In this paper the dynamic stability of the milling process is investigated through a single degree-of-freedom model by determining the regions where chatter (unstable) vibrations occur in the two-parameter space of spindle speed and depth of cut. Dynamic systems such as milling are modeled by delay-differential equations with time-periodic coefficients. A new approximation technique for studying the stability properties of such systems is presented. The approach is based on the properties of Chebyshev polynomials and a collocation expansion of the solution. The collocation points are the extreme points of a Chebyshev polynomial of high degree. Specific cutting force profiles and stability charts are presented for the up- and down-milling cases of one or two cutting teeth and various immersion levels with linear and nonlinear regenerative cutting forces. The unstable regions due to both secondary Hopf and flip (period-doubling) bifurcations are found, and an in-depth investigation of the optimal stable immersion levels for down-milling in the vicinity of where the average cutting force changes sign is presented.
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July 2009
Research Papers
Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels
Eric A. Butcher,
Eric A. Butcher
Department of Mechanical and Aerospace Engineering,
e-mail: eab@nmsu.edu
New Mexico State University
, Las Cruces, NM 88001
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Oleg A. Bobrenkov,
Oleg A. Bobrenkov
Department of Mechanical and Aerospace Engineering,
e-mail: chaalis@nmsu.edu
New Mexico State University
, Las Cruces, NM 88001
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Ed Bueler,
Ed Bueler
Department of Mathematics and Statistics,
e-mail: ffelb@uaf.edu
University of Alaska Fairbanks
, Fairbanks, AK 99775
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Praveen Nindujarla
Praveen Nindujarla
Department of Mechanical Engineering,
e-mail: ftpkn1@uaf.edu
University of Alaska Fairbanks
, Fairbanks, AK 99775
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Eric A. Butcher
Department of Mechanical and Aerospace Engineering,
New Mexico State University
, Las Cruces, NM 88001e-mail: eab@nmsu.edu
Oleg A. Bobrenkov
Department of Mechanical and Aerospace Engineering,
New Mexico State University
, Las Cruces, NM 88001e-mail: chaalis@nmsu.edu
Ed Bueler
Department of Mathematics and Statistics,
University of Alaska Fairbanks
, Fairbanks, AK 99775e-mail: ffelb@uaf.edu
Praveen Nindujarla
Department of Mechanical Engineering,
University of Alaska Fairbanks
, Fairbanks, AK 99775e-mail: ftpkn1@uaf.edu
J. Comput. Nonlinear Dynam. Jul 2009, 4(3): 031003 (12 pages)
Published Online: May 19, 2009
Article history
Received:
January 30, 2008
Revised:
September 26, 2008
Published:
May 19, 2009
Citation
Butcher, E. A., Bobrenkov, O. A., Bueler, E., and Nindujarla, P. (May 19, 2009). "Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels." ASME. J. Comput. Nonlinear Dynam. July 2009; 4(3): 031003. https://doi.org/10.1115/1.3124088
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