• Dynamic model developed for rotating cracked FGM beams using a torsional spring. • Governing equations derived via Hamilton’s principle and solved with Galerkin’s method. • Higher material gradation index decreases nondimensional natural frequencies. • Rotating speed and hub length increase frequencies; follower force reduces them. • Greater crack depth significantly lowers vibration frequencies. In the current study, we investigate the dynamic response of cracked rotating FG-beams with two variable boundary conditions. The crack is considered to be simulated by a massless torsional spring model. The beam motion equation is obtained based on Hamilton’s concept. In this study, a power-law exponent describes graded beam materials as they vary through the beam's thickness. The beam's natural frequencies are established by solving the vibration equations with the Galerkin method. The study investigates the effect of geometrical and material properties, rotating speed, distributed force, hub length, and crack parameters on these frequencies. The analysis shows that the power index decreases the dimensionless natural frequencies for all end conditions, with or without a crack. In the absence of cracks, the ratio of the reduction in frequency of the double-simply supported FG beam is 17.58%, and the ratio of the reduction in frequency of the clamped-free end conditions is 15.95%. The frequency decreases by 22.75% and 19.60% in S-S and C-F, respectively, with a crack. Also, the dimensionless vibration frequency decreases with increasing tangent follower force, by 14.97% in S-S and 10.65% in C-F. Also, the results exhibit that crack depth lowers the dimensionless vibration frequencies. Moreover, the analysis shows that the hub radius ratio raises the dimensionless vibration frequencies, irrespective of the presence of a crack, across all end conditions. The findings provide useful insight for the vibration analysis and design of rotating FG structures in practical applications.
Fadhel et al. (2026) studied this question.