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Research Papers

Identification of Plate Boundary Forces From Measured Displacements

[+] Author and Article Information
Simon Chesne

 Laboratoire Vibration Acoustique, INSA de Lyon, Bâtiment St. Exupéry, 25 bis, avenue Jean Capelle, 69621 Villeurbanne Cedex, Francesimon.chesne@insa-lyon.fr

Charles Pezerat, Jean-Louis Guyader

 Laboratoire Vibration Acoustique, INSA de Lyon, Bâtiment St. Exupéry, 25 bis, avenue Jean Capelle, 69621 Villeurbanne Cedex, France

J. Vib. Acoust 130(4), 041006 (Jul 01, 2008) (15 pages) doi:10.1115/1.2890398 History: Received April 05, 2007; Revised November 30, 2007; Published July 01, 2008

This work deals with the identification of forces at plate boundaries, by measuring displacements only. Shear force and bending moment directly depend on different spatial derivatives of displacement at plate boundaries that can be approximated from measured displacements (finite differences, modal approach, etc.), but two major difficulties occur: Derivatives are highly sensitive to measurement errors and the usual methods used to obtain them are not well adapted to boundary points. In this paper, a mathematical approach is proposed to identify shear force at boundaries without any direct calculation of the displacement derivatives. The method is based on the weak form of the plate equation of motion and a test function satisfying particular boundary conditions. Following the description of the technique and the definition of the test function that permits the identification at one boundary point, numerical simulation results, including the effects of noise on displacements, are provided in order to establish the spatial and frequency limits of this method.

Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Boundary conditions of test function η for the shear force reconstruction on a boundary σT.

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Figure 2

(a) Various localizations of elementary integration surfaces; (b) example of a rectangular integration surface S, and its boundaries σ and σT

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Figure 3

Example of a test function ηT for a rectangular integration surface (1×1) obtained from trigonometric expansion 11

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Figure 4

Example of a set of integration surfaces along a straight line

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Figure 5

Details of elementary consecutive integration surfaces

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Figure 6

Error levels εT for 15 different frequencies between 100Hz and 2000Hz, Δx=Δy=0.5cm

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Figure 7

Error level εT for 15 different frequencies between 100Hz and 2000Hz, Δx=Δy=0.1cm

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Figure 8

Comparison between the weighted average of T(w) (noted T¯(w)) computed using the analytical shear force (solid line) and the weighted average of T(w) computed using surface integrations (dotted line). f=1500Hz, Δ=0.5cm, integration surface: 15×15cm2

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Figure 9

Comparison between the analytical shear force (solid line) and reconstructed shear force (dotted line). (a) Modulus; (b) phases. f=1500Hz, Δ=0.5cm, elementary integration surface: 15×15cm2, truncation number r=70.

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Figure 10

Comparison between the weighted average of T(w) computed by using the analytical shear force (solid line) and the weighted average of T(w) computed by using surface integrations (dotted line) with noise displacements. f=1500Hz, Δ=0.5cm, integration surface: 15×15cm2.

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Figure 11

Comparison between an analytical shear force T(w) (solid line) and a reconstructed one (dotted line), by using noise displacements. f=1500Hz, Δ=0.5cm, integration surface: 15×15cm2, truncation number r=70.

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Figure 16

L curve. f=1500Hz, Δ=0.5cm, integration surface: 15×15cm2, optimum truncation parameter r=30.

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Figure 17

Comparison between an analytical bending moment M(w) (solid line) and a reconstructed one (dotted line). (a) Modulus; (b) phases. f=1500Hz, Δ=0.5cm, elementary integration surface: 15×15cm2, truncation number r=30.

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Figure 15

Example of test function ηM for rectangular integration surface (1×1) with trigonometric expansion 26

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Figure 14

Boundary conditions of the test function η for shear force reconstruction on a boundary σM

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Figure 13

Comparison between an analytical shear force T(w) (solid line) and a reconstructed one (dotted line). (a) Modulus; (b) phases. f=1500Hz, Δ=0.5cm, elementary integration surface: 15×15cm, truncation number r=35.

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Figure 12

L curve; f=1500Hz, Δ=0.5cm, integration surface: 1515cm2, optimum truncation parameter r=35

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