Design of a polynomially inhomogeneous bar with a tip mass for specified mode shape and natural frequency

In this study the closed form solution is obtained, apparently for the first time, for the free vibration inhomogeneous bar with a tip mass. It is remarkable that while a vibrations study the homogeneous bar with free tip leads to transcendental equation, here, in the case of the inhomogeneous bar, a closed form polynomial solution is obtained. If the mass ratio between the bar's mass and the concentrated mass is specified, numerical evaluation is needed for the axial rigidity coefficients' ratio; on the other hand, for specified mass ratios closed form solution is reported.