Skip to main content
SpringerLink
Log in

Thermo-mechanical dynamics of three-dimensional axially moving beams

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

Abstract

The thermo-mechanical nonlinear dynamics of a three-dimensional axially moving beam is examined numerically in this paper. Hamilton’s principle is employed to derive the nonlinear partial differential equations governing the motion of the system in the longitudinal, transverse, and lateral directions. The discretized equations of motion, which are in the form of second-order nonlinear ordinary differential equations, are obtained by applying the Galerkin technique to the nonlinear partial differential equations of motion. A change of variables is introduced to this set of equations, yielding a set of first-order nonlinear ordinary differential equations. The pseudo-arclength continuation technique is employed to solve this set of equations numerically so as to construct the frequency–response curves; this method allows continuation of both stable and unstable solution branches. The effect of modal interactions on the resonant dynamic behaviour of a fully symmetrical system (about the centreline) is also examined.

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

Similar content being viewed by others

References

  1. Ghayesh, M.H.: Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation. Int. J. Non-linear Mech. 45, 382–394 (2010)

    Article  Google Scholar 

  2. Mote Jr, C.D.: A study of band saw vibrations. J. Frankl. Inst. 279, 430–444 (1965)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  4. Zhang, N.-H., Chen, L.-Q.: Nonlinear dynamical analysis of axially moving viscoelastic strings. Chaos Solitons Fractals 24, 1065–1074 (2005)

    Article  MATH  Google Scholar 

  5. Chen, L.-Q., Yang, X.-D.: Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation. Chaos Solitons Fractals 27, 748–757 (2006)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Tang, Y.-Q., Chen, L.-Q., Yang, X.-D.: Nonlinear vibrations of axially moving Timoshenko beams under weak and strong external excitations. J. Sound Vib. 320, 1078–1099 (2009)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  10. 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  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  Google Scholar 

  14. Marynowski, K.: Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension. Chaos Solitons Fractals 21, 481–490 (2004)

    Article  MATH  Google Scholar 

  15. Pellicano, F., Vestroni, F.: Complex dynamics of high-speed axially moving systems. J. Sound Vib. 258, 31–44 (2002)

    Article  Google Scholar 

  16. Stylianou, M., Tabarrok, B.: Finite element analysis of an axially moving beam, part I: time integration. J. Sound Vib. 178, 433–453 (1994)

    Article  Google Scholar 

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

    Article  Google Scholar 

  18. Chakraborty, G., Mallik, A.K.: Stability of an accelerating beam. J. Sound Vib. 227, 309–320 (1999)

    Article  Google Scholar 

  19. Ravindra, B., Zhu, W.D.: Low-dimensional chaotic response of axially accelerating continuum in the supercritical regime. Arch. Appl. Mech. 68, 195–205 (1998)

    Article  MATH  Google Scholar 

  20. Ding, H., Zhang, G.C., Chen, L.Q.: Supercritical vibration of nonlinear coupled moving beams based on discrete Fourier transform. Int. J. Non-Linear Mech. 47, 1095–1104 (2012)

    Article  Google Scholar 

  21. Huang, J.-S., Fung, R.-F., Lin, C.-H.: Dynamic stability of a moving string undergoing three-dimensional vibration. Int. J. Mech. Sci. 37, 145–160 (1995)

    Article  MATH  Google Scholar 

  22. Chen, L.H., Zhang, W., Yang, F.H.: Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations. J. Sound Vib. 329, 5321–5345 (2010)

    Article  Google Scholar 

  23. Treyssède, F.: Prebending effects upon the vibrational modes of thermally prestressed planar beams. J. Sound Vib. 307, 295–311 (2007)

    Article  Google Scholar 

  24. Pradeep, V., Ganesan, N., Bhaskar, K.: Vibration and thermal buckling of composite sandwich beams with viscoelastic core. Compos. Struct. 81, 60–69 (2007)

    Article  Google Scholar 

  25. Sharnappa, Ganesan, N., Sethuraman, R.: Dynamic modelling of active constrained layer damping of composite beam under thermal environment. J. Sound Vib. 305, 728–749 (2007)

  26. Xiang, H.J., Yang, J.: Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction. Compos. Part B Eng. 39, 292–303 (2008)

