Skip to main content
SpringerLink
Log in

Nonlinear dynamic response of an Euler–Bernoulli beam under a moving mass–spring with large oscillations

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

This article presents a new approach for the nonlinear dynamic behavior of an Euler–Bernoulli beam under a moving mass. The governing equations for the behavior of the beam under a moving mass in large oscillations including the effect of horizontal and vertical beam displacements are considered via energy method. The systems of governing equations are solved in the condition of external and autoparametric resonance using Galerkin and perturbation methods. In order to validate the solution, the results are compared with a numerical solution and those available in the literature. The governing equations are used for the stability analysis of the beam in different points that will result in spectral responses in stable circumstances.

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

Similar content being viewed by others

References

  1. Frýba, L.: Vibration of Solids and Structures Under Moving Loads. Thomas Telford, Hardcover (1999)

    Book  Google Scholar 

  2. Wu, J.-S., Shih, P.-Y.: Dynamic responses of railway and carriage under the high-speed moving loads. J. Sound Vib. 236(1), 61–87 (2000). https://doi.org/10.1006/jsvi.2000.2959

    Article  Google Scholar 

  3. Humar, J.L., Kashif, A.H.: Dynamic response analysis of slab-type bridges. J. Struct. Eng. 121(1), 48–62 (1995). https://doi.org/10.1061/(ASCE)0733-9445(1995)121:1(48)

    Article  Google Scholar 

  4. Cifuentes, A., Lalapet, S.: A general method to determine the dynamic response of a plate to a moving mass. Comput. Struct. 42(1), 31–36 (1992). https://doi.org/10.1016/0045-7949(92)90533-6

    Article  MATH  Google Scholar 

  5. Uzal, E., Sakman, L.E.: Dynamic response of a circular plate to a moving load. AcMec 210(3), 351–359 (2010). https://doi.org/10.1007/s00707-009-0207-y

    Article  MATH  Google Scholar 

  6. Goel, R.P.: Vibrations of a beam carrying a concentrated mass. J. Appl. Mech. 40(3), 821–822 (1973). https://doi.org/10.1115/1.3423102

    Article  Google Scholar 

  7. Pipes LRH, L.A.: Applied Mathematics for Engineers and Physicists. McGraw-Hill Book Company Inc., New York (1970)

    Google Scholar 

  8. Prescott, J.: Applied Elasticity. Dover Publications, New York (1946)

    MATH  Google Scholar 

  9. Jeffcott, H.H.: Vi. On the vibration of beams under the action of moving loads. Lond. Edinb. Dublin Philos. Mag. J. Sci. 8(48), 66–97 (1929). https://doi.org/10.1080/14786440708564857

    Article  MATH  Google Scholar 

  10. Stanis̆ić, M.M., Hardin, J.C.: On the response of beams to an arbitrary number of concentrated moving masses. J. Frankl. Inst. 287(2), 115–123 (1969). https://doi.org/10.1016/0016-0032(69)90120-3

    Article  Google Scholar 

  11. Duffy, D.G.: The response of an infinite railroad track to a moving, vibrating mass. J. Appl. Mech. 57(1), 66–73 (1990). https://doi.org/10.1115/1.2888325

    Article  Google Scholar 

  12. Cheng, Y.S., Au, F.T.K., Cheung, Y.K., Zheng, D.Y.: On the separation between moving vehicles and bridge. J. Sound Vib. 222(5), 781–801 (1999). https://doi.org/10.1006/jsvi.1998.2134

    Article  Google Scholar 

  13. Akin, J.E., Mofid, M.: Numerical solution for response of beams with moving mass. J. Struct. Eng. 115(1), 120–131 (1989). https://doi.org/10.1061/(ASCE)0733-9445(1989)115:1(120)

    Article  Google Scholar 

  14. Nikkhoo, A., Rofooei, F.R., Shadnam, M.R.: Dynamic behavior and modal control of beams under moving mass. J. Sound Vib. 306(3–5), 712–724 (2007). https://doi.org/10.1016/j.jsv.2007.06.008

