Dynamic Response of a Damped Euler–Bernoulli Beam Having Elastically Restrained Boundary Supports

  • Kavikant Mahapatra
  • S. K. PanigrahiEmail author
Original Contribution


While doing vibration analysis in terms of the dynamic response of a beam, the end conditions, in most of the cases, are assumed to be fixed, simply supported, free or sliding imposing an ideal condition of end support displacement. But, in real engineering structures, the end supports are non-ideal (for example welded, riveted etc.) and allow certain degree of translational or rotational motion of the support ends. To simulate the effect of real engineering boundary states, a combination of linear and torsional springs has been considered at the two boundaries of the beam structure. The theoretical analysis of free undamped beam structure with elastically restrained end conditions has been undertaken using approximation of the modal displacement as a Fourier cosine series to generate natural frequencies and corresponding mode shape equations. The effect of variation in stiffness values of the boundary springs on the vibration characteristics (natural frequency, mode shapes and dynamic response) has been presented for vibration conditions for the free undamped beam structure. Thereafter, forced undamped and damped (taking combined effects of both viscous and structural damping) beam structures have been theoretically analysed to generate response equation. Finally, numerical assessment has been undertaken to present the effect of damping ratio, forcing frequency and force magnitude on the response of a nearly clamped beam and has been compared with the response of an undamped beam under similar forcing and boundary condition. The results indicate considerable effects of real boundary condition, on the vibration characteristics and dynamic response of beam structures thus cannot be ignored in design of real engineering structures. Further, the presence of damping does not change the vibration characteristics beam structures, but is useful in achieving steady-state condition of response for a harmonic force input. The technique presented provides a simplified and convenient tool for obtaining vibration response for real engineering boundary supports. Further, the beam model can be extended for analysis of plate structures considering non-ideal boundary conditions.


Beam structures Ideal/non-ideal boundary conditions Rotational stiffness Translational Stiffness Fourier cosine series Dynamic response Damping ratio Vibration characteristics 


  1. 1.
    S.C. Mittendorf, R. Greif, Vibrations of segmented beams by a fourier series component mode method. J. Sound Vib. 55(3), 431–441 (1977)CrossRefGoogle Scholar
  2. 2.
    C.K. Rao, S. Mirza, A note on vibration of generally restrained beams. J. Sound Vib. 130(3), 453–465 (1989)CrossRefGoogle Scholar
  3. 3.
    J.T.S. Wang, C.C. Lin, Dynamic analysis of generally supported beams using fourier series. J. Sound Vib. 196(3), 285–293 (1996)CrossRefGoogle Scholar
  4. 4.
    M.J. Maurizi, G.G. Robledo, A further note on the “dynamic analysis of generally supported beams using fourier series”. J. Sound Vib. 214(5), 972–973 (1998)CrossRefGoogle Scholar
  5. 5.
    W.L. Li, Free vibrations of beams with general boundary conditions. J. Sound Vib. 237(4), 709–725 (2000)CrossRefGoogle Scholar
  6. 6.
    H.K. Kim, M.S. Kim, Vibrations of beams with generally restrained boundary conditions using fourier series. J. Sound Vib. 245(5), 771–784 (2001)CrossRefGoogle Scholar
  7. 7.
    D. Wang, Optimal design of an intermediate support for a beam with elastically restrained boundaries. J. Vib. Acoust. 133(3), 1–8 (2011)CrossRefGoogle Scholar
  8. 8.
    S. Naguleswaran, Transverse vibration of an uniform Euler–Bernoulli beam under linearly varying axial force. J. Sound Vib. 275, 47–57 (2004)CrossRefGoogle Scholar
  9. 9.
    M. Bayat, I. Pakar, M. Bayat, On the large amplitude free vibrations of axially loaded Euler–Bernoulli beams. Steel Compos. Struct. 14(1), 78–83 (2013)CrossRefGoogle Scholar
  10. 10.
    X. Wang, G. Duan, Discrete singular convolution element method for static, buckling and free vibration analysis of beam structures. Appl. Math. Comput. 234, 36–51 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    H.Y. Lai, J.C. Hsu, C.K. Chen, An innovative eigenvalue problem solver for free vibration of Euler–Bernoulli beam by using the Adomian decomposition method. Comput. Math Appl. 56, 3204–3220 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    C.S. Manohar, S. Adhikari, Dynamic stiffness of randomly parametered beam. Prob. Eng. Mech. 13(1), 39–51 (1998)CrossRefGoogle Scholar
  13. 13.
    M.A. Foda, Z. Abduljabbar, A dynamic Green function formulation for the response of a beam structure to a moving mass. J. Sound Vib. 210, 295–306 (1998)CrossRefGoogle Scholar
  14. 14.
    M.H. Abu, H.S. Zibdesh, Vibration analysis of beams with general boundary conditions traversed by a moving force. J. Sound Vib. 229(2), 377–388 (2000)CrossRefGoogle Scholar

Copyright information

© The Institution of Engineers (India) 2018

Authors and Affiliations

  1. 1.DIAT (DU)Girinagar, PuneIndia

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