Fly Cutting Technology for Ultra-precision Machining (Precision Manufacturing) [1st ed. 2023] 9819907373, 9789819907373

Table of contents :
Series Preface
Volume Preface
Contents
About the Series Editor
About the Volume Editors
Contributors
1 Modeling of Surface Generation with Cutting Strategy Effect in Ultraprecision Raster Milling
Introduction
Development of Cutting Strategy
Tool Path Generation
Transformation Between Machine and Workpiece Coordinate System
Machining Time Evaluation
Evaluation of Tool Path Length
Real Feed Rates of Machine Axes
Cutting Time Estimation
Surface Generation Model
Material Removal Mechanism
Surface Roughness Topography Pattern
Effect of Tool-Interference on Surface Generation
Feed-Interval and Path-Interval Scallop Height Without Shift Length
Effect of Shift Length on Scallop Heights Without Tool-Interference
Effect of Shift Length on Scallop Heights with Tool-Interference
3D Kinematic Model of Surface Roughness Prediction
Experimental Verification
Experiment Design
Experimental Results
Results and Discussion
References
2 Modelling of Spindle Vibration and Cutting Mechanism in Ultra-precision Raster Milling
Introduction
Surface Formation in Ultra-Precision Raster Milling
Cutting Strategies
Cutting Directions
Fundaments of Modelling of Spindle Motions
Eulerian Angles
Newton-Euler Equations
Dynamic Modelling of an Aerostatic Bearing Spindle in UPRM
Dynamic System of a Spindle
Cutting Force Description
Dynamic Equations of Spindle Motions
Summary
References
3 Dynamic Characteristics of Spindle Vibration in Ultraprecision Raster Milling
Introduction
Dynamic Characteristics of Spindle Vibration in UPRM
Solutions of Spindle Motion Equations
Dynamic Responses of the Spindle Under Intermittent Cutting Forces
Influences of Intermittent Cutting Forces with Spindle Speed
Influences of Depths of Cut with Spindle Speed
Summary
References
4 Prediction and Optimization of Surface Generation Under Spindle Vibration in Ultra-precision Raster Milling
Introduction
Surface Generation in UPRM
Modeling of Surface Generation
Surface Topography
Shift Length of Tool Loci and Phase Shift of Vibration Influencing Surface Topography
Prediction of Vibration Effects on Surface Topography
Optimization of Vibration Effects on Surface Topography
Summary
References
5 Modelling and Optimization of Cutting Strategy on Freeform Machining in Ultra-precision Raster Milling
Introduction
Modeling of Surface Generation on Freeform Machining
Error Analyses on Freeform Machining
Effect of Cutting Strategy on Form Accuracy of Freeform Surface
Framework of Cutting Strategy Optimization
Effect of Cutting Strategy on Machining Time
Effect of Cutting Strategy on Surface Finishing
Effect of Cutting Feed Rate, Spindle Speed, and Path-Interval on Surface Roughness
Effect of Feed Direction on Surface Roughness
Effect of Shift Ratio and Depth of Cut on Surface Roughness
Optimization Methodologies of Cutting Strategy
The Integrated Optimization of Cutting Feed Rate and Path-Interval
Tool Path Optimization
Experimental Study of Effect of Cutting Strategy on Surface Roughness
Effect of Path-Interval on Surface Roughness
Effect of Depth of Cut on Surface Roughness
Effect of Entry Distance on Surface Roughness
Effect of Feed Direction on Surface Roughness
Verification of Cutting Strategy Optimization Methodologies
Quality-Optimal Cutting Strategy Optimization Methodology
Time-Optimal Cutting Strategy Optimization Methodology
Effect of Cutting Strategy on Form Error
Summary
References
6 Modelling and Characterization of Material Effect on Surface Generation in Ultra-precision Raster Milling
Introduction
Material Elastic Recovery Effect on Cutting Profiles
Modelling of Material Elastic Recovery on Surface Roughness
Calculation Model of the Local Axial Depth of Cut
Material Elastic Recovery Effects on Scallop Heights
Surface Roughness Prediction Model with Material Elastic Effects
Characterization of Material Effects on Surface Roughness
Experimental Study on Material Effects on Surface Generation
Experiment Design
Surface Roughness Prediction Under Material Elastic Recovery
Characterization of Material Effect on Surface Generation
Effects of Hard Particles on Surface Finishing in Machining Al6061
Summary
References
7 Tool Wear Characteristics in Ultra-precision Raster Milling
Introduction
Diamond Tool Wear Characteristics
Experimental
Experimental Setup
Experimental Procedures
Cutting Distance Calculation
Tool Wear Characteristics and Their Effects
Fracture Wear
Material Welding
Flank Wear Land Formation
Sub-Wear-Land Formation
Summary
References
8 Tool Wear Monitoring Method Using Cutting Force in Ultra-precision Raster Milling
Introduction
Experiments
Experimental Setup
Cutting Force Acquisition
Cutting Force Components
Modeling and Analysis of Cutting Force in UPRM
Modeling of Cutting Force Pulse
Investigation of Cutting Force Composition
Cutting-Force-Induced Free Vibration
Effects of Cutting Parameters on the Amplitude of Cutting Force
Cutting Force Evolution with Tool Wear Progress
Experimental Observations
Theoretical Interpretation
Power Spectrum Density Characteristics of Cutting Force Under Tool Wear
PSD Characteristics of Cutting Force
PSD of the Free Vibration of Dynamometer
PSD of the Modal Vibration of Workpiece
Summary
References
9 Tool Fracture Wear Evaluation Method Using Cutting Chips
Introduction
Mechanical Tool Wear Mechanisms
Thermal-Chemical Tool Wear Mechanisms
Tool Wear Suppressing Methods
Chip Formation in the Cutting Process
Relationship Between Tool Wear and Chip Formation
Experiment
Results and Discussion
Modeling of Cutting Chip and Virtual Cutting Edge
Cutting Chip Model
Virtual Cutting Edge Buildup Under Tool Fracture Wear
Modeling and Analysis of Surface Roughness Under Tool Fracture Wear
Surface Topography Simulation Under Tool Fracture Wear
Surface Roughness Evaluation Under Tool Fracture Wear
Summary
References
10 Tool Flank Wear Evaluation Method Using Cutting Chips
Introduction
Tool Wear Monitoring Methods
Direct Tool Wear Detection Methods
Indirect Tool Wear Monitoring Methods
Experiments
Results and Discussion
Modeling and Identification of Flank Wear Land
Chip Thickness Along the Feed Direction
Wear Land Angle Identification
Wear Land Width Calculation
Modeling and Analysis of Surface Roughness Under Tool Flank Wear
Surface Roughness Modeling Under Tool Flank Wear
Effects of Cutting Parameters on the Surface Roughness Under Tool Flank Wear
Summary
References
11 Ductile Machining of Brittle Materials by Ultra-precision Raster Milling
Introduction
Experiments and Methodology
Brittle-to-Ductile Transition in UPRM of Brittle Material
Characteristics of the Surface Topography
Cutting Forces in UPRM of Brittle Material
Summary
References
12 Ductile Machining Mechanism for Micro-structure by Ultra-precision Raster Milling
Introduction
Cutting Mechanism in UPFC of Micro-groove
Ductile Machining Model in UPFC of Micro-groove
Effect of Machining Parameters
Ductile Machining of Micro-structures by UPFC
Micro-grooves Fabricated by UPFC
Surface Micro-topography
Material Phase Transformation
Chip Morphology
Tool Wear
Summary
References
13 Ductile Machining Mechanism for Continuous Freeform Surface by Ultra-precision Raster Milling
Introduction
Cutting Mechanism for UPFC of Freeform Surface
Ductile Machining Model in UPFC of Freeform Surface
Modeling of Asymmetrical Chip Formation
Effect of Machining Parameters
UPFC of Freeform Surface on Brittle Materials
F-theta Lens
Machining Efficiency UPFC
Tool Wear in UPFC of Freeform Surface
Summary
References
14 One-Step Generation of Hybrid Structured Surface on Brittle Material by Ultra-precision Raster Milling
Introduction
Machining Principle of Hybrid Micro-optics
Optimal Toolpath Generation Strategy
Determination of SCPs
Selecting Criteria of Machining Parameters
Analysis of the Fabricated Hybrid Micro-optics
Hybrid Micro-lens Arrays
Hybrid Sinusoid Freeform Surface with Nano-gratings
Summary
References
15 Optimization Modeling of Biomimetic Structures for Self-Cleaning and Optical Performance in Ultra-precision Raster Milling
Introduction
Design Criteria of Surface Geometry
Self-Cleaning Performance Requirements
Optical Performance Requirements
Consideration of Machining Ability
Design Challenge
Structure Design
Mathematical Model of Micro-Directional Grooved Surfaces
Mathematical Model of Micro-pillar Surfaces
Experimental Procedures
Fabrication of Micro-Patterns Using Ultra-Precision Raster Milling
Surface Topography Measurement
Contact Angle Measurement
Results and Discussion
Static Contact Angle Measurement
Length of Water Droplet Elongation
Optical Testing of Biomimetic Structures
Summary
References
16 Wetting Characteristics of Micro-patterned Surfaces Fabricated by Ultra-precision Raster Milling
Introduction
Experimentation
Experimental Setup
Fabrication of Micro-patterned Surfaces
Micro-patterned Surface Topography
Results and Discussion
Comparison with the Theoretical Models
The Wenzel Model
The Cassie and Baxter Model
Static Contact Angle from the Experiment
Droplet Anisotropy
Droplet Contact Line
Contact Angle Hysteresis
Sliding Angle Measurement
Summary
References
17 Characterization of Intermediate Wetting States and Anisotropic Sliding on Micro-directional Grooved Surfaces
Introduction
Theoretical Analysis
Static Contact Angle
Contact Angle Hysteresis
Experimental Section
Results and Discussion
Surface Characterization
Observation of Droplet Contact Line
Validation of Theoretical Models
Sliding Angle Measurement
Contact Angle Hysteresis
Summary
References
18 Anisotropic Wetting of Micro-Micro Hierarchical Structures Fabricated by Ultra-precision Raster Milling
Introduction
Regular Micro-hierarchical Structures Fabricated by UPRM
Experimental Setup
Fabrication of Regular Micro-hierarchical Structures
Surface Topography Measurement
Results and Discussion
Irregular Micro-Micro Hierarchical Structures Fabricated by EFCS
Experimental Setup
Fabrication of Irregular Micro-Micro Hierarchical Structures
Surface Topography Measurement
Results and Discussion
Multi-level Hierarchical Structures Fabricated by EFC
Experimental Setup
Fabrication of Multi-level Hierarchical Structures
Surface Topography Measurement
Results and Discussion
Summary
References

