DOI QR코드

DOI QR Code

Response of forced Euler-Bernoulli beams using differential transform method

  • Catal, Seval (Department of Civil Engineering, Faculty of Engineering (Applied Mathematics), Dokuz Eylul University)
  • Received : 2011.12.12
  • Accepted : 2012.02.28
  • Published : 2012.04.10

Abstract

In this paper, forced vibration differential equations of motion of Euler-Bernoulli beams with different boundary conditions and dynamic loads are solved using differential transform method (DTM), analytical solutions. Then, the modal deflections of these beams are obtained. The calculated modal deflections using DTM are represented in tables and depicted in graphs and compared with the results of the analytical solutions where a very good agreement is observed.

Keywords

References

  1. Catal, S. (2006), "Analysis of free vibration of beam on elastic soil using differential transform method", Struct. Eng. Mech., 24, 51-62. https://doi.org/10.12989/sem.2006.24.1.051
  2. Catal, S. (2008), "Solution of free vibration equations of beam on elastic soil by using differential transform method", Appl. Math. Model., 32, 1744-1757. https://doi.org/10.1016/j.apm.2007.06.010
  3. Catal, S. and Catal, H.H. (2006), "Buckling analysis of partially embedded pile in elastic soil using differential transform method", Struct. Eng. Mech., 24, 24-268.
  4. Celebi, K., Keles, I. and Tutuncu, N. (2011), "Exact solutions for forced vibration of non-uniform rods by Laplace Transformation", Gazi University Journal of Science, 24, 347-353.
  5. Chen, C.K. and Ho, S.H. (1996), "Application of differential transformation to eigenvalue problem", J. Appl. Math. Comput., 79, 173-188. https://doi.org/10.1016/0096-3003(95)00253-7
  6. Chen, C.K. and Ho, S.H. (1999), "Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform", Int. J. Mech. Sci., 41, 1339-1356. https://doi.org/10.1016/S0020-7403(98)00095-2
  7. Chopra, A.K. (1995), Dynamics of Structures, Prentice-Hall Inc., New Jersey.
  8. Clough, R.W. and Penzien, J. (1993), Dynamics of Structures, 2nd Ed., McGraw-Hill Inc., Singapore.
  9. Demirdag, O. and Yesilce, Y. (2011), "Solution of free vibration equation of elastically supported Timoshenko columns with a tip mass by differential transform method", Adv. Eng. Softw., 42, 860-867. https://doi.org/10.1016/j.advengsoft.2011.06.002
  10. Fan, Z.J., Lee, J.H., Kang, K.H. and Kim, K.J. (1998), "The forced vibration of a beam with viscoelastic boundary supports", J. Sound Vib., 210, 673-682. https://doi.org/10.1006/jsvi.1997.1353
  11. Hassan, I.H.A.H. (2002a), "On solving some eigenvalue problems by using differential transformation", Appl. Math. Comput., 28, 513-525.
  12. Hassan, I.H.AH. (2002b), "Different applications for the differential transformation in the differential equations", Appl. Math. Comput., 129, 183-201. https://doi.org/10.1016/S0096-3003(01)00037-6
  13. Hilal, M.A. (2003), "Forced vibration of Euler-Bernoulli beams by means of dynamic Green functions", J. Sound Vib., 267, 191-207. https://doi.org/10.1016/S0022-460X(03)00178-0
  14. Jang, M.J. and Chen, C.L. (1997), "Analysis of the response of a strongly non-linear damped system using a differential transformation technique", Appl. Math. Comput., 88, 137-151. https://doi.org/10.1016/S0096-3003(96)00308-6
  15. Kai-yuan, Y., Xiao-hua, T. and Zhen-yi, J. (1992), "General analytic solution of dynamic response of beams with nonhomogenity and variable cross-section", Appl. Math. Mech., 13, 779-791. https://doi.org/10.1007/BF02481798
  16. Oniszczuk, Z. (2003), "Forced transverse vibrations of an elastically connected complex simply supported bouble-beam system", J. Sound Vib., 264, 273-286. https://doi.org/10.1016/S0022-460X(02)01166-5
  17. Ozgumus, O.O. and Kaya, M.O. (2006), "Flabse bending vibration analysis of rotating tapered cantilever Bernoulli- Euler beam by differential transform method", J. Sound Vib., 289, 413-420. https://doi.org/10.1016/j.jsv.2005.01.055
  18. Ozgumus, O.O. and Kaya, M.O. (2010), "Vibration analysis of rotating tapered Timoshenko beam by using DTM", Mechanica, 45, 33-42. https://doi.org/10.1007/s11012-009-9221-3
  19. Paz, M. (1997), Structural Dynamics, 4th Ed., Chapman & Hall, New York.
  20. Paz, M. and Dung, L. (1975), "Power series expansion of the general stiffness matrix for beam elements", Int. J. Numer. Meth. Eng., 9, 449-459. https://doi.org/10.1002/nme.1620090212
  21. Yesilce, Y. (2010), "Differential transform method for free vibration analysis of a moving beam", Struct. Eng. Mech., 35, 645-658. https://doi.org/10.12989/sem.2010.35.5.645
  22. Yesilce, Y. (2011), "DTM and DQEM for free vibration of axially loaded and semi-rigid-connected Reddy- Bickford beam", Int. J. Numer. Meth. Bio. Eng., 27, 666-693. https://doi.org/10.1002/cnm.1313
  23. Yesilce, Y. and Catal, H.H. (2011), "Solution of free vibration equation of semi-rigid connected Reddy-Bickford beams resting on elastic soil using the differential transform method", Arch. Appl. Mech., 81, 199-213. https://doi.org/10.1007/s00419-010-0405-z
  24. Yesilce, Y. and Catal, S. (2009), "Free vibration of axially loaded Reddy-Bickford beam on elastic soil using differential transform method", Struct. Eng. Mech., 31, 453-475. https://doi.org/10.12989/sem.2009.31.4.453
  25. Zhou, J.K. (1986), Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China.

Cited by

  1. Static and dynamic analysis of beam assemblies using a differential system on an oriented graph vol.155, 2015, https://doi.org/10.1016/j.compstruc.2015.02.021
  2. Dynamic stiffness approach and differential transformation for free vibration analysis of a moving Reddy-Bickford beam vol.58, pp.5, 2016, https://doi.org/10.12989/sem.2016.58.5.847
  3. Differential transform method and numerical assembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and rotary inertias vol.53, pp.3, 2015, https://doi.org/10.12989/sem.2015.53.3.537
  4. Investigation of bar system modal characteristics using Dynamic Stiffness Matrix polynomial approximations vol.180, 2017, https://doi.org/10.1016/j.compstruc.2016.10.015
  5. Study on modified differential transform method for free vibration analysis of uniform Euler-Bernoulli beam vol.48, pp.5, 2013, https://doi.org/10.12989/sem.2013.48.5.697
  6. Dynamic response of Euler-Bernoulli beams to resonant harmonic moving loads vol.44, pp.5, 2012, https://doi.org/10.12989/sem.2012.44.5.681
  7. Dynamic behavior of axially functionally graded simply supported beams vol.25, pp.6, 2012, https://doi.org/10.12989/sss.2020.25.6.669
  8. Closed-form solution for mode superposition analysis of continuous beams on flexible supports under moving harmonic loads vol.520, pp.None, 2022, https://doi.org/10.1016/j.jsv.2021.116587