Analytical solution for modal analysis of Euler-Bernoulli and Timoshenko beam with an arbitrary varying cross-section

In this article, the free vibrations of Euler-Bernoulli and Timoshenko beams with arbitrary varying cross-section are investigated analytically using the perturbation technique. The governing equations are linear differential equations with variable coefficients and the Wentzel, Kramers, Brillouin approximation is adopted for solving these eigenvalue equations and determining the natural frequencies and mode shapes. This method relates the solution of equations with the solving of some successive algebraic equations. A parametric study is performed and the effects of different profiles and different combinations of boundary conditions on the natural frequencies are investigated. To confirm the reliability of the present method, the analytical results are checked with those obtained from the finite elements method and other literatures which are found to be in a good agreement. The calculations show that the presented procedure is very effective to find the modal characteristics of the varying cross-sections beams.