Citation preview

Precision Manufacturing Series Editor: Liangchi Zhang

Suet To Sujuan Wang Editors

Fly Cutting Technology for Ultra-precision Machining

Precision Manufacturing Series Editor Liangchi Zhang, Southern University of Science and Technology (SUSTech), Shenzhen Key Laboratory of Cross-scale Manufacturing Mechanics, SUSTech Institute for Manufacturing Innovation, Department of Mechanics and Aerospace Engineering, Shenzhen, China

This series of handbooks covers a comprehensive range of scientific and technological matters in precision manufacturing. The proposed handbook series aims to bridge the gaps by a systematically designed strategy to cover the required range of knowledge and essential understanding, and hence provide researchers and engineers a vehicle for achieving the optimization of the intelligent manufacturing chain. The readers will understand their role and position in precision manufacturing chain and hence understand how they could progress more efficiently and effectively. Junior researchers and engineers could seek their starting points of career development more easily and grab essential knowledge more systematically with a clear direction.

Suet To • Sujuan Wang Editors

Fly Cutting Technology for Ultra-precision Machining With 316 Figures and 51 Tables

Editors Suet To State Key Laboratory of Ultra-Precision Machining Technology Department of Industrial and Systems Engineering The Hong Kong Polytechnic University Hong Kong SAR People’s Republic of China

Sujuan Wang State Key Laboratory of Precision Electronic Manufacturing Technology and Equipment School of Electromechanical Engineering Guangdong University of Technology Guangzhou, China

ISSN 2522-5464 ISSN 2522-5472 (electronic) Precision Manufacturing ISBN 978-981-99-0737-3 ISBN 978-981-99-0738-0 (eBook) https://doi.org/10.1007/978-981-99-0738-0 © Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Series Preface

Manufacturing has always been a major wealth-creating sector in developed economies and will remain the cornerstone of long-term economic growth. It is manufacturing that underpins the modern scientific and technological development, such as the advances in energy, bio, micro and nano technologies. Most of today’s complex and important technological problems are inseparably connected to manufacturing issues. Successful innovative solutions in almost all disciplines rely on manufacturing, because deep insights can only be obtained with the aid of properly manufactured instruments. Since the new century, the advances in electronics, optics, telecommunication, biology, medical surgery, energy generation, resource exploration and environment protection have brought about further challenges and have produced veritable onslaught of fundamentals and technologies that require the capacity of precision manufacturing. The production of precision components and systems is sensitive to many complex sets of conditions in which they are manufactured. Over the past decades, the design and manufacture of high-integrity systems have improved but are limited by the lack of understanding of the production processes as an integrated whole. The advancement of precision manufacturing requires that researchers and engineers master the fundamentals of the materials in use, the processing technologies for transforming such materials to functional products with minimized defects, the strategies and technologies to create machines that are accurate enough to realize precision production, and the solutions to environmental impact issues. Each aspect of the above plays a critical role in the precision manufacturing chain. The chain can be damaged if any of the aspects suffers from imperfections. If researchers and engineers do not have a comprehensive vision and understanding of the chain, optimization of precision manufacturing is difficult, which forms a major hurdle to realizing intelligent/smart manufacturing for a new industrial revolution. This has imposed severe restrictions on our ability to analyze the complex processes of precision manufacturing. It is because so much is now demanded of high-integrity systems that the slightest imperfection in their manufacture has become a serious matter. The objective of this book series is to redress the shortcomings in the isolated single aspect studies in the chain of precision manufacturing. The book will cover a comprehensive range of scientific and technological matters to bridge the gaps by a v

vi

Series Preface

systematically designed strategy to reinforce the required knowledge and essential understanding and hence provide the reader with a vehicle for achieving optimization of the intelligent manufacturing chain. Specific emphasis of the book series will be on the fundamentals of materials and mechanics for precision manufacturing, minimal damage and damage-free design in precision manufacturing, precision machines and their control, numerical simulation for precision manufacturing, precision forming, precision optics, precision additive manufacturing, precision biomedical manufacturing, precision sensing and measurement, non-traditional precision manufacturing, eco-technologies and re-manufacture of precision elements. The books are suitable for senior undergraduate students, postgraduate students, junior researchers, and engineers who are interested in or working in the field of precision manufacturing.

Volume Preface

Ultra-precision raster milling (UPRM) is an essential method to support a satisfactory solution for fabricating non-rotational freeform surfaces with nanometric surface roughness and sub-micrometric form accuracy without any subsequent polishing. Different from ultra-precision diamond turning (UPDT), UPRM is a typical intermittent cutting process which causes a relatively lower machining efficiency. Moreover, the intermittent cutting process of UPRM also results in distinctive surface generation mechanism, covering intermittent tool-workpiece relative motion, tool geometry imprinted into machined surface, and surface material separation and deformation. However, in contrast to the substantial advances those have been achieved in UPDT, relatively less research has been conducted on UPRM. In such background, this book is dedicated to conduct an in-depth study on the cutting mechanism of UPRM and its specialized applications as well as presents a novel viewpoint on tool wear characterization and its influences in UPRM. This book is divided into two parts: Part I, Surface Generation Mechanism and Tool Wear of UPRM and Part II, Applications of Ultra-precision Raster Milling Technology. The organization of the book is scheduled as: In ▶ Chap. 1, the effect of cutting strategy on surface generation and machining time is studied and a threedimensional surface roughness prediction model is built by considering the intermittent tool-workpiece relative motion in UPRM. In ▶ Chap. 2, a five-degree-offreedom dynamic model of an aerostatic bearing spindle is developed for spindle vibration with linearized Newtown-Euler equations under the excitation of intermittent cutting forces. Following ▶ Chap. 2, dynamic characteristics of spindle vibration is presented in ▶ Chap. 3 to provide mathematical solutions for its dynamic responses under the different cutting parameters in UPRM, and in ▶ Chap. 4, the surface generation model is further improved by adding dynamic responses of spindle vibration to make prediction and optimizations for UPRM. In ▶ Chap. 5, it is focused on the fabrication of freeform surface in UPRM. A new methodology for the integrated optimization of cutting parameters and tool path generation (TPG) is developed to improve the machining efficiency. In ▶ Chap. 6, material effect on surface generation in UPRM is studied to build a surface roughness prediction model by adding material elastic recovery on the model proposed in ▶ Chap. 1. A new method is also proposed in ▶ Chap. 6 to characterize material-induced surface roughness on the raster-milled surface. The investigation on tool wear characteristics vii

viii

Volume Preface

and their effects on machined surface quality in UPRM is unfolded in ▶ Chap. 7. In ▶ Chap. 8, an analytic cutting force model is established to monitor tool wear stages in UPRM. The tool fracture wear and tool flank wear and their effects on the machined surface are evaluated on machine by using cutting chips in ▶ Chaps. 9 and ▶ 10, respectively. In UPRM, diamond tool intermittently contacts and departs the workpiece surface with a large idle duration for each rotational cycle, which leads to totally different brittle-to-ductile transition phenomenon. In ▶ Chap. 11 of this book, the brittle-toductile transition phenomenon for single-crystal silicon and a novel ductile machining model to efficiently fabricate deep micro-structures on brittle materials is established in ▶ Chap. 12. Considering the existence of swing distance in UPRM, a novel ductile machining model in raster milling of freeform surfaces with large azimuthal height variation on brittle materials is proposed and UPRM is applied to fabricate hybrid structured surface on brittle materials in a ductile mode in ▶ Chap. 14. In ▶ Chap. 15, UPRM is applied to produce biomimetic structures for self-cleaning and optical performance, in which an optimization model is built to strike a balance between the wetting and optical performance. In ▶ Chap. 16, wetting characteristics of bare hydrophobic micro-patterned cyclic olefin copolymer surfaces machined by a one-step fabrication method in UPRM is described. A novel characterization method of droplet wetting states, remarkably influencing its sliding behavior on micro-grooves, is proposed in ▶ Chap. 17. In ▶ Chap. 18, anisotropic wetting of bare micro-micro hierarchical structured surfaces fabricated by UPRM, end-fly-cutting servo (EFCS), and end-fly-cutting (EFC) is described. This book contributes to enhancing the understanding of surface generation and chip formation mechanisms of UPRM in perspective of materials science and machining dynamics. This book also serves as a practical tool for the applications of UPRM on the fabrication and characterization of different types of functional freeform surfaces with carefully documented experimental techniques in years of the authors’ research endeavor. The authors would like to thank the Research Committee of the Department of Industrial and Systems Engineering of The Hong Kong Polytechnic University and State Key Laboratory of Ultra-precision Machining Technology (The Hong Kong Polytechnic University) for providing the financial support for this research work. Hong Kong SAR, People’s Republic of China Guangzhou, China August 2023

