Geomechanics and Engineering

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Nonlinear frequency analysis of beams resting on elastic foundation using max-min approach

  • Bayat, Mahmoud (Young Researchers and Elite club, Roudehen Branch, Islamic Azad University) ;
  • Bayat, Mahdi (Department of Civil Engineering, Roudehen Branch, Islamic Azad University) ;
  • Kia, Mehdi (Department of Civil and Environmental Engineering, University of Science and Technology of Mazandaran) ;
  • Ahmadi, Hamid Reza (Department of Civil Engineering, Faculty of Engineering, University of Maragheh) ;
  • Pakar, Iman (Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University)
  • Received : 2018.05.25
  • Accepted : 2018.07.10
  • Published : 2018.11.20
⟨ Previous Next ⟩

Abstract

In this paper, nonlinear vibration of Euler-Bernoulli beams resting on linear elastic foundation is studied. It has been tried to prepare a semi-analytical solution for whole domain of vibration. Only one iteration lead us to high accurate solution. The effects of linear elastic foundation on the response of the beam vibration are considered and studied. The effects of important parameters on the ratio of nonlinear to linear frequency of the system are studied. The results are compared with numerical solution using Runge-Kutta $4^{th}$ technique. It has been shown that the Max-Min approach can be easily extended in nonlinear partial differential equations.

