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Research Papers

Nonlinear Dynamical Analysis of the “Power Ball”

[+] Author and Article Information
Tsuyoshi Inoue, Kentaro Takagi

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

Kohei Okumura

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

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 13, 2016; final manuscript received June 1, 2017; published online July 12, 2017. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 12(5), 051027 (Jul 12, 2017) (8 pages) Paper No: CND-16-1621; doi: 10.1115/1.4037075 History: Received December 13, 2016; Revised June 01, 2017

The gyroscopic exercise tool called the “Power Ball,” used to train the antebrachial muscle, is focused on. The basin of attraction of the synchronous rolling motion in the state space of initial condition is investigated. The reduced model governing the synchronous rolling motion is used and its averaged equation is deduced. The first integral for the dynamical behavior of the synchronous rolling motion occurring in the power ball is obtained. The separatrix, which identifies the basin of attraction of the synchronous rolling motion, is derived, and the ranges of initial precession angle and the initial spin angular velocity for realizing the synchronous rolling motion are clarified. These theoretically obtained results are then experimentally confirmed. Furthermore, the influences of parameters to the basin of attraction are also clarified.

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References

Mishler , 1973, “ Gyroscopic Device,” Wornoto Inc., Willamina, OR, U.S. Patent No. US 3726146 A. http://www.google.co.in/patents/US3726146
Gulick, D. W. , and O'Reilly, O. M. , 2000, “ On the Dynamics of the Dynabee,” ASME J. Appl. Mech., 67(2), pp. 321–325. [CrossRef]
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Heyda, P. G. , 2000, “ Roller Ball Dynamics,” Math. Today, 36(9).
Petric, T. , Curk, B. , Cafuta, P. , and Zlajpah, L. , 2010, “ Modelling of the Robotic Powerball®: A Nonholonomic, Underactuated and Variable Structure-Type System,” Math. Comput. Modell. Dyn. Syst., 16(4), pp. 327–346.
Ishii, T. , Goto, Y. , Ogawa, T. , and Hosaka, H. , 2008, “ Development of Gyroscopic Generator,” J. Jpn. Soc. Precis. Eng., 74(7), pp. 764–768 (in Japanese). [CrossRef]
Yoshikawa, S. , Iwasaki, J. , Kishimoto, K. , Hosaka, H. , and Sasaki, K. , 2010, “ Transient Analysis of Gyroscopic Power Generator,” J. Jpn. Soc. Precis. Eng., 76(2), pp. 238–242 (in Japanese). [CrossRef]
Takahashi, T. , Iwasaki, J. , and Hosaka, H. , 2011, “ Passive Control of Gyroscopic Power Generator,” J. Rob. Soc. Jpn., 29(8), pp. 661–666 (in Japanese). [CrossRef]
Inoue, T. , Takagi, K. , Takezaki, Y. , and Ishikawa, M. , 2013, “ Dynamical Modeling and Stability Analysis of a Gyroscopic Exercise Tool Considering Contact/Separation and Slide Between the Rotor and the Case,” J. Syst. Des. Dyn., 7(1), pp. 52–65.
Inoue, T. , Takezaki, Y. , Takagi, K. , and Okumura, K. , 2015, “ Dynamical Modeling of the ‘Power Ball’ Considering the Transition Between the Rolling Mode and the Sliding Mode,” ASME J. Vib. Acoust., 137(6), p. 061010. [CrossRef]
Nayfeh, A. H. , and Mook, D. T. , 1995, Nonlinear Oscillations, Wiley Classics Library, Hoboken, NJ.
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Figures

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Fig. 1

Structure of Powerball®

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Fig. 2

Frames and angles used in the modeling of Powerball®

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Fig. 3

Phase plane ϕ,ϕ˙: (a) case of Eq. (8) and (b) case of averaged equation, Eq. (12)

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Fig. 4

Experimental and numerical results of time period Tω of the damped free vibration around the equilibrium point ϕ01

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Fig. 5

Energy distribution in the phase space

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Fig. 6

Energy curve in the phase space with φ˙=0

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Fig. 7

Orbits of phase space and their investigation using separatrix: (a) separatrix obtained from Eq. (18) and several energy curves obtained from Eq. (16) and (b) separatrix obtained from Eq. (18) and orbits from three initial conditions obtained from Eq. (12) (see color figure online)

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Fig. 11

Experimental result: relation between the initial precession angle α0 and the types of realized motions: (a) case with initial condition of Θ = 25 deg, Ω = 1.5 Hz, and γ˙0 = 2300 rpm (γ˙sync = 2354 rpm) and (b) case with initial condition of Θ = 20 deg, Ω = 1.3 Hz, and γ˙0 = 2000 rpm (γ˙sync = 2040 rpm)

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Fig. 10

Experimental result: relation between the initial spin angular velocity γ˙0 and the types of realized motions. Initial precession angle α0 was randomly set: (a) case with initial condition of Θ = 30 deg and Ω = 1.3 Hz and (b) case with initial condition of Θ = 25 deg and Ω = 1.5 Hz.

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Fig. 9

Experimental system

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Fig. 8

Entrainment region of synchronous motion (see color figure online)

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Fig. 12

Relationship between the basin of attraction and polar moment of inertia I1: (a) influence of polar moment of inertia I1 on the area J of basin of attraction and (b) basin of attraction of synchronous motion and subsynchronous motion for I1 = 60 μ kg m2 (input parameters: Θ  = 20 deg and Ω  = 1.3 Hz) (see color figure online)

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Fig. 13

Relationship between the basin of attraction and rotor shaft length Rt: (a) influence of rotor shaft length Rt on the area J of basin of attraction and (b) basin of attraction of synchronous motion and subsynchronous motion for Rt  = 25 mm (input parameters: Θ  = 20 deg and Ω  = 1.3 Hz)

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Fig. 14

Influence of rotor shaft radius Ra on the area J of basin of attraction

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