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On Galerkin Discretization of Axially Moving Nonlinear Strings

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Abstract

A computational technique is proposed for the Galerkin discretization of axially moving strings with geometric nonlinearity. The Galerkin discretization is based on the eigenfunctions of stationary strings. The discretized equations are simplified by regrouping nonlinear terms to reduce the computation work. The scheme can be easily implemented in the practical programming. Numerical results show the effectiveness of the technique. The results also highlight the feature of Galerkin’s discretization of gyroscopic continua that the term number in Galerkin’s discretization should be even. The technique is generalized from elastic strings to viscoelastic strings.

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References

  1. Wickert, J.A. and Mote, C.D. Jr., Current research on the vibration and stability of axially moving materials. The Shock and Vibration Digest, 1988, 20: 3–13.

    Article  Google Scholar 

  2. Chen, L.Q., Analysis and control of transverse vibrations of axially moving strings. ASME Applied Mechanics Reviews, 2005, 58: 91–116.

    Article  Google Scholar 

  3. Fung, R.F., Huang, J.S. and Chen, Y.C., The transient amplitude of the viscoelastic travelling string: an integral constitutive law. Journal of Sound and Vibration, 1997, 201: 153–167.

    Article  Google Scholar 

  4. Fung, R.F., Huang, J.S., Chen, Y.C., and Yao, C.M., Nonlinear dynamic analysis of the viscoelastic string with a harmonically varying transport speed. Computers and Structures, 1998, 66: 777–784.

    Article  Google Scholar 

  5. Chen, L.Q., Zhao, W.J. and Zu, J.W., Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law. Journal of Sound and Vibration, 2004, 278: 861–871.

    Article  Google Scholar 

  6. Pellicano, F., Catellani, G. and Fregolent, A., Parametric instability of belt: theory and experiments. Computers and Structures, 2004, 82: 81–91.

    Article  Google Scholar 

  7. Chen, L.Q. and Wu, J., Bifurcation in transverse vibration of axially accelerating viscoelastic strings. Acta Mechanica Solida Sinica, 2005, 26(1): 83–86(in Chinese).

    Google Scholar 

  8. Chen, L.Q. and Zhao, W.J., Iterative algorithm for axially accelerating strings with integral constitutive law. Acta Mechanica Solida Sinica, 2008, 21(5): 449–456.

    Article  Google Scholar 

  9. Chen, L.Q., Zhang, N.H. and Zu, J.W., The regular and chaotic vibrations of an axially moving viscoelastic string based on 4-order Galerkin truncation. Journal of Sound and Vibration, 2003, 261: 764–773.

    Article  Google Scholar 

  10. Zhang, L., Zu, J.W. and Zhong, Z., Transient response for viscoelastic moving belts using block-by-block method. International Journal of Structural Stability and Dynamics, 2002, 2: 265–280.

    Article  Google Scholar 

  11. Zhang, N.H., Dynamic analysis of an axially moving viscoelastic string by the Galerkin method using translating string eigenfunctions. Chaos, Solitons and Fractals, 2008, 35(2): 291–302.

    Article  MathSciNet  Google Scholar 

  12. Chen, L.Q. and Zhao, W.J., A computation method for nonlinear vibration of axially accelerating viscoelastic strings. Applied Mathematics and Computation, 2004, 162: 305–310.

    Article  Google Scholar 

  13. Mote, C.D.Jr., On the nonlinear oscillation of an axially moving string. ASME Journal of Applied Mechanics, 1966, 33: 463–464.

    Article  Google Scholar 

  14. Chen, L.Q. and Zu, J.W., Energetics and conserved functional of moving materials undergoing transverse nonlinear vibration. ASME Journal of Vibration and Acoustics, 2004, 126: 452–455.

    Article  Google Scholar 

  15. Yang, X.D. and Chen, L.Q., Non-linear forced vibration of axially moving viscoelastic beams. Acta Mechanica Solida Sinica, 2006, 19(4): 365–373.

    Article  Google Scholar 

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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, Weijia Zhao & Hu Ding

  3. Department of Mathematics, Qingdao University, Qingdao, 266071, China

    Weijia Zhao

Authors
  1. Liqun Chen
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  2. Weijia Zhao
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  3. Hu Ding
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Correspondence to Liqun Chen.

Additional information

Project supported by the National Outstanding Young Scientists Fund of China (No.10725209), the National Natural Science Foundation of China (No.10672092), Shanghai Subject Chief Scientist Project (No.09XD1401700), Shanghai Municipal Education Commission Scientific Research Project (No.07ZZ07), Shanghai Leading Academic Discipline Project (No.S30106), and Changjiang Scholars and Innovative Research Team in University Program (No.IRT0844).

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Chen, L., Zhao, W. & Ding, H. On Galerkin Discretization of Axially Moving Nonlinear Strings. Acta Mech. Solida Sin. 22, 369–376 (2009). https://doi.org/10.1016/S0894-9166(09)60286-X

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  • DOI: https://doi.org/10.1016/S0894-9166(09)60286-X

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