Suet To Sujuan Wang Editors

Contents

1

2

3

4

5

6

Modeling of Surface Generation with Cutting Strategy Effect in Ultraprecision Raster Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sujuan Wang and Suet To

1

Modelling of Spindle Vibration and Cutting Mechanism in Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shaojian Zhang, Li Zhang, and Suet To

45

Dynamic Characteristics of Spindle Vibration in Ultraprecision Raster Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shaojian Zhang, Zhipeng Wei, and Suet To

59

Prediction and Optimization of Surface Generation Under Spindle Vibration in Ultra-precision Raster Milling . . . . . . . . . . . Shaojian Zhang, Jiasong Cheng, and Suet To

75

Modelling and Optimization of Cutting Strategy on Freeform Machining in Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . Sujuan Wang and Suet To

89

Modelling and Characterization of Material Effect on Surface Generation in Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . Sujuan Wang and Suet To

131

7

Tool Wear Characteristics in Ultra-precision Raster Milling . . . . . Guoqing Zhang, Jianpeng Wang, and Suet To

8

Tool Wear Monitoring Method Using Cutting Force in Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guoqing Zhang, Jianpeng Wang, and Suet To

161

181

9

Tool Fracture Wear Evaluation Method Using Cutting Chips . . . . Guoqing Zhang, Jianpeng Wang, and Suet To

215

10

Tool Flank Wear Evaluation Method Using Cutting Chips . . . . . . Guoqing Zhang, Jianpeng Wang, and Suet To

253

ix

x

11

12

13

14

15

16

17

18

Contents

Ductile Machining of Brittle Materials by Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhanwen Sun and Suet To

291

Ductile Machining Mechanism for Micro-structure by Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhanwen Sun and Suet To

305

Ductile Machining Mechanism for Continuous Freeform Surface by Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . . . . . . . . . . Zhanwen Sun and Suet To

327

One-Step Generation of Hybrid Structured Surface on Brittle Material by Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . . Zhanwen Sun and Suet To

347

Optimization Modeling of Biomimetic Structures for Self-Cleaning and Optical Performance in Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cheung Tong Cheng and Suet To

369

Wetting Characteristics of Micro-patterned Surfaces Fabricated by Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . . . . . . . . . . Cheung Tong Cheng and Suet To

393

Characterization of Intermediate Wetting States and Anisotropic Sliding on Micro-directional Grooved Surfaces . . . . . . . . . . . . . . . Cheung Tong Cheng and Suet To

413

Anisotropic Wetting of Micro–Micro Hierarchical Structures Fabricated by Ultra-precision Raster Milling . . . . . . . . . . . . . . . . . Cheung Tong Cheng and Suet To

437

About the Series Editor

Liangchi Zhang is Chair Professor in the Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology (SUSTech), Director of Shenzhen Key Laboratory of Cross-scale Manufacturing Mechanics, and Director of SUSTech Institute for Manufacturing Innovation, Shenzhen, China. He is also the Fellow of the Australian Academy of Technological Science and Engineering (ATSE). Zhang obtained his B.Sc. and M.Eng. from Zhejiang University; Ph.D. from Peking University, China; and D.Eng. from the University of Sydney, Australia. Prior to joining SUSTech, he has worked at the University of New South Wales and University of Sydney, Australia; University of Cambridge, UK; National Mechanical Engineering Laboratory, MITI, Japan; and Zhejiang University, China. Zhang carries out research on both the fundamentals and industrial applications in the cross-disciplinary field of advanced manufacturing, advanced materials, tribology, nanotechnology, solid mechanics, and biomechanics. He has published extensively in his research areas, with some in multiple languages. His research outcomes have led to significant economic benefits for manufacturing industry. Zhang has been granted many awards and honors. He can be contacted at [email protected].

xi

About the Volume Editors

Prof. Sandy Suet To is a Professor in the Department of Industrial and Systems Engineering of The Hong Kong Polytechnic University and Associate Director of State Key Laboratory of Ultra-precision Machining Technology. Prof. To graduated from Kunming University of Science and Technology with a bachelor degree in Mechanical Engineering and obtained her M.Phil. degree in Materials Science from Brunel University in UK and Ph.D. in Ultra-precision Machining Technology from The Hong Kong Polytechnic University. She started her academic career as Assistant Professor in 2005 and was promoted to Associate Professor and Professor in 2010 and 2018, respectively. Since 1994, she began to engage in the fundamental and applied researches in ultra-precision machining, micro/nanostructured surface, difficult-to-cut materials, advanced optical manufacturing, precision injection molding technology, and material engineering. Her main research directions include the mechanism of ultra-precision machining, design and machining of micronanostructures, the influence of materials in ultraprecision machining, the processing mechanism of brittle materials and other materials, the machining strategy of multi-axis ultra-precision machining, the tool path generation, the online inspection of cutting tool and error compensation, and precision mold for plastic injection molding technology. Prof. To has successfully received grants from more than 20 research projects including General Research Fund from Hong Kong Research Grants Council, Innovation and Technology Fund of the Hong Kong Innovation and Technology Commission, National Natural Science Foundation of China, and the research projects xiii

xiv

About the Volume Editors

jointly organized by the European Commission/Hong Kong Research Grants Council, etc. She has published 4 books, and more than 280 international journals (SCI) and 150 international conference papers. She has been invited to deliver Keynotes and invited talks at international conferences for many times. According to Web of Science, the academic papers published by Prof. To rank first in the fields of ultra-precision machining. In addition, she successfully obtained 13 granted patents as the first inventor. Prof. To’s achievements in research and professional studies have been widely recognized locally and internationally. She has received the Second Class Technology Progress Award from China’s Ministry of Education in 2009 and the Second Class in Natural Science Award from China’s Ministry of Education in 2011, respectively. The developed project entitled “High Power LED Street Lighting System with a Modular Lamp Holder” beat 1,000 invention projects around the world and won three awards at the 36th Geneva International Invention and Innovative Technology and Products Exhibition in Switzerland, which include the Grand International Press Prize, the Gold Award, and the Award of High Scientific and Technological Level of Invention from the Romania Ministry of Education, Research and Youth at the 36th International Exhibition of Inventions, New Techniques & Products. The research project entitled “Micro-Nano Structure Surface by Rapid Prototyping Technology for Anticounterfeiting” won the Gold Award and the Outstanding Ultra-Precision Technology Invention Award in the Asia International Innovation and Invention Award in 2019. Prof. To holds various honorary positions in professional bodies including Board Member of the Asian Society for Precision Engineering and Nanotechnology (ASPEN), Committee Member of the Production Engineering Division of the Chinese Society for Mechanical Engineers (CSME), and Associate Member of the International Academy for Production Engineering (CIRP). She also serves as Associate Editor for Journal of Materials Processing Technology and Editorial Board member in several international journals, such as Journal of

About the Volume Editors

xv

Manufacturing and Materials Processing, Journal of Applied Optics, Journal of Nano-manufacturing and Metrology, International Journal of Extreme Manufacturing, and Chinese Journal of Mechanical Engineering. Dr. Grace Sujuan Wang is an Associate Professor in the School of Electromechanical Engineering in Guangdong University of Technology. Dr. Wang obtained her bachelor degree in Mechanical Engineering from Harbin University of Science Technology, M.Phil. degree in Mechanical Engineering from Harbin Institute of Technology, and Ph.D. degree in Ultra-precision Machining Technology from The Hong Kong Polytechnic University. Dr. Wang is one of the core members of Guangdong Innovative Research team, Communication Review Expert of the National Natural Science Foundation of China (NSFC), and Senior Member of China Society of Mechanical Engineering. Her major research areas include the ultra-precision machining, advanced optics manufacturing technology, and development of ultraprecision machine tools. She has published over 60 research papers in international journals and conferences, as well as obtained more than 10 patents in China. She has secured 5 NSFC projects and over 10 projects from Guangdong province and Guangzhou City.

