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Abstract

The determination of the natural frequencies of flexural vibrations of Euler-Bernoulli beams is a vital consideration in their analysis and design for dynamic loads. This paper presents the Sumudu transform method for the determination of the natural frequencies of Euler-Bernoulli beams under transverse free harmonic vibration for different boundary conditions. The end support conditions considered are: (a) simply supported at both bends, (b) clamped at both ends, (c) clamped-free ends (d) clamped-simply supported ends, and (e) simply supported-clamped ends. The governing partial differential equation is converted by the Sumudu transformation to an integral equation, which upon evaluation becomes an algebraic equation. The solution gives the dynamic modal displacement shape function in the Sumudu transform space V(u). Inversion gives the dynamic modal displacement function in the physical problem space V(x). The enforcement of boundary conditions for the end supports considered yielded systems of homogeneous equations. The condition for nontrivial solutions is used to determine the characteristic frequency equation for each considered boundary condition. It is found that the characteristic frequency equation has an infinite number of eigenvalues (roots or zeros) corresponding to the continuously distributed parameter model of idealization of the problem. The characteristic frequency equations obtained are solved for the n roots using computational software methods, Symbolic Algebra Software and Mathematica Software to obtain the eigenvalues (zeros or roots) for any (n) vibration mode. The eigenvalues are then used to obtain the eigenfrequencies or natural frequencies of flexural vibration for each considered boundary conditions. It is found that closed form solutions obtained are identical to the solutions in the literature; obtained by classical methods of separation of variables and eigenfunction expansion methods.