.10 0.00 0.33 0.00 0.00 0.00 0.00
II 0.00 0.00 0.26 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.26 0.00 0.00 0.00 0.00
12 0.00 .73 0.47 0.00 1.00 0.00 0.00 0.35 0.80 0.33 0.26
13 0.28 0.27 0.00 0.86 0.00 1.00 1.00 0.00 0.00 0.00 0.00 0.00
14 1.00 ().38 0.00 1.00 0.30 0.73 0.92 1.00 0.00 0.00 0.00 0.14 0.86
15 0.33 0.27 0.00 0.80 0.00 0.53 1.00 0.00 0.00 0.00 0.00 0.00 1.00 0.86
0.00 0.14 0.86 0.00 1.00 0.86 1.00 0.00 0.00 0.00
16 0100 i.00 0.00 0.00 1.00 0.00 0.00 0.00 1100 0.00 0.00 1.00 0.00 0.00 0.00
Table 4. Final Cellular Configuration Partition /Cell
1 2
....Individual Machines
1,4,6,7,8,11,13,14,15 ...... 2,3,5,9,10,12,16
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Note that the initial partitions have equal cell sizes, 8 machines in each cell. Using V=25%, the final two cells contain 9 and 7 machines respectively. Table 5. Initial Part-Machine Incidence Matrix MC No. MC Type I 2 3 4 5 6 7
Z
lO
~.,
12 13 14 15 16 17
1 I 2 ! 3 ! 4 15
i 6
I I 112121313
7 i 8 i 9 I I O ! I1 I i-2 i 13 1 1 4 1 1 5 l l6 1415151616l 71718 [ 9 [ 9
I
1 i 1
1 1 ! I
I I I 1
I
1 I
1 ! ! 1
I
I
I I
I 1 1 I I !
I I 1 1
! 1 1
1 I 1
I 1 I 1
I I I I I
I i I 1 1 1
I I
I I 1
I I
1 I 1
I 1
Table 6. Final Part-Machine Incidence Matrix MCNo.
,Z
1 1 2 _ 3 ' 1 4 ~ 7 I 8 1 9 1 I0 1
5 6
~-i~
[ 13 114 [
1 [ 4 I 6 I 7
MC rypel I i 2 1 3 I 1
1
14151 I
I I 1 1
1 ! I i i 1 I I
1 I 1
i8 19
1 I I
I I I I
2 [ 3 I 5 19-1
lo112116
1121315161719
11
1 I I 1
I I
FT]
ill
13 14 15 16
17
15
6 I 7 I 8 I 9
II I
I I
1 I I I 1 1 1 I
1
1 1 1
I 1
1 I 1 1 I
1
1 1
1 1
1 1 1
To visualize the effect of CFGP heuristic, the initial part-machine incidence matrix is given in Table 5 (this is not part of the CFGP input). The rows represent parts and the columns represent machines. An entry of 1 at the intersection of row r and column s indicates that part
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r requires an operation on machine s. The final part-machine incidence matrix is given in Table 6 (this is not part of the CFGP output). The two partitions that represent the two cells are clearly visible. This concludes the CFGP illustration. The proposed heuristic generates machine assignments to manufacturing cells, given the part-operations assignments to machines. 4. EVALUATION AND COMPARISON The CFGP heuristic, described in section 2, has been implemented in the C programming language and can partition a 1000-node graph. Fast data structures have been used in the C program to speed the search and looping operations such as single linked lists and double linked lists. In identifying the previous CF approaches for comparison, the Hamiltonian Path Heuristic (HPH) proposed by Askin et al [2] and the method proposed by Co and Araar [7] (CA) have been selected. The reasons for selecting these two methods are: (1)detailed solutions generated by these methods to one or more data sets are available and hence, there is no need to develop codes for operationalizing these procedures; and (2) The HPH method has shown to dominate previous matrix restructuring methods for the CF problem and thus, is a valid choice for comparison. CA method considers information similar to that of the CFGP method, which makes it a reasonable choice for comparison. Given that both the HPH and CA methods focus on the objective of minimizing intercell moves, Grouping Efficiency (GE) measure developed by Chandrasekharan and Rajagopalan [6] has been used as the basis of comparing the solutions generated by the HPH, CA, and CFGP