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TECHNICAL PAPERS

A Dynamical Systems Approach to Damage Evolution Tracking, Part 2: Model-Based Validation and Physical Interpretation

[+] Author and Article Information
Joseph P. Cusumano

Department of Engineering Science & Mechanics, Pennsylvania State University, University Park, Pennsylvania 16802e-mail: jpc@crash.esm.psu.edu

David Chelidze

Department of Mechanical Engineering & Applied Mechanics, University of Rhode Island, Kingston, Rhode Island 02881e-mail: chelidze@egr.uri.edu

Anindya Chatterjee

Department of Mechanical Engineering, Indian Institute of Science, Bangalore, 560012, Indiae-mail: anindya@mecheng.iisc.ernet.in.

J. Vib. Acoust 124(2), 258-264 (Mar 26, 2002) (7 pages) doi:10.1115/1.1456907 History: Received July 01, 2001; Revised December 01, 2001; Online March 26, 2002
Copyright © 2002 by ASME
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References

Moon,  F. C., and Holmes,  P., 1979, “A Magnetoelastic Strange Attractor,” J. Sound Vib., 65(2), pp. 275–296.
Cusumano,  J. P., and Kimble,  B., 1995, “A Stochastic Interrogation Method for Experimental Measurements of Global Dynamics and Basin Evolution: Application to a Two-Well Oscillator,” Nonlinear Dyn., 8, pp. 213–235.
Takens, F., 1981, “Detecting Strange Attractor in Turbulence,” Rand, D. A., and Young, L. S., eds., Dynamical Systems and Turbulence, Warwick, Springer Lecture Notes in Mathematics, Springer-Verlag, Berlin, pp. 336–381.
Sauer,  T., Yorke,  J. A., and Casdagli,  M., 1991, “Embedology,” J. Stat. Phys., 65(3–4), pp. 579–616.
Feeny,  B. F., Yuan,  C. M., and Cusumano,  J. P., 2001, “Parametric Identification of an Experimental Magneto-Elastic Oscillator,” Accepted, J. Sound Vib., 247(5), pp. 785–806.
Woodson, H. H., and Melcher, J. R., 1968, Electromechanical Dynamics, pt. 1: Discrete Systems, Wiley, New York.
Strauss,  E., Golodnitsky,  D., and Peled,  E., 1999, “Cathode Modification for Improved Performance of Rechargeable Lithium/Composite Polymer Electrolyte-Pyrite Battery,” Electrochem. Solid-State Lett., 2(3), pp. 115–117.
Cusumano,  J. P., and Chatterjee,  A., 2000, “Steps Towards a Qualitative Dynamics of Damage Evolution,” Int. J. Solids Struct., 37(44), pp. 6397–6417.
Sanders, J. A., and Verhulst, F., 1985, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York.

Figures

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Geometry of the mechanical subsystem: (left) single degree-of-freedom model of the constrained beam; (right) magnetic restoring force at beam tip
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Schematic diagram of the electromagnetic subsystem. The slowly decreasing open circuit battery voltage is Φ(εt), whereas the terminal voltage, which can be directly measured, is V.
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Typical plots of numerically computed theoretical tracking functions, e=‖ek‖, versus ϕ. Each figure is computed for a different randomly chosen initial condition. Curves are plotted for different prediction times, tp=mts, with m ranging from 1 to 10.
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Power spectra showing that the system is predominantly chaotic throughout the simulated experiment. 25 spectra are shown, each using 100,000 data points (Time is in units of 100,000/ts). Note the strong periodic window occurring during the 14th power spectrum calculation.
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Tracking results from numerical simulations: (left) Plot of ē4 versus ϕ̄2. Solid gray line shows a linear fit to the data. Top right corner sub-plot shows the mean fit error versus order of the fit. (right) Plot of the battery voltage ϕ̄ (solid gray line) vs. time, overlayed with the tracking metric ē4 (black dots). For both plots, ten consecutive data records of size 212 were used to calculate ē4 and its statistical fluctuation (gray background shows ± one standard deviation from the sample mean).
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Additional tracking results showing the effect of increasing the number of consecutive records NR used for ẽ4:(left)NR=20:(right)NR=40. See Fig. 5 for plot details. As in the experimental results, the different values of NR affect only the local fluctuations of the tracking, not the general trend. See text for additional discussion.
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Solutions to the slow flow equation: (left) Comparison of the slow flow solution Eq. (18), dashed line, to actual moving average ϕ̄, solid line, of the slow variable drift computed with the exact Eqs. (9), (10); (right) For comparison with the experiments, the unaveraged time series of the terminal voltage V (see Fig. 2) is plotted in light gray together with the slow flow solution. See text for additional details.
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Damage tracking results for the two well oscillator experiment with a crack growing to failure (see text for details). Black dots represent the tracking metric ē5, computed from NR=10 consecutive data records, plotted as a function of time. Light gray outline is drawn at ± one standard deviation of the local mean.

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