Perturbation Analysis of Spindle Speed Variation in Machine Tool Chatter

Abstract

Spindle speed variation has been shown to be an effective method for chatter control. In this paper, a single-degree-of-freedom regenerative type chatter equation is treated using perturbation methods. Rather than using the time coordinate, the angle of revolution is taken as the independent coordinate for maintaining a constant delay in the equations. The spindle speed is taken to be harmonically varying about a constant mean speed. Approximate analytical solutions are sought using the method of strained parameters, a perturbation technique. The amplitude of speed fluctuations (ε) is assumed to be small, and solutions are constructed using this parameter as the perturbation parameter. The stability lobes for constant spindle speeds are calculated exactly. By using the approximate perturbation analysis, the gain in stability is calculated for variable spindle speeds. The analysis is valid for (ε) values up to 0.02 (i.e., 2% of the constant mean speed). Solutions are verified using numerical simulations of the original equation.

Keywords

Perturbation methods machine tools chatter speed variation

Get full access to this article

View all access options for this article.

References

Arnold, R.N. , 1946, "The mechanism of tool vibration in the cutting of steel ," Proceedings, Instit. of Mech. Eng. 154, 261.

Chen, S.G. , Ulsoy, A.G. , and Koren, Y. , 1992, "Modeling, simulation and analysis of regenerative chatter in turning using the β domain formulation," Proceedings of the IFAC Workshop on Intelligent Manufacturing Systems, Dearborn, MI, October.

Hanna, N.H. and Tobias, S.A. , 1974, "A theory of nonlinear regenerative chatter," ASME J. Eng. Indust. 96, 247-255.

Hoshi , et al., 1977, "Study for practical application of fluctuating speed cutting for regenerative chatter control," Annals of the CIRP 25, 175-179.

Inamura, T. and Sata, T. , 1974, "Stability analysis of cutting under varying spindle speed ," Annals of the CIRP 23, 119-120.

Inamura, T. and Sata, T. , 1975, "Stability analysis of cutting under varying spindle speed ," J. Fac. Eng. Tokyo Univ. 33, 1.

Kondo, Y. , Kawano, O. , and Sato, H. , 1981, "Behavior of self-excited chatter due to multiple regenerative effect," ASME J. Eng. Indust. 103, 324-329.

Lin, J.S. and Weng, C.I. , 1990, "A nonlinear dynamic model of cutting," Int. J. Mach. Tools Manufact. 30, 53-64.

Lin, S.C. , DeVor, R.E. , and Kapoor, S.G. , 1990, "The effects of variable speed cutting on vibration control in face milling," ASME J. Eng. Indust. 112, 1-11.

10.

Nayfeh, A.H. , 1981, Introduction to Perturbation Techniques, Wiley-Interscience, New York.

11.

Nayfeh, A.H. , Chin, C.M. , and Pratt, J. , 1997, "Perturbation methods in nonlinear dynamics—applications to machining dynamics," J. Manufact. Sci. Eng. 119.

12.

Sexton, J.S. , Milne, R.D. , and Stone, B.J. , 1977, "A stability analysis of single-point machining with varying spindle speed," Appl. Math. Modeling 1, 310-318.

13.

Sexton, J.S. and Stone, B.J. , 1978, "The stability of machining with continuously varying spindle speed," Annals of the CIRP 27, 321-326.

14.

Sexton, J.S. and Stone, B.J. , 1980, "An investigation of the transient effects during variable speed cutting," J. Mech. Eng. Sci. 22, 107-118.

15.

Stoferle, T. and Grab, H. , 1972, "Vermeiden von Ratterschwingungen durch Periodische Drehzahlanderung ," Werkstatt und Betrieb 105, 727.

16.

Takemura, T. , Kitamura, T. , and Hoshi, T. , 1974, "Active suppression of chatter by programmed variation of spindle speed," Annals of the CIRP 23, 121-122.

17.

Tobias, S.A. , 1965, Machine Tool Vibration, John Wiley , New York.

18.

Tsao, T.C. , McCarthy, M.W. , and Kapoor, S.G. , 1993, " A new approach to stability analysis of variable speed machining systems," Int. J. Mach. Tools Manufact. 33, 791-808.

19.

Welbourn, D.B. and Smith, J.D. , 1970, Machine-Tool Dynamics: An Introduction, Cambridge University Press, Cambridge.