Fourier Sine Transform Method for the Free Vibration of Euler-Bernoulli Beam Resting on Winkler Foundation

Abstract

In this study, the fourth order homogeneous partial differential equation (PDE) governing the free vibrations of Euler-Bernoulli beams on Winkler foundation with prismatic cross-sections was solved using the finite Fourier sine integral transformation method. Euler-Bernoulli beam theory was used to model the beam while Winkler foundation model was used for the foundation. The beam of length l was assumed to be simply supported at the ends x = 0, and x = l. The PDE was decoupled by the assumption of harmonic vibration. Application of the finite Fourier sine integral transformation on the decoupled equation resulted in the transformation of the problem to an algebraic eigenvalue problem. The condition for non-trivial solutions resulted to the characteristic frequency equation which was expressed in terms of a non-dimensional frequency parameter . n  The frequency equation which was observed to be the exact frequency equation obtained in the literature using the Navier series method, was solved to obtain the non-dimensional frequencies. Numerical values of the non-dimensional frequencies were computed for the case where 4 4 1,  = l = 1, and for n = 1, 2, 3, 4, 5. It was found that exact values of the non-dimensional frequencies were obtained using the present method.