Skip to main content
SpringerLink
Log in

Nonlinear vibrations of spring-supported axially moving string

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this study, multi-supported axially moving string is discussed. Supports located at the ends of the string are simple supports. A support located in the middle section owns the features of a spring. String speed is assumed to vary harmonically around an average rate. Hamilton’s principle has been used to figure out the nonlinear equations of motion and boundary conditions. These equations and boundary conditions are dimensionless. Considering the nonlinear effects caused by the string extensions, nonlinear equations of motion are obtained. By using multi-timescaled method, which is one of the perturbation methods, approximate solutions have been found. The first term in the perturbation series causes the linear problem. With the solution of the linear problem, exact natural frequencies have been calculated for different locations of the supports on the middle, various spring coefficients and, with the spring support in the middle of the different location, different spring coefficient and axial speed values. Nonlinear terms on second order add correction terms to the linear problem. Effect of nonlinear terms on the natural frequency has been calculated for various parameters. The cases when the changing frequency of speed is equal to zero, close to zero and close to two times of the natural frequency have been analyzed separately. For each case, the stable and unstable areas in the solutions have been identified by stability analysis.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Ulsoy, A.G., Mote Jr, C.D., Syzmani, R.: Principal developments in band saw vibration and stability research. Holz Roh- Werkst 36, 273–280 (1978)

    Article  Google Scholar 

  2. Wickert, J.A., Mote Jr, C.D.: Current research on the vibration and stability of axially moving materials. Shock Vib. Dig. 20(5), 3–13 (1988)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  4. Pakdemirli, M., Ulsoy, A.G., Ceranoglu, A.: Transverse vibration of an axially acceleration string. J. Sound Vib. 169(2), 179–196 (1994)

    Article  MATH  Google Scholar 

  5. Pakdemirli, M., Ulsoy, A.G.: Stability analysis of an axially accelerating string. J. Sound Vib. 203, 815–832 (1997)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Wickert, J.A.: Response solutions for the vibration of a traveling string on an elastic foundation. J. Vib. Acoust. 116, 137–199 (1994)

    Article  Google Scholar 

  8. Nayfeh, A.H., Nayfeh, J.F., Mook, D.T.: On methods for continuous systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 3, 145–162 (1992)

    Article  Google Scholar 

  9. Öz, H.R., Pakdemirli, M., Özkaya, E.: Transition behaviour from string to beam for an axially accelerating material. J. Sound Vib. 215(3), 571–576 (1998)

    Article  Google Scholar 

  10. Özkaya, E., Pakdemirli, M.: Vibrations of an axially accelerating beam with small flexural stiffness. J. Sound Vib. 234(3), 521–535 (2000)

    Article  Google Scholar 

  11. Kural, S., Özkaya, E.: Vibrations of an axially accelerating, multiple supported flexible beam. Struct. Eng. Mech. 44(4), 521–538 (2012)

    Article  Google Scholar 

  12. Pellicano, F., Zirilli, F.: Boundary layers and non-linear vibrations in an axially moving beam. Int. J. Non-Linear Mech. 33, 691–711 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pakdemirli, M., Özkaya, E.: Approximate boundary layer solution of a moving beam problem. Math. Comput. Appl. 2(3), 93–100 (1998)

    Google Scholar 

  14. Öz, H.R., Pakdemirli, M.: Vibrations of an axially moving beam with time-dependent velocity. J. Sound Vib. 227(2), 239–257 (1999)

    Article  Google Scholar 

  15. Yurddas, a, Ozkaya, E., Boyaci, H.: Nonlinear vibrations and stability analysis of axially moving strings having nonideal mid-support conditions. J. Vib. Control 20, 518–534 (2012)

    Article  MathSciNet  Google Scholar 

  16. Yurddaş, a, Özkaya, E., Boyacı, H.: Nonlinear vibrations of axially moving multi-supported strings having non-ideal support conditions. Nonlinear Dyn. 73, 1223–1244 (2012)

    Article  Google Scholar 

  17. Öz, H.R.: On the vibrations of an axially travelling beam on fixed supports with variable velocity. J. Sound Vib. 239(3), 556–564 (2001)

    Article  Google Scholar 

  18. Özkaya, E., Öz, H.R.: Determination of stability regions of axially moving beams using artificial neural networks method. J. Sound Vib. 252(4), 782–789 (2002)

  19. Bağdatli, S.M., Özkaya, E., Ozyigit, H.A., Tekin, A.: Nonlinear vibrations of stepped beam systems using artificial neural networks. Struct. Eng. Mech. 33(1), 15–30 (2009)

    Article  Google Scholar 

  20. Öz, H.R., Pakdemirli, M., Boyacı, H.: Non-linear vibrations and stability of an axially moving beam with time dependent velocity. Int. J. Non-Linear Mech. 36, 107–115 (2001)

    Article  Google Scholar 

  21. Öz, H.R.: Non-linear vibrations and stability analysis of tensioned pipes conveying fluid with variable velocity. Int. J. Non-Linear Mech. 36, 1031–1039 (2001)

