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Research Papers

Spectral and Perturbation Analysis of First-Order Beams With Notch Damage

[+] Author and Article Information
N. Apetre1

School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332-0150nicole.apetre@gatech.edu

M. Ruzzene, S. Hanagud

School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332-0150

S. Gopalakrishnan

Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560 012, India

1

Corresponding author.

J. Appl. Mech 75(3), 031019 (May 02, 2008) (10 pages) doi:10.1115/1.2839904 History: Received January 08, 2007; Revised September 17, 2007; Published May 02, 2008

The influence of damage on waves propagating in beam structures is investigated through a numerical model formulated by combining spectral finite elements and perturbation techniques. The resulting numerical tool allows for an efficient computation of the wave propagation response and the analysis of the effects of localized damages of various extents and locations. The dynamic behavior of damaged beams is described through a first-order model, which couples bending and axial behavior, thus allowing the prediction of mode conversion phenomena. Damage is modeled as a small, localized reduction of the beam thickness which, allows for an application of perturbation theory. Numerical examples in the time and frequency domains are presented to illustrate the model capabilities.

Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 2

Spectral finite element with nodal displacements and loads

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Figure 3

(a) Schematic of the clamped-free beam with a longitudinal tip load, modeled using two spectral elements. (b) Modulated sinusoidal pulse load in time and frequency domains

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Figure 5

Comparison of ABAQUS and SFEM group velocities

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Figure 6

Schematic of the simply supported beam with a longitudinal load at the middle, used to compare the superposition of modes and SFEM results

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Figure 8

Displacements as a function of longitudinal coordinate at the same moments: (a) SFEM longitudinal displacement, (b) SM longitudinal displacement, (c) SFEM transverse displacement, and (d) SM transverse displacement

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Figure 9

Displacements as a function of time and horizontal coordinates: (a) longitudinal displacement and (b) transverse displacement. The length of the notch is Δl=0.01m.

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Figure 11

Longitudinal velocity at the midlength for undamaged beam and notched beams with a defect at xd=5L∕8 and xd=6L∕8

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Figure 4

Comparison between FEM and SFEM results: (a) longitudinal and (b) transverse displacements at the free end of the notched beam with a defect at xd=L∕2

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Figure 7

(a) Longitudinal and (b) transverse displacements at the midlength of notched beams with a defect at xd=3L∕4

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Figure 10

Displacements as a function of horizontal coordinate at the same moments: (a) longitudinal displacement and (b) transverse displacement

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Figure 17

Displacements at the tip of a cantilever beam in the frequency domain. Case I: horizontal load. (a) Longitudinal displacement and (b) transverse displacement.

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Figure 18

Displacements at the tip of a cantilever beam in the frequency domain. Case II: vertical load. (a) Longitudinal displacement and (b) transverse displacement.

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Figure 12

(a) Transverse velocity at the midlength of an undamaged beam and notched beams with a defect at xd=5L∕8 and xd=3L∕4. (b) Details of reflections caused by damage.

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Figure 13

(a) Longitudinal velocity at the midlength of notched beams with a defect length, Δl=0.001m, 0.005m, and 0.01m. (b) Details of reflections caused by damage.

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Figure 14

(a) Transverse velocity at the middle point of the beam for three values of damage length, Δl=0.001m, 0.005m, and 0.01m. (b) Details of reflections caused by damage.

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Figure 15

Displacements as a function of time and longitudinal coordinates: (a) longitudinal displacement and (b) transverse displacement. The length of the notch is Δl=0.001m.

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Figure 16

Displacements as a function of longitudinal coordinate at the same moments: (a) longitudinal displacement and (b) transverse displacement

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