• Received : 2015.01.21
  • Accepted : 2015.09.03
  • Published : 2016.03.25


In this study, the dynamic behaviour of a rotating Timoshenko beam when under the actions of a variable magnitude load moving at non-uniform speed is carried out. The effect of cross-sectional dimension and damping on the flexural motions of the elastic beam was neglected. The coupled second order partial differential equations incorporating the effects of rotary and gyroscopic moment describing the motions of the beam was scrutinized in order to obtain the expression for the dynamic deflection and rotation of the vibrating system using an elegant technique called Galerkin's Method. Analyses of the solutions obtained were carried out and various results were displayed in plotted curve. It was found that the response amplitude of the simply supported beam increases with an increase in the value of the foundation reaction modulus. Effects of other vital structural parameters were also established.


  1. L. Fryba, Vibrations of solids and structures under moving loads, Groningen: Noordhufff, (1972) 196.
  2. Y.H. Chang and Y. Li, Dynamic response of elevated high speed railway, Journal of Bridge Engineering, 5 (2000), 124-130.
  3. G.V. Rao, Linear Dynamics of an Elastic Beam Under Moving Loads, Journal of sound and vibration, 248(2) (2000), 267-288.
  4. T.O. Awodola, Flexural motions under moving concentrated masses of elastically supported rectangular plate resting on variable Winkler elastic foundation, Latin America Journal of Solid and Structures, 11(9) (2014), 1515-1540.
  5. P. Aleksandar and M.L. Dragan, A numerical method for free vibration analysis of beams, Latin America Journal of Solid and Structures, 11(8) (2014), 1432-1444.
  6. M.G. Yuwaraj and S. Rajneesh, A refined shear deformation theory for flexure of thick beams, Latin America Journal of Solid and Structures, 8(2) (2011), 183-195.
  7. N.A. Shahin and O. Mbakisya, Simply Supported Beam Response on Elastic Foundation Carrying Repeated Rolling Concentrated Loads, Journal of Engineering Science and Technology, 5(1) (2010), 52-66.
  8. J.C. Hsu, H.Y. Lai, and C. K. Chen, Free Vibration of non-uniform Euler-Bernoulli beams with general elastically end constraints using Adomian modified decomposition method, Journal of Sound and Vibration, 318 (2008), 965-981.
  9. M. Saravi, M. Hermaan, and K.H. Ebarahimi, The comparison of homotopy perturbation method with finite difference method for determination of maximum beam deflection, Journal of Theoretical and Applied Physics, 7(8) (2013), 1-8.
  10. R. Li, Y. Zhong, and M.L. Li, Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method, Proceedings of the Royal Society A, 46 (2013), 468-474.
  11. M.H. Hsu, Vibration Analysis of non-uniform beams resting on elastic foundations using the spline collocation method, Tamkang Journal of Science and Engineering, 12(2) (2009), 113-122.
  12. J.E. Akin and A. Mofid, Numerical Solution for response of beams with mass, Journal of the structural Engineering, 1159(1) (1989), 120-131.
  13. M.M. Stanisic, J.E. Euler, and S.T. Montgomeny, On a theory concerning the dynamical behaviour of structures carrying moving masses, Ing. Archiv, 43 (1974), 295-305.
  14. M.M. Stanisic, J.C. Hardin, and Y.C. Lou, On the response of plate to a moving multi-masses moving system, Acta Mechanica, 5 (1968), 37-53.
  15. J.A. Gbadeyan and S.T. Oni, Dynamic response to moving concentrated masses of elastic plates on a non-Winkler elastic foundation, Journal of sound and vibrations, 154 (1992), 343-358.
  16. W.L. Li, X. Zhang, J. Du, and Z. Liu, An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports, Journal of Sound and Vibrations, 321 (2009), 254-269.
  17. M.R. Shadnam, M. Mofid, and J.E. Akin, On the dynamic response of rectangular plate with moving mass, Thin-walled structures, 39 (2001), 797-806.
  18. A.N. Lowan, On transverse oscillations of beams under the action of moving variable load, Phil. Mag. Ser. 7, 19(127) (1935), 708-715.
  19. S.S. Kokhmanyuk and A.P. Filippov, Dynamic effects on a beam of a load moving at variable speed, Stroitel'n mekhanka i raschet so-oruzhenii, 9(2) (1967), 36-39.
  20. M.H.Huang and D.P Thambiratnam, Deflection response of plate onWinkler foundation to moving accelerated loads, Engineering Structures, 23 (2001), 1134-1141.
  21. S. Sadiku and H.E Leipohlz, On the Dynamics of Elastic System with Moving Concentrated Masses, Ing. Archiv. 57 (1981), 223-242.
  22. S.T. Oni and T.O. Awodola, Vibrations under moving non-uniform Rayleigh beam on variable elastic foundation. V Journal of the Nigerian Association of Mathematical Physics, 7 (2003), 191-206.
  23. V. Kolousek, S.P. Timoshenko, and J. Inglis, Civil Engineering Structures Subjected to Dynamic Loads, SVTI Brastislava, (1967).
  24. J. Kenny, Steady vibration of a beam on an elastic foundation for a moving load, Journal of Applied Mechanics, 76 (1954), 369-364.
  25. S.T. Oni, On the thick beam under the action of a variable travelling transverse load, Abacus Journal of Mathematical Association of Nigeria, 25(2) (1997), 531-546.
  26. A.L. Florence, Travelling force on a Timoshenko beam, Transactions of the ASME, Journal of Applied Mechanics, 32 (1965), 351-358.
  27. T.C. Huang, The effect of rotary inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions, Transactions of the ASME, Journal of Applied Mechanics, 28 (1961), 579-584.
  28. T.P. Chang, Deterministic and random vibration of an axially loaded Timoshenko beam resting on an elastic foundation, Journal of Sound and Vibration, 178 (1994), 55-66.
  29. S.A.A. Hosseini and S.E. Khadem, Vibration and reliability of a rotating beam with random properties under random excitation, International Journal of Mechanical Sciences, 49 (2007), 1377-1388.
  30. B. Omolofe, S.T. Oni, and J.M. Tolorunshagba, On the transverse motions of non-prismatic deep beam under the actions of variable magnitude moving loads, Latin American Journal of Solid and Structutures, 6 (2009), 153-157.
  31. Y.M. Wang, The dynamical analysis of a finite inextensible beam with an attached accelerating mass, International Journal of solid Structure, 35 (1998), 831-854.
  32. S. Eftekhar Azam, M. Mofid, and R. Afghani Khoraskanic, Dynamic response of Timoshenko beam under moving mass, Scientia Iranica A, 20(1) (2013), 50-56.

Cited by

  1. Dynamic Response of Uniform Rayleigh Beams on Variable Bi-parametric Elastic Foundation under Partially Distributed Loads vol.11, pp.4, 2018,
  2. Flexural Motions of Non-uniform Beams Resting on Exponentially Decaying Foundation vol.11, pp.4, 2018,