Acta Mechanica Sinica

, Volume 31, Issue 3, pp 406–415 | Cite as

A high-order full-discretization method using Hermite interpolation for periodic time-delayed differential equations

  • Yilong LiuEmail author
  • Achim Fischer
  • Peter Eberhard
  • Baohai Wu
Research Paper


A high-order full-discretization method (FDM) using Hermite interpolation (HFDM) is proposed and implemented for periodic systems with time delay. Both Lagrange interpolation and Hermite interpolation are used to approximate state values and delayed state values in each discretization step. The transition matrix over a single period is determined and used for stability analysis. The proposed method increases the approximation order of the semidiscretization method and the FDM without increasing the computational time. The convergence, precision, and efficiency of the proposed method are investigated using several Mathieu equations and a complex turning model as examples. Comparison shows that the proposed HFDM converges faster and uses less computational time than existing methods.


Full-discretization method Time delay Stability  Chatter 



This project was partially supported by a scholarship from the China Scholarship Council while Y.L. was visiting the University of Stuttgart. A.F. and P.E. would like to thank the German Research Foundation (DFG) for financial support within the Cluster of Excellence in Simulation Technology (EXC 310) at the University of Stuttgart.


  1. 1.
    Ruby, L.: Applications of the Mathieu equation. Am. J. Phys. 64, 39–44 (1996)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Insperger, T., Stépán, G.: Stability chart for the delayed Mathieu equation. Proc. R. Soc. Lond. Ser. A 458, 1989–1998 (2002)zbMATHCrossRefGoogle Scholar
  3. 3.
    Insperger, T., Stépán, G.: Stability of the damped Mathieu equation with time delay. J. Dyn. Syst. Meas. Control 125, 166–171 (2003)CrossRefGoogle Scholar
  4. 4.
    Morrison, T.M., Rand, R.H.: 2:1 resonance in the delayed nonlinear Mathieu equation. Nonlinear Dyn. 50, 341–352 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Altintas, Y., Engin, S., Budak, E.: Analytical stability prediction and design of variable pitch cutters. J. Manuf. Sci. Eng. 121, 173–178 (1999)CrossRefGoogle Scholar
  6. 6.
    Insperger, T., Stépán, G., Bayly, P.V., et al.: Multiple chatter frequencies in milling processes. J. Sound Vib. 262, 333–345 (2003)CrossRefGoogle Scholar
  7. 7.
    Segalman, D.J., Butcher, E.A.: Suppression of regenerative chatter via impedance modulation. J. Vib. Control 6, 243–256 (2000)CrossRefGoogle Scholar
  8. 8.
    Stépán, G.: Modelling nonlinear regenerative effects in metal cutting. Philos. Trans. R. Soc. Lond. Ser. A 359, 739–757 (2001)zbMATHCrossRefGoogle Scholar
  9. 9.
    Campbell, S.A., Ruan, S., Wei, J.: Qualitative analysis of a neural network model with multiple time delays. Int. J. Bifurcat. Chaos 9, 1585–1595 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Insperger, T., Stépán, G.: Stability improvements of robot control by periodic variation of the gain parameters. In: Proceedings of the 11th World Congress in Mechanism and Machine Science, 1816–1820 (2004)Google Scholar
  11. 11.
    Stépán, G.: Retarded Dynamical Systems: Stability and Characteristic Functions. Longman Scientific & Technical Marlow, New York (1989)zbMATHGoogle Scholar
  12. 12.
    Stépán, G., Kollár, L.: Balancing with reflex delay. Math. Comput. Model. 31, 199–205 (2000)zbMATHCrossRefGoogle Scholar
  13. 13.
    Insperger, T., Stépán, G.: Semi-discretization method for delayed systems. Int. J. Numer. Methods Eng. 55, 503–518 (2002)zbMATHCrossRefGoogle Scholar
  14. 14.
    Insperger, T., Stépán, G.: Updated semi-discretization method for periodic delay-differential equations with discrete delay. Int. J. Numer. Methods Eng. 61, 117–141 (2004)zbMATHCrossRefGoogle Scholar
  15. 15.
    Insperger, T., Stépán, G., Turi, J.: On the higher-order semi-discretizations for periodic delayed systems. J. Sound Vib. 313, 334–341 (2008)CrossRefGoogle Scholar
  16. 16.
    Ding, Y., Zhu, L., Zhang, X., et al.: A full-discretization method for prediction of milling stability. Int. J. Mach. Tools Manuf. 50, 502–509 (2010)CrossRefGoogle Scholar
  17. 17.
    Liu, Y., Zhang, D., Wu, B.: An efficient full-discretization method for prediction of milling stability. Int. J. Mach. Tools Manuf. 63, 44–48 (2012)CrossRefGoogle Scholar
  18. 18.
    Chanda, A., Fischer, A., Eberhard, P., et al.: Stability analysis of a thin-walled cylinder in turning operation using the semi- discretization method. Acta Mech. Sin. 30, 214–222 (2013)CrossRefGoogle Scholar
  19. 19.
    Fischer, A., Eberhard, P.: Improving the dynamic stability of a workpiece dominated turning process using an adaptronic tool holder. Theor. Appl. Mech. Lett. 3, 013008 (2013)CrossRefGoogle Scholar
  20. 20.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)zbMATHCrossRefGoogle Scholar
  21. 21.
    Insperger, T., Stépán, G.: Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications. Springer, New York (2011)CrossRefGoogle Scholar
  22. 22.
    Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press, Cambridge (2005)zbMATHCrossRefGoogle Scholar
  23. 23.
    Ambrósio, J.A.C.: Distributed deformation: a finite element method. In: Ambrósio, J.A.C., Eberhard, P. (eds.) Advanced Design of Mechanical Systems: From Analysis to Optimization, pp. 351–374. Springer, Berlin (2009)CrossRefGoogle Scholar
  24. 24.
    Wallrapp, O.: Standardization of flexible body modeling in multibody system codes, part I: definition of standard input data. Mech. Struct. Mach. 22, 283–304 (1994)CrossRefGoogle Scholar
  25. 25.
    Fehr, J.: Automated and error controlled model reduction in elastic multibody systems. Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart 21, Shaker Verlag, Aachen (2011)Google Scholar
  26. 26.
    Fischer, A., Eberhard, P.: Simulation-based stability analysis of a thin-walled cylinder during turning with improvements using an adaptronic turning chisel. Arch. Mech. Eng. 58, 367–391 (2011)Google Scholar
  27. 27.
    Fischer, A., Eberhard, P., Ambrósio, J.: Parametric flexible multibody model for material removal during turning. J. Comput. Nonlinear Dyn. 9, 011007 (2013)CrossRefGoogle Scholar
  28. 28.
    Henninger, C., Eberhard, P.: Improving the computational efficiency and accuracy of the semi-discretization method for periodic delay-differential equations. Eur. J. Mech. A/Solids 27, 975–985 (2008)zbMATHCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Yilong Liu
    • 1
    Email author
  • Achim Fischer
    • 2
  • Peter Eberhard
    • 2
  • Baohai Wu
    • 1
  1. 1.Key Laboratory of Contemporary Design and Integrated Manufacturing TechnologyNorthwestern Polytechnical UniversityXi’anChina
  2. 2.Institute of Engineering and Computational MechanicsUniversity of StuttgartStuttgartGermany

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