methods. The GE measure is defined as follows: G E = qO I + (1 - q)O 2 0 0, then the process has a simultaneous static and dynamic variation as illustrated in Fig. (l-b). Historical data and their statistical analysis are the means to find these key factors. In collecting historical data, screening suspended values or observations obtained in unusual circumstances, like measurement errors, is a must. Statistical control charts, if used, can be a good tool to find the range of the mean movement by noticing the changes in the center limit and find out its maximum and minimum values. Also, averaging these center limits gives the expected value of the global or long-term process mean; and consequently the process mean offset, Z~t, from its target value. Z~t
T= IJ
z
Fig. l-a. Process with DMV
[4
Z
Fig. l-b. Process with static and dynamic mean variation
4. PROCESSES WITH DYNAMIC MEAN VARIATION To find the capability index of processes with DMV, an equivalent static mean shiR is to be obtained. An equivalent mean shift is nothing more than a compensatory static offset in the mean which directly corresponds to dynamic inflation of the standard deviation such that the probability of nonconformance is the same in both cases. This provides us with comparable
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tail area when considering both types of occurrences. Based upon that notation of equivalent tail area, it would be reasonable to postulate equivalent capability ratio: CPK ---- CpD
(2)
where Cpg is the equivalent capability ratio of a process with static mean offset, and CPD is the dynamic capability ratio. More details about Cpg and other process capability indices can be found in [8]. Since the process with DMV has its mean bias toward the target value, the CpD capability ratio may be expressed as:
(3)
CPD IT - SL l/3 c osr --'--
where T is the process target mean, and SL is the closest specification limit to that target. Substituting about CPD and Cpg in Eq. (2),
I I T- SLI/3 osrl(1-K)
--
IT- SLI/3 c osr
(4)
Eq.(4) can be reduced to: K-
1 - I/C
(5)
Eq.(5) expresses the relationship between the process mean's static offset "K" (in units of oST) and the inflation factor C. The K parameter represents the tolerance ratio consumed by the process mean offset from the target value. The value of K is given by [6]: K= IT-~[/[T-SLI
(6)
Substituting Eq.(6) in Eq.(5) and dividing both sides by OST one can get the Z-transform that relates C to the process's mean shift from its target value:
I z~l
-- I Zsrl (1-1/c)
(7)
Eq.(7) translates C to an equivalent sustained mean shift (Zp) expressed in Z units of measure. According to the values of C and the process's instantaneous capability, the equivalent shift of the mean (Z~t) from its target value can be found. In Table (2), the equivalent mean shift, Z~t, is calculated for different values of short-term capabilities Cp ( where C p - ZST/3) at some values of the inflation factor C. At this point of analyses, the process with DMV can be calculated its capability value. Given the process's short-term capability and its equivalent mean offset Z~t, then the long term tolerance band in Z-form, ZLT, is defined as: ZLT ffi ZST- Zp = ZsT (l-K)
(8)
The capability index CpD call be found from: CPD = Z~T (I-K)/3
(9)
or
CrD = 7-,sT/3C
(10)
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Table 2: Equivalent mean offset (Z~t) at different values of C and
LCo
i r ->
1.2
~
0.501 1.002 1.25
1 2 3
"
1.4
1.6
0.857 . . . . 1.714 2.i43
...
