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Technical Brief

Parametric Instability of Dual-Ring Structure With Motionless and Moving Supports

[+] Author and Article Information
Zhifu Zhao

School of Mechanical Engineering,
Tianjin University,
Tianjin 300072, China;
Key Laboratory of Mechanism Theory and Equipment
Design of Ministry of Education,
Tianjin University,
Tianjin 300072, China;
Tianjin Key Laboratory of Nonlinear
Dynamics and Chaos Control,
Tianjin 300072, China

Shiyu Wang

School of Mechanical Engineering,
Tianjin University,
Tianjin 300072, China;
Key Laboratory of Mechanism Theory and Equipment
Design of Ministry of Education,
Tianjin University,
Tianjin 300072, China;
Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control,
Tianjin 300072, China
e-mail: Wangshiyu@tju.edu.cn

1Corresponding author.

Manuscript received August 31, 2013; final manuscript received March 10, 2015; published online August 12, 2015. Assoc. Editor: Carmen M. Lilley.

J. Comput. Nonlinear Dynam 11(1), 014501 (Aug 12, 2015) (9 pages) Paper No: CND-13-1209; doi: 10.1115/1.4030027 History: Received August 31, 2013

This work is to study parametric vibration of a dual-ring structure through analytical and numerical methods by focusing on the relationships between basic parameters and parametric instability. An elastic dual-ring model is developed by using Lagrange method, where the radial and tangential deflections are included, and motionless and moving supports are also incorporated. Analytical results imply that there are four kinds of parametric excitations, and the numerical results show that there exist stable and unstable areas separated by transition curves or straight lines, and even crossover points. The relationships are determined as simple expressions in basic parameters, including discrete stiffness number and wavenumber. Whether the parametric resonance can be excited or not depends on the values of support stiffness, rotating speed, and natural frequency. Vibrations at the crossover points are also addressed by using the multiscale method. Comparisons against the available results regarding ring structures with moving supports are also made. Extensions of this study, including the use of powerful Sinha method to deal with the parametric vibration, are suggested.

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Figures

Grahic Jump Location
Fig. 1

Schematic of a stationary dual-ring structure subjected to motionless and moving supports and coordinates

Grahic Jump Location
Fig. 2

Parametric instability of dual-ring systems for cases B–G, where white areas denote unstable

Grahic Jump Location
Fig. 3

Vibrations of dual-ring structures with discrete support stiffnesses, where the two rings exhibit uncoupled (a)–(j), or coupled (k)–(p) vibrations

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