Exact solution of vibrations of beams with arbitrary translational supports using shape function method

Abstract

In this paper, the exact and explicit solution of free vibrations of Euler–Bernoulli beams supported by an arbitrary number of translational springs at arbitrary positions under variable boundary conditions is obtained by the proposed shape function method. The differential equation governing the vibration is formulated by incorporating the Dirac’s delta function and the frequency equation and mode shape function are solved by the variational iteration method and the Laplace transform. The highest order of the frequency equation is four, which is independent of the number of springs, making this method more efficient than solutions in the literature. A total of 49 possible cases of boundary conditions (pinned, clamped, free, sliding, translational spring, rotational spring, and combined translational-rotational) are accommodated by a system of four homogeneous equations, which can be conveniently programmed into a computer to solve engineering problems. The validity of the proposed solutions is justified by comparing with results from the literature. A parametric study is carried out to investigate the influences of weakened stiffness on the vibration frequencies. The solutions provided in this work is mathematically complete and may be used as benchmarks for future research. The derived frequency equation and mode shape function are potentially useful for solving the inverse problem, i.e., identifying weakened elastic supports, in a future study.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

References

  1. Abdou, M., & Soliman, A. (2005). Variational iteration method for solving Burger's and coupled Burger's equations. JCoAM,181(2), 245–251.

    MathSciNet  MATH  Google Scholar 

  2. Abu-Hilal, M. (2003). Forced vibration of Euler-Bernoulli beams by means of dynamic Green functions. Journal of Sound and Vibration,267(2), 191–207.

    MATH  Google Scholar 

  3. Abulwafa, E., Abdou, M., & Mahmoud, A. (2006). The solution of nonlinear coagulation problem with mass loss. Chaos, Solitons and Fractals,29(2), 313–330.

    MathSciNet  MATH  Google Scholar 

  4. Banerjee, J. (1999). Explicit frequency equation and mode shapes of a cantilever beam coupled in bending and torsion. Journal of Sound and Vibration,224(2), 267–281.

    Google Scholar 

  5. Caddemi, S., Caliò, I., & Cannizzaro, F. (2015). Tensile and compressive buckling of columns with shear deformation singularities. Meccanica,50(3), 707–720.

    MathSciNet  MATH  Google Scholar 

  6. Carta, G., & Brun, M. (2015). Bloch-Floquet waves in flexural systems with continuous and discrete elements. Mechanics of Materials,87, 11–26.

    Google Scholar 

  7. Connolly, D. P., Kouroussis, G., Laghrouche, O., Ho, C., & Forde, M. (2015). Benchmarking railway vibrations—track, vehicle, ground and building effects. Construction and Building Materials,92, 64–81.

    Google Scholar 

  8. He, J. (1997). A new approach to nonlinear partial differential equations. Communications in Nonlinear Science and Numerical Simulation,2(4), 230–235.

    MathSciNet  Google Scholar 

  9. He, J.-H. (1998a). Approximate solution of nonlinear differential equations with convolution product nonlinearities. CMAME,167(1–2), 69–73.

    MATH  Google Scholar 

  10. He, J.-H. (1998b). Approximate analytical solution for seepage flow with fractional derivatives in porous media. CMAME,167(1–2), 57–68.

    MathSciNet  MATH  Google Scholar 

  11. He, J.-H. (1999a). Homotopy perturbation technique. CMAME,178(3–4), 257–262.

    MathSciNet  MATH  Google Scholar 

  12. He, J.-H. (1999b). Variational iteration method—a kind of non-linear analytical technique: some examples. International Journal of Non-Linear Mechanics,34(4), 699–708.

    MATH  Google Scholar 

  13. He, J.-H. (2000). Variational iteration method for autonomous ordinary differential systems. Applied Mathematics and Computation,114(2–3), 115–123.

    MathSciNet  MATH  Google Scholar 

  14. He, J.-H. (2006). Some asymptotic methods for strongly nonlinear equations. IJMPB,20(10), 1141–1199.

    MathSciNet  MATH  Google Scholar 

  15. He, J.-H., Wazwaz, A.-M., & Xu, L. (2007). The variational iteration method: Reliable, efficient, and promising. Computers & Mathematics with Applications, 54(7–8), 879–880.

    MathSciNet  MATH  Google Scholar 

  16. Kukla, S. (1991a). The Green function method in frequency analysis of a beam with intermediate elastic supports. Journal of Sound and Vibration,149(1), 154–159.

    Google Scholar 

  17. Kukla, S. (1991b). Free vibration of a beam supported on a stepped elastic foundation. Journal of Sound and Vibration,149(2), 259–265.

    Google Scholar 

  18. Kukla, S. (1997). Application of Green functions in frequency analysis of Timoshenko beams with oscillators. Journal of Sound and Vibration,205(3), 355–363.