Cp
l.s
1.125 2.25 2.813
2.0
1.332 2.664 .... 3.33
1.5 3.0 3.75
The practical utilities of the analysis above are important. A designer and/or a production manager can decide whether to use a process with DMV based on its expected production yield for certain tolerance bands. To illustrate, suppose that the required target production yield, in a part per million (ppm), is 1000, i.e., the number of out-of-specifications parts in a lot of million parts is 1000 (500 ppm at each side of the specifications). Then the corresponding ZLT (from the Normal Distribution Tables)should be 3.29 and, accordingly, CpD = 1.1. Now, if the process is historically characterized by having a DMV with, for example, C=1.5 (K=0.333), the specification limits should be set at least at ZST = 4.94 (from Eq. 10) to achieve the target value of the production yield (ppm = 1000). This means an equivalent potential capability of Cp = 1.65. The designer can decide whether he can allow for that value of tolerance, and if not, the production manager should look for a more precise (but usually expensive) process. On the other hand, if the designer insist on the tolerance he decided, the production manager can expect the production yield, and, if satisfied, he can go ahead and use the process. Table (3) introduces the production yield for processes with DMV (at some C values) for some selected values of tolerances (expressed in Z-form, ZST). The Table can help a production manager to determine the production yield given the process tolerance and the inflation factor of its standard deviation. Table 3: The expected ppm for processes with DMV at different tolerance bands
C 1.2 1.4 1.6 1.8 2.0
I ZsT " 9
3.0 32420 33360 60100 94920 133620
~
4.s 176 1326 4954 12420 24440
6.0 "
19 176 868 2700
7.0 2.8 30 176
Another practical utility is to find out the process robustness (CR) for the inflation parameter. As given by Richards [7], robustness is defined as the ability to resist or respond to perturbations without loss of performance as well as the ability to grow and expand. Since ZLr is the response of the process due to the dynamic mean behavior and Zsr is the original response without perturbations, one can apply the above definition, that is, the process robustness CR may be given by: CR- ZLT / ZST-- ZsT (l-K) / ZST = (l-K) -- I / C
(ii)
Expressed as a ratio, the process robustness may be given as Ca percent robust to perturbing influences of the mean variation, e.g., a process with C = 1.5 is 66.7 % robust due to that mean variation causing the inflation.
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5. VERIFICATION OF THE ANALYSIS A Monte-Carlo simulation program is developed to check out the derived capability index (in Eq. 10) associated with the processes with DMV. The simulation program, indeed, tests both the inflation factor and the process's expected production yield. The same procedure of generating normally distributed random variates with specified DMV, explained in Appendix A, is utilized. The out-of-specifications number of units for batches of 5000 (1000 data sets times 5-unit set) production units is counted. Ten replications are carried out for each one of the nine selected cases exhibited in Table (4) below. In each case, the calculated and simulation-resulted ppm are compared. The calculated ppm values are found through the same procedure used to develop Table (3).The simulation resulted ppm are found by counting the average out-of-specification number per batch and then applying the chi-square distribution method for best estimate detailed in [2]. Table 4: The expected ppm from calculations and simulation for some selected cases
Cv C Calculated ppm Simulatedppm
1'.4 33360 33506
1.0 1.6 60100 60424
1.8 94920 95234
1.4 1326 1384 .
.
1.5 1.6 4945 4880 .
.
.
1.8 1.4 12420111~' 20 12102 21
2.0 1.6 176 181
1.8 868 842
The analysis verification is based on comparing the calculated versus the simulation-resulted ppm considering the simulation results as the reference. As can be observed from the table, differences between the calculated and simulation-resulted ppm, in general, are not significant. The maximum difference is (324 / 106 %, at Cp = 1.0 and C = 1.6 in the Table). Also, there is no bias in the calculated ppm, and in turn in the developed capability index, since non of the calculated and simulation-based ppm values dominate the other. 6. PROCESSES WITH SIMULTANEOUS STATIC & DYNAMIC MEAN BEHAVIOR In some situations, both forms of static and dynamic mean behavior may come into play. In such a situation, the estimate of process capability and the corresponding production yield is more complicated because it should consider the process mean offset (static mean offset) as well as the dynamic influences that result over time. In this research, two approaches are used to evaluate a process's capability when it has a simultaneous static and dynamic mean behavior. The first approach, by Harry [2], computes the degrade in the process capability as the summation of the static and dynamic mean variation effects. The second approach, proposed by the author, bases the calculation of the capability indices for a process with a static mean shift on the inflated standard deviation to compensate for the dynamic mean behavior. Harry [2] suggested to add the effects of both the static and dynamic mean behavior. He proposed that the portion of mean variation due to dynamic variations (K') is given by: K" = 1 - I(ZsT + Z~ ) / Zs-r !
