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TECHNICAL PAPERS

# An Extended Karhunen-Loève Decomposition for Modal Identification of Inhomogeneous Structures

[+] Author and Article Information
U. Iemma, M. Diez, L. Morino

Mechanical and Industrial Engineering Department, University “Roma Tre,” Via Vasca Navale 79, 00146 Rome, Italy

$En$ is an $n$-dimensional $(n=1,2,3)$ Euclidean point space and $Vm$ is an $m$-dimensional $(m=1,2,3)$ vector space, with $n$ not necessarily equal to $m$; consider, for instance, the case of a bending beam $(n=1,m=2),$ or of a bending plate $(n=2,m=1)$.

Here, $(f,g)$ denotes the standard inner product in $L2(V)$, i.e., $(f,g)≔∫Vf∙gdx$.

Although the structure is capable of rigid-body motion, the theory presented is still applicable because we assume that only elastic modes are present in the response.

Feeny [12] applies the same procedure directly on the structural operator, so as to eliminate the density from the problem.

It may be noted that if $Δxi$ are not uniform, it is still possible to reduce the problem to a symmetric one, by using the unknown vector ${Δxjφ(xj)}$ and the matrix $[ΔxiRijΔxj]$ (this is analogous to what has been done for $ρ$). The issue of a nonuniform discretization of the structure domain has been addressed by Han and Feeny in Ref. 17.

J. Vib. Acoust 128(3), 357-365 (Dec 16, 2005) (9 pages) doi:10.1115/1.2172263 History: Received March 11, 2005; Revised December 16, 2005

## Abstract

An extension of the Karhunen-Loève decomposition (KLD) specifically aimed at the evaluation of the natural modes of $n$-dimensional structures $(n=1,2,3)$ having nonhomogeneous density is presented. The KLD (also known as proper orthogonal decomposition) is a numerical method to obtain an “optimal” basis, capable of extracting from a data ensemble the maximum energy content. The extension under consideration consists of modifying the Hilbert space that embeds the formulation so as to have an inner product with a weight equal to the density. This yields a modified Karhunen-Loève integral operator, whose kernel is represented by the time-averaged autocorrelation tensor of the ensemble of data multiplied by the density function. The basis functions are obtained as the eigenfunctions of this operator; the corresponding eigenvalues represent the Hilbert-space-norm energy associated with each eigenfunction in the phenomenon analyzed. It is shown under what conditions the eigenfunctions, obtained using the proposed extension of the KLD, coincide with the natural modes of vibration of the structure (linear normal modes). An efficient numerical procedure for the implementation of the method is also presented.

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## Figures

Figure 1

Free rectangular plate. First test case.

Figure 4

Fifth structural mode evaluated through the KLD for first test case

Figure 5

Comparison between third and fifth modes evaluated through the KLD and their corresponding structural modes at y=0.16m for first test case

Figure 6

First ten KLD eigenvalues for first test case

Figure 7

Free rectangular plate. Second test case.

Figure 8

Mass per unit area versus x for second test case

Figure 9

First structural mode evaluated through the KLD for second test case

Figure 10

Fourth structural mode evaluated through the KLD for second test case

Figure 11

Comparison between first and fourth modes evaluated through the KLD and their corresponding structural modes at y=0.11m for second test case

Figure 2

Mass per unit area versus x for first test case

Figure 3

Third structural mode evaluated through the KLD for first test case

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