Skip to main content
SpringerLink
Log in

Complex modes and traveling waves in axially moving Timoshenko beams

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

Complex modes and traveling waves in axially moving Timoshenko beams are studied. Due to the axially moving velocity, complex modes emerge instead of real value modes. Correspondingly, traveling waves are present for the axially moving material while standing waves dominate in the traditional static structures. The analytical results obtained in this study are verified with a numerical differential quadrature method.

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. Wickert, J. A. and Mote, C. D. Classical vibration analysis of axially moving continua. ASME Journal of Applied Mechanics, 57, 738–744 (1990)

    Article  MATH  Google Scholar 

  2. Lin, C. C. Stability and vibration characteristics of axially moving plates. International Journal of Solids and Structures, 34, 3179–3190 (1997)

    Article  MATH  Google Scholar 

  3. Lee, U., Kim, J., and Oh, H. Spectral analysis for the transverse vibration of an axially moving Timoshenko beam. Journal of Sound and Vibration, 271, 685–703 (2004)

    Article  MATH  Google Scholar 

  4. Wickert, J. A. Non-linear vibration of a traveling tensioned beam. International Journal of Non-Linear Mechanics, 27, 503–517 (1992)

    Article  MATH  Google Scholar 

  5. Pellicano, F. and Vestroni, F. Nonlinear dynamics and bifurcations of an axially moving beam. ASME Journal of Vibration and Acoustics, 122, 21–30 (2000)

    Article  Google Scholar 

  6. Chen, L. Q. and Zhao, W. J. A conserved quantity and the stability of axially moving nonlinear beams. Journal of Sound and Vibration, 286, 663–668 (2005)

    Article  Google Scholar 

  7. Ding, H., Chen, L. Q., and Yang, S. P. Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load. Journal of Sound and Vibration, 331, 2426–2442 (2012)

    Article  Google Scholar 

  8. Ding, H. and Zu, J. W. Steady-state responses of pulley-belt systems with a one-way clutch and belt bending stiffness. ASME Journal of Vibration and Acoustics, 136, 041006 (2014)

    Article  Google Scholar 

  9. Yan, Q. Y., Ding, H., and Chen, L. Q. Nonlinear dynamics of axially moving viscoelastic Timoshenko beam under parametric and external excitations. Applied Mathematics and Mechanics (English Edition), 36, 971–984 (2015) https://doi.org/10.1007/s10483-015-1966-7

    Article  MathSciNet  MATH  Google Scholar 

  10. Sahoo, B., Panda, L. N., and Pohit, G. Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam. Nonlinear Dynamics, 82, 1721–1742 (2015)

    Article  MathSciNet  Google Scholar 

  11. Michon, G., Manin, L., Remond, D., Dufour, R., and Parker, R. G. Parametric instability of an axially moving belt subjected to multifrequency excitations: experiments and analytical validation. ASEM Journal of Applied Mechanics, 75, 041004 (2008)

    Article  Google Scholar 

  12. Öz, H. R., Pakdemirli, M., and Boyaci, H. Non-linear vibrations and stability of an axially moving beam with time-dependent velocity. International Journal of Non-Linear Mechanics, 36, 107–115 (2001)

    Article  MATH  Google Scholar 

  13. Öz, H. R. and Pakdemirli, M. Vibrations of an axially moving beam with time-dependent velocity. Journal of Sound and Vibration, 227, 239–257 (1999)

    Article  Google Scholar 

  14. Ding, H., Zhang, G. C., Chen, L. Q., and Yang, S. P. Forced vibrations of supercritically transporting viscoelastic beams. ASME Journal of Vibration and Acoustics, 134, 051007 (2012)

    Article  Google Scholar 

  15. Li, H. Y., Li, J., and Liu, Y. J. Internal resonance of an axially moving unidirectional plate partially immersed in fluid under foundation displacement excitation. Journal of Sound and Vibration, 358, 124–141 (2015)

    Article  Google Scholar 

  16. Tang, Y. Q., Zhang, D. B., and Gao, J. M. Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. Nonlinear Dynamics, 83, 401–418 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ghayesh, M. H. and Amabili, M. Nonlinear dynamics of an axially moving Timoshenko beam with an internal resonance. Nonlinear Dynamics, 73, 39–52 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ding, H., Huang, L. L., Mao, X. Y., and Chen, L. Q. Primary resonance of traveling viscoelastic beam under internal resonance. Applied Mathematics and Mechanics (English Edition), 38, 1–14 (2017) https://doi.org/10.1007/s10483-016-2152-6

