The information dimension, D(0), of attractors associated with orthogonal turning is determined from experimental tool-workpiece relative acceleration data. Let E≡dimension of a delay coordinate space, n≡number of generic data points and m≡ number of reference points on the attractor. It is shown that properties of D(0) as a function of E, denoted by D(0):E, are unchanging, invariant, over large intervals of n and m. The qualitative properties of D(0):E discriminate between various cutting cases. This discrimination can be based on relatively small data sets. The computation of D(0) is shown to be robust in the sense that estimated values of D(0) are invariant or slowly varying over intervals of n and m.
Issue Section:
Research Papers
1.
Aleksic´
Z.
1991
, “Estimating the Embedding Dimension
,” Physica D
, Vol. 52
, pp. 361
–368
.2.
Badii
P.
Politi
A.
1985
, “Statistical Description of Chaotic Attractors: The Dimension Function
,” Journal of Statistical Physics
, Vol. 40
, No. 516
, pp. 725
–750
.3.
Berger, B. S., Minis, I., and Rokni, M., 1991, “The Dimension of Attractors Associated With Metal Cutting Dynamics,” Sensors, Controls and Quality Issues in Manufacturing, Winter Annual Meeting of the ASME, Atlanta, GA, PED-Vol. 55, pp. 333–343.
4.
Berger, B. S., Rokni, M., and Minis, I., 1993, “Complex Dynamics of Metal Cutting,” Quarterly Journal of Applied Mathematics (to appear).
5.
Broggi, G., 1988(a), “Numerical Characterization of Experimental Chaotic Signals,” PhD Dissertation, Institut fur Theoretische Physik der Universitat Zurich, Zurich, Switzerland.
6.
Broggi
G.
1988
(b), “Evaluation of Dimensions and Entropies of Chaotic Systems
,” Journal of the Optical Society of America B
, Vol. 5
, No. 5
, pp. 1020
–1028
.7.
Casdagli
M.
Eubank
S.
Farmer
J. D.
Gibson
J.
1991
, “State Space Reconstruction in the Presence of Noise
,” Physica D
, Vol. 51
, pp. 52
–98
.8.
Eckmann
J. P.
Ruelle
D.
1985
, “Ergodic Theory of Chaos and Strange Attractors
,” Review of Modern Physics
, Vol. 57
, No. 3
, Part 1, pp. 617
–656
.9.
Farmer, J. D., 1981, “Order Within Chaos,” PhD Dissertation, University of California at Santa Cruz, Santa Cruz, California.
10.
Farmer
J. D.
Ott
B.
Yorke
J. A.
1983
, “The Dimension of Chaotic Attractors
,” Physica D
, Vol. 7
, pp. 153
–180
.11.
Farmer
J. D.
Sidorowich
J. J.
1991
, “Optimal Shadowing and Noise Reduction
,” Physica D
, Vol. 47
, pp. 373
–392
.12.
Fraser
A. M.
Swinney
H. L.
1986
, “Independent Coordinates for Strange Attractors From Mutual Information
,” Physical Review A
, Vol. 33
, No. 2
, pp. 1134
–1140
.13.
Grabec
I.
1986
, “Chaos Generated by the Cutting Process
,” Physics Letters
, Vol. 117
, pp. 384
–386
.14.
Grabec
I.
1988
, “Chaotic Dynamics of the Cutting Process
,” International Journal of Machine Tools and Manufacturing
, Vol. 28
, pp. 275
–280
.15.
Grassberger
P.
Procaccia
I.
1983
, “Measuring the Strangeness of Strange Attractors
,” Physica D
, Vol. 9
, pp. 189
–208
.16.
Guckenheimer, J., and Holmes, P., 1983, “Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,” Springer-Verlag.
17.
Hanna
N. J.
Tobias
S. A.
1969
, “The Nonlinear Dynamic Behavior of a Machine Structure
,” International Journal of Machine Tool Design and Research
, Vol. 9
, pp. 293
–307
.18.
Kim
K. J.
Eman
E. F.
Wu
S. M.
1984
, “Identification of Natural Frequencies and Damping Ratios of Machine Tool Structures by the Dynamic Data System Approach
,” International Journal of Machine Tool Design and Research
, Vol. 24
, pp. 161
–170
.19.
Klamecki
B. E.
1989
, “On the Effects of Turning Process Asymmetry on Process Dynamics
,” ASME Journal of Engineering for Industry
, Vol. III
, pp. 193
–198
.20.
Koenigsberger, I., and Tlusty, J., 1971, Structure of Machine Tools, Pergamon Press.
21.
Kostelich
E. J.
Swinney
H. L.
1989
, “Practical Considerations in Estimating Dimension From Time Series Data
,” Physica Scripta
, Vol. 40
, pp. 436
–441
.22.
Lin
J. W.
Weng
C. I.
1990
, “A Nonlinear Dynamic Model of Cutting
,” International Journal of Machine Tools and Manufacturing
, Vol. 30
, pp. 53
–64
.23.
Mandelbrot, B. B., 1993, The Fractal Geometry of Nature, W. H. Freeman and Co., New York.
24.
Marteau
P. F.
Abarbanel
H. D. I.
1991
, “Noise Reduction in Chaotic Time Series Using Scaled Probabilistic Methods
,” Journal of Nonlinear Science
, Vol. 1
, pp. 313
–343
.25.
Minis, I., 1988, “Prediction of Machine Tool Chatter in Turning,” PhD Dissertation, University of Maryland, College Park, MD.
26.
Minis
I.
Magrab
E.
Pandelidis
I.
1990
, “Improved Methods for the Prediction of Chatter in Turning, Part 1: Determination of the Structural Response Parameters
,” ASME Journal of Engineering for Industry
, Vol. 112
, pp. 12
–20
.27.
Sauer, T., Yorke, J., and Casdagli, M., 1991, “Embedology,” Technical Report, Dept. of Math. Sciences, George Mason University, Fairfax, VA 22030.
28.
Smith
L. A.
Fournier
J. D.
Spiegel
E. A.
1986
, “Lacunarity and Intermittency in Fluid Turbulence
,” Physics Letters
, Vol. 114A
, No. 8, 9
, pp. 465
–468
.29.
Takens, F., 1981, “Detecting Strange Attractors in Turbulence,” Lecture Notes in Mathematics, No. 898, Springer-Verlag.
30.
Tlusty
J.
1978
, “Analysis of the State of Research in Cutting Dynamics
,” CIRP Annals
, Vol. 27
, pp. 583
–589
.31.
Van De Water
W.
Schram
P.
1989
, “Oscillatory Scaling Functions of Near-Neighbor Distances in Fractal Sets
,” Physics Letters A
, Vol. 140
, No. 4
, pp. 173
–178
.32.
Week, M., 1985, Handbook of Machine Tools, Vol. 4, John Wiley, New York.
33.
Wu
D. W.
Liu
C. R.
1985
, “An Analytical Model of Cutting Dynamics
,” Parts 1 and 2, ASME Journal of Engineering for Industry
, Vol. 107
, pp. 107
–118
.34.
Wu
D. W.
1989
, “A New Approach to Formulating the Transfer Function for Dynamic Cutting Processes
,” ASME Journal of Engineering for Industry
, Vol. III
, pp. 37
–47
.
This content is only available via PDF.
Copyright © 1995
by The American Society of Mechanical Engineers
You do not currently have access to this content.
