Abstract
In this paper, we present a model of turning operations with state-dependent distributed time delay. We apply the theory of regenerative machine tool chatter and describe the dynamics of the tool-workpiece system during cutting by delay differential equations. We model the cutting force as the resultant of a force system distributed along the rake face of the tool, which results in a short distributed delay in the governing equation superimposed on the large regenerative delay. According to the literature on stress distribution along the rake face, the length of the chip–tool interface, where the distributed cutting force system is acting, is function of the chip thickness, which depends on the vibrations of the tool-workpiece system due to the regenerative effect. Therefore, the additional short delay is state dependent. It is shown that involving state-dependent delay in the model does not affect linear stability properties, but does affect the nonlinear dynamics of the cutting process. Namely, the sense of the Hopf bifurcation along the stability boundaries may turn from sub- to supercritical at certain spindle speed regions.
Similar content being viewed by others
References
Altintas, Y.: Manufacturing Automation-Metal Cutting Mechanics, Machine Tool Vibrations and CNC Design, 2nd edn. Cambridge University Press, Cambridge (2012)
Astakhov, V.P., Outeiro, J.C.: Modeling of the contact stress distribution at the tool–chip interface. Mach. Sci. Technol. 9(1), 85–99 (2005)
Bachrathy, D., Stépán, G., Turi, J.: State dependent regenerative effect in milling processes. J. Comput. Nonlinear Dyn. 6(4), 041002 (2011)
Bagchi, A., Wright, P.K.: Stress analysis in machining with the use of sapphire tools. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 409(1836), 99–113 (1987)
Barrow, G., Graham, W., Kurimoto, T., Leong, Y.F.: Determination of rake face stress distribution in orthogonal machining. Int. J. Mach. Tool Des. Res. 22(1), 75–85 (1982)
Bélair, J., Mackey, M.C.: Consumer memory and price fluctuations in commodity markets: an integrodifferential model. J. Dyn. Differ. Equ. 1(3), 299–325 (1989)
Buryta, D., Sowerby, R., Yellowley, I.: Stress distributions on the rake face during orthogonal machining. Int. J. Mach. Tools Manuf. 34(5), 721–739 (1994)
Chandrasekaran, H., Kapoor, D.V.: Photoelastic analysis of chip–tool interface stresses. ASME J. Eng. Ind. 87, 495–502 (1965)
Childs, T.H.C., Mahdi, M.I.: On the stress distribution between the chip and tool during metal turning. CIRP Ann. Manuf. Technol. 38(1), 55–58 (1989)
Cooke, K.L., Huang, W.: On the problem of linearization for state-dependent delay differential equations. Proc. Am. Math. Soc. 124(5), 1417–1426 (1996)
Dombóvári, Z., Stépán, G.: The effect of helix angle variation on milling stability. J. Manuf. Sci. Eng. Trans. ASME 134(5), 051015 (2012)
Dombóvári, Z., Wilson, R.E., Stépán, G.: Estimates of the bistable region in metal cutting. Proc. R. Soc. A Math. Phys. Eng. Sci. 464, 3255–3271 (2008)
Driver, R.D.: A two-body problem of classical electrodynamics: the one-dimensional case. Ann. Phys. 21(1), 122–142 (1963)
Engelborghs, K., Luzyanina, T., Roose, D.: Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. 28(1), 1–21 (2002)
Germay, C., Denoël, V., Detournay, E.: Multiple mode analysis of the self-excited vibrations of rotary drilling systems. J. Sound Vib. 325(1–2), 362–381 (2009)
Hajdu, D., Insperger, T.: Demonstration of the sensitivity of the Smith predictor to parameter uncertainties using stability diagrams. Int. J. Dyn. Control (2014). doi:10.1007/s40435-014-0142-1
Hartung, F.: Linearized stability in periodic functional differential equations with state-dependent delays. J. Comput. Appl. Math. 174(2), 201–211 (2005)
Hartung, F., Krisztin, T., Walther, H.O., Wu, J.: Functional differential equations with state-dependent delays: theory and applications. In: Drábek, P., Cañada, A., Fonda, A. (eds.) Handbook of Differential Equations: Ordinary Differential Equations, pp. 435–545. Elsevier/North-Holland, Amsterdam (2006)
Hartung, F., Turi, J.: Stability in a class of functional differential equations with state-dependent delays. In: Corduneanu, C. (ed.) Qualitative Problems for Differential Equations and Control Theory, pp. 15–31. World Scientific, Singapore (1995)
Hartung, F., Turi, J.: On differentiability of solutions with respect to parameters in state-dependent delay equations. J. Differ. Equ. 135(2), 192–237 (1997)