    Article  MathSciNet  MATH  Google Scholar 

  19. Mao, X. Y., Ding, H., Lim, C.W., and Chen, L. Q. Super-harmonic resonance and multi-frequency responses of a super-critical translating beam. Journal of Sound and Vibration, 385, 267–283 (2016)

    Article  Google Scholar 

  20. Mao, X. Y., Ding, H., and Chen, L. Q. Parametric resonance of a translating beam with pulsating axial speed in the super-critical regime. Mechanics Research Communications, 76, 72–77 (2016)

    Article  Google Scholar 

  21. Zhang, G. C., Ding, H., Chen, L. Q., and Yang, S. P. Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds. Journal of Sound and Vibration, 331, 1612–1623 (2012)

    Article  Google Scholar 

  22. Liu, Y., Zhao, Z., and He, W. Boundary control of an axially moving system with high acceleration/deceleration and disturbance observer. Journal of the Franklin Institute, 354, 2905–2923 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. He, W., He, X. Y., and Sun, C. Y. Vibration control of an industrial moving strip in the presence of input deadzone. IEEE Transactions on Industrial Electronics, 64, 4680–4689 (2017)

    Article  Google Scholar 

  24. Zhao, H. and Rahn, C. D. On the control of axially moving material systems. ASME Journal of Vibration and Acoustics, 128, 527–531 (2006)

    Article  Google Scholar 

  25. Ghayesh, M. H. and Balar, S. Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams. Applied Mathematical Modelling, 34, 2850–2859 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tang, Y. Q., Chen, L. Q., and Yang, X. D. Natural frequencies, modes and critical speeds of axially moving Timoshenko beams with different boundary conditions. International Journal of Mechanical Sciences, 50, 1448–1458 (2008)

    Article  MATH  Google Scholar 

  27. Tang, Y. Q., Chen, L. Q., and Yang, X. D. Nonlinear vibrations of axially moving Timoshenko beams under weak and strong external excitations. Journal of Sound and Vibration, 320, 1078–1099 (2009)

    Article  Google Scholar 

  28. Chen, L. Q., Tang, Y. Q., and Lim, C. W. Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams. Journal of Sound and Vibration, 329, 547–565 (2010)

    Article  Google Scholar 

  29. Yan, Q. Y., Ding, H., and Chen, L. Q. Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam. Nonlinear Dynamics, 78, 1577–1591 (2014)

    Article  MathSciNet  Google Scholar 

  30. Ding, H., Tan, X., Zhang, G. C., and Chen, L. Q. Equilibrium bifurcation of high-speed axially moving Timoshenko beams. Acta Mechanica, 227, 3001–3014 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wickert, J. A. and Mote, C. D. Classical vibration analysis of axially moving continua. ASME Journal of Applied Mechanics, 57, 738–744 (1990)

    Article  MATH  Google Scholar 

  32. Yang, X. D., Yang, S., Qian, Y. J., Zhang, W., and Melnik, R. V. N. Modal analysis of the gyroscopic continua: comparison of continuous and discretized models. ASME Journal of Applied Mechanics, 83, 084502 (2016)

    Article  Google Scholar 

  33. Yang, X. D., Chen, L. Q., and Zu, J. W. Vibrations and stability of an axially moving rectangular composite plate. ASME Journal of Applied Mechanics, 78, 011018 (2011)

    Article  Google Scholar 

  34. Ghayesh, M. H. and Amabili, M. Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support. Mechanism and Machine Theory, 67, 1–16 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai, 201418, China

    Youqi Tang & Erbao Luo

  2. Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Engineering, College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, China

    Xiaodong Yang

Authors
  1. Youqi Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erbao Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaodong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaodong Yang.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11672007 and 11672186), the Training Scheme for the Youth Teachers of Higher Education of Shanghai (No.ZZyyy12035), and the “Chen Guang” Project (No. 14CG57)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, Y., Luo, E. & Yang, X. Complex modes and traveling waves in axially moving Timoshenko beams. Appl. Math. Mech.-Engl. Ed. 39, 597–608 (2018). https://doi.org/10.1007/s10483-018-2312-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-018-2312-8

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation