Skip to main content
SpringerLink
Log in

Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions

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

Abstract

In this paper, parametric and 3:1 internal resonance of axially moving viscoelastic beams on elastic foundation are analytically and numerically investigated. The beam is restricted by viscous damping force. The beam’s material obeys the Kelvin model in which the material time derivative is used. The governing equations of coupled planar vibration and the associated boundary conditions are derived from the generalized Hamilton principle. The effects of the nonhomogeneous boundary conditions due to the viscoelasticity are highlighted, while the boundary conditions are assumed to be homogeneous in previous studies. In small but finite stretching problems, the equation is simplified into a governing equation of transverse nonlinear vibration. It is a nonlinear integro-partial differential equation with time-dependent and space-dependent coefficients. The dependence of the tension on the finite axial support rigidity is also modeled. The method of multiple scales is directly applied to establish the solvability conditions. The nonlinear steady-state oscillating response along with the stability and bifurcation of the beam is investigated. A detailed study is carried out to determine the influence of the viscoelastic coefficient and the viscous damping coefficient on dynamic behavior of the system. The numerical calculations by the differential quadrature scheme confirm the approximate analytical results.

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

Similar content being viewed by others

References

  1. Öz, H.R., Pakdemirli, M., Boyaci, 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 

  2. Chen, L.Q., Ding, H., Lim, C.W.: Principal parametric resonance of axially accelerating viscoelastic beams: multi-scale analysis and differential quadrature verification. Shock Vib. 19, 527–543 (2012)

    Article  Google Scholar 

  3. 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 

  4. Sahoo, B., Panda, L.N., Pohit, G.: Parametric and internal resonances of an axially moving beam with time-dependent velocity. Model. Simul. Eng. 2013, 919517 (2013)

    Google Scholar 

  5. Ghayesh, M.H., Balar, S.: Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams. Int. J. Solids Struct. 45, 6451–6467 (2008)

    Article  MATH  Google Scholar 

  6. Tang, Y.Q., Chen, L.Q., Yang, X.D.: Parametric resonance of axially moving Timoshenko beams with time-dependent speed. Nonlinear Dyn. 58, 715–724 (2009)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Ghayesh, M.H., Balar, S.: Non-linear parametric vibration and stability analysis for two dynamic modes of axially moving Timoshenko beams. Appl. Math. Model. 34, 2850–2859 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

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

    Article  Google Scholar 

  11. Ding, H., Chen, L.Q.: Nonlinear dynamics of axially accelerating viscoelastic beams based on differential quadrature. Acta Mech. Solida Sin. 22, 267–275 (2009)

    Article  Google Scholar 

  12. Chen, L.H., Zhang, W., Yang, F.H.: Nonlinear dynamics of higher dimensional system for an axially accelerating viscoelastic beam. J. Sound Vib. 329, 5321–5345 (2010)

    Article  Google Scholar 

  13. Chen, L.Q., Tang, Y.Q.: Parametric stability of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. ASME J. Vib. Acoust. 134, 011008 (2012)

    Article  Google Scholar 

  14. Chen, L.Q., Tang, Y.Q.: Combination and principal parametric resonances of axially accelerating viscoelastic beams: recognition of longitudinally varying tensions. J. Sound Vib. 330, 5598–5614 (2011)

    Article  Google Scholar 

  15. Tang, Y.Q., Chen, L.Q., Zhang, H.J., Yang, S.P.: Stability of axially accelerating viscoelastic Timoshenko beams: recognition of longitudinally varying tensions. Mech. Mach. Theory 62, 31–50 (2013)

    Article  Google Scholar 

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

    Article  Google Scholar 

  17. 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 

  18. Chen, L.Q., Tang, Y.Q., Zu, J.W.: Nonlinear transverse vibration of axially accelerating strings with exact internal resonances and longitudinally varying tensions. Nonlinear Dyn. 76, 1443–1468 (2014)

    Article  MATH  Google Scholar 

  19. Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances. Nonlinear Dyn. 49, 9–30 (2007)

    Article  MATH  Google Scholar 

  20. Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances. J. Sound Vib. 309, 375–406 (2008)

  21. 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 

  22. Chen, S.H., Huang, J.L., Sze, K.Y.: Multidimensional lindstedt-poincaré method for nonlinear vibration of axially moving beams. J. Sound Vib. 306, 1–11 (2007)

    Article  Google Scholar 

  23. Huang, J.L., Su, R.K.L., Li, W.H., Chen, S.H.: Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. J. Sound Vib. 330, 471–485 (2011)

    Article  Google Scholar 

  24. Ghayesh, M.H.: Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance. Int. J. Mech. Sci. 53, 1022–1037 (2011)

    Article  Google Scholar 

  25. Ghayesh, M.H., Kafiabad, H.A., Reid, T.: Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam. Int. J. Solids Struct. 49, 227–243 (2012)

    Article  Google Scholar 

  26. Ghayesh, M.H., Kazemirad, S., Amabili, M.: Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance. Mech. Mach. Theory 52, 18–34 (2012)

    Article  Google Scholar 

  27. Tang, Y.Q., Chen, L.Q.: Nonlinear free transverse vibrations of in-plane moving plates: without and with internal resonances. J. Sound Vib. 330, 110–126 (2011)

    Article  Google Scholar 

  28. Tang, Y.Q., Chen, L.Q.: Primary resonance in forced vibrations of in-plane translating viscoelastic plates with 3:1 internal resonance. Nonlinear Dyn. 69, 159–172 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tang, Y.Q., Chen, L.Q.: Parametric and internal resonances of in-plane accelerating viscoelastic plates. Acta Mech. 223, 415–431 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, Y.Q., Liang, L., Guo, X.H.: Internal resonance of axially moving laminated circular cylindrical shells. J. Sound Vib. 332, 6434–6450 (2013)

    Article  Google Scholar 

  31. Mockensturm, E.M., Guo, J.: Nonlinear vibration of parametrically excited, viscoelastic, axially moving strings. ASME J. Appl. Mech. 72, 374–380 (2005)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  33. Chen, L.Q., Zu, J.W.: Solvability condition in multi-scale analysis of gyroscopic continua. J. Sound Vib. 309, 338–342 (2008)

    Article  Google Scholar 

  34. Bert, C.W., Malik, M.: The differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49, 1–28 (1996)

    Article  Google Scholar 

  35. Malik, M., Bert, C.W.: Implementing multiple boundary conditions in the DQ solution of higher-order PDE’s: application to free vibration of plates. Int. J. Numer. Meth. Eng. 39, 1237–1258 (1996)

    Article  MATH  Google Scholar 

  36. Bracewell, R.N.: The Fourier Transform and Its Applications. McGraw-Hill, New York (2000)

    MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Project No. 11202135), Training scheme for The Youth Teachers of Higher Education of Shanghai (No. ZZyyy12035), Training Programs of Innovation and Entrepreneurship for Undergraduates of Shanghai Institute of Technology (No. DCX2014114), and Professional and Comprehensive Reform Program of Shanghai (No. 1021NH141142-085). Chen Guang Project was supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (No. 14CG57).

Author information

Authors and Affiliations

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

    You-Qi Tang, Deng-Bo Zhang & Jia-Ming Gao

Authors
  1. You-Qi Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Deng-Bo Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jia-Ming Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to You-Qi Tang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, YQ., Zhang, DB. & Gao, JM. Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. Nonlinear Dyn 83, 401–418 (2016). https://doi.org/10.1007/s11071-015-2336-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2336-2

Keywords

Search

Navigation