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Research Papers

Analytical Prediction for Free Response of Rotationally Ring-Shaped Periodic Structures

[+] Author and Article Information
Dongsheng Zhang, Jianping Liu

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

Shiyu Wang

School of Mechanical Engineering,
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.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 17, 2013; final manuscript received May 4, 2014; published online June 2, 2014. Assoc. Editor: Walter Lacarbonara.

J. Vib. Acoust 136(4), 041016 (Jun 02, 2014) (12 pages) Paper No: VIB-13-1110; doi: 10.1115/1.4027630 History: Received April 17, 2013; Revised May 04, 2014

The in-plane wave motion is analytically examined to address the stationary deflection, natural frequency splitting, and mode contamination of the rotationally ring-shaped periodic structures (RRPS). The governing equation is developed by the Hamilton's principle where the structure is modeled as a thin ring with equally-spaced particles, and the centrifugal effect is included. The free responses are captured by the perturbation method and determined as closed-form expressions. The results imply that the response of stationary RRPS is characterized as standing wave, and the natural frequencies can split when the wave number n and particle number N satisfying 2n/N = int. Also the splitting behavior is determined by the relative angle between the particle and wave antinode. The coefficients of the mode contamination are also obtained. For rotating RRPS, the invariant deflections due to the centrifugal force are estimated at different rotating speeds. It is found that, for certain waves satisfying 2n/N = int, the natural frequency exceeds that of the corresponding smooth ring at the critical speed, and furthermore, the critical speed of the backward traveling wave is lower than that of the forward one. The contamination coefficients of the two kinds of waves are also obtained and they have different magnitudes. All results verify that the splitting and contamination can be determined by the relationship among the mode order, wave number, particle number, and relative position between the particle and antinode. Numerical examples and comparisons with the existing results in the literature are presented.

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Figures

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Fig. 1

Schematic of RRPS and coordinates

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Fig. 2

Centrifugal force for N = 4

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Fig. 3

Time-invariant deflection

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Fig. 4

Invariant deflection for n = 2 and n = 3 when Ωv1 = 1, Ωv2=1/3, and Ωv3=1/10, respectively

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Fig. 5

Variations of ωn with θ0 for n = 2 and N = 2, 3, and 4, respectively

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Fig. 6

Two phases of particles and corresponding frequencies for n = 2 and N = 4

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Fig. 7

Natural frequencies of forward traveling waves (lower curves) and backward ones (higher curves) for N = 3

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Fig. 8

Natural frequencies of forward (lower curves) and backward (higher curves) traveling waves for N = 4

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