(12)
Since K is given by {1 - (ZLT / ZST )}, from Eq. (8), the effect of the static mean offset is simply given by ( K - K" ). To illustrate, suppose that ZST =5.0 and ZLT = 3.75. The K value will be given by 1- (3.75/5.0) = 0.25. Suppose that the mean offset Z~ = 1.0, then by applying Eq. (12), K* will be 0.05. The conclusion is that the dynamic mean behavior
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consumes 5.0 percent of the semi-tolerance zone while (0.25 - 0.05) 100 = 20.0 percent of the same zone would be consumed by the static mean offset. The long-term capability index (C*pK) is given by [2]: CpK = Cp(ZLT / ZST) =Cp[l - (K + KST)I
(13)
The author here introduces another approach for dealing with processes with static and dynamic mean behavior. This approach makes use of the inflation factor given in Table (1) and the well known process capability indices. As given in [8], the most popular capability indices for processes with a static mean shift from its target value are Cpr and Cpm where: CpK = Cp ( l - K ) ,
and
Cpm = Cp / 3/1 +(It -oT)2
In terms of the notations used in this research:
CpK -- TZST
(1 - K )
(14)
Applying the inflated standard deviation, the capability indices for a process with simultaneous static and dynamic mean behavior will be:
Cpm---~ ,~Jlq-Z~
(15)
CpmD
= ZST / 3/1 + Z 2 3C ~t
(17)
Ceua~
= zsT (I - K) 3C
(16)
7. COMPARISON BETWEEN DIFFERENT APPROACHES In this section, a Monte-Carlo simulation program is developed to evaluate the two approaches introduced in the previous section. The Harry's approach to compute the capability index for processes with simultaneous static and dynamic mean variation (Eq. 13) and the author's approach for the same purpose (Eqs. 16,17) are used to compute the expected out-of-specifications percentage for some selected combinations of static and dynamic mean variations (presented in Table (5) below). The simulation program is used to generate normally distributed random variates with a specified offset of their mean from its target value. The same technique applied in section (5) is used to cause the effect of the dynamic variation; and the specified mean's offset from the target value is added to the generated variates to reflect the static variation of the mean. As in section (5), ten replications are conducted for every case, each case includes a batch of 5000 variates. The average
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percentages of out-of-specifications for the ten replications (shown in the last row of Table (5)) are used to compare their corresponding values calculated using the two approaches. Table 5: The expected out-of-specifications percentages for processes with simultaneous static and dynamic mean variation ZsT C
C'PK 0.5
Cpr,D CPmD simulation
C'pK l.O
Cpgn
CPmD Simulation
1.5
C*PK CPKD CPmD Simulation
....
6'0
4.5
3.0
1.4 3.14 3.67 2.74 3.61 5.81 7.6 6.4 7.45 10.02 i4.2 11.7 14.31
,,
1.8 6.92 8.23 6.81 8.39 10.37 13.3 11.9 12.95 15.8 20.3 17.9 20.35
1.4 .16 0.21 0.21 0.19
0.54 0.62 1.16
6.71 1.13 1.62 3.75 1.85
1.8 1.19 1.32 1.25 1.35 2.2 2.6 3.8 2.9 4.42 4.7 8.2 5.61
1.4 3.9e-3 4.4e-3 6.4e-3 4.2e-3 1.5e-2 1.8e-2 0.12 3.1e-2 5.7e-2 6.6e-2 0.86 0.09
1.8 .09 0.11 0.14 0.12 0.23 0.28 0.91 0.37 0.55 0.62 3.2 0.95
As can be observed, Harry's approach always underestimates the out of specification percentage or, in other words, overestimates the capability of the process. The author's approach of using the well known indices and basing their calculations on the inflated standard deviation seems more reasonable; and the simulation results comply to a high extent with their corresponding values of either CpKD and CpmD. Comparing the results of Cp~ and CpmD, one can observe that the CpmD -- based values are generally greater than those of the simulation results especially in high values of Ztt and ZST. However, the CpmDis better than C'pr as an index of process capability but is not as good as CpKD. The CpgD-based out of specification values are the closest to the simulation results, with no bias, such that it is recommended to be used as a capability index in characterizing processes with simultaneous static and dynamic variation. Table (5) can also help a production manager deciding whether to use the process based on the expected out of specification percentage. 