This work aims to investigate the effectiveness of the Gram-Schmidt orthogonalization in conjunction with the Rayleigh-Ritz method for estimating transverse natural frequencies of a rotor system. For this purpose, the Gram-Schmidt orthogonalization process was used to estimate the mode shapes of the rotor system, which satisfied its geometric boundary conditions. The relationship between the spatial error distribution of mode shape and the accuracy of estimated transverse natural frequencies was explored in this work. Through comparison of the approximate model with the finite element model of a rotor system, it was observed that the spatial error distribution of each mode shape influenced the accuracy of the associated natural frequency estimation differently, with the first mode being more sensitive to the ratio of rotor thickness to the shaft length, compared to the second mode. It was shown that for a particular range of rotor thickness-to-shaft length ratio, the use of the Gram-Schmidt orthogonalization with the Rayleigh-Ritz method could provide a sufficiently accurate estimation of transverse natural frequencies of a rotor system, which could be used for a rotor system with a variety of boundary conditions.