Mathematical Modeling
Vol. 2 (2018), Issue 4, pg(s) 122-125

Mixed variational properties for some fourth-order beam problems

  • 1 Technical University of Gabrovo, Bulgaria
  • 2 Institute of Information and Communication Technologies – BAS, Bulgaria

Abstract

In this paper we study eigenvalue problems for fourth-order ordinary differential equations. These boundary problems usually describe the bending vibrations of a homogeneous beam. Our aim here is to present mixed variational forms depending on a wide class of boundary conditions. In particular, we show when the symmetry in variational formulations is available. This property ensures real spectrum of the corresponding problem. The effect of the theoretical results is illustrated by some realistic examples.

Keywords

References

  1. W. Weaver, S. P. Timoshenko, D. H. Young. Vibration problems in engineering. John Wiley & Sons, 1990.
  2. I. Babuška, J. Osborn, Eigenvalue Problems, In Handbook of Numerical Analysis, Vol. II, (Eds. P. G. Lions and Ciarlet P.G.), Finite Element Methods (Part 1) North-Holland, Amsterdam, 641- 787, 1991.
  3. M. Racheva, Mixed variational formulation for some fourth order beam problems, Comp. rend. Acad. bulg. Sci. Tome 64, No 11, 2011, 1525-1532.
  4. M. R. Racheva, A. B. Andreev, Variational aspects of onedimensional fourth-order problems with eigenvalue parameter in the boundary conditions, Sib. JNM, No4(5), 373 – 380, 2002.
  5. L. Collatz, Eigenwerteprobleme und ihren numerische Behandlung, Chelsea, New York, 1948.
  6. B. B. Bolotin. Vibrations in technique: Handbook in 6 volumes, The vibrations of linear systems, I, Engineering Industry, Moscow, 1978.
  7. M. Roseau. Vibrations in Mechanical Systems. Springer-Verlag, Berlin, New York, 1987.
  8. Z. S. Aliyev, F. M. Namazov, Spectral properties of a fourthorder eigenvalue problem with spectral parameter in the boundary conditions. Electronic Journal of Differential Equations, 2017(307), 1-11.
  9. S. B. Guliyeva, Basis properties of the system of eigenfunctions of a fourth order eigenvalue problem with spectral parameter in the boundary conditions, Transactions of NAS of Azerbaijan, Issue Mathematics, 37 (4), 42–48 (2017).
  10. N. B. Kerimov, Z. S. Aliev, On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in the boundary condition, Differential Equations 43.7, 905-915 (2007).
  11. J. B. Amara, Sturm theory for the equation of vibrating beam. Journal of Mathematical Analysis and Applications, 349(1), 1-9 (2009).
  12. J. B. Amara, Fourth order spectral problem with eigenvalue in the boundary conditions, North-Holland Mathematics Studies. Vol. 197, North-Holland, 49-58 (2004).
  13. J. B. Amara, A. A. Vladimirov, On a fourth-order problem with spectral and physical parameters in the boundary condition, Izvestiya: Mathematics 68.4 (2004): 645. (in Russian)

Article full text

Download PDF