Abstract
In this paper, a simulation method called the differential transform method (DTM) is employed to predict the vibration of an Euler–Bernoulli and Timoshenko beam (pipeline) resting on an elastic soil. The DTM is introduced briefly. DTM can easily be applied to linear or nonlinear problems and reduces the required computational effort. With this method exact solutions may be obtained without any need for cumbersome calculations and it is a useful tool for analytical and numerical solutions. To clarify and illustrate the features and capabilities of the presented method, various problems have been solved by using the technique and solutions have been compared with those obtained in the literature.
Similar content being viewed by others
References
Avramidis I.E. and Morfidis K. (2006). Bending of beams on three-parameter elastic foundation. Int. J. Solids Struct. 43: 357–375
De Rosa M.A. (1995). Free vibration of Timoshenko beams on two-parameter elastic foundation. Comput. Struct. 57(1): 151–156
Matsunaga H. (1999). Vibration and bucklig of deep beam-columns on two-parameter elasti foundations. J. Sound Vib. 228(2): 359–376
El-Mously M. (1999). Fundamental frequencies of Timoshenko beams mounted on Pasternak foundation. J. Sound Vib. 228(2): 452–457
Chen C.N. (2000). Vibration of prismatic beam on an elastic foundation by the differential quadrature element method. Comput. Struct. 77: 1–9
Chen C.N. (2002). DQEM vibration analyses of non-prismatic shear deformable beams resting on elastic foundations. J. Sound Vib. 255(5): 989–999
Coşkun İ. (2003). The response of a finite beam on a tensionless Pasternak foundation subjected to a harmonic load. Eur. J. Mech. A Solids 22: 151–161
Chen W.Q., Lü C.F. and Bian Z.G. (2004). A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. Appl. Math. Model. 28: 877–890
Maheshwari P., Chandra S. and Basudhar P.K. (2004). Response of beams on a tensionless extensible geosynthetic-reinforced earth bed subjected to moving loads. Comput. Geotech. 31: 537–548
Auciello N.M. and De Rosa M.A. (2004). Two approaches to the dynamic analysis of foundation beams subjected to subtangential forces. Comput. Struct. 82: 519–524
Elfelsoufi Z. and Azrar L. (2005). Buckling, flutter and vibration analyses of beams by integral equation formulations. Comput. Struct. 83: 2632–2649
Ruta P. (2006). The application of Chebyshev polynomials to the solution of the nonprismatic Timoshenko beam vibration problem. J. Sound Vib. 296: 243–263
Zhou J.K. (1986). Differential Transformation and its Application for Electrical Circuits. Wuhan, Huazhong University Press, China
Chen C.K. and Ho S.H. (1999). Solving partial differential equations by two-dimensional differential transform method. Appl. Math. Comput. 106: 171–179
Ayaz F. (2003). On the two-dimensional differential transform method. Appl. Math. Comput. 143: 361–374
Ayaz F. (2004). Solutions of the system of differential equations by differential transform method. Appl. Math. Comput. 147: 547–567
Arıkoğlu A. and Özkol İ. (2005). Solution of boundary value problems for integro-differential equations by using differential transform method. Appl. Math. Comput. 168: 1145–1158
Özdemir Ö. and Kaya M.O. (2006). Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method. J. Sound Vib. 289: 413–420
Kaya M.O. (2006). Free vibration analysis of rotating timoshenko beam by differential transform method. Aircr. Eng. Aerosp Technol. 78(3): 194–203
Özdemir Ö. and Kaya M.O. (2006). Flapwise bending vibration analysis of double tapered rotating Euler–Bernoulli beam by using the differential transform method. Meccanica 41(6): 661–670. doi:10.1007/s11012-006-9012-z
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Balkaya, M., Kaya, M.O. & Sağlamer, A. Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method. Arch Appl Mech 79, 135–146 (2009). https://doi.org/10.1007/s00419-008-0214-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-008-0214-9