Exponential solutions for a longitudinally vibrating inhomogeneous rod

A special class of closed form solutions for inhomogeneous rods is investigated, arising from the following problem: for a given distribution of the material density, find the axial rigidity of an inhomogeneous rod so that the exponential mode shape serves as the vibration mode.Specifically, for a rod clamped at one end and free at the other, the exponentially varying vibration mode is postulated and the associated semi-inverse problem is solved.This yields distributions of axial rigidity which, together with a specific law of material density, satisfy the governing eigenvalue problem.The results obtained can be used in the context of functionally graded materials for vibration tailoring, that is, for the design of a rod with a given natural frequency according to a postulated vibration mode.