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Technical Briefs

A Robust Semi-Analytical Method for Calculating the Response Sensitivity of a Time Delay System

[+] Author and Article Information
Mohammad H. Kurdi1

 University of Florida, Gainesville, FL 32611mhkurdi@gmail.com

Raphael T. Haftka, Tony L. Schmitz

 University of Florida, Gainesville, FL 32611

Brian P. Mann

 Duke University, Durham, NC 27708

1

Present address: Air Force Research Laboratory, Wright-Patterson Air Force Base, 2210 8th Street, B146, Wright-Patterson AFB, OH 45433.

J. Vib. Acoust 130(6), 064504 (Oct 22, 2008) (6 pages) doi:10.1115/1.2981093 History: Received October 24, 2007; Revised May 24, 2008; Published October 22, 2008

It is often necessary to establish the sensitivity of an engineering system’s response to variations in the process/control parameters. Applications of the calculated sensitivity include gradient-based optimization and uncertainty quantification, which generally require an efficient and robust sensitivity calculation method. In this paper, the sensitivity of the milling process, which can be modeled by a set of time delay differential equations, to variations in the input parameters is calculated. The semi-analytical derivative of the maximum eigenvalue provides the necessary information for determining the sensitivity of the process stability to input variables. Comparison with the central finite difference derivative of the stability boundary shows that the semi-analytical approach is more efficient and robust with respect to step size and numerical accuracy of the response. An investigation of the source of inaccuracy of the finite difference approximation found that it is caused by discontinuities associated with the iterative process of root finding using the bisection method.

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Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

(a) Schematic of 2DOF milling tool. (b) Identification of key variables. (c) Various types of milling operations.

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Figure 2

Stability boundary calculated using the cutting conditions listed in Table 1. Since the stable boundary is determined by the maximum magnitude of an eigenvalue of the dynamic map, a slope discontinuity occurs at the cusp where two eigenvalues swap places as the one with the largest magnitude.

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Figure 3

The logarithmic derivative of axial depth with respect to spindle speed versus step size percentage at a spindle speed of 10,000rpm.—Semi-analytical, ε=1×10−2. ○ Central finite difference, ε=1×10−2; ▷ central finite difference, ε=1×10−4; × central finite difference, ε=1×10−7. The semi-analytical method provides a large accurate step size range even with a larger value of ε.

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Figure 4

The logarithmic derivative of axial depth with respect to spindle speed versus step size percentage near a derivative discontinuity (11,800rpm). —Semi-analytical, ε=1×10−2; ––Semi-analytical, ε=1×10−4. ○ Central finite difference, ε=1×10−2; ▷ central finite difference, ε=1×10−4; × central finite difference, ε=1×10−7. The semi-analytical results are minimally affected by the slope discontinuity. For the central finite difference, however, only a tolerance error of 1×10−7 gives a stable derivative calculation. Furthermore a smaller step size range than Fig. 3(1×10−6–2×10−5) is obtained.

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Figure 5

Source of numerical error in sensitivity computation. (a) Smooth variation of λmax. (b) Staircase variation in blim:—number of iterations, n=15; ⋅ε=1×10−2.

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Figure 6

Comparison between the calculated sensitivity using overall finite difference and semi-analytical methods for the same step size h=1×10−3 at a spindle speed of 10,000rpm.—Semi-analytical; ○ central finite difference. The semi-analytical method gives accurate calculation of blim sensitivity even for large ε.

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Figure 7

Comparison between the calculated sensitivity using central finite difference and semi-analytical methods.—Semi-analytical; ⋅ central finite difference. Good agreement is observed provided the conditions are carefully selected (see Table 2).

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