A fifth-order polynomial that serves as both buckling and vibration mode of an inhomogeneous structure

This paper studies vibration of inhomogeneous beams in the presence of the distributed axial loads. We postulate the mode shape to be a fifth-order polynomial that represents the static displacement of the associated homogeneous beam under the linear loading. The beam is reconstructed such that it possesses the postulated modes of vibration and of loss of stability. It must be noted that the reconstruction is possible in most of the considered cases.