Keywords

References

  1. Akgoz, B. and Civalek, O. (2015), "Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity", Compos. Struct., 134, 294-301 https://doi.org/10.1016/j.compstruct.2015.08.095
  2. Al-Hosani, K., Fadhil, S. and El-Zafrany, A. (1999), "Fundamental solution and boundary element analysis of thick plates on Winkler foundation", Comput. Struct., 70(3), 325-336. https://doi.org/10.1016/S0045-7949(98)00171-0
  3. Arikoglu, A. and Ozkol, I. (2006), "Solution of differential-difference equations by using differential transform method", Appl. Math. Comput., 181(1), 153-162. https://doi.org/10.1016/j.amc.2006.01.022
  4. Auersch, L. (2008), "Dynamic interaction of various beams with the underlying soil-finite and infinite, half-space and Winkler models", Eur. J. Mech. A Solids, 27(5), 933-958. https://doi.org/10.1016/j.euromechsol.2008.02.001
  5. Azrar, L., Benamar, R. and White, R.A. (1999), "Semi-analytical approach to the non-linear dynamic response problem of S-S and C-C beams at large vibration amplitudes part I: General theory and application to the single mode approach to free and forced vibration analysis", J. Sound Vib., 224(2), 183-207. https://doi.org/10.1006/jsvi.1998.1893
  6. Bayat, M., Pakar, I. and Cao, M.S. (2017c), "Energy based approach for solving conservative nonlinear systems", Earthq. Struct., 13(2),131-136 https://doi.org/10.12989/EAS.2017.13.2.131
  7. Bayat, M. and Pakar, I. (2017a), "Accurate semi-analytical solution for nonlinear vibration of conservative mechanical problems", Struct. Eng. Mech., 61(5), 657-661. https://doi.org/10.12989/sem.2017.61.5.657
  8. Bayat, M., Pakar, I. and Bayat, M. (2017b), "Nonlinear vibration of multi-body systems with linear and nonlinear springs", Steel Compos. Struct., 25(4), 497-503. https://doi.org/10.12989/SCS.2017.25.4.497
  9. Bayat, M., Bayat, M. and Pakar, I. (2018), "Nonlinear vibration of oscillatory systems using semi-analytical approach", Struct. Eng. Mech., 65(4), 409-413 https://doi.org/10.12989/SEM.2018.65.4.409
  10. Ghannadiasl, A. and Mofid, M. (2015), "An analytical solution for free vibration of elastically restrained Timoshenko beam on an arbitrary variable Winkler foundation and under axial load", Lat. Am. J. Solids Struct., 12(13), 2417-2438. https://doi.org/10.1590/1679-78251504
  11. Gorbunov-Posadov, M.I., Malikova, T.A. and Solomin, V.I. (1973), The Design of Structures on an Elastic Foundation, Stroiizdat, Moscow, Russia.
  12. Gupta, U., Ansari, A., and Sharma, S. (2006), "Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation", J. Sound Vib., 297(3-5), 457-476. https://doi.org/10.1016/j.jsv.2006.01.073
  13. Hadji, L., Daouadji, T.H. and Bedia, E.A. (2015), "A refined exponential shear deformation theory for free vibration of FGM beam with porosities", Geomech. Eng., 9(3), 361-372. https://doi.org/10.12989/gae.2015.9.3.361
  14. He, G., Li, X. and Lou, R. (2016), "Nonlinear FEA of higher order beam resting on a tensionless foundation with friction", Geomech. Eng., 11(1), 95-116. https://doi.org/10.12989/gae.2016.11.1.095
  15. He, J.H. (2008), "Max-min approach to nonlinear oscillators", Int. J. Nonlin. Sci. Numer. Simul., 9(2), 207-210. https://doi.org/10.1515/IJNSNS.2008.9.2.207
  16. Kacar, A., Tan, H.T. and Kaya, M.O. (2011), "Free vibration analysis of beams on variable winkler elastic foundation by using the differential transform method", Math. Comput. Appl., 16(3), 773-783.
  17. Liu, Y. and Gurram, C.S. (2009), "The use of He's variational iteration method for obtaining the free vibration of an Euler-Bernoulli beam", Math. Comput. Model., 50(11-12), 1545-1552. https://doi.org/10.1016/j.mcm.2009.09.005
  18. Lohar, H., Mitra, A. and Sahoo, S. (2016), "Free vibration analysis of axially functionally graded linearly taper beam on elastic foundation", IOP Conf. Series Mater. Sci. Eng., 149(1), 012130. https://doi.org/10.1088/1757-899X/149/1/012130
  19. Mirzabeigy, A. (2014), "Semi-analytical approach for free vibration analysis of variable cross-section beams resting on elastic foundation and under axial force", Int. J. Eng. Trans. C Aspects, 27(3), 385-394.
  20. Motaghian, S., Mofid, M. and Alanjari, P. (2011), "Exact solution to free vibration of beams partially supported by an elastic foundation", Scientia Iranica, 18, 861-866.
  21. Niknam, H. and Aghdam, M.M. (2015), "A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation", Compos. Struct., 119, 452-62 https://doi.org/10.1016/j.compstruct.2014.09.023
  22. Ozis, T. and Yildirim, A. (2007), "Determination of the frequency-amplitude relation for a duffing-harmonic oscillator by the energy balance method", Comput. Math. Appl., 54(7), 1184-1187. https://doi.org/10.1016/j.camwa.2006.12.064
  23. Ozis, T. and Yildirim, A. (2009), "Generating the periodic solutions for forcing van der Pol oscillators by the iteration perturbation method", Nonlin. Anal. Real World Appl., 10(4), 1984-1989.
  24. Rabia, B., Hassaine, D., Said, M. and Hadji, L. (2016), "Free vibration analysis of FG plates resting on the elastic foundation and based on the neutral surface concept using higher order shear deformation theory", Comptes Rendus Mecanique, 10(5), 1033-1048.
  25. Shariyat, M. and Alipour, M.M. (2011), "Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations", Arch. Appl. Mech., 81(9), 1289-1306. https://doi.org/10.1007/s00419-010-0484-x
  26. Shou, D.H. (2009), "The homotopy perturbation method for nonlinear oscillators", Comput. Math. Appl., 58(11-12), 2456-2459. https://doi.org/10.1016/j.camwa.2009.03.034
  27. Soldatos, K. and Selvadurai, A. (1985), "Flexure of beams resting on hyperbolic elastic foundations", Int. J. Solid. Struct., 21(4), 373-388. https://doi.org/10.1016/0020-7683(85)90062-9
  28. Tsiatas, G.C. (2010), "Nonlinear analysis of non-uniform beams on nonlinear elastic foundation", Acta Mech., 209(1-2), 141-152. https://doi.org/10.1007/s00707-009-0174-3
  29. Xing, J.Z. and Wang, Y.G. (2013), "Free vibrations of a beam with elastic end restraints subject to a constant axial load", Arch. Appl. Mech., 83(2), 241-252. https://doi.org/10.1007/s00419-012-0649-x
  30. Xu, L. (2007), "He's parameter-expanding methods for strongly nonlinear oscillators", J. Comput. Appl. Math., 207(1), 148-154. https://doi.org/10.1016/j.cam.2006.07.020
  31. Ying, J., Lu, C. and Chen, W. (2008), "Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations", Compos. Struct., 84(3), 209-219. https://doi.org/10.1016/j.compstruct.2007.07.004
  32. Zahedinejad, P. (2016), "Free vibration analysis of functionally graded beams resting on elastic foundation in thermal environment", Int. J. Struct. Stability Dyn., 16(7), 1550029. https://doi.org/10.1142/S0219455415500297

Cited by

  1. A Prediction Method for Acoustic Intensity Vector Field of Elastic Structure in Shallow Water Waveguide vol.2020, pp.None, 2020, https://doi.org/10.1155/2020/5389719