Skip to main content
SpringerLink
Log in

Natural frequencies of nonlinear vibration of axially moving beams

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Axially moving beam-typed structures are of technical importance and present in a wide class of engineering problem. In the present paper, natural frequencies of nonlinear planar vibration of axially moving beams are numerically investigated via the fast Fourier transform (FFT). The FFT is a computational tool for efficiently calculating the discrete Fourier transform of a series of data samples by means of digital computers. The governing equations of coupled planar of an axially moving beam are reduced to two nonlinear models of transverse vibration. Numerical schemes are respectively presented for the governing equations via the finite difference method under the simple support boundary condition. In this paper, time series of the discrete Fourier transform is defined as numerically solutions of three nonlinear governing equations, respectively. The standard FFT scheme is used to investigate the natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results are compared with the first two natural frequencies of linear free transverse vibration of an axially moving beam. And results indicate that the effect of the nonlinear coefficient on the first natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results also illustrate the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mote, C.D. Jr.: A study of band saw vibrations. J. Franklin Inst. 276, 430–444 (1965)

    Article  Google Scholar 

  2. Mote, C.D. Jr., Naguleswarn, S.: Theoretical and experimental band saw vibrations. ASME J. Eng. Ind. 88, 151–156 (1966)

    Google Scholar 

  3. Wickert, J.A., Mote, C.D. Jr.: Classical vibration analysis of axially moving continua. J. Appl. Mech. 57, 738–744 (1990)

    Article  MATH  Google Scholar 

  4. Öz, H.R., Pakdemirli, M.: Vibrations of an axially moving beam with time dependent velocity. J. Sound Vib. 227, 239–257 (1999)

    Article  Google Scholar 

  5. Öz, H.R.: On the vibrations of an axially traveling beam on fixed supports with variable velocity. J. Sound Vib. 239, 556–564 (2001)

    Article  Google Scholar 

  6. Özkaya, E., Öz, H.R.: Determination of natural frequencies and stability regions of axially moving beams using artificial neural networks method. J. Sound Vib. 254, 782–789 (2002)

    Article  Google Scholar 

  7. Öz, H.R.: Natural frequencies of axially travelling tensioned beams in contact with astationary mass. J. Sound Vib. 259, 445–456 (2003)

    Article  Google Scholar 

  8. Kong, L., Parker, R.G.: Approximate eigensolutions of axially moving beams with small flexural stiffness. J. Sound Vib. 276, 459–469 (2004)

    Article  Google Scholar 

  9. Chen, L.Q., Yang, X.D.: Vibration and stability of an axially moving viscoelastic beam with hybrid supports. Eur. J. Mech. A-Solids 25, 996–1008 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ghayesh, M.H., Khadem, S.E.: Rotary inertia and temperature effects on non-linear vibration, steady-state response and stability of an axially moving beam with time-dependent velocity. Int. J. Mech. Sci. 50, 389–404 (2008)

    Google Scholar 

  11. Wang, L., Ni, Q.: Vibration and stability of an axially moving beam immersed in fluid. Int. J. Solids Struct. 45, 1445–1457 (2008)

    Article  MATH  Google Scholar 

  12. Tang, Y.Q., Chen, L.Q., Yang, X.D.: Natural frequencies, modes and critical speeds of axially moving Timoshenko beams with different boundary conditions. Int. J. Mech. Sci. 50, 1448–1458 (2008)

    Article  Google Scholar 

  13. Özkaya, E., Sarigul, M., Boyaci, H.: Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass. Acta Mech. Sin. 25(6), 871–882 (2009)

    Article  MathSciNet  Google Scholar 

  14. Wickert, J.A.: Non-linear vibration of a traveling tensioned beam. Int. J. Non-linear Mech. 27, 503–517 (1992)

    Article  MATH  Google Scholar 

  15. Thurman, A.L., Mote, C.D. Jr.: Free, periodic, nonlinear oscillation of an axially moving strip. J. Appl. Mech. 36, 83–91 (1969)

