Advertisement

Transformations from Variable Delays to Constant Delays with Applications in Engineering and Biology

Chapter
First Online:
Part of the Advances in Delays and Dynamics book series (ADVSDD, volume 7)

Abstract

The transformation from systems with time-varying delays or state-dependent delays to systems with constant delays is studied. The transformation exists if the delay is defined by a transport with a variable velocity over a constant distance. In fact, time-varying or state-dependent delays, which are generated by such a mechanism, are common in engineering and biology. We study two paradigmatic time delay systems in more detail. For metal cutting processes, or more precisely turning with spindle speed variation, we show that the analysis of the machine tool vibrations via constant delays in terms of the spindle rotation angle is more advantageous than the conventional analysis of the vibrations with time-varying delays in the time domain. In a second example, motivated by the McKendrick equation modeling structured populations, we show that systems with variable delays and the equivalent systems with constant delays are, in addition, equivalent to partial differential equations with moving and constant boundaries.

Keywords

Spindle Speed Chip Thickness Time Delay System Delay Differential Equation Variable Delay 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Ahmed, A., Verriest, E.: Estimator design for a subsonic rocket car (soft landing) based on state-dependent delay measurement. IEEE Conf. Decis. Control 5698–5703 (2013)Google Scholar
  2. 2.
    Altintas, Y., Weck, M.: Chatter stability of metal cutting and grinding. CIRP Ann. 53, 619–642 (2004)CrossRefGoogle Scholar
  3. 3.
    Ambika, G., Amritkar, R.: Anticipatory synchronization with variable time delay and reset. Phys. Rev. E 79, 056,206 (2009)Google Scholar
  4. 4.
    Bellman, R., Cooke, K.: On the computational solution of a class of functional differential equations. J. Math. Anal. Appl. 12(3), 495–500 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bocharov, G., Rihan, F.: Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math. 125, 183–199 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bresch-Pietri, D., Chauvin, J., Petit, N.: Prediction-based stabilization of linear systems subject to input-dependent input delay of integral-type. IEEE Trans. Autom. Control 59(9), 2385–2399 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bresch-Pietri, D., Petit, N.: Implicit integral equations for modeling systems with a transport delay. In: Witrant, E., Fridman, E., Sename, O., Dugard,L. (eds.) Recent Results on Time-Delay Systems: Analysis and Control, pp. 3–21. Springer (2016)Google Scholar
  8. 8.
    Ghil, M., Zaliapin, I., Thompson, S.: A delay differential model of enso variability: parametric instability and the distribution of extremes. Nonlinear Proc. Geophys. 15, 417–433 (2008)CrossRefGoogle Scholar
  9. 9.
    Hale, J., Lunel, S.: Introduction to functional differential equations. Appl. Math. Sci. No. 99 (1993). SpringerGoogle Scholar
  10. 10.
    Hartung, F., Krisztin, T., Walther, H.O., Wu, J.: Functional differential equations with state-dependent delays: theory and applications. In: Canada, A., Drabek, P., Fonda, A. (eds.) Handbook of Differential Equations: Ordinary Differential Equations, pp. 435–545. North-Holland (2006)Google Scholar
  11. 11.
    Insperger, T., Stépán, G.: Stability analysis of turning with periodic spindle speed modulation via semi-discretization. J. Vib. Control 10(12), 1835–1855 (2004)CrossRefzbMATHGoogle Scholar
  12. 12.
    Insperger, T., Stépán, G.: Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications. Springer (2011)Google Scholar
  13. 13.
    Insperger, T., Stépán, G., Turi, J.: State-dependent delay in regenerative turning processes. Nonlinear Dyn. 47, 275–283 (2007)CrossRefzbMATHGoogle Scholar
  14. 14.
    Jüngling, T., Gjurchinovski, A., Urumov, V.: Experimental time-delayed feedback control with variable and distributed delays. Phys. Rev. E 86, 046,213 (2012)Google Scholar
  15. 15.
    Keyfitz, B., Keyfitz, N.: The mckendrick partial differential equation and its uses in epidemiology and population study. Math. Comput. Model. 26(6), 1–9 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kuang, Y.: Delay Differential Equations: With Applications in Population Dynamics. Academic Press (1993)Google Scholar
  17. 17.
    Kyrychko, Y., Hogan, S.: On the use of delay equations in engineering applications. J. Vib. Control 16, 943–960 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lakshmanan, M., Senthilkumar, D.: Dynamics of Nonlinear Time-Delay Systems. Springer (2011)Google Scholar
  19. 19.
