A technique is developed to find the axial mass distribution of rods and beams based upon desired properties of its first vibrational mode deflection function. This is an inverse Sturm‐Liouville problem for rods. The cases of particular interest are when a specified characteristic function does not satisfy the boundary conditions of the problem or continuity conditions. For example, a uniformly strained fixed‐free rod vibrating axially cannot exist because the strain must be zero at the free end. Using calculus of variations, a method is developed whereby the rod or beam is found whose first characteristic function can be as close as possible to the specified unattainable function. This is done for linearly elastic axially nonuniform materials vibrating axially or transversely. Examples include the application of the technique to the design of ultrasonic resonators.