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Research Papers

Free Vibrations and Energy Transfer Analysis of the Vibrating Piezoelectric Gyroscope Based on the Linear and Nonlinear Decoupling Methods

[+] Author and Article Information
Wei Li

Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Engineering,
College of Mechanical Engineering,
Beijing University of Technology,
Beijing 100124, China
e-mail: webbli@163.com

Xiao-Dong Yang

Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Engineering,
College of Mechanical Engineering,
Beijing University of Technology,
Beijing 100124, China
e-mail: jxdyang@163.com

Wei Zhang

Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Engineering,
College of Mechanical Engineering,
Beijing University of Technology,
Beijing 100124, China
e-mail: sandyzhang0@163.com

Yuan Ren

Department of Aeronautical and Astronautical Science,
Aerospace Engineering University,
Beijing 101416, China
e-mail: renyuan_823@aliyun.com

Tianzhi Yang

Department of Mechanics,
Tianjin University,
Tianjin 300072, China
e-mail: yangtz@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 August 25, 2018; final manuscript received February 26, 2019; published online May 10, 2019. Assoc. Editor: Miao Yu.

J. Vib. Acoust 141(4), 041015 (May 10, 2019) (11 pages) Paper No: VIB-18-1366; doi: 10.1115/1.4043062 History: Received August 25, 2018; Accepted February 28, 2019

The present research is concerned with the free vibrations and energy transfer of a vibrating gyroscope, which is composed of a flexible beam with surrounded piezoelectric films in a rotating space. The governing equations involve nonlinear curvature, and rotary inertia of an in-extensional rotating piezoelectric beam is obtained by using the transformation of two Euler angles and the extended Hamilton principle. The gyroscopic effect due to the rotating angular speed is investigated in the frame of complex modes based on the invariant manifold method. The effects of angular speed, initial values, and electrical resistance to the nonlinear natural frequencies of a rotating piezoelectric beam are studied by both linear and nonlinear decoupling methods. The results reveal that the rotation causes one nonlinear frequency to bifurcate into a pair of frequencies: one forward and one backward nonlinear frequencies. The variation of the frequency with the angular speed is used to measure the angular speed. Finally, the energy transfer due to nonlinear coupling under 1:1 internal resonance condition and the energy transfer due to the linear gyroscopic decoupling are investigated.

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Figures

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Fig. 1

Schematic of a rotating beam with surrounded four piezoelectric films

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Fig. 2

The snapshots of the modal motions: (a) the mode ϕv when Ω = 0.5 and (b) the mode ϕw when Ω = 0.5

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Fig. 3

Nonlinear frequencies (ω1,NL, ω2,NL) versus angular speed Ω. The solid lines represent the natural frequencies obtained from the linearly decoupled system. The asterisks represent the natural frequencies obtained from the numerical method. The circles represent the natural frequencies obtained from the nonlinearly decoupled system.

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Fig. 4

Nonlinear frequencies (ω1,NL, ω2,NL) versus electrical resistance Z/Z0. The solid lines represent the natural frequencies obtained from the linearly decoupled system. The asterisks represent the natural frequencies obtained from the numerical method. The circles represent the natural frequencies obtained from the nonlinearly decoupled system.

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Fig. 5

Nonlinear frequencies (ω1,NL, ω2,NL) versus initial amplitude. The solid lines represent the nonlinear natural frequencies. The dashed lines represent the linear natural frequencies.

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Fig. 6

Nonlinear frequency difference (ω2,NL-ω1,NL) versus angular speed Ω

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Fig. 7

Time history for angular speed Ω = 0: (a) initial displacements and velocities q(0) = 0.02, p(0) = 0.02, q˙(0) = 0, and p˙(0) = 0 and (b) initial displacements and velocities q(0) = 0.04, p(0) = 0.04, q˙(0) = 0, and p˙(0) = 0

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Fig. 8

Time history for initial displacements and velocities q(0) = 0.02, p(0) = 0.02, q˙(0) = 0, and p˙(0) = 0: (a) angular speed Ω = 0.2 and (b) angular speed Ω = 0.4

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