Non-stationary motion of a cracked rotating shaft with accelerating or decelerating angular velocity Ω through a critical speed is studied. The shaft has a breathing transverse crack and a disk. There are two parts, which are the investigation of flexural response, neglecting the torsional vibrations, and the investigation of flexural-torsional response. In both studies the longitudinal vibration and the influence of shear deformation are neglected. The boundary conditions of the supports are simply supported for the transverse displacements and fixed-free in relation to torsion (for the flexural-torsional response only). The transverse surface crack, which causes a geometric discontinuity, is replaced by generalized moments at the crack location. The equations of motion follow the formulation of Wauer. Galerkin’s method and numerical integration are used to obtain approximate solutions. The maximum responses are determined. The effects of the acceleration and deceleration rate and the different parameters of the breathing cracked rotating shaft, such as crack depth, crack location, disk location, disk eccentricity, disk eccentricity angle, and disk mass, are studied. The influence of internal damping, external damping, and torsional external damping are investigated. Comparisons with an open cracked rotating shaft and an uncracked rotating shaft are also presented. The influence of torsional deformation is analyzed. The results are presented in tables and figures. / Master of Science