Contributors

Cheung Tong Cheng State Key Laboratory of Ultra-precision Machining Technology, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, People’s Republic of China Jiasong Cheng Key Laboratory of Ultra-precision Machining Technology, School of Advanced Manufacturing, Nanchang University, Nanchang, Jiangxi, People’s Republic of China Zhanwen Sun State Key Laboratory of Precision Electronic Manufacturing Technology and Equipment, Guangdong University of Technology, Guangzhou, Guangdong, China Suet To State Key Laboratory of Ultra-precision Machining Technology, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, People’s Republic of China Jianpeng Wang Shenzhen Key Laboratory of High Performance Nontraditional Manufacturing, College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen, China Sujuan Wang State Key Laboratory of Precision Electronic Manufacturing Technology and Equipment, School of Electromechanical Engineering, Guangdong University of Technology, Guangzhou, China Zhipeng Wei Key Laboratory of Ultra-precision Machining Technology, School of Mechatronics Engineering, Nanchang University, Nanchang, Jiangxi, P.R. China Guoqing Zhang Shenzhen Key Laboratory of High Performance Nontraditional Manufacturing, College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen, China Li Zhang Key Laboratory of Ultra-Precision Machining Technology, School of Advanced Manufacturing, Nanchang University, Nanchang, P. R. China Shaojian Zhang Key Laboratory of Ultra-precision Machining Technology, School of Advanced Manufacturing, Nanchang University, Nanchang, Jiangxi, People’s Republic of China xvii

1

Modeling of Surface Generation with Cutting Strategy Effect in Ultraprecision Raster Milling Sujuan Wang and Suet To

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of Cutting Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tool Path Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transformation Between Machine and Workpiece Coordinate System . . . . . . . . . . . . . . . . . . . . . Machining Time Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of Tool Path Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real Feed Rates of Machine Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cutting Time Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Generation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Removal Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Roughness Topography Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Tool-Interference on Surface Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feed-Interval and Path-Interval Scallop Height Without Shift Length . . . . . . . . . . . . . . . . . . . . . . Effect of Shift Length on Scallop Heights Without Tool-Interference . . . . . . . . . . . . . . . . . . . . . . Effect of Shift Length on Scallop Heights with Tool-Interference . . . . . . . . . . . . . . . . . . . . . . . . . . 3D Kinematic Model of Surface Roughness Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 7 11 13 13 14 15 18 18 20 22 23 25 27 31 34 34 34 34 42

S. Wang (*) State Key Laboratory of Precision Electronic Manufacturing Technology and Equipment, School of Electromechanical Engineering, Guangdong University of Technology, Guangzhou, China e-mail: [email protected] S. To (*) State Key Laboratory of Ultra-precision Machining Technology, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, People’s Republic of China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 S. To, S. Wang (eds.), Fly Cutting Technology for Ultra-precision Machining, Precision Manufacturing, https://doi.org/10.1007/978-981-99-0738-0_1

1

2

S. Wang and S. To

Abstract

This chapter studies the effect of cutting strategy on surface generation and machining time in ultraprecision raster milling (UPRM) to build a holistic three-dimensional (3D) surface roughness prediction model. The established surface roughness prediction model adds two new factors (the shift length and tool-interference) on surface generation and therefore not only takes into account the influences of cutting parameters, geometries of cutting tool, but also considers the effects of tool path generation (TPG), workpiece size, and the machine characteristics. The calculation methodology for the tool-interference in UPRM and evaluation models for shift length and machining time are also proposed in this chapter. A series of cutting experiments has been conducted to verify the proposed surface generation model. The experimental results agree well with the predicted results from the model. Keywords

Surface generation · Cutting strategy · Raster milling · Tool path · Cutting parameters

Introduction The development of ultraprecision machining is due to the industry’s demand for high-quality components. The achievement of a high-quality surface is therefore the main goal of ultraprecision machining. Surface roughness is predominantly considered as the most important feature of practical engineering surface due to its crucial influence on the mechanical and physical properties of a part and, in many applications, the surface roughness is vital for the functionality and performance of the product, such as the optical (Benardos and Vosniakos 2003), the polarization (Bennett and Porteus 1961; Lonardo and Micheletti 1974), superhydrophobicity (Yang et al. 2006), fatigue resistance (Proudhon et al. 2005), heat transfer (Sodtke and Stephan 2007), adhesion (Müser 2016), and reflectance (Bennett and Porteus 1961). Nowadays, freeform optics with the continuous freeform surface or the discontinuous freeform surface (microstructured surface) becomes more and more popular for various advanced applications to fulfill different optical functions, such as the printing, scanning, ophthalmic, and the head-up displays. However, the fabrication of freeform optics is difficult and time consuming because of its profile complexity and high-quality requirement. Ultraprecision raster milling (UPRM) with single-crystal diamond tools can directly produce freeform surfaces with submicrometric form accuracy and nanometric surface finishing, but with a relatively lower machining efficiency. The milling process becomes notoriously time consuming for the fabrication of large freeform surfaces, for example, for the machining of freeform reflectors and freeform headlights in the automotive industry. At present, the control on surface quality still relies on the time-consuming and uneconomical trial and error approach.

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

3

Therefore, it is necessary to find an efficient way to produce high-quality products by UPRM which indicates that a better understanding on surface generation and a precise surface roughness prediction model are much needed in UPRM. The surface roughness prediction in machining has been studied by different approaches. Benardos and Vosniakos (2003) classified the methods used to predict surface roughness in the turning and milling process into four categories: the machining theory, experimental investigation, designed experiments, and artificial intelligence (AI). Lu (2008) divided the approaches of surface roughness prediction into three major categories: pure modeling-based approach, signal-based approach, and AI-based approach. Meanwhile, lots of studies have been conducted on building surface generation model by investigating the factors affecting surface roughness in turning (Sata et al. 1985; Mital and Mehta 1988; Basu et al. 1993; Lin and Chang 1998; Abouelatta and Mádl 2001; Miller et al. 1983), end milling (Kline et al. 1982; Wang and Chang 2004; Omar et al. 2007; Fan and Loftus 2007; Li and Feng 2004), face milling (Kyun Baek et al. 1997), and high-speed machining (Ozcelik and Bayramoglu 2006; Lee 2003; López De Lacalle et al. 2002). In these surface generation models for conventional metal cutting process, the factors being considered include cutting parameters (i.e., spindle speed and feed rate), cutter geometry, tool wear, cooling strategy, spindle error motion, and machine characteristics. The method of surface roughness prediction based on the cutting mechanics or machining theory is generally applied to develop analytical models or computer algorithms to represent the machined surface quality. The cutting kinematics (Albrecht 1961; Tai et al. 1980; Kim and Kim 1998), cutting tool properties, cutting force (Lee et al. 2001), and chip formation mechanism are involved in developing the model. In fact, it is relatively difficult to accurately predict the achieved surface quality only based on these models. Therefore, some additional factors such as vibration (Jiang et al. 2008), cutter runout (Kyun Baek et al. 1997; Franco et al. 2004), and back cutting effects (Ryu et al. 2006; Franco et al. 2008) were introduced to improve the accuracy of prediction and simulation. Many studies combined different factors into surface roughness prediction models. Kim and Chu (Kim and Chu 1999) presented the texture superposition method to develop a three-dimensional (3D) surface topography model by studying the maximum height of the effective scallops, which considered the mixed effects of cutter marks, runouts, and conventional scallops as well as the filled radius of different end milling cutters (ball, filleted, and flat). Xu et al. (2003) developed a surface generation model under the consideration of the static and dynamic properties of the cutter in end milling process. Chen et al. (2005) presented a model to describe the path-interval and feedinterval scallops generating mechanism in ball-end milling process by adding the effects of tool radius, feed/pick ratio, initial cutting-edge entrance angle, and toolaxis inclination angles. Some researchers studied surface generation and roughness prediction in ultraprecision machining based on the mechanistic process (Li et al. 2002; Cheung and Lee 2002; Lee and Cheung 2001; To et al. 2001), the finite element method (FEM) (Dai et al. 2004), and molecular dynamic (MD) simulation (Shimada et al. 1993). Nevertheless, the surface generation mechanism of UPRM is much more complex

4

S. Wang and S. To

than that of SPDT, therefore the surface generation model of SPDT is not applicable for raster milling. Due to the wider and wider applications of UPRM in the fabrication of high-quality freeform surface, recent investigations have been found on UPRM (Cheung et al. 2006, 2008), while these existing theoretical models are oversimplified and did not consider the tool paths, materials removal mechanisms of UPRM, and their effect on the generation of 3D surface topography of the machined surfaces. As known to all, the achieved surface quality in metal cutting process is governed by the tool geometry and the relative movement between the cutting tool and the workpiece, which depends on the controllability of machine tool, and the transfer characteristics of cutting tool profile to the machined workpiece. Cutting strategy refers to the methodology used to schedule the relative movement between the cutting tool and the workpiece according to the desired surface quality and the machined profile geometries, as well as the employed ultraprecision machine, so as to generate the numerical control (NC) program to manufacture the product. Therefore, in this chapter, a comprehensive 3D surface generation model is developed by considering the effects of shift length and tool-interference on surface generation in UPRM. Moreover, the quantitative relationship between cutting strategy and surface roughness in UPRM is presented. Factors taken into consideration in the kinematic model of surface roughness prediction include tool path generation (TPG), cutting parameters, tool geometry, machine feed rate module, and workpiece size. An optimization method for shift ratio is built by studying the effect of cutting strategy on shift length. A model-based integrated system is developed to select the cutting parameters, planning tool paths, and automatically generate NC program for UPRM with the function of predicting the total machining time and surface roughness. The prediction model and the performance of the integrated system are verified by cutting experiment.