    Article  MathSciNet  Google Scholar 

  15. Kiani, K., Nikkhoo, A., Mehri, B.: Prediction capabilities of classical and shear deformable beam models excited by a moving mass. J. Sound Vib. 320(3), 632–648 (2009). https://doi.org/10.1016/j.jsv.2008.08.010

    Article  Google Scholar 

  16. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)

    MATH  Google Scholar 

  17. Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. Part I. CMAME 49(1), 55–70 (1985). https://doi.org/10.1016/0045-7825(85)90050-7

    Article  MATH  Google Scholar 

  18. Reissner, E.: On one-dimensional finite-strain beam theory: the plane problem. Zeitschrift für angewandte Mathematik und Physik ZAMP 23(5), 795–804 (1972)

    Article  Google Scholar 

  19. Meier, C., Popp, A., Wall, W.A.: Geometrically exact finite element formulations for slender beams: Kirchhoff–Love theory versus Simo–Reissner theory. Arch. Comput. Methods Eng. 26(1), 163–243 (2019)

    Article  MathSciNet  Google Scholar 

  20. Crisfield, M.A., Jelenić, G.: Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 455(1983), 1125–1147 (1999)

    Article  MathSciNet  Google Scholar 

  21. Romero, I.: The interpolation of rotations and its application to finite element models of geometrically exact rods. CompM 34(2), 121–133 (2004)

    MathSciNet  MATH  Google Scholar 

  22. Zavodney, L.D., Nayfeh, A.H.: The non-linear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment. Int. J. Non-Linear Mech. 24(2), 105–125 (1989). https://doi.org/10.1016/0020-7462(89)90003-6

    Article  MATH  Google Scholar 

  23. Dwivedy, S.K., Kar, R.C.: Nonlinear dynamics of a cantilever beam carrying an attached mass with 1:3:9 internal resonances. Nonlinear Dyn. 31(1), 49–72 (2003). https://doi.org/10.1023/A:1022128029369

    Article  MATH  Google Scholar 

  24. Dias, C.A.N.: General exact harmonic analysis of in-plane Timoshenko beam structures. Lat. Am. J. Solids Struct. 11(12), 2171–2202 (2014)

    Article  Google Scholar 

  25. Dias, C.A.N., Alves, M.: A method to solve the nonlinear eigenvalue problem of Timoshenko plane frames with rigid offsets and end releases. J. Sound Vib. 332(5), 1372–1387 (2013). https://doi.org/10.1016/j.jsv.2012.10.029

    Article  Google Scholar 

  26. Venkateswara Rao, G., Meera Saheb, K., Ranga Janardhan, G.: Fundamental Frequency for large amplitude vibrations of uniform Timoshenko beams with central point concentrated mass using coupled displacement field method. J. Sound Vib. 298(1–2), 221–232 (2006). https://doi.org/10.1016/j.jsv.2006.05.014

    Article  Google Scholar 

  27. Esmailzadeh, E., Jalili, N.: Parametric response of cantilever Timoshenko beams with tip mass under harmonic support motion. Int. J. Non-Linear Mech. 33(5), 765–781 (1998). https://doi.org/10.1016/S0020-7462(97)00049-8

    Article  MathSciNet  MATH  Google Scholar 

  28. Malaeke, H., Moeenfard, H.: Analytical modeling of large amplitude free vibration of non-uniform beams carrying a both transversely and axially eccentric tip mass. J. Sound Vib. 366, 211–229 (2016). https://doi.org/10.1016/j.jsv.2015.12.003

    Article  Google Scholar 

  29. Auclair, S.C., Sorelli, L., Salenikovich, A., Fafard, M.: The effect of rotatory inertia on the natural frequencies of composite beams. J. Sound Vib. 366, 230–247 (2016). https://doi.org/10.1016/j.jsv.2015.12.004

    Article  Google Scholar 

  30. Wu, Y.-F., Xu, R., Chen, W.: Free vibrations of the partial-interaction composite members with axial force. J. Sound Vib. 299(4–5), 1074–1093 (2007). https://doi.org/10.1016/j.jsv.2006.08.008

