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Infinite dimensional slow modulations in a well known delayed model for cutting tool vibrations

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Abstract

We apply the method of multiple scales (MMS) to a well-known model of regenerative cutting vibrations in the large delay regime. By “large” we mean the delay is much larger than the timescale of typical cutting tool oscillations. The MMS up to second order, recently developed for such systems, is applied here to study tool dynamics in the large delay regime. The second order analysis is found to be much more accurate than the first order analysis. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy in that plotted solutions of moderate amplitudes are visually near-indistinguishable. The advantages of the present analysis are that infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space; lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS; the strong sensitivity of the slow modulation dynamics to small changes in parameter values, peculiar to such systems with large delays, is seen clearly; and though certain parameters are treated as small (or, reciprocally, large), the analysis is not restricted to infinitesimal distances from the Hopf bifurcation.

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Author notes

  1. K. Nandakumar

    Present address: University of Aberdeen, Scotland, UK

  2. Anindya Chatterjee

    Present address: Indian Institute of Technology Kharagpur, Kharagpur, India

Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Science, Bangalore, India

    K. Nandakumar & Anindya Chatterjee

  2. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India

    Pankaj Wahi

Authors
  1. K. Nandakumar
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  2. Pankaj Wahi
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  3. Anindya Chatterjee
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Correspondence to K. Nandakumar.

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Nandakumar, K., Wahi, P. & Chatterjee, A. Infinite dimensional slow modulations in a well known delayed model for cutting tool vibrations. Nonlinear Dyn 62, 705–716 (2010). https://doi.org/10.1007/s11071-010-9755-x

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