We study the combined effect of different noise sources on systems with delay. In a model for machine tool vibrations, specifically regenerative chatter, variation of material parameters or external (additive) noise can amplify vibrations via coherence resonance, while random variation of delay can suppress these vibrations. A key feature is the interplay of the noise with several different inherent time scales in the system: the short period of the intrinsic oscillations, the long time scale of their envelope, and the intermediate time interval for the variation of the delay. Optimizing the suppression of vibrations relies on understanding how these time scales and the bifurcation structure influence the combined effect of the noise sources. Furthermore, this multiple-scales viewpoint provides predictions for related types of stochastic dynamics in other systems with delay.
Altintas, Y. and Budak, E., 1995, "Analytical prediction of stability lobes in milling," Annals CIRP44, 357-362.
2.
Bayley, P.V., Lamar, M.T., and Calvert, S.B., 2001, "Low frequency regenerative vibrations and the formation of lobed holes in drilling," Journal of Manufacturing Science and Engineering124, 275-285.
3.
Beuter, A., Belair, J., and Labrie, C., 1993, "Feedback and delays in neurological diseases: a modeling study using dynamical systems," Bulletin of Mathematical Biology55, 525-541.
4.
Buckwar, E., Kuske, R., L’Esperance, B., and Soo, T., 2006, "Noise-sensitivity in machine tool vibrations," International Journal of Bifurcation and Chaos16, 2407-2416.
5.
Cabrera, J.L., Gorronogoitia, J., and de la Rubia, F.J. , 1999, "Noise-correlation time-mediated localization in random nonlinear dynamical systems," Physical Review Letters82, 2816-2819.
6.
Engin, S. and Altintas, Y., 2001, "Mechanics and dynamics of general milling cutters: part I: helical end mills," International Journal of Machine Tools and Manufacture41, 2195-2212.
7.
Engin, S. and Altintas, Y., 2001, "Mechanics and dynamics of general milling cutters: part II: inserted cutters," International Journal of Machine Tools and Manufacture41, 2213-2231.
8.
Garcia-Ojalvo, J. and Roy, R., 1996, "Noise amplification in a stochastic Ikeda model," Physics Letters A224, 51-56.
9.
Insperger, T. and Stépán, G., 2004, "Stability analysis of turning with periodic spindle speed modulation via semidiscretization," Journal of Vibration and Control10, 1835-1855.
10.
Insperger, T. and Stépán, G., 2004, "Vibration frequencies in high-speed milling processes or a positive answer to Davies, Pratt, Dutterer and Burns," Journal of Manufacturing Science and Engineering126, 481-487.
11.
Insperger, T., Mann, B.P., Stépán, G., and Bayley, P.V., 2003, "Stability of up-milling and down-milling, part 1: alternative analytical methods," International Journal of Machine Tools and Manufacture43, 25-34.
12.
Kevorkian, J. and Cole, J.D., 1985, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York.
13.
Kim, S., Park, S.H., and Pyo, H.-B., 1999, "Stochastic resonance in coupled oscillator systems with time delay," Physical Review Letters82, 1620-1623.
14.
Klamecki, B., 2000, "End milling process vibration signal enhancement by stochastic resonance," ASME Manufacturing Engineering Division11, 119-126.
15.
Klosek, M.M. and Kuske, R., 2005, "Multiscale analysis of stochastic delay-differential equations," SIAM Multi-scale Modeling and Simulation3, 706-729.
16.
Kuske, R., 2006, "Multiple-scales approximation for a coherence resonance route to chatter," Computing Science and Engineering Magazine8, 35-43.
17.
Lei, J. and Mackey, M., 2006, "Stochastic differential delay equations, moment stability, and the application to hematopoietic regulation system," SIAM Journal of Applied Mathematics67, 387-407.
18.
Longtin, A., Milton, J.G., Bos, J., and Mackey, M., 1990, "Noise and critical behavior of the pupil light reflex at oscillation onset," Physical Review A41, 6992-7005.
19.
Mackey, M.C. and Glass, L., 1977, Science 197, 287.
20.
Mann, B.P., Insperger, T., Bayley, P.V., and Stépán, G., 2003, "Stability of up-milling and down-milling, part 2: experimental verification," International Journal of Machine Tools and Manufacture43, 35-40.
Namachchivaya, N. Sri and Doyle, M., 2006, Private communication .
23.
Namachchivaya, N. Sri , and Beddini, R., 2003, "Spindle speed variation for the suppression of regenerative chatter," Journal of Nonlinear Science13, 265-288.
24.
Namachchivaya, N. Sri , and Van Roessel, H.J., 2003, "A center manifold analysis of variable speed machining," Journal of Dynamical Systems18, 245-270.
25.
Ohira, T. and Sato, Y., 1999, "Resonance with noise and delay," Physical Review Letters82, 2811-2815.
Stépán, G., 2001, "Modelling nonlinear regenerative effects in metal cutting ," Philosophical Transactions of the Royal Society London A359, 739-757.
28.
Stépán, G. and Kalmar-Nagy, T., 1997, "Nonlinear regenerative machine tool vibrations," in Proceedings ASME DETC 17th Biennial Conference on Mechanical Vibration and Noise, Sacramento, DETC97/VIB-4021, pp. 1-11.
29.
Stone, E. and Askari, A., 2002, "Nonlinear models of chatter in drilling processes," Dynamical Systems17(1), 65-85.
30.
Stone, E. and Campbell, S.A., 2004, "Stability and bifurcation analysis of a nonlinear DDE model for drilling," Journal of Nonlinear Science14, 27-57.
31.
Tlusty, J. and Spacek, L., 1954, Self-Excited Vibrations on Machine Tools, Nakl. CSAV, Prague.
Wallace, M., Sieber, J., Neild, S., Wagg, D., and Krauskopf, B., 2005, "Delay differential equation models for real-time dynamic substructuring," in IDETC/CIE 2005, ASME.
34.
Yilmaz, A., Al-Regib, E., and Ni, J., 2002, "Machine tool chatter suppression by multi-level random spindle speed variation," Transactions of ASME124, 208.
