Apparently first closed-form solutions for inhomogeneous vibrating beams under axial loading

Closed-form solutions are presented for fundamental natural frequencies of inhomogeneous vibrating beams under axially distributed loading. The mode s hape is postulated as coinciding with the static deflection of the associat ed homogeneous beam without distributed axial loading. Then the inverse pro blem of determining the stiffness and mass density distributions, producing the above mode shape, is solved. To describe these variations, the family of polynomial functions is used. Several sets of boundary conditions are co nsidered. It is shown that the natural frequency vanishes when the intensit y of the axially distributed loading equals the critical buckling value. A linear relationship is established between the square of the natural freque ncy and the load ratio for all reported sets of boundary conditions, in con trast to uniform beams where the exact linear relationship holds for column s with simply supported and/or sliding ends.