    Article  Google Scholar 

  22. Pellicano, F.: On the dynamic properties of axially moving system. J. Sound Vib. 281, 593–609 (2005)

    Article  Google Scholar 

  23. Pakdemirli, M., Öz, H.R.: Infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations. J. Sound Vib. 311(3–5), 1052–1074 (2008)

    Article  Google Scholar 

  24. 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(1), 37–50 (2005)

    Article  Google Scholar 

  25. Chen, L.Q., Yang, X.D.: Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed. J. Sound Vib. 284, 879–891 (2005)

    Article  Google Scholar 

  26. Yang, X.D., Chen, L.Q.: Bifurcation and chaos of an axially accelerating viscoelastic beam. Chaos Solitons Fractals 23(1), 249–258 (2005)

    Article  MATH  Google Scholar 

  27. 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  MathSciNet  MATH  Google Scholar 

  28. Lewandowski, R.: Nonlinear free vibrations of multispan beams on elastic supports. Comput. Struct. 32(2), 305–312 (1989)

    Article  MATH  Google Scholar 

  29. Gürgöze, M., Özgür, K., Erol, H.: On the eigenfrequencies of a cantilevered beam with a tip mass and in-span support. Comput. Struct. 56, 85–92 (1995)

    Article  MATH  Google Scholar 

  30. Gürgöze, M., Erol, H.: Determination of the frequency response function of a cantilevered beam simply supported in-span. J. Sound Vib. 247, 372–378 (2001)

    Article  Google Scholar 

  31. Özkaya, E., Bağdatlı, S.M., Öz, H.R.: Nonlinear transverse vibrations and 3:1 internal resonances of a beam with multiple supports. J. Vib. Acoust. 130, 021013 (2008)

    Article  Google Scholar 

  32. Bağdatli, S.M., Öz, H.R., Özkaya, E.: Non-linear transverse vibrations and 3:1 internal resonances of a tensioned beam on multiple supports. J. Math. Comput. Appl. 16(1), 203–215 (2011)

    Google Scholar 

  33. Tekin, A., Özkaya, E., Bağdatlı, S.M.: Three-to-one internal resonance in multiple stepped beam systems. Appl. Math. Mech. 30, 1131–1142 (2009)

    Article  MATH  Google Scholar 

  34. Özkaya, E., Tekin, A.: Non linear vibrations of stepped beam system under different boundary conditions. Struct. Eng. Mech. 27(3), 333–345 (2007)

    Article  Google Scholar 

  35. Ghayesh, M.H.: Coupled longitudinal-transverse dynamics of an axially accelerating beam. J. Sound Vib. 331(23), 5107–5124 (2012)

    Article  Google Scholar 

  36. Ghayesh, M.H., Amabili, M., Païdoussis, M.P.: Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis. Nonlinear Dyn. 70(1), 335–354 (2012)

    Article  Google Scholar 

  37. Bağdatli, S.M., Özkaya, E., Öz, H.R.: Dynamics of axially accelerating beams with an intermediate support. J. Vib. Acoust. 133, 031013 (2011)

    Article  Google Scholar 

  38. Bağdatli, S.M., Özkaya, E., Öz, H.R.: Dynamics of axially accelerating beams with multiple supports. Nonlinear Dyn. 74, 237–255 (2013)

    Article  MATH  Google Scholar 

  39. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)

    MATH  Google Scholar 

  40. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)

    MATH  Google Scholar 

  41. Chakraborty, G., Mallik, A.K., Hatwal, H.: Non-linear vibration of a travelling beam. Int. J. Non-Linear Mech. 34, 655–670 (1999)

    Article  MATH  Google Scholar 

  42. Pellicano, F., Zirilli, F.: Boundary layers and non-linear vibrations in an axially moving beam. Int. J. Non-Linear Mech. 33, 691–711 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  43. Chen, L.-Q., Zhang, W., Zu, J.W.: Nonlinear dynamics for transverse motion of axially moving strings. Chaos Solitons Fractals 40, 78–90 (2009)

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

    Article  MATH  Google Scholar 

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

Download references

Acknowledgments

This work is supported by TUBITAK (The Scientific and Technological Research Council of Turkey) under Project Number MAG107M302.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Hitit University, 19030, Corum, Turkey

    A. Kesimli

  2. Department of Mechanical Engineering, Faculty of Engineering, Celal Bayar University, 45140, Yunusemre, Manisa, Turkey

    E. Özkaya & S. M. Bağdatli

Authors
  1. A. Kesimli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Özkaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. M. Bağdatli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. M. Bağdatli.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kesimli, A., Özkaya, E. & Bağdatli, S.M. Nonlinear vibrations of spring-supported axially moving string. Nonlinear Dyn 81, 1523–1534 (2015). https://doi.org/10.1007/s11071-015-2086-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2086-1

Keywords

Search

Navigation