Development of Cutting Strategy The development of a cutting strategy involves the selection of cutting parameters and tool path generation (TPG) to define the speed and locus of the tool-work’s relative movement, as shown in Fig. 1. In UPRM, the main cutting parameters include spindle speed, cutting feed rate, rapid feed rate, and depth of cut. The tasks of TPG in the milling process includes: the selection of feed direction, the confirmation of tool path topology pattern, tool path linking method, and selection of path interval. After developing the cutting strategy, the numerical control (NC) program will be generated for the actual machining. Accordingly, based on the study of the effect of the cutting strategy on surface quality and machining efficiency, the achieved surface quality and the used machining time can be predicted. In general, the NC program used in CNC machining contains the information of machine motions, such as the speed of axes and the movement locus of axes, which all belongs to the machine coordinate system (XM, YM, ZM), while the tool path

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

5

Fig. 1 The framework of developing cutting strategy in UPRM

planning is conducted on the workpiece coordinate system. As shown in Fig. 2, there are four main steps in the generation of a NC program for raster milling: (1) confirmation of tool geometry (swing distance and tool-nose radius); (2) planning tool paths to generate cutter contact (CC) points in the workpiece coordinate system (XW, YW, ZW) and transforming CC points into the machine coordinate system (XM, YM, ZM); (3) the generation of cutter location (CL) points in the machine coordinate system (XM, YM, ZM); and (4) selection of appropriate cutting parameters. After these four steps, the NC program with standard “G-code” can be provided to the actual machining process. In this study, the tool paths planning in the workpiece coordinate system (XW, YW, ZW) comprises the following steps: (a) select the feed direction of cutting paths (parallel to XW-axis or YW-axis); (b) design tool path topology pattern; (c) confirm path interval; and (d) select the tool path linking method to confirm the clearance height (hc), the entry, and retreat distance (le, lr). The cutter contact (CC) points in the workpiece coordinate system (XW, YW, ZW) are then generated in order to select the feed direction of the diamond tool in the machine coordinate system so as to achieve the transformation matrix between the workpiece coordinate system and the machine

6

S. Wang and S. To

Fig. 2 Flow chart of developing cutting strategy to generate NC program

coordinate system. In UPRM, the diamond tool on the spindles generally feeds parallel to the XM-axis or YM-axis. After the transformations, the CL points in the machine coordinate system can be generated and translated to the NC program by post-processing (Fig. 2).

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

7

Tool Path Generation Tool path generation (TPG) in NC machining is a critical task, which can be divided into two aspects: the tool path layout strategy and the tool path plan technique. The tool path layout strategy is to select the tool path feed direction, to design the tool path topology pattern as well as the tool path linking method. The tool path plan technique refers to the algorithm of calculating the distances between two cutter contact (CC) points in feed direction and step direction. The linear distance between two CC points in the feed direction is called feed interval while the distance between two CC points in the step direction is called path interval or step size. Confirmation of these two factors must take into consideration the acceptable machining efficiency and the desired surface quality, since each of them is one of the crucial factors affecting the surface quality and machining efficiency in ultraprecision machining. During the NC milling process, the cutting tool traces a sequence of CC points to achieve the desired surface, and the tracing pattern is known as the tool path topology pattern (Marshall and Griffiths 1994). Tool path topology patterns for the CNC multiaxis milling are classified into four types (Choi and Jerard 1998): serial pattern (Fig. 3a–c), radial pattern (Fig. 3d), strip pattern, and contour pattern. Different tool path topology patterns are suitable for different machining cases and depend on the geometry of the surface boundary and the cutting conditions (El-Midany et al. 2006). The serial and radial patterns are usually applied for

Fig. 3 Tool path topology patterns

8

S. Wang and S. To

machining an area; the contour pattern is for vertical or slant wall machining; and the strip pattern is suitable for machining cleanup areas (Ren et al. 2004). In a serial-pattern tool path, path segments are parallel to a predefined line (Fig. 3a). Proper selection of the reference line (angel θ) directly affects the generated path length. This line could be parallel (θ ¼ 0  or 180 , Fig. 5b) or normal (θ ¼ 90  or 270 , Fig. 5c) to the boundary of the machined area. The radial-pattern tool paths (Fig. 3d) are constructed by the boundary curves of the machined surface, and each tool path is an offset of the boundary curves. Comparing these two patterns, the radial-pattern tool paths are more computationally expensive and results in more frequent changes in the cutting direction and resulting in more dynamic problems (Lasemi et al. 2010). The serial pattern is selected in this study for planning the tool paths for UPRM. Lw and Ww represent the length and width of the machined area, respectively. le and lr refer to the entry and retreat distance in the feeding direction (Fig. 3a, b); we and wr is the entry and retreat distance in the raster direction. In Fig. 3, the dash line represents the actual machining area. The presence of entry and retreat distances ensures machining safety, which makes the actual machining area larger than the workpiece size. When the tool path feeding direction is parallel to the XW-axis (Fig. 3b), the area of the actual machining (AM) can be represented as:

Fig. 4 The design of tool path linking method

Feed per revolution Path-interval Raster direction

Feed direction

Workpiece

Workpiece

Raster direction

Feed direction YM

YM XM

Path-interval

XM

Feed per revolution

ZM

ZM (a) Horizontal cutting

(b) Vertical cutting

Fig. 5 Two selections of tool feed direction for serial-pattern tool path

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

9

AM ¼ (Lw + le + lr)  (Ww + we + wr). Otherwise, if the tool paths is planned to feed along the YW-axis, in Fig. 3c, the actual machining area (AM) is: AM ¼ (Ww + le + lr)  (Lw + we + wr). The tool path linking method, being one part of TPG, is generally classified into two types, i.e., one way and zigzag. In Fig. 4, hCj is the clearance height of the jth step. The one way method contains retreats on the ZW-axis for each tool path (Fig. 4a), while the zigzag method has no retreat expect for the first step and the last one throughout the tool paths (Fig. 4b). Comparing the paths generated by these two linking methods, the paths by one-way method contains more no-cutting paths, therefore it needs more no-cutting time. This indicates that the zigzag method is more efficient. However, tool paths generated by the one-way method conduct the cuttings in one direction thereby making fewer changes in the cutting direction, which results in less machine tool dynamic problems. Two types of tool paths exist in NC machining: one is the rapid path which does not cause material removal and the other one is the cutting path where materials are removed from the workpiece by the cutting tool (Fig. 4). In the actual machining, the feed rate for the rapid paths, such as the feed rates of the clearance height and the linking paths between two adjacent steps, as shown in Fig. 6, can be set larger than that for the cutting paths. And FR and FC refer to the rapid feed rate and the cutting feed rate, respectively. In this study, the rapid feed rate is set equal to the maximum feed rate of machine. The motions of the diamond tool in UPRM include the rotation around the spindle and linear movement along the ZM-axis and the YM-axis of the machine coordinate system. At the same time, the workpiece on the machine table moves linearly along

Fig. 6 Tool tip movement locus for two cutting strategies

10

S. Wang and S. To

the XM-axis. In UPRM, diamond tool generally feeds parallel to the XM-axis or YM-axis, which leads to two possible cutting strategies for the serial-pattern tool paths: horizontal cutting and vertical cutting, as shown in Fig. 5. In horizontal cutting, feed direction is parallel to the XM-axis and the raster direction is along the YM-axis, while the ZM-axis is employed to control contour for achieving the desired surface profile (Fig. 5a). In Fig. 5b, vertical cutting is done by changing the functions of the YM-axis and the XM-axis. Figure 6 shows the locus gi, j(xgi, j, ygi, j, zgi, j) of tool tip movement in horizontal cutting and vertical cutting. It shows that the locus of the tool tip movement in horizontal cutting is a 2D spiral curve (Fig. 6a), while the locus in vertical cutting is a 3D spiral curve (Fig. 6b). In Fig. 8, Oi, j(xoi, j, yoi, j, zoi, j) represents the tool center location of the ith feed in the jth step; j ¼ 1. . .N, where N is the total number of steps in the whole machining cycle. Table 1 Tool-workpiece relative movement patterns in horizontal cutting Pattern no. Tool motion (XM-axis and YM-axis)

Workpiece motion (ZM-axis) Cutting strategy Tool-part relative movement locus model

H-1 Rotation (clockwise) Linear movement (+XM-axis) No linear movement in YM-axis Linear movement (ZM-axis) Up-milling xg ¼ xO þ L cosðθ0 þ wt Þ zg ¼ zO  L sinðθ0 þ wt Þ Ft xO ¼ xO, 0 þ 60 2πS w¼ 60

H-2 Rotation (clockwise) Linear movement (XM-axis) No linear movement in YM-axis Linear movement (ZM-axis) Down-milling xg ¼ xO þ L cosðθ0 þ wt Þ zg ¼ zO  L sinðθ0 þ wt Þ Ft xO ¼ xO, 0  60 2πS w¼ 60

Table 2 Tool-workpiece relative movement patterns in vertical cutting Pattern no. Tool motion (XM-axis and YM-axis)

