natural frequencies and internal resonance of beams with arbitrarily distributed axial loads

An exact analytical solution for transversal free vibrations of a beam subjected to an arbitrary distributed axial load and a tip tension is obtained by means of a power series representation, whose coefficients are determined recursively in an easy way. the dependence on the natural frequencies on the load is then investigated, and the buckling load (corresponding to vanishing frequency) is also discussed. next, the 1:3 internal resonance between the first and the second mode is deeply studied, and an interesting (and unexpected) property is found for linearly distributed axial loads