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Internal-external resonance of beams on non-linear viscoelastic foundation traversed by moving load

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Abstract

Vibration of a finite Euler–Bernoulli beam, supported by non-linear viscoelastic foundation traversed by a moving load, is studied and the Galerkin method is used to discretize the non-linear partial differential equation of motion. Subsequently, the solution is obtained for different harmonics using the Multiple Scales Method (MSM) as one of the perturbation techniques. Free vibration of a beam on non-linear foundation is investigated and the effects of damping and non-linear stiffness of the foundation on the responses are examined. Internal-external resonance condition is then stated and the frequency responses of different harmonics are obtained by MSM. Different conditions of the external resonance are studied and a parametric study is carried out for each case. The effects of damping and non-linear stiffness of the foundation as well as the magnitude of the moving load on the frequency responses are investigated. Finally, a thorough local stability analysis is performed on the system.

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References

  1. Fryba, L.: Vibration of Solids and Structures under Moving Loads. Thomas Telford, London (1999)

    Book  Google Scholar 

  2. Esmailzadeh, E., Ghorashi, M.: Vibration analysis of Timoshenko beams subjected to a traveling mass. J. Sound Vib. 199(4), 615–625 (1997)

    Article  Google Scholar 

  3. Andersen, L., Nielsen, S.R.K., Kirkegaard, P.H.: Finite element modeling of infinite Euler beams on Kelvin foundations exposed to moving loads in convected co-ordinates. J. Sound Vib. 241(4), 587–604 (2001)

    Article  Google Scholar 

  4. Yu, Y., Nian-Guan, T., Yian-Feng, T.: Vibration analysis of a simply-supported beam traversed by uniform distributed moving loads. J. Shanghai Jiaotong Univ. 38, 155–160 (2004)

    Google Scholar 

  5. Gurav, S.: Dynamic analysis of bridges under moving loads. In: Proceedings, Responding to Tomorrow’s Challenges in Structural Engineering, Budapest, Hungary, 13–15 September (2006)

  6. Ouyang, H., Mottershead, J.E.: Vibration of a beam excited by a moving flexible body. In: Proceedings, Sixth International Conference on Modern Practice in Stress and Vibration Analysis, Bath, UK, pp. 457–464, 5–7 September (2006)

  7. Kim, S.M., Cho, Y.H.: Vibration and dynamic buckling of shear beam-columns on elastic foundation under moving harmonic loads. Int. J. Solids Struct. 43, 393–412 (2006)

    Article  MATH  Google Scholar 

  8. Martinez-Castro, A.E., Museros, P., Castillo-Linares, A.: Semi-analytic solution in the time domain for non-uniform multi-span Bernoulli–Euler beams traversed by moving loads. J. Sound Vib. 294, 278–297 (2006)

    Article  Google Scholar 

  9. Garinei, A.: Vibrations of simple beam-like modeled bridge under harmonic moving loads. Int. J. Eng. Sci. 44, 778–787 (2006)

    Article  Google Scholar 

  10. Ouyang, H., Mottershead, J.E.: A numerical-analytical combined method for vibration of a beam excited by a moving flexible body. Int. J. Numer. Methods Eng. 72, 1181–1191 (2007)

    Article  Google Scholar 

  11. Stancioiu, D., Ouyang, H., Mottershead, J.E.: Vibration of a beam excited by a moving oscillator considering separation and reattachment. J. Sound Vib. 310(4–5), 1128–1140 (2008)

    Article  Google Scholar 

  12. Kiral, Z., Goren Kiral, B.: Dynamic analysis of a symmetric laminated composite beam subjected to a moving load with constant velocity. J. Reinf. Plast. Compos. 27(1), 19–32 (2008)

    Article  Google Scholar 

  13. Coskun, I., Engin, H.: Non-linear vibrations of a beam on an elastic foundation. J. Sound Vib. 223(3), 335–354 (1999)

    Article  Google Scholar 

  14. Yanmeni Wayou, A.N., Tchoukuegno, R., Woafo, P.: Non-linear dynamics of an elastic beam under moving loads. J. Sound Vib. 273(4–5), 1101–1108 (2004)

