On Approximate Solution of the Euler-Bernoulli Beam Equation via Galerkin Method

Bu çalışmada Euler-Bernoulli kiriş denklemi olarak bilinen dördüncü mertebeden bir adi diferansiyel denklem için sınır değer probleminin sayısal çözümü sunulmuştur. İlgili denklem büyük binalar, büyük nehirler arasındaki uzun köprüler, uçaklar ve arabalar gibi mühendislik alanlarında yaygın olarak kullanılmaktadır. Ele alınan problemin yaklaşık çözümü, sınır koşullarını sağlayan temel fonksiyonlar ile Galerkin metodu kullanılarak elde edilmektedir. Önerilen yöntemin doğruluğu Maple® programı yardımıyla iki nümerik örnek üzerinden gösterilmektedir.

Anahtar Kelimeler:

Euler-Bernoulli kiriş denklemi, Galerkin metodu, Adi diferansiyel denklemler

On Approximate Solution of the Euler-Bernoulli Beam Equation via Galerkin Method

In this paper, numerical solution of a boundary value problem for a fourth-order ordinary differential equation, known as the Euler-Bernoulli beam equation, is presented. The related equation is used extensively in engineering areas such as huge buildings, long bridges across big rivers, planes and cars. The approximate solution of the problem considered is obtained by using the Galerkin method with basis functions that satisfy the boundary conditions given. The accuracy of the proposed method is given through two numerical examples with the help of Maple®program.

Keywords:

Euler-Bernoulli beam equation, Galerkin method, Ordinary differential equations,

PDF

___

Al-Omari, A., Schüttler, H.B., Arnold, J. Taha, T. 2013. Solving nonlinear systems of first order ordinary differential equations using a Galerkin finite element method. IEEE Access, 1, 408-417.
Biot, M.A. 1937. Bending of an infinite beam on an elastic foundation. Journal of Applied Mechanics, 2, 165-184.
Dubeau, F., Ouansafi, A., Sakat, A. 2003. Galerkin methods for nonlinear ordinary differential equation with impulses. Numerical Algorithms, 33, 215-225.
Gunakala, S.R., Comissiong, D.M.G., Jordan, K. Sankar, A. 2012. A finite element solution of the beam equation via Matlab. International Journal of Applied Science and Technology, 2(8), 80-88.
Han, S.M., Benaroya, H., Wei, T. 1999. Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration, 225 (5), 935-988.
Hunter, J.K., Narchtergaele, B. 2000. Applied Analysis, World Scientific, Singapore.
Kostadinova, S., Buralieva, J.V., Hadzi, K., Saneva, V. 2013. Wavelet-Galerkin solution of some ordinary differential equations, Proceedings of the XI International Conferences ETAI, Ohrid, Republic of Macedonia.
Ladyzhenskaya, O.A. 1985. Boundary Value Problems in Mathematical Physics, Springer, New York.
Lesnic, D. 2006. Determination of the flexural rigidity of a beam from limited boundary measurements. J. Appl. Math. And Computing, 20(1-2), 17-34.
Musa, A.E.S. 2017. Galerkin method for bending analysis of beams on non-homogeneous foundation. Journal of Applied Mathematics and Computational Mechanics, 16(3), 61-72.
Peradze, J. 2009. A numerical algorithm for a Kirchhoff-type nonlinear static beam. Journal of Applied Mathematics, Article ID 818269, 12 p.
Peradze, J. 2016. On the approximate solution of Kirchhoff type static beam equation. Transactions of A. Razmadze Mathematical Institute, 170, 266-271.
Smith, R.C., Bowers, K.L., Lund, J. 1992. A fully Sinc-Galerkin method for Euler-Bernoulli beam models. Numerical Methods for Partial Differential Equations, 8(2), 171-202.
Subaşı, M., Şener, S.Ş., Saraç, Y. 2011. A procedure for the Galerkin Method for a vibrating system. Computers and Mathematics with Applications, 61, 2854-2862.
Şener, S.Ş., Saraç, Y., Subaşı, M. 2013. Weak solutions to hyperbolic problems with inhomogeneous Dirichlet and Neumann boundary conditions. Applied Mathematical Modelling, 37, 2623-2629.
Younesian, D., Saadatnia, Z., Askari, H. 2012. Analytical solutions for free oscillations of beams on nonlinear elastic foundations using the variational iteration method. Journal of Theoretical and Applied Mechanics, 50 (2), 639-652.
Thankane, K.S., Stys, T. 2009. Finite difference method for beam equation with free ends using Mathematica. Southern Africa Journal of Pure and Applied Mathematics, 4, 61-78.