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TECHNICAL PAPERS

# Reducing the Impact of Measurement Errors When Reconstructing Spatial Dynamic Forces

[+] Author and Article Information
Yi Liu

The University of Alabama, Department of Mechanical Engineering, 290 Hardaway Hall, Box 870276, Tuscaloosa, AL 35487

W. Steve Shepard

The University of Alabama, Department of Mechanical Engineering, 290 Hardaway Hall, Box 870276, Tuscaloosa, AL 35487sshepard@eng.ua.edu

J. Vib. Acoust 128(5), 586-593 (Mar 30, 2006) (8 pages) doi:10.1115/1.2202162 History: Received March 23, 2005; Revised March 30, 2006

## Abstract

Inferring external spatially distributed dynamic forces from measured structural responses is necessary when direct measurement of these forces is not possible. The finite difference method and the modal method have been previously developed for reconstructing these forces. However, the accuracy of these methods is often hindered due to the amplification of measurement errors in the computation process. In order to analyze these error amplification effects by using the singular value decomposition approach, the mathematic expressions for these two force reconstruction methods are first transformed into a certain linear system of equations. Then, a regularization method, the Tikhonov method, is applied to increase computational stability. In order to achieve a good regularized result, the L-curve method is used in conjunction with the Tikhonov method. The effectiveness in reducing the influence of the measurement errors when applying the regularization method to the finite difference method and the modal method is investigated analytically and numerically. It is found that when the regularization method is appropriately applied, reliable computational results for the reconstructed force can be achieved.

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## Figures

Figure 3

Reconstructed external dynamic forces with the force modulus ℘f=0.069N found using the finite difference method with exact virtual measurements

Figure 12

Computation results from the regularized modal method with polluted measured responses. Note the force modulus ℘mreg here is 0.061N and the regularization parameter α is 5.3×10−10.

Figure 1

Rectangular plate showing layout of the measured response locations for finite difference method, used to find F(i,j)

Figure 2

Exact force distribution with the force modulus ℘exact=0.04N on a square plate of 1m×1m

Figure 4

Reconstructed external dynamic forces with the force modulus ℘m=0.049N using the modal method with exact virtual measurements

Figure 5

Reconstructed external dynamic forces with the force modulus ℘ferror=0.20N from the polluted measured responses using the finite difference method

Figure 6

Reconstructed external dynamic forces with the force modulus ℘merror=6.60N from the polluted measured responses using the modal method

Figure 7

Picard condition plot for the finite difference method. (a) Clean bf vector and (b) polluted bf vector (∙, singular value si; ×, ∣uiTbf∣; O, ∣uiTbf∣∕si).

Figure 8

L-curve plot for selection of optimal regularization parameter α to be used in the finite difference method

Figure 9

Computation results from the regularized finite difference method with polluted measured responses. Note the force modulus ℘freg here is 0.075N and the regularization parameter α is 4.6×10−10.

Figure 10

Picard condition plot for the modal method, (a) clean bm vector and (b) polluted bm vector (∙, singular value si; ×, ∣uiTbm∣; O, ∣uiTbm∣∕si)

Figure 11

L-curve plot for selection of optimal regularization parameter α, modal method

## Errata

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