8. APPLICATION The process of "Assembling Water Housing and Water Cover" in Helwan Company for Metallic Appliances is selected to demonstrate how to find out capability and other performance measures of processes with mean variation. In this application, the focus will be on the boring process that is used to produce the water housing cover. This process, as claimed by the quality control manager in the company, is experienced to have repeated mean shift from time to time. The inner diameter of the cover has a nominal size of 88.8 mm and a tolerance zone of :1:0.4 mm around the nominal size. Measurements of the inner diameter over about one year have been collected. Gauge and operator errors have been neglected. A 180 samples, five-unit each, have been plotted their means (and also their individuals) on a chart where the horizontal axis represented the sample number in a time sequence. The chart included also the process's target value (88.8 mm) and specification limits. Samples seem to have high variation are excluded. The plotted samples represented clusters of different means and run lengths, and that proves that the process has a
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dynamic mean variation. Although the exact timing of process mean shift is hard to be accurately determined, the clusters of measurements guided us to visually figure out these shifts. The process analysis has been conducted through the following steps: 1. The range and the mean of each sample are calculated. The average range is calculated and then the short-term standard deviation is computed and found to be 0.08 mm. The process's grand average (long-term average) is calculated by averaging the samples' means (after excluding samples with abnormal variations) and found to be 88.68 mm. 2. The maximum and minimum cluster means are addressed and found to be 88.85 and 88.53 mm respectively, i.e., about • 2 Csr around the mean value. The corresponding value of the inflation parameter (Table 1) is C=1.4. 3. The process's short-term, or potential, capability index is calculated as: Cr
= Z s x / 3 = ( S L - T ) / 3 o s r = ( 8 9 . 2 - 8 8 . 8 ) / 3 x 0 . 0 8 --- 1 . 6 7
4.
Since the process's grand average (88.68 mm) deviates from the target value (T = 88.8 mm) and the mean has random shifts (from the samples' plotting), then the process is considered to have a simultaneous static and dynamic mean variations, and the Corn should be calculated. 5. The degree of mean offset (K) can be calculated as: K = IT - 6 1 / I S L -
6.
TI = 0 . 3
Applying Eq. (16), the Cp~a) can be obtained as: Cer~ = (Zsr / 3C )(1 - K) = 0.83
The corresponding production yield (from the Normal Dist Tables) is expected to be 99.38 percent. 9. CONCLUSIONs Some manufacturing processes have natural shifts and drifts of their mean that happen randomly across their specification limits. This phenomenon is called dynamic mean variation if the process's long-term average coincides with its target value. Otherwise, the process is considered to have simultaneous static and dynamic mean variation. A process with DMV can be translated into an equivalent process with the same mean but having an inflated standard deviation. The inflation factor depends on the range of the random process's mean shift. This inflation can be related to a process static mean offset and, accordingly, the process's degraded capability can be calculated. In case of simultaneous static and dynamic mean variation, the problem gets more complicated. In this research, two approaches are introduced, one by Harry [2] and the other by the author, to compute the capability index. The two approaches are assessed by comparing their results of the expected out-of-specification percentages against other percentages resulted from a simulation program developed for this purpose. It was found that, within the tested range of parameters, the results of the author's approach comply to high extent with the simulation results; while those of Harry's approach are always less than the simulation results. REFERENCES 1. Gunter, B.H., "The Use and Abuse of CPK, Part 2.", Quality Progress, pp. 108, (1989). 2. Harry, M. J., and Lawson, J. R., "Six Sigma Producibility Analysis and Process Characterization", Motorola Inc., (1992).
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3. 4. 5. 6. 7. 8. 9. 10. 11.
12.