Workpiece motion (ZM-axis) Tool-part relative movement locus model

V-1 Rotation (clockwise) Linear movement (+YM-Axis) No linear movement in XM-axis Linear movement (ZM-axis) xg ¼ xO þ R cosðθ0 þ wtÞ zg ¼ zO  R sinðθ0 þ wt Þ Ft yg ¼ yO,0 þ 60 2πS w¼ 60

V-2 Rotation (clockwise) Linear movement (YM-axis) No linear movement in XM-axis Linear movement (ZMaxis) xg ¼ xO þ R cosðθ0 þ wt Þ zg ¼ z  OR sinðθ0 þ wtÞ Ft yg ¼ yO,0  60 2πS w¼ 60

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

11

Meanwhile, the locii of the tool tip movement changes with the relationship between the feed direction and the rotation direction of the diamond tool. Tables 1 and 2 present the tool-workpiece relative movement patterns in horizontal cutting and vertical cutting. In UPRM, the diamond tool on the spindle can rotate in a clockwise or counterclockwise direction, while a diamond tool generally rotates in one direction during one machining process. In Tables 1 and 2, l _ Pj is the path interval between the ( j  1)th and the jth step; L is the swing radius (mm); F is defined feed rate (mm/min); w is the rotation speed, w ¼ 2πS 60 ; and S is the spindle speed (rpm). Note that tool feed direction in UPRM is different from the tool path feed direction. The selection of tool path feed direction is conducted in the workpiece coordinate system; an optimal selection of tool path feed direction contributes to the shortest tool path length and the shortest machining time. The transformation matrix between the workpiece coordinate system and the machine coordinate system is dependent on the selections of tool path feed direction and cutting tool feed direction. On the other hand, the selection of cutting tool feed direction affects the surface generation mechanism in UPRM, thereby influencing the achieved surface roughness.

Transformation Between Machine and Workpiece Coordinate System The tool path points are a series of successive coordinates (cutter contact points or CC points) in the workpiece coordinate system (XW, YW, ZW). The freeform machine is guided by axial commands carrying the three spatial linear axes and two rotation axes in the machine coordinate system (XM, YM, ZM). This means that the coordinate data for the machine coordinate system, instead of CL data for the workpiece coordinate system, have to be input to the control system of the machine. The NC program provides the coordinate information of each position in the machine coordinate system. Figure 7 shows the transformations from the CC points in the workpiece coordinate system to calculate CL points in the machine coordinate system (Freeform 705G, Precitech, USA). In this figure, (X, Y, Z) is the global coordinate system; (XW, YW, ZW) is the workpiece coordinate system, which is regarded as fixed to the global coordinate system; (XM, YM, ZM) is the machine coordinate system; PW(xP_W, yP_W, zP_W) are the coordinates of the current CC point in the machined surface; PM(xP_M, yP_M, zP_M) are the coordinates of the current CC point in the machine coordinate system; QM(xQ_M, yQ_M, zQ_M) are the coordinates of the tool tip center in the machine coordinate system; OM(xO_M, yO_M, zO_M) are the coordinates of the current CL point in the machine coordinate system; and n refers to the unit normal vector in the position PM(xP_M, yP_M, zP_M) of the machined surface. Figure 8 presents the coordinate systems for the calculation of transformation matrix. It shows that, to calculate the coordinates of the CL points in the machine

12

S. Wang and S. To

Fig. 7 Calculations of CL points in raster milling

Fig. 8 Coordinate systems for calculating transformation matrix

coordinate system, the coordinates of CC point PW(xP_W, yP_W, zP_W) in the workpiece coordinate system are firstly translated into the machine coordinate systemPM (xP_M, yP_M, zP_M): xP yP zP

M M M

¼

cos β sin β 0

sin β cos β 0

0 0 1

xP yP zP

W W

(1)

W

where β is the angel between the XW-axis and XM-axis. For the serial-pattern tool paths, when the tool path feed direction is parallel to the XW-axis and the tool feed direction is parallel to the XM-axis, the tool path feed direction is parallel to the YW-axis and the cutter feed direction along YM-axis, β ¼ 0; in the case where the tool path feed direction is parallel to the XW-axis and the cutter feed direction is parallel

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

13

to the YM-axis, the tool path feed direction is parallel to the YW-axis and the cutter feed direction is along the XM-axis, β ¼ 90 . After the coordinates of CC points are transformed into the machine coordinate system, the CC point PM(xP_M, yP_M, zP_M) is transformed into the center of tool tip point QM(xQ_M, yQ_M, zQ_M), and later to the CL point OM(xO_M, yO_M, zO_M) as shown in Fig. 10. Equation (3) represents the multiplication of transformation matrices from the CC point PW(xP_W, yP_W, zP_W) to the CL point OM(xO_M, yO_M, zO_M ) as: ðL  r Þ xO yO zO

¼

M M M

cos β sin β 0

¼

xQ yQ zQ sin β cos β 0

M

n2 x þ n2 z 0

þ

M M

0 0 0 1

xP yP zP

W W

0

0

0

0 ðL  r Þ

0 rþ

þ

W

nx ny nz

n2 x þ n2 z ðL  r Þ 0 n2 x þ n2 z 0 r 0

0 rþ

(2) 0 0 ðL  r Þ

nx ny nz

n2 x þ n2 z

where, nx, ny, or nz is the projection of the unit normal vector (n) in position PM on the axis XM, YM, and ZM of the machine coordinate system, respectively. The unit normal vector (n) depends on the profile of the machined surface. The ULTRAPATH™ CONTROL SYSTEM of the Freeform machine (Freeform 705G, Precitech, USA) combines flexibility and simplicity in both its programming and operation. The control system accepts standard “G-code” commands, and as a result, after the cutting parameters are confirmed, the CL points calculated by Eq. (3) can be translated to the standard “G-code” commands so as to generate a NC program for the actual machining.

Machining Time Evaluation Evaluation of Tool Path Length The evaluation of the machining efficiency contains two main tasks: the confirmation of tool paths length and the real feed rate of each tool path. Referring to Figs. 5 and 6, to machine a specified area, the tool path topology pattern and tool path linking method, as well as the cutting parameters and path interval, affect the total length of the tool paths. In Fig. 5, the length of CC points along the jth tool path (LCj) under the cutting feed rate (FC) is determined by the size of the workpiece, tool path feed direction, and the entry and retreat distances: Lc j ¼ lj þ le j þ lr j ¼

Lw þ le j þ lr j ðθ ¼ 0 or 180 Þ W w þ le j þ lr j ðθ ¼ 90 or 270 Þ

where lj is the length of the jth step with material removed from the workpiece.

ð3Þ

14

S. Wang and S. To

The length of rapid CC path of the jth step (LRj) under rapid feed rate (FR) is affected by the tool path linking method (Fig. 4): one way : LR j ¼

Lc j 2 þ l p2 j þ 2 hC j þ aep

zig zag : LR j ¼ l pj

ð4Þ

The total length of the CC point paths (Lcc) in the whole machining therefore can be confirmed as: N

Lcc ¼

Lc j þ LR j

ð5Þ

j¼1

where, N is the number of tool paths in the whole machining and depends on the value of path interval (l _ Pj), the tool path topology pattern (Fig. 3), and the size of the workpiece (Lw and Ww) as well the entry and retreat distance in the raster direction (we, wr). In Fig. 3, if θ ¼ 0  or 180 , N ¼ (Ww + we + wr)/l _ P; if θ ¼ 90  or 270 , N ¼ (Lw + we + wr)/l _ P. Equations (4), (5), and (6) show the relationship between the length of the CC point path and cutting strategy. Therefore, according to Eqs. (3), (4), and (5), the length of tool paths (CL point) are achieved to calculate the tool paths length in the machining of freeform surface. Note that when machining a flat surface, the length of the CL point paths is equal to that of the CC point paths (Eq. 3).

Real Feed Rates of Machine Axes The real feed rate in NC machining depends on not only the defined feed rates in NC program but also the machine’s characteristics, such as the power of the motion axes and the maximum accelerations and speeds. Precitech Freeform 705G utilizes G10 mode to establish the standard program acceleration and deceleration time setting for programmed moves in the NC program. In this mode, the ac/deceleration time (ta/td) is set to 10 ms, which means that any slide or rotary axis will accelerate or decelerate to its next position within 10 ms. Generally, the generated tool path in the actual machining contains two types of tool paths: cutting paths and no-cutting paths, as shown in Fig. 4. The two paths employ different values of feed rates: the rapid feed rate is generally set to a constant value and equal to the maximum feed rate of the CNC machine, while the cutting feed rate depends on the machining efficiency and surface finish. As a result, the NC program contains the rapid motion blocks and cutting motion blocks. Figure 9 shows the feed rate profiles of machine axes in horizontal cutting while vertical cutting only needs to shift the feed rate profiles of the XM-axis with YM-axis. The acceleration/ deceleration time are equal to 10 ms (t0 ¼ 10 ms).

