Investigation of Parametric Instability in Milling Simulated by Screw Chasing Operation

  • G. Droubi
  • M. M. Sadek


The parametric instability in milling, resulting from the equivalent periodic stiffness variation of the machine structure is investigated, using a special machining set-up which eliminates regeneration. A theoretical model has been developed to predict this type of instability for a specially designed workpiece of a single degree of freedom. The principal axis of vibration of this system rotates in relation to the cutting force orientation. The analysis led to a Mathieu’s equation incorporating a damping term. Good correlation has been achieved between the predicted results and those experimentally obtained from cutting tests.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. M. Sadek and S. A. Tobias. Comparative dynamic acceptance tests for machine tools applied to horizontal milling machines, Proc. Inst. Mech. Engrs. (1970–71) 185, London.Google Scholar
  2. 2.
    G. Sweeney and S. A. Tobias. An algebraic method for the determination of the dynamic stability of machine tools, Inst. Res. in Prod. Eng. Conf., Pittsburg, p. 475, 1963.Google Scholar
  3. 3.
    G. Droubi. Instability During Continuous and Interrupted Horizontal Milling, Ph.D. Thesis, The University of Birmingham, 1972.Google Scholar
  4. 4.
    G. Droubi, M. M. Sadek and S. A. Tobias. The effect of the instantaneous force orientation on the stability of horizontal milling, to be submitted to the Inst. Mech. Engrs., London, 1974.Google Scholar
  5. 5.
    R. Sridhar, E. R. Hohn and G. W. Long. A stability algorithm for a special case of the milling process—contribution to Machine Tool Chatter Research, Part 6, Journal of Engineering for Industry, Trans. A.S.M.E. Series B (1968) 90, 325.Google Scholar
  6. 6.
    N. W. McLachlan. Theory and Application of Mathieu Functions, Oxford University Press, 1947.Google Scholar
  7. 7.
    N. W. McLachlan. Ordinary Non-linear Differential Equations, 2nd Edition, Oxford Press, 1956.Google Scholar
  8. 8.
    C. Hayashi. Non-linear Oscillations in Physical Systems, McGraw-Hill Book Co. New York, 1964.Google Scholar
  9. 9.
    E. T. Whittaker. General solution of Mathieu’s equation, Proc. Edinburgh Math. Soc., (1913–14) 32, 75.CrossRefGoogle Scholar
  10. 10.
    E. T. Whittaker and G. N. Watson. Cambridge University Press, London, 1935.Google Scholar
  11. 11.
    A. W. Young. Quasi-periodic solutions of Mathieu’s Equation, Proc. Edinburgh Math. Soc. (1913–14) 32, 81.CrossRefGoogle Scholar
  12. 12.
    W. A. Knight and S. A. Tobias. Torsional vibrations and machine tool stability, Proc. 10th Int. M.T.D.R. Conf. Manchester, p. 299, 1969.Google Scholar
  13. 13.
    M. M. Nigm, M. M. Sadek and S. A. Tobias. Prediction of dynamic cutting coefficients from steady state cutting data, Proc. 13th M.T.D.R. Conf. Birmingham, 1972.Google Scholar
  14. 14.
    G. Droubi, M. M. Sadek and S. A. Tobias. Determination of the dynamic cutting coefficients for milling. Proc. 13th M.T.D.R. Conf. Birmingham, 1972.Google Scholar

Copyright information

© Macmillan Publishers Limited 1975

Authors and Affiliations

  • G. Droubi
    • 1
  • M. M. Sadek
    • 1
  1. 1.Mechanical Engineering DepartmentUniversity of BirminghamUK

Personalised recommendations