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Bender, A., "Statistical Tolerances as it Relates to Quality Control and the Designer", Society of Automotive Engineers, SAE, Paper No. 75049, pp. 412-419, (1975). Hussein, M., "Analysis of Quality Characteristics for Processes with Dynamic Mean Variation", Eng. Research Journal, Helwan University, pp. 46-52, Dec. (1998). Evans, D. H., "Statistical Tolerancing: The State of the Art, Part II: Methods for Estimating Moments", Journal of Quality Technology, Vol.7, No. 1, pp. 1-12, (1975). Montgomery, D. C., "Introduction to Statistical Quality Control", 2"~ John Wiley & sons Inc., (1991 ). Richards, L.D., "Analysis of Robustness in the Formulation of Technology Strategy", ASEM Proceedings, pp. 7-16, (1996). Sweeny, P. J., and Mohammed, H. M., "Capability Indices and Process Evaluation", ASME Conference Proceedings, pp. 157-164, (1996). Woo et al, "Tolerance Synthesis for Nonlinear Systems Based on Nonlinear Programming", IIE Transactions, Vol. 25, No. 1, pp. 51-61, (1993). Balakrishnan, N., "A Multiple-Choice Knapsack Model for Tolerance Allocation in Mechanical Assemblies", IIE transactions, Vol. 25, No. 4, pp. 13-14, (1993). Cheng, B.W., et al, "Optimization of Mechanical Assembly Tolerances Incorporating Taguchi's Quality Loss Function", Journal of Manufacturing Systems, Vol. 14, No. 4, pp. 264-276, (1995). AI-Sultan, K.S., and AI-Fawzan, M.A., "Variance reduction in a process with random linear drift", Int. J. Prod. Res., Vol. 35, No. 6, pp. 1523-1533, (1997).
APPENDIX A ESTIMATING THE INFLATION FACTOR USING SIMULATION A computer program, encoded in the C-language, is developed to estimate the values of the inflation factor (C) at different ranges of mean variation. The selected ranges are exhibited in Table 1. For each of these ranges, the program is run 250 times (250 replications). The average values and their corresponding confidence intervals of C are given in Table 1. In each run, the program is initiated by generating N=I000 data set, each contains 5 measurements distributed normally N(0, l). The mean and standard deviation of each set are computed; and the cumulative standard deviation is progressively calculated for the 1000 set. Nonrandom perturbations in the mean parameter are calculated as follows. A uniform random distribution is used to create a series of mean shift vectors, one for each data set. A positive vector called for positive shift and vice versa. The two vectors are similar for each range and have values range from zero to the specific mean variation range (five equidistant va!ues).The extent of the mean shift is determined by a random uniform number. Each of these numbers is multiplied by the corresponding vector. To test the inflation in the process standard deviation, the cumulative standard deviation is calculated progressively across all the 1000 data sets. A ratio is calculated of the cumulative standard deviation for each shifted subgroup to that of the respective unshifted group; resulting in the value of C for that replication. The expected value and confidence intervals for the inflation parameter C are based on the 250 replication-resulted C values.
Current 4dvances in Mechanical Design and Production Seventh Cairo University International MDP Conference Cairo, February 15-17, 2000
631
SHORT TERM MANAGEMENT OF WATER RESOURCE SYSTEMS
Faye, R. M. 3, Sawadogo, S. 3, Gonzalez-Rojo,
S. 1'4 and
Mora-Camino, F. t'2
ILAAS du CNRS, 7 Av. du Colonel Roche 31077 Toulouse France 2Ecole Nationale de l'Aviation Civile 7, Av. Edouard Belin 31055 Toulouse - France 3Ecole Sup6rieure Polytechnique B.P. 10 Thins- S6n6gal 4Instituto Tecnologico de Chihuahua, Av. Tecnologico N~ C.P 31310, Mexico
ABSTRACT This paper addresses the problem of short term management of Water Resource Systems (WRS). This problem is coped through an adaptive generation of a recurrent optimization problem which is solved by a specialized primal linear programming technique. A fuzzy approach is used in each instance of the recurrent optimization problem to express accurately the demand constraints and the terminal level stock constraints. The proposed approach, applied to a WRS of medium complexity, shows that it can be adapted to the management of particular situations like flood management, management in presence of failed devices. KEYWORDS Water Resource Systems, Hybrid Dynamics, Intelligent Management Systems, Fuzzy Logic, Linear Programming. 