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TECHNICAL PAPERS

Identification of Boundary Forces in Beams From Measured Displacements

[+] Author and Article Information
S. Chesne, C. Pezerat, J. L. Guyader

 Laboratoire Vibration Acoustique de l’INSA de Lyon, 20 avenue A. Einstein, 69621 Villeurbanne Cedex, France

J. Vib. Acoust 128(6), 757-771 (Jan 19, 2006) (15 pages) doi:10.1115/1.2202171 History: Received July 11, 2005; Revised January 19, 2006

This paper deals with shear force and bending moment identification in a beam from measured displacement. The proposed approach, using the weak form of the equation of motion, is based on the extraction of shear force or bending moment from integral equation and choice of test functions, associated to each boundary quantity of interest. After the theorical description, numerical simulation results are shown in order to clarify limits of the method and to stress its self-regularization. Two experimentations are described, showing very good accuracy of shear force and bending moment reconstruction in comparison with direct measurements.

Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 6

SFREL ϵT between the simulation and the analytic shear force for (a) trapezoid integration method and (b) Gauss-Legendre integration method, using 10 (dashed line), 14 (crosses), or 20 (solid line) points. The driving frequency is 2500Hz and the caracteristics of the beam are: L=2.5m, E=2.1011(1+j10−2)N∕m2, l=0.06m, h=0.01m, and ρ=7800kg∕m3.

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Figure 7

SFREL ϵT between the simulation and the analytic shear force for trapezoid integration method using 20 points for 500 different driving frequencies between 10Hz and 5000Hz. The characteristics of the beam are: L=2.5m, E=2.1011(1+j10−2)N∕m2, l=0.06cm, h=0.0m, and ρ=7800kg∕m3.

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Figure 18

Identification of the shear force with trapezoid integration method

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Figure 19

Identification of the shear force with Gauss-Legendre integration method

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Figure 20

Second experimental setup used for the bending moment reconstruction. Steel beam length L=1.5m, width l=6cm, and height h=1cm.

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Figure 21

Identification of the bending moment with trapezoid integration method

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Figure 22

Identification of the bending moment with Gauss-Legendre integration method

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Figure 9

BMREL between the simulation and the analytic bending moment for (a) trapezoid integration method and (b) Gauss-Legendre integration method, using 10 (dashed line), 14 (crosses), or 20 (solid line) points. The driving frequency is 2500Hz, and the characteristics of the beam are: L=2.5m, E=2.1011(1+j10−2)N∕m2, l=0.06m, h=0.01m, and ρ=7800kg∕m3.

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Figure 8

Clamped free beam of length L excited by an harmonic force located at Xf=0.8L

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Figure 10

SFREL between the simulation and the analytic shear force (a) trapezoid integration method and (b) Gauss-Legendre integration method, using 20 points with noisy displacements (multiplicative noise). SFREL using exact displacements: solid line, using noisy displacements: crosses. The driving frequency is 2500Hz and the characteristics of the beam are: L=2.5m, E=2.1011(1+j10−2)N∕m2, l=0.06m, h=0.01m, and ρ=7800kg∕m3.

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Figure 11

SFREL between the simulation and the analytic shear force (a) trapezoid integration method and (b) Gauss-Legendre integration method, using 20 points with an error on the location of the points. SFREL using exact displacements: solid line, using noisy displacements: crosses. The driving frequency is 2500Hz and the characteristics of the beam are: L=2.5m, E=2.1011(1+j10−2)N∕m2, l=0.06m, h=0.01m, and ρ=7800kg∕m3.

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Figure 16

Experimental setup for the shear force reconstruction. Steel beam of length L=1.5m, width l=6cm, and height h=1cm.

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Figure 1

Function η(x) for x∊[a;b], used for shear force calculation at the left boundary of the integration domain

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Figure 2

Function ∂4η∕∂x4forx∊[a;b]and∥ab∥=1, used for shear force calculation at the left boundary of the integration domain

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Figure 3

Function η(x) for x∊[a;b], used for bending moment calculation at the left boundary of the integration domain

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Figure 4

Function ∂4η∕∂x4 for x∊[a;b] and ∥ab∥=1, used for bending moment calculation at the left boundary of the integration domain

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Figure 5

Simply supported beam of length L excited by a harmonic force located at Xf=0.8L

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Figure 17

Modulus of the FRF in terms of displacement at the excited boundary of the beam. Steel beam of length L=1.5m, width l=6cm, and height h=1cm.

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Figure 12

SFREL between the simulation and the analytic shear force (a) trapezoid integration method and (b) Gauss-Legendre integration method, using 20 points with noisy displacements (additive noise). SFREL using exact displacements: solid line, using noisy displacements: crosses. The driving frequency is 2500Hz, and the characteristics of the beam are: L=2.5m, E=2.1011(1+j10−2)N∕m2, l=0.06m, h=0.01m, and ρ=7800kg∕m3.

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Figure 13

BMREL between the simulation and the analytic bending moment for (a) trapezoid integration method and (b) Gauss-Legendre integration method, using 20 points with noisy displacements (multiplicatve noise). BMREL using exact displacements: solid line, using noisy displacements: crosses. The driving frequency is 2500Hz and the characteristics of the beam are: L=2.5m, E=2.1011(1+j10−2)N∕m2, l=0.06m, h=0.01m, and ρ=7800kg∕m3.

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Figure 14

BMREL between the simulation and the analytic bending moment for (a) trapezoid integration method and (b) Gauss-Legendre integration method, using 20 points with an error on the location of the points. BMREL using exact displacements: solid line, using noisy displacements: crosses. The driving frequency is 2500Hz and the characteristics of the beam are: L=2.5m, E=2.1011(1+j10−2)N∕m2, l=0.06m, h=0.01m, and ρ=7800kg∕m3.

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Figure 15

SFREL between the simulation and the analytic shear force (a) trapezoid integration method and (b) Gauss-Legendre integration method, using 20 points with noisy displacements (additive noise). BMREL using exact displacements: solid line, using noisy displacements: crosses. The driving frequency is 2500Hz and the characteristics of the beam are: L=2.5m, E=2.1011(1+j10−2)N∕m2, l=0.06m, h=0.01m, and ρ=7800kg∕m3.

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