Hartung, F., Turi, J.: Linearized stability in functional-differential equations with state-dependent delays. In: Proceedings of the Conference Dynamical Systems and Differential Equations, Added Volume of Discrete and Continuous Dynamical Systems, pp. 416–425 (2000)
Insperger, T., Barton, D.A.W., Stépán, G.: Criticality of hopf bifurcation in state-dependent delay model of turning processes. Int. J. Non-Linear Mech. 43(2), 140–149 (2008)
Insperger, T., Stépán, G., Turi, J.: State-dependent delay in regenerative turning processes. Nonlinear Dyn. 47(1–3), 275–283 (2007)
Kato, S., Yamaguchi, K., Yamada, M.: Stress distribution at the interface between tool and chip in machining. ASME J. Eng. Ind. 94(2), 683–689 (1972)
Kilic, D.S., Raman, S.: Observations of the tool-chip boundary conditions in turning of aluminum alloys. Wear 262(7–8), 889–904 (2007)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)
Lehotzky, D., Turi, J., Insperger, T.: Stabilizability diagram for turning processes subjected to digital PD control. Int. J. Dyn. Control 2(1), 46–54 (2014)
Liu, X., Vlajic, N., Long, X., Meng, G., Balachandran, B.: Nonlinear motions of a flexible rotor with a drill bit: stick-slip and delay effects. Nonlinear Dyn. 72(1–2), 61–77 (2013)
Liu, X., Vlajic, N., Long, X., Meng, G., Balachandran, B.: State-dependent delay influenced drill-string oscillations and stability analysis. J. Vib. Acous. 136(5), 051008 (2014)
Manitius, A.Z., Olbrot, A.W.: Finite spectrum assignment problem for systems with delays. IEEE Trans. Autom. Control AC–24(4), 541–553 (1979)
Molnár, T.G., Insperger, T.: On the robust stabilizability of unstable systems with feedback delay by finite spectrum assignment. J. Vib. Control (2014). doi:10.1177/1077546314529602
Molnár, T.G., Insperger, T.: On the effect of distributed regenerative delay on the stability lobe diagrams of milling processes. Period. Polytech. Mech. Eng. 59(3), 126–136 (2015)
Molnár., T.G., Insperger, T., Hogan., S.J., Stépán, G.: Investigating multiscale phenomena in machining: the effect of cutting-force distribution along the tool’s rake face on process stability. In: Proceedings of the ASME International Design Engineering Technical Conferences, DETC2015-47165, Boston, MA (2015)
Sieber, J., Engelborghs, K., Luzyanina, T., Samaey, G., Roose, D.: DDE-BIFTOOL v. 3.0 Manual-Bifurcation analysis of delay differential equations (2014). arxiv.org/abs/1406.7144
Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer, New York (2011)
Stépán, G.: Delay-differential equation models for machine tool chatter. In: Moon, F.C. (ed.) Dynamics and Chaos in Manufacturing Processes, pp. 165–192. Wiley, New York (1997)
Stépán, G.: Delay, nonlinear oscillations and shimmying wheels. In: Moon, F.C. (ed.) New Applications of Nonlinear and Chaotic Dynamics in Mechanics, pp. 373–386. Kluwer, Dordrecht (1999)
Stépán, G., Kalmár-Nagy, T.: Nonlinear regenerative machine tool vibrations. In: Proceedings of DETC’97. In: ASME Design and Technical Conferences, Sacramento, CA, USA, pp. 1–11 (1997)
Sutter, G., Molinari, A., List, G., Bi, X.: Chip flow and scaling laws in high speed metal cutting. ASME J. Manuf. Sci. Eng. 134(2), 021005 (2012)
Szalai, R., Orosz, G.: Decomposing the dynamics of heterogeneous delayed networks with applications to connected vehicle systems. Phys. Rev. 88(4), 040902 (2013)
Takács, D., Orosz, G., Stépán, G.: Delay effects in shimmy dynamics of wheels with stretched string-like tyres. Eur. J. Mech. A/Solids 28(3), 516–525 (2009)
Toropov, A., Ko, S.L.: Prediction of tool–chip contact length using a new slip-line solution for orthogonal cutting. Int. J. Mach. Tools Manuf 43(12), 1209–1215 (2003)
Usui, E., Takeyama, H.: A photoelastic analysis of machining stresses. ASME J. Eng. Ind. 82, 303–307 (1960)
Woon, K.S., Rahman, M., Neo, K.S., Liu, K.: The effect of tool edge radius on the contact phenomenon of tool-based micromachining. Int. J. Mach. Tools Manuf. 48(12–13), 1395–1407 (2008)
Yang, X., Liu, C.R.: A new stress-based model of friction behavior in machining and its significant impact on residual stresses computed by finite element method. Int. J. Mech. Sci. 44(4), 703–723 (2002)
Acknowledgments
This work was supported by the Hungarian National Science Foundation under Grant OTKA- K105433 and OTKA-K101714. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Advanced Grant Agreement No. 340889.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Molnár, T.G., Insperger, T. & Stépán, G. State-dependent distributed-delay model of orthogonal cutting. Nonlinear Dyn 84, 1147–1156 (2016). https://doi.org/10.1007/s11071-015-2559-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-015-2559-2