    Mahaffy, J.M., Bélair, J., Mackey, M.C.: Hematopoietic model with moving boundary condition and state dependent delay: applications in erythropoiesis. J. Theor. Biol. 190(2), 135–146 (1998)CrossRefGoogle Scholar
  20. 20.
    Margolis, S., O’Donnell, J.: Rigorous treatments of variable time delays. IEEE Trans. Electron. Comput. 12(3), 307–309 (1963)CrossRefGoogle Scholar
  21. 21.
    Martínez-Llinàs, J., Porte, X., Soriano, M.C., Colet, P., Fischer, I.: Dynamical properties induced by state-dependent delays in photonic systems. Nat. Commun. 6, 7425 (2015)CrossRefGoogle Scholar
  22. 22.
    Masoller, C., Torrent, M.C., García-Ojalvo, J.: Dynamics of globally delay-coupled neurons displaying subthreshold oscillations. Philos. Trans. R. Soc. A 367(1901), 3255–3266 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    McKendrick, A.G.: Applications of mathematics to medical problems. Proc. Edinb. Math. Soc. 44, 98–130 (1925)CrossRefGoogle Scholar
  24. 24.
    Orosz, G., Moehlis, J., Murray, R.M.: Controlling biological networks by time-delayed signals. Philos. Trans. R. Soc. A 368(1911), 439–454 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Otto, A., Radons, G.: Application of spindle speed variation for chatter suppression in turning. CIRP J. Manuf. Sci. Technol. 6(2), 102–109 (2013)CrossRefGoogle Scholar
  26. 26.
    Otto, A., Radons, G.: The influence of tangential and torsional vibrations on the stability lobes in metal cutting. Nonlinear Dyn. 82, 1989–2000 (2015)CrossRefGoogle Scholar
  27. 27.
    Pyragas, V., Pyragas, K.: Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay. Phys. Lett. A 375(44), 3866–3871 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39(10), 1667–1694 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rosin, D.P., Rontani, D., Gauthier, D.J.: Synchronization of coupled boolean phase oscillators. Phys. Rev. E 89, 042,907 (2014)Google Scholar
  30. 30.
    Schley, D., Gourley, S.: Linear stability criteria for population models with periodically perturbed delays. J. Math. Biol. 40, 500–524 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Schuster, H.G., Schöll, E.: Handbook of Chaos Control. Wiley-VCH (2007)Google Scholar
  32. 32.
    Seddon, J., Johnson, R.: The simulation of variable delay. IEEE Trans. Comput. C-17, 89 –94 (1968)Google Scholar
  33. 33.
    Sieber, J.: Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations. Discrete Contin. Dyn. Syst. 32(8), 2607–2651 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Smith, H.: Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study. Math. Biosci. 113(1), 1–23 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Smith, H.: An Introduction to Delay Differential Equations With Applications to the Life Sciences. Springer (2010)Google Scholar
  36. 36.
    Soriano, M.C., García-Ojalvo, J., Mirasso, C.R., Fischer, I.: Complex photonics: dynamics and applications of delay-coupled semiconductors lasers. Rev. Mod. Phys. 85, 421–470 (2013)CrossRefGoogle Scholar
  37. 37.
    Sugitani, Y., Konishi, K., Hara, N.: Experimental verification of amplitude death induced by a periodic time-varying delay-connection. Nonlinear Dyn. 70(3), 2227–2235 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Tsao, T.C., McCarthy, M.W., Kapoor, S.G.: A new approach to stability analysis of variable speed machining systems. Int. J. Mach. Tools Manuf. 33(6), 791–808 (1993)CrossRefGoogle Scholar
  39. 39.
    Verriest, E.I.: Inconsistencies in systems with time-varying delays and their resolution. IMA J. Math. Control Inf. 28(2), 147–162 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Voronov, S., Gouskov, A., Kvashnin, A., Butcher, E., Sinha, S.: Influence of torsional motion on the axial vibrations of a drilling tool. J. Comput. Nonlinear Dyn. 2(1), 58–64 (2007)CrossRefGoogle Scholar
  41. 41.
    Walther, H.O.: On a model for soft landing with state-dependent delay. J. Dyn. Differ. Equ. 19(3), 593–622 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zatarain, M., Bediaga, I., Muñoa, J., Lizarralde, R.: Stability of milling processes with continuous spindle speed variation: analysis in the frequency and time domains, and experimental correlation. CIRP Ann. 57(1), 379–384 (2008)CrossRefGoogle Scholar
  43. 43.
    Zenger, K., Niemi, A.: Modelling and control of a class of time-varying continuous flow processes. J Proc. Control 19(9), 1511–1518 (2009)CrossRefGoogle Scholar
  44. 44.
    Zhang, F., Yeddanapudi, M.: Modeling and simulation of time-varying delays. In: Proceedings of TMS/DEVS, p. 34. San Diego, CA, USA (2012)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Physics, Chemnitz University of TechnologyChemnitzGermany

Personalised recommendations

Cite chapter

Buy options