Vibration analysis of a tapered beam with exponentially varying thickness resting on Winkler foundation using the differential transform method

  • Authors

    • Sayyad Boreyri Department of Marine Technology Amirkabir University of Technology,Hafez Ave., Tehran, Iran
    • Pouya Mohtat Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
    • Mohamad Javad Ketabdari Department of Marine Technology, Amirkabir University of Technology, Hafez Ave., Tehran, Iran
    • Ali Moosavi Department of Mathematics, Sharif University of Technology, Tehran, Iran
    2014-04-07
    https://doi.org/10.14419/ijpr.v2i1.2152

  • Abstract

    In this paper, free vibration of a new type of tapered beam, with exponentially varying thickness, resting on a linear foundation is analyzed. The solution is based on a semi-analytical technique, the differential transform method (DTM). Applying DTM, nonlinear partial differential equations of the varying thickness beam are transformed into algebraic equations, which are then solved to obtain the solution. An Euler-Bernoulli beam with a number of boundary conditions and different exponential factor is taken into account. Results have been compared to the 4th order Runge-Kutta, and where possible with DQEM and analytical solution. These comparisons prove the preciseness of this method, based on which DTM can be considered as a powerful framework for eigenvalue analysis of new type of tapered beams.

     

    Keywords: Free Vibration, Exponential (Tapered) Beam, Winkler Foundation, Differential Transform Method (DTM).

  • References

    1. Y.K. Lin, “Free vibrations of a continuous beam on elastic supports”, International Journal of Mechanical Sciences, Vol.4, No.5, pp.409–423 (1962). Available online: http://www.sciencedirect.com/science/article/pii/S0020740362800277.
    2. A.J. Valsangkar, R. Pradhanang, “Vibration of beam-columns on two-parameter elastic foundations”, Earthquake Engineering and Structural Dynamics, Vol.16, pp.217–225 (1988). Available online: http://onlinelibrary.wiley.com/doi/10.1002/eqe.4290160205/abstract.
    3. C. Franciosi, A. Masi, “Free vibrations of foundation beams on two-parameter elastic soil”, Computers & Structures, Vol.47, No.3, pp.419–426 (1993). Available online: http://www.sciencedirect.com/science/article/pii/0045794993902378.
    4. M.A. De Rosa, “Free vibration of Timoshenko beams on two-parameter elastic foundation”, Computers & Structures, Vol.57, No.1, pp.151–156 (1995). Available online: www.sciencedirect.com/science/article/pii/004579499400594S.
    5. P. Ruta, “The application of Chebyshev polynomials to the solution of the nonprismatic Timoshenko beam vibration problem”, Journal of Sound and Vibration, Vol.296, pp.243–263 (2006). Available online: http://www.sciencedirect.com/science/article/pii/S0022460X06002100.
    6. C.N. Chen, “Vibration of prismatic beam on an elastic foundation by the differential quadrature element method”, Computers & Structures, Vol.77, No.1, pp.1–9 (2000). Available online: http://www.sciencedirect.com/science/article/pii/S0045794999002163.
    7. W.Q. Chen, C.F. Lü, Z.G. Bian, “A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation”, Applied Mathematical Modelling, Vol.28, No.10, pp.877–890 (2004). Available online: http://www.sciencedirect.com/science/article/pii/S0307904X04000605.
    8. R.J. Hosking, S.A. Husain, F. Milinazzo, “Natural flexural vibrations of a continuous beam on discrete elastic supports”, Journal of Sound and Vibration, Vol.272, No.1-2, pp.169–185 (2004). Available online: www.sciencedirect.com/science/article/pii/S0022460X03003250.
    9. J.K. Zhou, Differential Transform and Its Applications for Electrical circuits, (in chinese), Huazhong University press, Wuhan China, 1986.
    10. S. Çatal, “Solution of free vibration equations of beam on elastic soil by using differential transform method”, Applied Mathematical Modelling, Vol.32, No.9, pp.1744–1757 (2008). Available online: http://www.sciencedirect.com/science/article/pii/S0307904X07001357.
    11. M. Balkaya, M.O. Kaya, A. Sağlamer, “Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method”, Archive of Applied Mechanics, Vol.79, No.2, pp.135–146 (2009). Available online: http://link.springer.com/article/10.1007/s00419-008-0214-9.
    12. T.O. Awodola, “Variable Velocity Influence on the Vibration of Simply Supported Bernoulli - Euler Beam under Exponentially Varying Magnitude Moving Load”, Journal of Mathematics and Statistics, Vol.3, No.4, pp.228–232 (2007). Available online: http://www.thescipub.com/abstract/10.3844/jmssp.2007.228.232.
    13. Q. Mao, S. Pietrzko, “Free vibration analysis of a type of tapered beams by using Adomian decomposition method”, Applied Mathematics and Computation, Vol.219, No.6, pp.3264-3271 (2012). Available online: http://www.sciencedirect.com/science/article/pii/S0096300312009812.
    14. F. Ayaz, “On the two-dimensional differential transform method”, Applied Mathematics and Computation, Vol.143, No.2-3, pp.361–374 (2003). Available online: http://www.sciencedirect.com/science/article/pii/S0096300302003685.
    15. F. Ayaz, “Solutions of the system of differential equations by differential transform method”, Applied Mathematics and Computation, Vol.147, No.2, pp.547–567 (2004). Available online: http://www.sciencedirect.com/science/article/pii/S0096300302007944.
    16. Arikoglu, I. Ozkol, “Solution of differential-difference equations by using differential transform method”, Applied Mathematics and Computation, Vol.174, No.2, pp.1216-1228 (2006). Available online: http://www.sciencedirect.com/science/article/pii/S0096300305005436.

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  • How to Cite

    Boreyri, S., Mohtat, P., Ketabdari, M. J., & Moosavi, A. (2014). Vibration analysis of a tapered beam with exponentially varying thickness resting on Winkler foundation using the differential transform method. International Journal of Physical Research, 2(1), 10-15. https://doi.org/10.14419/ijpr.v2i1.2152

    Received date: 2014-03-07

    Accepted date: 2014-03-29

    Published date: 2014-04-07