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TECHNICAL PAPERS

# Combined Expansion and Orthogonalization of Experimental Modeshapes

[+] Author and Article Information
Y. Halevi1

Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering,  Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0261yoramh@vt.edu

C. A. Morales2

Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering,  Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0261cmorales@vt.edu

D. J. Inman

Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering,  Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0261dinman@vt.edu

1

On sabbatical from the Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel.

2

On sabbatical from Departamento de Mecánica, Universidad Simón Bolívar, Apdo. 89000, Caracas 1080 A, Venezuela.

J. Vib. Acoust 127(2), 188-196 (Jun 04, 2004) (9 pages) doi:10.1115/1.1891817 History: Received December 22, 2003; Revised June 04, 2004

## Abstract

The paper describes a method of combined expansion and orthogonalization (CEO) of experimental modeshapes. Most model updating and error localization methods require a set of full length, orthogonal with respect to the mass matrix, eigenvectors. In practically every modal experiment, the number of measurements is less than the order of the model, and hence modeshape expansion, i.e., adding the unmeasured degrees of freedom, is required. This step is then followed by orthogonalization with respect to the mass matrix. Most current methods use two separate steps for expansion and orthogonalization, each one optimal by itself, but their combination is not optimal. The suggested method combines the two steps into one optimization problem for both steps, and minimizes a quadratic criterion. In the case of an equal number of analytical and experimental modeshapes, the problem coincides with the Procrustes problem and has a closed form solution. Otherwise the solution involves nonlinear equations. Several examples show the advantage of CEO, especially in cases where the measurements are limited either in number or in space, i.e., not spanned through the entire structure.

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## Figures

Figure 9

The structure of example 3. All the measurement points are at 14, 12, and 34 of their substructures.

Figure 8

Case D—the modeshapes error (absolute value). CEO (solid), TSM (dashed), analytical (dash dot).

Figure 7

Case C—the modeshapes error (absolute value). CEO (solid), TSM (dashed), analytical (dash dot).

Figure 6

Case B—the modeshapes error (absolute value). CEO (solid), TSM (dashed), analytical (dash dot).

Figure 5

Case A—the modeshapes error (absolute value). CEO (solid), TSM (dashed), analytical (dash dot).

Figure 4

The true (solid) and the analytical (dashed) modeshapes in example 2

Figure 3

The structure of example 2. Point 1 is at L∕3, points 2–8 are equally spaced between L∕2 and L.

Figure 2

The system of example 1

Figure 1

Schematic diagram of the optimization process

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