1. INTRODUCTION WRS can be viewed as hybrid dynamics systems subject to continuous operations broken by discrete events when new goals must be defined on the short term either to guarantee an efficient use of the upstream water reserves or to enforce security [ 1]. Until very recently little has been developed on methodological grounds to face at the tactical level the large uncertainties that impair the operations of such systems. So in this communication, this problem is coped with two levels 9first, through an adaptive optimization approach where initial conditions and constraint levels are updated according to the last available information [3] and second, through a reactive approach when the occurrence of significant events leads to the definition of new short term goals and constraints for the current optimization problem. Fuzzy Logic is used in each instance of the recurrent optimization problem to express accurately the demand constraints and the level of the terminal stock constraints, then a Linear Programming solution is developed. The proposed approach is illustrated by its application to a water resource system of medium complexity. 2. MODELLING OF WRS FOR SHORT TERM MANAGEMENT The structure of WRS can be considered as ordered graphs composed of upstream reservoirs, sequences of interconnected reaches, end users pumping stations and outlet sections (Fig. 1). It
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appears that to cope efficiently with the management of WRS on the short term, stock and transfer dynamics must be taken into account. This leads to the following unified dynamic representation [2]" Z i ( t + A t ) = Z i ( t ) + [ ~ y,.hiJ .Qi(x)_ E Q ~ ( t ) - ~ P ~ ( t ) + a i ( t ) - d i ( t ) ] A t / o i (1) jEA~
't
'-'
J
je~
jePi
where Ai is the set of convergent canals towards reach i, Si is the set of canals originated at reach i, Pi is the set of pumping stations at reach i, 9is the transfer delay, Zi(0 is the downstream water level in reach or dam i at time t, Q~(z) is the mean inflow to reach or dam i during period [~, ~+Az], ai(t), Q~(t), di(t) and P)(t) are the respective mean values of natural inflows, controlled outflows, overflows and pumpings from, or to, reach i or dam i during period [t, t+At], m are shape parameters for the storage components and hiJt-~ are transfer coefficients representative of the flow dynamics along the reaches.
Overflow Inflows
--~ Losses
v~
~-~[ v-] Pumpings /
intermediateOutlet ",~jDam v-
Fig. 1. Components and structure of a WRS
3. THE PROPOSED APPROACH FOR SHORT TERM MANAGEMENT OF WRS Like in the case of many other large scale distributed systems, the management of water resource systems can be organized in three levels according to the span of time considered [1 ]: long term, short term and very short tenn. At the long term level, WRS is managed in order to gumantee availability of the resource and efficiency in its use over the whole annual cycle. At the short term level, reference values and control parameters are optimized to adequate long term goals with the actual operational state of the system. At this level, discrete decisions must also be taken to enforce security of operations and equity in deliveries. Following an adaptive approach, it is proposed in this communication to identify on a hourly basis the current operational situation and to reprogrmn the pumpings and releases over a chosen period of time [4] (Fig. 2). When no terminal event is expected for this period, its span should be taken longer than the maximum transfer delay present in the system. However, since the short term demand forecasts present a decaying accuracy with an increasing horizon of prediction, this period of time should be restricted to a few days. The planned hourly releases and pumpings are then obtained by the resolution of a recurrent optimization problem (ROPh) taking into account the current conditions of the system (water level in the reaches and dams, operational state of the actuators) and short term demand forecasts.
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Current Advances tn Mechanical Design and Production, MDP-7
Fig. 2. Proposed short term WRS management structure
4. ADAPTIVE GENERATION OF THE RECURRENT OPTIMIZATION PROBLEM At first, a general linear formulation is considered for the ROPh problem : h+H-I
max
Y'.)".
~
P~(t),Qij (t)i r162
(PJ(t)-Di(t))
(2)
t=h
with the dynamics constraints 9 ,.
Zi(t+l)_Zi(t)_[E
~ h 0 .Qi(x)_ ~ Q](t)- ~ P j ( t ) + a i ( t ) - d i ( t ) ] A t / c i=O x_(l-~,i)[E(EP~ max+ EQ~max). H]
(20)
where a~(t) is the forecast realized at time of inflows for dam i during period t which is h, ~i is a positive real number small w.r.t, unity. 5.2 Management with failed devices When pumping station jo(io) goes down during period [ho-l,ho] with a predicted repair time iho of some hours (iho