Constructing rods with given natural frequencies

• We present a method for constructing rods having given natural frequencies.
• Rods can be constructed explicitly by means of closed-form expressions.
• The analysis is based on the determination of quasi-isospectral rods.
• The key tool of the construction is a lemma by Darboux.

In this paper we present a new method for constructing axially vibrating rods having prescribed values of the first N natural frequencies, under a given set of boundary conditions. The analysis is based on the determination of the so-called quasi-isospectral rods, that is rods which have the same spectrum as a given rod, with the exception of a single eigenvalue which is free to move in a prescribed interval. The reconstruction procedure needs the specification of an initial rod whose eigenvalues must be close to the assigned eigenvalues. The rods and their normal modes can be constructed explicitly by means of closed-form expressions.