Identification of the parameters of a rod with a longitudinal rectangular groove using two spectra of natural frequencies of bending vibrations

The inverse coefficient problem involves determining the geometric parameters of a longitudinal rectangular groove based on the natural frequencies of the bending vibrations of a rectangular rod. It is assumed that the groove does not extend along the entire length of the rod, but rather from a certain point to the right end. To solve the problem, the rod with the longitudinal groove is modeled as two sections: the first section without a groove and the second section with a groove.
Mating conditions are applied at the connection point, where deflection values, rotation angles, bending moments, and shear forces are equated. The behavior of the natural frequencies of bending vibrations when changing the length of the groove was investigated. A solution method is proposed that allows for determining the required parameters based on a finite number of natural frequencies of bending vibrations. It is shown that the solution is unambiguous when using frequency spectra with respect to mutually perpendicular axes.