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Research Papers: Gas Turbines: Structures and Dynamics

Vibration Analysis of a Nonlinear System With Cyclic Symmetry

[+] Author and Article Information
Aurélien Grolet1

Laboratoire de Tribologie et Dynamique des Systèmes 36, École Centrale de Lyon, Avenue Guy de Collongue, 69134 Ecully Cedex, Franceaurelien.grolet@ecl2009.ec-lyon.fr

Fabrice Thouverez

Laboratoire de Tribologie et Dynamique des Systèmes 36, École Centrale de Lyon, Avenue Guy de Collongue, 69134 Ecully Cedex, France

1

Author to whom correspondence should be addressed.

J. Eng. Gas Turbines Power 133(2), 022502 (Oct 26, 2010) (9 pages) doi:10.1115/1.4001989 History: Received April 12, 2010; Revised April 26, 2010; Published October 26, 2010; Online October 26, 2010

This work is devoted to the study of nonlinear dynamics of structures with cyclic symmetry under geometrical nonlinearity using the harmonic balance method (HBM). In order to study the influence of the nonlinearity due to large deflection of blades, a simplified model has been developed. It leads to nonlinear differential equations of the second order, linearly coupled, in which the nonlinearity appears by cubic terms. Periodic solutions in both free and forced cases are sought by the HBM coupled with an arc length continuation and stability analysis. In this study, specific attention has been paid to the evaluation of nonlinear modes and to the influence of excitation on dynamic responses. Indeed, several cases of excitation have been analyzed: punctual one and tuned or detuned low engine order. This paper shows that for a localized, or sufficiently detuned, excitation, several solutions can coexist, some of them being represented by closed curves in the frequency-amplitude domain. Those different kinds of solution meet up when increasing the force amplitude, leading to forced nonlinear localization. As the closed curves are not tied with the basic nonlinear solution, they are easily missed. They were calculated using a sequential continuation with the force amplitude as a parameter.

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Figures

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Figure 1

Retained model and system of coordinates for a rectangular plate

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Figure 2

Backbone curves and deformed shapes for natural NNMs with (a) 0 or (b) three nodal diameters

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Figure 3

Backbone curves and deformed shapes for natural NNMs with (a) one and (b) two nodal diameters

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Figure 4

Backbone curve of NNM3D (–) and its localized bifurcations (–.–): (a) one sector, (b) two sectors, (c) three sectors, and (d) four sectors

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Figure 5

Backbone curves of traveling wave motions and time series of the two nodal diameters localized traveling wave

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Figure 6

Nonlinear forced response for a low engine order with zero nodal diameter (–, stable; …, unstable; –.–, linear response)

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Figure 7

Nonlinear forced response for a low engine order with zero nodal diameter with af=1/4 (–, stable; …, unstable; –.–, linear response)

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Figure 8

Nonlinear forced response for a localized excitation on sector 1 (–, stable; …, unstable; –.–, linear response)

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Figure 9

Nonlinear forced response for a localized excitation on sector 1 (–, stable; …, unstable; –.–, linear response) and principle of the closed curve detection

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Figure 10

Nonlinear forced response for a detuned excitation with ε1=1% (–, stable; …, unstable; –.–, linear response)

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Figure 11

Nonlinear forced response for a detuned excitation (–, stable; …, unstable; –.–, linear response)

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