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Research Papers

Stochastic Analysis of the Eigenvalue Problem for Mechanical Systems Using Polynomial Chaos Expansion— Application to a Finite Element Rotor

[+] Author and Article Information
E. Sarrouy1

 École Centrale de Lyon Laboratoire de Tribologie et Dynamique des Systèmes (UMR CNRS 5513),36 avenue Guy de Collongue, 69134 Ecully, Cedex, Franceemmanuelle.sarrouy@ec-lyon.fr

O. Dessombz

 École Centrale de Lyon Laboratoire de Tribologie et Dynamique des Systèmes (UMR CNRS 5513),36 avenue Guy de Collongue, 69134 Ecully, Cedex, Franceolivier.dessombz@ec-lyon.fr

J.-J. Sinou

 École Centrale de Lyon Laboratoire de Tribologie et Dynamique des Systèmes (UMR CNRS 5513),36 avenue Guy de Collongue, 69134 Ecully, Cedex, Francejean-jacques.sinou@ec-lyon.fr

1

Address all correspondence to this author.

J. Vib. Acoust 134(5), 051009 (Jun 05, 2012) (12 pages) doi:10.1115/1.4005842 History: Received May 26, 2011; Revised September 29, 2011; Published June 04, 2012; Online June 05, 2012

This paper proposes to use a polynomial chaos expansion approach to compute stochastic complex eigenvalues and eigenvectors of structures including damping or gyroscopic effects. Its application to a finite element rotor model is compared to Monte Carlo simulations. This lets us validate the method and emphasize its advantages. Three different uncertain configurations are studied. For each, a stochastic Campbell diagram is proposed and interpreted and critical speeds dispersion is evaluated. Furthermore, an adaptation of the Modal Accordance Criterion (MAC) is proposed in order to monitor the eigenvectors dispersion.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

FE rotor: plan and node numbers

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Figure 2

FE rotor: deterministic Campbell diagram

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Figure 3

MAC-like plot arrangement

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Figure 4

Case 1: relative error on eigenvalues (%); (a) mean and (b) standard deviation; light blue, order 1 PC and dark red, order 2 PC. Each group of values is relative to a mode. For one mode, each bar represents a different speed Ω = 0, 5, …, 70 Hz. (Please check the online version for color figures)

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Figure 5

Case 2: relative error on eigenvalues (%); (a) mean and (b) standard deviation. Light blue, order 1 PC and dark red, order 2 PC. Each group of values is relative to a mode. For one mode, each bar represents a different speed Ω = 0, 5, …, 70 Hz. (Please check the online version for color figures)

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Figure 6

Case 3: relative error on eigenvalues (%); (a) mean and (b) standard deviation. Light blue, order 1 PC and dark red, order 2 PC. Each group of values is relative to a mode. For one mode, each bar represents a different speed Ω = 0, 5, …, 70 Hz. (Please check the online version for color figures)

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Figure 7

Case 1: relative error on eigenvectors (%); (a) mean and (b) standard deviation. Light blue, order 1 PC and dark red, order 2 PC. Each group of values is relative to a mode. For one mode, each bar represents a different speed Ω = 0, 5, …, 70 Hz. (Please check the online version for color figures)

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Figure 8

Case 2: relative error on eigenvectors (%); (a) mean and (b) standard deviation. Light blue, order 1 PC and dark red, order 2 PC. Each group of values is relative to a mode. For one mode, each bar represents a different speed Ω = 0,5, …, 70 Hz. (Please check the online version for color figures)

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Figure 9

Case 3: relative error on eigenvectors (%); (a) mean and (b) standard deviation; light blue, order 1 PC and dark red, order 2 PC. Each group of values is relative to a mode. For one mode, each bar represents a different speed Ω = 0,5, …, 70 Hz. (Please check the online version for color figures)

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Figure 10

Stochastic Campbell diagrams for (a) case 1, (b) case 2 and (c) case 3; (x.1): histograms, colorbar indicates the number of occurrences; (x.2): mean value (—) and mean value ±3 × standard deviation (light blue patch) (Please check the online version for color figures)

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Figure 11

Critical speeds dispersion: number of occurrences versus critical speed value (Hz) for (a) case 1, (b) case 2 and (c) case 3

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Figure 12

Eigenvectors dispersion: (a) mean and (b) standard deviation of stochastic weights of deterministic modes 1 to 12 and conjugates for PC expansion of order 2 for uncertain case 1

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Figure 13

Eigenvectors dispersion: (a) mean and (b) standard deviation of stochastic weights of deterministic modes 1 to 12 and conjugates for PC expansion of order 2 for uncertain case 2

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Figure 14

Eigenvectors dispersion: (a) mean and (b) standard deviation of stochastic weights of deterministic modes 1 to 12 and conjugates for PC expansion of order 2 for uncertain case 3

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Figure 15

Critical modeshapes dispersion: — mean shape (interpolated), | mean ±3 × standard deviation at nodes; (a) case 1, (b) case 2 and (c) case 3

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