Lobes and Lenses in the Stability Chart of Interrupted Turning

[+] Author and Article Information
Róbert Szalai

 Budapest University of Technology and Economics, Budapest, H-1521, Hungaryszalai@mm.bme.hu

Gábor Stépán1

 Budapest University of Technology and Economics, Budapest, H-1521, Hungarystepan@mm.bme.hu


Corresponding author.

J. Comput. Nonlinear Dynam 1(3), 205-211 (Nov 15, 2003) (7 pages) doi:10.1115/1.2198216 History: Online November 15, 2003; Online November 21, 2003; Received August 20, 2005; Revised February 15, 2006

In this paper, a new method for the stability analysis of interrupted turning processes is introduced. The approach is based on the construction of a characteristic function whose complex roots determine the stability of the system. By using the argument principle, the number of roots causing instability can be counted, and thus, an exact stability chart can be drawn. In the special case of period doubling bifurcation, the corresponding multiplier 1 is substituted into the characteristic function, leading to an implicit formula for the stability boundaries. Further investigations show that all the period doubling boundaries are closed curves, except the first lobe at the highest cutting speeds. Together with the stability boundaries of Neimark-Sacker (or secondary Hopf) bifurcations, the unstable parameter domains are formed from the union of lobes and lenses.

Copyright © 2006 by American Society of Mechanical Engineers
Topics: Stability , Turning , Cutting
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Figure 1

Mechanical model of interrupted turning

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Figure 2

Structure of stability chart for turning (ζ=0.02), unstable region shaded

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Figure 3

Structure of stability chart for highly interrupted cutting (ζ=0.0038), unstable region shaded

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Figure 7

Period-2 boundaries (small scale) (ρ=0.1, ζ=0.02)

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Figure 8

Period-2 boundaries at zero damping (ρ=0.1)

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Figure 4

Stability chart (ρ=0.1, ζ=0.0038), stable region shaded

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Figure 5

Stability charts containing all unit circle crossings for ζ=0.0038 and (a) ρ=0.05, (b) ρ=0.1, (c) ρ=0.2, stable regions shaded

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Figure 6

Period-2 boundaries (large scale) (ρ=0.1, ζ=0.02)



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