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Nonstationary vibration of a rotating shaft with nonlinear spring characteristics during acceleration through a critical speed (A critical speed of a summed-and-differential harmonic oscillation)

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Abstract

Nonstationary vibration of a flexible rotating shaft with nonlinear spring characteristics during acceleration through a critical speed of a summed-and-differential harmonic oscillation was investigated. In numerical simulations, we investigated the influence of the angular acceleration λ, the initial angular position of the unbalance ψn and the initial rotating speed ω on the maximum amplitude. We also performed experiments with various angular accelerations. The following results were obtained: (1) the maximum amplitude depends not only on λ but also on ψn and ω: (2) when the initial angular position ψn changes. the maximum amplitude varies between two values. The upper and lower bounds of the maximum amplitude do not change monotonously for the angular acceleration: (3) In order to always pass the critical speed with finite amplitude during acceleration. the value of λ must exceed a certain critical value.

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Abbreviations

O-xyz :

rectangular coordinate system

θ, θ1, θ1 :

inclination angle of rotor and its projections to thexy- andyz-planes

I r :

polar moment of inertia of rotor

I :

diametral moment of inertia of rotor

i r :

ratio ofI r toI

τ:

dynamic unbalance of rotor

Ψ:

directional angle of τ fromx-axis

c :

damping coefficient

δ:

spring constant of shaft

N nt ,N nt :

nonlinear terms in restoring forees in θ1 and θ1 directions

τ4 :

representative angle

ε:

a small quantity

V. V u .V N :

potential energy and its components corresponding to linear and nonlinear terms in the restoring forees

ϕ:

directional angle

ɛn :

coefficients of asymmetrical nonlinear terms

βn :

coefficients of symmetrical nonlinear terms

ɛ:

coefficients of asymmetrical nonlinear terms experessed in polar coordinates

β:

coefficients of symmetrical nonlinear terms expressed in polar coordinates

ω:

rotating speed of shaft

t :

time

ψn :

initial angular position of τ att=0

p :

natural frequency

p 1.p t :

natural frequencies of forward and backward precessions

θω, θ1, θ1 :

total phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation

β, δ1, δ1 :

phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation

P, R t ,R b :

amplitudes of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation

ψ:

difference between phases (ψ = δf − δu)

λ:

acceleration of rotor

ω:

initial rotating speed

t ′ t ,r ′ b :

amplitudes of nonstationary oscillation during acceleration

(r ′ t )max, (r ′b )max :

maximum amplitudes of nonstationary oscillation during acceleration

(r 11 )max, (r 1b )max :

maximum value of angular acceleration of non-passable case

λ0 :

critical value over which the rotor can always pass the critical speed

p 1,p 2,p 3,p 4 :

natural frequencies of experimental apparatus

References

  1. Mitropol'skii, Yu. A.,Problems of the Asymptotic Theory of Nonstationary Vibrations (English Translation), D. Davey & Co., New York, 1965.

    Google Scholar 

  2. Evan-Iwanowski, R. M., ‘Nonstationary Vibration of Mechanical Systems’,Applied Mechanics Reviews 22, 1969, 213–219.

    Google Scholar 

  3. Agrawal, B. N. and Evan-Iwanowski, R. M., ‘Resonances in Nonstationary, Nonlinear, Multidegree-of-Freedom Systems’.AIAA Journal 11 (7), 1973, 907–912.

    Google Scholar 

  4. Ishida, Y., Ikeda, T., and Yamamoto, T.: ‘Transient Vibration of a Rotating Shaft with Nonlinear Spring Characteristies during Acceleration through a Major Critical Speed’,JSME International Journal 30 (261–III), 1987, 458–466.

    Google Scholar 

  5. Ishida, Y., Ikeda, T., Yamamoto, T., and Murakami, S., ‘Nonstationary Vibration of a Rotating Shaft with Nonlinear Spring Characteristics during Acceleration through a Critical Speed (A Critical Speed of a 1/2-Order Subharmonic Oscillation)’,JSME International Journal 32 (4–III), 1989, 575–584.

    Google Scholar 

  6. Yamamoto, T. and Ishida, Y., ‘Theoretical Discussion on Vibration of a Rotating Shaft with Noninear Spring Characteristics’,Ingenieur-Archiv 46 (2), 1977, 125–135.

    Google Scholar 

  7. Yamamoto, T. and Ishida, Y., ‘Oscillations of Rotating Shaft with Symmetrical Nonlinear Spring Characteristics’,Bulletin JSME 18 (123), 1975, 965–975.

    Google Scholar 

  8. Ishida, Y., Ikeda, T., and Yamamoto, T., ‘Effects of Quartic Nonlinear Restoring Forces on 1/2-Order Subharmonic and Summed-and-Differential Harmonic Oscillations of a Rotating Shaft (Variations of Resonance Curves and Occurrence of Unstable Vibrations)’,Bulletin JSME 29 (258), 1986, 4326–4333.

    Google Scholar 

  9. Evan-Iwanowski, R. M.,Resonance Oscillation in Mechanical Systems, Elsevier Scientific Pub. Co., New York, 1976, 132.

    Google Scholar 

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Authors and Affiliations

  1. School of Engineering, Nagoya University, Furo-cho. Chikusa-ku, 464-01, Nagoya, Japan

    Yukio Ishida & Shin Murakami (Postgraduate)

  2. Department of Science and Technology, Meijo University, Shiogamaguchi Tenpaku-ku, 468, Nagoya, Japan

    Toshio Yamamoto

  3. School of Engineering, Hiroshima University, Saijo-cho, 424-05, Higashihiroshima, Japan

    Takashi Ikeda

Authors
  1. Yukio Ishida
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  2. Toshio Yamamoto
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  3. Takashi Ikeda
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  4. Shin Murakami
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Ishida, Y., Yamamoto, T., Ikeda, T. et al. Nonstationary vibration of a rotating shaft with nonlinear spring characteristics during acceleration through a critical speed (A critical speed of a summed-and-differential harmonic oscillation). Nonlinear Dyn 1, 341–358 (1990). https://doi.org/10.1007/BF01893168

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  • DOI: https://doi.org/10.1007/BF01893168

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