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TECHNICAL PAPERS

Parameter Reconstruction Based on Sensitivity Vector Fields

[+] Author and Article Information
Bogdan I. Epureanu2

Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125

Ali Hashmi

Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125

2

Corresponding author.

J. Vib. Acoust 128(6), 732-740 (Jun 20, 2006) (9 pages) doi:10.1115/1.2346692 History: Received August 31, 2004; Revised June 20, 2006

A novel approach to determine very accurately multiple parameter variations by exploiting the geometric shape of dynamic attractors in state space is presented. The approach is based on the analysis of sensitivity vector fields. These sensitivity vector fields describe changes in the state space attractor of the dynamics and system behavior when parameter variations occur. Distributed throughout the attractor in state space, these fields form a collection of snapshots for known parameter changes. Proper orthogonal decomposition of the parameter space is then employed to distinguish multiple simultaneous parametric variations. The parametric changes are reconstructed by analyzing the deformation of attractors which are characterized by means of the sensitivity vector fields. A set of basis functions in the vector space formed by the sensitivity fields is obtained and is used to successfully identify test cases involving multiple simultaneous parametric variations. The method presented is shown to be robust over a wide range of parameter variations and to perform well in the presence of noise. One of the main applications of the proposed technique is detecting multiple simultaneous damage in vibration-based structural health monitoring.

Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Overlapping healthy and damaged attractors and the grid used to construct the sensitivity vector fields

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Figure 2

Sensitivity vector q, constructed by interrogating the system in state space

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Figure 3

The zooming effect on sensitivity vector magnitude

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Figure 4

Coupled oscillators

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Figure 5

Sketch of a vibrating plate

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Figure 6

Representative shapes of the sensitivity vector fields; 2D state space projection shown: x1 and y1 (left) and x2 and y2 (right)

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Figure 7

Representative shapes of the sensitivity vector fields; 2D state space projection shown: x1 and y2 (left) and x2 and y1 (right)

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Figure 8

Absolute values of the eigenvalues of the snapshot correlation matrix are shown from largest to smallest

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Figure 9

Error in parameter reconstruction is shown for variations of G2—note that the best performance was achieved with a basis of fewer than the maximum number of available eigenvectors (m=k=24)

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Figure 10

The parameter reconstruction error with m=6 is shown for the various system parameters over increasing noise levels: (a) λ1, (b) G1, (c) Rx1, (d) λ2, (e) G2, and (f) Rx2

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Figure 11

The parameter reconstruction error with m=9 is shown for the various system parameters over increasing noise levels

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Figure 12

Average residual error in capturing the sensitivity vector field and parameter reconstruction errors for several test cases by using an increasing number of basis functions

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Figure 13

Standard deviation of the residual error in capturing the sensitivity vector field and parameter reconstruction errors by using an increasing number of basis functions

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Figure 14

The parameter reconstruction error for 6 levels of variation of G2 and increasing number m of basis functions

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Figure 15

The parameter reconstruction error for 6 levels of variation of G2 and increasing number m of basis functions

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