    MATH  Google Scholar 

  16. Tabarrok, B., Leech, C.M., Kim, Y.I.: On the dynamics of an axially moving beam. J. Franklin Inst. 297, 201–220 (1974)

    Article  MATH  Google Scholar 

  17. Wang, K.W., Mote, C.D. Jr.: Vibration coupling analysis of Band/wheel mechanical systems. J. Sound Vib. 109, 237–258 (1986)

    Article  Google Scholar 

  18. Hwang, S.J., Perkins, N.C.: Supercritical stability of an axially moving beam Part I: Model and equilibrium analysis. J. Sound Vib. 154, 381–396 (1992)

    Article  MATH  Google Scholar 

  19. Hwang, S.J., Perkins, N.C.: Supercritical stability of an axially moving beam Part II: Vibration and stability analysis. J. Sound Vib. 154, 397–409 (1992)

    Article  Google Scholar 

  20. Hwang, S.J., Perkins, N.C.: High speed stability of coupled band/wheel systems: Theory and experiment. J. Sound Vib. 169, 459–483 (1994)

    Article  Google Scholar 

  21. Riedel, C.H., Tan, C.A.: Coupled, forced response of an axially moving strip with internal resonance. Int. J. Non-linear Mech. 37, 101–116 (2002)

    Article  MATH  Google Scholar 

  22. Kong, L., Parker, R.G.: Coupled belt-pulley vibration in serpentine drives with belt bending stiffness. J. Appl. Mech. 71, 109–119 (2004)

    Article  MATH  Google Scholar 

  23. Sze, K.Y., Chen, S.H., Huang, J.L.: The incremental harmonic balance method for nonlinear vibration of axially moving beams. J. Sound Vib. 281, 611–626 (2005)

    Article  Google Scholar 

  24. Chen, L.Q., Ding, H.: Steady-state transverse response in coupled planar vibration of axially moving viscoelastic beams. J. Vib. Acoust. 132(1), 011009 (2010)

    Article  Google Scholar 

  25. Ding, H., Chen, L.Q.: On two transverse nonlinear models of axially moving beams. Sci. China Ser. E 52, 743–751 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Ding, H., Chen, L.Q.: Equilibria of axially moving beams in the supercritical regime. Arch. Appl. Mech. DOI:10.1007/s00419-009-0394-y (2009)

  27. Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. Wiley, New York (2004)

    Book  MATH  Google Scholar 

  28. Chen, L.Q., Ding, H.: Steady-state responses of axially accelerating viscoelastic beams: approximate analysis and numerical confirmation. Sci. China Ser. G 51, 1707–1721 (2008)

    Article  Google Scholar 

  29. Chen, L.Q., Yang, X.D.: Steady-state response of axially moving viscoelastic beams with pulsating speed: Comparison of two nonlinear models. Int. J. Solids Struct. 42, 37–50 (2005)

    Article  MATH  Google Scholar 

  30. Chen, L.Q., Yang, X.D.: Nonlinear free vibration of an axially moving beam: Comparison of two models. J. Sound Vib. 299, 348–354 (2007)

    Article  Google Scholar 

  31. Boresi, A.P., Chong, K.P., Saigal, S.: Approximate Solution Methods in Engineering Mechanics, 2nd edn. Wiley, New York (2003)

    Google Scholar 

  32. Duhamel, P., Vetterli, M.: Fast Fourier transforms: A tutorial review and a state of the art. Signal Process. 19, 259–299 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China

    Hu Ding & Li-Qun Chen

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

    Li-Qun Chen

Authors
  1. Hu Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li-Qun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hu Ding.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ding, H., Chen, LQ. Natural frequencies of nonlinear vibration of axially moving beams. Nonlinear Dyn 63, 125–134 (2011). https://doi.org/10.1007/s11071-010-9790-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-010-9790-7

Keywords

Search

Navigation