Abstract
A milling-process model with a variable time delay associated with each cutting tooth is presented in this article. The source of this variable time delay is the feed rate. The effect of the feed motion on the entry cutting angle, the exit cutting angle, and the amplitude of feed mark is also discussed. Loss-of-contact effects are also considered. The system dynamics is described by a set of delay differential equations with periodic coefficients and variable time delays. A semi-discretization scheme is presented for analyzing the stability of periodic orbits of this system. The analysis provides evidence of period-doubling bifurcations and secondary Hopf bifurcations. Good agreement is found between the numerical results obtained from this work and the results of related experimental studies.
Similar content being viewed by others
References
Martellotti, M.E.: An analysis of the milling process. Trans. ASME 63, 677–700 (1941)
Martellotti, M.E.: An analysis of the milling process. Part II: Down Milling. Trans. ASME 67, 233–251 (1945)
Sridhar, R., Hohn, R.E., Long, G.W.: A stability algorithm for the general milling process. ASME J. Eng. Ind. 90, 330–334 (1968)
Balachandran, B.: Nonlinear dynamics of milling processes. Phil. Trans. R. Soc. Lond. A 359, 793–819 (2001)
Balachandran, B., Zhao, M.X.: A mechanics based model for study of dynamics of milling operations. Meccanica 35, 89–109 (2000)
Zhao, M.X., Balachandran, B.: Dynamics and stability of milling process. Int. J. Solids Struc. 38, 2233–2248 (2001)
Long, X.-H., Balachandran, B.: Milling model with variable time delay. In: Proceedings of ASME International Mechanical Engineering Congress and RD&D Expo, Anaheim, CA, Paper No. IMECE2004-59207 pp. 13–19 (2004)
Balachandran, B., Gilsinn, D.: Nonlinear oscillations of milling. Math. Comp. Model. Dynam. Syst. 11, 273–290 (2005)
Tlusty, J., Polacek, M.: The stability of the machine tool against self-Excited vibration in machining. In: Proceedings of the Conference on International Research in Production Engineering, Pittsburgh, PA, pp. 465–474 (1963)
Tobias, S.A.: Machine-tool vibration. Wiley, New York (1965)
Opitz, H., Dregger, E.U., Roese, H.: Improvement of the dynamic stability of the milling process by irregular tooth pitch. In: Proceedings of the 7th International MTDR Conference, Pergamon Press, New York (1966)
Hanna, N.H., Tobias, S.A.: A Theory of nonlinear regenerative chatter. ASME J. Eng. Industry 96, 247–255 (1974)
Minis, I., Yanushevsky, R.: A new theoretical approach for the prediction of machine tool chatter in milling. ASME J. Eng. Industry 115, pp. 1–8 (1993)
Altintas, Y., Budak, E.: Analytical prediction of stability lobes in milling. Ann. CIRP 44, 357–362 (1995)
Mann, B.P., Young, K.A., Schmitz, T.L., Dilley, D.N.: Simultaneous stability and surface location error predictions in milling. ASME J. Manufactur. Sci. Eng. 127, 446–453 (2005)
Insperger, T., Stépán, G.: Semi-discretization method for delayed systems. Int. J. Numer. Methods Eng. 55, 503–518 (2002)
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)
Long, X.-H., Balachandran, B.: Stability analysis for milling process. Nonlinear Dyn., accepted for publication. (2004)
Insperger T., Stépán, G., Hartung, F., Turi, J.: State dependent regenerative delay in milling processes. In: Proceedings of ASME 2005 International Design Engineering Technical Conference & Computers and Information in Engineering Conference, Long Beach, CA (Sept. 24–28, 2005)
Insperger T., Stépán, G.: State-dependent delay in regenerative turning processes. Non-Linear Dyn., submitted for publication. (2005)
Liu, Z., Liao, L.: Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays. J. Math. Anal. Appl. 290, 247–262 (2004)
Hahn, W.: On difference-differential equations with periodic coefficients. J. Math. Anal. Appl. 3, 70–101 (1961)
Farkas, M.: Periodic Vibrations. Springer-Verlag, (1994)
Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, New York (1995)
Insperger, T., Mann, B.P., Stépán, G., Bayly, P.V.: Stability of up-milling and down-milling, part 1: Alternative analytical methods. Int. J. Mach. Tools Manufact. 43, 25–34 (2003)
Insperger, T., Stépán, G., Bayly, P.V., Mann, B.P.: Multiple chatter frequencies in milling processes. J. Sound Vib. 262, 333–345 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Long, XH., Balachandran, B. & Mann, B.P. Dynamics of milling processes with variable time delays. Nonlinear Dyn 47, 49–63 (2007). https://doi.org/10.1007/s11071-006-9058-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-006-9058-4