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Research Papers

Spectral Element Approach to Wave Propagation in Uncertain Composite Beam Structures

[+] Author and Article Information
V. Ajith1

Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, Indiaajith@aero.iisc.ernet.in

S. Gopalakrishnan

Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, Indiakrishnan@aero.iisc.ernet.in

1

Corresponding author.

J. Vib. Acoust 133(5), 051006 (Jul 26, 2011) (19 pages) doi:10.1115/1.4003945 History: Received September 21, 2010; Revised March 14, 2011; Published July 26, 2011; Online July 26, 2011

This paper presents a study of the wave propagation responses in composite structures in an uncertain environment. Here, the main aim of the work is to quantify the effect of uncertainty in the wave propagation responses at high frequencies. The material properties are considered uncertain and the analysis is performed using Neumann expansion blended with Monte Carlo simulation under the environment of spectral finite element method. The material randomness is included in the conventional wave propagation analysis by different distributions (namely, the normal and the Weibul distribution) and their effect on wave propagation in a composite beam is analyzed. The numerical results presented investigates the effect of material uncertainties on different parameters, namely, wavenumber and group speed, which are relevant in the wave propagation analysis. The effect of the parameters, such as fiber orientation, lay-up sequence, number of layers, and the layer thickness on the uncertain responses due to dynamic impulse load, is thoroughly analyzed. Significant changes are observed in the high frequency responses with the variation in the above parameters, even for a small coefficient of variation. High frequency impact loads are applied and a number of interesting results are presented, which brings out the true effects of uncertainty in the high frequency responses.

Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Nondimensional wavenumber (kh) for [012] glass-epoxy laminated composite. (a) Euler–Bernoulli beam and (b) Timoshenko beam.

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Figure 2

Nondimensional group speed (Cg/C0) for [012] glass-epoxy laminated composite (C0=(E11/ρ)1/2). (a) Euler–Bernoulli beam and (b) Timoshenko beam.

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Figure 3

Comparison of SFEM and FEM, where E11 is considered uncertain normal random variable with COV 10%, with 10,000 MCS.

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Figure 4

Time required for uncertain analysis, using FEM and SEM, as a function of number of Monte Carlo simulations for different loading frequencies: (a) 10 kHz (1024 FFT points) and (b) 50 kHz (512 FFT points)

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Figure 5

Normalized time required for the NE-MCS analysis (time required by NE-MCS/Time required by direct MCS) with the increase in the order of MCS

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Figure 6

Wavenumber distribution, obtained using direct MCS analysis and NE-MCS, for different order of expansion (here, all material properties are considered as uncertain). Variation of flexural wavenumber at 100 kHz (maximum frequency used) for different COVs, distributions, numbers of layers, lay-up sequences, and layer thicknesses: (a) COV 1%, normal, 12 layers (00), 0.2 mm thick; (b) COV 10%, normal, 12 layers (00), 0.2 mm thick; (c) COV 10%, normal, 24 layers (00), 0.2mm thick; (d) COV 10%, normal, 12 layers (00), 0.8 mm thick; (e) COV 10%, normal, lay-up sequence 0/0/0/90/90/90, 0.4 mm thick; (f) COV 10%,normal, lay-up sequence 0/90/0/0/90/0, 0.4 mm thick; (g) COV 10%, normal, 12 layers (00), 0.8 mm thick; and (h) COV 10%, Weibul, 12 layers (00), 0.8 mm thick. Variation of axial wavenumber: (i) COV 10%, normal, 12 layers (00), 0.2 mm thick.

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Figure 7

COV of wavenumber (flexural), when all the material parameters are uncertain and are represented by normal random variables, with (a) COV of input parameters and (b) frequency.

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Figure 8

COV of wavenumber (flexural)with variation of (a) COV of input parameters E11 and E22. Variation of COV with frequency for (b) E11 variation and (c) E22 variation.