    Article  Google Scholar 

  31. Nguyen, Q.-H., Hjiaj, M., Le Grognec, P.: Analytical approach for free vibration analysis of two-layer Timoshenko beams with interlayer slip. J. Sound Vib. 331(12), 2949–2961 (2012). https://doi.org/10.1016/j.jsv.2012.01.034

    Article  Google Scholar 

  32. Jiang, J.-Q.: Transient responses of Timoshenko beams subject to a moving mass. J. Vib. Control 17(13), 1975–1982 (2011). https://doi.org/10.1177/1077546310382808

    Article  MathSciNet  MATH  Google Scholar 

  33. Mackertich, S.: Dynamic response of a supported beam to oscillatory moving masses. Modal Anal. 9(9), 1083–1091 (2003). https://doi.org/10.1177/107754603030681

    Article  MATH  Google Scholar 

  34. Karimi, A., Ziaei-Rad, S.: Nonlinear coupled longitudinal–transverse vibration analysis of a beam subjected to a moving mass traveling with variable speed. Arch. Appl. Mech. 85(12), 1941–1960 (2015)

    Article  Google Scholar 

  35. Siddiqui, S., Golnaraghi, M., Heppler, G.: Dynamics of a flexible beam carrying a moving mass using perturbation, numerical and time-frequency analysis techniques. J. Sound Vib. 229(5), 1023–1055 (2000)

    Article  Google Scholar 

  36. Siddiqui, S., Golnaraghi, M., Heppler, G.: Large free vibrations of a beam carrying a moving mass. Int. J. Non-Linear Mech. 38(10), 1481–1493 (2003)

    Article  Google Scholar 

  37. Khalily, F., Golnaraghi, M.F., Heppler, G.R.: On the dynamic behaviour of a flexible beam carrying a moving mass. Nonlinear Dyn. 5(4), 493–513 (1994). https://doi.org/10.1007/BF00052456

    Article  Google Scholar 

  38. Siddiqui, S.A.Q., Golnaraghi, M.F., Heppler, G.R.: Dynamics of a flexible beam carrying a moving mass using perturbation, numerical and time-frequency analysis techniques. J. Sound Vib. 229(5), 1023–1055 (2000). https://doi.org/10.1006/jsvi.1999.2449

    Article  Google Scholar 

  39. Siddiqui, S.A.Q., Golnaraghi, M.F., Heppler, G.R.: Large free vibrations of a beam carrying a moving mass. Int. J. Non-Linear Mech. 38(10), 1481–1493 (2003). https://doi.org/10.1016/S0020-7462(02)00084-7

    Article  MATH  Google Scholar 

  40. Siddiqui, S.A.Q., Golnaraghi, M.F., Heppler, G.R.: Dynamics of a flexible cantilever beam carrying a moving mass. Nonlinear Dyn. 15(2), 137–154 (1998). https://doi.org/10.1023/A:1008205904691

    Article  MATH  Google Scholar 

  41. Becker, E.B., Carey, G.F., Oden, T.J.: Finite Elements on Introduction (1981)

  42. Bajer, C.I., Dyniewicz, B.: Numerical modelling of structure vibrations under inertial moving load. Arch. Appl. Mech. 79(6–7), 499–508 (2009)

    Article  Google Scholar 

  43. Shampine, L.F., Reichelt, M.W.: The matlab ode suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Structural Engineering Department, Road, Housing and Urban Development Research Center (BHRC), Hekmat Ave, Noori Highway, Tehran, Iran

    Amir Jahangiri & Nader K. A. Attari

  2. Department of Civil Engineering, University of Science and Culture, Tehran, Iran

    Ali Nikkhoo

  3. Department of Civil Engineering, Shahed University, Tehran, Iran

    Zakariya Waezi

Authors
  1. Amir Jahangiri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nader K. A. Attari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ali Nikkhoo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zakariya Waezi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nader K. A. Attari.

Ethics declarations

Conflict of interest

All authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jahangiri, A., Attari, N.K.A., Nikkhoo, A. et al. Nonlinear dynamic response of an Euler–Bernoulli beam under a moving mass–spring with large oscillations. Arch Appl Mech 90, 1135–1156 (2020). https://doi.org/10.1007/s00419-020-01656-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-020-01656-9

Keywords

Search

Navigation