    MATH  Google Scholar 

  19. Kukla, S., & Posiadala, B. (1994). Free vibrations of beams with elastically mounted masses. Journal of Sound and Vibratiom,175(4), 557–564.

    MATH  Google Scholar 

  20. Legault, J., Mejdi, A., & Atalla, N. (2011). Vibro-acoustic response of orthogonally stiffened panels: The effects of finite dimensions. Journal of Sound and Vibration,330(24), 5928–5948.

    Google Scholar 

  21. Leissa, A. W., & Qatu, M. S. (2011). Vibrations of continuous aystems. New York: McGraw-Hill.

    Google Scholar 

  22. Mead, D., & Pujara, K. (1971). Space-harmonic analysis of periodically supported beams: response to convected random loading. Journal of Sound and Vibration,14(4), 525–541.

    Google Scholar 

  23. Mohamad, A. (1994). Tables of Green's functions for the theory of beam vibrations with general intermediate appendages. IJSS,31(2), 257–268.

    MATH  Google Scholar 

  24. Momani, S., & Odibat, Z. (2006). Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Physics Letters A,355(4–5), 271–279.

    MATH  Google Scholar 

  25. Munjal, M., & Heckl, M. (1982). Vibrations of a periodic rail-sleeper system excited by an oscillating stationary transverse force. Journal of Sound and Vibration,81(4), 491–500.

    Google Scholar 

  26. Rao, S. S. (2019). Vibration of continuous systems. Hoboken: Wiley.

    Google Scholar 

  27. Romeo, F., & Luongo, A. (2002). Invariant representation of propagation properties for bi-coupled periodic structures. Journal of Sound and Vibration,257(5), 869–886.

    Google Scholar 

  28. Rončević, G. Š., Rončević, B., Skoblar, A., & Žigulić, R. (2019). Closed form solutions for frequency equation and mode shapes of elastically supported Euler-Bernoulli beams. Journal of Sound and Vibration,457, 118–138.

    Google Scholar 

  29. Shabana, A. A. (1991). Theory of vibration. Berlin: Springer.

    Google Scholar 

  30. Sun, L. (2001). A closed-form solution of a Bernoulli-Euler beam on a viscoelastic foundation under harmonic line loads. Journal of Sound and Vibration,242(4), 619–627.

    Google Scholar 

  31. Wang, L., Zhang, Y., & Lie, S. T. (2017). Detection of damaged supports under railway track based on frequency shift. Journal of Sound and Vibration,392, 142–153.

    Google Scholar 

  32. Wazwaz, A.-M. (2007a). The variational iteration method for rational solutions for KdV, K (2, 2), Burgers, and cubic Boussinesq equations. JCoAM,207(1), 18–23.

    MathSciNet  MATH  Google Scholar 

  33. Wazwaz, A.-M. (2007b). The variational iteration method for solving two forms of Blasius equation on a half-infinite domain. Applied Mathematics and Computation,188(1), 485–491.

    MathSciNet  MATH  Google Scholar 

  34. Wazwaz, A.-M. (2007c). The variational iteration method for exact solutions of Laplace equation. Physics Letters A,363(4), 260–262.

    MathSciNet  MATH  Google Scholar 

  35. Wazwaz, A.-M. (2007d). The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations. Computers and Mathematics with Applications,54(7–8), 926–932.

    MathSciNet  MATH  Google Scholar 

  36. Wazwaz, A.-M. (2007e). The variational iteration method: a powerful scheme for handling linear and nonlinear diffusion equations. Computers and Mathematics with Applications,54(7–8), 933–939.

    MathSciNet  MATH  Google Scholar 

  37. Wazwaz, A.-M. (2007f). The variational iteration method for a reliable treatment of the linear and the nonlinear Goursat problem. Applied Mathematics and Computation,193(2), 455–462.

    MathSciNet  MATH  Google Scholar 

  38. Wazwaz, A.-M. (2007g). A comparison between the variational iteration method and Adomian decomposition method. JCoAM,207(1), 129–136.

    MathSciNet  MATH  Google Scholar 

  39. Wazwaz, A.-M. (2008). A study on linear and nonlinear Schrodinger equations by the variational iteration method. Chaos, Solitons and Fractals,37(4), 1136–1142.

    MathSciNet  MATH  Google Scholar 

  40. Zhao, X. (2019). Free vibration analysis of cracked Euler-Bernoulli beam by Laplace transformation considering stiffness reduction. Romanian Journal of Acoustics and Vibration,16(2), 166–173.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees for valuable comments and suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xingzhuang Zhao.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is 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

Verify currency and authenticity via CrossMark

Cite this article

Chang, P., Zhao, X. Exact solution of vibrations of beams with arbitrary translational supports using shape function method. Asian J Civ Eng (2020). https://doi.org/10.1007/s42107-020-00275-7

Download citation

Keywords

  • Laplace transform
  • Shape function
  • Frequency equation
  • Variational iteration method
  • Weakened elastic support