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Figure 9

COV of wavenumber (flexural), when G44, G55, and G66 vary as normal random variables, with (a) COV of input parameters and (b) frequency

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Figure 10

COV of wavenumber (flexural) with (a) COV of input parameter density and (b) with the change in lay-up sequence

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Figure 11

COV of wavenumber (axial) with variation of COV of input parameters for (a) different input parameters, (b) different fiber orientations and the change in COV of wavenumber, and (c) with the change in lay-up sequences.

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Figure 12

Histogram of distribution of wavenumber (flexural) at different frequencies. All the material parameters varies as normal random variables with COV 10% (layer thickness=0.2 mm). (a) 1 kHz and (b) 100 kHz.

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Figure 13

Histogram of distribution of wavenumber (flexural) at different frequencies. All the material parameters varies as Weibul random variables with COV 10%(layer thickness=0.2 mm). (a) 5 kHz and (b) 100 kHz.

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Figure 14

Histogram of distribution of wavenumber (flexural) at different frequencies. All the material parameters varies as normal random variables with COV 10% (layer thickness=0.8 mm). (a) 5 kHz and (b) 100 kHz.

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Figure 15

Histogram of distribution of wavenumber (flexural) at different frequencies. All the material parameters varies as Weibul random variables with COV 10% (layer thickness=0.8 mm). (a) 5 kHz and (b) 100 kHz.

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Figure 16

Histogram of distribution of wavenumber (axial). All the material parameters varies with COV 10% (layer thickness=0.2 mm): (a) normal and (b)Weibul.

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Figure 17

Input force used in simulation: (a) broad band pulse and (b) narrow band modulated (at 20 kHz) pulse

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Figure 18

Axial velocity response for an axial impact load using normal random variables(layer thickness=0.2 mm) (a) E11 varies with COV 1%, (b) E22 varies with COV 1%, (c) E11 varies with COV 10%, and (d) E22 varies with COV 10%

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Figure 19

Axial velocity response for an axial impact load using normal random variables (layer thickness=0.2 mm) with COV 10%: (a) G66 variations and (b) density variations

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Figure 20

Transverse velocity response for an impact load using normal random variables (layer thickness=0.2 mm): (a) E11 varies with COV 1%, (b) E11 varies with COV 10%, and (c) E22 varies with COV 10%.

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Figure 21

Transverse velocity response for an impact load using normal random variables. Density varies with COV 1% and COV 10% with fiber direction (layer thickness=0.2 mm): (a) 0 deg, (b) 60 deg, and (c) 90 deg.

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Figure 22

Transverse velocity response for an impact load using normal random variables, with COV 10%: (a) G55 variations and (b) density variations.

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Figure 23

Histogram of distribution of time of first reflection (flexural). All the material parameters vary as random variables with COV 10% (layer thickness=0.2 mm). (a) Normal and (b) Weibul.

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Figure 24

Histogram of distribution of time of first reflection (flexural). All the material parameters varies as normal random variables with COV 10% (layer thickness=0.8 mm).

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Figure 25

Histogram of distribution of time of first reflection (flexural). Density varies as normal random variables with COV 10% (layer thickness=0.2 mm).

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Figure 26

Histogram of distribution of time of first reflection (flexural). E11 varies as normal random variables with COV 10% (layer thickness=0.2 mm).

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Figure 27

Histogram of distribution of time of first reflection(flexural). E22 varies as normal random variables with COV 10% (layer thickness=0.2 mm).

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Figure 28

Histogram of distribution of time of first reflection(flexural). G55 varies as normal random variables with COV 10% (layer thickness=0.2 mm).

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Figure 29

Histogram of distribution of time of first reflection (flexural). G44, G55, and G66 vary as normal random variables with COV 10% (layer thickness=0.2 mm).

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Figure 30

Variation of cutoff frequency (shear mode), COV fixed to 10% considering input normal random variables. (a) G55, 0.4 mm layer thickness, (b) G55, 0.8 mm layer thickness; (c) density, 0.4 mm layer thickness; and (d) density, 0.8 mm layer thickness.

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