Turbo generator shafts are often subjected to cyclic torsion resulting in formation of large longitudinal cracks as well as circumferential cracks. The presence of these cracks could greatly impact the shaft resonance frequencies. In this paper, dynamic response of a shaft with longitudinal and circumferential cracks is investigated through a comprehensive analytical study. The longitudinally cracked section of the shaft was modeled as an uncracked shaft with reduced torsional rigidity. Torsional rigidity correction factor (i.e., the ratio of torsional rigidity of the cracked shaft to that of the uncracked shaft) was obtained from finite element analysis and was shown to be only a function of crack depth to the shaft radius. The resonance frequency and frictional energy loss of a shaft with a longitudinal crack were found little affected by the presence of the crack as long as the crack depth was less than $20%$ of the shaft radius even if the entire shaft is cracked longitudinally. Moreover, we showed that the longitudinal crack location could be more conveniently identified by monitoring the slope of the torsional response along the shaft. The circumferential crack was modeled as a torsional spring with a torsional damping. The torsion spring and damping constants were obtained using fracture mechanics. For a shaft with both a longitudinal crack and a circumferential crack, the resonance frequency was governed by the longitudinal crack when the circumferential crack depth was less than $30%$ of the shaft radius.

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