An asymptotic solution to transverse free vibrations of variable-section beams

The transverse free vibration of a class of variable-cross-section beams is investigated using the Wentzel, Kramers, Brillouin (WKB) approximation. Here the governing equation of motion of the Euler–Bernoulli beam including axial force distribution is utilized to obtain a singular differential equation in terms of the natural frequency of vibration and a WKB expansion series is applied to find the solution. Based on this formulation, a closed form solution is obtained for determination of natural vibration mode shapes and the corresponding frequencies. The first four terms of this asymptotic solution are simplified for homogenous beams to give a compact third-order WKB approximation. Next, the resulting solution is employed to determine the natural frequencies and mode shapes of some examples with and without axial force distribution. The results are then been compared with those in the literature and very good agreement is achieved.