Geometrical theory of cutting stock with torus end mills in five-axis CNC machining and its applications in machining simulation

  • Zhiyong Chang
  • Junjie Qian
  • Zezhong C. ChenEmail author
  • Neng Wan
  • Dinghua Zhang


We have established the geometric theory of cutting stock with flat end mills in five-axis CNC machining and an accurate and efficient approach to 3D geometric modeling of un-deformed chips in this machining (Chang et al. Comput Aided Des 88:42–59, 2017). The work has laid a theoretical foundation for geometrical and physical simulation for five-axis milling. However, fillet and ball end mills are more popular than flat end mills. The equations of the instantaneous cutting edges of a fillet end mill are more complicated in the geometric theory, and the boundary of the area covered by the cutting edges is more difficult to determine. Therefore, this article is to formulate the instantaneous cutting edges and their critical points in five-axis milling and to illustrate how to use the boundary construction diagram to determine the boundary with two examples of the geometric and the physical simulations. This work is verified with the valid results of the examples, and the results indicate that this model can take into account the errors of cutter and machine tool. Therefore, the geometric theory is viable, and its application on the five-axis milling simulation is feasible for high-performance machining.


Geometric theory of cutting stock Un-deformed chip geometry In-process workpiece model Machining simulation Five-axis CNC milling 


Funding information

This study received financial support from the National Natural Science Foundations of China (Grant No. 51775445), the Aeronautical Science Foundation of China (Grant No. 2017ZE53053), the major R&D project of Shaanxi Province (Grant No. 2019GY-64), and the National Science and Engineering Research Council of Canada.


