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TECHNICAL PAPERS

Analysis of Piecewise Linear Systems With Boundaries in Displacement and Time

[+] Author and Article Information
Mikio Nakai

Department of Precision Engineering, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501 Japan

Shinji Murata

Japan Electric Power Information Center, Inc., 4-15-33 Shibaura Minato-ku, Tokyo, 108-0023 Japan

Seiji Hagio

Kawasaki Heavy Industry Ltd., 1-1, Kawasaki-cho, Akashi-shi, Hyogo, 673-8666 Japan

J. Vib. Acoust 124(4), 527-536 (Sep 20, 2002) (10 pages) doi:10.1115/1.1502668 History: Received November 01, 2000; Revised February 01, 2002; Online September 20, 2002
Copyright © 2002 by ASME
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References

Nakai, M., Etoh, H., and Murata, S., 1997, “An Analytical Method for Piecewise Linear Systems,” Proceedings of Asia-Pacific Vibration Conference ’97, Vol. II, pp. 928–933.
Komuro,  M., 1988, “Normal Forms of Continuous Piecewise Linear Vector Fields and Chaotic Attractors Part I: Linear Vector Fields with a Section,” Japan Journal of Applied Mathematics, 5(2), pp. 257–304.
Komuro,  M., 1988, “Normal Forms of Continuous Piecewise Linear Vector Fields and Chaotic Attractors Part II: Chaotic Attractors,” Japan Journal of Applied Mathematics, 5(3), pp. 503–549.
Blankenship,  G. W., and Kahraman,  A., 1995, “Steady State Forced Response of a Mechanical Oscillator With Combined Parametric Excitation and Clearance Type Non-linearity,” J. Sound Vib., 185(5), pp. 743–765.
Kahraman,  A., and Blankenship,  G. W., 1996, “Interactions Between Commensurate Parametric and Forcing Excitations in a System with Clearance,” J. Sound Vib., 194(3), pp. 317–336.
Padmanabhan,  C., and Singh,  R., 1996, “Analysis of Periodically Forced Nonlinear Hill’s Oscillator with Application to a Geared System,” J. Acoust. Soc. Am., 99(1), pp. 324–334.
Shaw,  S. W., and Holmes,  P. J., 1983, “A Periodically Forced Piecewise Linear Oscillator,” J. Sound Vib., 90(1), pp. 129–155.
Kuroda,  M., Nakai,  M., Hikawa,  T., and Matsuki,  Y., 1996, “Bifurcations of Higher Subharmonics and Chaos in a Forced Vibratory System with an Asymmetric Restoring Force,” JSME Int. J., Ser C, 39(4), pp. 753–766.
Sato,  K., Yamamoto,  S., and Fujishiro,  S., 1988, “Dynamical Distinctive Phenomena in a Gear System (Bifurcation Sets of Periodic Solutions and Chaotically Transitional Phenomena),” Trans. Jpn. Soc. Mech. Eng., Ser. C, 54(507), C, pp. 2735–2740.
More,  J. J., and Cosnard,  M. Y., 1979, “Numerical Solution of Nonlinear Equations,” ACM Trans. Math. Softw., 5(1), pp. 64–85.

Figures

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Approximated stiffness k(τ) of a pair of gears as a rectangular wave
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Meshing behavior of a pair of gears in two time and three displacement regions
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Defining nomenclatures used in this methodology
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The necessary condition for periodic solution
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Restoring forces in two time regions
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Nomenclatures for this methodology
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Phase portraits and time histories for a stable period-one orbit at the point B (ω=0.7) in Fig. 6(a)
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Phase portraits and time histories for a stable period-two orbit at the point B (ω=1.5) in Fig. 6(a)
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Phase portraits and time histories of x calculated with our present method (2×2 matrix) and numerical results at the point B (ω=1.5) in Fig. 6(a)
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Solution curves (solid line: stable solution, thick solid line: stable solution in bifurcation diagrams of Figs. 6(a) and (b), thick dotted: directly unstable solution, dotted line: inversely unstable solution, all at τ [mod(2π/ω)]=0)
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Phase portraits and time histories of x calculated with our present method (2×2 matrix) and numerical results for a stable period-three orbit at point C (ω=0.598) in Fig. 6(b)

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