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Energetics and conserved quantity of an axially moving string undergoing three-dimensional nonlinear vibration

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Abstract

Nonlinear three-dimensional vibration of axially moving strings is investigated in the view of energetics. The governing equation is derived from the Eulerian equation of motion of a continuum for axially accelerating strings. The time-rate of the total mechanical energy associated with the vibration is calculated for the string with its ends moving in a prescribed way. For a string moving in a constant axial speed and constrained by two fixed ends, a conserved quantity is proved to remain unchanged during three-dimensional vibration, while the string energy is not conserved. An approximate conserved quantity is derived from the conserved quantity in the neighborhood of the straight equilibrium configuration. The approximate conserved quantity is applied to verify the Lyapunov stability of the straight equilibrium configuration. Numerical simulations are performed for a rubber string and a steel string. The results demonstrate the variation of the total mechanical energy and the invariance of the conserved quantity.

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Authors and Affiliations

  1. Department of Mechanics, Shanghai University, Shanghai, 200444, China

    Liqun Chen

  2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai, 200072, China

    Liqun Chen & Hu Ding

  3. Department of Construction and Building, City University of Hong Kong, Hong Kong, China

    C. W. Lim

Authors
  1. Liqun Chen
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  2. C. W. Lim
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  3. Hu Ding
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Correspondence to Liqun Chen.

Additional information

The project supported by the National Natural Science Foundation of China (10472060), Research Grants Council of the Hong Kong Special Administrative Region (9041145), Shanghai Municipal Education Commission Scientific Research Project (07ZZ07) and Shanghai Leading Academic Discipline Project (Y0103).

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Chen, L., Lim, C.W. & Ding, H. Energetics and conserved quantity of an axially moving string undergoing three-dimensional nonlinear vibration. Acta Mech. Sin. 24, 215–221 (2008). https://doi.org/10.1007/s10409-007-0110-5

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  • DOI: https://doi.org/10.1007/s10409-007-0110-5

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