    Article  Google Scholar 

  27. Wu, G.Y.: The analysis of dynamic instability and vibration motions of a pinned beam with transverse magnetic fields and thermal loads. J. Sound Vib. 284, 343–360 (2005)

    Article  Google Scholar 

  28. Ghayesh, M.H., Amabili, M., Farokhi, H.: Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed. Chaos Solitons Fractals 52, 8–29 (2013)

    Article  MathSciNet  Google Scholar 

  29. Farokhi, H., Ghayesh, M.H., Amabili, M.: In-plane and out-of-plane nonlinear dynamics of an axially moving beam. Chaos Solitons Fractals 54, 101–121 (2013)

    Article  MathSciNet  Google Scholar 

  30. Ghayesh, M.H.: On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: parametric study. Acta Mech. Solida Sin. 24, 373–382 (2011)

    Article  Google Scholar 

  31. Ghayesh, M.H., Kazemirad, S., Reid, T.: Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: A general solution procedure. Appl. Math. Model. 36, 3299–3311 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  32. Farokhi, H., Ghayesh, M.H., Amabili, M.: Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory. Int. J. Eng. Sci. 68, 11–23 (2013)

    Article  MathSciNet  Google Scholar 

  33. Ghayesh, M.H., Kazemirad, S., Darabi, M.A.: A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions. J. Sound Vib. 330, 5382–5400 (2011)

    Article  Google Scholar 

  34. Ghayesh, M.H., Amabili, M.: Steady-state transverse response of an axially moving beam with time-dependent axial speed. Int. J. Non-Linear Mech. 49, 40–49 (2013)

  35. Ghayesh, M.: Subharmonic dynamics of an axially accelerating beam. Arch. Appl. Mech. 82, 1169–1181 (2012)

    Article  MATH  Google Scholar 

  36. Ghayesh, M.H., Farokhi, H., Amabili, M.: Nonlinear behaviour of electrically actuated MEMS resonators. Int. J. Eng. Sci. 71, 137–155 (2013)

    Article  MathSciNet  Google Scholar 

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

  38. Ghayesh, M.H., Farokhi, H.: Nonlinear dynamics of a microscale beam based on the modified couple stress theory. Compos. Part B Eng. 50, 318–324 (2013)

    Article  Google Scholar 

  39. Ghayesh, M.H., Amabili, M., Farokhi, H.: Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams. Int. J. Eng. Sci. 71, 1–14 (2013)

    Article  MathSciNet  Google Scholar 

  40. Ghayesh, M.H., Amabili, M., Farokhi, H.: Coupled global dynamics of an axially moving viscoelastic beam. Int. J. Non-Linear Mech. 51, 54–74 (2013)

    Article  Google Scholar 

  41. Kazemirad, S., Ghayesh, M.H., Amabili, M.: Thermo-mechanical nonlinear dynamics of a buckled axially moving beam. Arch. Appl. Mech. 83, 25–42 (2013)

    Article  MATH  Google Scholar 

  42. Ghayesh, M.H., Kazemirad, S., Darabi, M.A., Woo, P.: Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system. Arch. Appl. Mech. 82, 317–331 (2012)

    Article  MATH  Google Scholar 

  43. Doedel, E., Paffenroth, R., Champneys, A., Fairgrieve, T., Kuznetsov, Y.A., Oldeman, B.: AUTO-07P: continuation and bifurcation software for ordinary differential equations. Concordia University, Montreal (2007)

    Google Scholar 

Download references

Acknowledgments

The financial support to this research by the start-up grant of the University of Wollongong is gratefully acknowledged.

Author information

Authors and Affiliations

  1. School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, Wollongong, NSW, 2522, Australia

    Mergen H. Ghayesh

  2. Department of Mechanical Engineering, McGill University, Montreal, QC, H3A 0C3, Canada

    Hamed Farokhi

Authors
  1. Mergen H. Ghayesh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hamed Farokhi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mergen H. Ghayesh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghayesh, M.H., Farokhi, H. Thermo-mechanical dynamics of three-dimensional axially moving beams. Nonlinear Dyn 80, 1643–1660 (2015). https://doi.org/10.1007/s11071-015-1968-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-1968-6

Keywords

Search

Navigation