Coupled systems mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Linear and nonlinear vibrations of inhomogeneous Euler-Bernoulli beam

  • Bakalah, Ebrahim S. (Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals) ;
  • Zaman, F.D. (Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals) ;
  • Saleh, Khairul (Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals)
  • Received : 2018.03.19
  • Accepted : 2018.08.27
  • Published : 2018.10.25
⟨ Previous Next ⟩

Abstract

Dynamic problems arising from the Euler-Bernoulli beam model with inhomogeneous elastic properties are considered. The method of Green's function and perturbation theory are employed to find the deflection in the beam correct to the first-order. Eigenvalue problems appearing from transverse vibrations of inhomogeneous beams in linear and nonlinear cases are also discussed.

Keywords

References

  1. Abu-Hilal, M. (2003), "Forced vibration of Euler-Bernoulli beams by means of dynamic Green functions", J. Sound Vibr., 267(2), 191-207. https://doi.org/10.1016/S0022-460X(03)00178-0
  2. Asghar, S., Zaman, F.D. and Ahmad, M. (1991), "Field due to a point source in a layer over an inhomogeneous medium", Nuovo Cim., 5, 569-573.
  3. Burlon, A., Failla, G. and Arena, F. (2016), "Exact frequency response analysis of axially loaded beams with viscoelastic dampers", Int. J. Mech. Sci., 115-116, 370-384.
  4. Duque, D. (2015), "A derivation of the beam equation", Eur. J. Phys., 37(1).
  5. Ebrahimi, F. and Barati, M.R. (2018), "Stability analysis of porous multi-phase nanocrystalline nonlocal beams based on a general higher-order couple-stress beam model", Struct. Eng. Mech., 65(4).
  6. Failla, G. (2016), "An exact generalized function approach to frequency response analysis of beams and plane frames with the inclusion of viscoelastic damping", J. Sound Vibr., 360, 171-202. https://doi.org/10.1016/j.jsv.2015.09.006
  7. Ghosh, M.L. (1970), "Love waves due to a point source in an inhomogeneous medium", Gerlands Beitr. Geophys., 79, 129-141.
  8. Graef, J.R., Henderson, J. and Yang, B. (2009), "Positive solutions to a fourth order three-point boundary value problem", Discret. Contin. Dyn. Syst., 269-275.
  9. Graef, J.R., Yang, B. and Yang, B. (1999), "On a nonlinear boundary value problem for fourth order equations", Appl. Anal., 72(3-4), 439-448. https://doi.org/10.1080/00036819908840751
  10. Gupta, C.P. (1988), "Existence and uniqueness theorems for the bending of an elastic beam equation", Appl. Anal., 26(4), 289-304. https://doi.org/10.1080/00036818808839715
  11. Han, S.M., Benaroya, H. and Wei, T. (1999), "Dynamics of transversely vibrating beams using four engineering theories", J. Sound Vibr., 225(5), 935-988. https://doi.org/10.1006/jsvi.1999.2257
  12. Huang, Y., Chen, C.H., Keer, L.M. and Yao, Y. (2017), "A general solution to structural performance of pre-twisted Euler beam subject to static load", Struct. Eng. Mech., 64(2).
  13. Kukla, S. and Zamojska, I. (2007), "Frequency analysis of axially loaded stepped beams by Green's function method", J. Sound Vibr., 300, 1034-1041. https://doi.org/10.1016/j.jsv.2006.07.047
  14. Li, X.Y., Zhao, X. and Li, Y.H. (2014), "Green's functions of the forced vibration of Timoshenko beams with damping effect", J. Sound Vibr., 333, 1781-1795. https://doi.org/10.1016/j.jsv.2013.11.007
  15. Lindell, I.V. and Olyslager, F. (2001), "Polynomial operators and green functions", Prog. Electromagn. Res., 30, 59-84. https://doi.org/10.2528/PIER00031305
  16. Logan, J.D. (2007), Pure and Applied Mathematics in An Introduction to Nonlinear Partial Differential Equations, 2nd Edition, Hoboken, John Wiley & Sons, New Jersey, U.S.A.
  17. Mohammadimehr, M. and Alimirzaei, S. (2016), "Nonlinear static and vibration analysis of Euler-Bernoulli composite beam model reinforced by FG-SWCNT with initial geometrical imperfection using FEM", Struct. Eng. Mech., 59(3).
  18. Morrison, S.M. (2007), Application of the Green's Function for Solutions of Third Order Nonlinear Boundary Value Problems, Tennessee, Knoxville, U.S.A.
  19. Nejad, M.Z., Hadi, A. and Farajpour, A. (2017), "Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials", Struct. Eng. Mech., 63(2).
  20. Orucoglu, K. (2005), "A new green function concept for fourth-order differential equations", Electron. J. Differ. Equat., (28), 1-12.
  21. Pietramala, P. (2011), "A note on a beam equation with nonlinear boundary conditions", Bound. Value Probl., 1-14.
  22. Rizov, V.I. (2017), "Fracture analysis of functionally graded beams with considering material non-linearity", Struct. Eng. Mech., 64(4).
  23. Stakgold, I. and Holst, M.J. (2011), Green's Functions and Boundary Value Problems, John Wiley & Sons, New York, U.S.A.
  24. Stuwe, H.C. and Werner, P. (1996), "A Green's function approach to wave propagation and potential flow around obstacles in infinite cylindrical channels", Math. Meth. Appl. Sci., 19(8), 607-638. https://doi.org/10.1002/(SICI)1099-1476(19960525)19:8<607::AID-MMA785>3.0.CO;2-O
  25. Teterina, O.A. (2013), The Green's Function Method for Solutions of Fourth Order Nonlinear Boundary Value Problem, University of Tennessee, Knoxville, U.S.A.
  26. Truesdell, C. (1983), Rational Mechanics, Academic Press, New York, U.S.A.
  27. Webb, J.R.L., Infante, G. and Franco, D. (2008), "Positive solutions of nonlinear fourth-order boundaryvalue problems with local and non-local boundary conditions", Proceedings of the Royal Society of Edinburgh.
  28. Zaman, F.D., Asghar, S. and Ahmad, M. (1990), "Love-type waves due to a line source in an inhomogeneous layer trapped between two half spaces", Boll. Di Geofis. Teor. Ed Appl., XXXII(N), 127-128.
  29. Zaman, F.D. and Al-Zayer, R.M. (2004), "Dispersion of Love waves in a stochastic layer", Nuovo Cimento-Socista Ital. Di Fis. Sez., C(27), 143-154.
  30. Zaman, F.D., Masood, K. and Muhiameed, Z. (2006), "Inverse scattering in multilayer inverse problem in the presence of damping", Appl. Math. Comput., 176(2), 455-461. https://doi.org/10.1016/j.amc.2005.09.034