Inhomogeneous beams that may possess a prescribed polynomial second mode

The following problem is posed and solved in this study: Reconstruct the stiffness variation of an inhomogeneous Bernoulli–Euler beam, whose second shape expressed as a polynomial function possesses a node at any pre-selected location. It is shown that for polynomial mode shapes there are regions of allowable position of the node for three specific cases studied in detail: (1) a beam with constant mass density; (2) linear mass density and (3) parabolically varying mass density.