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Research Papers

# Improved Experimental Ritz Vector Extraction With Application to Damage Detection

[+] Author and Article Information
Stuart G. Taylor

Engineering Institute, MS-T001, Los Alamos National Laboratory, Los Alamos, NM 87545staylor6@uh.edu

David C. Zimmerman

Department of Mechanical Engineering, University of Houston, Houston, TX 77004–4006

$E=70 GPa$, $ρ=2700 kg/m3$.

$E=210 GPa$, $ρ=7850 kg/m3$.

J. Vib. Acoust 132(1), 011012 (Feb 01, 2010) (10 pages) doi:10.1115/1.4000762 History: Received January 22, 2007; Revised November 01, 2009; Published February 01, 2010; Online February 01, 2010

## Abstract

Load-dependent Ritz vectors, or Lanczos vectors, are alternatives to mode shapes as a set of orthogonal vectors used to describe the dynamic behavior of a structure. Experimental Ritz vectors are extracted recursively from a state-space system realization, and they are orthogonalized using the Gram–Schmidt process. In addition to the Ritz vectors themselves, the associated nonorthogonalized vectors are required for application to damage detection. First, this paper presents an improved experimental Ritz vector extraction algorithm to correctly extract the nonorthogonalized Ritz vectors. Second, this paper introduces a Ritz vector accuracy indicator for use with noisy data. This accuracy indicator is applied as a tool to guide the deflation of a state-space system realization identified from simulated noisy data. The improved experimental Ritz vector extraction algorithm produces experimental nonorthogonalized Ritz vectors that match the analytically computed vectors. The use of the accuracy indicator with simulated noisy data enables the identification of a state-space realization for Ritz vector extraction from which damage location and extent are correctly estimated. The improved Ritz vector extraction algorithm improves the application of Ritz vectors to damage detection, more accurately estimating damage location and extent. The accuracy indicator extends the application of Ritz vectors to damage detection in noisy systems as well.

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## Figures

Figure 1

32DOF truss

Figure 2

Left singular vectors

Figure 3

Stiffness update matrix plots

Figure 4

8DOF truss

Figure 5

SSTI plots for the noise-free case

Figure 6

SSTI plots with 5% RMS noise

Figure 7

SSTI plots with 10% RMS noise

Figure 8

10% Noise versus analytical MAC plot

Figure 9

SSTI plots for the full ERA-identified system

Figure 10

SSTI plots for the first deflated ERA-identified system

Figure 11

SSTI plots for the second deflated ERA-identified system

Figure 12

First left singular vectors

Figure 13

Stiffness update for 10% RMS noise

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