Some unexpected results in vibration of non-homogeneous beams on elastic foundation

A special class of inverse vibration problems is studied in this investigation. We pose the following problem: Find the spatial distribution of the material density and the elastic stiffness of an inhomgeneous beam on elastic formulation so that the vibration mode shape constitutes a preselected function, namely the static deflection of the associated homogeneous beam under some simple loading conditions. Four sets of boundary conditions are considered, in context of polynomial representation of the material density and stiffness. It turns out unexpectedly, that irrespective of the boundary condition, the natural frequency is given by the same analytical formula.