The spectral data i.e. eigenvalues (natural-frequencies) and eigenvectors (mode-shapes), characterizes the dynamics of
the system. The dynamic analysis of physical systems leads to certain direct and inverse eigenvalue problems. The direct
eigenvalue problem deals in evaluating the spectral behavior of structures for given distributions of physical parameters
such as mass, area, stiffness etc. whereas, the estimation of these physical parameters form the spectral data is known as
inverse eigenvalue problem. The detection of minuscule (small) changes in the stiffness and mass of the structure, by
solving certain inverse eigenvalue problems, is addressed here by considering a grooved axially vibrating rod. In solving
direct problems, we have considered two types of eigenvalue problem: (i) traditional algebraic eigenvalue problems and
(ii) transcendental eigenvalue problems associated with the continuous system. In conclusion, we have (a) obtained the
eigenvalues of damaged rod, (b) analyzed the behavior of the spectral data due to minuscule change in the physical
parameters, and (c) determined the different type of spectral data that are required for detecting damage parameters.
Several numerical examples are solved here demonstrating the feasibility and accuracy of the identification technique by
solving Transcendental Inverse Eigenvalue Problems.