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

15

Fig. 9 Feed rate variations in horizontal cutting for different tool path linking methods

The time used in one path between the jth and the ( j + 1)th step (tj) is calculated as: tj ¼

tC j þ

le þ lr 1 þ t0 2 VC

þ tR j ¼

lj þ le þ lr 1 þ t0 2 VC

þ tR j

ð6Þ

where tCj is the cutting time and represents the time used in the jth tool path where the cutter removes materials from the workpiece. VR and VC (mm/s) refer to the rapid and cutting feed, respectively. V R ¼ FR=60 and V C ¼ FC=60. tRj refers to the time used to finish the rapid motions which is different from the tool path linking methods: 2 hC j þ aep þ 3t one way : t j ¼ 0 þ 2 l pj t zig zag : tR j ¼ 0 þ 2 VR R

l pj 2 þ lj þ le þ lr VR

2

ð7Þ

The total machining time (T ) is the summation of the time used in each tool path: N

T¼

tj

ð8Þ

j¼1

where N is the number of tool paths in the whole machining.

Cutting Time Estimation When machining a freeform surface by UPRM, the feed interval and the cutting feed rate must be selected in consideration of the desired surface quality, since these two factors greatly affect the surface roughness and form accuracy. Meanwhile, the two factors also affect the length of tool paths as well as the machining efficiency. In

16

S. Wang and S. To

Fig. 10 The cutting path curve and axes movement in horizontal cutting

Fig. 11 The feed rate profiles of two machine axes in horizontal cutting

Fig. 10, as the diamond tool moves from the position Oi, j to the next position (Oi + 1, j) then to the following one (Oi + 2, j), the feeding vector (fi,j) changes. From this figure, the real length of cutting path (lj) is as follows: M

Horizontal cutting : lj ¼

XOiþ1, j  XOi, j

2

þ ZOiþ1, j  ZOi, j

2

ð9Þ

i¼1 M

Vertical cutting : lj ¼

YOiþ1, j  YOi, j

2

þ ZOiþ1, j  ZOi, j

2

ð10Þ

i¼1

where M is the number of feed components in one cutting path which depends on the feed interval and the size of the machined workpiece, the entry and retreat distance in the feeding direction, and the selection of tool paths feed direction. Figure 11 shows the variations of the feed rates of the machine axes in horizontal cutting. In the first path segment (* Oi, j Oiþ1, j ), as the workpiece moves along

Modeling of Surface Generation with Cutting Strategy Effect. . .

1

17

XM-axis from point A to point Bx, it firstly moves in a uniform target velocity (FC), and then decelerates to the end point B. Before the workpiece reaches to the position Bx, the feed rate of XM-axis starts the deceleration so as to reach the target velocity in position Bx. Therefore, there exists uniform velocity, acceleration, and deceleration in one cutting path (Fig. 11a). The number of accelerations and decelerations depends on the number of feed components. FCX _ i, j and FCZ _ i, j represent the real feed rate of XM-axis and ZM-axis when the machine axes move from position (Oi,j) to the next position (Oi + 1,j): FCX FCZ

i,j

i,j

¼ ¼

XOiþ1, j  XOi, j XOiþ1, j  XOi, j

2

þ ZOiþ1, j  ZOi, j

2

ð11Þ

ZOiþ1, j  ZOi, j XOiþ1, j  XOi, j

2

þ ZOiþ1, j  ZOi, j

FC

2

FC

The ac/deceleration distance (LdX_i, j, LdZ_i, j) in the XM-axis and ZM-axis direction when machine axes move from position Oi, j to position Oi+1, j can be calculated by the following equations: LdX LdZ

i,j

¼

i,j

¼

FCX FCZ

þ FCX 120 i, j þ FCZ 120 i, j

iþ1, j iþ1, j

t0

ð12Þ

t0

The cutting time (tCj) used to finish the cutting path can be represented as:

t

C

M j

¼

t0 þ i¼1

60

XOiþ1, j  XOi, j

2

þ ZOiþ1, j  ZOi, j

2



LdX

i, j

2

þ LdZ

i, j

2

FC

ð13Þ Equations (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), and (13) represent quantitative relationships between the cutting strategy, the size and geometrical profile of the workpiece, the total tool path length (L ), and the machining time (T ). Referring to Eqs. (7), (9), and (14), the developed machining time evaluation model considers several factors, such as the distribution of NC blocks, the cutting feed rate, the rapid feed rate, the acceleration and deceleration constants of ultraprecision machine, and the geometrical information of the machined workpiece.

18

S. Wang and S. To

Surface Generation Model Material Removal Mechanism In raster milling, the motions of the diamond tool on the spindle include the rotations around the spindle and linear movement, and the workpiece on the table also makes linear movement. The relative movement between the diamond tool and workpiece removes materials from the workpiece so as to achieve the desired surface profile and surface quality. As shown in Fig. 12, when the diamond tool rotates around the spindle, the tool tip makes contact with the workpiece and the materials are removed from the workpiece, then the tool-edge profiles are generated on the machine surface (Fig. 13a). In the other direction, the tool tip engages into the workpiece and the tool tip profile is replicated onto the machined workpiece, which forms the tool-nose profiles (Fig. 13b). In Figs. 12 and 13, LX and LY are the lengths of tool-edge profile and tool-nose 2

2

profile in one single revolution. LX ¼ 2 L2  L  aep ; LY ¼ 2 r 2  r  aep . LF and LP are the contact lengths of the tool tip and the workpiece in the feed direction and the raster direction. For horizontal cutting, LF ¼ LX ¼ 2 2L  aep  aep 2 ; LP ¼ LY ¼ 2 2r  aep  aep 2 ; in vertical cutting, LF ¼ LY ¼ 2 2r  aep  aep 2 ; LP ¼ LX ¼ 2 2L  aep  aep 2 . The ratio (e) of the contact length in feed direction to the length in raster direction is defined as: e¼

LF ¼ LP

Fig. 12 Material removal mechanism in UPRM

2R1  aep 2R2  aep

ð14Þ

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

19

Fig. 13 Surface roughness profiles in two directions

where R1 refers to swing distance (L ) and R2 refers to tool-nose radius (r) in horizontal cutting; while for vertical cutting, R1 is the tool-nose radius (r) and R2 is the swing distance (L ). In horizontal cutting, e > 1; while in vertical cutting, e < 1. The ratio (e) is an important factor affecting surface quality in UPRM, which will be discussed in detail later. The volume of removed materials per revolution is the product of the engaged tool area (Ak) and the arc length of locus (dsg) where the tool tip contacts with the workpiece in one revolution: V ¼ Ak  dsgk ¼

T 0

Ak 

2

2

2

ðxgk Þ0 þ ðyg k Þ0 þ ðzg k Þ0 dt

ð15Þ

where dsg refers to the arc length of tool-part contact point locus where the tool makes contact with the workpiece in one revolution. The machining time (T ) refers to the time when the diamond tool keeps contacts with materials being cut: T¼

L  aep 30 π  2 sin 1 πS L

ð16Þ

Ak refers to the local contact area where the tool tip contacts with the workpiece (Fig. 15b): Ak ¼ r 2 cos 1

r  ak  ðr ak Þ r

2rak  ak 2

ð17Þ

20

S. Wang and S. To

Fig. 14 The tool tip locus in one revolution for two cutting strategies

where ak is the local axial depth of cut in the current position Pk which is determined by the defined depth of cut (aep), the swing distance of diamond tool (L), and the position angel (δk): ak ¼ L 

L  aep sin δk

ð18Þ

Figure 14 shows the tool-workpiece contact locus in one revolution for horizontal cutting and vertical cutting. It is found that the contact locus is different for different cutting strategies and the axial depth of cut ak changes in one revolution: firstly increasing from zero to aep then decreasing from aep to zero.

Surface Roughness Topography Pattern The volume of material removal per revolution in UPRM presented in Eq. (16) is suitable for one feed and one step, while the generated surface roughness in the process of milling is the combinations of several steps and several feeds. Theoretically, the generated surface quality in NC machining depends on cutting parameters (feed rate, spindle speed, etc), tool geometries (swing distance, tool-nose radius, and tool-nose waviness), and machine axes movement accuracy. However, the diamond tool removes materials by making discontinuous contacts with the workpiece in UPRM. This kind of material removal mechanism generates a series of surface topography patterns on the machined workpiece, the effects of which on surface quality cannot be ignored in ultraprecision machining. Figure 15 shows two different scenarios of the generation of the surface roughness topography patterns in UPRM (with and without shift length). Δlj refers to the distance of two start positions where the diamond tool starts removing materials from workpiece between the jth step and the ( j + 1)th step, which is called the shift length between the jth step and the ( j + 1)th step. They clearly describe that the existence or inexistence of shift length (Δlj) changes the patterns of surface topography.

