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Research Papers

Dynamical Modeling of the “Power Ball” Considering the Transition Between the Rolling Mode and the Sliding Mode

[+] Author and Article Information
Tsuyoshi Inoue, Yuki Takezaki, Kentaro Takagi, Kohei Okumura

Department of Mechanical Science and Engineering,
Nagoya University,
Furo-cho, Chikusa-ku,
Nagoya 464-8603, Japan

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 27, 2015; final manuscript received July 9, 2015; published online August 4, 2015. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 137(6), 061010 (Aug 04, 2015) (14 pages) Paper No: VIB-15-1142; doi: 10.1115/1.4031066 History: Received April 27, 2015

The gyroscopic exercise tool which utilizes a gyroscopic effect caused by the whirling motion of a high-speed rotating body to train the antebrachial muscle is considered. When an input motion of 3–5 Hz is added to the case, the rotor spins at thousands of rpm whirling with a precession motion which is synchronous to the input motion given to the case. This tool utilizes a contact phenomenon between the rotor and the case to generate this high-speed spin motion. This paper develops its dynamical model considering the transition among noncontacting, rolling, and sliding conditions. The dynamical characteristics of its motions are numerically investigated and are also confirmed in the experiment.

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Figures

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Fig. 1

Model of gyroscopic exercise tool power ball

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Fig. 2

Definition of frames e,et,ep, and ef

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Fig. 3

Contact between rotor and track: (a) configuration at contact and (b) contact model using equivalent spring and damper (view from origin O to ef1 direction)

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Fig. 4

Modes of contact (view from origin O to ef1 direction): (a) floating, (b) rolling, and (c) sliding

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Fig. 5

Model of the rotor as HDS (seven modes) (view from origin O to ef1 direction)

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Fig. 6

Possible motion patterns: (a) uniform rotational motion and (b) periodic reversing motion

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Fig. 7

Time history of rotational angle γ·, precession angle α·, nutation angle δ, and transition of modes in the rotational mode (HDS model): (a) angular velocity of the shaft rotation, (b) precession angle, (c) nutation angle, (d) transition of modes, (e) transition of modes t = 0–10, and (f) experimental result of spin angular velocity γ·

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Fig. 8

Time history of rotational angle γ·, angle α·, angle δ, and transition of modes in the periodic reversing motion (HDS model): (a) angular velocity of the shaft rotation, (b) precession angle, (c) nutation angle, (d) transition of modes, (e) transition of modes t = 20–30, (f) transition of modes t = 30–40, and (g) experimental result of spin angular velocity γ·

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Fig. 9

Time history of procession angle α and transition of modes in the periodically reverse mode (HDS model): (a) precession angle, (b) transition modes, and (c) nutation angle

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Fig. 10

Time history of rotational angle γ·, angle α·, angle δ, and transition of modes in the periodic reversing subsynchronous precession motion: (a) angular velocity of the shaft rotation, (b) precession angle, (c) nutation angle, (d) transition of modes, (e) transition of modes t = 30–40, and (f) experimental result of spin angular velocity γ·

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Fig. 11

Time history of procession angle α and transition of modes in the periodic reversing subsynchronous precession motion: (a) precession angle, (b) transition modes, and (c) nutation angle

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Fig. 12

Experimental setup

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