Can a trigonometric function serve both as the vibration and the buckling mode of axially graded structures?

It is well known that for homogeneous beams that have simply supported or guided ends a trigonometric function serves both as a vibration and a buckling mode. The research in this paper shows that, remarkably, the same result is valid for some axially graded beams. Specifically, three cases of harmonically varying vibration modes are postulated and the associated semi-inverse problems that result in the distributions of elastic modulus that together with a specific variation of material density and axial load distribution satisfy the governing eigenvalue problem for the inhomogeneous Bernoulli-Euler beam are solved. In all cases the closed-form solutions are obtained for the natural frequency. Those closed-form solutions are contrasted with approximate solutions based on appropriate polynomial functions, serving as trial functions in an energy method.