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Research Papers

Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner’s Equation

[+] Author and Article Information
Eric A. Butcher

Department of Mechanical and Aerospace Engineering,  New Mexico State University, Las Cruces, NM 88003

Brian P. Mann

Department of Mechanical Engineering and Materials Science,  Duke University, Durham, NC 27708

J. Comput. Nonlinear Dynam 7(1), 011004 (Aug 09, 2011) (10 pages) doi:10.1115/1.4004468 History: Received August 27, 2010; Revised June 10, 2011; Published August 09, 2011; Online August 09, 2011

A method for obtaining analytic bounds for period doubling and cyclic fold instability regions in linear time-periodic systems with piecewise constant coefficients and time delay is suggested. The method is based on the use of transition matrices for Meissner’s equation corresponding to the desired type of instability. Analytic expressions for the disconnected regions of fold and flip instability for two- and three-segment coefficients including both complex and real eigenvalues in Meissner’s equation are obtained. The proposed method when applied to the example of two-segment interrupted turning with complex eigenvalues in each segment yields the same results as those obtained recently for the boundaries of the flip regions (Szalai and Stepan, 2006, “Lobes and Lenses in the Stability Chart of Interrupted Turning,” J Comput. Nonlinear Dyn., 1 , pp. 205–211). Next, the period-doubling instability regions for a particular delay differential equation related to the damped Meissner’s equation and the fold instabilities for a model of delayed position feedback control are analytically obtained. Finally, we extend the method to a single degree-of-freedom milling model with a three-piecewise-constant-segment approximation to the true specific cutting force in which lower bounds for and horizontal locations of the regions of flip instability are obtained. The analytic results are verified through numerical stability charts obtained using the temporal finite element method. Conditions for the existence of islands of instability are also obtained.

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Figures

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Figure 1

Mechanical model of interrupted turning

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Figure 6

Stability of the Meissner equation 8 with ζ=0,ŵ1=1,ŵ2=-1 is given by the condition |cos⁡τ2|≤1/cosh⁡τ2. The flip instability points are given by the circled intersections at 2πρ=1.8751 and 4.6941.

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Figure 7

Stability of the DDE in Eq. 2 where h¯(t) switches between the constants 0 for time length τ1=2π(1-ρ) and −1 for time length τ2=2πρ. The shaded region is unstable and the unshaded region is stable. The red curve is a region of flip instability which is bounded by the blue and green curves of Eqs. 25,25,25, and the values of τ2=2πρ where it intersects with ζ=0 are identical to the circled locations in Fig. 6.

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Figure 8

Stability of the DDE in Eq. 4 where k¯(t) switches between the constants 1.2 for time length τ1=2π(1-ρ) and −1.2 for time length τ2=2πρ and the boundaries of Eqs. 26,26. The shaded region is unstable and the unshaded region is stable. The red curve corresponds to cyclic fold instability.

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Figure 9

Illustrations of (a) up-milling and (b) down-milling. For full immersion these two types of milling are identical.

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Figure 10

Specific cutting force profile for full immersion milling for a single cutting tooth. The averages over the principal period (long-dashed line) and separately over the positive and negative regions (short-dashed line) are shown.

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Figure 11

Example of regions of minima of Eq. 27 from Eqs. 29,30 where flip instability may occur.

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Figure 12

Stability charts for (a) milling for the specific cutting force in Fig. 1 with m = 2.573 kg, ζ = 0.007, ωn  = 920.04 rad/s, and (b) the three-segment approximation in Fig. 1 with the curves of Eqs. 31,31. The shaded region is unstable and the unshaded region is stable. The red curves correspond to flip instability.

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Figure 5

Stability chart for interrupted turning with ρ=0.2,ζ=0.02. The shaded region is unstable and the unshaded region is stable. The red curves are regions of flip instability which are bounded by the blue and green curves of Eqs. 20,20,20.

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Figure 4

Stability chart for interrupted turning with ρ=0.1,ζ=0.02. The shaded region is unstable and the unshaded region is stable. The red curves are regions of flip instability which are bounded by the blue and green curves of Eqs. 20,20,20.

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Figure 3

Stability chart for interrupted turning with ρ=0.1,ζ=0.007. The shaded region is unstable and the unshaded region is stable. The red curves are regions of flip instability which are bounded by the blue and green curves of Eqs. 20,20,20.

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Figure 2

Regions of minima of Eq. 16 from Eqs. 18,19 where flip instability may occur

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