Abstract
In this paper, the finite difference and the finite element methods are applied to evaluate natural frequencies of non-prismatic and non-homogeneous beams, with different boundary conditions and resting on variable Winkler foundation. The finite difference method is used for solving differential equation of motion, especially with variable coefficients. This technique requires a lesser computing effort and is used in situation where the exact solution is very difficult to obtain. The main idea of this method is replacing derivatives present in the free vibration equation and boundary condition equations with finite difference expressions. The natural frequencies are determined by solving the eigenvalue problem of the obtained algebraic system resulting from finite difference method. In order to illustrate the correctness and performance of the method, a comprehensive numerical example of non-prismatic beams is presented. The results are compared with the finite element results using ABAQUS software and other available numerical and analytical solutions.
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