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

21

Fig. 15 Surface generation pattern in two conditions

As can be seen from Fig. 15b, the shift length (Δlj) depends on the relationship between the time (tj ) and the period of spindle rotation (Tcircle): Δlj ¼ MOD

tj T circle

 T circle  FC ¼ MOD

tj T circle

 f i,j

ð19Þ

where tj is the time used to finish one step (the jth step) and can be calculated according to Eqs. (8) and (9); Tcircle is the period of spindle rotation (T circle ¼ 60=S); S is the spindle speed (rpm); and fi, j is the feeding length per revolution which is equal to the cutting feed rate (FC) in mm per min/spindle speed (S) in rpm: f i,j ¼ FSC. The MOD function returns the remainder after a number is divided by a divisor. For tj ) is an integer, the remainder of the ratio example, if the value of the ratio (T circle MOD(μj) is equal to zero. Referring to Eqs. (8) and (9), when machining a flat surface with a rectangle machining area, the real length of cutting path (lj) meets: lj ¼ lj + 1( j ¼ 1,    N), the time used to finish one step: tj ¼ tjþ1 ðj ¼ 1,    N Þ. The cutting feed rate (FC) is kept

22

S. Wang and S. To

constant during the whole machining process, which is f i,j ¼ f i,jþ1 ¼ f iþ1,j ¼ f ¼ FC=S . Therefore, the shift length Δlj in the fabrication of flat surface meets the conditions: (1) Δl1 ¼ 0; (2) Δlj ¼ Δlj + 1 ¼ Δl( j ¼ 2,    N  1). Referring to Eq. (20), the shift ratio (η) defined as the ratio of shift length (Δl) to the feed length per revolution ( f ) can be represented as: η¼

tj Δl ¼ MOD μj ¼ MOD f T circle

ð20Þ

From Eqs. (8), (9), and (20), the shift length (Δl) depends on the size of workpiece, the cutting parameters including the cutting feed rate, the rapid feed rate, and spindle speed, the acceleration/deceleration time (t0) of the ultraprecision machine, and TPG including the path interval (l _ P), the tool path linking method, the selection of feed direction, the entry and retreat distance in feed direction (lej, lrj), and the clearance height (hcj). This means that the shift length can be controlled by developing a cutting strategy for machining a specified part, so as to achieve the desired surface topography pattern.

Effect of Tool-Interference on Surface Generation From Fig. 15, the generated surface topography pattern in UPRM depends on the relationships between the feeding length per revolution ( f ), the path interval (l _ P), the shift length, and the contact lengths (LF, LP) in the feed direction and raster direction. The feeding length per revolution ( f ) and the path interval (l _ P) meet: LP f  LF 2 and l P  2 , then the formation of the surface roughness topography profile is clear and regular. This infers that, for large feeding length per revolution ( f ) and large path interval (l _ P), the diamond tool cuts sequentially and individually for each revolution, which is referred to as the “noninterference of tool” in this study. However, in the ultraprecision machining process, a high spindle speed and a fine feed rate together with a small path interval are usually adopted to ensure the highquality requirement. When the shift length is zero, under the small feeding length per revolution and path interval, the surface topography pattern remains clear and regular as shown in Fig. 18a. In Fig. 18b, due to the existing shift length, the small values of feeding length per revolution ( f ) and path interval (l _ P) result in irregular surface topography patterns on the machined surface since some areas left by the previous cuttings are removed by the preceding several cuttings, which are referred to as the “tool-interference” in this study. Therefore, the investigation on the scallop generation mechanism in UPRM is divided into three parts: (1) shift length is equal to zero (Δl ¼ 0); (2) Δl 6¼ 0 and no tool-interference; and (3) Δl 6¼ 0 with tool-interference (Fig. 16).

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

23

Fig. 16 Surface generation topography patterns for small values of feeding length and path interval

Feed-Interval and Path-Interval Scallop Height Without Shift Length In UPRM, two scallops exist in both feed direction and raster direction, which are called the feed-interval and path-interval scallops, respectively. Figure 17 shows the generations of feed-interval and path-interval scallops in the absence of shift length. It shows that, if Δl ¼ 0, the theoretical values of the feed-interval scallop height (h _ fi, j) and the path-interval scallop height (h _ pi, j) in the ith feeding of the jth step are described as: h f iþ1,j ¼ h f i,j ¼ h f 0 ¼ R1  h pi,jþ1 ¼ h pi,j ¼ h p0 ¼ R2 

f i,j 2 4 l Pj 2 R2 2  4

R1 2 

ð21Þ

24

S. Wang and S. To

Fig. 17 Scallop heights calculation model in the absence of shift length

where R1 refers to the swing distance (L ) and R2 refers to the tool-nose radius (r) in horizontal cutting; while in vertical cutting, R1 refers to the tool-nose radius (r) and R2 refers to the swing distance (L ). In Fig. 17, Lfi, j and Lf 0i, j refer to the lengths of the generated cutting profile in the feed direction, while Wpi, j and Wp0i, j are the widths of the generated cutting profile in raster direction and depend on the path interval which only depend on the feeding length per revolution:

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

1 Lf i, j ¼ Lf 0i, j ¼ f 2 1 0 Wpi, j ¼ Wpi, j ¼ l P 2

25

ð22Þ

The maximum peak-to-valley heights in feed and raster directions can be represented as follows: Rt Rt

feed

raster

¼ h f 0 ¼ R1  ¼ h p 0 ¼ R2 

R1 2  R2

2

f2 4

l P2  4

ð23Þ

Effect of Shift Length on Scallop Heights Without Tool-Interference The existence of shift length has an important effect on surface finishing in UPRM, which in this study is termed the “shift effect.” As mentioned, if the feeding length LP and path interval meet two conditions: f  LF 2 and l P  2 , no tool-interference exists in both feed direction and raster direction. Under these conditions, the feedinterval scallop height is the same with that shown in Eq. (22), while the path interval height is affected by the presence of shift length (Fig. 18). The existence of shift length generates unparallel surface topography patterns on the machined surface, which results in uneven surface profiles in raster direction by generating a shift height (Δhpj) in the ZM-axis direction of the machine coordinate system. The shift height (Δhpj) in the ZM-axis direction can be described as:

Fig. 18 Generation of shift height in raster direction for shift length

26

S. Wang and S. To

Fig. 19 Calculation model of path-interval scallop height for the existence of shift length

Δhp1 ¼ 0 Δhpj ¼

R1 2  ððj  1ÞΔlÞ2 

Δhpj ¼

R1 2  ððj  1Þ  ΔlÞ2 

R1 2  ðj  ΔlÞ2 ; j ¼ 2,   , u 2

R1 2  f i,j  j  Δl ; j ¼ u þ 1,   , N ð24Þ

where R1 in horizontal cutting refers to swing distance (L ), while in vertical cutting, 1 R1 is tool-nose radius (r). u ¼ INT 2η ; η is the shift ratio. The INT function returns the integer portion of a number. Figure 19 shows the calculation model of path-interval scallop height and the maximum peak-to-valley height in raster direction under the consideration of shift effect. It shows that the path-interval scallop height (h p0i,j ) between the jth and ( j + 1)th steps not only depends on the tool geometry and path interval, but also is affected by the generated shift height induced by the existence of shift length: if : j ¼ 1,   u  1 ) h p0i,j ¼

j k¼1

Δhpk þ R2 þ

1 Δhpjþ1  Δhpj  2

R2 2 1þ

Δhpjþ1 Δhpj l P

2



l P2 4

h p0i,u ¼ uþ1 k¼1

Δhpk þ R2 þ

1 Δhpuþ1  Δhpu  2

R2 2 1þ

Δhpuþ1 Δhpu l P

2



l P2 4 ð25Þ

1

Modeling of Surface Generation with Cutting Strategy Effect. . .

27

where the factor u is the same with that in Eq. (24). The widths of each generated surface profile in the raster direction (Wpi, j and Wp0i,j ) are changed as follows: R2 2  Δhpj 2

1 Wp0i,j ¼ l P  2

Δhpj 2 þ l P 2

R2  Δhpjþ1

1 Wpi,j ¼ l P þ 2

2

uþ1 raster

¼

Δhpk þ R2 þ k¼1

¼ R2 þ R1  

_ raster)

1 Δhpu  Δhpuþ1  2

R1 2  ðu þ 1Þ2  Δl2 þ

1 2

R1 2  u2  Δl2 

2



Δhpjþ1 4

ð26Þ

2

in the raster direction therefore

R2 2  l P2 l P2  2 2 4 l P þ Δhpu

R1 2  u2  Δl2 

R2 2  l P2 l P2 þ

Δhpj 2 4

2

Δhpjþ1 þ l P

The maximum peak-to-valley height (Rt can be represented as follows: Rt

 2

R1 2  ðf  ðu þ 1Þ  ΔlÞ

2



R1 2  ðf  ðu þ 1Þ  ΔlÞ2

l P2 4

ð27Þ

Effect of Shift Length on Scallop Heights with Tool-Interference Figure 20 illustrates the tool-workpiece contact loci when both the shift length and tool-interference exist in the feed direction. It is found that a small value of path interval generates a shift height (Δhfj) in feed direction between the jth step and the ( j  1)th step: Δhf j ¼ R2 2  ðj  1Þ2 l P2  R2 2  ðj  l PÞ2 ( j ¼ 1,   , N ). When the shift height in feed direction is smaller than the feed-interval scallop height, that is, Δhfj + 1 < h _ fi, j, the materials left by the previous steps (Ci, j, Ci + 1, j) will be removed by the following steps (Ci, j + 1, Ci + 1, j + 1, Ci, j + 2, Ci + 1, j + 2). This results in tool interference in feed direction and produces irregular patterns on the machined surface. Accordingly, the tool-interference will appear in feed direction if the developed cutting strategy and tool geometry meet the three conditions simultaneously: (1) Δl 6¼ 0; (2) l P