    Article  Google Scholar 

  15. Kang, B., Tan, C.A.: Non-linear response of a beam under distributed moving contact load. Commun. Non-linear Sci. Numer. Simul. 11, 203–232 (2006)

    Article  MATH  Google Scholar 

  16. Huang, W., Zou, Y.: The dynamic response of a viscoelastic Winkler foundation-supported elastic beam impacted by a low velocity projectile. Comput. Struct. 52(3), 431–436 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ayoub, A.: Mixed formulation of non-linear beam on foundation elements. Comput. Struct. 81, 411–421 (2003)

    Article  Google Scholar 

  18. Patel, B.P., Ganapathi, M., Touratier, M.: Non-linear free flexural vibrations/post-buckling analysis of laminated orthotropic beams/columns on a two parameter elastic foundation. Compos. Struct. 46, 189–196 (1999)

    Article  Google Scholar 

  19. Naidu, N.R., Rao, G.V.: Free vibration and stability behavior of uniform beams and columns on non-linear elastic foundation. Comput. Struct. 58(6), 1213–1215 (1996)

    Article  MATH  Google Scholar 

  20. Lenci, S., Tarantino, A.M.: Chaotic dynamics of an elastic beam resting on a Winkler-type soil. Chaos Solitons Fractals 7(10), 1601–1614 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Coskun, I.: The response of a finite beam on a tensionless Pasternak foundation subject to a harmonic load. Eur. J. Mech. A/Solids 22, 151–161 (2003)

    Article  MATH  Google Scholar 

  22. Sheinman, I., Adan, M., Altus, E.: On the role of the displacement function in non-linear analysis of beams on an elastic foundation. Thin-Walled Struct. 15(2), 109–125 (1993)

    Article  Google Scholar 

  23. Kuo, Y.H., Lee, S.Y.: Deflection of non-uniform beams resting on a non-linear elastic foundation. Comput. Struct. 51(5), 513–519 (1994)

    Article  MATH  Google Scholar 

  24. Santee, D.M., Goncalves, P.B.: Oscillations of a beam on a non-linear elastic foundation under periodic loads. Shock Vib. 13, 273–284 (2006)

    Google Scholar 

  25. Dahlberg, T.: Dynamic interaction between train and non-linear railway track model. In: Proceedings, Fifth European Conference on Structural Dynamics (EURODYN 2002), Munich, Germany, vol. 2, pp. 1155–1160, 2–5 September (2002)

  26. Kargarnovin, M.H., Younesian, D., Thompson, D.J., Jones, C.J.C.: Response of beams on non-linear viscoelastic foundations to harmonic moving loads. Comput. Struct. 83, 1865–1877 (2005)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  28. Abe, A.: On non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities. Int. J. Non-Linear Mech. 41, 873–879 (2006)

    Article  MATH  Google Scholar 

  29. Iwnicky, S.: Handbook of Railway Vehicle Dynamics. Taylor and Francis, New York (2007)

    Google Scholar 

  30. Wu, T.X., Thompson, D.J.: The effects of track non-linearity on wheel/rail impact. Proc. Inst. Mech. Eng., F., J. Rail Rapid Transit 218(1), 1–15 (2004)

    Article  MathSciNet  Google Scholar 

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Author notes

  1. M. Ansari

    Present address: Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada

  2. D. Younesian

    Present address: School of Railway Engineering, Iran University of Science & Technology, Tehran, Iran

Authors and Affiliations

  1. Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, 2000 Simcoe St. North, Oshawa, Ontario, L1H 7K4, Canada

    M. Ansari, E. Esmailzadeh & D. Younesian

Authors
  1. M. Ansari
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  2. E. Esmailzadeh
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  3. D. Younesian
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Correspondence to E. Esmailzadeh.

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Ansari, M., Esmailzadeh, E. & Younesian, D. Internal-external resonance of beams on non-linear viscoelastic foundation traversed by moving load. Nonlinear Dyn 61, 163–182 (2010). https://doi.org/10.1007/s11071-009-9639-0

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