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Research Papers

Multi-Dimensional Harmonic Balance With Uncertainties Applied to Rotor Dynamics

[+] Author and Article Information
J. Didier

e-mail: jerome.didier@ec-lyon.fr

J.-J. Sinou

e-mail: jean-jacques.sinou@ec-lyon.fr
LTDS,
UMR-CNRS 5513,
Ecole Centrale de Lyon,
69134, France

B. Faverjon

LaMCoS,
UMR-CNRS 5259,
INSA Lyon,
69621, France
e-mail: beatrice.faverjon@insa-lyon.fr

Contributed by the Design Engineering Division of ASME for publication in the Journalof Vibrationand Acoustics. Manuscript received June 24, 2011; final manuscript received March 12, 2012; published online September 10, 2012. Assoc. Editor: Paul C.-P. Chao.

J. Vib. Acoust 134(6), 061003 (Sep 10, 2012) (17 pages) doi:10.1115/1.4006645 History: Received June 24, 2011; Revised March 12, 2012

This paper describes the coupling of a Multi-Dimensional Harmonic Balance Method (MHBM) with a Polynomial Chaos Expansion (PCE) to determine the dynamic response of quasi-periodic dynamic systems subjected to multiple excitations and uncertainties. The proposed method will be applied to a rotor system excited at its support. Uncertainties considered include both material and geometrical parameters as well as excitation sources. To demonstrate the effectiveness and validity of the proposed numerical approach, the results that include mean, variation of the response, envelopes of the Frequency Response Functions and orbits will be systematically compared to a classical Monte Carlo approach.

Copyright © 2012 by ASME
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References

Figures

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Fig. 1

Rotor system with the cross section of the asymmetric shaft (a) rotor with asymmetric shaft (b) uncertainty on the shaft section

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Fig. 2

Shaft finite element

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Fig. 3

Hypertime concepts related to a bi-periodic motion

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Fig. 4

Frequency Response Function via PCE (case 1): (a) mean; (b) variation coefficient

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Fig. 6

Frequency Response Function (case 1 and fb = 100 Hz), global quasi-periodic response, envelope PCE (red solid lines), MCS (dotted gray lines), deterministic response (dashed-dotted black line)

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Fig. 7

Mean and variance of the FRFs (case 1 and fb = 100 Hz), PCE (red solid lines), MCS (dashed black lines), order FRFs [k1,k2] (a) [0,0], (b) [0,1], (c) [1,0], (d) [2,-1], (e) [2,0], (f) [2,1], (g) [3,0], (h) [4,0]

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Fig. 8

Mean and variance of the FRFs (case 1 and fb = 100 Hz), global quasi-periodic response, PCE (red solid lines), MCS (dashed black lines)

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Fig. 9

Frequency Response Function (case 2 and fb = 100 Hz); envelope PCE (red solid lines), MCS (dotted gray lines), deterministic response (dashed-dotted black lines); order [k1,k2] (a) [0,0], (b) [0,1], (c) [1,0], (d) [2,-1], (e) [2,0], (f) [2,1], (g) [3,0], and (h) [4,0]

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Fig. 10

Frequency Response Function (case 2 and fb = 100 Hz); global quasi-periodic response, envelope PCE (red solid lines), MCS (dotted gray lines), deterministic response (dashed-dotted black line)

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Fig. 16

Case 1: (a), (c), and (e) orbits at fr = 12.5 Hz and fb = 200 Hz; (b), (d), and (f) orbits at fr = 64 Hz and fb = 100 Hz; envelope PCE (a),(b) (green); MC samples (c), (d) (gray lines); deterministic response (all) (blue lines), mean PCE (a), (b), (e), and (f) (red lines), mean MCS (c), (d), (e), and (f) (dashed black lines)

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Fig. 11

Case 1: (a), (c), and (e) orbits at fr = 12.5 Hz and fb = 100 Hz; (b), (d), and (f) orbits at fr = 64 Hz and fb = 100 Hz; envelope PCE (a),(b) (green); MC samples (c),(d) (gray lines); deterministic response (all) (blue lines), mean PCE (a), (b), (e), (f) (red lines), mean MCS (c), (d), (e), and (f) (dashed black lines)

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Fig. 5

Frequency Response Function (case 1 and fb = 100 Hz); envelope PCE (red solid lines); MCS (dotted gray lines); deterministic response (dashed-dotted black lines); order [k1,k2] (a) [0,0], (b) [0,1], (c) [1,0], (d) [2,-1], (e) [2,0], (f) [2,1], (g) [3,0], and (h) [4,0]

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Fig. 12

Frequency Response Function (case 2 and fb = 100 Hz); envelope PCE (red solid lines), MCS (dotted gray lines), deterministic response (dashed-dotted black lines); order [k1,k2] (a) [0,0], (b) [0,1], (c) [1,0], (d) [2,-1], (e) [2,0], (f) [2,1], (g) [3,0], and (h) [4,0]

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Fig. 13

Frequency Response Function (case 2 and fb = 100 Hz); global quasi-periodic response, envelope PCE (red solid lines), MCS (dotted gray lines), deterministic response (dashed-dotted black line)

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Fig. 14

Frequency Response Function (case 2 and fb = 200 Hz); envelope PCE (red solid lines), MCS (dotted gray lines), deterministic response (dashed-dotted black lines); order [k1,k2] (a) [0,0], (b) [0,1], (c) [1,0], (d) [2,-1], (e) [2,0], (f) [2,1], (g) [3,0], and (h) [4,0]

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Fig. 15

Frequency Response Function (case 2 and fb = 200 Hz); global quasi-periodic response, envelope PCE (red solid lines), MCS (dotted gray lines), deterministic response (dashed-dotted black line)

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