  1. 1.
    Chang Z, Chen ZC, Mo R, Zhang DH, Deng Q (2017) An accurate and efficient approach to geometric modeling of undeformed chips in five-axis CNC milling. Comput Aided Des 88:42–59MathSciNetCrossRefGoogle Scholar
  2. 2.
    Choi BK, Jerard RB (1998) Sculptured surface machining: theory and applications. Kluwer Academic, Dordrecht, LondonCrossRefGoogle Scholar
  3. 3.
    Yau HT, Tsou LS (2009) Efficient NC simulation for multi-axis solid machining with a universal APT cutter. Trans ASME J Manuf Sci Eng 9(2):021001–021001Google Scholar
  4. 4.
    Zhou Y, Chen ZC, Yang X (2015) An accurate, efficient envelope approach to modeling the geometric deviation of the machined surface for a specific five-axis CNC machine tool. Int J Mach Tools Manuf 95:67–77CrossRefGoogle Scholar
  5. 5.
    Joy J, Feng HY (2017) Frame-sliced voxel representation: an accurate and memory-efficient modeling method for workpiece geometry in machining simulation. Comput Aided Des 88:1–13CrossRefGoogle Scholar
  6. 6.
    Arai W, Tanaka F, Onosato M (2018) Error estimation of machined surfaces in multi-axis machining with machine tool errors including tool self-intersecting motion based on high-accuracy tool swept volumes. Int J Autom Technol 5:680–687CrossRefGoogle Scholar
  7. 7.
    Li ZL, Zhu LM (2018) An accurate method for determining cutter-workpiece engagement in five-axis milling with a general tool considering cutter runout. Trans ASME J Manuf Sci Eng 140(2):021001CrossRefGoogle Scholar
  8. 8.
    Aras E Five axis swept profiles of torus like cutters via separation of inner and outer characteristic curves. Proceedings of the ASME Design Engineering Technical Conference, 1A-2018.Google Scholar
  9. 9.
    Machchhar J, Plakhotnik D, Elber G (2017) Precise algebraic-based swept volumes for arbitrary free-form shaped tools towards multi-axis CNC machining verification. Comput Aided Des 90:48–58CrossRefGoogle Scholar
  10. 10.
    Bo PB, Barton M, Plakhotnik D, Pottmann H (2016) Towards efficient 5-axis flank CNC machining of free-form surfaces via fitting envelopes of surfaces of revolution. Comput Aided Des 79:1–11CrossRefGoogle Scholar
  11. 11.
    Voelcker HB, Hunt WA (1981) Role of solid modelling in machining-process modelling and NC verification. SAE PreprintsGoogle Scholar
  12. 12.
    Spence AD, Altintas Y (1994) A solid modeler based milling process simulation and planning system. Trans ASME J Eng Ind 116:61–69CrossRefGoogle Scholar
  13. 13.
    Lee P, Altintas Y (1996) Prediction of ball-end milling forces from orthogonal cutting data. Int J Mach Tools Manuf 36:1059–1072CrossRefGoogle Scholar
  14. 14.
    Yucesan G, Altintas Y (1996) Prediction of ball end milling forces. Trans ASME J Eng Ind 118:95–103CrossRefGoogle Scholar
  15. 15.
    El Mounayri H, Spence AD, Elbestawi MA (1998) Milling process simulation - a generic solid modeler based paradigm. Trans ASME J Manuf Sci Eng 120:213–221CrossRefGoogle Scholar
  16. 16.
    Imani BM, Sadeghi MH, Elbestawi MA (1998) Improved process simulation system for ball-end milling of sculptured surfaces. Int J Mach Tools Manuf 38:1089–1107CrossRefGoogle Scholar
  17. 17.
    Altintas Y (2000) Manufacturing automation: metal cutting mechanics, machine tool vibrations, and CNC design. Cambridge University Press, New YorkGoogle Scholar
  18. 18.
    Kim GM, Cho PJ, Chu CN (2000) Cutting force prediction of sculptured surface ball-end milling using Z-map. Int J Mach Tools Manuf 40:277–291CrossRefGoogle Scholar
  19. 19.
    Feng HY, Su N (2001) A mechanistic cutting force model for 3D ball-end milling. Trans ASME J Manuf Sci Eng 123:23–29CrossRefGoogle Scholar
  20. 20.
    Zhu R, Kapoor SG, DeVor RE (2001) Mechanistic modeling of the ball end milling process for multi-axis machining of free-form surfaces. J Manuf Sci Eng 123:369–379CrossRefGoogle Scholar
  21. 21.
    Yip-Hoi D, Huang X (2006) Cutter/workpiece engagement feature extraction from solid models for end milling. Trans ASME J Manuf Sci Eng 128:249–260CrossRefGoogle Scholar
  22. 22.
    Ferry W, Yip-Hoi D (2008) Cutter-workpiece engagement calculations by parallel slicing for five-axis flank milling of jet engine impellers. Trans ASME J Manuf Sci Eng 130(5):051011–051011CrossRefGoogle Scholar
  23. 23.
    Guo DM, Ren F, Sun Y (2010) An approach to modeling cutting forces in five-axis ball-end milling of curved geometries based on tool motion analysis. Trans ASME J Manuf Sci Eng 132(4):041004Google Scholar
  24. 24.
    Wei ZC, Wang MJ, Zhu JN, Gu LY (2011) Cutting force prediction in ball end milling of sculptured surface with Z-level contouring tool path. Int J Mach Tools Manuf 51(5):428–432CrossRefGoogle Scholar
  25. 25.
    Boz Y, Erdim H, Lazoglu I (2011) Modeling cutting forces for five axis milling of sculptured surfaces. Adv Mater Res 223:701–712CrossRefGoogle Scholar
  26. 26.
    Sullivan A, Erdim H, Perry RN, Frisken SF (2012) High accuracy NC milling simulation using composite adaptively sampled distance fields. Comput Aided Des 44(6):522–536CrossRefGoogle Scholar
  27. 27.
    Li ZL, Wang XZ, Zhu LM (2016) Arc-surface intersection method to calculate cutter-workpiece engagements for generic cutter in five-axis milling. Comput Aided Des 73:1–10CrossRefGoogle Scholar
  28. 28.
    Comak A, Altintas Y (2017) Mechanics of turn-milling operations. Int J Mach Tools Manuf 121:2–9CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Zhiyong Chang
    • 1
  • Junjie Qian
    • 1
  • Zezhong C. Chen
    • 2
    Email author
  • Neng Wan
    • 1
  • Dinghua Zhang
    • 1
  1. 1.The Key Laboratory of Contemporary Design and Integrated Manufacturing Technology of Ministry of Education, Department of Advanced Manufacturing Engineering, School of Mechanical EngineeringNorthwestern Polytechnical UniversityXi’anChina
  2. 2.Department of Mechanical and Industrial EngineeringConcordia UniversityMontrealCanada

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