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Research Papers

Complete and Simplified Models for Estimating Vibration Instability of Cyclically Symmetric Ring Structures: Comparison and Verification

[+] Author and Article Information
Shiyu Wang

School of Mechanical Engineering,
Tianjin University,
Tianjin 300072, China;
Tianjin Key Laboratory of Nonlinear
Dynamics and Control,
Tianjin 300072, China

Penghui Zhang

School of Mechanical Engineering,
Tianjin University,
Tianjin 300072, China

Wenjia Sun

China Automotive Technology and
Research Center,
Tianjin 300399, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 1, 2017; final manuscript received November 5, 2017; published online January 10, 2018. Assoc. Editor: Massimo Ruzzene.

J. Comput. Nonlinear Dynam 13(3), 031006 (Jan 10, 2018) (14 pages) Paper No: CND-17-1346; doi: 10.1115/1.4038446 History: Received August 01, 2017; Revised November 05, 2017

In-plane vibration of cyclically symmetric ring structures is examined with emphasis on the comparison of instabilities estimated by complete and simplified models. The aim of this paper is to understand under what conditions and to what degree the simplified models can approach the complete model. Previous studies develop time-variant models and employ perturbation method by assuming weak support. This work casts the rotating-load problem into a nonrotating load problem. A complete model with time-invariant coefficients is developed in rotating-support-fixed frame, where the bending and extensional deformations are incorporated. It is then reduced into two simplified ones based on different deformation restrictions. Due to the time-invariant effect observed in the rotating-support-fixed frame, the eigenvalues are formulated directly by using classical vibration theory and compared based on a sample structure. The comparisons verify that the two types of models are comparable only for weak support. Furthermore, the simplified models cannot accurately predict all unstable behaviors in particular for strong support. The eigenvalues are different even in comparable regions. For verification purpose, the time-invariant models are transformed into time-variant ones in the inertial frame, based on which instabilities are estimated by using Floquét theory. Consistence between the time-invariant and -variant models verifies the comparisons.

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Figures

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Fig. 1

Schematic of a cyclically symmetric ring structure in (a) rotating-support-fixed frame and (b) inertial frame

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Fig. 2

Comparisons of eigenvalues versus support stiffness using different models

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Fig. 3

Comparisons of eigenvalues versus wavenumbers using different models

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Fig. 4

Comparisons of unstable regions versus speed and stiffness of rotating support for different wavenumbers

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Fig. 5

Comparisons of unstable regions versus support stiffness and speed for different wavenumbers based on analytical eigenvalue

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Fig. 6

Comparisons of unstable regions versus rotating stiffness and speed for different wavenumbers based on Floquét theory

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Fig. 7

Comparison of real parts versus support stiffness and speed for different wavenumbers based on Floquét theory

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